Beam Shaping and Control withNonlinear Optics

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Beam Shaping and Control withNonlinear OpticsEdited by

F . KajzarCommissariat a l’Energie AtomiqueGif-sur-Yvette, France

and

R. ReinischInstitut National Polytechnique de GrenobleGrenoble, France

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PREFACE

The field of nonlinear optics, which has undergone a very rapid development since thediscovery of lasers in the early sixties, continues to be an active and rapidly developing re-search area. The interest is mainly due to the potential applications of nonlinear optics: di-rectly in telecommunications for high rate data transmission, image processing andrecognition or indirectly from the possibility of obtaining large wavelength range tuneablelasers for applications in industry, medicine, biology, data storage and retrieval, etc.

New phenomena and materials continue to appear regularly, renewing the field. Thishas proven to be especially true over the last five years. New materials such as organics havebeen developed with very large second- and third-order nonlinear optical responses. Impor-tant developments in the areas of photorefractivity, all optical phenomena, frequency conver-sion and electro-optics have been observed. In parallel, a number of new phenomena havebeen reported, some of them challenging the previously held concepts. For example, solitonsbased on second-order nonlinearities have been observed in photorefractive materials andfrequency doubling crystals, destroying the perception that third order nonlinearities are re-quired for their generation and propagation. New ways of creating and manipulating nonlin-ear optical materials have been developed. An example is the creation of highly nonlinear(second-order active) polymers by static electric field, photo-assisted or all-optical poling.Nonlinear optics involves, by definition, the product of electromagnetic fields. As a conse-quence, it leads to the beam control. This includes amplitude or phase modulation, genera-tion of new laser frequencies and altering the propagation of beams either in space or time.Different nonlinear optical interactions and mechanisms lead to a large variety of functions.

The time thus seemed appropriate to us to bring all these new developments into focusin a summer school format. This was the main objective of the NATO Advanced Science In-stitute held in Cargese (Corsica, France) August 4–16, 1997. A good understanding of non-linear optical phenomena and their dependence on wavelength, electronic structure,structural properties, etc. requires not only an excellent knowledge of the basic laws of phys-ics, governing the nonlinear optical phenomena, but also a good knowledge of materials andthe laws governing the interaction between light beams in different forms of matter.

The lectures given at the school covered the following topics: nonlinear optical phe-nomena and their applications, temporal and spatial solitons, third-order effects, organic ma-terials, organic and inorganic multiple quantum wells, hybrid excitons and microcavityeffects, cascading effects and applications, parametric processes, applications of opticalparametric oscillators and second harmonic generators, the latest developments in nonlinearmagneto-optics, new techniques for molecule orientation, ato-optics, light-induced kineticeffects in gases, light upconversion to the blue, nonlinear waveguiding optics, photorefrac-tive effects and photorefractive solitons, χ(2) spatial solitons. The subjects covered by theschool underline the importance of the ever improving fundamental research and continuingtechnological developments. We hope that this book will contribute to the dissemination of

v

the theoretical and experimental results concerning this fascinating field of all-optical inter-actions.

Organization of the school would be impossible without financial support. We arehighly indebted to its main sponsor: the NATO Scientific Affairs Division. The financial con-tributions from other sources such as Centre National de la Recherche Scientifique, Direc-tion des Systemes de Forces et de la Prospective de la DGA, Institut National Polytechniquede Grenoble, LETI-Saclay, and Centre National d’Etude des Telecommunications were alsovery helpful and we would like to thank these organizations for their support. We would likealso to acknowledge the Scientific Committee members V. M. Agranovich, G. Assanto, C.Flytzanis, S. Kryszewski and G. Stegeman for their suggestions and help in the organizationof the school. Thanks are also due to Ms. Amelie Kajzar for her assistance in the organiza-tional tasks and in the preparation of these proceedings. Finally, many thanks are due to thestaff of the Institut Scientifique de Cargese for its efficiency and kindness from which webenefited during the school, and more generally to all the lecturers and students for their con-tribution in making this meeting very pleasant and successful.

François KajzarRaymond ReinischSaclay and Grenoble

vi

CONTENTS

Introduction to Nonlinear Optics: A Selected Overview . . . . . . . . . . . . . . . . . . . . . . . . 1G. I. Stegeman

Introduction to Ultrafast and Cumulative Nonlinear Absorption and NonlinearRefraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37

E. W. Van Stryland

From Dipolar Molecular Engineering to Multipolar Photonic Engineering inNonlinear Optics 7 7

J. Zyss and S. Brasselet

Molecule Orientation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101F. Kajzar and J.-M. Nunzi

Nonlinear Pulse Propagation along Quantum Well in a Semiconductor Microcavity . . 1 3 3V. M. Agranovich, A. M. Kamchatnov, H. Benisty, and C. Weisbuch

Some Aspects of the Theory of Light Induced Kinetic Effects in Gases . . . . . . . . . . . 1 4 9S. Kryszewski

Temporal and Spatial Solitons: An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 8 3A. Boardman, P. Bontemps, T. Koutoupes, and K. Xie

Spatial Solitons in Quadratic Nonlinear Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 2 9L. Torner

Photorefractive Spatial Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5 9M. Segev, B. Crosignani, P. Di Porto, M.-F. Shih, Z. Chen, M. Mitchell, and

G. Salamo

Sub-Cycle Pulses and Field Solitons: Near- and Sub-Femtosecond EM-Bubbles . . . . . 2 9 1A. Kaplan, S. F. Straub, and P. L. Shkolnikov

Nonlinear Waveguiding Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1 9R. Reinisch

Quadratic Cascading: Effects and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4 1G. Assanto

vii

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Nonlinear Optical Frequency Conversion: Material Requirements, EngineeredMaterials, and Quasi-Phasematching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

M. Fejer

Low-Power Short Wavelength Coherent Sources: Technologies and Applications . . . . 407D. Ostrowsky

Artificial Mesoscopic Materials for Nonlinear Optics . . . . . . . . . . . . . . . . . . . . . . . . . . 427C. Flytzanis

Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .465

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

viii

I N T R O D U C T I O N T O N O N L I N E A R O P T I C S : A S E L E C T E DOVERVIEW

C.R.E.O.L., University of Central Florida4000 Central Florida Blvd., Orlando, FL 32816-2700, USA

George I. Stegeman

INTRODUCTION

Historical Perspective

Nonlinear optics (NLO) has enjoyed great success as a discipline for over 30 years now.Although it was a relative newcomer to nonlinear wave sciences, other examples beingnonlinear fluid dynamics, acoustics, plasmas etc, it has contributed many new phenomena.Since its inception in 1962, nonlinear optics has passed through many phases and differenttopics have been “hot” at any given time. 1,2 One of the fascinating features of the nonlinearoptics field is its regenerative power to develop new topics over the years. The first ten yearswitnessed demonstration of many of the fundamental interactions such as second harmonicgeneration (SHG), sum and difference frequency generation, stimulated Raman, Brillouin andRayleigh scattering, self-focusing etc. And many more interesting phenomena were predictedtheoretically, some having to wait two decades before experimental confirmation wasforthcoming. This “novelty” trend continued into the second decade with the development ofmultiple nonlinear spectroscopies and their applications to materials science, phaseconjugation, bistability leading to concepts of all-optical signal processing, the beginnings ofnonlinear optics in fibers, etc. The third phase, from about the mid 1980s to the present has hadits own highlights such as the development of nonlinear guided wave optics, especially infibers where a whole spectrum of new propagation effects and light induced non-centrosymmetric effects were found, the exciting development of efficient, widely tunablesources through optical parametric oscillators, temporal solitons and their potential for long-haul communications, a surprising variety of spatial solitons, terahertz sources, femtosecondpulses, generation of tens of higher harmonics in gases etc. One of the exciting recentdevelopments is the blurring of the roles of second and third order nonlinear optics, namelythe creation of second order nonlinear effects via third order nonlinearities, and the use ofsecond order effects to mimic third order phenomena. Many of these topics will be discussedin this book.

The key to applications of nonlinear optics is, has been and always will be theavailability of appropriate materials. The initial stages of the field which focused ondemonstrating and understanding new effects utilized the materials available at that time. For

Beam Shaping and Control with Nonlinear OpticsEdited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002 1

example, for second harmonic generation, materials developed for piezoelectric applicationswhich also require non-centrosymmetric media were used first. In the case of third order,much of the early work was done with liquids. Ultimately, the search for better materials wasdriven by the realization that in order for any applications to be practical, nonlinear optics hadto move forward from the era in which high power lasers were almost exclusively needed toobserve nonlinear phenomena. Compact semiconductor lasers with 100s of mW power levelsdrove the need for sub-watt nonlinear optics. The search for better materials gainedmomentum in the mid to late 1970s and continues unabated to the present day. In the case ofsecond order materials, there have been multiple goals including doubling into the UV regionof the spectrum, widely tunable sources via parametric interactions, inexpensive sources in theblue, etc. The exciting concept of all-optical processing has fueled third order nonlinear opticsfor many years.

Formalism

Traditionally nonlinear optics has been discussed in terms of the nonlinear polarizationinduced in a nonlinear medium by the mixing of one or more intense electromagnetic waves.2,3

Typically multiple beams with different frequencies are incident onto a nonlinear medium,either modifying the linear optical properties of the medium or leading to the generation of newwaves at new frequencies. As a matter of notation, the incident fields E(r,t) of frequency ω iand wavevector ki for propagation along the z-axis can be written in the form:

(1)

For plane waves, ei is the electric field unit vector, fi (x,y) = 1 and ai (z), the slowly varying(complex) amplitude, is normalized so that |a (z) | ²i is the intensity in units of W/cm² . Whenthe interacting beams are of finite extent, for example in a waveguide, the fi (xy) describe thetransverse field profiles. The nonlinear polarization

(2)

induced by the mixing of the optical fields has the general form:²

(3)

where E(r,t) is the total electric field in the medium and χ(n) is the n’th order nonlinearity.Here the r - r i allow for a spatially non-local response, e.g. carrier diffusion and t - t i allow fora polarization field at time t to be generated by fields at an earlier time ti , for example due toa finite carrier recombination time. For a total field of the form E(r,t) = ∑ ½ E (r;ω i ) exp[i(ωit- ki r)] + c.c., i.e. an expansion in terms of its Fourier components, the induced polarizationcan be written as a Taylor’s expansion in the Fourier components of the mixing fields,²

2

(4)

Here kp = k1 + k 2 + k3 + … is the wavevector of the induced polarization, with the number ofterms determined by the order of the nonlinearity. Furthermore if the complex conjugate termparticipates in the mixing, the corresponding field term E( ω i ) in the field product appears asthe complex conjugate, i.e. E*(ωi). The polarization field, and any electromagnetic field thatit subsequently generates, oscillates at the frequency ω = ω1 ± ω 2 ± ω3 ± … where the minussign corresponds again to the conjugate terms, if appropriate. The Di (ω ), the degeneracyfactors unique to each nonlinear interaction, essentially “count” the number of equivalent termswhich contribute to the polarization field for a given set of input fields.² For example, D3 (ω )takes the values 1, 3, 6 and 6 for third harmonic generation (THG), an intensity-dependentrefractive index (n2), electric field induced second harmonic generation (EFISH) anddegenerate four wave mixing (DFWM) respectively.

The nonlinear polarization source term now drives electromagnetic fields at thefrequency ω. This part of the nonlinear optics problem is a relatively straightforwardapplication of Maxwell’s equations, leading to the usual polarization driven wave equation.These fields can be both radiating and non-radiating in nature. Usually, only the radiativefields are of interest because they can under certain conditions grow to be usefully large. Thiswill be clear with subsequent specific examples.

It is important to note that the well-known polarization expansion, given above, is notcomplete and was originally meant to describe nonlinear phenomena which involve purelyelectronic nonlinearities. As such, the crystal symmetry (or lack of it) plays a key role indetermining which tensor components are zero, and which are non-zero. However, in themodern context this polarization expansion is used to describe any and all processes whichinvolve optically induced nonlinearities, including charge excitation in semiconductors,thermal effects etc. It is important to note that the physics of the nonlinearity, in concert withthe crystal symmetry determines the non-zero coefficients and their inter-relationships.²Higher order terms can be and sometimes are included, but to date only the fifth order termχ(5) (-ω; ω1 , ω2 , ω3 , ω4 ) in the limit ω = ω1 = -ω2 = ω3 = -ω4 has proven to be important. Itoccurs as a correction to an intensity-dependent refractive index which itself is proportionalto χ(3) (-ω; ω1 , -ω1, ω1 ). Furthermore, additional terms are typically added to include interactionswith other types of excitations such as magnetic or acoustic waves, polaritons etc.

As implied by their arguments, the nonlinear susceptibility coefficients all undergodispersion, i.e. are frequency dependent.² There is resonant enhancement of the coefficientsinherent in the spectral content of the nonlinear coefficients, the exact form depending on thephysics of the interaction of the electromagnetic fields with matter. For example, it can reflectthe interaction of radiation with multi energy level molecules, virtual and real transitions ofelectrons in semiconductors etc. Near such resonances the χ(n) are complex, far from suchresonances they are real. This resonant enhancement can be very useful. For example,nonlinear spectroscopy such as multi photon absorption accesses transitions which are not onephoton active, for example two photon states in centrosymmetric molecules. Furthermore, thelarge nonlinearities can be used for bistable logic elements.

A common oversight made in nonlinear optics involves the dispersion of nonlinearitieswith frequency. It is incorrect to assume that the nonlinearity χ(3) measured with a specifictechnique at some frequency is the same as the value used in a different interaction withdifferent frequency inputs. Consider a simple case of dispersion frequently derived intextbooks, χ(2) based on the anharmonic electron oscillator model.² Specifically,

3

(5)

where the constant K contains the details of the anharmonic forces, etc. The key point is thateven this simple susceptibility contains two resonance denominators which produce sizeabledispersion and large enhancements.

Until recently, the roles played by χ(2) and χ (3) were well defined. Second orderphenomena were used for frequency conversion, usually second harmonic generation, sum anddifference frequency mixing, optical parametric oscillators etc.² Also in this class is theelectro-optic effect which has been investigated from the 1970s for modulators.

On the other hand, third order phenomena are the origin of many effects. For example,the intensity-dependent refractive index derived from χ(3) (-ω:ω,-ω,ω), has been applied to all-optical signal processing, spatial and temporal solitons (discussed in later chapters).² Otherinteresting phenomena such as phase conjugation, electric field induced second harmonicgeneration, and nonlinear spectroscopies such Raman gain, coherent Anti-Stokes Ramanscattering etc. have all been investigated..

One of the most interesting developments of the last five years is the use of χ(2) t omimic effects well known in χ(3) , and vice-versa. For example, optically induced electric fieldslead to charge migration and permanent second order nonlinearities for second harmonicgeneration in glass fibers.5 Conversely, the strong energy exchange between a fundamental andits second harmonic leads to nonlinear phase shifts, and even to solitons.6 This area will beexplored in subsequent chapters.

SECOND ORDER NONLINEAR PHENOMENA

Second Harmonic Generation

It is a useful exercise to work through the formalism of second harmonic generationin detail. It is typical of a wide spectrum coherent interactions.² The total incident field is

( 6 )

which corresponds to Type I SHG with one fundamental input field and one harmonic field.The dominant terms in the field product EjEk are:

( 7 )

which give the following nonlinear polarization terms

(8a)

(8b)

These terms are then substituted into the polarization driven wave equation

( 9 )

4

which gives, for example at the frequency ω,

(10)

Note that it has been assumed that the so-called slowlyvarying phase and amplitude approximation.² Since the fields are eigenmodes of the waveequation, which gives

(11)

with ∆k = k3 - 2k1 . A similar analysis gives the second coupled mode equation at 2ω :

(12)

with where the ei are the electric field unit vectors.Furthermore, Kleinman symmetry was assumed so that deff (-ω ) = deff (-2ω). Assuming thesimplest case of negligible depletion of the fundamental,

(13)

There are a number of parameters which are important for second harmonic generation.1. SHG Figure of Merit: the smaller ofHere α is the dominant loss mechanism, usually at the second harmonic frequency.2. Phase-matching condition: ∆φ = 0where This is a generalization to the case where there aretwo orthogonally polarized fundamental beams with wavevectors k1 ( ω ) and k2 (ω).3. Minimize beam cross-sectional area A (which can be achieved in waveguides)4. Minimize absorption (because it can lead to the thermal detuning of the phase-matchingcondition and reduce the effective length of the medium)5. Environmental and chemical long term stability6. Good mechanical properties (for polishing)7. Material homogeneity8 . . . . . . . . . . .

SHG Phase-Matching: Bulk Media

The dispersion of refractive index with frequency in the wavelength range 300nm < λ <2000nm makes it essentially impossible to phase-match second harmonic generation in bulkcrystals with co-polarized fundamental and second harmonic beams without the artificialintroduction of additional dispersion, for example through a grating.² In general, the refractiveindex decreases with increasing wavelength due to electronic resonances in the UV and nearUV visible part of the spectrum. The standard phase matching techniques involve apolarization change between one of the fundamental inputs and the second harmonic.

5

Type I SHG: Here there is only a single fundamental input beam so that ∆φ = ½ [k3(2ω) -2k1(ω)]L and the fundamental and harmonic are orthogonally polarized. Writing k3= 2ωn3/cand k1 = ωn2/c, n1=n3 is required. This can be achieved in a birefringent crystal, for exampleas shown in Figure 1a in which no> ne, i.e. the “ordinary” polarized refractive index is largerthan the “extraordinary” polarized index.

Type II SHG: Here there are two orthogonally polarized fundamental beams with differentrefractive indices, for example n0 = n1 > ne = n2. One photon from each polarization is usedto form a harmonic photon. Therefore now ∆φ = ω[n3 -½(n1+n2)]L/c so that n3= ½(n1+n2)is required for phase-matching. This case is shown schematically in Figure 1b.

Very recently a new approach has been developed for bulk crystals called Quasi-Phase-Matching (QPM) SHG.7 It has proven possible to use periodic electric fields in ferroelectriccrystals like lithium niobate and lithium tantalate to periodically reverse the ferroelectricdomains and hence the nonlinear coefficient d(2)

333. For a domain period Λ = 2 π/ κ, the phasemismatch becomes ∆φ = ½ [k3(2ω) - 2k1 (ω) ± κ]L. Therefore, choosing the period to producewavevector conservation gives ∆φ = 0.

SHG Phase-Matching in Waveguides

The best conversion efficiency occurs when a minimum beam cross-sectional area canbe maintained over the full length of the sample. This can be achieved in fiber or channelwaveguides whose cross-sectional areas can be of order a few λ2 8. The usual geometry usedis a channel waveguide by which we mean a waveguide whose shape does not have circularsymmetry but is more rectangular in nature. There is a “core” guiding region surrounded bymedia of lower refractive index. Typically a discrete number of modes with propagationwavevectors βm, n (along z) are allowed for a given index difference between the core guidingregion and the dimensions and a given shape of the guiding region where m and n are integersstarting from 0. The corresponding field distributions fm n(x,y) are oscillatory across the twochannel dimensions with m zeros across the x-dimension and n along y. There are two sets oforthogonal modes, each of which contains electric fields along all three axes, including the

6

Figure 1 Phase-matching conditions for (a) Type I SHG and (b) Type II SHG in birefringent mediawhere e and o identify the e- and o-polarized beams.

propagation direction. However, for each set, usually one field component dominates, TEmn(TMmn) with the dominant electric field parallel (perpendicular) to the long channel cross-sectional dimension (y) which is also usually parallel to the air-material interface. An exampleof the x-dependence of the TEmn field distributions, identified as TEm is shown in Figure 2.

m.

Wavevector matching in waveguides requires finding conditions for which ∆φ = ½[βmn(2 ω) - 2βm’n’(ω )]L 0. (Phase-matching in waveguides usually involves only onefundamental input.). Because there are in principle mn values for the propagation wavevectorsβ at each frequency, one for each mode, there are more degrees of freedom in achieving phase-matching than there are in bulk media. The phase matching condition can be expressed moreconveniently in terms of the waveguide mode effective indices, Neff = β /kvac as ∆φ =kvac( ω)[Nmn(2ω) - Nm’n’(ω)]L 0. As will be discussed later, there is a price to be paid for thisflexibility in terms of efficiency.

In addition to the Type I and Type II SHG phase-matching approaches, there are otherswhich are unique to waveguides and they will now be discussed:(i) Modal Dispersion Phase Matching (MDPM)

Here phase matching is implemented from a lower order fundamental mode to a higherorder second harmonic mode, typically with both modes having the same polarization,although the mixed mode case is of course also possible. The modal dispersion curves inFigure 3 show by a circle the MDPM case for the simpler example of a slab waveguide. (Thechannel case would require a three dimensional plot with two axes, one for each transversedimension.) First note that at a given frequency, the effective indices undergo dispersion withincreasing waveguide dimension, from cut-off where Neff = ns, the substrate index for the slabcase (air above), to Neff = nf, the guiding film index. Furthermore, there is dispersion withfrequency in the refractive indices of both the substrate and film which in the visible and nearinfrared requires phase-matching from a lower to higher order mode. For the channel case, theMDPM condition needed is in a three-dimensional plot. The possible multiplicity of suchMDPM conditions in multi-mode waveguides shows the flexibility of using waveguides versusbulk media.

The loss in efficiency inherent to using modes of different orders is clear from theexpression for SHG in a slab waveguide:

(14)

where the integral term is called the “overlap integral”. It effectively projects the secondharmonic polarization onto the second harmonic field at every point in the waveguide andsums the product. For the case shown in Figure 3, the corresponding fields are shown inFigure 4a and it is clear that the overlap integral is small. However, a number of solutions tothis problem have been successfully demonstrated. For example, in Figure 4b about one half

7

Figure 2 Representative field distributions for some lowest order slab waveguide modes TE

of the guiding film is made from a material with d (2)eff =0 so that no interference occurs between

the two halves of the waveguide.9 Another solution has been to reverse the sign of thenonlinearity at the position that the harmonic field reverses sign so that the overlap integral isagain maximized, Figure 4c.10 Both schemes have recently been implemented in poledpolymer channel waveguides. 10,11

(ii) Quasi-Phase-Matching (QPM)This technique is very powerful because it allows phase-matching between the lowest

order, co-polarized fundamental and harmonic modes.12 Only a brief description is given herebecause a subsequent chapter will deal with this concept in greater detail. A grating isintroduced with periodicity Λ = 2π/κ so that ∆φ = ½ [β00(2ω) - 2β 00(ω) - κ]L 0. In termsof effective indices, ∆φ = kvac(ω )[N00(2ω) - N00(ω) - κ/2kvac (ω)]L 0. As shown in Figure3, this corresponds to a vertical translation between the dispersion curves and Λ can be chosento minimize the effects of fabrication parameters on phase-matching. This modulation can beproduced by either a linear grating, for example by modulating the waveguide dimensions, orby a nonlinear grating achieved by periodically reversing the nonlinearity d(2) with distance

8

Figure 3 Dispersion in effective indices in a slab waveguide and the phase matching conditions forMDPM (circle), QPM (solid vertical arrow) and Cerenkov (range given by horizontal arrow).

Figure 4 (a) Field overlap condition for overlap integral for a TE0(ω) and a TE1(2 ω). Waveguide with(b) one half of the core linear and one half nonlinear and (c) with nonlinearity reversed at the harmonicfield reversal plane.

down the waveguide. Certainly the second approach is much more effective in optimizing theSHG efficiency. It has been implemented with various techniques in ferroelectric media suchas LiNbO3 and LiTaO3.12,13 There has also been some work on polymer waveguides, but theefficiencies have been disappointing.14

(iii) Cerenkov Phase Matching (CPM)This is an “automatic” wavevector-matching condition (parallel to the interfaces) made

possible by the dispersion of material refractive index with wavelength in the visible and nearinfrared. l5 In Figure 3, note that there is a region of waveguide dimension “h” in which ns(2 ω)> Neff(ω). Because in the waveguide the projection of the wavevectors of the interactingelectromagnetic modes must be conserved, then radiation fields are generated into the bulk atan angle θ from the surface given by ns(2ω)cosθ = N eff(ω).

The above discussion has been facilitated by considering only a slab waveguide, butclearly the concepts can be extended easily to the two dimensional confinement case.

Other d(2) Parametric Processes

SHG is only one of many possible parametric d(2) processes. Clearly sum anddifference frequency generation also require phase-matching and they have been implementedin bulk and waveguide media. In fact, the application of difference frequency mixing to theshifting of the wavelength of communications signals has been investigated. An example ofshifting from the 1310 nm communications band to the 1550 nm is to mix the 1310 signal witha laser at 708 nm to produce a signal 1540 nm.16 Alternatively, a shift in the signal frequencywithin the erbium amplifier band can be achieved.17

9

Figure 5 The range of a single OPO as compared to the range of numerous common narrowbandwidth sources.

One of the most powerful applications of second order processes is to opticalparametric generators (OPG), amplifiers (OPA) and oscillators (OPO).2 This is essentially adown-conversion process in which a pump beam at frequency ωp breaks up into two beams atfrequencies ω i and ω i, i.e a pump photon breaks up into a signal and ans so that ωp = ωs + ωidler photon. The choice of the signal and idler frequencies is determined by the wavevectormatching condition kp = ks + ki. The wavevector matching condition can be tuned either bychanging the temperature or the direction of the wavevectors. When either the signal or theidler is used to seed the interaction, this is called an OPA. When noise is used to supply theseed, this is an OPG, and when the crystal is placed in a cavity (in order to enhance the output)the device is called an OPO. An example of the tuning range of an OPO based on the crystalBBO is shown in Figure 5 for comparison with the output of a number of discrete lasersystems. 18 Tunability from 300 to 2400 nm is achievable with a pump beam derived from thefourth harmonic of a Nd:YAG laser. There have been reports of many other impressive OPOsystems, with pulse widths varying from cw right down to tens of femtoseconds.19

Second Order Materials

Materials are the key to successful applications of nonlinear optics. By its very nature,nonlinear optics requires high intensities for efficient implementation. However, the larger thenonlinearity, the smaller the intensity required. This is true for both the second and third ordernonlinearities. Many of the most impressive recent advances in nonlinear optics have involvedthe development of both new second order materials and new techniques for phase-matchingthem as well as existing materials.

Until about ten years ago, the only source of second order materials was single crystalswhich could be grown in non-centrosymmetric lattices. At a molecular level, the potential wellin which the electrons sit must be anharmonic, i.e. the potential well should be of the form Ax2

+ Bx3. For example, this rules out symmetric molecules. As a result, for a given level ofexcitation, the electron displacement is larger in one direction than the other so that theinduced dipole has a second harmonic component. Such molecules can be packed in a latticein which these induced polarizations cancel out, leading to a macroscopically centrosymmetriccrystal. Alternatively, other lattice configurations can be non-centrosymmetric and thereforeshow macroscopic second order nonlinearities.

10

Figure 6 Electric field poling of polymers.

Recently it has proven possible to take non-centrosymmetric molecules and artificiallyarrange them into non-centrosymmetric configurations. One example is Langmuir-Blogettfilms which are deposited one monolayer at a time.20 More promising seem to be poledpolymers.21 The poling process is shown in Figure 6. The molecules (chromophores) requirea permanent dipole moment in addition to a large molecular second order nonlinearity to beincorporated into a polymer. When the polymer is heated to its glass transition temperature,it becomes soft and the chromophores can be oriented towards the direction of a strong appliedelectric field. This partial alignment leads to a macroscopic d(2) when the polymer is cooleddown to room temperature and the external field is removed. Such materials have been usedfor both frequency conversion and electro-optical effects. 21,22

Three classes of materials are usually considered for their d(2) activity.23 Of theinorganic crystals, ferroelectrics have shown nonlinearities in the 10s of pm/V range.Semiconductors, for example in the GaAs-based family, have typical nonlinearities in the 150pm/V range. Organic media have nonlinearities ranging from 20 pm/V (LB films, poledpolymers, crystals) to 100 pm/V (poled polymers), to 600 pm/V (the crystal DAST). Asdiscussed, candidate materials must also have a good transparency range and be phase-matchable.

Table 1 Representative list of d(2) active materials, their largest diagonal and off-diagonalnonlinear coefficients and their transparency window.20,23

Material d ij (pm/V) d ii(pm/V) Transparency (nm)

BaB2O 4 1.6 190 → 2500

LiB 3O5 1.1 0.06 180 → >2800

LiNbO3* 5.8 30 350 → >3000

KTiOPO4* 4.4 18.5 350 → 3000

KNbO 3* -18 -27 400 → >2500

GaAs* 125 0 900 →

NPP 84 30 480 → 1800

DMNP * 90 30 450 → 1700?

DAST 30 600 750 → 1700

†DANS* 100 500 → 1800

‡BAMSAB* 21 450 → 1700?* - waveguides † - poled polymer ‡ - Langmuir Boldgett filmNPP: N-(4-nitrophenyl)-L-prolinolDMNP: 3,5-dimethyl-1-(4-nitrophenyl)pyrazoleDAST: Dimethyl-amino-4-N-methylstilbazolium tosylateDANS: 4-dimethylamino-4|nitrostilbeneBAMSAB: 4-(dimethylamino)-4|-(methylsulfonyl)azobenzene

A representative list of materials is given above, along with their transparency ranges.23

The trends are quite clear. Materials for operation into the UV region of the spectrum, forexample the borates, have in general quite modest nonlinearities, of order 1 pm/V. Theferroelectrics have larger nonlinearities, 10s of pm/V, but their transparency window extends

11

from the just below the visible to around 3000-4000 nm. Organic materials have nonlinearitiesin the 100s of pm/V range, but their transparency window requires operation above 500 nm,and below 1800 nm (set by vibrational overtones). The large nonlinearity of semiconductorslimits their utility to wavelengths beyond the near infrared, although it is noteworthy that theyare transparent well into the infrared. Unfortunately, the known trends indicate that the largerthe nonlinearity, the more limited the transparency window.

Waveguide SHG Results

The progress in producing efficient SHG in channel waveguides has been spectacular.It is summarized in Table 2. It is important to note that this table only contains the figure ofmerit and in some cases the absolute conversion efficiency is limited by high waveguide losses,for example in the case of the semiconductor and polymer work quoted where the losses were10's of dB/cm.10,17

There are two noteworthy developments in the application of semiconductors to SHG.Asymmetric quantum wells (AQW) lead to asymmetric wave functions for the electrons (seeFigure 7a for an example), and hence a non-centrosymmetric response for a single quantumwell. By stacking quantum wells, a d (2)

xzx = 10 pm/V has been obtained.24 An alternativescheme uses the d (2)

xyz ~ 150 pm/V component of GaAlAs semiconductors which normally isvery difficult to implement in a phase-matched geometry. Using a combination of waferbonding, selective etching and organometallic vapor deposition, it has proven possible to growthe QPM structure shown in Figure 7b.17 An effective nonlinearity of ~ 90 pm/V was obtainedwhen the fundamental was propagated along the [1,1,0] direction. However, although thelosses were high, 33 dB/cm at the harmonic frequency, improvements in growth technologycould reduce these losses significantly.

Table 2 FOM for waveguide SHG in terms of the figure of merit for cases in which boththe fundamental and harmonic are guided and for η' =

for the Cerenkov case. Note η ∝ λ - 4.

Phase-Match Waveguide η %/[W-cm2] λ (nm)

Birefringent SiO2/Ta2O5 /KTP 25 960 830

QPM KTP 26 800 830

QPM LiTaO 327 1500 830

QPM LiNbO328 44 1550

MDPM DR110 44 1550

QPM AlGaAs17 76 1550

η ' %/[W-cm]

CPM DMNP core fiber 29 40 830

CPM SiO2/Ta2 O5/KTP25 130 830

12

Figure 7 (a) An example of the quantum well structure and the electron wave functions for an

AQW. 24 (b) QPM AlGaAs structure fabricated for SHG. 17

Electro-optic Effect

Although not usually discussed in such terms, the electro-optic effect is also a χ (2)

phenomenon. 2 The nonlinear polarization can be written in the form:

(15)

where ω>>Ω , i.e. E(Ω) is a DC or slowly varying field. Furthermore, ri j k is the electro-opticcoefficient = χ(2)

i j k /n4 . For the simplest case this leads to ∆ni = -½ninj2 rijk Ek( Ω ). Such an electricfield induced index change can be used in, for example, a Mach-Zehnder switch of the typeshown below in Figure 8. The applied voltage leads to changes in the effective indices of thechannel waveguides which in turn leads to the phase difference ∆φ ∝ n3 r eff V/d L in the lightpropagated through the channels. When ∆φ π , the throughput is switched from maximumto zero etc.

Such applications require a variety of different material properties. Some of theimportant ones, and their values for three different materials options are listed in Table 3. Ofthese, two are key: the space-bandwidth product which gives a modulator’s bandwidth for a1 cm long device; and the voltage-length product which predicts the voltage required to switcha 1 cm long device. The very large bandwidth of electro-optic polymer devices has recentlybeen verified (110 Ghz).22

Photorefractive Effect

This phenomenon leads to optically induced changes in the refractive index and isclassified as second order because the electro-optic effect plays a key role. The materialrequirements are as follows:30

13

Figure 8 Geometry (upper) for an electro-optic modulator in which the index change is different inthe two channels, as shown in the lower figure.

Table 3 Figures of merit for electro-optic materials

Figure of Merit GaAs LiNbO3 EO Polymer

EO coefficient reff (pm/V) 1.5 31 10-60

Dielectric Constant ∈ 10 28 4

Refractive Index n 3.5 2 . 2 1.6

n3r (pm/V) 64 248 123

n3r/∈ 6.4 8.7 31

Loss (dB/cm @ λ = 1300 nm) 2 0.2 0.5-2.0

Space-Bandwidth product (GHz-cm) 10 1 0 120

Voltage-Length product (V-cm) 0.3-1.0 5 5-10

1. Electrons are promoted from trap states into a “conduction band” by the absorptionof light. Therefore some absorption at a suitable wavelength is required.2. Conduction of electrons in response to Coulomb forces, diffusion and/or local orexternal electric fields.

14

3. Refractive index change via the electro-optic effect.4. Subsequent optical beams in the medium are affected by the refractive index changes.Therefore the best candidate materials are electro-optic materials with trap states due to theincorporation of selected impurities, for example KNbO3 , LiNbO3 , SBN etc.30

The unique properties of the interaction of light with photorefractive materials isillustrated in Figure 9. Consider the interference between two optical beams (9a). Electronsare promoted to the conduction band in the bright regions of the interference pattern. Due toCoulomb repulsion, a charge separation is created by electron conduction as indicated in Figure9c. This leads to a space charge field as shown in Figure 9d, and in turn to an index changevia the electro-optic effect, Figure 9e. Note that the resulting index modulation is shifted byπ/2 relative to the intensity modulation introduced by the interference pattern. This has a largeeffect on any optical beams traveling in the medium. For example, power can be interchangedbetween the two beams. This and other effects will be discussed in more detail in anotherchapter in this book.

Figure 9 Generation of phase-shifted gratings via the photorefractive effect. Details described in

text.30

Third Order Nonlinear Phenomena

A large number of effects derive from the χ (3) susceptibility tensor. Starting from thenonlinear polarization they can for convenience be classedinto four categories:2

1. Incoherent All-Optical EffectsThis leads to self and cross phase modulation, an intensity-dependent refractive index

∆n(I), and applications such as all-optical switching, logic, bistability, and temporal andspatial solitons2. Coherent Frequency Degenerate Effects

Degenerate Four-wave Mixing phase conjugation

15

3. Nonlinear SpectroscopyThis leads to the generation of new frequencies and spectroscopies such as third

harmonic generation, coherent Stokes (Anti-stokes) Raman spectroscopy, and Raman gainspectroscopy.4. Stimulated Scattering

Stimulated Raman, Brillouin and Rayleigh ScatteringHere examples of these categories will be discussed, in the reverse order to that given

above.

Example of Nonlinear Spectroscopy: Coherent Anti-Stokes Raman Spectroscopy (CARS)

This technique can be used to investigate the normal modes of materials, for examplevibrations, rotation etc.2 This CARS example deals specifically with vibrational modes.

Consider two waves of frequency ω1 , wavevector k1 and frequency ω2, wavevector k2with ω > ω incident onto a material with a molecular vibration at the frequency ω1 2 q ω1-ω2 . If the mixing field E1E

* modulates the molecular vibration, it coherently excites the2 (vibration in space with the spatial modulation exp[i(k - k ). r ]. Now the ω (ω2) beam is2 1 1scattered by the excited vibration producing an output CARS field at the frequency ω3 = 2ω1 -ω 2 (2ω2- ω1 , coherent Stokes Raman Scattering) with wavevector k3. Therefore this processinvolves the mixing of three incident fields and hence is a χ(3) process. Under appropriatewavevector matching conditions, i.e. k3 + k2 - 2k1 = 0, a strong CARS signal is measured.

Formally, this process can written in terms of a nonlinear polarization as

(16)

where it has been assumed for convenience that all the beams are y-polarized so that χ(3)

χ( 3 )yyyy (- ω ; -ω ). This leads to the coupled mode equation at ω2,ω1,ω

eff =1 3 :3

(17)

Because of the usual dispersion of the refractive index in the visible, |k2 | + |k3| > 2|k1| so thatwavevector matching requires that the beams be non-collinear. Usually the angle between k2

and k1 is only a few degrees. When the difference frequency ω - ω ω , the nonlinear2 1 q

susceptibility has the form

(18)

where the background susceptibility χ (3) contains all of the other contributions and χ (3)b q

describes the interaction with the q’th vibrational mode. Because of the resonance at ω2 - ω 1

ω , the signal is enhanced and the location of ω can be identified by tuning ω - ω In fact,2 .q q 1monolayers have been investigated in waveguide geometries.31

Example of a Stimulated Process: Stimulated Raman Scattering (SRS)

This was one of the first effects observed in the early days of nonlinear optics.2 Astrong pump beam (Ep) at frequency ωp leads via a stimulated process to the generation of aStokes beam (Es) at ω sshifted from the pump by a vibrational frequency ωq = ω p - ω s .

16

Although it is initiated by Raman scattering from vibrations excited by thermal fluctuationsat some unknown distance into a material, the analysis is the easiest to perform if one assumesboth the incident and scattered signals are propagating in the medium. Just as in the CARScase, the mixing of the pump and Stokes beams excites vibrations with amplitude Q0 in themolecules. The distortion in the “electron cloud” leads to a modulation of the molecularpolarizability so that α = α ∂0 + [∂α/ q]q where Thedetailed calculation of the vibrational amplitude Q0 is tedious, can be found inhere only the result is given for ωq = ωp -ω s :2

textbooks and

(19)

where µ red is the reduced mass of the vibration. The total polarization induced in a mediumwith a density of molecules N is given by

(20)

The nonlinear polarization terms describing the evolution of the pump and Stokes beams are:

Applying coupled mode theory gives:

(21)

(22)

As long as the pump is undepleted, these equations clearly lead to exponential growth(stimulated emission) of the Stokes beam. The gain coefficient for the Stokes power is

(23)

Note that it depends on the pump power. Taking the combination dI/dz ∝ E* dE/dz + E dE* /dzleads to the Manley Rowe relations for the energy flow between the two beams, i.e.

(24)

This means that as the Stokes grows by one photon, the pump beam depletes by one photon.Since the energy difference goes into exciting the vibrations which are in turn

17

dissipated to the lattice.The other well-known stimulated process is Stimulated Brillouin Scattering.2 In this

case, the stimulated Stokes beam travels in the opposite direction to the pump beam so thatacoustic phonons of wavevector kp + kp are required. The corresponding frequency shift is ωp- ωB = Ω s = 2kp /vs where Ωs is the phonon frequency and vs is the sound wave velocity.

Degenerate Four Wave Mixing2

As indicated below, in this interaction there are two counter-propagating pump beamsand an incident signal beam which collectively generate a fourth, conjugate beam whichpropagates backwards to the signal beam. If the pump beams are misaligned, then theconjugate no longer retraces along the path of the signal but the process etc. still occurs withperhaps a reduced efficiency due to the reduced overlap volume of the beams.

(27)

Figure 10 Degenerate four wave mixing geometry.

The input field in this case is:

(25)

and clearly there are a great number of terms in the product χ(3) EEE. In fact it is the behaviorof the signal and conjugate beams that is of prime interest, usually in the regime of negligibledepletion of the pump beams. The key nonlinear polarization terms are:

It is a straight-forward process to evaluate the coupled mode equations for the signal andconjugate beams:

(26b)

(26a)

18

Solving gives:

(28)

There are two regimes of interest. The first involves significant gain for both beams.One photon from each of the pump beams provides gain for the signal and the conjugate.Furthermore, the conjugate beam is actually the complex conjugate of the signal beam at theinput, hence the nomenclature phase conjugation. The second frequent application is to theevaluation of the third order susceptibility χ(3). Note that for L<< lcr , the peak power in theconjugate beam is proportional to |χ(3) | 2 .

All-Optical Devices and Material Figures of Merit

One of the most active fields of research recently has been the application of anintensity-dependent refractive index to all-optical switching and processing devices, solitonsetc. 5 , 3 2 Solitons, both spatial and temporal, will be discussed in a number of other chapters inthis volume. Here a specific all-optical switching device will be used to provide some insightinto material requirements for this whole class of devices.

Consider a nonlinear Mach-Zehnder interferometric switch as shown below:32

Figure 11 An al l-optical Mach-Zehnder switch in which one of the channels has a larger area overa distance L.

If the two arms of the interferometer were identical, the phase shifts φ1 and φ2 would be equal,resulting in constructive interference and the input signal will be recovered. However, if theintensity in the two arms is high and the channel cross-sectional areas are different, then I 1 >I2 . Assume for simplicity a Kerr-law nonlinearity, i.e.

π, destructive interference occursWhen the nonlinearly induced phase difference isand the output signal is zero. The important point is that the nonlinear phase shift

dz is the key parameter, and for this device a minimum value of π is needed. Thisminimum value depends on the particular device of interest. Probably the most useful all-optical device reported to date is a nonlinear directional coupler for which a φ N L >2 π isrequired of the nonlinear material.32

Based on the discussion above, it proves useful to compare materials via nonlinearphase shift based material Figures Of Merit (FOM).32 Consider a nonlinear material describedby a intensity dependent refractive index change ∆n = n2 I so that the nonlinear phase shift φ NL

= kv a c n2 ILeff where the effective length Le f f can be limited by material absorption (or scattering)losses. Here we assume that one, two or three photon absorption can occur, i.e. α = α1+ α2 I+ α3I2 and limit Leff to approximately α-1 .

19

Limit #1: One Photon absorption dominates α1 >> α2 I, α3 I2

(29)

Limit #2: Two Photon absorption dominates α2 I >> α1 , α3I 2

Limit #3: Three Photon absorption dominates α3I2 >> α2 I, α1

These three FOM assess the potential of a material to produce a 2 π phase shift in the presenceof various sources of loss. Note that in the presence of both one and two photon absorption,the appropriate combined FOM is W/[1 + WT] > 1. The problem of finding suitable materialsis now one of characterizing promising materials over broad spectral ranges. A subsequentchapter in this volume will deal with this problem.

Material Systems and Nonlinearities

A change in the refractive index of a material due to the propagation of a high powerbeam through it can occur due to a large variety of physical mechanisms. The most interestinginvolve resonant interactions between an electromagnetic field and some state or excitationover a limited frequency range. In this limit, the nonlinear index change can usually be “turnedon” very quickly if enough energy is put into the material, and usually decays with somecharacteristic relaxation time for the “excitations” created. Detuned from these resonances,virtual transitions occur, and the response time ∆t is typically limited by ∆ω∆t ~ 1 where ∆ωis the frequency detuning from the resonance. In this limit, most nonlinear mechanismsresemble an ideal Kerr nonlinearity, namely that ∆n = n2 I. Furthermore, now a χ (3) formalismis usually a useful description.

It was found that there are usually trade-offs between the nonlinearity n2 due to aspecific mechanism, the absorptive loss α associated with it, and the characteristic relaxationtime τ. Namely n2 /ατ ~ a constant to within a couple of orders of magnitude over a wide rangeof materials and mechanisms. 33

Although it is not always a valid description of a nonlinear index change for a specificmechanism, it is useful to define an effective n2 in order to compare the magnitudes of variousnonlinearities. A table of such comparisons is given below.

For any material, an index change can be induced onto one beam by itself or by anotherbeam of a different frequency and/or polarization. The general relationship between thedifferent nonlinear refractive indices can be given in terms of χ(3) tensor elements wheneversuch a formalism is valid. In general

The first term gives the usual refractive index n0 = [1 + χ( 1 ) ]½.in terms of intensity-dependent refractive indices, namely

The second term can be written

(30)

where ω can be either ω2 or ω1 (orthogonally polarized case), I1 is the intensity of beam at ω1for which the index is changed and I2 that of the intensity of a second beam at ω2 or ω1 ( ⊥polarization).

20

NONLINEAR MECHANISMS |n 2 (cm2 /W)|

5.

6 .

7 .

8 .

(31)

1.

2.

3.

4.

9.

10.

11.

Electronic (∑ atoms)

Electrostriction

non-resonant

Electronic (conjugation) non-resonant

resonant

Charge transfer molecules non-resonant

resonant

Cascading (χ ( 2 )χ( 2 )) non-resonant

Electronic, excited states resonant

Carrier generation (semiconductors) resonant

Semiconductor Kerr in-gap

below ½ gap

Exciton bleaching (semiconductors) resonant

Liquid Crystals (reorientation) non-resonant

Thermal (material and heat sinkingpulse width etc. dependent)

Case I: Self-phase modulation (1 beam only)

2x10- 1 4 →10 -16

3x10-14 →10-16

10 -11 → 10-13

10- 5

2x10-13 → 10-14

10- 8

10- 9 → 10 -14

10-10 → ??

10 -6

10 -11

10 -13 →0

10 - 5

<10 - 6

10- 3 → 0

Case II: Cross-phase modulation (2 beams with the second orthogonally polarized beam at ω1)

(32)

Case III: Cross-phase modulation (2 beams, second co-polarized beam at ω2)

(33)

Case IV: Cross-phase modulation (2 beams, second beam orthogonally polarized at ω 2)

(34)

For isotropic media, far from any resonances, there is a simple relationship between thedifferent χ(3) components, namely;

21

Molecular Nonlinearities The nonlinearities associated with molecules have theirorigin in transitions between molecular states, usually starting from the ground state. Inmolecular (more than one atom) systems there is a series of energy levels, which in somespectral regions are discrete and in others overlapping. At high enough energies, these discretelevels become sufficiently dense that they form a quasi-continuum, at least at roomtemperature.

(35)

Electric dipole-allowed transitions between the electronic energy levels Ei and Ej areresponsible for the complex dielectric constant and hence the refractive index and absorptioncoefficient. For example, for input photon energies approximately equal to an allowedtransition energy, E - Ej i = i j , a peak occurs in the absorption spectrum, accompaniedby dispersion in the refractive index, Figure 12 at low intensities. Normally the lower state forthe transition is the ground state (i ≡ g). For electric-dipole transitions to be allowed, a changein the symmetry of the eigenfunctions between the lower and upper states is necessary. Thelarger the oscillator strength of the transition, the larger the absorption maximum and thestronger the dispersion in refractive index. The wavelength dispersion in the linear opticalproperties is a superposition of absorption maxima and index dispersion due to all of theallowed one photon transitions from the ground state.

The multiphoton optical response depends on dipole allowed transitions involvingmany different electronic states. Here, transitions can occur multiple times between the groundstate and the same excited state within the lifetime of the excited state, or between multiple(>2) states, either simultaneously, or sequentially. Within the scope of this review, it is notfeasible to discuss all the contributions to the nonlinearities and neglected are transitionsbetween vibronic sub-levels which give a fine structure to the transitions between electroniclevels, electric quadrapole and magnetic dipole contributions etc.

The scenarios which govern the gross nonlinear response are shown schematically inFigure 12a and 12b for the interaction of two photons with an oversimplified molecularsystem. These processes are normally associated with third order nonlinear optics. One photonchanges the optical properties of the medium which can then be experienced by the second (orsubsequent) photon(s). The first case is for electric dipole allowed transitions from the groundstate to a one photon allowed state (opposite parity to ground state), Figure 12a. This occursfor molecules with well-defined parity states, or with mixed parity states. For high enoughincident intensities, the excited state population becomes large enough so that the probabilityof the absorption of subsequent photons is reduced, i.e. α is reduced from it’s low intensityvalue α1 to α = α 1 + α 2I with α2 < 0. For intensities approaching the saturation intensity, thenext higher order term α 3 I2 will be positive etc. (α goes to zero when the ground and excitedstate populations are equal.) This is called “saturable absorption”. Simultaneously thedispersion in the refractive index is “bleached out”, as also shown in Figure 12a. This can bewritten as n = n1 + n

2I where n2 >0 for ω > ω i j and n2 <0 for ω < ω i j . For photons far off

resonance with the transition, the nonlinearity can be considered as a contribution to thesusceptibility χ(3)( -ω; ω,- ω,ω). For photon energies near the transitions’s resonance, thenonlinearity is enhanced, but is also accompanied by absorption. Because the saturationevolves during a pulse of duration shorter than the excited state lifetime, the response of n2

and α2 is fluence dependent, evolving over the duration of the pulse. For pulses much longerthan the excited state lifetime, n2 and α2 can be considered “instantaneous”.

22

Figure 12 (a) The change in the absorption and refractive index with increasing intensity near anallowed one photon transition. (b) The same for a two photon transition (see text for details).

The second two process in Figure 12b also leads to n(I) and α(I), this time via theparticipation of an even symmetry excited state in a centrosymmetric molecule, and a changein the permanent dipole moment from the ground to excited state for asymmetric molecules(which have mixed parity, i.e. both even and odd). In either case, an absorption linear in theintensity is induced which results in the population of the two photon active state. If indeed thetwo photon state can be populated close to saturation via two photon absorption, then theabsorption decreases with further increase in intensity so that α = α

2I + α

3I

2 + … with α

2 > 0

and α3 < 0. Although this case is relatively rare, it has been observed recently in thepolydiacetylene PTS. 4 The corresponding variation in n with intensity, n = n0 + n2 I + n3 I2 , isalso quite different from the one photon case. For example, for frequencies less than theresonance frequency, n2 > 0 and n3 < 0 where-as, for ω > ω ij, n 2 < 0 and n3 > 0 . Nearsaturation the response of n is not instantaneous over an exciting pulse and hence they3 and α3

are fluence dependent. Note that the signs for α2 are different for the one and two photon

cases. This implies that at certain wavelengths they could cancel leading to an effective α2 →0. Standard nomenclature is that the nonlinear response within a few linewidths of theabsorption maximum is called the "resonan" response. "Near-resonant" refers to the spectralregion where there is a strong spectral dependence to the nonlinearity n2 and "non-resonant"is defined by the long wavelength limit where n2 becomes essentially independent ofwavelength.

Multiphoton processes involving the simultaneous interaction of three, four etc.photons with a molecular system can also occur. In general they require very high intensitiesto interfere with the one and two photon contributions. The three photon case leads to threephoton absorption and dispersion, i.e. to terms with ∆n ∝ n3I 2 + n4

I3etc. and ∆α ∝ α3 I 2 + I3 α 4

etc. where terms in I3 and beyond are due to saturation of the three photon process.Intensity dependent changes in the refractive index and absorption are linked by the

Kramers-Kronig relationship, even if the exact physical origin of the nonlinearity is unknown.

23

where P.V. is the principal value of the integral. For example, in a pump probe experiment toevaluate ∆n(ω,Ω) where the probe is at frequency ω and the pump (strong beam) is at Ω, it isnecessary to measure ∆α with the same strong beam (Ω ), and a tunable probe (frequency over a broad spectral range. Although it is preferable to make measurements of the absorptionchange over as wide a frequency range as possible, the most important spectral ranges are those

(36)

containing resonances ω ij.

Semiconductor Nonlinearities Semiconductors have many physical mechanisms inwhich an intensity-dependent refractive index can be induced, i.e. ∆n = f(intensity, carrierdensity). Note that the index change is also a function of the carrier density in the conductionband. The dominant mechanisms are due to the absorption or emission of photons movingelectrons across the band gap or creating excitonic states. (Some of the energy involved isdissipated in the lattice, leading to strong thermo-optic effects for long pulses or cw excitation.)

It proves convenient to classify the nonlinearities as passive (conduction band initiallypopulated in thermal equilibrium with the valence band) and active nonlinearities in which thepopulation of the conduction band is initially out of thermal equilibrium due to electrical oroptical pumping.Passive Nonlinearities:34,35

1. Bandfilling- associated with photon absorption → carrier generation- all parameters depend on detuning from band gap- 0 < |n2 | < 10 - 6 cm2 /W | ∆nsat | < 0.1 τ ~ 10 ns

2. Exciton bleaching- associated with photon absorption- parameters depend on detuning from exciton line- 0 < |n2| < 10 -5 cm2 /W |∆nsat | <0.1 τ ~ 10 ns (ends up in carrier generation)

3. Bound electron effects- ultrafast two photon Kerr nonlinearity- sign of n2 depends on detuning- 0 < |n 2 | < 10-11 cm 2 /W |∆ns a t | < 0.01 τ - 10 fs

4 . Thermo-optic effect- absorption ∆T → ∆n (via dn/dT)- large [n

2 ~ α/σ], slow (τ ~ µs)

Active Amplifier Nonlinearities: (electrical/photon pumping)36,37

1. Ultrafast nonlinearities near transparency- two photon Kerr nonlinearity- spectral hole burning- carrier heating

2 . Carrier nonlinearities- stimulated emission (gain)The passive nonlinearities due to carrier generation and exciton bleaching are quite well

understood in both bulk and quantum well semiconductors.3 4 An example for the transitionfrom bulk to narrow quantum wells is reproduced from reference 34 in Figure 13. Note theincrease in both ∆n and α1 which occurs in going from bulk to multiple quantum well (MQW)structures. As shown in Figs 14 for W and T, the FOM are well understood for bulksemiconductors. 32 Note that the combination of W and T leaves only a narrow spectralwindow below the band gap in which bulk semiconductors satisfy both FOMs. The situation

24

in MQWs is similar but with the spectral window depending on the details of the quantumwells.

Figure 13 The intensity dependent change in the absorption and refractive index change for GaAs inbulk and quantum well form (well widths given in figure).

Another useful spectral region is for photon energies just below one half thesemiconducting band gap where two photon absorption rapidly goes to zero.35 Here in theAlGaAs system with composition tuned for operating in the communications band at 1550 nm,n2 ≈2x10 - 1 3 cm2 /W with the FOM W > 10 and the FOM T < 0.2. Furthermore, for propagationalong [1,-1,0], the self and cross phase-matching coefficients and the nonlinearities for bothpolarizations are essentially equal. In fact, this material resembles very closely a Kerr-lawmedium and has been used to demonstrate many all-optical switching devices, and spatialsoliton phenomena.

Active interband (between valence and conduction band) nonlinearities have provenvery useful for various communications functions such as demultiplexing, clock recovery,wavelength shifting and others.36 Electrons are “pumped” into the conduction band either bydirect electrical injection (via electrodes) or by absorption of radiation. When in the gainregime, an incident photon stimulates an electron transition from the conduction to the valenceband producing device gain. Furthermore, the change in population in the two bands inducesan index change which has the opposite sign to that in Figure 14. A 3 picojoule input pulseenergy is sufficient for creating a π phase shift! However, the recovery time is in the nsec

25

range. It has been shortened by maintaining strong pumping into the conduction band so thatthe switching process only utilizes a small fraction of the total available phase shift, reducingthe recovery time down to the 10 psec range.

Figure 14 (a) FOM W for semiconductors versus normalized (to the exciton linewidth) detuning fromthe band gap. (b) FOM T for bulk semiconductors for the two photon nonlinearity.

There are also multiple ultrafast mechanisms which occur near and at the transparencypoint, i.e. the point at which stimulated emission and absorption are just balanced. 36,37 (Notethat is not the lossless case!) As indicated schematically in Fig 15, an incident photon flux“burns” a hole via stimulated emission into the equilibrium electron distribution in theconduction band. This occurs on a sub-100 fsec time scale. Over a time period of 100s offemtoseconds the electron distribution is redistributed to a new equilibrium distribution by“carrier heating”. This second mechanism produces a relatively large index change, i.e.effective n2 .

26

Figure 15 The time sequence of the electron distribution, “hole burning” and “carrier heating”, in theconduction band due to stimulated emission.

Table 4 The properties of the ultrafast mechanisms are summarized in the table below.

Mechanism

Ultrafast Kerr (40 fsec)

Carrier Heating (carriers thermalize inband, 100s fsec)

Spectral Hole Burning (photons burn"hole" in conduction band, <100 fsec)

λµm

1.5

1.5

n2

cm2/W

-3x10 -12

4.5x10-12

small

αcm-1

40

40

W

0.5

0.75

T

4

3

Glasses Glasses have been under development for nonlinear optics since the earliestdays of their applications to long distance transmission fibers. Although silica glasses havevery small nonlinearities, ~2.4x10 -16 cm2/W, they have proven very useful for nonlinear opticsbecause of their very low loss in the communications windows which allows very largenonlinear phase shifts to be easily accumulated. However, the long lengths required (>10m)have made them awkward to work with, leading to the development of new, more nonlinearglasses as indicated in the table below.

Table 5 Nonlinear glasses

Glass

SiO 2

RN39

VA38

As.38 S.6240

As.37 S.6241

Absorption Edge (nm) λ (nm) α n 2 cm2/W

300

480

450

640

640

1550

1250

1250

1300

1550

<10-6 cm-1

<10-2 cm-1

6.6 dB/m

2 dB/m

2.4x10 -16

7x10-15

2.3x10-14

4x10-14

2x10-14

RN - PbO3 (40%); Ga2O3 (25%); Bi2O3(35%)VA - La2S3 (35%); Ga2S3 (65%)

27

Figure 16 Wavelength dispersion in the real and imaginary parts of χ(3)(-ω;ω,−ω,ω) for VA glass.

28

Table 6 Tabular Summary of Nonlinear Materials

Material

Kerr AIGaAs λGAP = 0.79 µm

Kerr AlGaAsλGAP = 0.75 µm35

CH InGaAsPλ GAP = 1.55 µm 37

Kerr InGaAsPλGAP = 0.75 µm42

Organics

PTS (crystal)43

DANS (side-chain polymer)44

DEANST (20% solution)45

Glasses (fibers)

SiO2

As0.38 As0.6240

RN39

†Assumed intensity = 1 GW/cm2

n2 cm 2/W

-4x10 -12

2x10-13

4.5x10 -12

-3x10-12

2.2x10-12

8x10-14

6x10-14

2.4x10-16

4.2x10-14

1.3x10 -14

αcm-1

18

0.1

40

40

0.8

<0.2

<10-2

10-6

0.02

0.01

W†

2.5

>8

0.75

0.5

>10

>5.0

>40

>103

16

13

T†

0.9

<0.1

3

4

<0.1

0.2

<1

<<1

<2

<0.1

λµm

0.81

1.55

1.55

1.55

1.6

1.32

1.06

>1.06

>1.32

1.06

An excellent example of the dispersion in the nonlinear properties of glasses is shownabove in Figure 16. Note that the non-resonant n2 region occurs after the effect of two photonabsorption on the response becomes negligible.

EFFECTIVE χ (2) FROM χ (3) , AND χ (3) FROM χ(2)

In the traditional evolution of nonlinear optics, χ(2) has been used to implementfrequency conversion, with great success. However, it has been known from the early days ofnonlinear optics (1967) that χ(2) :χ(2) generates nonlinear phase shifts in a fundamental beamduring SHG, an effect now called “cascading”. 46 It has recently been shown that this effect canproduce large effective third order nonlinearities.6 Furthermore, self-focusing and theexistence of spatial solitons during SHG were predicted back in 1976. 47 Such “quadratic”solitons have recently been observed.6 These effects are both normally associated with χ( 3 )

nonlinearities.On the other hand, χ (3) is usually associated with all-optical effects such as switching

devices, solitons etc. However, EFISH, Electric Field Induced Second Harmonic generationis a well-known phenomenon by which a DC electric field is used to generate a non-centrosymmetric medium via χ(3) , i.e. χ (2) (-2ω; ω, ω) = χ (3) (-2ω; ω, ω, 0) E DC . 2 One of themost exciting recent developments in nonlinear optics has been the use of optical fields toproduce automatically phase-matched SHG in both glass optical fibers and polymers.5,48,49

29

Cascading Effects

When a fundamental beam generates a second harmonic signal in a χ(2)-active crystalunder near phase-matching conditions, the fundamental amplitude can be partially depleted andits phase changed:

(37)

where the nonlinear phase shift is given by φ NL. 6 This phenomenon mimics many (but not all)χ (3) effects, for example n2 in ∆n = n2I. The anticipated values of an effective n2 nonlinearityare listed in the table below for selected materials.

Table 7

LiNbO3

LiNbO3

NPP

DAST

dii pm/V

36

600

diipm/V

5.8

84

n (effective) cm22 /W

2x10 -11

5x10-13

2x10 -10

6x10 -9

Note however that achieving such large effects requires that the material be phase-matchable,not yet the case for DAST! 50

The magnitude of the effect can easily be gotten from the coupled mode equations.Writing the fundamental and second harmonic as a1 exp[i(ωt - k1 z)] and a2 exp[i(2ωt - k 2z)]respectively with Γ∝χ (2) , ∆k = k2 - 2k1 and a2(0) = 0 (no SH input). Then, the previouslydiscussed coupled mode equations (Eqns 11 and 12). are:

(38)

Assuming now negligible depletion, | a1 (z)| = a 1(0), gives

(39)

i.e. an effect proportional to the fundamental intensity |a1(0)|2 . To identify the effective n2 , the

(40)

equivalent equation for χ(3) is reproduced below:

(41)

30

From this result, the following properties of cascading are clear:1. It is a non-local nonlinearity (requires propagation)2. a (z)= | a (z)| exp[-i NL φ NL

1 φ ]1 is the key parameter3. The sign determined by ∆k, +ve for n (2ω) > n (ω)2 1

4 . It is accompanied by “loss” to SHG

Further details are given in a later chapter in this book.

Figure 17 The spatial profile of the second harmonic due to up-conversion, and the regeneratedfundamental due to down-conversion. Input and diffracted fundamental (after propagation) - solidheavy line; regenerated fundamental - dashed line; total fundamental - light solid line

Another aspect of the χ (2) :χ (2) interaction is the existence of quadratic solitons, and theyhave indeed been observed experimentally in both slab waveguides and bulk media.6 The self-trapping mechanism can be understood with the help of the SHG coupled mode equations andFigure 17 as follows. Because da2 /dz is proportional to a1

2, a2 is narrower in space than a1 .Furthermore, because da then a1 can also be narrowed provided that a1 has not1/dz ∝ a2 a1 *,spread significantly due to diffraction. As long as the parametric gain length Lpg = [ Γpga 1 (0)]-1

is shorter than the diffraction length L , then self-trapping can occur. For non-zero phase-dmatching, the cascaded nonlinear phase shift will also affect mutual self-trapping via eitherself-focusing or de-focusing. The details will be discussed in a later chapter.

All-Optical Poling

There are now two examples of the creation of a χ(2) -active medium from an initiallyisotropic, amorphous medium by the application of optical fields. In both cases, optical fields at ω E (3)

1 exp[i( ωt - k 1z)] and 2 ω E 3 exp[i(2ωt-k 3z)] are mixed via χ (0;-ω , -ω , 2ω) toproduce a product term, <E3 > = E* exp[i(2k1

2E3 1 - k 3 )z] whose time average is non-zero. Allthe other terms have a zero time average. Formally this is written as a polarization PNL (0) =ε (3)

0 χ (0; -ω, -ω, 2ω)E*12 E3 exp[i(2k

1- k3)z] which generates a DC electric field proportional

to P NL (0). Note that the wavevector produced is exactly that needed for QPM if a d(2) ∝exp[i(2k

The mechanisms needed for poling polymers are different from those for fibers.1- k 3) z] can be induced by the physics of the material.48

49 I nthe polymer case, charge transfer molecules with trans-cis isomerization are used. Suchmolecules can have either a trans (linear) or cis (bent) geometry. The trans state is the moststable, and can have a large d(2) where-as the cis state has a small d(2) . Since the molecules havemixed parity ground and excited states, electrons can be raised from the ground state by either

31

one or two photon absorption. When both ω and 2ω beams undergo absorptionsimultaneously, the absorption ∝ C1 cos2θ + C2cos4 θ + <E*1

2 E3>cos3 θ where θ is the polar

angle defined by E·µ ∝cosθ. As a result, there is an anisotropic spatial distribution of transmolecules in the first trans excited state. That is, there are more excited molecules for θ 0than for θ π . While in the excited state, the trans molecules can efficiently undergo astructural transition to the cis form (trans-cis isomerization) which is more compact andspherical. While in this state, the molecules can rotate before they reconvert back to the morestable transform. As a result, during the duration of the optical fields, the trans population (inthe ground + the excited state) along θ 0 is preferentially partially depleted and thisdistribution is subsequently locked-in when the fields are turned off. Since the re-orientationof trans molecules in the ground state is very, very slow, there is a net orientation of transmolecules produced (more pointing along θ π), and hence a net d .(2)

The same interference between one and two photon absorption from impurity “trap”states occurs in optical fibers.5 The DC electric field created leads to the electrons beingemitted preferentially along the field direction This leads to charge separation inthe steady state with the electrons trapped by empty states along the beam periphery. Thisspatially anisotropic charge separation, has a periodicity of ~20µm. In the steady state, the

DC field is balanced by a Coulomb field EDC which continues to exist after thewriting beams are turned off, i.e. a form of EFISH. Thisexp[i(2k )z]. When a fundamental beam alone is now incident onto the fiber, there is a1 - k3nonlinear polarization created of the formSubstituting now for EDC gives

(42)

This looks like a χ (5) process and leads to phase-matched second harmonic generation!

SUMMARY

The field of nonlinear optics has continued to expand and become richer in phenomenasince its inception in the 1960s. Furthermore, new materials and new ways of using oldmaterials have continuously become available, reducing the power levels necessary forefficient interactions. Provided here has been a brief and incomplete overview, as anintroduction to the more detailed chapters that follow in this book.

This research was supported by the US National Science Foundation and the Air Forceof Scientific Research.

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periodic domain grating by electron beam writing and its application to nonlinear optics”, inNonlinear Optics, ed. S. Miyata, (Elsevier Science Publishers, Amsterdam, 1992)8. D. Marcuse (1974) Theory of Dielectric Optical Waveguides, Academic Press, NewYork; numerous articles in (1990) T. Tamir (ed.) Guided-Wave Optoelectronics, 2nd ed., Vol26 in Electronics and Photonics, Springer-Verlag, Berlin.9. H. Ito and H. Inaba, ,,Efficient phase-matched second-harmonic generation method infour-layered optical-waveguide structure“ Opt. Lett. 2: 139-141 (1978).10. W. Wirges, S. Yilmaz, W. Brinker, S. Bauer-Gogonea, S. Bauer, M. Jaeger, G.I.Stegeman, M. Ahlheim, M. Stahelin, B. Zysset, F. Lehr, M. Diemeer and R. Flipse, “PolymerWaveguide for Modal Dispersion Phase Matched Second-Harmonic Generation”, Appl. Phys.Lett.., 70,3347-9 (1997)11. M. Jaeger, G.I. Stegeman, M. Diemeer, C. Flipse and G. Mohlmann, “ModalDispersion Phase-Matching over 7 mm Length in Overdamped Polymeric ChannelWaveguides”, Appl. Phys. Lett., 69:4139-41 (1997)12. for example: M.M. Fejer, G.A. Magel, D.H. Jundt and R.L. Byer, "Quasi-phase-matched second harmonic generation", IEEE, J. Quant. Electron., 28, 2631-54 (1992); T.Suhara and H. Nishihara, IEEE J. Quant. Electron, 26:1265-84, 199013. K. Mizuuchi, K. Yamamoto and T. Taniuchi, "Second harmonic generation of bluelight in a LiTaO3 waveguide", Appl. Phys. Lett., 58, 2732-4 (1991)14. Y. Shuto, T. Watanabe, S. Tomura, I. Yokohama, M. Hikita and M. Amano, “Quasi-Phase-Matched Second Harmonic Generation in Diazo-Dye-Substituted Polymer ChannelWaveguides”, IEEE J. Quant. Electron., 33:349-357 (1997)15. for example, G. Tohmon, J. Ohya, K. Yamamoto and T. Taniuchi, “Generation ofUltraviolet Picosecond Pulses by Frequency Doubling of Laser Diode in Proton-ExchangedMgO: LiNbO3 Waveguide”, IEEE Phot. Techn. Lett., 2, 629-31 (1990)16. C.Q Xu, H. Okayama and M. Kawahara, “Wavelength conversions between the twosilica fibre loss windows at 1.31` and 1.55 µm using difference frequency generation”,Electron. Lett., 30, 2168-9 (1994)17. C.Q. Xu, H. Okayama, K. Shinozaki, K. Watanabe and M. Kawahara, “WavelengthConversion Around 1.5 µm by Difference Frequency Generation in Periodically Domain-Inverted LiNbO3 Channel Waveguides”, Appl. Phys. Lett., 63, 1170-2 (1994); S.J.B. Yoo, C.Caneau, R. Bhat, M.A. Koza, A. Rajhel and N. Antoniades, “Wavelength conversion bydifference frequency generation in AlGaAs waveguides with periodic domain inversionachieved by wafer bonding”, Appl. Phys. Lett., 68, 2609-11 (1996)18. see for example W.R. Bosenberg, L.K. Cheng and C.L. Tang, “Ultraviolet opticalparametric oscillator in β-B a B 2O4", Appl. Phys. Lett., 54, 13-5 (1989); W.R. Bosenberg, W.S.Pelouch and C.L. Tang, “High efficiency and narrow linewidth operation of a two crystal β-BaB2O4 optical parametric oscillator”, Appl. Phys. Lett., 55, 1952-4 (1989)19. for example, D.E. Spence, S. Wielandy, C.L. Tang, C. Bosshard and P. Gunter, “High

33

average power, high-repetition rate femtosecond pulse generation in the 1-5 µm region usingan optical parametric oscillator”, Appl. Phys. Lett., 68, 452-4 (1996)20. T.L. Penner, H.R. Motschmann, N.J. Armstrong, M.C. Ezenyilimba, and D.J. Williams,“Efficient phase-matched second-harmonic generation of blue light in an organic waveguide”,Nature 367:49-51 (1994).21. G. Khanarian, R.A. Norwood, D. Haas, B. Feuer, and D. Karim, " Phase-matchedsecond-harmonic generation in a polymer waveguide. Applied Physics Letters, 57: 977-979,(1990)22. D. Chen, H.R. Fetterman, A. Chen, W.H. Steier, L.R. Dalton, W. Wang and Y. Shi,“Demonstration of 110 Ghz electro-optic polymer modulators”, 70,3335-7 (1997)23. extensive material listing by S. Singh, “Organic and Inorganic Materials”, in CRCHandbook of Laser Science and Technology, Supplement 2: Optical Materials, M.J. Webbereditor (CRC Press, Ann Arbor, 1995) pp147-26624. E. Rosencher, A. Fiore, B. Vinter, V. Berger, Ph. Bois and J. Nagle, “QuantumEngineering of Optical Nonlinearities”, Science, 271, 168-173 (1996)25. T. Doumuki, H. Tamada and M. Saitoh, “Phase-Matched Second-Harmonic Generationin Ta O2 5/KTP Waveguide”, Appl. Phys. Lett., 65,251-219 (1994).26. D. Eger, M. Oron, M. Katz and A. Zussman, “Highly efficient blue light generation inKTiO 4waveguides, Appl. Phys. Lett., 64,3208-10 (1994).27. S-Y Yi, S-Y Shin, Y-S Jin and Y-S Son, “Second-harmonic generation in a LiTaO 3waveguide domain-inverted by proton exchange and masked heat treatment”, Appl. Phys. Lett.,68,2493-5 (1996)28. M.A. Arbore and M.M. Fejer, “Singly Resonant Optical Parametric Oscillation inPeriodically Poled Lithium Niobate Waveguides”, Opt. Lett., 22: 151-3 (1997),29. A. Harada, Y. Okazaki, K. Kamiyama and S. Umegaki, “Generation of blue coherentlight from a continuous-wave semiconductor laser using an organic crystal-cored fiber”, Appl.Phys. Lett., 59, 1535-7 (1991).30. P. Gunter and J-P. Huignard, Photorefractive Effects and Materials”, in PhotorefractiveMaterials and Their Applications I, Fundamental Phenomena, Volume 61 in series Topics inApplied Physics, P. Gunter and J.-P. Huignard editors, (Springer-Verlag, Berlin, 1988) pp 7-7331. W.M. Hetherington III, Z.Z. Ho, E.W. Koenig, R.M. Fortenberry and G.I. Stegeman,"Surface CARS spectroscopy of pyridine and phenol on ZnO optical waveguides", Chem.Phys. Lett., 128, 150-5 (1986)32. G.I. Stegeman and A. Miller, “Physics of all-optical switching devices”, book chapterin Photonic Switching, Vol I, ed. J. Midwinter, (Academic Press, Orlando, 1992), 81-146(1993).33. D.H. Austin +22 co-authors, “Research on Nonlinear Optical Materials”, Appl. Opt. 26 ,211-234 (1987)34. reviewed in Peyghamberian, N. and Koch, S.W. (1990) ‘Semiconductor NonlinearMaterials’, in H.M. Gibbs, G. Khitrova and N. Peyghamberian (eds.), Nonlinear Photonics,Springer-Verlag, New York, pp. 7-6035. J. U. Kang, G. I. Stegeman, D. C. Hutchings, J. S. Aitchison and A. Villeneuve, ”TheNonlinear Optical Properties of AlGaAs at the Half Band Gap”, IEEE J. Quant. Electron., 33 ,341-8 (1997)36. M.J. Adams, D.A.O. Davies, M.C. Tatham, and M.A. Fisher, “Nonlinearities insemiconductor amplifiers”, Optical and Quantum Electronics 27, 1-13 (1995)37. A. D’Ottavi, E. Iannone, A.Mecozzi, S. Scotti, P. Spano, R. Dall’Ara, G. Guekos, andJ. Eckner, “Terahertz four wave mixing spectroscopy of InGaAsP semiconductor amplifiers”Appl. Phys. Lett. 65,2633-35 (1994)38. I. Kang, T.D. Krauss, F.W. Wise, B.G. Aitken and N.F. Borerelli, “Femtosecondmeasurement of enhanced optical nonlinearities of sulfide glasses and heavy-metal-dopedoxide glasses”, J. Opt. Soc. Am. B, 12,2053-9 (1995)

34

39. D. W. Hall, M.A. Newhouse, N.F. Borrelli, W.H. Dumbaugh, and D.L. Weidman,“Nonlinear optical susceptibilities of high-index glasses”, Appl. Phys. Lett. 54, 1293-5 (1989)40. M. Asobe, K. Suzuki, T. Kanamori and K. Kubodera, “Nonlinear refractive indexmeasurement in chalcogenide-glass fibers by self-phase modulation”, Appl. Phys. Lett., 60,1153-4 (1992)41. M. Asobe, H. Kobayashi, and H. Itoh, “Laser-diode driven ultrafast all-opticalswitching by using highly nonlinear chalcogenide glass fiber”, Opt. Lett. 18, 1056-8 (1993)42. Ippen and Anderson; K.L. Hall, A.M. Darwish, E.P. Ippen, U. Koren, G. and Rayborn,“ Femtosecond index nonlinearities in InGaAsP optical amplifiers”, Appl. Phys. Lett. 62, 1320-2 (1993)43. B. Lawrence, M. Cha, J.U. Kang, W. Torruellas, G.I. Stegeman, G. Baker, J. Meth andS. Etemad, "Large Purely Refractive Nonlinear Index of Single Crystal P-Toluene Sulfonate(PTS) at 1600 nm", Electron. Lett., 30:447-8 (1994)44. D.Y. Kim, M. Sundheimer, A. Otomo, G.I. Stegeman, W.G.H. Horsthuis and G.R.Mohlmann, “Third order nonlinearity of DANS waveguides at 1319 nm”, Appl. Phys. Lett.,63, 290-2 (1993).45. H. Kanbara, H. Kobayashi, K. Kubodera, T. Kurihara and T. Kaino, “Optical KerrShutter Using Organic Nonlinear Optical Materials in Capillary Waveguides”, IEEE Phot.Techn. Lett., 3, 795-7 (1991)46. L. A. Ostrovskii, "Self-action of light in crystals", JETP Lett. 5, 272-5 (1967); Chr.Flytzanis, in Quantum Electronics, ed. H. Rabin and C.L. Tang, (Academic, NY, 1975), Vol1, Part A.47. Yu. N. Karamzin, A. P. Sukhorukov, "Mutual focusing of high-power light beams inmedia with quadratic nonlinearity", Zh.Eksp.Teor.Phys 68, 834-40 (1975) (Sov. Phys.-JETP41, 414-20 (1976))48. R.H. Stolen and H. W.K. Tom, “Self-organized phase-matched harmonic generation inoptical fibers”, Opt. Lett., 12, 585-7 (1987); N.B. Baranova, B.Ya. Zel’dovich, “Physicaleffects in optical fields with nonzero average cube, <E3 > ≠ 0", JOSAB 8, 27-32 (1991)49. F. Charra, F Devaux, J.M. Nunzi and P. Raimond, “Picosecond light-inducednoncentrosymmetricity in a dye solution”, Phys. Rev. Lett., 68, 2440-2 (1992); C. Fiorini, F,Charra, J.M. Nunzi and P. Raimond, “Quasi-permanent, all-optical encoding ofnoncentrosymmetry in azo-dye polymers”, J. Opt. Soc. Am. B, 1984-2003 (1997)50. for example: Ch. Bosshard, K, Sutter, R. Schlesser and P. Gunter, "Electro-optic effectsin molecular crystals”, J. Opt. Soc. Am. B, 10, 867-85 (1993); G. Knopfle, R. Schlesser, R.Ducret and P. Gunter, “Optical and Nonlinear Optical Properties of 4'-dimethylamino-N-methyl-4-stilbazolium tosylate (DAST) crystals”, Nonlin. Opt., 9:143-9 (1995)

35

INTRODUCTION TO ULTRAFAST AND CUMULATIVENONLINEAR ABSORPTION AND NONLINEAR REFRACTION

Eric W. Van Stryland

Center for Research and Education in Optics and Lasers, CREOLUniversity of Central FloridaOrlando, Florida 32816-2700

INTRODUCTION

We introduce several of the basic processes that lead to third-order nonlinearresponses. In general these nonlinearities range from thermally-induced index of refractionchanges to the ultrafast bound-electronic nonlinearities of two-photon absorption and itsassociated nonlinear refraction. While it is relatively easy to distinguish between thermaland ultrafast nonlinearities, there are often ambiguities in single experiments that requirecareful studies to unravel the basic physical processes. For example, two-photon absorptionis easily confused with excited-state absorption where there is a real rather than virtualintermediate state. We give examples of experimental methods to distinguish such processesalong with some details of the nonlinear mechanisms. Specifically, we give examples ofnonlinear refraction and absorption in semiconductors and wide gap dielectric materials.We introduce the concept of causality and how this relates the absorption and refractionspectra for both linear and nonlinear systems. However, for nonlinear systems, thenondegenerate nonlinearity is needed as obtained, for example, by pump-probespectroscopy. We also briefly discuss higher-order nonlinearities associated with freecarriers (excited states) being generated by two-photon absorption. This leads us to discussexcited-state nonlinearities where the excitation is via linear absorption. These appear asthird-order responses but are associated with a cascading of linear susceptibilities, i.e.χ(1):χ (1) where χ is an electric susceptibility. For these nonlinearities it is more convenientto define cross sections than to use the usual expansion in terms of higher-ordersusceptibilities. If a χ(3) is defined for a cumulative nonlinearity, it will not be a materialconstant but will depend on the illumination parameters such as the laser pulsewidth. Wego on to look at two-beam interactions which can lead to interesting phenomena such as“weak-wave retardation” and two-beam coupling whose description is useful for theunderstanding of nonlinear light scattering.

Beam Shaping and Control with Nonlinear Optics37Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

“THIRD-ORDER"NONLINEARITIES

The textbook derivation of nonlinear optics takes the wave equation;

(4)

(1)

describing the interaction of light with matter through the polarization driving term andexpands the polarization P in a Taylor series in the electric field E. Ignoring the vectornature of P and E, nonlocality, the tensorial nature of the susceptibilities and the spatial zdependence, this expansion is;¹

(2)

where χ(n) is defined as the nth-order time-dependent response function or time-dependentsusceptibility. Here, we take the field as

(3)

with E a complex, slowly varying function of time and space, i.e. it contains both amplitudeand phase information. As an example, for harmonic generation (second harmonic from χ(2)

or third from χ(3)) the nonlinearity by necessity follows the rapidly varying field. The onlymaterial response capable of this is the ultrafast bound-electronic response, i.e. the so-called"instantaneous" response. Self-action effects come from the odd order susceptibilities andcan be caused by a variety of nonlinearities with response times from "instantaneous" forbound-electronic NLR in silica fibers, to seconds as for some photochromic effects (e.g. inphotodarkening sunglasses).

As is usually done, Eq. 2 is Fourier transformed to give frequency dependentfunctions. This gives P(ω). However, for pulsed inputs, as are usually used in nonlinearoptics, the electric fields are narrow functions of frequency and they are still allowed toslowly vary in time (the slowly varying envelope approximation).2 Similarly the polarizationis allowed to slowly follow the field envelope. Therefore P(ω) becomes a slowly varyingfunction of time, Pω(t). For a single frequency input at ω, and looking only at self-action,this leads to a slowly varying polarization at ω given by

There is a also an implied slow variation with propagation direction, z, not explicitly shown.Writing this Fourier transform as a function of t, as done here and elsewhere, can besomewhat confusing. It assumes that changes of the field, and thus polarization, are so

38

slow that the material response is the same as for a cw input (i.e. delta function infrequency). Here we ignore the degeneracy factors and the tensor and polarizationproperties of these nonlinearities. Thus, for example, χ (3) is an effective nonlinearsusceptibility.

In order to demonstrate how this nonlinearity leads to nonlinear absorption, NLA,and nonlinear refraction, NLR, we return to the rapidly varying field and polarization, insertthis nonlinearity into the wave equation and neglect diffraction effects i.e. ∇2 is replaced by∂2/∂z2. The neglect of diffraction effects is a very useful approximation that allows theseparation of absorptive and refractive effects. It is a good approximation under theconditions that the linear optics depth of focus of the beam (Z

20=πw0

2/λ) is much greaterthan the sample thickness L (w0 is the half width at the 1/e maximum in the irradiance,HW1/e 2M), and the input beam profile is unaffected by the phase distortion induced bynonlinear interactions (i.e. the induced phase distortion is much less than Z0/L). This regimeis called "external self-action".3 The wave equation with Eq. 3 keeping only the mostrapidly varying terms can then be reduced to4

and

(9)

(8)

where the effect of the linear index has been included. Transforming to coordinatestraveling with the wave, τ =t-zn/c and z'=z, leads to the simplified equation4

(5)

(6)

where E and P are functions of z' and τ and we have only included up to third-orderresponses. Looking at the magnitude and phase of E by defining

(7)

leads to separate equations describing loss (or gain) and phase shifts;

Given that the irradiance I is proportional to E02, Eqs. 8 and 9 (or 6) clearly show how the

real and imaginary parts of χ(3) lead to irradiance dependent phase shifts and lossrespectively.

The "i" in Eq. 6 is important. It shows the π/2 phase shift between polarization andfield. Thus, the polarization can be viewed as having a real part which is in phase with thedriving electric field leading to a change of field phase, Eq. 9, and an imaginary part which

39

leads to a change in the field amplitude, Eq. 8, (index and absorption respectively). Wereturn to this point later when we discuss two-beam coupling.

Rewriting Eqs. 8 and 9 in terms of the irradiance, I, and considering only thenonlinearly induced phase φ results in,

(12)

(10)

(11)

and with k=ω/c,

where β is defined as the two-photon absorption (2PA) coefficient (in m/W) and n2 i sdefined as the nonlinear refractive index (in m2/W). Two-photon absorption requires thesimultaneous absorption of two quanta of energy Nonlinear refraction associated withthis process is known as the bound electronic Kerr effect characterized by n2. In theliterature, n is often used to discuss everything from thermal and reorientatioal (eg. for2

CS ) index changes, to changes in index from saturation of absorption to ultrafast χ2(3)

nonlinearities. Here we restrict the use of n2 to describe the ultrafast index change.Gaussian units (esu) are often used for n2, and a useful relation is

where the right hand side is all in MKS units (SI).The nonlinear optical properties of materials range from the index change due to the

ultrafast interaction of light with bound electrons to the index change caused by therelatively slow thermal expansion of a liquid due to linear absorption. The effects caused bythese nonlinear interactions with matter range from the reduction of transmittance fromincreasing absorption with increasing irradiance (e.g. two-photon absorption) to beamspreading from self-defocusing to the ultimate nonlinear interaction of laser-induceddamage. The textbook analysis in terms of χ is most convenient for describing ultrafastresponses. It can also be useful for describing nonlinear interactions where the materialresponse time is short compared with the time scales of the experiments, e.g. compared withthe laser pulsewidth. An example here is the reorientation of molecules of CS2 leading toself focusing where most often optical pulses are much longer than the ≅2 ps reorientationaltime. However, it may not be the most convenient way to describe all nonlinearinteractions, for example, those we refer to as cumulative nonlinearities, i.e. ones that buildup during the pulse and whose response time is longer than, or on the order of, thepulsewidth. Examples here include thermal nonlinearities, better described by the indexchange with temperature, dn/dT, and excited state nonlinearities where the relaxation timeof the excited state is longer than the pulsewidth. In the latter case, the use of crosssections is more convenient than the use of electric susceptibilities. These parameters arematerials constants, i.e. independent of the irradiation conditions.

We discuss methods for measuring these different nonlinear responses along withconvenient ways to describe them. We concentrate on two types of related nonlinear

40

interactions; bound-electronic and free-carrier or excited-state nonlinearities. We alsodescribe the intrinsic link that exists between nonlinear refraction and nonlinear absorptiondue to causality. This linkage allows us to write nonlinear Kramers-Kronig relationsrelating the nonlinear refraction and nonlinear absorption analogous to the relations relatingthe dispersion of linear refraction to the linear absorption spectrum.

KRAMERS-KRONIG RELATIONS

Mathematically, the complex response function of any linear, causal system obeys adispersion relation that relates the real and imaginary parts of the response function viaHilbert transform pairs. In optics, Kramers-Kronig (KK) relations are dispersion relationsrelating the frequency dependent refraction, n(ω) to an integral over all frequencies of theabsorption α(ω) and vice-versa. Toll5 gave an interesting way of viewing the necessity ofthese dispersion relations as illustrated in Fig. 1. The electric field of an optical pulse in time(a), consisting of a superposition of many frequencies, arrives at an absorbing medium. Ifone frequency component (b) is completely absorbed we could naively expect that theoutput should be given by the difference between (a) and (b) as shown in (c). However,this would obviously violate causality since there is an output signal occurring at timesbefore the incident wave train arrives. In order for causality to be satisfied, the absorption

Figure 1. Illustration of the need for index dispersion.; a) input pulse electric field, b) monochromaticabsorption in time, c) “expected” output without dispersion.

41

of one frequency component must be accompanied by phase shifts of all of the remainingcomponents in just such a manner that the field prior to the pulse vanishes. Such phaseshifts result from the index of refraction and its dispersion.

As will be seen, the real and imaginary parts of χ (3) , or n2 and β , are related throughcausality by Kramers-Kronig relations in much the same way as n and α are related forlinear optics.6,7,8,9 This may appear somewhat surprising at first glance since Kramers-Kronig relations are derived from linear response theory. However, we treat the material inthe presence of a bright light source as a new “linear” system and then apply causality onthis changed system and obtain relations between the changes in absorption, ∆α, a n dchanges in refraction, ∆n. An important difference here with what might first be written forthis relation is that the connection is between nondegenerate nonlinearities. That is, thematerial plus light beam is held constant so that in the integral relation for the nonlinearabsorption, it is the change in absorption at frequency ω due to the presence of a strongbeam at ω e that is needed; ∆α(ω;ωe) or β (ω;ωe ). The above concept is illustrated in Fig. 2.Thus, using this relation requires a knowledge of the spectrum of the nondegeneratenonlinear absorption as obtained, for example, from pump-probe spectra. Therefore ,∆n(ω;ωe) or n2 (ω;ωe) and ∆α(ω;ωe ) or β (ω;ωe )describe the change in refractive index andabsorption coefficient, respectively, for a weak optical probe of frequency ω when a strongpump of fixed frequency ωe is applied as illustrated in Fig. 2. The following shows how tomathematically cancel this precursor field in both linear and nonlinear interactions.

Figure 2. Illustration of Kramers-Kronig relations for linear and nonlinear systems.

42

Linear Kramers-Kronig relations

( 1 8 )

(17)

In a dielectric medium the linear optical polarization, P(t) as given in Eq. 2 is,

(13)

The response function, χ(τ), is equivalent to a Green's function, as it gives the response(polarization) resulting from a delta function input (electric field). This equation is oftenstated in terms of its Fourier transform, where the convolution in Eq. 13 is transformed intoa product

(14)

where χ(ω) is the frequency dependent susceptibility defined by,

(15)

Causality states that the effect cannot precede the cause requiring that E(t-τ) cannotcontribute to P(t) for t < (t -τ). Therefore, in order to satisfy causality, χ(τ ) = 0 for τ < 0 sothat the integral in Eq. 13 need only be performed for positive times. An easy way to seethis is to consider the response to a delta function E(τ) = E0 δ(τ), where the polarizationwould then follow χ( t). The usual method for deriving the KK relation from this point is toconsider a Cauchy integral in the complex frequency plane. However, in the Cauchyintegral method, the physical principle from which dispersion relations result (namelycausality) is not obvious. The principle of causality can be stated mathematically as

(16)

i.e., the response to an impulse at t = 0 must be zero for t < 0. Here Θ(t) is the Heavisidestep function defined as Θ(t) =1 for t > 0 and Θ(t) = 0 for t < 0. Upon Fourier transformingthis equation, the product in the time domain becomes a convolution in frequency space

which is the KK relation for the linear optical susceptibility indicates principle value).Thus, the KK relation is simply a restatement of the causality condition (Eq. 16) in thefrequency domain. Taking the real part with χ=χ’+ i χ”, we have,

It is more usual to write the optical dispersion relations in terms of the more familiarquantities of refractive index, n(ω), and absorption coefficient, α(ω). These relations arederived in Ref. 8 using relativistic arguments. However, if we assume dilute media withsmall absorption and indices, we obtain the identical result. By setting n(ω )-I=χ’ /2 and

43

α(ω) =ωχ”(ω)/c, we obtain

(19)

Since E(t) and P(t) are real, n(-ω) = n(ω) and α(-ω) = α(ω), which when transforming theintegral in Eq. 19 to 0 to ∞ gives the final result of

(20)

Nonlinear Kramers-Kronig Formalism

Clearly causality holds for nonlinear systems as well as for linear systems, however,confusion has existed about the application of causality to nonlinear optics. As statedpreviously, the usual KK relations are derived from linear dispersion theory, so it wouldappear impossible to apply the same logic to a nonlinear system. The simplest way to viewthis process is to first linearize the problem by viewing the material plus strong perturbinglight beam as a new linear system upon which we apply causality, i.e. the light interactionresults in a new absorption spectrum for the material as illustrated in Fig. 2. Thus, both theNLA and the NLR are equivalent to pump-probe spectra with a fixed pump frequency andvariable probe frequency.

Here we discuss the Kramers-Kronig relation used to calculate the change inrefractive index from the change in third-order absorption. The third-order susceptibility canbe determined by integration over positive times only as,

(21)

It is now possible to use the same method used earlier for the linear susceptibility in order toderive a dispersion relation for χ(3). For example, we can write

( 2 2 )

where j can refer to any of the three indices and then calculate the Fourier transform of thisequation. We could also use two or three step functions, however, the simplest result isobtained by taking just one. This gives us a generalized nonlinear Kramers-Kronig relationfor a non-degenerate third-order nonlinear susceptibility (here choosing j=1);

(23)

This integral is over only one frequency argument, Ω , and all other frequencies are heldconstant. Thus, we do not obtain a relationship between the degenerate Kerr coefficient,n2 (ω), and the degenerate two-photon absorption coefficient, β (ω).

Using an analogous definition for the nondegenerate n2 and β , defined by Eqs. 10and 11;

44

(24)

(25)

As discussed in Refs. 10 and 11 there is a factor of two difference between the degenerateand nondegenerate definitions. This accounts for cross-modulation or "beating" terms thatresult for the nondegenerate case for nonlinearities that can respond fast enough. This issometimes referred to as "weak -wave retardation" as discussed later in this chapter.12,13

The definition of the nondegenerate β now includes all possible nonlinear mechanismsincluding 2PA, Raman and AC-Stark effects as discussed later in the next section and inRef. 7.

We can relate these quantities by choosing ω1 = -ω 2 with ω2 = ω e to be theperturbing field frequency (or excitation field) in Eq. 23 and convert to an integral from 0 to∞ to give

(26)

which with the definitions of Eqs. 24 and 25 leads to

(27)

Although the calculation as illustrated above gives the nondegenerate NLR for a specificpair of frequencies, in most cases we set ω=ωe (after the integral is performed) and considerself-refraction. This gives two times the degenerate nonlinear refractive index n2 (the factorof two difference from weak-wave retardation).

The origin of the nonlinearity need not be optical but of any external perturbation.For example this method has been used to calculate the refractive index change resultingfrom an excited electron-hole plasma and a thermal shift of the band edge. For cases wherean electron-hole plasma is injected, the subsequent change of absorption gives the plasmacontribution to the refractive index. In such cases, the excitation in Eq. 27 is not necessarilyan optical frequency but can be taken as a general excitation. Thus, ωe can represent, forexample, the sample temperature change. For nonlinearities due to an existing plasma, ω ise

taken as the change in plasma density. Van Vechten and Aspnes14 obtained the lowfrequency limit of n2 from a similar KK transformation of the Franz-Keldysh electro-absorption effect where, in this case, ωe is the DC field. It is important to note that we mustfirst perform the integral before setting ω=ω e . This has been a source of confusion inconsidering saturation of a two-level atomic system as discussed below.

This form of calculation of the refractive index for nonlinear optics is often moreuseful than the analogous linear optics relation since absorption changes (which can beeither calculated or measured) usually occur only over a limited frequency range and, thus,the integral in Eq. 27 need only be calculated over this finite frequency range. Incomparison, for the linear KK calculation, absorption spectra tend to cover a very largefrequency range and it is necessary to take account of this full range in order to obtain aquantitative fit for the dispersion. For both linear and nonlinear systems the refractive index

45

changes are usually quite extensive in frequency. Therefore, a calculation of absorptionchanges from refractive index changes is seldom performed.

Example: Two-Level Atom

The familiar saturable "two-level atom" problem is illustrative of the application ofthe nonlinear Kramers-Kronig relation. The absorption spectrum is given by; 2,15

(28)

As noted, for example in Yariv’s text 15 , this does not obey causality. The problem is that asω is tuned the excitation is tuned, and the saturation changed for this degenerate form of theabsorption. What is needed is the nondegenerate, or pump-probe, absorption spectrum;

Here ωe is fixed and α clearly obeys causality since now the absorption spectrum is identicalin shape to the linear absorption but simply reduced in amplitude by homogeneoussaturation. (Note that we have ignored population pulsations and other more exotic effectsin this oversimplified description of the 2-level atom).2 We next present experimental dataand methods for determining the nonlinear coefficients in semiconductors.

ULTRAFAST NONLINEARITIES IN SEMICONDUCTORS AND DIELECTRICS

The nonlinear optical properties of semiconductors are used for a variety ofapplications (e.g. optical switching and short pulse production). Some of the largestnonlinearities ever reported have been in semiconductors and involve near-gap excitation.Unfortunately, these resonant nonlinearities, by their nature, involve significant linearabsorption. Here we discuss the nonlinear response in the transparency range, i.e. forphoton energies far enough below the band-gap energy Eg that bound-electronicnonlinearities either dominate the nonlinear response or are responsible for initiating free-carrier nonlinearities (e.g. two-photon absorption created free carrier nonlinearities). Thebound-electronic nonlinearities due to the anharmonic response of bound, valence electronshave been extensively studied in the past 16,7

(29)

. The response time for these nonlinearities hasbeen estimated as on the order of 1 femtosecond or less. This ultrafast response time hasbeen exploited in applications such as soliton propagation in glass fibers and recently in thegeneration of femtosecond pulses in solid-state lasers (Kerr-lens mode-locking). Anothersignificant application is the development of ultrafast all-optical-switching (AOS) devices.

46

Our interest here is to utilize the Kramers-Kronig relations to determine thedispersion of the bound-electronic n2 from the spectrum of nondegenerate nonlinearabsorption. We begin by looking at semiconductor nonlinearities. However, we will findthat this also provides a good description of nonlinearities in wide-gap dielectrics. In thetransparency range for photon energies greater than half the bandgap energy (i.e.

two-photon absorption (2PA) dominates the nonlinear losses.17,18 In a similarway, three-photon absorption (3PA) should dominate for Othernonlinearities, usually optical damage, make it difficult to observe four-photon absorptionand higher nonlinear absorption. Wherrett19 has shown via second-order perturbationtheory for the 2PA transition rate using only one dipole allowed with one dipole forbiddentransition with two parabolic bands, that the 2PA scales as Eg

-3. The forbidden transition(as shown in Fig. 3) is what is referred to as a self transition either within the valence bandor within the conduction band that essentially couples the s-like part of the wave function tothe p-like part. For semiconductors this coupling depends on the electron or holemomentum, κ , going to zero at κ =0. Third-order perturbation theory for allowed-allowed-allowed transitions shows that 3PA scales as Eg

-7. The identical result for 2PA wasobtained using a Keldysh tunneling model by Brandi et. al.20

The traditional theoretical approach for calculating n2 and β involves direct quantummechanical calculation of the complex χ (3) using second-order perturbation theory. Anotherapproach, more suited for absorptive processes, uses transition rate calculations to arrive atβ via diagrams as in Fig. 3. In order to calculate the nonlinear refraction, we must performa Kramers-Kronig integral of the nondegenerate nonlinear absorption over all frequencies.This integral, therefore, includes frequencies above the bandgap where linear absorption ispossible. In this situation two nonlinear absorption processes in addition to two-photon

Figure 3. The two-parabolic-band model of a semiconductor showing an allowed followed by a forbiddentransition (left) and a forbidden followed by an allowed transition (middle), promoting an electron from thevalence to the conduction band via two-photon absorption. The drawing on the right depicts three-photonabsorption via an allowed-allowed-allowed transition.

47

absorption become possible, electronic Raman and the AC-stark effects. Both of thesephenomena contribute significantly to the overall nonlinear refractive index. Theseprocesses are most easily modeled by using a Keldysh tunneling type model with a“dressed” final state and only keeping the first-order perturbation term in the expansion. 21

Using a two-parabolic band model for the semiconductor, this method gives identical resultsto second-order perturbation theory for 2PA as given by Wherrett19 in the degenerate case,but includes these extra nonlinear absorption contributions and directly gives thenondegenerate results as well. Calculations of the nondegenerate nonlinear absorption inthe two-parabolic bad model are given in Ref. 22. Both methods give the same scaling ofnonlinear absorption with linear index and bandgap energy. In addition, the scaling for n2 isalso obtained either from the Kramers-Kronig integral or from the scaling of χ(3) in the two-band model as given below.

An important goal of nonlinear optical materials characterization is to determinetrends from which scaling laws can be developed to give a predictive capability and checktheories. After experiments (to be discussed) have been carefully performed and analyzed toextract β and n2 , we can compare the results to theory. It would be best to make thiscomparison with the nonlinear spectra for a given material. Unfortunately there are fewmaterials for which nonlinear spectra are known. One reason for this is that tunable sourceswith the required irradiance, pulsewidth and beam quality are not typically available.Instead we use simple scaling relations to scale out the material dependence. Wherrett19 hasshown that the third-order nonlinear susceptibility, χ (3), in inorganic solids should scale as

(30)

where the complex function ƒ depends only on the ratio ω/Eg(i.e. upon which states areoptically coupled). This yields;

(31)

(32)

where the defined functions F and G are band structure dependent. Therefore, F gives the2PA spectrum and G gives the dispersion of n2 . One method to test the above scalingrelations is to scale the experimental data to obtain the experimental functions (designatedby the superscript e);

(34)

where βe and n2e are experimental values of β and n2 , and K and K' are proportionality

constants. Here Ep is the Kane energy and is nearly material independent with a value near

(33)

48

21 eV.23 Figure 4 plots the scaled data for β as a function of ω/Eg, with the predicteddependence from the two-parabolic band model given from perturbation theory or theKeldysh tunneling model. The value of K is fit to the data shown for medium gapsemiconductors and has the value K=3100 in units so that Ep and Eg are in eV and β is incm/GW and K’ comes directly frm the Kramers-Kronig integration (see Eq. 36).18,8 Figure 4shows 2PA turning on sharply at half the band-gap energy (there are many data below Eg/2with β ≅0 not shown) and then slowly decreasing for photon energies approaching the bandgap. While this data is for degenerate 2PA, the theory for nondegenerate 2PA has a similarshape but turns on at (ω1+ω2)=Eg. The functional dependence of the nondegenerate 2PAis given by,

(35)

where Once this functional dependence is known, the extent of variation ofthe magnitude of β can better be seen in a log-log plot of β scaled by the spectral responsefunction F(x

28

1 =x2) versus Eg as in Fig. 5. Here data for other materials including wide gapdielectric materials are included.

With the inclusion of Raman and AC-Stark contribution to the NLA in Eq. 35, wecan perform the Kramers-Kronig integral and obtain the dispersion of the nondegenerateNLR. This calculation, in agreement with the scaling relation of Eq. 32 yields for n

Figure 4. Scaled values of β (according to Eq. 33) as a function of ω /Eg along with the theoretical plot ofthe function F (see Eq. 35 with x1 =x2). Data from Ref. 18.

49

(36)

(37)

while the nondegenerate dispersion function G is given by

Scaling the data as in Eq. 34 (ω1 =ω2) gives the plot of Fig. 6 showing a small,positive, nearly dispersionless n2 for much less than Eg, reaching a peak near Eg /2(where 2PA turns on) and then decreasing, reaching negative values as ω approaches theband edge. The curve is the result of the Kramers-Kronig integral of Eq. 27. This curve issimilar to the behavior of the linear index in a solid which has its peak value at the bandedge, where linear absorption turns on, and then rapidly turning down toward smaller valuesas ω increases. Note that in order to obtain the degenerate n2, we set ω1 = ω2 =ω anddivide the result by two to account for “weak-wave retardation”.

Again, a large variation in the magnitude of n2, including a change in sign, is hiddenby this scaling and is better seen in a log-log plot of n2 scaled by the dispersion function Gas given in Fig. 7. At the same time it shows the hidden Eg

–4 scaling (see Fig. 6) of the

Figure 5. A log-log plot of scaled data for β versus Eg (closed circles from Ref. 18, open triangles form Ref.24, and open circles for InSb from Ref. 26).

50

Figure 6. a. A plot of experimental values of the nonlinear refractive index, n2e, scaled according to Eq. 34versus The solid line is the two-parabolic-band-model prediction for the dispersion functionThe values for semiconductors (squares) were obtained from Z-scan measurements at 1.06 and 0.532 µm.Also shown are n2 measurements of large-gap optical materials16 (solid circles). From Ref. 7.

nonlinear index that leads to a wide variation of n2 : 2.5x10 –14 esu for MgF2 at 1.06 µm, –2x10–9 esu for AlGaAs at 810 nm, and 2.7x10 –10 esu for Ge at 10.6 µm. The straight line istheory showing the Eg

–4 dependence. It is seen that the scaling law holds over a 5 orders-of-magnitude variation in the modulus of n2 . Also note that although the measured values ofn2 for ZnSe at 1.06 and 0.532 µm have different signs, both measurements are consistentwith the scaling law. More recent data on a series of UV transmitting materials at theharmonics of the Nd:YAG laser are shown in Fig. 8.25

We next describe a few of the experimental techniques used to measure ∆α and ∆ nfrom which the physical processes can be determined. Adding to the complexity of analysis,Eqs. 10 and 11 adequately describe material interactions only when β and n2 are solelyresponsible for the nonlinearity and only when diffraction can be ignored within the material(external self action). As we discuss below, other nonlinear mechanisms must often beincluded.

EXPERIMENTAL METHODS

There are a number of difficulties that need to be addressed when attempting todetermine the value of β or n2 from experiment. An examination of the literature on valuesof β for the semiconductor GaAs show well over a two order of magnitude change in the

51

Figure 7. A log-log plot showing the Eg 4 dependence of n2 . The data points are identical to those in Fig. 6,but are scaled by the dispersion function G. The solid line is the function Eg 4

line of slope –4 on the log-log plot (adapted from Ref. 8).which appears as a straight

reported value over the past three decades as shown in Fig. 9. It is illustrative to lookbriefly at the reasons behind the trend toward smaller values with time shown in this figure.In general the reasons boil down to poorly characterized laser beam parameters andcompeting nonlinearities, i.e. experimental technique and interpretation. In the early yearsof these measurements laser pulses were often multimode in either space or time or both,leading to irradiance fluctuations resulting in larger losses from nonlinear absorption thansmooth pulsed beams. This leads to an overestimation of β . More importantly long pulseswere used which results in the dominant nonlinear absorption process of free-carrierabsorption from the 2PA generated carriers, again resulting in an overestimation of β (this isdiscussed in more detail later under “excited-state nonlinearities). An additional problem isthat nonlinear refraction can cause beam size deviation either within the sample or after thesample. If this occurs within the sample the irradiance is changed and, thus the loss ischanged. Changes in beam size after the sample can cause errors if some fraction of the

52

Figure 8. A similar plot to that of Fig. 7. That is, a log-log plot of n2 data scaled by the dispersion functionG versus taken with harmonics of a picousecond Nd:YAG laser on wide bandgap materials. Thesolid line is the function Eg

–4which appears as a straight line of slope –4 on the log-log plot (adapted from

Ref. 8)

transmitted light beam is not collected. This also results in an overestimation of β . Insemiconductors such as GaAs the dominant nonlinear refraction is usually self-defocusingfrom the 2PA generated carriers. The longer the pulse for a given irradiance the larger theenergy and, thus, the greater the carrier density produced and the larger both free-carrierabsorption and free-carrier refraction become (see later section on “exited-statenonlinearities.

The solution to the three problems mentioned after they have been identified isrelatively simple; use smooth beam profiles, e.g. TEM0,0, carefully characterize the output,use short pulses (for most semiconductors 30 ps is short enough to nearly eliminate free-carrier absorption effects, and the carrier defocusing is reduced to a manageable level),carefully collect all the transmitted light (e.g., place a large area detector directly at the backof the sample), and be sure to use samples short enough and irradiance low enough to be inthe external self-action regime.

Beam Propagation

As an example of beam distortion due to external self-action, Fig. 10 shows the farfield energy distribution of a picosecond pulse after transmittance through 2.5 cm of NaClat low and high irradiance.27 The curves were normalized to coincide at the center of thebeam. The optical path-length change at the center of the beam and at the peak of the pulsedue to this bound-electronic n2 as shown in the figure is ≅λ/2.0. The sensitivity of thismethod is limited to the order of a π/4 peak phase distortion with the sample placed at thebeam waist of a Gaussian input beam (100 µm HW1/e2 M was used for the data in Fig. 10).

53

Figure 9. The two-photon absorption coefficient, β, plotted as a function of year published in the literature(note the semilogarithmic scale). The data is from Ref. 26.

However, the sensitivity to induced phase distortion is minimized by placing the sample atthe beam waist, i.e. from linear optics, a phase mask or lens has little effect on a beam whenplaced at the waist. One way to take advantage of the increased effect of a lens on a beamwhen moved away from the waist is the Z-scan method.

Z-Scan and EZ-Scan

We first describe the use of these techniques for measuring NLR. We then describetheir use for measuring NLA, and finally describe how NLR can be measured in thepresence of NLA. With the development of this method, accurate measurements of n2 in alarge number of semiconductors and optical solids in various spectral regions have beenobtained.28,29 The Z-scan has the advantage of easily providing the sign of the nonlinearity,an important factor for the comparison of experiment with theory presented here.Techniques such as degenerate four-wave mixing (DFWM), for example, are sensitive to|χ(3)|2 so that ∆α and ∆n effects are not readily distinguished.

Using a single Gaussian laser beam in a tight focus geometry, as depicted in Fig. 11,we measure the transmittance of a nonlinear medium through a finite aperture (Z-scan) oraround an obscuration disk (EZ-scan30 ), both positioned in the far field, as a function of thesample position Z measured with respect to the focal plane. The following example

54

Figure 10. A one dimensional beam scan of a picosecond pulse at 1.06 µm having an initial Gaussianspatial profile after traversing a NaCl sample and propagating to the far field. Left; low irradiance whereself-focusing is negligible. Right; high irradiance. From Ref. 27.

55

sample.qualitatively describes how such data (Z-scan or EZ-scan) are related to the NLR of the

Figure 11. The experimental setup for performing Z-scans (or EZ-scans by replacing the aperture with adisk.

Figure 12. Typical closed aperture Z-scans of a material showing only nonlinear refraction for positive(solid line) and negative (dotted line) ∆n.

Assume, for example, a material with a positive nonlinear refractive index. Startingthe scan from a distance far away from the focus (negative Z) the beam irradiance is lowand negligible NLR occurs; hence, the transmittance remains relatively constant, and thenormalized transmittance is unity as shown in Fig. 12. As the sample is brought closer tofocus, the beam irradiance increases leading to self-focusing in the sample. This positiveNLR moves the focal point closer to the lens leading to a larger divergence in the far field,thus reducing the transmittance. Moving the sample to behind focus (Z>0), the self-focusing helps to collimate the beam increasing the transmittance of the aperture. Scanningthe sample farther toward the detector returns the normalized transmittance to unity. Thus,

56

the valley followed by peak signal shown by the solid line in Fig. 12 is indicative of positiveNLR, while a peak followed by valley shows self-defocusing.

The EZ-scan can be described in nearly identical terms except we monitor thecomplementary information of what light leaks past the obscuration disk, or eclipsing disk.Since in the far field, the largest fractional changes in irradiance occur in the wings of aGaussian beam (see Fig. 10), the EZ-scan can be more than an order-of-magnitude moresensitive than the Z-scan. Figure 13 demonstrates the sensitivity of this method bycomparing a Z-scan and an EZ-scan on neat toluene with nanosecond 532 nm pulses underidentical experimental parameters (only replacing an aperture by a disk). Note that thevertical scale for the Z-scan is expanded by a factor of 10, and the signal is inverted for theEZ-scan since what is transmitted by an aperture is blocked by a disk. Using this methodwe have observed a peak optical path length change of as small as λ/2200 with a signal-to-noise ratio greater than 5 (∆Φ0=2π/2200, where ∆Φ0 is defined as the integrated peak-on-axis phase shift).30

Figure 13. A Z-scan and EZ-scan on toluene. From Ref. 30.

It is an extremely useful feature of the Z-scan (or EZ-scan) method that the sign ofthe nonlinear index is immediately obvious from the data. In addition the methods aresensitive and simple single beam techniques. We can define an easily measurable quantity∆ Tpv as the difference between the normalized peak and valley transmittance: Tp - Tv. Thevariation of ∆T pv is found to be linearly dependent on the temporally averaged inducedphase distortion, defined here as ∆Φ0 (for a bound-electronic n2, ∆Φ0 involves a temporalintegral of Eq. 11). For example, in a Z-scan using an aperture with a transmittance of≅ 40%;

(38)

With experimental apparatus and data acquisition systems capable of resolving transmissionchanges ∆Tpv≅1%, Z-scan is sensitive to less than λ/225 wavefront distortion (i.e.,∆ Φ0=2π/225). The Z-scan has a demonstrated sensitivity to a nonlinearly induced opticalpath length change of nearly λ/103 while the EZ-scan has shown a sensitivity of λ/104.

57

In the above picture we assumed a purely refractive nonlinearity with no absorptivenonlinearities (such as multiphoton or saturation of absorption). Qualitatively, multiphotonabsorption suppresses the peak and enhances the valley, while saturation produces theopposite effect. If NLA and NLR are simultaneously present, a numerical fit to the data canextract both the nonlinear refractive and absorptive coefficients. The NLA leads to asymmetric response about Z=0, while the NLR leads to an asymmetric response (if ∆Tpv isnot too large), so that the fitting is unambiguous. In addition, noting that the sensitivity toNLR in a Z-scan is entirely due to the aperture, removal of the aperture completelyeliminates the effect. In this case, the Z-scan is only sensitive to NLA. Nonlinearabsorption coefficients can be extracted from such “open aperture” experiments. A furtherdivision of the apertured Z-scan (referred to as “closed aperture” Z-scan) data by the openaperture Z-scan data gives a curve that for small nonlinearities is purely refractive in nature.In this way we can have separate measurements of the absorptive and refractivenonlinearities without the need of computer fits with the Z-scan. Figure 14 shows such aset of Z- scans for ZnSe. Separation of these effects without numerical fitting for the EZ-scan is more complicated.

The single beam Z-scan can be modified to give nondegenerate nonlinearities byfocusing two collinear beams of different frequencies into the material and monitoring onlyone of the frequencies (different polarizations can be used for degenerate frequencies). 31

This “2-color Z-scan” can separately time resolve NLR and NLA by introducing a temporaldelay in the path of one of the input beams. This method is particularly useful to separatethe competing effects of ultrafast and cumulative nonlinearities.

Figure 14. Z-scans of ZnSe showing closed aperture (upper left), open aperture (lower left) and closeddivided by open aperture data (upper right). The solid lines are theoretical fits. Adapted from Ref. 7.

58

Pump-Probe Z-scan

Pump-probe (or excite-probe) techniques in nonlinear optics have been commonlyemployed in the past to deduce information that is not accessible with a single beamgeometry. The most significant application of such techniques concerns the ultrafastdynamics of the nonlinear optical phenomena. There has been a number of investigationsthat have used Z-scan in pump-probe scheme. The general geometry is shown in Fig. 15where collinearly propagating excitation and probe beams are used. After propagationthrough the sample, the probe beam is then separated and analyzed through the far-fieldaperture. Due to collinear propagation of the pump (excitation) and probe beams, we areable to separate them only if they differ in wavelength or polarization. The time-resolvedstudies can be performed in two fashions. In one scheme, Z-scans are performed at variousfixed delays between excitation and probe pulses. In the second scheme, the sampleposition is fixed (e.g. at the peak or the valley positions) while the transmittance of theprobe is measured as the delay between the two pulses is varied. The analysis of the 2-colorZ-scan is naturally more involved than that of a single beam Z-scan. The measured signal,in addition to being dependent on the parameters discussed for the single beam geometry,will also depend on parameters such as the excite-probe beam waist ratio, pulsewidth ratioand the possible focal separation due to chromatic aberration of the lens. However, thesecan easily be handled theoretically. Figure 16 shows a temporally-resolved, 2-color Z-scanfor ZnSe using 30 ps, 532 nm pulses as the excitation source and 40 ps, 1.064 µm pulses asthe temporally delayed probe.32

Figure 15. The experimental setup for performing a time-resolved, 2-color Z-scan.

Figure 16. The results of a time-resolved, 2-color Z-scan performed on ZnSe using a 532 nm pump and1.06 µm probe showing nonlinear refraction vs. time (left) and nonlinear absorption vs. time (right).32

59

Degenerate Four-Wave Mixing

Another commonly used method to determine the dynamics of the nonlinearresponse of a material is time-resolved degenerate four-wave mixing (DFWM).33 Oneimplementation of this technique is shown in Fig. 17 where the interference of thetemporally and spatially coincident forward pump (irradiance If) and probe (Ip) sets up anonlinearity that is examined by the backward pump (Ib) as a function of its temporal delay.One interpretation is that If and Ip set up a grating whose dynamics is investigated by Ibscattering off this grating into the detector shown in Fig. 17 (often referred to as the“conjugate” direction).34 While this does not adequately describe the signal within thepulsewidth, it gives a reasonable picture of the longer time response of this method. Figure18 shows the response in this experiment performed on ZnSe using 30 ps, 532 nm pulses.35

Clearly there is a fast response following the pulse shape and a slower response with adecay time of 100’s of picoseconds (dominated by carrier diffusion washing out thegrating). While this technique gives information about the dynamics of the nonlinearresponse, absorptive and refractive nonlinearities both contribute to the signal and theireffects are difficult to separate. That is, in the grating picture, both absorptive andrefractive gratings scatter the backward pump into the detector.

Figure 17. Experimental setup for time-resolved degenerate four-wave mixing (DFWM). From Ref. 35.

Figure 18. DFWM experiment performed on a sample of ZnSe at 532 nm. From Ref. 35.

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Streak Camera Imaging

The dynamics of the nonlinear lensing effects can be dramatically demonstrated bymonitoring the spatial beam profile in real time with the use of a streak camera. Figure 19shows the spatial profile at low and high input energies for 30 ps, 532 nm pulses incident onZnSe after propagation in the relatively near field.36 At high inputs the 2PA creates carrierswhich are long lived with respect to the pulsewidth used and build up in time with theintegrated energy. Thus, the defocusing from these carriers is observed in Fig. 19 to getstronger with increasing time in the pulse.

Figure 19. Spatial scans at different times (separated by 7.4 ps) in the pulse after transmission throughZnSe and propagation to the near field as measured using a streak camera/vidicon combination. The leftside shows low energy (i.e. the input shape) and the right side shows the defocusing at high input. FromRef. 36.

EXCITED-STATE NONLINEARITIES

Both Figs. 16 and 18 show a fast response mimicking the input pulse (i.e. a responsetime less than the input pulsewidth) along with another more slowly respondingnonlinearity. The rapid response is either a combination of degenerate 2PA, β(2ω) andn2 (2ω) effects for the DFWM experiment (see Fig. 18), or the separate effects ofnondegenerate 2PA, β(ω;2ω), for the 2-color Z-scan and nondegenerate n2(ω ;2ω), see Fig.16. Here ω correspond to a wavelength of 1.064 µm. The longer time response in Figs. 16and 18 (and 19) is due to the nonlinear absorption and refraction induced by 2PA generatedcarriers. The generation rate for these carriers of density N is given by

(39)

The absorption from these carriers is referred to as free-carrier absorption, FCA, and therefraction as free carrier refraction, FCR and both effects are linear in the carrier density.While the FCA and FCR depend on the carrier density independent of their generationmechanism, when the carriers are generated via 2PA these effects appear as fifth order

61

nonlinearities or, as an effective, pulsewidth dependent χ (5).37 However, they are bestdescribed in terms of absorptive (σ a) and refractive (σr) cross sections as;

(40)

Sometimes k is included in the phase equation changing the units of σr to length cubed. Thesign of σa is intrinsically positive while, in principle σr can have either sign. In fact,however, for below gap excitation, σr is always negative leading to self-defocusing. Onceexcited, these carriers can undergo a variety of processes including several types ofrecombination and diffusion which have not been included in Eq. 40. For short pulseexcitation (e.g. ps) with pulsewidths less than recombination and diffusion times Eq. 39 isadequate to describe the response within the duration of the pulse. Combining the 3rd and 5th

order responses gives,

(41)

The dynamics of these carriers is seen in the time-resolved Z-scan of Fig. 16, the DFWMdata of Fig. 18 and the streak camera imaging of Fig. 19. In the Z-scan data carrierrecombination dominates the decay while in the DFWM experiment carrier diffusionbetween peaks and valleys of the grating dominates. These decays would need to beincluded in Eq. 39 to describe these dynamics. The order of the response is seen in theDFWM data as the inset of Fig. 18 where the signal at two time delays is plotted as afunction of the input irradiance I (all three input irradiances varied simultaneously). At zerodelay the slope of the signal versus I is three (third-order, χ(3) response) while at a 200 psdelay (well past the overlap of the pulses), the slope is five indicating the fifth-orderresponse.

Figure 20. A plot of the index change, ∆n, divided by the irradiance, I, as a function of I. From Ref. 37.

62

This higher-order response for nonlinear refraction is also observed in Z-scansperformed at different irradiance inputs of 532 nm picosecond pulses. At low inputs thethird-order bound-electronic response dominates while at higher inputs the fifth-order self-defocusing from the 2PA generated free carriers becomes important. Figure 20 shows theindex change divided by the peak-on-axis input irradiance, I0, in ZnSe at 532 nm, as afunction of I0 . The index change is calculated from the measured ∆Φ0. For a purely third-order response, ∆n=n2I0, and this figure would show a horizontal line. The slope of the lineshown in Fig. 20 shows a fifth-order response while the intercept gives n2, the negativebound-electronic defocusing at 532 nm. This helps explain some of the discrepancies ofmeasured values of 2PA coefficients in GaAs (see Fig. 9). These higher order nonlinearitiesseen in semiconductors give some indication of the importance of the carefulcharacterization needed to interpret the measured nonlinear loss and phase.

It would seem reasonable that σr and σa would be related by causality throughKramers-Kronig relations. However, after excitation there can be rapid redistribution ofthe carriers within the bands due to various mechanisms. This redistribution leads to so-called band-filling nonlinearities, and for the time scales of picoseconds used in theexperiments shown, this prohibits the use of Kramers-Kronig relations for the cross sections(note that after excitation and redistribution the absorption and refraction due to thesecarriers are related by Kramers-Kronig relations).

EXCITED-STATE NONLINEARITIES VIA ONE-PHOTON ABSORPTION

As discussed in the previous section, excited carriers can lead to NLA and NLR. Inother materials such as molecular systems, the creation of excited states can lead toanalogous nonlinearities described by identical equations (Eqs. 40) where N is interpreted asthe density of excited states. Again, how they are generated is unimportant. If the carriersor excited states are created by 2PA the resulting nonlinearities are fifth order, i.e., aneffective χ(5). Depending on the absorption spectra, these states can also be created by linearabsorption where, neglecting decay within the pulse,

(42)

Concentrating on molecular nonlinearities we refer to these nonlinearities as excited-statenonlinearities, ESA and ESR in analogy to FCA and FCR respectively. Assuming thatdepletion of the ground state can be ignored (i.e., no saturation),

By temporal integration of Eqs. 43 with 42 we find;

(43)

(44)

where F is the fluence (i.e., energy per unit area). This equation is exactly analogous to Eq.10 which describes 2PA, except that the irradiance is replaced by the fluence and the 2PAcoefficient, β is replaced by ασa/2 ω. Thus, experiments such as Z-scan will monitor athird-order nonlinear response that could easily be mistaken for 2PA. However, there mustbe some linear absorption present, however small, for ESA to take place. Two-photon-

63

absorption does not require linear loss. Unfortunately this is not enough to differentiate theprocesses as there can be linear absorption present in 2PA materials unrelated to the NLAprocess, e.g. from impurities or other absorbing levels. A temporally resolved measurement,such as DFWM or time-resolved Z-scan, would also show the excited-state lifetimeassuming the pulsewidth was short compared to this lifetime. Another way to determine themechanism is to measure the nonlinear response for different input pulsewidths, againassuming the pulses can be made shorter than the excited state lifetime. Figure 21 showsthis measurement performed on a solution containing chloro-aluminum phthalocyanine(CAP).38 While the energy in the pulses was held fixed while the irradiance was changed bya factor of two by changing the pulsewidth, the nonlinear transmittance remained the samein the open aperture Z-scans. This clearly indicates that the NLA is fluence rather thanirradiance dependent and, therefore, must be described by a real state population, i.e., ESA.In CAP, at 532 nm, the ESA cross section σa is considerably larger than the ground-statecross section. This type of absorber is referred to as a reverse-saturable absorber since theabsorption increases with increasing input. Such effects are useful in optical limiting.39 Forlarge inputs the ground state can become depleted reducing the overall NLA.

Figure 21. Z-scans performed on a sample of CAP at 532 nm. The left shows open aperture Z-scans forpulsewidths of 29 ps (squares) and 61 ps (triangles) and the right shows closed aperture Z-scans (afterabsorption is divided out) for the same pulsewidths. Adapted from Ref. 38.

Associated with ESA is ESR as given by the second term in Eq. 41, which is simplydue to the redistribution of population from ground to excited state. This is analogous tothe index change in a laser from gain saturation which leads to frequency pulling of thecavity modes.2 Ground state absorbers are being removed and excited state absorbers arebeing added. Depending on the spectral position of the input with respect to the peak linearand peak excited-state absorption, the NLR can be of either sign. For reverse saturableabsorbing materials the NLR is most likely controlled by the addition of excited-stateabsorbers, and their spectrum since the cross section is larger. Thus N is determined by Eq.42. Figure 21 also shows the NLR in CAP for two different pulsewidths demonstrating thatit is also fluence dependent and, thus, dependent on real state populations.

TWO-BEAM INTERACTIONS

Here we give examples of “nondegenerate” nonlinearities, where here the breakingof degeneracy is not just frequency, but propagation direction, e.g. 2-beam coupling. There

64

are many phenomena that can occur when two beams are coincident in space and time on anonlinear sample. Among these are cross-phase modulation where a pump beam modulatesthe phase of a probe through the optical Kerr effect, 2PA induced on the probe, excitedspecies created by the pump affecting the probe or temperature changes induced by thepump changing the index seen by the probe. In these cases, if the probing beam, p, is muchweaker than the pump (exciting beam e), the changes in the weak probe beam can be twiceas large as the changes in the strong beam (the strong beam is unaffected by the weakprobe). This factor of two comes from the cross term or grating term in the nonlinearinteraction and is sometimes referred to as weak-wave retardation.12 From Eq. 4 with twoinput fields and keeping only the

third-order self-action term (ignoring their vector nature other than keeping track of thewave-vector dependence) gives;

(45)

We plug this equation into the reduced wave equation given in the slowly varying amplitudeand phase approximation by Eq. 6. Looking just at the terms with k vectors in the pumpexp , and probe, exp , directions separately, we have;

(46)

where it has been assumed that Ep <<Ee.. This explicitly shows the factor of two differencein the nonlinearity seen by the pump and by the weak probe. The |E|2 acts like an irradianceor time averaging of the field squared. This averaging can have important consequences ifthe nonlinearity has a finite response time compared to the pulsewidth. For example, let’sassume that the pump pulse creates excited states that change the index of refraction (orabsorption) as seen by the probe pulse. We then write the slowly varying nonlinearpolarization at ω for the two beams separately by observing their separate k dependence as;

(47)

65

where τL is the excited-state lifetime and τG is the grating lifetime, both assumed here tohave exponential decays. The quotation marks are placed around χ(3) since this is reallydescribing a χ(1): χ(1) effect. The second term in Pp is called the grating term because itarises from the pump scattering off the material grating induced in the sample by theinterference of the pump and probe beams. The grating decay is in general shorter than theexcited-state decay since it includes the latter but also decays via diffusion which smoothesthe grating in time. Note that if the pulsewidth is long compared to the grating decay timebut short compared to the excited-state lifetime, the grating term will not contributesignificantly to the nonlinearity and the pump and probe will see the same nonlinearresponse, both seeing only the direct effect of the pump.

Figure 22. Vidicon images of the spatial irradiance distribution of an originally Gaussian spatial profilebeam after traversing a Si sample and propagating to the far field A) probe without pump, B) probe withpump C) pump beam. From Ref. 12.

Such a grating can or cannot cause energy to be transferred between the two beams(2-beam energy transfer- or 2-beam coupling) depending on the relative phase of the gratingproduced with respect to the irradiance modulation. We’ll get back to this point later. Avery graphic example of this weak-wave retardation can be seen by looking at the self-defocusing produced in silicon by the generation of free-carriers by linear absorption.

12

Once carriers are generated (excited) the index of refraction is lowered (band-blocking).This appears as a third-order nonlinearity (χ(1) : χ(1) rather than χ(3) ). Figure 22 shows thetransmitted 1.06 µm beam profiles in the far field after propagating through 270 µm of

66

silicon. Silicon is an indirect gap semiconductor and 1.06 µm falls below the direct gap butabove the indirect gap so that the transmittance for this sample length is ≅25% includingFresnel reflection losses. The 65 ps (FWHM) pulses are short compared to carrier lifetimesand diffusion times assuring that the nonlinearity including the grating did not decay withinthe pulsewidth so that the nonlinearity accumulates throughout the pulse without decay. Asshown in the figure, the weak probe beam undergoes more self-defocusing (more phasedistortion) than the excitation beam (≅2x). By temporally delaying the probe with respectto the pump we can see how this nonlinearity builds up from carriers excited by the pumpwith time as shown in Fig. 23. The scattering of light off the grating is graphically shownby moving the vidicon to the near field as seen in Fig. 24 where both beams can besimultaneously observed. Here the pump and probe pulse have equal energy to show theextra scattered beams on either side of the pump and probe. The beams diffracted in theouter directions are the other order beams (i.e. there is a +1 and -1 order diffraction) in thethin grating region. This thin grating or Raman-Nath limit is appropriate in this experimentwhere the sample is thin and the angle between beams is small ≅1.20. The extra beam (whenthe probe is small) is often called the forward scattered conjugate beam. In the case ofsilicon, where the primary nonlinear interaction is the index change due to the creation offree carriers (the excited states), there is not a direct transfer of energy from the pump tothe probe since the phase of the scattered light is π/2 out of phase, i.e. an index effect. Thisπ/2 phase shift is seen as the i in the field change, ∂E/∂z, equations, e.g. see Eq. 46. Itindicates that the phase of E is normally changed by Reχ. A shift in the phase of thegrating can lead to amplitude changes in the probe from Reχ. There is, however, energytransferred to the forward conjugate beams since in these directions there is no light to“interfere” with. We discuss energy transfer or 2-beam coupling more in what follows.

Figure 23. The spatial profile profile of the probe beam of Fig. 22B as a function of the time delay between thepump and probe.

In order to have energy transferred, the phase term in the ∂E/∂z equation given byi=exp(iπ/2) must be altered. Clearly, if χ (3) is complex, loss or gain of the field amplitudebecomes possible, but this can be caused by material absorption (or gain) and not actualtransfer of energy from the pump beam. In what follows we assume a purely refractivenonlinearity so that energy conservation in the light beams can be invoked. The question ofwhether or not energy is transferred between beams due to the grating produced by theinterference between two beams is illustrated in Fig. 25. This figure is meant to give aphysical picture of why the grating results in no net energy transfer if the phase of thegrating is unshifted with respect to the irradiance modulation. The arrows point in thedirection normal to the phase front (i.e. rays) and their size indicates the irradiance - for

67

equal input beams there is an equal irradiance in the direction of either beam, no net energytransfer. For unequal inputs, a little more thought shows that the initial imbalance ismaintained. On the other hand, Figure 25 shows that a π/2 phase shift between the incidentirradiance modulation and the material grating results in a net transfer of energy from onebeam to the other (the direction depending on the sign of the phase shift). Deviation from aπ/2 phase shift reduces the magnitude of the effect. The question remains of how toproduce this phase shift.

Figure 24. The spatial profiles of equal energy “pump” and “probe” beams in the near field along withextra scattered beams at high irradiance as a function of the time delay between equal energy pump andprobe beams (inset shows the pump and probe at low irrandiance).

Figure 25. An illustration of the physical reason for needing a π /2 phase shift between the interferencepattern and the material refractive grating in order to have energy transfer.

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The photorefractive effect, utilizing charge migration and the electro-optic effect, isthe usual example of how this phase shift is obtained.40 In pulsed experiments the transientenergy transfer that occurs from absorptive gratings leads to what are commonly referred toas coherent artifacts.41,42 A phase shift is guaranteed for absorption gratings in the transientregime. 43,44,45,46 These coherent artifacts occur near zero temporal delay between the pulses(within the temporal coherence time) and transfer of energy from the stronger to the weakerbeam. However, for purely refractive gratings, no energy transfer occurs even in thetransient regime. In the example of silicon shown in Figs. 22-24 the nonlinearity wasdominated by the refractive grating from the free-carriers. If, on the other hand, there is afrequency difference between the pump and probe, and the material response is non-instantaneous, a phase lag can occur between the refractive index grating and the movingirradiance interference pattern leading to energy transfer or 2-beam coupling. We give thefollowing as an example of how the grating term can lead to energy transfer between beamsfor a purely refractive nonlinearity.

Purely Refractive 2-Beam, Transient Energy Transfer

Here the non-instantaneous nonlinearity is the optical Kerr effect due toreorientation of the cigar-shaped CS2 molecules. The phase shift is produced by the phaselag of the material grating with respect to a moving interference pattern. The interferencepattern is made to move by interfering two slightly different frequency beams, This resultsin transient energy transfer or two-beam coupling in CS2 or other transparent Kerr liquids.The frequency difference results form using an initially chirped pulse which is split intopump and probe. The frequency difference then depends on the relative time delay betweenthe two beams. Energy can be transferred from either beam to the other depending only onthe relative delay (this changes the sign of the phase lag) with no transfer at zero time delaysince then the grating is stationary (no frequency difference). Of course this scattering onlyoccurs within the coherence time of the pulse (here picoseconds) where an interferencegrating is produced. Figure 26 shows the energy transfer as a function of time delay. Thisparticular type of scattering is referred to as stimulated Rayleigh-wing scattering (SRWS).RWS is caused by thermal fluctuations of the macroscopic polarization due to fluctuationsof the orientation of the individual dipoles. For these pulses the frequency increases withtime in the pulse (i.e positive chirp). Thus, the first pulse always loses energy while thesecond pulse gains this energy.

Figure 26. Energy transfer into and out of the probe beam as a function of time delay with respect to thepump beam (dashed line, theory for linear chirp - solid line, theory for actual chirp).

69

The only parameters needed for the theoretical fittings are the nonlinear index n2, itsrelaxation time, τ, and the linear chirp of the laser pulse. The first two are well known forCS2 and the laser chirp can be independently measured using first and second orderautocorrelation measurements. The chirp parameter, C, gives a measure of how rapidly thefrequency changes per unit time within the pulse. Assuming a linear chirp the electric fieldis given by;

(48)

where C is the linear chirp coefficient, and the coherence time is determined from, where τ p is the pulsewidth (HW1/eM). In this case we find the frequency

difference, Ω, between the two beams is linear in the time delay between pulses τ, i.e.. This leads to a simple expression for the signal, S;

(49)

where and τrot is the reorientation relaxation time. Even simpler, if the aboveexpression is differentiated to find the peak and valley, the total change in transmittancebetween peak and valley, ∆Tpv (not related to a Z-scan) is

(50)

where here Figure 26 shows Eq. 49 fit to the signal obtained for CS2.Given the nonlinear refractive index, irradiance and chirp, the lifetime can be

determined from this measurement. An interesting feature of the Eqs. 49 and 50 and theexperiment on CS2 is that lifetimes considerably shorter than the pulsewidth can bedetermined. In the case of CS2, 24 ps (FWHM) pulses were used to measure a lifetime 10times shorter, and the only limitation is how small a total change in transmittance can bemeasured. With high repetition rate femtosecond lasers using high frequency modulationmethods ∆T pv of 10-5 can be detected. With 100 fs pulses and n 2 values typical oftransparent dielectrics, this should allow measurements of lifetimes of the order of 0.1 fs.This is of the estimated order of bound-electronic nonlinearities for these materials (so-called instantaneous nonlinearities).

47

ALL OPTICAL SWITCHING

An important application of the nonlinearities discussed in this chapter is switchinguse all-optical means (i.e. all-optical switching, AOS). The theory of n2 and β allows directdetermination of the ideal operating point of a passive optical switch. Optical switchdesigners have established a figure-of-merit (FOM) for candidate materials, defined by theratio k0n2/β.48 The goal of maximizing the FOM clearly shows the need for a largenonlinear phase shift (πn 2/λ) while keeping the 2PA loss (β) small. Using Eqs. 33 for β and36 for n2, along with Eq. 37 relating the dispersion function G to F for 2PA, the FOM canbe determined as shown in Fig. 27. Here the absolute value of the FOM is shown as the

70

solid line. Figure 27 also compares experimental data for several semiconductors to thistheory. Note that the data here is the ratio of two experimental values, β and n2 for eachmaterial. The remarkable agreement between theory and experiment indicates that thisquantity is indeed a fundamental property of semiconductors, depending only on thenormalized optical frequency ω /Eg).

The two horizontal lines in Fig. 27 represent the minimum acceptable FOM fornonlinear directional couplers (NLDC) and Fabry-Perot (FP) interferometers. Although itdemands a larger FOM, the NLDC scheme is the preferred practical geometry. From Fig.27 we see that the FOM requirement is satisfied either just below the 2PA edge or very nearresonance ω ≈ E g). Since n2 ∝ Eg

–4 , a low switching threshold at a given wavelengthdemands a material with the smallest possible bandgap energy. The theory then suggeststhat the ideal operating region is just below the bandgap. However, linear loss due to band-tail absorption makes this scheme unworkable at present. Operation near to but above thehalf bandgap where there is a small “resonance” in n2 requires increased irradiance due tothe reduced absolute magnitude of n2 , resulting in detrimental 2PA as discussed above. Onthe other hand, operation just below the Eg/2 eliminates 2PA with only a small reduction inn2 . It has recently been suggested that by using semiconductor laser amplifiers (SLA),parasitic linear loss can be mitigated, making near-gap operation a practical possibility.49,50

Fig. 27. All-optical switching figure of merit for passive optical switches. Adapted from Ref. 7.

OPICAL LIMITING

Another application of the nonlinearities discussed above is for sensor protectionagainst laser pulses. Devices for this purpose are called optical limiters. The ideal opticallimiter has the characteristics shown in Fig. 28. It has a high linear transmission for lowinput (e.g. energy E or power P), a variable limiting input E or P, and a large dynamic rangedefined as the ratio of the E or P at which the device damages (irreversibly) to the limitinginput. Such devices can also be used as power or energy regulators. However, since theprimary application of the optical limiter is for sensor protection, and damage to detectors isalmost always determined by fluence or irradiance, these are usually the quantities ofinterest for the output of the limiter. Getting the response of Fig. 28 turns out to be possibleusing a wide variety of materials; however, it is very difficult to get the limiting threshold as

71

Figure 28. The ideal optical limiter input-output response characteristics.

low as is often required and at the same time have a large dynamic range. Because hightransmission for low inputs is desired, we must have low linear absorption. These criterialead to the use of two-photon absorption (2PA) and nonlinear refraction. Such deviceswork well for picosecond inputs. For example, a monolithic ZnSe device limits at inputs aslow as 10 nJ (300 W), and has a dynamic range greater than 10

4for 0.53 µm, 30 ps

(FWHM) pulses.36 The liming effect from such a device is a combination of nonlinear lossfrom 2PA and free-carrier absorption and defocusing of the beam, which reduces thetransmitted fluence, from free-carrier refraction. The lensing from n2 is usually a smallereffect except when quite short pulses are used. Unfortunately, except in the IR, 2PAcoefficients of inorganic solids are too small for most of these applications which look toprotect against nanosecond sources. Organic materials have the potential for largernonlinearities and are being actively investigated. In addition, if small linear absorption canbe tolerated, reverse saturable absorbers can be effective.51 Here the transmitted fluence isreduced so that the energy of a long pulse is limited as well as a short pulse as long as thepulsewidth is less than excited-state decay times.

CONCLUSION

It should now be clear that the interpretation of NLA and NLR measurements isfraught with many pitfalls. Great care must be taken. In extensive studies of a wide varietyof materials it is found that there is seldom a single nonlinear process occurring. Oftenseveral processes occur simultaneously, sometimes in unison, sometimes competing. Forexample we have given the example of potential confusion between “instantaneous” two-photon absorption and excited-state absorption. Such processes as reorientational,electrostrictive, thermal, saturation and excited-state nonlinearities can be thought of as twostep processes, or cascaded χ (1) :χ(1) nonlinearities. For example, for excited-stateabsorption, light first induces a transition creating an excited state (an Imχ(1) process) andthen the excite state absorbs (a second Imχ(1) process), i.e., two linear absorption

72

processes. For these types of slow cumulative nonlinearities, the irradiance (or field) may nolonger be the important input parameter. It makes sense to describe the ultrafast process of2PA by χ (3) or β (sometimes α2 is also used in analogy to n2) and the cumulative process ofESA or ESR by an absorptive or refractive cross section. Fourier transformation of theresponse function (see Eq. 1) results in the usually quoted frequency dependentsusceptibility χ(n)(ω1,ω2,…,ωn). Memory, which was previously explicitly included in theresponse function, is lost in the dispersion. Thus, irradiance, I, and fluence, F, dependenciesare treated equally. This can lead to confusion as the two processes (ultrafast andcumulative) are indistinguishable for pulses long compared to relevant relaxation processes.The usual expansion in terms of susceptibilities is useful for fast nonlinear responses but isoften not the most convenient description and has been overused. Recently, studies of so-called χ2:χ2 cascading of second-order nonlinearitites have shown they can also mimic third-order responses of both β and n2.

52 It is necessary to experimentally distinguish and separatethese various processes in order to understand and model the interactions. There are avariety of methods and techniques for determining the nonlinear optical response, each withits own weaknesses and advantages. In general, it is advisable to use as manycomplementary techniques as possible over a broad spectral range in order tounambiguously determine the active nonlinearities. Numerous techniques are known formeasurements of NLR and NLA in condensed matter. Nonlinear interferometry, degeneratefour-wave mixing (DFWM), nearly-degenerate three-wave mixing, ellipse rotation, beamdistortion, beam deflection, and third-harmonic generation, are among the techniquesfrequently reported for direct or indirect determination of NLR. Z-scan is capable ofseparately measuring NLA and NLR. Other techniques for measuring NLA includetransmittance, calorimetry, photoacoustic, and pump-probe methods.

ACKNOWLEDGEMENT

I gratefully acknowledge the contributions of many of my colleagues, postdoctoralassociates and graduate students, many of whom are included in the references. I alsoacknowledge the financial support of the National Science Foundation and DARPA overmany years.

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Absorption Coefficients at 355 and 266 nm”, Phys. Rev. B17, 4620 (1978).25. Richard DeSalvo, A. Said, D. Hagan, and E. Van Stryland, “Infrared to Ultraviolet Measurements of 2-

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27. W. E. Williams, M. J. Soileau, and E. W. Van Stryland, "Simple direct measurement of n2”, NBSSpecial pub. 688, 552-531, (1983).

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30. T. Xia, D.J. Hagan, M. Sheik-Bahae, and E.W. Van Stryland, “Eclipsing Z-Scan Measurement ofλ /104 Wavefront Distortion”, Opt. Lett. 19, 317-319 (1994).

31. M. Sheik-Bahae, J. Wang, J.R. DeSalvo, D.J. Hagan and E.W. Van Stryland, “Measurement ofNondegenerate Nonlinearities using a 2-Color Z-Scan”, Opt. Lett., 17, 258-260 (1992).

32. J. Wang, M. Sheik-Bahae, A.A. Said, D.J. Hagan, and E.W. Van Stryland, “Time-Resolved Z-ScanMeasurements of Optical Nonlinearities”, JOSA B11, 1009-1017 (1994).

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75

FROM DIPOLAR MOLECULAR ENGINEERING TO MULTIPOLARPHOTONIC ENGINEERING IN NONLINEAR OPTICS

J.Zyss (a,b) and S. Brasselet (b)

(a) France Telecom, CNET, Laboratoire de Bagneux (URA CNRS 250)196 avenue Henri Ravera, BP107, 92225 Bagneux, FRANCE(b) Laboratoire de Photonique Quantique et MoléculaireEcole Normale Supérieure de Cachan61 av. du Président Wilson, 94235 Cachan, FRANCE

I. INTRODUCTION

The development of molecular nonlinear optics1 has been inspired over the last twodecades by the early recognition of intramolecular charge transfer conjugated systems as thebest candidate-templates towards further material and device developments2 . The centralparadigm of the domain is the coupling of dipolar molecular diode structures, such asparanitroaniline, to an externally applied poling electric field at thermal equilibrium insolution for the molecular scale measurement of the first hyperpolarizability tensor β via theElectric Field Induced (EFISH) method 3, or in viscous media towards modulation andswitching devices in guest-host electrooptic polymers4 . Moreover, a two-level quantummodel 9 whereby molecular-field interactions are accounted for by a single dominant chargetransfer transition, reinforces the relevance of dipolar molecular structures towardsquadratic nonlinear optics.

The more recent extension to a broader variety of two- and three-dimensionalmultipolar molecular systems has demonstrated the potentiality of new directions inmolecular engineering, and in particular the possibility to take advantage of the fulltensorial properties of the β tensor, traditionally constrained to the vectorial and one-dimensional scheme in the beginning of quadratic nonlinear optics in organics.

Early studies based on group theory helped to approach more general multipolarstructures 5 , and in particular to reveal the interest of octupolar molecules, which aredeprived of ground-state dipolar moment6,7 . Quantum mechanic studies permittedfurthermore to described such systems by a degenerated two-level model accounting for theoctupolar symmetry properties8 . In a more general model, multipolar complex structureshas shown the need to call upon n-level quantum models with n ≥ 3, to complete the simpletwo-level model9 assigned to one-dimensional molecules. The understanding of linear and

Beam Shaping and Control with Nonlinear Optics77Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

nonlinear optical properties was helped by the development of a rotationally invariantformalism, which has shown to greatly ease basic operations on tensors as molecularrotation and statistical averaging, permitting therefore to retrieve underlying invariantcombinations of molecular and field observables.

Full tensorial characterization of molecular nonlinear properties and perspectives inmacroscopic ordering of multipolar systems and in particular octupolar ones, neededhowever to by-pass constraints of the electric field coupling drawbacks imposed bytraditional methods.

This Chapter will review the successive theoretical and experimental advances thathave helped develop a full fledged multipolar molecular engineering approach in the realmof nonlinear optics, which can be shown to embed the earlier dipolar approach as a specialcase. Vectorial coupling imposed by the EFISH measurement for nonlinear characterizationhas been overcome by use of the incoherent Light scattering experiment10 , accessible tovanishing dipole moment molecules as octupolar structures, and more generally multipolarand ionic species11 . We will review molecular properties in Section II and show thatpolarized harmonic scattering experiments permit to sort-out the full rotational spectrum of

the β tensor12 and independently reach both dipolar βJ=1and octupolar β J = 3 components.

At the macroscopic scale, a different approach, whereby all-optical photoinducedprocesses are called-upon opens-up new possibilities towards photonic related issuesotherwise unamenable to the traditional molecular engineering approach. It is indeedpossible to imprint transient as well as permanent polar order in media subject to coherent

multiwavelength irradiation with non-zero averaged cube of the total field13 (⟨E3⟩ ≠ 0).

Such a situation is met by interfering single photon and two-photon absorption process, ashad been initially discussed as early as 1967 by Glauber in the realm of resonantabsorption14 , or later with reference to one- and two-photon induced photoionization

processes by Baranova et al.13

. A decisive step in the experimental implementation of ⟨E3 ⟩

related phenomena was the demonstration of unseeded15 and seeded16 second-harmonicgeneration in silica fibers. This approach has been extended to side-chain polymers withdipolar nonlinear side-groups and lead to the demonstration of second-harmonic generationin polymer films with efficiencies comparable to that of electrically poled films17 . It wasthen realized that this method being based on the interference between one and two-photontransitions did not require permanent dipoles and could be used to impart transient orpermanent non-centrosymmetric order to octupoles respectively in solution and in polymerfilms 18,19 . In contrast with the electric field induced dipolar schemes whereby the dipolarnature of the external perturbation restricts the induced tensorial properties to a dipolarcomponent, the broader rotational spectrum of the « write » field tensor can drive themacroscopic symmetry pattern to any desired tensorial symmetry12c,20 , as showntheoretically and experimentally in Section IV. In conclusion, we point-out some newpossibilities emerging from such photonic engineering of nonlinear optical materials andstructures which enlarge and complement the more restricted scope of the earlier molecularengineering approach.

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II. MULTIPOLAR SYMMETRY PATTERN AT MOLECULAR SCALE

Irreducible decomposition of the β β tensor

Nonlinear molecules with enhanced quadratic efficiency follow in general the samegeneric pattern: a strongly one-dimensional rod-like structure consisting of a single donor-acceptor substituent pair linked by a conjugated π electron system. Such highly anisotropicpolarizability features tend to reduce the β tensor to its one-dimensional projectionβ = β x x x (x ⊗ x ⊗ x ) along the main charge transfer axis x. However, for less anisotropicsystems such as 2-D or 3-D multiply substituted conjugated structures, other non-diagonalout-of axis coefficients may appear. Rotational manipulations such as required to evaluatethe macroscopic ensemble averaged response of individual molecules are cumbersome andwill scramble individual cartesian components of β following tedious, frameworkdependent calculations. The resulting expressions will lack the desired invariance that isneeded towards a framework independent physical discussion aiming at the unambiguous,framework independent identification of those molecular parameters emerging at themacroscopic scale. The irreducible tensor formalism provides a relevant framework tosatisfy elementary invariance requirement with respect to rotations and related statisticaloperations 21-23 . In this scheme, the molecular hyperpolarizability β tensor is decomposed asfollowing:

where the β Jm coefficients are the spherical components of the β tensor, with 0 ≤ J ≤ 3,

which can eventually be related to its cartesian components in a given framework. The CJm

basis set vectors are related to the normalized spherical harmonic functions21 . In non-resonant conditions, Kleinman relations are valid and all permutations of indexes areallowed for non-diagonal β coefficients. The spherical components can then be limited tothe dipolar order J = 1 and the octupolar order J = 3. Close to resonance, additional pseudo-tensorial terms J = 0 (scalar) and J = 2 (quadrupolar), which are antisymmetric with respectto index permutations may however appear. As the J order of the spherical harmonic

Y mj

( θ, ϕ) functions is preserved throughout rotations, the tensorial rotation will not

scramble β jm

spherical components with different J' s. In particular, rotation of the β tensor

from the molecular framework to the macroscopic one, designated by RΩ (β ) , whereby the

Euler angles Ω = ( θ,ϕ,ψ ) characterize a random molecular orientation with respect to thelaboratory frame, can be directly expressed with the Wigner matrix coefficients23 , whichfollow an important orthogonality property.

This representation permits to discuss the implications of symmetry properties onspherical components. Whereas multipolar molecules correspond to weaker symmetryconditions compatible with both non vanishing J = 1 and J = 3 components, octupolar

molecules are defined in the situation whereby

component, and the opposite « dipolar » situation correspond to a ratio

maximal. It can be indeed easily checked from the relation between spherical ands cartesiancomponents that in this latter situation the octupolar component does not vanish, whichrules-out the concept of a purely dipolar molecule as far as a rank three tensorial property isconcerned1 2 c . In intermediate situations, the multipolar character of a given molecule can be

is the only non-vanishing

(1)

βJ = 3

79

quantified by the relative balance between the dipolar and octupolar contributions to the β

tensor as measured by the nonlinear molecular anisotropy ratio5a Aswill be exemplified throughout this work, spherical decomposition is particularly useful inthe context of quadratic nonlinear optics, leading in particular to considerable simplificationof eventually complex molecules and poling field coupling expressions in both coherentand scattering regimes. Spherical components can be easily determined in terms of cartesiancoefficients, and in particular for C2v planar molecules whereby only two diagonal βxx x andoff-diagonal βxyy are non-vanishing, with x the charge transfer axis and y the perpendicular

direction to x in the molecular plane. It can be shown in this case5a that planar octupolarmolecules will satisfy the relation βx x x = –βx y y , and « quasi-dipolar » ones will follow the

relation βx x x = 3βx y y .

Experimental determination of off-diagonal β coefficients, independently fromdiagonal ones, is however unamenable to the traditional electric field poling (EFISH)measurement, whereby the purely vectorial interaction between the electric field and themolecule permits to retrieve only the projection on the ground state dipole moment of the"dipolar" J = 1 spherical order. It can be shown furthermore, based on symmetry argumentsas detailed in section III, that other possible polarization configurations in the EFISHexperiment cannot provide additional information about the tensorial nature of β, andespecially about its octupolar J = 3 component. In order to by-pass this constraint imposedby the poling symmetry, measurements can be performed in the scattering regime using theincoherent Harmonic Light Scattering technique, which will be shown in the next section tobe polarization sensitive in contrast with EFISH and thus permitting access to the full βrotational spectrum.

Rotationally invariant expression of the polarized harmonic light scattering intensity

Off-axis harmonic scattering induced by thermal fluctuations can be observed whenirradiating a centrosymmetric solution by a laser beam at frequency ω. We assume in thefollowing that randomness of the solution originates mainly from isotropic orientational

fluctuations. The scattered harmonic intensity I , conveniently collected at 90° away from2ω

the incident beam, is then proportional to the ⟨β ⊗ β⟩Ω squared tensorial product averaged

over all equi-probable molecular orientations defined by the Euler angle Ω = (θ,φ,ψ) wi threspect to the macroscopic (X,Y,Z) framework. Analysis of the different incident andscattered polarization configurations shows that for an incoming beam polarized in the

(X,Y) plane with a polarization angle defined by as depicted in Fig. 1, one

can retr ieve only two independent macroscopic coeff ic ients12b , namely

⟨ β ⊗ β ⟩ X X X X X X = ⟨ β2X X X ⟩ (measured for parallel ω incident and 2ω ana lyzed

polarizations) and ⟨β ⊗ β⟩ ZXXZXX = ⟨ β 2ZXX ⟩ (measured for perpendicular ω incident and

2ω analyzed polarizations), as from the following relations :

(2)

80

where (resp. is the scattered intensity analyzed along the X (resp. Z) harmonicpolarization direction, G being an experimental calibration correction factor accounting forspecific set-up parameters such as instrumental response, shape factors or local fieldcorrections and N standing for the molecular density.

stands for the laboratory framework.

Figure 1. Different polarization configurations used in the Harmonic Light Scattering experiment in order to

sort-out the two independent and molecular nonlinear scattering cross-sections. (X,Y,Z)

The two macroscopic and molecular nonlinear scattering cross-.

sections can be expressed as complex quadratic functions of the products

following adequate averaging of cartesian projection factors24 . It is then in general

impossible to retrieve the full β tensor from the measurement of the sole accessible

and terms. The tensor can however be fully resolved in simple cases, such asplanar C2v symmetry, whereby symmetry conditions allow for only two non-zero two β x x x

and β xyy coefficients. Invariant expressions of the scattered harmonic intensity permits

however to avoid specific discussions for different symmetry cases as exemplified here inthe case of planar C2v systems and provide further physical basis for a more generaldiscussion. Application of the irreducible tensorial formalism in the context of HarmonicLight Scattering experiments shows12c that the transposition the of the φ dependence of Eq.2

in its spherical decomposition form provides the two invariant and

irreducible components, which can be defined for any molecular symmetry.

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Experimental evidence of non-diagonal β coefficients

Polarized Harmonic Light Scattering experiments were performed on some prototypemolecular systems in chloroform solutions, as depicted in Fig. 2. In each case, the harmonicwavelength was located far from resonances. Calibration of a pure chloroform samplepermits to estimate the background solvent depolarization. Among the molecular structuresdepicted in Fig. 2, the well-known Disperse red One dye (DR1) is a typical representativeof a rod-like system of cylindrical C∞ v symmetry, whereby the diagonal βxxx coefficient isclearly dominant. In the case of the triphenylmethane Crystal Violet dye (tri-(p-dimethylamino-phenyl)methane), which is assumed here in a first approximation to displayan average planar structure, D3h symmetry imposes the βxxx = –βxyy relation between the

two non vanishing β components. The (1,5)-Dinitro,(2,6)-Di(N,N)-n-butylAminoBenzene(DNDAB) molecule is representative of an intermediate system between the quasi-dipolarand octupolar symmetry, whereby C2v symmetry imposes the existence of only two nonvanishing βxxx and βxyy coefficients.

Figure 2. Chemical representation of prototype molecules measured by the polarized Harmonic LightScattering experiment at the 1.064 µm incident wavelength for the DNDAB, and 1.34 µm in the case of themore absorbing DR1 and Crystal Violet, and assuming that the harmonic detected wavelength stays withinthe transparency region of the molecules.

D =

norms.

In those three cases, the relatively high level of symmetry permits to independentlyretrieve both β xxx and β x y y microscopic coefficients by polarized Harmonic Light

Scattering. Results are summarized in Fig. 3, where the HLS depolarization ratio defined by

is plotted as a function of the microscopic cartesian anisotropic

ratio12c u = βx y y / βxxx . Based on the more general framework of irreducible representation,

one is furthermore able to retrieve for all symmetry configurations the nonlinear anisotropy

ratio defined by and expressed in terms of irreducible component

82

Figure 3. Dependence of the macroscopic depolarization D and the squared nonlinear anisotropy ρ2 as

functions of the cartesian nonlinear anisotropy u = βx y y / β xxx related to the in-plane anisotropy of the β

tensor in the case of C2v symmetry, and values for the various prototype molecules in Fig. 2.

As seen from Fig. 3, two possible values of u can be retrieved from the evaluation ofD. In general, one of these solutions can be ruled-out by simple physical arguments. Ascould be predicted from the highly anisotropic feature of the DR1 molecule, theexperimental ratio u = –0.05 is quasi vanishing, in contrast with the more isotropic CrystalViolet molecule, leading to two possible values, namely u = –0.79 or u = –1.3, closelysurrounding the -1 value corresponding to a pure octupole. The departure from the idealu = –1 value may originate from out-of-plane distortions of this molecule. A significantoff-diagonal coefficient has been measured in the DNDAB case, showing the importance ofthe βx y y component for this type of multiply substituted molecules. As seen in Fig. 5, the

spherical counterparts of the cartesian coefficients confirm the octupolar character of theCrystal Violet molecule, with a high anisotropy ratio ρ = 10.2, the octupolar contributionbeing still significant in the case of the DNDAB molecule, with ρ = 3.7. In the DR1 case,we could futhermore evidence an octupolar contribution of the same size as the dipolarone, with ρ = 0.9, thus confirming that strongly anisotropic quasi-one-dimensional systemsare indeed compatible with significant octupolar contributions. This last anisotropy value is

closed to the theoretical ρ = = 0.82 value.

III. MULTIPOLAR FIELD TENSORS: A UNIFIED FRAMEWORK FORTHERMALLY EQUILIBRATED AND PHOTOINDUCED PROCESSES

Orientation of molecules in diluted media is traditionally performed in thermallyequilibrated conditions at temperature T, the normalized orientational distribution beingthen governed by a Maxwell-Boltzmann statistics of the type:

83

(3)

where W(Ω) is the coupling energy of a randomly oriented molecule, with the externalordering perturbation defined by an adequate set of angular variables Ω. In the case of anexternally applied electric field, such as commonly used in solution for the EFISHexperiment or close to the glass transition temperature for electrooptic poled polymers, W istaken as:

(4)

where RΩ is the rotation operator from the molecular to the macroscopic framework, µ(E )being the induced dipole, Q and O the quadrupolar rank two and octupolar rank threetensorial molecular moments, ∇E and ∇∇E the rank two field gradient and rank three fieldcurvature tensor with cartesian components ∇ i Ej and ∇ i∇ jE k. Introduction of contractedtensorial products in Eq. 4 permits to avoid tedious index notations such as

In the context of orientational processes, it is generally

sufficient to expand the induced dipole moment up to the linear polarizability α, namely:

µ(E) = µ 0 + α :E (5)

where µ0 is the permanent dipole.We consider the case where the medium is exposed to the combined influence of a

static poling field and an off-resonance coherent laser field at frequency ω with

E = E0 + E ω + E ω *in complex notations. We can now combine Eqs. 4 and 5 and retain

only those terms which lead to non-zero time average due to the finite inertia of moleculeswhich will require times of the order of many optical cycles before thermal orientationalequilibrium is reached, leading to:

(6)

Neglecting the nonlinear terms in the expansion in Eq. 5 amounts to neglecting in thecurrent context weaker higher order orientational non-resonant terms such as

The first contribution is the well-known dipole-static field Langevin coupling energyresponsible for the break-up of centrosymmetry, the second and third terms are respectivelythe static and dynamical Kerr effects both leading to axial orientation as a result of thepolarizability anisotropy. The last terms are associated to spatial inhomogeneities of thestatic field and are respectively known as static multipolar coupling terms.

From Eq. 6, the different coupling mechanisms, hereafter labeled by the index i, areassociated to coupling energies which can be cast in the same general pattern of a tensorialcontraction between a molecular charge distribution or polarizability tensor Ti and anorienting field tensor Ei , namely R Ω( Ti ) • Ei . The tensorial rotation operator RΩ is meantto rotate the molecular property (e.g. charge distribution moment, dipole or polarizability)

84

from the molecular framework where it is readily expressed as Ti onto the macroscopicframework where it is natural to express the orienting field(s) following common sensepolarization considerations. One can therefore express W in the following synthetic form:

(7)

The specific form of the tensorial rotation operator will depend on the rank of T and will bei

seen to be most conveniently expressed in the irreducible formalism. Assuming furthermorethe case of weak interactions, namely W (Ω) /kT << 1, it is then sufficient to retain the firstorder in Eq. 3, leading to:

(8)

χ ( 2) or χ ( 3)) can be accounted for by statistical averaging over the unit volume thecorresponding molecular property s (for example α, β or γ) attached to a randomly orientedmolecule defined by the angular variable set Ω. With adequate local field factor correctionssummarized here by the factor F and performing the averaging in the macroscopic axis:

In the oriented gas approximation, a macroscopic optical property S (for example χ(1),

(9)

The weak field approximation and subsequent linearization of the Maxwell-Boltzmanndistribution as in Eq. 8 leads to:

(10)

It is possible to considerably simplify this expression by introducing the sphericalirreducible tensorial decomposition of tensors s and T whereby rotations are expressed bymeans of the canonical Wigner matrix and by using the related orthogonality relations23.One of the major virtues of this formalism lies in the possibility to somehow « permute »the field Ei and the molecular optical property s, in the integral on the right side of Eq. 10,albeit with different factors in different J irreducible spaces.

In order to ease notations, we remove in the following the i label representative of a

specific coupling mechanism which can be readily reintroduced if necessary. Callingthe irreducible component of S (J up to the rank of S or s and – m ≤ J ≤ m ) :

(11)

where the are the usual normalized irreducible tensor behaving like sphericalharmonics, and :

(12)

85

with

The first term in Eq. 12 corresponds to isotropic averaging (zero order expansion ofthe Maxwell-Boltzmann distribution) and will be canceling for odd-rank tensor (dipole,

(2)χ , etc.) which are well-known to vanish in centrosymmetric media. The second termcorresponds to more anisotropic multipolar symmetry averaging patterns and permits todiscuss in a physically relevant invariant framework the respective influence on themacroscopic property of tensors T, s and E. Odd J orders will contribute to quadratic

processes ( , χ( 2 ), χ (4) etc.) and related multipolar non-centrosymmetric patterns (e.g.

dipolar, octupolar etc.) whereas even J will contribute to cubic processes (χ(1) , χ ( 3 ) etc.)and related centrosymmetric patterns (e.g. isotropic, quadrupolar, hexadecapolar etc.).

Exp. 12 furthermore points-out the central role of the poling field tensor which limitsthe order J in the multipolar expansion of S : indeed, the rank of E can be smaller than thatof S (or s), which leads to a corresponding truncation in the S expansion. Such truncation isa formalized expression of the Curie principle which expresses the symmetry reduction

imposed by perturbation upon a physical system. Moreover the T J • s J contractionimposes obvious compatibility conditions between the molecular poling tensor T and themicroscopic optical property s. Assuming that T and s are both obeying index permutationsymmetry valid for non-resonant conditions, non-zero TJ ( o r s J) components willcorrespond to Jm a x – 2, J m a x – 4, etc., Jm a x being the rank of T (or s). In order to haveat least one J value corresponding simultaneously to non-zero values of TJ or sJ, the rank ofT and s will have to be of same parity. For example dipolar coupling of rank 1 (T = ) will

permit observation of χ( 2 ) properties (rank 3) whereas Kerr orientation (T = α, of rank 2)

will not induce a χ( 2 ) tensor.Whereas those results are familiar in the case of thermally equilibrated processes, the

same formalism can be shown to be applicable to the less intuitive case of all opticalphotoinduced processes. Contrary to the thermal electric field poling process which takesplace at the vicinity of the glass transition temperature, all-optical poling can be performedat room temperature and permanent statistical orientation is achieved by purely optical

means. Simultaneous irradiation of the sample by a strong fundamental field Eω

a n d a

weaker harmonic field E2ω

breaks the centro-symmetry of the medium, as a result of the

symmetry properties of the non-zero average ⟨E2ω* ⊗ E ω ⊗ Eω ⟩ t tensorial product which

plays the role of the poling static field13

. The time scale of the time averaging has to besmall compared to the laser pulse duration but large in comparison with the life-time of theexcited states involved in the process so as to allow for a cumulative process. The 2ωfrequency is indeed located near the absorption region of the molecule, which facilitatesreorientation following angularly selective electronic excitation1 7 , and photoisomerizationcycles or more complex randomization processes via adequate vibrational states. The mainadvantage of this all-optical process is the possibility to orient either dipolar or octupolarmolecules1 9, the latter being deprived of permanent dipole moment : transition dipolemoments only are involved in the quantum absorption cross-sections whereas, contrary tolower order poling mechanisms, permanent ground or excited states dipoles contributionsare optional. All-optical poling has been analyzed in terms of simultaneous resonant one-step absorption of a 2ω beam and two-photon absorption at ω, with a crucial cross-interference term imparting non-centrosymmetry to the medium13 . Calling respectively

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the one-photon, two-photon and joint transition

probability from the ground state |0⟩ to the resonantly excited state |1⟩ in a two-levelsystem, time-dependent perturbation theory for a two-level system will provide simpleexpressions which amount to tensorial contraction of a relevant nonlinear(hyper)polarizability tensor with the corresponding tensor field, namely :

(13)

α, β and γ stand respectively for the linear polarizability, first and second-orderhyperpolarizability tensors. The tensorial combination of the « write » fields in Eq. 13 aresuch that time averaged tensorial coefficients are non vanishing so as to ensure cumulativebuild-up of photoinduced excitations over many cycles. The joint probability only exhibitsa spatial phase dependence due to a multiwavelength tensor field. Periodic spatial

modulation follows with a n d ∆ ø = ø 2ω

– 2øω , ø ω (resp. ø 2ω)

standing for a constant phase factor for the fundamental (resp. harmonic) wave. Among the

contributions, the last one only is capable to induce a

non-centrosymmetric pattern.Eq. 13 has been established in the particular case of the two-level resonant system,

applied in general for one-dimensional molecules for which the β hyperpolarizability

appearing in is reveals in i ts quantum form proport ional to

under resonance conditions, whereby µ0 1 i s t h e

transition dipole moment between the ground state and the excited state |1⟩, and ∆µ 1 is thedifference between the excited state dipole moment and the ground state one.

In the more general case of multipolar molecules however, other non-negligible andpossibly significant coefficients such as β x y y cannot coexist within the two-level with the

usual diagonal βx x x coefficient. In the case of C2v symmetry for example, a three-levelmodel has been introduced12b , whereby transition moments of the two excited levels |1⟩ and

|2⟩ have perpendicular polarizations so as to account for the non-diagonal coefficientβ xyy , according to molecular symmetry elements. In a general spectroscopic scheme

consisting of two neighboring excited levels |1⟩ and |2⟩ quasi-resonantly addressed bymutually coherent one- and two-photon absorption steps, the various transition probabilities

involved therein take then more complex expressions, involving the second

excited level in the µ0 2 ⊗ (∆µ 2 ⊗ µ0 2 + µ0 2 ⊗ ∆µ2 ) term, and the coupling terms

µ 0 1 ⊗(µ0 2 ⊗ µ2 1 + µ2 1 ⊗ µ0 2) a n d µ 02 ⊗(µ 01 ⊗ µ 12 + µ 12 ⊗ µ 01). The resultingquantum expression, in parallel with the quantum expression of β, show finally that Eq. 13is even applicable in this condition, permitting therefore to consider here multipolarmolecules as well one-dimensional ones.

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At this stage, it is possible to recognize that the expression of the overall transition

probability is strikingly similar to the linearized

orientational distribution in Eq. 8, namely :

(14)

with the Ti and Ei tensors are provided by Eqs. 13. Following the « in-place » electronicphotoexcitation by these three mechanisms, molecules are orientationally randomized witha randomization cross-section scaling with the excitation probability, the scaling factorbeing furthermore assumed to be isotropic. Following reorientation, the initially isotropicangular distribution will be decreased by angular holes with depth proportional to P0 1 ( Ω ) ,namely :

(15)

with f(Ω) has been normalized in this expression.It is then straightforward to express any optical property S of the corresponding molecularproperty s(Ω ) with the statistical distribution f(Ω ) :

(16)

where N is the density of molecules per unit volume, F an adequate local field factor. Thisexpression is strikingly similar to Exp. 10, and by use of a similar irreducible tensorrepresentation formalism and application of the same permutation scheme between s and Ei ,albeit in irreducible spaces of order J :

(17)

Expression 17 is analogous to expression 12 with 1/kT replaced by –κ, T J referring nowto the Jth order irreducible component of a molecular (hyper)polarizability α, β or γ, and EJ

to the projection of the corresponding « write » field tensor in the irreducible space of same

order J with 2J+1 components In expression 17, the index i has been dropped to easenotations In particular, the quadratic susceptibility can be further expressed as :

(18)

8 8

where the factors are the squared norms of the first order hyperpolarizability

irreducible components and F is a relevant local field factors depending on the « read » and

« write » frequencies. Note that the phase factor in χ( 2 ) has been remove for the sake of

simplicity.

The χ (2) rotational spectrum encompasses both J = 1 dipolar and J = 3 octupolar

irreducible tensorial spaces, a situation in contrast with the more restrictive purely dipolarspectrum of the electrically poled systems at equilibrium, as from Eq. 12 with

namely:

(19)

Confrontation of Eqs. 18 and 19 illustrates the increased potential of optical poling in terms

of the extended photoinduced χ (2) rotational spectrum. In both expressions, we have left

respectively κ, a post-excitation randomization cross-section factor for photoexcitedmolecules and the thermal energy 1/kT to point-out the distinctive physical origins of bothprocesses.

It is possible to cover the full E spectrum from a purely dipolar (e.g. the write fieldnonlinear anisotropy reaches its minimum ρ E = 1/2) to a purely octupolar (e.g. ρ E = ∞

with E J= l vanishing) write field configuration by keeping the E ω polarization circular

and varying the ellipticity of the E2 ω

beam. Such ellipsometric engineering of the write Etensor will be further developed in the next Section.

In conclusion to this section, Exp. 18 embodies the respective influences of themolecular structure via the ρ molecular anisotropy parameter and that of the write fieldconfiguration via the ρ E write field anisotropy. It opens-up the possibility to monitor the

χ (2) symmetry pattern following either a molecular engineering approach based on the

connection between molecular structure and molecular anisotropy or by a less conventionalphotonic engineering approach based on the ellipsometric engineering of an adequate writefield tensorial anisotropy as will be further elaborated in the coming Section.

IV. DUAL MOLECULAR AND PHOTONIC ENGINEERING BY ALL-OPTICALPHOTOINDUCED POLING

The rich tensorial features of multiphoton absorption processes governing opticalpoling tend to devote this experiment to complex molecular and field tensor structures, anddemands that the possibly complex spectroscopic structure of multipolar molecules beaccounted for. Depending on the molecular symmetry, and on the associated quantummodel, the excitation probability will exhibit different symmetry features, and we will in

particular concentrate on the two-level model expression in the case of one-

dimensional molecules, and on the three-level model expression in the case C 2 v

molecules satisfying a three-level with 2ω located at the vicinity of the excited levels |1⟩and |2⟩. This will be studied in the following section, where the influence of different

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molecular symmetries are investigated under the constant background of a fixed « write »field tensor configuration.

Eq. 18 suggests the possibility to pattern a given photoinduced χ(2) tensor via the βtensor (e.g. molecular parameters) which pertains to the more traditional molecularengineering approach, or via the field tensor (e.g. ellipsometric parameters) which can be

referred to as a photonic engineering approach. As the molecular tensorial components β J

are directly coupled with the corresponding field components E J of the same order J, onemay want to exploit the opportunity of filtering the spherical orders J of the rotational

spectrum of χ(2) by adequate engineering of β or E. In a first approach, a given optical

field tensor can be engineered by adequate choice of the polarizations of the E ω and E 2 ω

fields, each of the spherical components EJ

being weighted by corresponding molecularspherical properties of the same order J. Fine tuning of the dipolar and octupolar

contributions to the final χ( 2 ) tensor can thus be achieved. As an example, poling a pure

octupolar molecule will result in a purely octupolar macroscopic susceptibility χ( 2 ),

provided the write field tensor accommodates a non vanishing the octupolar component.

Contrary to the molecular engineering approach whereby the accessible χ(2) tensorsymmetry is somewhat limited by the choice of molecular structures, the ellipsometricengineering of the « write » beams permits to reach any desired macroscopic anisotropy.

Molecular engineering approach

We consider hereafter a given write field tensor E exhibiting reasonably balanceddipolar and octupolar contributions, as provided for instance in the configuration whereby

the E ω and E 2ω fields are linearly polarized along the same direction Z. The final

write field tensor leads to comparable EJ= 3

and E J = 1

contributions with In such given optical configuration, the

consequence of molecular symmetry on the patterned χ( 2 )symmetry can be approached by

the study of the angular dependence of the excitation probability which governs,

via Eq. 13, the sole non-centrosymmetric contribution to the orientational distributionfunction in the medium and the subsequent macroscopic order.

In the case of polar rod-like molecules which satisfy the spectroscopic two-levelmodel, the dipolar transition moment and the difference dipole are both polarizedalong the x direction. The optical field and molecular configuration impose cylindricalsymmetry around the Z axis and the angular distribution functions are then amenable to asingle angular parameter θ = whereby x is the molecular charge transfer axis. Eq. 13

permits to retrieve the angular dependencies of which is plotted in Fig. 4.

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Figure 4. Angular plot of the joint one- and two-photon excitation probability contribution as a function of

the θ angle, for a one-dimensional molecule satisfying the two-level model, the 2ω and ω beams being bothlinearly polarized along Z. The continuous line is representative of the positive contribution to the

(θ) function, and the dashed line to the negative one.

As seen on Fig. 4, the molecular symmetry and corresponding one-dimensionaltransition and difference dipole moments lead to a polar distribution with increased cross-section for molecules oriented “upward” (i.e. in the x>0 hemisphere) and decreased for

molecules “downward” (i.e. in the x<0 hemisphere). The negative value of ( θ )

represents a perturbation to the positive dominant (θ) one-photon absorption cross-

sections while the two-photon perturbation (θ) is always positive. As a consequence,

the overall transition probability remains, as

required for a probability density, positive for all molecular orientations.In the case of multipolar molecules, the two-level model must be extended to a more

comprehensive quantum model as invoked before. In the case of C2v molecules, both the |1⟩and |2⟩ excited levels participating to the three-level model have been shown to beaccessible from the ground state via specifically x or y polarized transitions12b : as in theone-dimensional case, the vectors and are polarized along the x direction,

whereas the and transition dipole moments are polarized along the perpendicularin-plane y direction. In those conditions, a randomly oriented molecule is defined by a setof three angles : the previous θ angle characterizing the tilt angle of the charge transfer axisx with respect to the optical axis Z ; φ the second spherical angle defining a rotation of the xvector around the Z axis, and ψ the rotation angle of y around the molecular axis x. Such aconfiguration obviously follows a cylindrical symmetry and all x directions of a circularbasis cone of axis Z and half-angle θ will be equivalent. The angle φ will furthermore beabsent from the statistical distribution but will have to be taken into account in thestatistical summation. In the case of planar C2v molecules, is along x and along y

91

as previously discussed. ψ = 0 will correspond to molecules with plane parallel to the writebeam polarization, ψ = π/2 to a molecular plane at an angle π/2 – θ with respect to Z.

We consider the case of a strongly two-dimensional system whereby the excited states

|1⟩ and |2⟩ are located close-by, namely , w i t h, a n d t h e o r t h o g o n a l y-polarized transition with

dominates over the x-polarized transition with µ01 = 1 (a.u.). Theprevailing contributions from the µ02 and µ12 transition dipole moments promote animportant off-diagonal βxyy linear coefficient as compared to the diagonal βxxx one as

discussed previously for the DNDAB molecule. This exemplary case corresponds inparticular to a highly anisotropic C2v molecule with βxyy = 12 βxxx . We consider here the θdependence of the joint one- and two-photon probability contribution in the radialconfiguration whereby ψ = 0, the molecular plane being then parallel to Z. In this case, the

two transition dipole moments and are in the (Z,X) plane and can interact with thewrite optical fields. The joint one- and two-photon contribution to the excitation probability

is depicted in Fig. 5. For such a system, the angular dependence of the

term tends to be more isotropic than in the one-dimensional case, with

two pairs of antisymmetric lobes instead of one in the one-dimensional case (Fig. 4). Theangular spread between the two lobes can be linked to the relative magnitudes of y over xpolarized transition dipoles.

Figure 5. Angular plot of the joint one- and two-photon excitation probability contribution

as a function of the θ angle for a molecule satisfying the three-level model with

, the ω and 2ω beam being linearly polarizedalong Z. The continuous (resp. dashed) portion indicates a positive (resp. negative) contribution to the

function.

92

In the tangential ψ = π/2 case configuration, is still in the (Z,X) plane buthas been rotated in the orthogonal direction. The corresponding polar plot exhibits thereforea pronounced axial feature similar of Fig. 4 and reminiscent of the simple x-polarized two-level system, as the |0⟩ → | 2⟩ transition remains perpendicular to Z for all θ and istherefore dipolar forbidden.

In the boundary of octupolar molecules of D3 h symmetry, the excited level can beshown from symmetry arguments to be doubly degenerated9, the second excited state being

then now referred to as in order to specify the degeneracy, with ω1 = ω2 = and

. The joint one- and two-photon contribution for octupolar

molecules, depicted in Fig. 6, is representative of a purely octupolar distribution, as in thiscase the sole octupolar component of the write field tensor is coupled to the molecularhyperpolarizability tensor, in the absence of dipolar component in the molecular tensor.

Figure 6. Angular plot of the joint one- and two-photon excitation probability

contribution as a function of the θ angle, for a molecule satisfying a doubly degenerated-level model, the ωand 2ω beam being linearly polarized along Z. The continuous (resp. dashed) portion indicates a positive

(resp. negative) contribution to the function.

As in the previous case, the ψ = π/2 situation is similar to that of the one-dimensionalcase.

We have thus shown in principle the possibility to monitor the linear and nonlinearabsorption cross-sections that govern the final macroscopic symmetry pattern by adequatetuning of the molecular properties. Molecular multipolar symmetry and moreover therelative amplitudes of transition dipoles along different molecular directions are seen toplay a central role in modeling nonlinear induced joint one- and two-photon probability,reflecting the consistent implementation of both geometric and quantum dimensionalityconsiderations.

93

Photonic engineering approach

The macroscopic χ(2) susceptibility symmetry can be conversely governed by the writefield tensor symmetry properties for a given molecular structure, thus following a more

photonic engineering drive. As evidenced in Eq. 18, the βJ spherical components are thencoupled to spherical components of the same order J of the write field tensor. In general, asthe writing process is performed in resonant conditions, Kleinman permutation symmetry isno more valid and J = 0 and J = 2 antisymmetric orders appear in the β decomposition.Such pseudo-tensorial terms depend on non-diagonal β coefficients, and will vanish in theone-dimensional case, with β = βxxx (x ⊗ x ⊗ x ). As discussed in the previous sections andexemplified experimentally via polarized HLS in the case of the prototype DR1 molecule,the magnitude of the octupolar and dipolar components are of comparable amplitudes with

. Such one-dimensional structures provide therefore an adequate

choice for discussing the influence of the write field pattern symmetry filtering with no apriori restriction as a result of molecular symmetry constraints.

A one-dimensional molecule with comparable octupolar β J=3 and dipolar β J=1

components may then be coupled with a write field tensor correspondingly exhibiting both

E J=1 and E J = 3 components. These contributions can be tuned via different sets of

polarizations for the two incident E ω and E 2ω write beams12c . Considering firstly linearlypolarized beams, it can be shown that varying the relative angle between the two ω and 2ω

polarizations directly reflects on the anisotropy ratio of the write

field tensor, which permits to cover a broad range of ρ E , but falls short of reaching theboundary cases of dipolar and octupolar symmetries12c. Using the angular tunable parameter

θ defining the tilt angle of the linear polarization E ω relative to the fixed E 2ω linear

polarization, this procedure leads to a variation of the write field squared anisotropy ρ2E as

a function of θ ranging from ρ E = 2 (e.g. E 2ω / /E ω ) to (e.g. E 2ω ⊥E ω ) .

Purely octupolar (e.g. ρ E = ∞ ) or dipolar (e.g. ρ E= 1/2) field tensor are thus notencompassed within this range and cannot be reached a purely linear polarization basedconfiguration. The use of elliptically polarized fields permits however to extend thisanisotropy range and encompass octupolar and dipolar light tensor configurations. It has

been shown12c in particular that if E 2ω is elliptical of ellipticity ε and E ω circular,adjustment of the 2ω beam ellipticity via the δ tunable parameter defined as ε = tan δpermits to extend the ρ E variation from the octupolar ρ E = ∞ value (with δ = – π /4, that

is E ω and E 2ω counter-circularly polarized) to the dipolar ρ E = 1/2 value (with

δ = π /4, that is E ω and E 2 ω co-circularly polarized).Such possibilities have been demonstrated experimentally as shown in the following.

Different write configurations were tried experimentally on a thin film made of quasi-one-dimensional Dispersed Red One grafted to a MethylMetacrylate matrix in a 30:70 massproportion. The DR1 rod-like chromophore is furthermore photoisomerizable at the 532 nmexcitation wavelength as a result of the reversible trans → cis → trans photoinduced motionalong the diazo -N=N- backbone. A quasi-co-propagating geometry has been set-up, asdepicted in Fig. 7. The fundamental IR laser source at 1.064 µm is a transverse andlongitudinal single-mode Nd 3+:Yag laser, with pulses of 30 ps duration at 30 Hz repetition.

rate. The E 2ω harmonic writing field is generated by frequency doubling of the incident

94

E ω fundamental field in a phase-matched type I KDP crystal. A small angular deviationbetween the fundamental and harmonic write beams permits to control independently therespective ellipticity or polarization angle of both ω and 2ω polarizations with independenthalf- and quarter-wave plates. The detection direction is set along the write harmonic beampropagation direction, as defined by wave-vector (e.g. photon momentum) conservationconditions for a six-wave mixing process. A variable delay line on the fundamental beampathway permits to ensure temporal overlap of the ω and 2ω pulses. Following a writing

step of approximately ten minutes duration, the χ (2) symmetry is probed after relaxation

by second harmonic generation of a linearly polarized E ω fundamental incoming beam.The incident polarization is varied by use of a half-wave plate, permitting to probe themacroscopic polarization response of the photoinduced susceptibility at a variable angle

polarization incidence.

Figure 7. Experimental set-up for a quasi co-propagative all-optical poling configuration. (a) writing step: P:Glan polarizer. F1, F2: infrared and visible band-pass filters. M1, M2: mirrors. L: converging lens. (b) read-out step: rotation of the half-wave plate on the fundamental beam path. F2: interferential filter to select theharmonic 532 nm wavelength. PM: Photomultiplier tube.

To express the measured nonlinear intensity as a function of the read polarizationangle φ, we introduce the spherical decomposition of the « read » F field tensor defined

as F = e 2ω⊗ F ω ⊗ F ω , where e 2ω is a unit vector along the analyzed polarization

and F ω is the vector amplitude of the incoming fundamental beam. This « read » field

95

tensor is quite similar to the field tensor introduced in a crystalline engineering context25,whereby the study of propagation properties are crucial. In the present configuration, the

resulting macroscopic harmonic polarization amplitude ⟨P⟩ Ω = χ (2) • F , where thebracket notation refers to the orientational averaging with respect to the orientationaldistribution function invoked previously, can then be expressed as:

(20)

where the F J components depend on the polarization configuration which has beenchosen to probe the nonlinear medium. In the present context of a co-propagatingconfiguration, we chose to use the incoming read fundamental field as the write

fundamental beam, so F ω= E ω .Linear polarizations configurations have been first tested and found in a good

agreement with the expected φ-dependence derived from the general formulation inEq.20, adapted to the specific write/read polarization configuration. The configuration in

Fig. 7 corresponds to the overall nonlinear intensity

analyzed along the X harmonic polarization direction, with and different

angular parameters for = 0°, 45° and 90°. The different SHG patterns

clearly evidence the dependence of the photoinduced χ(2) symmetry going from a more

anisotropic outlook for θ = 0 to a more isotropic φ dependence for larger θ values. From

the fits of the various polar plots in Fig. 8, which depend on the sole β J = 1 and β J = 3

irreducible molecular components theoretically appearing in the one-dimensional DR1

molecule, a value of can be consistently inferred for the DR1-

MMA film. This ratio is in relatively satisfactory agreement with the expected value of

ρ = = 0.82 for rod-like systems which has been firstly proposed on the basis ofpolarized HLS measurements in solution.

96

Figure 8. Experimental and corresponding fits of the harmonic intensity analyzed along the X direction

for a DR1-MMA sample, as a function of the read polarization angle φ, in the case of linearly polarized

writing fields = 0°, 45° and 90°, corresponding to variable write field tensor anisotropies.

Beyond the demonstration of the control of the χ (2) symmetry by linearly polarizedwrite field polarizations, other write configurations with Eω circularly and E2ω ellipticallypolarized can be advantageously implemented and probed. The demonstration of thepossibility to achieve a purely octupolar configuration is indeed of particular interest, inview of its unique ability to generate an outgoing harmonic intensity independent of the φincident polarization angle, in the absence of polarization analysis. We have implementedboth co-circular (e.g. δ = 45°) and counter-circular (e.g. δ = -45°) configurations asdescribed previously, corresponding respectively to a purely dipolar and a purely octupolarwrite field multipolar symmetry. The overall φ-dependence of the nonlinear intensity

was measured without analyzer, and the resulting polar plots are

presented in Fig. 9. From the different polarization response patterns, we evidence thepossibility to control the anisotropy of the photoinduced nonlinear response by a purelyellipsometric optical approach. The resulting response of the dipolar (Fig. 9a, obtained forco-circular write polarizations) and octupolar (Fig. 9c, for counter-circular writepolarizations) write configurations can be clearly distinguished from their remarkablydifferent anisotropic features: the response of the dipolar macroscopic configurationexhibits a pronounced anisotropy, in contrast with the octupolar response which isindependent of the incident polarization angle. The anisotropic feature in Fig. 9a, results

indeed in the coupling of a purely dipolar write field tensor and the βJ = 1 dipolar molecular

component. In the octupolar case, the sole β J = 3 octupolar component of the molecular βtensor of the DR1 molecule is being coupled to the octupolar write field. Any intermediate

97

situation between dipolar and octupolar χ(2)symmetry configurations can be obtained by

varying the tan δ ellipticity parameter of the harmonic field, following the dependencedepicted in Fig.9b.

The purely octupolar configuration furthermore demonstrates the possibility tophotoinduce a SHG pattern displaying polarization independence, an important prerequisitein practical situations where the nonlinear optical function has to operate at a randomincoming polarization.

The write field tensor structure has been shown to complement that of the molecularsusceptibility, demonstrating the possibility to drive the photoinduced nonlinear tensor toany desired pattern by means of different write field configurations. Ellipsometric control ofthe all-optical poling scheme complements therefore the advantages of a molecularengineering approach, offering the possibility to reach any desired anisotropy features ofthe second harmonic response by adequate optical engineering of the write beampolarization.

Figure 9. Experimental (squared dots) and theoretical plots (full line) of the harmonic intensity I2 ω(φ)without analyzer for a DR1-MMA thin film sample, as a function of the read polarization angle φ. (a) :

dipolar configuration, E ωand E 2 ω

co-circular. (b) : multipolar configuration, E ωcircular and E 2ω

elliptic. (c) : octupolar configuration, E ωand E 2 ω counter-circular.

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V. PERSPECTIVES AND CONCLUSION

One may envision the last two decades of molecular nonlinear optics as a progressiveshift from an educated, however relatively « passive » and moderately predictive approachto a more « active » stage whereby participation of externally controllable physicalprocesses increasingly permit to monitor and predictively drive the elaboration process tomore refined structural targets. The first decade or so of studies on the nonlinear opticalproperties of organic molecules and crystals were based on the predictive discriminationamong the unlimited pool of available structures of those systems exhibiting the requiredfeatures at microscopic and macroscopic levels towards enhanced efficiency . This26

sequential approach whereby nonlinear optics comes into play after long and eventuallytedious fabrication steps entailed considerable drawbacks among which the limited abilityto control, beyond some crude semi-predictive guidelines, the sophistication and intricacyof internal forces that govern the structure of materials, or the long time lapse reachingsometimes many years between the early proposition of a molecular engineering conceptdown to its validation by crystal optics experiments27 as illustrated for example in thedecade long maturing of NPP crystals from molecular studies to Optical ParametricOscillator.

In that respect, a key turning point has been in the mid eighties the implementation ofelectric field poling techniques in the realm of functionalized polymer films. Such polingtechniques had been known before the advent of nonlinear optics28 and applied to solutionsin the context of the EFISH experiment, but came to full fruition with the recognition ofviscous phases of amorphous polymers as practically relevant poling media4 towards solidstate applications. Nevertheless, this technology entails limitations which may ultimatelycurtail some of its application perspectives, like the drawbacks due to electrode depositionin terms of charge injection, the difficulty to obtain quasi-phasematched structures, or theanisotropy of the polarization response in waveguides.

As summarized in this Chapter, all-optical poling scheme is presented as a third nextstep in the advance towards predictive control of the structural properties of opticalmolecular materials, and give the possibility to push back previous limitations ofelectrically poled polymers, of although many problems remain to be clarified, by tailoringmultipolar symmetry patterns using the diversity of ellipsometric configurations,

controlling the χ ( n ) spatial distribution of linear and nonlinear optical properties andmaking then periodic modulation and polarization independent octupolar nonlinear thinfilms possible.

Beyond early demonstrations descussed here, further exploration of the applicationpotential of this approach, particularly in waveguiding and microcavity formats, is currentlyunder way. A better understanding of the photoinduced reorientation mechanism, inparticular its time dependence, and optimization of the relevant underlying chemical andphysical parameters, once clearly identified, are challenging goals of a more fundamentalnature which are no less crucial towards applicative developments.

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and J. Zyss, Nonlinear Optics 14, 23 (1995), (c) J. Zyss, T. Chauvan, C. Dhenaut and I. Ledoux,Chem. Phys. 177, 281(1995)

[12] (a) J. Zyss and I. Ledoux, Chem. Rev. 94, 77 (1994), (b) S. Brasselet, J. Zyss, Journal of Nonlinear Phys.and Mater. Vol. 5, n°4, 671 (1996), (c) S. Brasselet and J. Zyss, J. Opt. Soc, Am. B, feature issue onOrganic and Polymeric Nonlinear Optical Material (to be published, January 1998)

[13] N.B. Baranova, B.Ya Zel’dovich, J. Opt. Soc. Am. B. 8, 1, 27 (1991)[14] R.J. Glauber, in « Quantum Optics, Proceedings of E. Fermi International School in Physics » p. 15, R.J.

Glauber, Ed. (Academic, New York, 1967)[15] (a) Y. Sasaki, Y. Ohmori, Appl. Phys. Lett., 39, 466 (1981), (b) U. Osterberg and W. Margulis, Opt. Lett.

12, 57 (1987)[16] R. H. Stolen and H.W.K. Tom, Opt. Lett. 12, 585 (1987)[17] C Fiorini, F. Charra, J.M. Nunzi, J. Opt. Soc. Am. B. 11, 12, 2347 (1994)[18] (a) J.M. Nunzi, F. Charra, C Fiorini, J. Zyss, Chem. Phys. Lett. 219, 349 (1994), (b) J. Zyss, I.D.W.

Samuel, C Fiorini, F. Charra and J.M. Nunzi, proceedings of the OSA/ACS conference on “OrganicThin Films for Photonics Applications”, Technical Digest Series Vol. 2, pp. 264-267, Portland(1995)

[19] C Fiorini, F. Charra, J.M. Nunzi, I.D.W. Samuel and J. Zyss, Optics Lett. 20, 24, 2469 (1995)[20] S. Brasselet and J. Zyss, « Control of the polarization dependence of optically poled nonlinear polymer

films » to be published in Opt. Lett. (Oct. 1997)[21] R, Bonneville, Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. B: Nonlinear Optics. 2, 159 (1992)[22] J. Jerphagnon, D.S. Chemla and R. Bonneville, Adv. in Phy. 27, 609 (1978)[23] E.P. Wigner, Group Theory, Academic Press, New York (1959)[24] (a) S.G. Cyvin, J.E. Rauch and J.C. Decius, J. Chem. Phys. 43, 4083 (1965), (b) R. Bersohn, Y.H. Pao

and H.L. Frisch, J. Chem. Phys. 45, 3184 (1966)[25] (a) B. Boulanger and G. Marnier, Opt. Comm. Vol. 79, n°1, 2 (1990), (b) B. Boulanger, J. Zyss, Chap.I.8

in International Tables for Crystallography, Vol. D: Physical properties of crystals, A. Authier Ed.,to be published by the International Union of Crystallography (Kluwer Academic Publisher,Dordrecht, Netherlands, 1997)

[26] J. Zyss and D.S. Chemla, in “Nonlinear Optical Properties of Organic Molecules and Crystals”, Vol. 1,D.S. Chemla and J. Zyss Eds., Academic Press, Orlando (1987)

[27] (a) J. Zyss, JF Nicoud and M. Coquillay, J. Chem. Phys. 81,4160 (1984), (b) S.X. Dou, D. Josse, J.Zyss, J. Opt. Soc. Am. B 10, 1708 (1993)

[28] Benoit, Ann. Phys. (Paris) 6, 561 (1951)

100

MOLECULE ORIENTATION TECHNIQUES

François Kajzar and Jean - Michel Nunzi

CEA, LETI - Technologies Avancées, DEIN/SPE/GCO, Centre d’Etudes deSaclay, France

INTRODUCTION

Under an external forcing field the medium polarization is varied. Within the dipolarapproximation this variation can be developed in the power series of the external forcingfield E

(1)

where the development coefficients χ(n) are three dimensional (n+1) rank tensors describinglinear (χ(1)) and nonlinear optical (NLO) response (χ (2) , (χ (3)

, …) in the laboratory referenceframe. Similar development is valid on the molecular level. Again, under the action of theexternal forcing field the molecule dipole moment may be also developed in its series giving

(2)

where, similarly as before, the development coefficients α, β , γ , … are tensors describing theresponse of molecules to the forcing field E. In that case the field E exerted on molecule isnot the same as the applied external field (E in Eq. (1) ) because of the screening by internalfield.

The factors Ki appearing in Eqs. (1) - (2) depend on conventions and system of unitsused and involve also degeneracy factors, which depend on the NLO process underconsideration and are defined below (cf. Table 1).

As it is seen from Eqs. (1) - (2) the values of molecular hyperpolarizabilities andmacroscopic susceptibilities, within a given system of units, depend directly on theconventions used. Therefore, it is important when comparing data coming from differentdeterminations or with theoretical calculations to ensure what kind of conventions wereused.

For centrosymmetric materials, all odd rank tensors χ(2)

, χ (4), … ≡ 0 and for

centrosymmetric molecules, all odd rank molecular hyperpolarizabilities µ, β, … ≡ 0. Itfollows from the inversion symmetry. Indeed, if we inverse the time and consequently the

Beam Shaping and Control with Nonlinear Optics101Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

direction of electric field E we will have for the macroscopic polarization vector P (or,similarly, molecular dipolar moment µ on the microscopic level) the following relation

and consequently

(3)

(4)

with the only possibility : χ (2) = 0

Fig. 1. Laboratory (XYZ) and molecular (xyz) reference frames. The charge transfer axis of NLO

molecules (PNA in the present case) is directed along the z - axis.

The same consideration is valid for the other odd rank tensors. Equations (3)-(4) show thatin order to get non-zero even order (χ(2), χ (4), …) NLO susceptibility tensors, a polar axis issufficient.

For device applications important is the value of the macroscopic second order NLOsuceptibility χ(2). This is determined not only by the microscopic NLO response of individualmolecules, but also by the way how these molecules have been assembled into the bulkmaterial. In an extreme case one can obtain a macroscopically centrosymmetric structurewith no second order NLO response, despite the fact that the constituent molecules arenoncentrosymmetric and exhibit a large β value. This is often the case of dipolar moleculeswhich tend to align antiparallel in order to minimize the ground state energy. Obviously suchstructure will be macroscopically centrosymmetric, with all χ

(2)tensor components equal to

zero.For a single crystal and for any second order NLO process (cf. Table 1) involving

three photons into interaction with frequencies ω1, ω 2, ω 3, where ω 1, ω 2 denotes thefrequencies of input and ω3 of output (ω1 + ω 2 = ω 3 ) photons, respectively, within theoriented gas model, there exists a simple relationship between the macroscopic χ(2)

I J K

susceptibility and molecular first hyperpolarizability βijk tensor components:

102

(5)

degeneracyfactor

where n numbers molecular species with corresponding number densities N(n), a’s are

Wigner rotation matrices transforming molecular reference frame (ijk) to the laboratorysystem (IJK) (cf. Fig. 1) and ƒ’s are local field factors taking account of the external fieldscreening exerted on the molecules by the internal field. For a spherical symmetry moleculethis is given by the Lorentz-Lorenz formula

(6)

where ε(n)ω= (n(n)ω)² is the dielectric constant and n is the refractive index of the mediumunder consideration at the optical frequency ω.

Table 1: Second order NLO processes and degeneracy as well as convention factors

2nd order NLO process

Nondegenerate three wavemixingSecond harmonic generationLinear (Pockets) electro-opticeffectOptical rectification

χ(2)

susceptibility

χ(2)(-ω3;ω1 ,ω2)

χ (2) (-2ω;ω,ω)χ (2) (-ω;ω, 0)

χ(2)(0; ω, -ω)

2

12

2

Multiplicativefactor conv I

Kconv I2

2

12

2

Multiplicativefactor conv II

K conv II2

1

½2

1

Relation (5) can be used when, for a given crystal and molecular symmetry, thenumber of χ(2) or β tensor components is reduced, to calculate the bulk χ(2) susceptibility ifmolecular hyperpolarizability is known4-5. Otherwise it happens to be too complicated.

In other cases (e.g. poled polymers), the second order NLO susceptibility implies anaveraging over all molecular orientations and is given by

(7)

where for the sake of simplicity and as it is usually the case, we have assumed onemolecular, active, species and F is the global local field factor

(8)

The averaging in Eq. (7) is performed on all molecular orientations.Two conventions for the Fourier transform of polarization and electric fields are

usually used:

Convention I:

(9)

103

(10)

Convention II:

(11)

(12)

The multiplication factors K’s (cf. Eqs. (1)-(2)) resulting from these two conventions andtaking into account the degeneracy factors (the number of non-equivalent permutations ofelectric fields referred to input photons) are listed in Table 1. Throughout this text we usethe esu reference system and convention II. In SI system the factors K listed in Table 1should be multiplied by ε0. More information on different conventions can be found in refs. 1-3 .

The envisaged noncentrosymmetric structures may find applications in an importantclass of devices, such as frequency doubling (blue conversion to enhance the data storagecapacity, microlithography, medicine, biology), tunable light sources (optical parametricoscillators), electro-optical modulation for high rate (tens to hundreds of GHz) signaltransmission, tetrahertz electric pulse generation, not achievable by purely electric circuitry,etc. For this kind of applications the macroscopically non-centrosymmetric, and highlyoptically nonlinear structures are required. Moreover, the majority of applications aretargeted in the integrated structures and in waveguiding configurations. Therefore it is ofimportance to fabricate non-centrosymmetric thin films with non-centrosymmetry in adesired direction. For these reasons the achievement of macroscopically noncentrosymmetricthin films is an important challenge mobilizing a lot of effort. Between different approachesdeveloped up to now let us note the following ones:

- Langmuir - Blodgett X, Z or alternate layers build - up technique8-9

- Poled polymers (for a review see Refs. 10 - 11)- Monocrystalline (epitaxy or molecular epitaxy)12 or oriented thin film growth

(heteroepitaxy) 13

- Intermolecular charge transfer complexes 14.In this paper we limit the discussion and description to poled polymers only. Differenttechniques of chromophore orientations, their yields and kinetics will be described anddiscussed in next chapters. A special attention will be paid to newly introduced polarorientation methods such as optical poling techniques: the photo - assisted15-16 or all opticalpoling17-18 techniques. These techniques have been developed with functionalized polymersexhibiting good propagation properties and large NLO responses.

104

Fig. 2. Schematic representation of different ways of making functionalized polymers for second order NLOapplications: (a) - guest - host systems, (b) - side chain polymers, c - main chain polymers, (d) - photo - orthermally crosslinking polymers Arrows represent chromophore dipole moments orientation.

Fig. 3. Chemical structure of side chain (a), matrix and active chromophore which cross link under heating(b) or UV irradiation (c).

105

CHROMOPHORE ORIENTATION TECHNIQUES

Fig. 4. Schematic representation of a charge transfer molecule. A and D are electron accepting (e.g. NO2 ,NO, CN, CHO, SO3 , ) and electron donating groups (NH2 , OC H3, OH, N(CH 3)2, ), respectively.

Fig. 5. Chemical structure of some charge transfer molecules.

As already mentioned functionalized polymers marry excellent optical quality ofpolymer backbone with a large nonlinear optical response of chromophores. Chromophores

106

can be either introduced to the polymer matrix by a simple dissolution (guest-host systems),attached to the polymer matrix (side chain polymers) or incorporated to the polymer mainchain (main chain polymers), as it is shown schematically in Fig. 2 In order to enhance theorientation stability the thermally19 or photo crosslinking2 0 systems have been also invented.In that case the chromophores are used to bind different polymer chains or polymer chainsegments (cf. Fig. 2(d)) under heating (thermal crosslinking) or UV irradiation (photo-crosslinking). Such procedure leads to the increase of the glass transition temperature and abetter stability of the induced orientation. Obviously the thermal - or photo crosslinkingprocedure is done under the poling field. Examples of thermally and photocrosslinkingpolymers are shown in Fig. 3.The following chromophore orientation techniques have been elaborated:

(i) static field poling(ii) photoassisted poling(ii) all optical poling

STATIC FIELD POLING TECHNIQUES

Application of a static field to dipolar molecules leads to the alignment of theirdipolar moments in the external electric field direction. Let us consider a quasi 1-D chargetransfer molecule (cf. Fig. 4). The transmitter (usually a conjugated π electron backbone)serves to transfer the charge from the electron donating to the electron accepting group.Some examples of charge transfer molecules are shown in Fig. 5.

The idea of using solid solutions and large electric fields in order to orient imbeddeddipolar chromophores is relatively old 6-7. Classically the required orientation of activemolecules is obtained by heating polymers to the glass transition temperature, at which theycan rotate, and by applying a sufficiently large external DC field (typically 100÷200 V/µm).The obtained orientation is frozen by cooling polymer to room temperature under theapplied external field (cf. Fig. 6). An additional chemical modification of the polymernetwork is performed, through thermal or photo-crosslinking, in order to increase thetemporal stability of induced orientation (cf. Fig. 2).The following techniques of DC poling have been developed:

(i) contact (electrode) poling(ii) corona poling21-22

(iii) photothermal poling23-24

(iv) electron beam poling 11,25

Fig. 6. Static field poling of dipolar molecules. Before applying electric field the dipolar moments arerandomly distributed. At higher temperatures the dipole moments are mobile and orient in the direction ofthe applied external field.

107

In the first case a large poling field is created through electrodes with the polymer thin filmplaced in between (cf Fig. 7). The film is heated up to the glass transition temperature andthe poling voltage is applied through electrodes. The main drawback of this techniqueconsists in the limited voltage which can be applied due to the dielectric breakdown insurrounding air. For this reason the technique usually is used in a high vacuum chamber.

Fig. 7. Schematic representation of an electrode poling set-up.

In corona poling technique the high poling field is obtained by deposited charges onthe film surface, which are created by the gas ionization in surrounding discharge needleatmosphere. The film is again heated to the glass transition temperature using a heatingblock. Sometimes a metallic grid (cf. Fig. 8) is used in order to control well the polingcurrent and to get a better homogeneity of poling. While in electrode poling technique thepoling occurs at any applied voltage (although its efficiency depends on it), the coronapoling is a threshold phenomenon and requires usually application of relatively high voltages(around 6-8 kV) with needle electrode distant by 2-2.5 cm from the thin film surface.

Fig. 8. Schematic representation of a corona poling set-up.

Phothermal poling23-24 is a simple modification of the electrode poling technique. Theonly difference consists in the use of a laser beam, with wavelength lying in the materialabsorption band, to heat thin film. The main advantage of this technique consists in a verylocalized poling. It has been used for fabrication of bidirectionally poled polymer films24.

In the electron beam poling technique the constant current, created in material by amonoenergetic electron beam with energy ranging between 2 and 40 keV11,25, is used toorient chromophores. The dipoles are oriented by the polarization field, created by trappedin bulk decelerated electrons. In this sense the technique is very similar to the corona polingtechnique, with the difference that in the former one the poling is due to the field created by

108

surface while in the last to the bulk charges. The technique allows also poling very smallareas and has been used for fabrication of periodical structures for quasi phase matched(QPM) second harmonic generation26 applications. In guest-host polymer systems PMMA -DR #1 similar poling efficiency as using the corona poling technique has been reported27 .

In all cases the use of electrode is required, with all possible negative aspects such ascharge injection and light absorption. As consequence, it implies the necessity of using bufferlayers, in such applications like frequency conversion in periodically poled systems30, where,otherwise it is unnecessary. Moreover, the poling fields are limited due to the micro-circuitsconnected with the point effect. This will lead also to unwanted and prohibitory increase ofoptical propagation losses.

In all techniques the chromophore orientation is frozen by cooling poled film to theroom temperature under the applied external field or by thermal or photo-crosslinkingduring the poling procedure. In the case of thermal cross-linking it is important to controlwell the poling temperature and to increase it simultaneously with the increasing cross -linking rate, as the glass transition temperature is also increasing. Shining with UV lightduring poling usually leads to an unwanted and uncontrollable increase of the poling current,reason why it is recommended to make poling alternatively with photo-crosslinking.Similarly as in the case of thermal cross-linking, the glass transition temperature alsoincreases. Therefore it is also important to increase the poling temperature during the photo-crosslinking process.

Variation of the linear absorption spectrum

With the poling field applied perpendicularly to the thin film surface the dipolarmoments of NLO chromophores will orient preferentially parallel to the field. This can beevidenced easily by observing UV - VIS absorption spectrum during poling31-32. At thenormal incidence, with electric field parallel to the thin film surface a net decrease of thelinear absorption is observed as it is seen in Fig. 9 for a side chain liquid crystalline polymerfunctionalized with the cyanobiphenyl chromophore33 . This variation may be used tocalculate the order parameter <P2> as it will be described below.

Fig. 9. Linear absorption spectrum of a side chain liquid crystalline polymer thin film, functionalized withthe cyanobiphenyl chromophore, before (solid line) and after (dashed line) poling (after Ref. 28).

109

In order to get closer into the DC field poling process let us consider the simplestcase of a linear, CT molecule with axial symmetry and dipolar moment directed along the z-axis (cf. Fig. 4). The linear polarizability α ij components of such a molecule in the principalaxes of the molecular reference frame are given by

αxx (ω) = α yy (ω) = α⊥ (ω) (13)

and

α z z (ω) = α || (ω) (14)

The dielectric constants in the principal axes of the laboratory reference frame will be givenby:

(15)

and

(16)

where θ is angle between molecular axis and poling field direction (cf. Fig. 10) and N is themolecule density number.The birefringence induced by applying poling field is given by

(17)

and

(18)

where the subscripts pol and isotr refer to the orientation average of poled films andisotropic films. With

(19)

Fig. 10. Molecular axis orientation with respect to the poling field.

110

one gets the well unknown formulas linking the variations of the optical dielectric constantswith the order parameter <P2 >

(20)

and

(21)

It follows from Eqs. (20) - (21) that in poled polymers the εZZ component of the dielectric

constant (and consequently the extraordinary index of refraction) increases during polingwhile ε X X (and as consequence the ordinary index of refraction) decreases. From Eqs. (20)- (21) one gets also the induced variation of the refractive index parallel and perpendicular tothe poling field direction

(22)

which in the case of a moderate poling can be rewritten as

(23)

Neglecting the change of the real part of refractive index and using the relation linking theimaginary part of refractive index with the optical density one obtains the following relationpermitting the determination of the order parameter

Fig. 11. Experimental geometry for measurement of the variation of linear absorption spectrum due topoling.

111

(24)

where A| |

and A⊥ (A +2A| | ⊥ = 3A0) are absorbances (optical densities) measured with the

incident light polarization parallel and perpendicular to the poling field direction,respectively (cf. Fig. 11). Typically the order parameters <P2 > take the values between 0.1and 0.25 for isotropic polymers, 0.5 - 0.6 for nematic liquid crystals and 0.9 - 1 for smecticliquid crystals, respectively. It is worthy to note that determined in this way the orderparameter <P2> does not discriminate between polar and axial order. The only way to getsuch a discrimination is by using NLO techniques, sensitive to the polar order. The build upof the order parameter with poling time t may be well described by a monoexponentialfunction (cf. Refs. 42, 43)

(25)

where <P2 > ∞ is the saturation value and the time constant of the orientation process τdepends on the poling temperature and poled polymer itself (viscosity of polymer, mobilityof chromophores, etc).

Fig. 12. Build up of the order parameter <P2 > for a side chain liquid crystal polymer at different poling

temperatures: - T = T + 5°C, - T = T , - T = T - 5°C, - T = T - 10°C, - T = T - 15°C (after Ref.g g g g g

43).

In order to link the order parameter <P2 > to the poling field it is important to knowthe orientation distribution function G(θ) giving the probability of finding the molecular axisin the direction defined by angle θ (for poled polymers the azimuthal symmetry is obvious).The average value of the function f(θ ) is given then by the following expression

(26)

where N A is the normalization constant

112

(27)

At higher temperatures (above or slightly below the room temperature), when moleculesmay rotate (free gas model) one uses for G(θ) the Gibbs - Boltzmann distribution functiongiven by

(28)

where Ep is the poling field. For dipolar molecules the ordering energy U is given by

(29)

where µ is the dipole moment of molecule. First expression on RHS of Eq. (29) describesthe dipolar interaction energy with poling field and the second one that due to the induceddipole moment by the applied field. Neglecting the induced dipole moment interactionenergy with the poling field (isotropic model) one can develop the orientation distributionfunction into the Legendre polynomials obtaining

(30)

where the expansion coefficients can be expressed in terms of the modified spherical Besselfunctions

34

with the parameter

expressing the ratio of the dipolar interaction ordering energy to the thermal randomizationenergy. The following recurrent relations obey for the modified spherical Bessel functions

where

Similarly for Legendre polynomials

(31)

(32)

(33)

(34)

(35)

113

with

(36)

and

(37)

(38)

(39)

and

(40)

Introducing the development (Eq. (30) into Eqs. (15)-(16)) one can calculate the inducedbirefringence by the poling field which is depicted in Fig. 13. It is clearly seen that the polingprocess, with poling field directed perpendicular to the thin film surface, increases theextraordinary index of refraction and decreases the ordinary one. The variation depends onthe x parameter.

The second observation seen in the variation of the linear absorption spectrum (cf.Fig. 9)) is the shift of the maximum absorption wavelength towards larger (red shift) orsmaller (blue shift) wavelengths (see. Fig. 9 and Refs. 31, 32). This shift is due to the DCStark effect. The polar orientation induced by the poling field leads to establishment of ahigh DC field experienced by molecules. The corresponding shift of the maximumabsorption wavelength may be related to the difference ∆µ = µ 11 – µ 00 between moleculedipole moments in excited (1) and fundamental (0) states

32

(41)

where is the Planck constant divided by 2π and c is the light velocity. Thus depending onthe sign of ∆µ one observes a red or blue shift in the optical absorption spectrum.

POLAR ORDER

As already mentioned, the measure of the variation of the linear absorption spectrum givesinformation on axial order, although, in some cases, especially in DC poling, this isconfounded with the polar order. One can get the true information on polar order by usingNLO techniques, such as SHG or linear electro-optic effect, sensitive to the non-centrosymmetry. For poled polymers with point symmetry ∞mm, functionalized with 1Dcharge transfer active molecules, characterized by an enhanced first hyperpolarizability β zzzin the CT direction z and provided that the Kleinman conditions are satisfied (usually this isdone far from the absorption band) there exist two non zero χ (2) (-ω , ω3 ;ω1 2) tensorcomponents (cf. Eq. (7))

(42)

114

and

Fig. 13. Calculated variation of the induced birefringence of optical dielectric constant with x = µE p / k T

(43)

Similarly as before the capital letters (X,Y,Z) refer to the laboratory reference framewhereas lower case to the molecular system.

Fig. 14. Calculated variation of the diagonal (χ(2)zzz) and off diagonal (χ (2) zzz) tensor components with x =

µEp /kT.

115

Again by using the orientation distribution function (cf. Eq. (30)) one can express thecorresponding tensor components in terms of the averages <Pn(cosθ)>. As consequence oneobtains

(44)

and

(45)

Both diagonal and off diagonal χ(2) tensor components increase at small poling field E p (cf.Fig. 14). At higher values of the ratio x (cf. Eq. (32) the diagonal tensor components χ(2)zzzis still increasing while the off diagonal χ(2)

xxz starts to decrease. The efficiency of the polarordering is given by the ratio

( 4 6 )

which varies between 1 and ∞. The last value is reached for perfectly ordered structures (alldipole moments pointing in the same direction). The parameter is a = 3 for a free electrongas (isotropic model). For side chain liquid crystalline polymers a values as high as 18 havebeen obtained35.

Case of liquid crystals

The formalism described above for polar order applies to polymers with no initialorder. However, in some cases, like in liquid crystals (LC’s) and side chain liquid crystallinepolymers (SCLCP’s) such initial axial order is present. In that case the model proposed bySinger, Kuzyk and Sohn (SKS model)36 or by Vand der Vorst and Picken37-38 , the last basedon Maier - Saupe theory39,40 applies (MSVP model).

Orientation mechanism

Singer, Kuzyk and Sohn (SKS) model36. In order to take into account the pre-existing axial order such as found in LC’s and described by the order parameters <P2> and<P 4> Singer et al36 neglected the interaction energy with induced dipolar moment and addedanother term to the molecular energy in Eq. (29) describing the tendency of rod-likemolecules to have their long axes more or less parallel to a preferred direction. They did notintroduce any explicit analytical expression for this term. Instead, they assumed that the axialorder of the LC « host » is transferred to the NLO « guest » molecules. In this model, thedipolar energy is first expanded into Taylor series up to first order in the poling field Ep.Then, the Gibbs-Boltzmann distribution is expanded in terms of Legendre polynomials (cf.Eq. (30)) with <P1> coefficients given by Eq. (31). In this case, the configurational averagevalues of <cos³θ> and <sin θ cos θ>/2 (cf. Ref. 35) are linear functions of the poling fieldstrength through the term x (cf. Eq. 32) and depend on the intrinsic axial order of the LCthrough the microscopic order parameters <P2> and <P4>.

116

²

MSVP model 37,38. In the SKS model, the transfer of axial order from the mesogenic rods tothe NLO molecules assumes implicitly the existence of latent mesogenic properties in theNLO molecules. In the MSVP model, the intrinsic mesogenic properties of the NLOmolecules themselves are taken explicitly into account.

The natural tendency of rod-like molecules to align mutually parallel can bedescribed by the effective single particle energy U0 (Θ), introduced originally by Maier andSaupe 39-40

(49)

(47)

where ξ is a parameter describing the strength of the anisotropic interactions. It takes aconstant value for a given liquid crystal. In the Maier - Saupe theory this parameter isproportional to the clearing temperature Tc (at zero field strength) through the followingrelation T c)

(48)

The order parameter <P2> has to be determined selfconsistently by using numerical methods.This value, injected into Eq. (29) (with neglect of U2 (θ), cf. Eqs. (47) and (29)) allows todetermine the orientational averages <cos³θ> and <sin² θ cosθ>/2.

Van der Vorst and Picken37-38,41 extended this model to the case of poling of singlerod systems. In addition to U0( θ) and U1(θ) they introduced the dipolar moment interactionenergy term, induced by poling field, similar to that in Eq. (29)

where ∆α is the anisotropy of molecular linear polarizability of the rod-like NLO molecules.The resulting orientational averages <cos³θ > and <sin² θcosθ>/2 are given in Table 2.

Table 2. rientational averages of <cos³ θ > and <sin² θ cos θ>/2 in different statistical orientational models(after Ref. 35, for details see text).

a) u = µ0Ep/kTb) The order parameters <P2> and <P 4> are assumed to be independent of the electric field strength andare to be determined before the poling field has been applied.c) The order parameter <P 2> has to be determined selfconsistently by using numerical methods. Thisvalue, injected into Eq. (47) allows to determine the orientational averages <cos³ θ > and <sin² θ cosθ>/2.

Indeed, the SHG experiments performed on a series of isotropic and side chain liquidcrystalline polymers with different axial order parameters show a correlation between thesecond order NLO susceptibility χ(2) as well as the a parameter (cf. Eq. (46)) on one sideand the order parameter <P2> on the second, respectively.

As the degree of orientation depends on the product µE, the DC poling techniquelimits the kind of molecules to which it can be applied. As already mentioned it applies to

117

dipolar molecules only, which always show a tendency to aggregate, thus increasing thepropagation losses. This effect limits the amount of active molecules in polymer matrix,which on other side, is so important for macroscopic nonlinear optical response (see Eqs.(42)-(43)). Moreover heating to glass transition temperature may lead to unwanted,thermally induced chemical reactions. For practical applications thermostable polymers arerequired, with high glass transition temperature (200-250°C) ensuring a better temporalstability of the induced polar order.

In situ second harmonic generation measurements

The induced polar order has been studied by in situ SHG measurements. Figure 15shows a typical temporal growth of SHG signal for an isotropic polymer36-37 . This growthcan be described by a triexponential function

(50)

where l is the thin film thickness, τ‘s are the time constants and P’s are the maximumcontributions of the different mechanisms taking part in molecular orientation process,respectively. The sum of the poling saturation limits: P = P1 + P 2 + P 3 represents the overallpoling efficiency under the experimental conditions (temperature, geometry, poling field,atmosphere). The superscripts p and r in Eq. (24) refer to polymer thin film and referenceharmonic intensities, respectively. Both the time constants and the overall poling efficiency

Fig. 15. Temporal growth of the SHG signal during poling. Points depict experimental data whereas dashed,broken and solid lines show least square fit with a mono - , two- and three - exponential quations,respectively (after Ref. 35).

118

PHOTOASSISTED POLING

Dumont and coworkers 15,16,44,45 have observed that shining doped (or functionalized)polymer thin films, with non-centrosymmetric dipolar chromophores, in the chromophoreabsorption band induces a significant increase of electro-optic coefficient, corresponding toa better , polar orientation of chromophores. The measurements have been done using theattenuated total reflection technique, and the optical field polarization was perpendicular tothe applied low frequency external electric field to the thin film (cf. Fig. 16). A betterstability of induced orientation was observed in the case of functionalized polymers than inguest - host system, as it is usually the case with the static field poled polymers. Thechromophores orient with dipolar moments perpendicular to the optical field (and parallel tothe applied static (or low frequency) field. As it will be discussed later, the chromophoreorientation is going through trans - cis izomerization process (cf. Fig. 17).

Fig. 16. Attenuated total reflection set - up (a) and temporal growth of the electro-optic coefficient asfunction of laser illumination at 514.5 nm: (b) - experiment, (c) - theory (courtesy of M. Dumont).

Figure 16 shows the experimental set-up used by Dumont and coworkers togetherwith the observed and calculated dependence of the electro-optic coefficient on the external

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light illumination (green Argon laser line). The studied, poled film is placed between twoelectrodes, one of them being a silver electrode deposited on the side of a prism. Thecondition to excite surface polaritons in thin film depends on its refractive index andthickness as well as on the refractive indices of electrode material and prism. By varyingincidence angle it is possible to satisfy this condition, what is observed as a dip in theintensity of reflected beam. By varying refractive index of thin film with applied voltage onechanges the resonance conditions (or in other words the coupling angle). The technique isvery sensitive to thin film thickness and refractive index variation. As the variation ofrefractive index depends on thin film electro-optic coefficient the technique serves todetermine it with a high precision. The variation of refractive indices is measured with alock-in amplifier. By shining the studied film with HeNe laser emitting at 632.8 nm or withan Argon laser at 514.5 nm parallel to the poling field (optical electric field perpendicular toit) Dumont et al observed a noticeable increase of the measured electro-optic coefficient (cf.Fig. 16 ). The effect was significantly larger in the case of grafted polymer than with a guest- host system. Switching off the light source leads to a decrease of electro-optic coefficient,which is connected with the decrease of the polar order. The relaxation of polar order isfaster in guest - host system than in side chain polymer, as it is usually observed.

Fig. 17. Light induced reorientation of DR #1 molecule in the presence of the static field andphotoizomerization process. From almost parallel orientation to the exciting optical field the molecule dipolemoments reorient to an almost perpendicular direction.

A possible explanation of this effect is shown schematically in Fig. 17. The activemolecule, which is disperse red #1 (DR #1) (cf. Fig. 18) undergoes the trans - cisizomerization (cf. Fig. 19) under light illumination. The double N=N bond is flexible atexcited state and through rotation or translation, molecule may change configuration fromtrans to cis form. This process is very fast in liquids 46, 47 and is significanly slower in solidstate (Kajzar et a1

48estimated it for a grafted polymer as taking about 150 ns). The inverse

transformation (although sometimes the process is irreversible) from cis to truns form maygo only through nonradiative channels and is very slow (of the order of few seconds insolids). Molecules can return to the previous configuration, but will be again excited byincoming light. Thus the only stable position will be obtained if molecule orients with thedipole moment perpendicular to the incident light polarization, thus parallel to the appliedelectric field. As consequence, the light induced molecular re-orientation will lead to anincrease of the electro-optic coefficient as it was observed experimentally15,16,44-45 ( see a l soFig. 16). Using a simple rate equation for the trans - cis izomerization process Dumont etal 45 describe well the observed temporal behavior of electro-optic coefficient duringattenuated total reflection measurements (cf. Fig. 16).

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Fig. 18. Chemical structure of side-chain copolymer of DR1 in PMMA with 35% molar content andits optical absorption spectrum.

Fig. 19. Light (or heat) induced photoizomerization process (a) and electron transitions betweenfundamental and excited singlet states diagram (b) with ωC

>ωT. The trans to cis transition time is muchfaster than the reverse one and depends on molecular environment (usually much faster in liquids than insolids). In cis form, which often is unstable, the volume of the molecule is smaller, thus easing its rotation.

ALL OPTICAL POLING

The first observation of a polar orientation of chromophores in a grafted PMMA -DR1 polymer has been done by Charra et al 50 in a four wave mixing geometry with twopicosecond pump beams at 1.064 µm and a probe beam at doubled frequency (cf. Fig. 20).The observed signal at 0.532 µm rose slowly with time, up to a saturation value. Aspontaneous SHG was observed after switching off the probe beam, with a fast relaxation

1 2 1

component at the beginning48 . The maximum second order nonlinear optical susceptibilityvalue obtained in these experiments was of 3 pm/V

50.This experiment has shown that using

purely optical fields one can obtain a polar orientation of chromophores in a functionalizedor a doped polymer film.

Fig. 20. Four wave mixing geometry leading to the observation of all optical poling in a side chain polymerPMMA - DR1. Pump beams I1 and I 2 are at 1.064 µm while probe (I 3 ) and signal (I4 ) beams at 0.532 µm(after Ref. 50).

Fig. 21: Experimental set-up for collinear seeding geometry. A fast photodiode synchronizes the sampler.SHG intensity is measured with the photo-multiplier tube (PMT). P: polarizers; F: interferential filter at532 nm; SFS : spatial filtering system; R: dielectric mirror for fundamental rejection; S: shuttersynchronised with insertion of the green blocking RG670 Schott filter. BK7 glass plate is fixed on a rotatingstage for phase adjustment between ω and 2ω beams.

Significantly larger χ(2) value was obtained in seeding geometry using two collinearpicosecond beams at 1.064 µm and 0.532 µm. The experimental set-up used for opticalpreparation of polymers is shown in Fig. 21. It consists in a pulsed picosecond Nd:YAGlaser delivering both fundamental (1064 nm) and harmonic (532 nm) wavelengths at 10 Hzrepetition rate. Energies are 500 µJ and 2.5 µJ, respectively at 1064 nm and 532 nm, andbeam diameter is 2 mm at sample location. With the same polymer as that used in the fourwave mixing geometry the best obtained value for susceptibility was of 76 pm/V, closeto that obtained in corona poling 51-52 . This geometry has been already used for opticalpoling of glass fibers, where also an efficient SHG has been obtained after a seedingprocedure53-54 .

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Fig 22. Schematic view of the orientation mechanism in all optical poling with azo-dye molecules (a) andelectron transition diagram between fundamental and excited singulet states (b). In the case of azo dyes, thetrans -cis izomerization is achieved through both one and 2 photon excitations.

Poling field <E3>t ≠ 0

The mechanism of creation of χ(2) grating in glass fibers has been described byBaranova and Zeldovich 5 5 in terms of a polychromatic interference of input fields at ω and2ω frequencies, leading to a non-zero temporal average field

(51)

where ∆φ is the relative phase difference between Eω and E 2 ω fields. This interference leadsto a non-zero poling field, as it is shown schematically in Fig. 23. However the microscopicmechanism in polymers is different than that in glass fibers, where color centers and defectsare at origin of the created polarization. Similarly as in photoassisted poling and in the caseof azo dyes, the polar orientation of dipole moments can be explained by the trans-cisizomerization. Contrariwise to the photoassisted poling, where this izomerisation is inducedby a one photon transitions (cf. Fig. 17) in all optical poling this is excited simultaneously by2 photons with frequency ω and one photon with frequency 2ω (cf. Fig. 22).

As already mentioned, the used azo dyes absorb strongly light if the exciting opticalfield is parallel to the dipole transition moment. The double N=N bond in excited state ismobile and molecule changes conformation to cis-form, with a smaller volume. Subsequentlythe molecule relaxes slowly to the trans form through the non-radiative channels. Inphotoassisted poling a stable orientation of chromophore is achieved when the dipoletransition moment is perpendicular to the optical, exciting field. A given polar orientation isimposed by the applied DC field.

In all optical poling, due the existence of a non-zero temporal average “poling”field (cf. Eq. (51)) a stable chromophore orientation is obtained when the dipole momentchange is directed oppositely to the exciting field). There is a fundamental differencebetween photoassisted poling and all optical poling in the obtained chromophore orientation.In the first case chromophores orient perpendicularly to the optical field, whereas in thesecond - parallel (to the resulting E

3 poling field).

Dependence on phase mismatch between ω ω an 2ω ω beams

As it is seen from Eq. (51) the sign of the temporal average of cubic interference isdetermined by the relative phase between ω and 2ω beams. As in condensed materials,fundamental and harmonic frequency propagation velocities are different, the optical polarityalternates between positive and negative extremes. This can be easily verified by changingthe phase between ω and 2ω fields. It has been done by introducing a BK7 slab in opticalpath of fundamental and harmonic beams (cf. Fig.21) and rotating it along an axis

123

perpendicular to the beam propagation direction. Because of BK7 dispersion of refractiveindex the relative phase between ω and 2ω fields is varied continuously. Figure 24 shows anexperimental verification of this interference. A good agreement is observed between theobserved and calculated dependence of SHG signal on phase mismatch between ω and 2 ωfields.

Fig. 23. Polar cubic interference between optical electric fields at fundamental and harmonic frequencies.Double arrows are the excited molecules.

Fig. 24. Experimental dependence of the SHG intensity induced after 20-minutes preparation-time, on therelative phase ∆Φ of the ω and 2ω beams. Solid line corresponds to a theoretical dependence with∆ n = n(2ω ) – n (ω ) = 0.3. Sample was 0.1 µm thick with 0.3-optical density at 532 nm.

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Fig. 25 : Experimental real-time growth and decay dynamics of typical second harmonic signal amplitudes( α |χ(2) |) in a side chain polymer DR #1 - PMMA with 35% molar dye content seeded at 1064+532 nm.Negative times correspond to the seeding preparation process, positive times correspond to χ (2)decay. Insertgives the different seeding-beam polarizations checked (after 62).

Growth of polar order.

The growth and decay kinetics of the induced polar order was studied by Fiorini et alfor a guest host system and side chain polymer using the apparatus shown in Fig. 21. Theseeding procedure was alternated with reading procedure by using a mobile Schott filterRG670-glass. The filter was alternatively placed or removed from the input beams path.When it was placed in, the green beam was blocked and fundamental ω beam generated 2ωbeam from earlier created noncentrosymmetry. In the other case seeding procedure tookplace. The filter removal (or placing) was controlled by computer and consequently thelecture of induced SHG signal (corresponding to the moments when filter was in place). Thetemporal growth of SHG intensity (and consequently of the induced non-centrosymmetry) isshown in Fig. 25 for a grafted polymer PMMA-DR #1 with 35 mol% of chromophorecontent. The d33(-2ω;ω,ω) susceptibility reaches a value of 45 pm/V after 2 hours ofpreparation.52 It is as high as what is currently obtained using corona electric-fieldpoling 58, 59 . After seeding-type preparation, the decay dynamics of the induced d33 is thesame as with the same polymer prepared using corona poling. This corresponds to aninfrared electrooptic coefficient r33 close to 10 pm/V in the plane of the film.60

Spatial profile of the induced polar order.

The spatial profile of the induced second-harmonic efficiency has been studied byusing a small diameter probe beam at fundamental frequency and scanning the poled area bytranslating thin film. The result is shown in Fig. 26. Unlike what is currently observed inseeded bulk glasses,53-54 the spatial profile along the polymer-film plane is uniform. Thisconfirms that optical poling of polymers results in a local effect. Our preparation techniquethus permits patterning of micro structures using the laser spot.

In this respect, an interesting question concerns the effect of the accumulated space-charge on the nonlinear optical coefficient. Space-charge fields are indeed known todominate SHG in seeded optical glass-fibers53-57 . Considering that we get the samenonlinearity as using corona-poling, we can expect internal fields as large as 1 MV/cm in the

125

optically poled region. We thus prepared a polymer film using various angles of incidence. Itresults that SHG signal is independent on the angle of incidence, proving that space-chargefield effects are inefficient with this polymer.

Fig. 26. Spatial profile of the second harmonic generation intensity along the film plane.

Symmetry of the induced χχ(2) susceptibility.

The symmetry of the induced χ (2) susceptibility was studied on a seeded film by SHGmeasurements placing it between two polarizers, as shown in Fig. 27. The SHG intensitywas collected when rotating the analyzer. The result, displayed in Fig. 28 shows that theinduced χ(2) susceptibility has similar symmetry to that observed in corona poled polymerswith where X is the poling field direction(polarization direction of ω and 2ω beams) as for moderately poled polymers (isotropicmodel). The main difference consists in the direction of the induced net polarization. It isperpendicular to the thin film plane in corona poling and in plane (parallel to the electric field

Fig. 27. Schematic representation of experimental arrangement for the study of the symmetry ofinduced χ(2) susceptibilty.

126

polarization of collinear ω and 2ω beams).

Fig. 28: Polarisation dependence of seeded SHG coefficient. Angle θ is measured with respect to polarisationaxis around the beam-propagation direction.

Seeding power dependence.

The optically induced χ (2) is proportional to

Fig. 29. SHG-amplitude growth at different seed intensity-ratios (I 2 ω /I ω ×2000).

An optimized poling efficiency also requires a good interference between one and two-photon absorption effects in order to maximize the polar E3 -term. It requires equalization of

one and two-photon absorptions; that is . Obviously, χ(2) -initial

growth-rate is proportional to E 3 which is also I ω × I 2 ω1/2 .χ(2) s a t u r a t e s a s

127

when poling becomes as efficient as loss of polar

orientation resulting from axial excitation processes. Such behavior is observed in Figure 29in which I2ω / I ω -ratio was varied while keeping sample and phase-∆Φ constant. It is clearthat optimization of optical-preparation conditions can lead to orders of magnitude gain onpoling efficiency. Of course, optical-poling conditions must be optimized as concernsstationary parameters such as phase and relative intensities of the beams,61 but also asconcerns dynamic parameters such as excitation and orientation diffusion rates.62

One of the important parameters determining practical applicability of poledpolymers is the stability of the induced polar orientation. This is studied usually through thetemporal behavior of the χ (2) susceptibility or of the electro-optic coefficient r at elevatedtemperatures. Relaxation studies have been almost done for static field poled polymers. Thetemporal decay with t of the χ(2) susceptibility is usually described by the Kohlrausch -Williams-Watt (KWW) stretched exponential function

(52)

where τ is the relaxation time constant, depending on temperature and β (0<β<1) describesthe width of relaxation (departure from a monoexponential behavior).63

The relaxation of poled polymers (isotropic or liquid crystalline may be also well

(53)

described by a biexponential function

RELAXATION

Fig. 30. Temporal decay of SHG intensity for a polyacrylate functionalized with cyanobiphenylchromophore. Dash-dotted and solid lines have been computed using one and biexponential functions,respectively (after Ref. 35).

128

where 1 is the thin film thickness, is the second harmonic intensity of the studied thinfilm (p) and of reference r, respectively. The constant C in Eq. 53 characterizing the residualorientation at the experimental time scale which is important for practical applications, R’sare relaxation rates and τ 1 and τ 2 are relaxation times of different processes contributing tothe molecular disorientation, respectively. All these parameters depend on the measurementtemperature. Closer the glass transition temperature is to the measurement temperature,smaller are time constants τ and larger are relaxation rates. This behavior is true for bothisotropic and liquid crystalline polymers, as it was observed by Dantas de Morais et al.28 .Fig. 30 shows an exemple of such a biexponential fit of χ(2) relaxation curve.

CONCLUSIONS

As already mentioned, poled polymers have known a very important development inlast few years, not only from the point of view of chemical synthesis and characterization butalso from device applications. They exhibit the most important progress, concerning theobtained values of second order NLO susceptibilities, thermal and orientational stability aswell as light propagation properties. High rate, 100 Ghz signal modulation has beendemonstrated with close commercialization of electro-optic modulators64. Stacked electro-optic modulators have been also fabricated for parallel signal processing.65 , which cannot beanticipated with single crystals like the standard material LiNbO3 . A successful integration ofpoled polymer based electro-optic modulators with semiconductor technology has been alsodemonstrated. 66 . One can reasonably expect their forthcoming, commercial application inhigh rate electro-optic modulation..

Among the herein presented oriented poled polymer film preparation methods abetter adapted poling efficiency is expected with photo-assisted and all optical poling. Inparticular, the all optical poling technique represents several important advantages, withrespect to the classical static field poling method. Among them, we quote :

(i) the absence of electrodes and consequently no charge injection, which usuallyleads to dielectric breakdown(ii) a higher damage threshold observed with optical fields E, allowing consequentlyhigher poling efficiencies, with no “ point effect ”, observed sometimes withelectrodes(iii) automatic phase matching, assured by the seeding procedure and useful for thefrequency conversion(iv) possibility of auto-regeneration, as frequency conversion is done simultaneouslywith seeding. It allows an automatic χ(2) grating adjustment for harmonic conversionto e.g. refractive index variation due to e.g. heating(v) poling can be done at any temperature, especially at room (“ cold poling ”) or atany operation temperature(vi) possibility of poling other than dipolar molecules as it was shown with octupolarmolecules. 67 Although the poling mechanism is still not as clear as with azo dyes, it isthe only technique which allows to orient such interesting molecules.

ACKNOWLEDGMENTS

This paper describes results of numerous collaborations and discussions. The authors wouldlike to thank warmly all people who contributed to it, and in particular Fabrice Charra,Pierre-Alain Chollet, Michel Dumont, Céline Fiorini, Maryanne Large, Claudine Noël andPaul Raimond.

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NONLINEAR PULSE PROPAGATION ALONG QUANTUM WELLIN A SEMICONDUCTOR MICROCAVITY

V.M. Agranovich1, A.M. Kamchatnov1, H. Benisty2, and C. Weisbuch2

1 Institute of Spectroscopy of Russian Academy of Sciences,Troitsk, Moscow obl. 142092 Russia

2 Ecole Polytechnique, Laboratoire P.M.C.,91128 Palaiseau cedex France

1 INTRODUCTION

The possibility of propagating stable nonlinear optical pulses near resonances stem-ming from Wannier-Mott as well as organic Frenkel excitons has been the subject ofintense theoretical effort for more than two decades. Attractive stabilizing mechanismsare (i) self-induced-transparency (SIT) and (ii) Kerr nonlinearity which allows soliton-type solutions. The former require very short pulses ( p< T 2) of given “area” (2π−pulse)whereas the latter applies to longer pulses ( p > T 2 ). Such pulses were considered topropagate either in bulk materials [1]–[13] or along surfaces and waveguides [14]–[17].Renewed interest recently arose from hopes to use such pulses in the field of high-bitrate communication and ultrafast optical devices. Such hopes have been recently sub-stantiated by first experimental observations: Ref. [18] demonstrated the distortion-freepropagation of polariton pulses near (below) excitonic resonance in CuCl due to giantKerr nonlinearity. These pulses are stabilized by the anomalous nonlinear dispersionassociated with the biexciton two-photon transition and are restricted to narrow fre-quency domains. They thus differ from polariton solitons considered in the frameworkof Ref. [12], based on a dispersionless Kerr nonlinear constant. In a second example[19] propagation of pulses of self-induced transparency was achieved taking advantageof local impurity states (donor-bound exciton) in a CdS platelet. There is no doubtthat these pioneering experiments will trigger an increased interest in this field in thenext few years and applications to a variety of systems.

In this paper, we propose to lay a theoretical basis for the understanding of non-linear pulses propagating along a quantum well in semiconductor microcavities, whichseems good candidates not only for observing such phenomena, as explained below butalso for tailoring them through cavity parameters. Imbedding quantum wells (QWs)in semiconductor microcavity (MC) allows interaction of confined excitons and cavityphotons with specific dispersion law. In a planar microcavity, upon increasing mirrorsreflectivity, the continuum of photon modes condensates into Fabry-Perot and waveg-

Beam Shaping and Control with Nonlinear Optics133Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

It seems therefore timely to investigate the formation and propagation of nonlinearsoliton pulses along a planar microcavity due to nonlinear interaction of cavity modeswith QW excitonic transition. Excitons in QWs are more stable than excitons in bulkmaterials raising reasonable hopes to experimentally observe the phenomena which wedescribe here. We chose a configuration where the same quantum well supports boththe (linear) polaritons mode and the nonlinear process for simplicity. But the degrees offreedom of state-of-the-art molecular deposition techniques (MBE, etc.) allow a varietyof related configurations where for example the host cavity material or a second, differ-ent QW would sustain the nonlinearity whereas the light-coupled polaritons would stemfrom the first QW. Combining these and others functionalities in a microcavity opensthe road to engineering of photonic properties for applications. For such configurations,the formalism presented here can be easily extended as long as active layer thicknessesare much smaller than the working wavelength.

uide quasimodes yielding, for small in-plane wavevectors, an increase of light electricfield by more than two orders of magnitude. In this situation, we can expect opticalprocesses such as three or four-wave mixing and stable nonlinear pulse formation maybe realized for smaller incident light intensity. The same applies of course to light scat-tering processes like Brillouin and especially Raman scattering, whose investigation hasbegun with some success [20]. Saturation effects have also been observed for QW inMC [21].

In the next section we introduce a semiclassical formalism to obtain the field responseto QW polarization in a microcavity. We explicit both TM and TE field polarization forcompleteness, and show that only the latter lends itself to efficient coupling schemes. Inthe following section, we apply the above results to very short pulses of SIT propagatingalong a quantum well in a microcavity and point out the specificities due to photonconfinement. In the fourth section, we turn to third–order nonlinearity, at longer timescale. For this case, we first take into account the linear polarization leading to thesplit of cavity polariton branches. On this basis, we include nonlinearity and show thatit leads, in some approximation, to nonlinear Schrödinger equations and soliton-typesolutions. We also make estimates of the pulse energy required to form such solitonsfor realistic cavity parameters. The last section is devoted to a conclusion.

2 GENERAL EXPRESSIONS FOR ELECTROMAGNETIC FIELD ANDPOLARIZATION IN QUANTUM WELL

We suppose that a single QW is located in the middle (z = 0) of the cavity (–L /2≤ z ≤ L/2) with dielectric constant ε, surrounded by a cladding medium (|z | > L /2)with dielectric constant We assume for simplicity that all interfaces are plane (seefigure 1); generalization on cylindrical geometry is straightforward. The QW is treatedas a thin film (its thickness l is much smaller than L and the medium wavelength)having polarization P, which can be taken into account by means of modification ofthe boundary conditions at z = 0 for fields in cavity (see, e.g., [14, 15, 16]).

For a given k = (k, 0, 0), the three directions of P give rise to L, T or Z quantum wellexciton branches with respectively P||x, P ||y, P ||z. The first two cases respectively cou-ple to TM and TE modes. The Z case can be treated by combining the TM approachwith the appropriate boundary conditions given in [16]. Close to normal incidence,however, only L and T QW branches couple well to outside waves making the Z branchless suited to nonlinear effects at low intensities. Hence next sections will focus only onthe L and T cases. Following sections will exemplify the TE case for simplicity.

134

2.1 TM MODES

Let us consider at first the TM mode propagating along x axis: H = (0, Hy , 0),E = ( Ex , 0, Ez ). We assume that these field components do not depend on y coordi-nate. Then the solution of the wave equation leads to the following dependence of Hy

component on z after Fourier transform from space–time coordinates (x , t ) to the wavevector k and frequency ω :

(1)

where q = and similar solutions for Ex and Ey .We have taken into account that only outgoing fields components must be present atz → ±∞. Let H >, E > denote fields above some boundary and H<, E < below it. Thenat z = ± L/2 we have usual boundary conditions of continuity of tangential electric andmagnetic fields:

(2)

However at z = 0 occurs a discontinuity of H ≡ H y because of the surface currentsalong the QW:

(3)

With the use of Maxwell equations we find immediately from (1) and (2) that

(4)

where

(5)

is the value which evidently is related to reflection coefficient of surfaces z = ± L /2.Then the equations (3) can be written as

(6)

For the cavity normal modes (when Px = 0) we find i.e. anti-symmetric (r = –1) H ~ cos β z, E ~ sin β z, or symmetric (r = 1) H ~ sin β z, E ~cos βz modes.

Elimination of A3 from (6) gives

and the equation permitsus to connect Ex and Px in the following way:

(7)

135

where

and r is given by eq. (5). In space–time representation we have the equation

(9)

(8)

Using the material nonlinear relation between Px and E x , we are able to obtain theequation for E x which describes the propagation of pulses along our system.

2.2 TE MODES

Now let us consider the TE modes with E = (0, E y, 0), H = (H x , 0, Hz). Again thegeometry of the system is shown in figure 1. The solution of the wave equation has thesame form as in eq. (1), but now it is more convenient to write it for the Ey component:

(10)

where q and β have the same meaning as before. At z = ±L/2 the electric and magneticfields must satisfy usual boundary conditions (2) which again yield (4) but now with

(11)

Figure 1. Scheme of the cavity model with a quantum well inside.

136

At z = 0 the boundary conditions have the form

(12)

and, on inserting Eq. (10), we obtain

(13)

In this case the normal modes (at Py = 0) correspond to A2 = ± A3 , r = 1 i.e. wehave symmetrical E ~ cos βz ( r = –1) and antisymmetric E ~ sin β z ( r = 1) modes.In the same way as for TM modes we obtain the equation

(14)

where now

and r is given by eq. (11). In space–time representation we have

(15)

(16)

In the following, we consider only the TE configuration for which we shall proceed tosome applications of the derived equations.

3 SELF–INDUCED TRANSPARENCY PULSES

We consider here intense short pulses with duration , where is the relax-ation time of donor–bound QW excitons. We assume exact resonance of cavity modefrequency and donor–bound exciton transition, so that the polarizability of the QW,the same as in [19] can be modelled as a two–level medium [1,2]. Considering boundQW excitons in this way we neglect exciton–exciton screening effects which usuallycan be taken into account in the frame of optical Maxwell-Bloch equations. But forQW thicknesses small in comparison not only with wavelength but also with boundexciton radius the effect of screening is small and can be safely neglected. To have anon–vanishing electric field at z = 0, we must take the symmetric TM (r = 1) and T E( r = –1) modes when E ~ cos βz. In both cases the frequency and the wave vector ofthe plane wave solution

(17)

satisfy the dispersion relation for normal modes:

F( Q, Ω ) = 0. (18)

We can look for the solution of equations (9) or (16) in the form of a pulse:

(19)

where the envelope amplitude changes slowly enough,

137

where

and Q and Ω satisfy the above dispersion relation (18). Then we have

(20)

(21)

(22)

(25)

Assuming that the function F(k, w) has first-order changes in the spectral width of thepulse, we substitute its expansion

into equations (9) and (16). For example, in the case of TE mode (16), taking intoaccount (18) and (21), we obtain the equation

(23)

is the group velocity of the pulse and the amplitude of polarization Py (x, t) has beenintroduced according to

In the case under consideration of exact resonance betweentransition frequencies we have [1, 2]

(24)

the wave and QW exciton

where µ is the dipole moment of the transition, n is the concentration of donors (two–level “atoms”) per unit area of the QW, and the argument in the sine is

(26)

so that equation (23) reduces to the well–known sine–Gordon equation for the functionΨ ( x, t). For Ψ (x, t) depending only on = t – x/u, u being the pulse velocity, weobtain the equation

(27)

where

(28)

For real p — what we assume here and will check below — equation (27) has thesolution

138

corresponding to the self–induced transparency (SIT) soliton described by the electricfield

(29)

that is has a meaning of the pulse duration. A similar result can be obtained for TMmodes.

Let us calculate for TE cavity modes of a cavity confined by perfectly reflectingwalls. This corresponds to and according to equation (11), we get

(30)

For a symmetric mode with r = – 1, the wavevector component β is quantized accordingto

βL = π N, N = 1, 3,... (31)

Easy calculation give the required derivatives

and from Eq. (28) we have,

Thus, we evidence the possibility of propagating SIT pulses along the QW at the reso-nance condition wlaser = wb o u n d e x c i t o n . It is important to note that the pulse velocity is

(32)

necessarily smaller than the group velocity of the bare wave.Let us provide some quantitative estimates using typical parameters of the CdS

system where SIT pulses have already been observed in bulk CdS. To calculate thequantity

(33)

we must know Ω, l, ε, L, µ 2 and the concentration of donors per unit area n (n = n owhere n o is a bulk concentration of donors and l the QW thickness). Going to estimates,we take from [19], µ 5 ⋅10–29 C ⋅ m, a frequency Ω = 4 ⋅ 1015 s –1 , and we find that fora CdS QW of thickness l = 100 with a bulk donor concentration n o = 1016 cm –3 in aL = 1µm= 10 –4 cm cavity of dielectric constant ε = 10 and a pulse duration = 5 pswe obtain a pulse velocity

(34)

evidencing that the soliton moves much slower than the group velocity.The amplitude of the soliton electric field, given by

139

(35)

has the following order of magnitude

ε0 26 CGSE = 8.103 V / cm, (36)

a cavity cross section of S 100 µ m2 is found in theFinally, the pulse energy forpicojoule range according to

If the pulse energy falls below this critical value, then it fails to satisfy the well–known

(37)

“area theorem”. It is unstable and transforms during propagation into linear waves.It is interesting to compare the properties of SIT pulses in bulk and in cavity.

Comparing Eq. (33) to its counterpart for bulk material, we see that only the relationbetween and u changes: in the bulk case, the difference between vg and u can bemuch larger due to n o l being replaced by n oL.

Note that the cavity may be made still narrower and still more the wave (guide)cross sections, which would allow to lower the soliton critical energy and would shortenpulses. To get more flexible design, we could go to the case where the cavity polaritonsand optical nonlinearity arise from different QWs, allowing separate choice of e.g. vg

and soliton frequency w 0 More generally, the degrees of freedom provided by presentday growth techniques allow full tailoring of cavity and QW parameters, and thusobtaining pulses of given energy and duration as long as the first criterion issatisfied, that is for rather short pulses. Still, the possibility to generate in this rangecalibrated solitons together with the simple and selective excitation scheme providedby microcavity quasi-modes at reasonable angles of incidence is attractive. For longerpulses we have to exploit the possibly slower third-order Kerr nonlinearity, as explainedin the following section, focusing again on the TE case.

4 NONLINEAR SCHÖDINGER EQUATION FOR POLARITON SOLI-TONS IN WELL

Let us consider long pulses sustained by TE waves with duration is therelaxation time of QW excitons), when the QW polarizability can be expressed as a

sum of linear and nonlinear terms, here supported by the same well (see comments inthe introduction):

(38)

where the linear part

(39)

140

is responsible for the so-called vacuum Rabi splitting of cavity-polaritons. Before wecan take the nonlinear part into account, our first task is to include these linear effects,as made in the next few paragraphs. We thus modify the general formalism by includingthe linear polarizability in the bare cavity response. We will thus give the bare cavity

(40)

photon dispersion relation and deduce the modified relation associated to the modifiedresponse function Using this, we will be able below to take properly into accountthe nonlinear term.

It is straightforward to obtain from Eq. (16) the new form:

if we just define(41)

The modified dispersion relation giving rise to linear cavity polaritons is given by thezeroes of this new function:

(42)

Let us consider again, as for SIT, the symmetric TE mode in the case of cavity confinedby the ideally reflecting walls, so that the bare function

(43)

vanishes for modes with r –1. The value β = of the N = 1 mode and the definitionof β lead to the dispersion law for the bare cavity

(44)

where

We assume that the dielectric susceptibility has a resonant behavior

(45)

where A is proportional to the oscillator strength per unit area. Now we can findthe dispersion law for cavity polaritons near the resonance ω ≈ Ω (k) ≈ ω0, using theexpansion of Eq. (43). Then Eq. (41) leads to the dispersion relation

(46)

141

where the first term of the expansion vanishes for bare cavity photon ω = Ω (k). In thelimit >> ε we have from (30)

(47)

Thus, from Eq. (46) we obtain the expected quadratic equation with respect to ω :

which yields two (upper and lower) cavity polariton modes,

separated by the vacuum Rabi splitting

(48)

(49)

In dimensionless units the dispersion relation (48) takes the form

(50)

where δ = (ω0 – Ω ) /Ωc is a relative detuning andc

is a dimensionless Rabi splitting. This “modified” dispersion law (48) will be consideredfrom now on as known and we shall write it in a general form

(51)

Now we are ready to include nonlinear effects.As is well-known [22]-[24], for long enough pulses the nonlinear part of susceptibility

of the QW is determined mainly by the phase–space filling effect when the excitonsand the electron–hole pairs created by the pulse prevent further excitation due to theexclusion principle. The lifetime of the excitons at room temperature is determined bytheir ionization due to collisions with thermal LO phonons and is about 0.1–0.4 ps. Theresulting electron–hole pairs live for very long time ~ 20ns and produce approximatelythe same blocking effect as the excitons. Therefore, for rough estimation, we supposethat the electromagnetic pulse creates only one long–living species (electron-hole pairsin bound or free state) which number N is determined by the rate equation

(52)

where is the decay time of pairs. In the experiment [22] it depends on recombinationand diffusion time and, as was mentioned above, has the order of magnitude 2 0ns.For very long pulses with duration greater than we have the stationary state withsaturated plasma density

142

According to [22]–[24], the nonlinear part of the QW polarizability can be written as

(55)

that is the steady–state nonlinear susceptibility is given by

(53)

and for AsGa QW has the experimental (balk) value about

(54)

of time and is proportional toFor short pulses with duration the nonlinear susceptibility becomes the function

For rough estimate, we approximate this integral by 2 Im being the pulseduration, and obtain for the nonlinear susceptibility the value

that is we replace the variable nonlinear susceptibility by its mean value and take intoaccount the quantum well width l. For p = 1 ps and l = 10 –6 cm we find numericalvalue

(56)

So we can proceed as if the QW were characterized by the “Kerr type” nonlinearity:

(57)

Then, as is well known, such a nonlinearity can be compensated by the dispersion ofgroup velocity, i.e., we must take into account the quadratic terms in the expansion of

(k, w) :

(58)

where all derivatives are taken at k = Q, ω = Ω. As above, we shall look for the.solution of equation (40) in the form of a pulse:

(59)

where the envelope amplitude changes slowly enough as stated by Eq. (20), and Q andΩ satisfy the above dispersion relation (47). Then we have eqs. (21) and analogousformulas for higher derivatives. On inserting of (57) and (58) into (40), we obtain withthe use of (21) the following equation for the envelope function ε (x, t) :

(60)

143

Introducing the group velocity and its dispersion

and neglecting the terms which correspond to the cubical corrections to the expansion(58), we arrive at the well known nonlinear Schrödinger (NLS) equation [25]

in a moving coordinate system defined by ξ = x, η = t – x /υg , where

(61)

(62)

Now we have to calculate It follows from eq.(20),(26) that (see e.g. eq.(24) and (22)) and to first order in ∆/ω0 , we have = –2i k L c2 /ω2. It means that

(63)

The sign of k" plays a crucial role in the solution of the NLS equation. We have plottedin figure 2 the inverse dispersion relation k(ω) for a typical splitting of two percent.It shows that both signs of the second derivative can be obtained and a large range ofmagnitude of k" is possible. The near-resonance trends would however be smeared outwithin the system’s absorption linewidth. Let us discuss solutions for both signs of k".If k" < 0, (as it takes place for ω above the anticrossing region), then this equation, inthe case of positive χ (3) , takes the form of the “focusing” NLS equation

(64)

and has the soliton solution [25]. As in Ref. [12], we shall write here the solutionpropagating in the laboratory frame with the group velocity of linear approximationwhich is enough for qualitative estimations. Thus, the soliton solution of Eq. (64) hasthe form

(65)

where the amplitude ε0 depends on the pulse duration p according to

(66)

If k" > 0, then the equation (61) can be transformed to the form

144

Figure 2. Dispersion relation in the form k(ω): reduced wave number = k L/π is a function

of the reduced frequency

and again has the soliton solution for < 0, where now

If k" < 0, < 0 or k" > 0, > 0, we come to the “defocusing” NLS equation whichdoes not have soliton solutions (but permits the so called dark soliton solutions).

Supposing that χ > 0, we shall estimate the field strength at k = 5 ⋅ 1 04 cm–1 forupper branch of the dispersion curve, where k" –0.9 ⋅ 10–23 CGS. Then we obtain

The energy of the pulse is given by the formula

(67)

where S is the cross-section of the cavity.

145

5 CONCLUSIONWe proposed a formalism enables us to describe accurately the interaction between

the electromagnetic field and linear as well as nonlinear polarizations from a quantumwell in a microcavity, considering on the same footing either true guided modes orFabry-Perot quasi-modes. We showed that stable pulses propagating along quantumwells in such a microcavity can be sustained by cavity modes for two kinds of nonlinearmechanisms at different time scales.

The first mechanism is ultrafast two-level unharmonicity leading to the self-inducedtransparency for 2π-pulses. In this case, we could derive soliton velocity as well as otherpulse characteristics and compare them to bulk ones. We found pulse energies in thepicojoule range. We gave some hint of the advantages of quantum wells and possibleways to flexibly design a system in view of given pulse requirements.

The second mechanism is the slower third-order nonlinearity. In this case the linearpolarizability of the quantum well giving rise to the so-called vacuum Rabi splittinghas to be included and new split modes have to be obtained before nonlinearity istaken into account. The obtainment of the soliton solution is found to depend criticallyon the sign and magnitude of the curvature of the new dispersion relation for whichsuggestive examples are given. In view of experiments, laser pulses for self-inducedtransparency seem available. For Kerr nonlinearity, the detailed effect of inhomogeneityon splitting and linewidth has to be studied in more detail to evaluate the potential ofthis mechanisms in novel systems for soliton generation.

ACKNOWLEDGEMENTSV.M.A. is thankful to Laboratoire PMC URA 1254 du CNRS for hospitality and

support as well as to DRET, Grant 93 811-62/A000. He is also thankful for partialsupports through INTAS Grant 93-461, Grant 96-0334049 of Russian Foundation ofBasic Research and Grant 1-044 from Russian Ministry of Science and Technology

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[19] M. Jütte, H. Stolz, and W. van der Osten, Phys. Stat. Sol., (b) 188:327 (1995); J.Opt. Soc. Am., B13:1205 (1996); J. Luminescence, 67:45 (1996)

[18] K. Ema and M. Kuwata-Gonokami, Phys. Rev. Lett., 75:224 (1995).

[17] A. Boardman, G.C. Cooper, A.A. Maradudin, and T.P. Shen, Phys. Rev.,B34:8273(1986).

[16] V.M. Agranovich, V.I. Rupasov, and V.Y. Chernyak, Sov. Phys. Solid State,24:1693 (1983).

[15] V.M. Agranovich, V.Y. Chernyak, and V.I. Rupasov, Opt. Commun., 37:363(1981).

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147

SOME ASPECTS OF THE THEORY OF

LIGHT–INDUCED KINETIC EFFECTS

IN GASES

University of Gda skInstitute of Theoretical Physics and Astrophysicsul. Wita Stwosza 57, 80-952 Gda sk, POLANDE-mail: fizsk@univ.gda.pl

Stanislaw Kryszewski

Firstly, we shall briefly discuss the LIKE introducing basic notions necessary to describethese effects. They occur due to the light-induced modifications of the velocity distributionsof active atoms (or molecules) interacting collisionally with perturbers and with incominglight. For example, light-induced drift (LID) is possible when the active particles immersedin the much denser buffer gas are excited in a velocity-selective manner. This induces oppo-sitely directed fluxes of ground and excited state atoms. The velocity selectivity, however, is

One of the most frequently used experimental setups of quantum optics consists ofa container with gaseous mixture irradiated by external laser radiation. The number ofphenomena occurring in such a system is enormous. It is impossible to cover all of them evenin a large monograph. Vast amount of work is devoted to such studies, it is therefore, quiteimpossible even to list all relevant literature, except a few books reviewing the subject [1]–[5] .

It is necessary to restrict attention to some more specific cases. First of all, we specify thegas in the container to be a mixture of two species. One kind of atoms (molecules) is calledactive (A), these atoms are coupled to the incident radiation field. The second species (P) isinert, not coupled to the light field. P atoms serve as perturbers, or a thermal bath for thesystem since they interact collisionally with active ones and constitute a reservoir of energyand momentum. Let us observe, quite generally, that two basic kinds of physical phenomena,namely the collisional and radiative ones, occur in the discussed case. Therefore, we mayexpect that the physical effects and properties of active atoms will depend mainly on theinterplay between collisional and radiative processes. The aim of this lecture is to presentone of the possible approaches to the description of such effects. Light-induced kinetic effects(LIKE) in gases are one of the examples of the phenomena in which such an interplay playsan essential role. The applications of the proposed theoretical methods to LIKE will thusserve as an illustration of the usefulness of our model.

L I G H T A S A T H E R M O D Y N A M I C F O R C E

Beam Shaping and Control with Nonlinear Optics149Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

usually not sufficient to observe macroscopic fluxes. When the atoms in either of the statessuffer different diffusive friction, the two fluxes do not cancel and the macroscopic drift isobserved. LID is just one of the manifold of LIKE in atomic or molecular gases. Such effectsare reviewed by Hermans [ 6 ] , de Lignie and Eliel [7,8] . The examples given further in thispaper do not exhaust the variety of situations in which kinetic effects, connected with thetranslational degrees of freedom, are coupled with or influenced by external irradiation. Muchmore complete reviews can be found in the above given references (see also the materials inRef. [9]).

Secondly, we will introduce some theoretical methods necessary to describe the inter-connections between collisional and radiative processes. Generally, collisions influence theinternal state of active atoms and their translational degrees of freedom, or, in other words,velocities. The first type of collisions is a typical subject of investigations within the classiclineshape theory. We shall call these collisions dephasing ones. They can be dealt with withina simple impact approximation in which the homogeneous linewidth may be accounted forby a phenomenologically adopted linewidth and lineshift. The other type of collisions arefrequently called velocity-changing collisions (VCC). They are responsible, for instance, forthermalization of the atomic velocity distributions, that is due to these collisions the systemattains and preserves thermal equilibrium. When the system is light irradiated, then ac-tive atoms undergo simultaneously radiative and collisional processes and the incident lightmodifies the velocity distributions of the active atoms both in the ground and in the excitedstate. The considered system finds itself in the steady state which often differs from thermalequilibrium.

The aim of this work is to give a theoretical method for modelling the VCC occurring inthe described irradiated mixture of gases and to show how it can be applied to describe someof the physical effects due to VCC. We will present a theoretical model which incorporatesVCC explicitly. The necessary assumptions essential for the applicability of the model will beintroduced and discussed. Let us note that a fully quantum-mechanical treatment is possible.However, the classical kinetic theory of gases seems to provide tools sufficient for our aims.Thus, discussing collisional effects due to VCC we will confine ourselves to the modifiedclassical methods. On the other hand, consistent description of radiative processes obviouslyrequires a quantum approach. This is easily obtained within the model of optical Blochequations. Combining the two kinds of physical processes we arrive at the Bloch-Boltzmannequations which are the fundamental theoretical framework for our main aim – study of theinterconnections between radiative and collisional processes affecting the active atoms. Itshould be, however, stressed that the presented approach is just one of the alternatives. Wewill try to show that this alternative is an attractive one, attractive in the sense that it seemsto allow a satisfactory description of a wide scope of physical phenomena occurring in thediscussed physical system.

Finally, we will present some specific applications of our theoretical method based uponthe Bloch-Boltzmann equations. Since the presentation of the method itself is central to thiswork, we will briefly give only some results, mainly concerning LIKE. We shall, however, alsoindicate other possible fields in which our method seems to be useful. We will discuss somepossible generalizations as well as limitations. The prospective areas of future investigationswill be indicated and discussed.

We will try to avoid many of the technical or mathematical details which can be found inthe original papers. In the Appendix we will briefly summarize some main results concerningthe formal solutions to the Boch-Boltzmann equations. They seem to be relevant for thepresentation of the applications for which such solutions are essential.

Let us also observe that the process of photon absorption results in the recoil effect,atomic momentum changes by amount with being the wave vector of incident radiation.

Similarly, the emission process changes the atomic momentum by – , where in the

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stimulated process, but not in the spontaneous one. Recoil effects play an important role inmany physical phenomena such as radiation pressure or radiative cooling [10] . The changes ofatomic momentum due to interatomic collisions are at least by an order of magnitude greater.Therefore, in the case when VCC play an important role the effects connected with photonrecoil can usually be neglected. In our discussion of the collisional processes in the gaseousmixture we will leave the photon recoil out of the picture. Such an approximation seems tobe well justified and frequently used in literature.

Light–induced kinetic effects in gases

simple two-level ones, with |1 ⟩ and |2 ⟩ denoting the lower and upper states, respectively.When such atoms are irradiated slightly off-resonance, the detuning ∆ = ω L – ω21 ≠ 0(ω is the atomic frequency). Then, onlyL is the frequency of the laser radiation, while ω21

only the atoms possessing velocity determined by the relation kvr = ω L – ω21 (where k is

creation of the Bennett hole in the velocity distribution of the ground-state atoms (see e.g.,Ref.[3]). The number of the ground-state atoms with velocities close to vr is depleted, whilethere is a certain number of excited atoms with the same velocities. We may say that theradiation selects atoms, the velocity of which coincides with resonance one vr . The scheme

Light-induced kinetic effects (LIKE) in gases started to receive considerable attentionin the early eighties, with the first theoretical predictions by Gel’mukhanov and Shalagin [11]

and then experimental demonstration of light-induced drift (LID) of active atoms by Antsiginet al [12]. LID is reviewed in great detail by Eliel [7,8] . Many various aspects of LIKE ingases were then investigated both theoretically [13], and experimentally [9].

To understand the essential aspects of LIKE let us consider active atoms modelled by

those atoms which are Doppler-shifted into resonance may be excited. This implies that

the magnitude of the wave vector of the radiation) can appreciably absorb. This results in

for such a velocity selection is given in Fig.1 which presents the corresponding distributions.

Figure 1. The scheme of one-dimensional velocity distributions for two-level atoms irradiated alongthe z –axis. The light field is tuned in the blue wing of the Doppler-broadened absorption profile, i.e.,∆ > 0, or equivalently ωL > ω

i = 1) and for excited-state ones (i = 2). W (v21 . ρ ii (vz ) are the velocity dependent populations for ground-state

atoms ( z ) is the Maxwellian distribution. vr i s theDoppler-selected velocity, as in the text above.

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From Fig. 1 it is evident that radiation introduces asymmetries into the velocity distribu-tions and may, therefore, result in nonvanishing macroscopic fluxes and flows. The situationis especially interesting when the sum of two velocity distributions ρ11 (v) + ρ 22 (v) = f ( v )does not equal the Maxwellian W (v). Such a case occurs when the collisional cross section forcollisions is state dependent, that is when it is different for ground- and excited-state atoms.Let us also note, that for exact resonance (ωL = ω21) the curves similar to those in Fig.1 aresymmetric with respect to the vertical axis, and the selected velocity vr = 0.

Off-resonance velocity selectivity results in asymmetric distributions and, in effect, wehave nonvanishing fluxes since partial average velocities do not vanish. This is, however,usually insufficient to observe macroscopic flows. The two opposite fluxes cancel each other.The situation changes when diffusive friction is state dependent. It is reasonable to expectthat excited atoms have larger collisional cross section. Hence, they suffer larger diffusivefriction, or in other words, have smaller mobility and smaller diffusion coefficient. This resultsin the fluxes which do not cancel. Net macroscopic flow is then observed. It may be saidthat the LIKE in gases are due to radiation-induced nonequilibrium velocity distributionscombined with state-dependent collisions. In the forthcoming considerations we will brieflydescribe some of kinetic effects occurring in such a situation. Further on we will present atheoretical model which allows the consistent description of these phenomena.

It is well known from the kinetic theory of gases [14] that the collisions produce en-tropy. Only in thermal equilibrium the entropy is constant and the velocity distributions ofthe considered particles are Maxwellian. Light irradiating a sample, which was initially inequilibrium, changes the situation. Steady irradiation forces the considered system into anew state which is stationary but nonequilibrium. Since the ordered macroscopic fluxes arepredicted, we may expect some decrease of entropy. The problem of entropy production wasconsidered in great detail by van Enk and Nienhuis [15] . They have shown that the entropyof the matter (i.e., of active atoms and/or perturbers) may decrease. This confirms the no-tion of radiation introducing some order in the system. On the other hand, stimulated andspontaneous radiative processes are also sources of entropy. Entropy of photons increases byseveral orders of magnitude more, so that the total entropy increases, as it should. Radiationis the source of these effects. Hence, following the usual convention[14, 15] we may call lightto be a thermodynamic force.

To show that light indeed acts as a thermodynamic force let us discuss one [16] ofthe approaches to LIKE. These effects are determined both by the radiative and collisionalphenomena, the interplay of which results in macroscopic flows. Obviously, the former effectsare microscopic and, as such, occur on the time scale much faster than the flows. This allowsone to separate the usual Liouville–von Neumann equation for the density operatorof active atoms into two parts. The resulting equation is formally written as

with (1)

Operator L1 gives the free (macroscopic) flow term and operates on the slower time scale. Itgives rise to slow diffusive flows between macroscopically separated positions. Operator L0

governs the evolution of active atoms due to radiative processes and collisions. The specificform of operator L0 is unimportant for the present purposes, it will be discussed in muchdetail in further sections. Let us, however, note that L0 depends on the intensity of incidentlight and, therefore, may depend implicitly on the position within the sample due to possiblespatial variations of intensity, but we assume that it does not contain derivatives with respectto Moreover, L0 contains radiative effects, so it must depend on the Doppler shiftHence, we expect this operator to be anisotropic in the velocity space.

Since L 0 describes the processes which occur on the rapid time scale (of the order ofseveral atomic lifetimes) we expect that L0 drives the density operator at any position to its

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local steady state before the diffusive flow, expressed by L1, has had any chance to produceappreciable changes of the local atomic density n( , t ) which is defined as

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(2)

It is therefore, reasonable to seek a solution to Eq.(1) by elimination of rapid variables. Thismay be done by assuming the density operator in the form where

may be viewed as the local steady state solution to the equation

(3)

Local stationary density operator is thus specified by fast processes, radiative and collisional.depends on position only parametrically (implicitly) via the position-dependent light

intensity. Moreover, for any position we impose the normalization condition

(4)

which is consistent with (2). Employing the well-known technique of elimination of rapidvariables we have obtained [16] the general diffusion equation for light-induced effects in thefollowing form

(5)

with the atomic flux given by three terms

(6)

Technical details of the derivation of the equation (5) are given in Ref.[16]. Thereforewe proceed to physical discussion of the terms appearing in the atomic flux (6). The firstterm contains velocity (0) which is related to local density operator by

(7)

The next quantity in the atomic flux (6) is the light-modified diffusion tensor It isgiven by the fairly complicated formal expression

It is just the average velocity of active atoms in the state which is stationary with respectto L0 . Velocity (0) may be called drift velocity and be associated with light-induced drift.In the forthcoming sections we will show that this is indeed the case and we will discussits properties in more detail. Now, we only note that when the density operator is asymmetric function of velocity (as is the case on resonance) then the drift velocity is expectedto vanish. In absence of light certainly corresponds to the Maxwellian because all atomsare in the ground state and the system is in thermal equilibrium, so there is no drift.

(8)

Operator L 0 possesses a stationary solution, hence it has an eigenvalue zero. It can beshown that the second term in the brackets ensures that the integrand in the time integralapproaches zero on the rapid time scaleand the integral converges. Light-modified diffusiontensor depends on local light intensity (via and it is thus an implicit function of Sinceoperator L0 is anisotropic, due to its dependence on the light wave vector , the diffusiontensor must also be anisotropic and it must have the cylindrical symmetry. It is, therefore,

determined by two constants, a transverse diffusion constant Dt, and the parallel one Dp , sothat it has matrix elements

(9)

describing active atoms in either of the atomic states.

in the perturber gas, while they do not depend on the intensity of the light. We expect1

The last term appearing in (6) contains velocity (1) which we call gradient velocity.It is a correction to the effective drift velocity, and it arises from the variation of the localdensity operator with position. The formal expression for gradient velocity is

This clearly illustrates that light indeed modifies the kinetic properties of the consideredsystem [17] . It is important to distinguish between the light-modified diffusion tensor (8)and diffusion coefficients D1 and D 2The constants D1 and D 2 depend on collisional cross sections (or mobility) of active atoms

that light-modified diffusion tensor is a complicated combination of constants D and D 2.Obviously, for zero light intensity the diffusion tensor must reduce to the isotropic one andhas the strength of the diffusion constant for the ground state atoms in the buffer gas.

(10)

The integrand in the time integral again vanishes on the rapid time scale. Since and(0) depend on the position exclusively through the dependent intensity, we may express(1) in the form , with the cylindrically symmetric second-rank tensor

(11)

The general diffusion equation (5) with the flux (6) thus describes some of LIKE. Eachof the components of the atomic flux is driven by the light intensity or its gradient. Weconclude that light field is indeed a thermodynamic force. The presented results, althoughquite formal, give the essential insights into the LIKE in gases. Obviously, any concretecalculations require one to specify the local evolution operator L0. We shall do that in the

LIKE. Before doing so, we will briefly discuss the physical effects connected with the formalresults. The full discussion of the methods to compute the above given quantities is givenin Ref.[16] which also gives the relevant references. We also note that the obtained light-modified diffusion tensor and gradient velocity are closely related to Green–Kubo relationsfor Navier–Stokes transport coefficients [14] . It seems necessary to stress that the presentedapproach is one of the several possibilities. Another theoretical framework to describe gaskinetics in a light field is due to van Enk and Nienhuis [18]–[21] .

Gradient velocity defined formally in Eq.(10) plays an essential role in the effect of light-induced diffusive pulling.

next section where we will present the theoretical methods allowing consistent modelling of

Light–induced drift

Let us assume that a sample of active and perturber atoms is irradiated by monochro-matic radiation detuned into the blue side of resonance, that is ∆ = ωL – ω21 > 0. Let usalso assume that radiation wave vector determines positive direction of the z axis of thelaboratory coordinate frame. Resonance velocity vr is then positive, as it follows from theDoppler shifted resonance condition = ∆ , (this is the case depicted in Fig.1). Then, asevident from Fig.1, the average partial velocity of ground-state atoms is directed in the neg-ative direction of z axis, i.e., towards the light source. At the same time the partial averagevelocity of the excited-state atoms has positive direction. We have two oppositely directed

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fluxes of atoms in either of the states. When the incident radiation is tuned to the red wing ofthe resonance (ωL <ω 21) then directions of the fluxes are reversed, while irradiation exactlyon resonance gives rise to no fluxes, since then the corresponding velocity distributions aresymmetric and the average partial velocities are zero.

If the collisions do not discriminate between the atomic states, then the fluxes of ground-and excited-state atoms will be equal, as it follows from the particle number conservation.Then, there will be no macroscopic flows. On the other hand, when collisional cross sectionsare state dependent, we may expect that due to differences in the diffusive friction the netflow will occur. Excited atoms have larger collisional cross section, so they diffuse moreslowly. This situation is schematically presented in Fig.2.

Figure 2. Light-induced drift. Active atoms are irradiated at the blue side of resonance. Excitedatoms are pushed by light, while the ground-state ones are pulled towards the light source. Since theexcited atoms suffer greater diffusive friction the drift with velocity (0) arises. The net effect is thatactive atoms are pulled by light.

The essence of the LID, may be summarized by a simple proportionality

(12)

Each term in this relation has a clear physical meaning. The resonance velocity is positivefor irradiation in the blue side of the resonance, negative for the red wing case and zero forexact resonance. The next fraction in (12) is the ratio of the density of excited-state atoms tothe total density of the active atoms. Below saturation level this fraction is proportional tothe intensity of the incoming light, so it vanishes in absence of light. In the saturation regimethe situation is more complicated and must be considered in more detail. The last term inEq.(12) may be viewed as the relative change of the collision cross section upon excitation.Usually, we expect σ2 > σ1 , as discussed above. When the interatomic interaction does notdistinguish between atomic states (i.e., when σ1 = σ2) then LID vanishes as expected. Thegeneral expression (7) for drift velocity must incorporate the features discussed here in anintuitive manner. We will show that this is indeed the case.

It seems worth noting that an interesting phenomenon occurs when the sample of activeatoms and perturbers is optically thick. Then, irradiation of the sample in the red wing ofthe Doppler absorption profile results in pushing the atoms from the entrance window. Theactive atoms in the part of the container close to the entrance window are pushed by lightrelatively strongly. When the penetration depths increases, the light intensity decreases dueto absorption and the drift becomes weaker. Hence, a cloud of active atoms gathers inthe vicinity of the entrance. The atomic density is locally higher than in the other partsof the capillary. As a result of the drift effect, a cloud of active atoms travels from the

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entrance window towards the other end. This is the effect of optical piston demonstrated inan experiment by Werij et al [22] and described theoretically by Nienhuis [23] . The behaviorof the atomic density n( , t) may be given a soliton-like interpretation.

The LID effect is still studied and used in many practical applications, see for exampleRefs.[24, 25, 26]. The implications or applications of LID in other fields as isotope separation,investigations of molecular properties, solid-state physics or astrophysics are very interesting.However, it is not our purpose to review these questions, so we refer to the paper by Eliel [8]

for a review of experiments and theory concerning LID effect.

Light–induced diffusive pulling

Light irradiating the sample of active and perturber atoms usually does not fill the wholecross section of the cell, or its intensity is not constant across the beam. Then, in the regionof higher intensity there are more atoms in the excited state than in the nearby regions oflower intensity. The excited atoms have larger cross section for collisions, hence, they diffuseat a rate slower than the ground state ones. Therefore, the diffusive flows of active atomsout of the region of higher intensity is, on average, smaller than the flux of ground-stateatoms into the considered region. In other words, the higher mobility of ground state atomsinduces a diffusive flux of ground state atoms towards high-intensity regions that more thancounterbalances the opposite diffusive flux of excited atoms. As a result, the concentrationof active atoms inside the high intensity region is higher than outside. This is the effect oflight-induced diffusive pulling (LIDP) [27, 28, 29].

F i g u r e 3 . Light-induced diffusive pulling. The light beam (indicated by gray) does not fill thecontainer’s cross section and its intensity varies across the beam (white curve). Active atoms inground state (small circles) and in excited state (larger, filled circle) diffuse out of the beam, thelatter ones at a slower rate. Ground-state atoms from outside diffuse into the beam. The fluxes donot balance, more atoms enter the beam than flow out. Concentration inside the beam is higher thanoutside.

It is evident that the LIDP effect depends on two factors: firstly, the diffusion coef-ficients of the ground-state and excited atoms must be different; secondly, there must besome gradients of light intensity. The intuitive considerations can be easily put into moreformal, though approximate interpretation [16] . The light-modified diffusion tensor (8) maybe viewed as a weighted average of the diffusion constants of atoms in the ground and excitedstates. To illustrate that, let us assume that velocity autocorrelation functions for atoms in

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either of the states depend negligibly on the shape of the stationary velocity distributions.We, however, assume these correlation functions to be state dependent. This approximate as-sumption allows the conclusion that diffusion tensor becomes isotropic, because anisotropiesof the velocity distributions are no longer important (are neglected). Hence, the diffusiontensor can be expressed as a weighted average of diffusion constants D1 and D 2 correspondingto atoms in two states. Then we have

(13)

where

(14)

are the fractions of atoms in each state. The same approximation applied to gradient velocity(10) yields

(15)

The gradients of the populations result from intensity variation with respect to position.Since an obvious condition p1 + p 2 = 1 must hold, we get As itwas argued, we expect that D1 > D 2. So we conclude that gradient velocity gives rise to adiffusive-like flow of atoms towards the region of larger p2 , i.e., towards region of higher lightintensity where the number of excited-state atoms is greater. This also explains the termgradient velocity.

Within the discussed approximation the atomic flux (6) becomes

(16)

This relation includes the basic features of LID and of diffusive pulling. To see that moreclearly, let us assume that atoms are irradiated at resonance, so that the drift velocity vanishesand only the diffusive part remains in Eq.(16). In the closed cell, the steady state flux mustvanish and then Eq.(16) implies = const., or, equivalently

(17)

Since D1 > D 2 we see that in the higher intensity region (where p2 is appreciable) theatomic density n must be larger than in the darker regions. This is the essence of thelight-induced diffusive pulling effect. Consequently, the term is essential in physicaldescription of LIDP. Previous derivations [13, 23] of the diffusion equation for light-inducedeffects did not include the LIDP. This was due to another approximation method concentratedrather on the difference between collisional cross sections for two atomic states. This did notpermit to account for the light-modifications of the diffusion tensor. Hence, the diffusive flowconnected with gradient velocity was also absent. The approach given here includes the effectof collisions to all orders and generalizes earlier studies.

This qualitative discusion of the light-induced diffusive pulling gives some basic explana-tion of the phenomena which are due to the joint effects of radiation and collisions. The otherlight-induced effects are also possible, but their presentation goes beyond the scope of thiswork, we only refer to already mentioned literature. To proceed further we need an operatorL0 necessary to find the coefficients of the diffusion equation, drift velocity (7), light-modifieddiffusion tensor (8) and the gradient velocity (10) which are expressed by integrals over timedependent correlation functions, time evolution of which is governed by the fast operator L0 .

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THEORETICAL FRAMEWORK

Discussing LIKE in gases we have associated an operator L0 with radiative and colli-sional effects which occur on the rapid time scale, and the free-flow operator L1 with the slowmacroscopic fluxes. We have introduced the local stationary density operator whichparametrically depends on position within the sample. We eliminated the fast variables andobtained the diffusion equation (5). This section is, therefore, devoted to establishing theexplicit form and the properties of operator L0 . To construct operator L0 we must accountfor radiative and collisional processes. Firstly we briefly recall the theoretical description ofradiative ones.

Optica l B loch equat ions

Realistic atoms used in the experiments on LIKE usually possess complicated internalstructure. For theoretical purposes, giving all essential insights it is, however, frequentlysufficient to consider a simple two-level model with ground state denoted by | 1 ⟩ and theexcited state by | 2 ⟩ with being the energy difference. The density operator for thisatom is a 2 × 2 matrix and in general, its matrix elements are functions of: position withinthe cell, velocity and time t. Hence, we write for a, b = 1,2. Timeevolution of the density operator is governed by the standard optical Bloch equations. Sincethese equations are very well documented in literature [5, 30, 31, 32] , it seems that they requireno comments or justification. Bloch equations allow many generalizations and applicable toa variety of physical problems. The Bloch equations include the atom-light-field coupling andthe effects of spontaneous emission. The standard optical Bloch equations in the rotating-wave approximation read

(18a)

(18b)

(18c)

with The notation and the physical meaning of the terms appearing in Eqs.(18)is as follows. denotes Rabi frequency, defined as where isthe local intensity of the electric field of the incoming light, is the electric dipole momentof the two-level atom, which without loss of generality, may be assumed to be real. Rabifrequency may depend on position within the sample, because of the possible variations oflight intensity, for example due to absorption. A is the Einstein coefficient for spontaneousemission from upper to the lower level. The velocity-dependent detuning includes the Dopplershift and is given as with ω L and being the frequency and thewave vector of the incident monochromatic radiation.

When the duration of the collision is short as compared to any other characteristic timescale then it can be shown [5, 31] , that in many practical cases, the effect of collisions on theinternal atomic state, is accounted for by the following contribution

(19)

with γ ph being the homogeneous collisional linewidth and δ is the corresponding collision-induced lineshift [5] . Both of these quantities may be considered as position and velocity

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independent. The meaning of Eq.(19) is that the effect of collisions on atomic coherencesconsists in the dephasing of the oscillations of the atomic dipole. This is typical to theclassic theory of pressure broadening of spectral lines and it is the essence of the impactapproximation, which we assume to hold. Such a situation arises, for example, when the loweratomic state is much less polarizable than the upper one [33]. The presence of dephasing leavesthe populations unchanged [31]. We assume that collisional lineshift δ is already included inthe atomic frequency ω21, while the homogeneous linewidth ΓC = A /2 + γ ph .

The solutions to optical Bloch equations depend on velocity only implicitly, via theDoppler shift. Since we are interested in the study of the interplay between radiative andcollisional effects we look for velocity distributions. Therefore, we must augment equations(18) by suitable terms describing VCC occurring in the system. In order to do so, we willadopt a series of assumptions which facilitate the theoretical descriptions, but which are notvery restrictive, and are applicable to a large class of experimental situations.

Description of velocity-changing collisions

In the active-atom (A) and perturber (P) mixture three kinds of collisions occur, namelythe A–A, A–P and P–P ones, Henceforward, we will assume that the density NA of the activeatoms is much lower than the density of the perturbers NP , i.e.,: N A P . P–P collisions << Nare much more frequent than A–P ones. A–A collisions are still much more rare than theformer ones, and as being extremely rare, can be neglected and dropped out of consideration.P–P collisions occur frequently enough so that the thermalization of the perturber’s velocitydistribution is much more rapid than that for active atoms. Thus, we approximate the P-atomvelocity distribution by the equilibrium one, that is, by the spatially uniform Maxwellian

(20)

with = 2k BT / mP being the square of the most probable velocity of perturbers at tem-perature T. We conclude that only A–P collisions are considered relevant and need to bediscussed. This may seem quite restrictive but it is not really the case, since it correspondsto a fairly common experimental situation (see, for example Refs.[1]-[5] and [8, 9, 11, 12]).Furthermore, we assume that the average thermal energy of the active atoms 3k B T /2 is muchsmaller than the energy separation . Moreover, the energy of the first excited level ofthe perturber is taken to be much larger than Therefore, the energy transfer fromthe excited atom A to the perturber is impossible. There is no non-radiative deexcitation(quenching) of atom A during the collision. Finally, we take the density of perturbers asnot too high so that the A–P collisions can be treated within the binary collision regime.Stated conditions are well satisfied in great majority of typical experiments, so they may beconsidered reasonable and justified.

When these assumptions are met, the collisions are accounted for by means of thequantum-mechanical Boltzmann equation. Such an approach was proposed by Snider [34] .Berman [33, 35], elaborated further on the subject in the context similar to one discussedhere. Rautian and Shalagin [3] also give an extensive and modern discussion of the quantum-mechanical Boltzmann equation. The rate of change of the matrix elements of the active-atomdensity operator due to collisions is formally written as

(21)

Fully quantum-mechanical expressions for collision kernels and rates are quite complicatedsince they involve scattering amplitudes for active atoms in both states. On the other hand,as it is generally accepted, classical kinetic theory is quite appropriate for determination of

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the velocity distributions of not too dense gases. The investigated gas can be considered asa mixture of three species: perturbers, active atoms in the ground and in the excited state.Due to our assumptions, when applying the classical kinetic theory [36, 37, 38] we take thevelocity distribution of the perturbers to be a Maxwellian (20), and we arrive at the linearBoltzmann equations for the populations ρaa which are formally the same as Eq.(21). Thecollision kernel for populations is then expressed in a simple, intuitive form, namely

(22)

The delta functions ensure momentum and conservation. The corresponding collisionalenergyrate γaa is then given as

(23)

where (or are the velocities of an active atom, and (or of the perturber after(or before) collision. (or are the relative velocities. m A , m P

and µ denote the masses of the active atom, perturber and the reduced mass, respectively.Expressions (22) and (23) instead of quantum-mechanical scattering amplitudes now containdifferential cross section for the active-atom-perturber scattering in the center-of-mass frame,with χs being the scattering angle. It should be stressed that the cross sections for variousatomic levels are in general different, due to differences in the interaction potentials.

It seems necessary to clarify terminological distinction between linearized and linearBoltzmann equations. When we assume that the velocity distribution only slightly differsfrom the Maxwellian, we can take it as Then, in the Boltzmannequation we can retain only the terms linear in the correction φ This leads to thelinearized Boltzmann equation. On the other hand, when we consider a mixture of gases withone component much denser, then we can assume that denser component thermalizes rapidlyand its velocity distribution is Maxwellian. Due to this assumption the Boltzmann equationbecomes linear in the velocity distribution of the dilute component and attains the form ofEq.(21). Thus, our model corresponds to the linear Boltzmann equation.

The approach via classical kinetic theory to the influence of VCC on coherences isquestionable, if at all possible. First of all, we note that the kernel and rate defined inEqs.(22-23) are real. This is consistent with the probabilistic interpretation of populations.The coherences are complex, and so are the corresponding quantum-mechanical kernels andrates. Thus, they do not have classical analogues. Nevertheless, classical approach givesuseful insights in describing the effect of VCC, especially when the influence on coherencesmay be reduced to dephasing only. We shall return to the problems raised by the coherencecollision kernels and rates later, after discussing quantities corresponding to populations.

First of all, we note that the number of particles must be conserved during the collisions.This requirement implies that the integral over d of both sides of Eq.(21) must yield zeroand entails the general requirement

(24)

The integral term in Eq.(21) is the gain one and it gives the number of particles which changevelocity from before, to after the collision. Hence, the collision kernel Kaa is ameasure of transition probability between and velocity groups. The term with the rateγaa is the loss one, and it gives the number of particles escaping from velocity interval

to any other one. γaa can be also viewed as the collision frequency, and its

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inverse can be interpreted as the average time between collisions. Hence,the names: collision rate and frequency, can be used interchangeably. The given probabilisticinterpretation of the kernel and frequency is fully consistent with requirement (24).

Since the kernel gives the transition probability between various velocity groups, itsatisfies the detailed balance condition in equilibrium [36] :

(25)

(26)

where WA is the Maxwellian distribution for active atoms

with u ²0 = 2kBT / m A being the square of the most probable velocity. Integrating requirement(25) over and using Eq.(24) we obtain

(27)

So, the Maxwellian WA must be the stationary solution to the kinetic equation. It isequivalent to say that WA is the eigenfunction of the right-hand side of Eq.(21) with theeigenvalue zero. This result is also evident from the notion of the equilibrium distribution(which is known to be Maxwellian) as the stationary solution to the kinetic equation (21).With the aid of the particle number conservation requirement it can be shown [36] that WA

is a unique steady-state solution. Hence the discussed zero eigenvalue is nondegenerate.It is perhaps of interest to note that the linearized Boltzmann equation has five scalar

invariants (eigenfunctions) corresponding to the zero eigenvalue of the collisional integral.They are 1, and v² and correspond to mass (or equivalently, particle number), momentumand kinetic energy conservation, respectively. In our case, we consider the linear Boltzmannequation and relation (24) corresponds to mass (particle number) conservation while theperturber gas serves as the reservoir of momentum and kinetic energy.

Furthermore, it can be shown [36] that the collision kernel Kaa and the corre-sponding collision rate γ aa satisfy the relation

(28)

This inequality does not have any simple interpretation. It is, however, important for theforthcoming considerations.

The above given relations summarize the most important properties of the collisionkernels and rates. Although these properties are simple and physically understandable, theydo not facilitate the computation of the kernel and rate. The presented expressions arevery general and difficult to deal with in practice, either in quantum or semiclassical case.Finding the collision kernels and rates requires the knowledge of the scattering amplitudesor, equivalently, the corresponding T-matrix elements [3, 39] in the quantum case. Moreover,real atom possesses spatially degenerate levels, which is still another complication, becausethen the T-matrix will have off-diagonal elements between various m-states. In the classicalcase, we need the collisional cross sections. They can be found, at least in principle, whenthe interatomic potentials of the A–P interaction are known, which is rarely the case. Theintroduction of realistic potentials usually leads to intractable analytic expressions for thekernels and requires extensive numerical calculations [40, 41] . Thus, their determination, evenfor a simple two-level atom, is a formidable task.

Hence, in practical investigations one has to resort either to approximate methods, orto some models. The latter approach is adopted by many authors and seems to be quite

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effective. When an analytical model kernel is adopted, the rate follows by Eq.(24). Thesimplest model, called the strong collision (SC) one, is based on the assumption that evena single collision fully thermalizes the velocity distribution of active atoms. In such a case,Eq.(21) reduces to

(29)

where γ (sc) is a constant, while ƒA is an arbitrary velocity distribution for active atoms.The other useful model was introduced in 1952 by Keilson and Storer [42]. Although it

is not derivable from any physically reasonable interatomic potential, it seems to be the mostfrequently used model for VCC, see for example [43, 44] . The kernel for this model is

(30)

where γ (ks) is a constant, while and parameter αks ∈ (0,1). It is straight-forward to check that both given models satisfy all the necessary requirements. The Keilson-Storer model possesses one interesting property, which allows relating the average velocityafter the collision to the average velocity before the collision

(31)

Therefore, αks is interpreted as the persistence ratio for an atom having velocity andis related to the masses of the active atoms and perturbers 1 – αks = 2mA /( mA + mP ).Nevertheless, in practical applications the parameter αks is often treated as adjustable. Forexample, when αks → 0, the Keilson-Storer model reduces to the SC one. Another interestingfeature of the KS model is, that it has a set of eigenvalues and eigenfunctions [43] . Theseeigenfunctions coincide with the ones of the kernel found from (22) for Maxwell molecules,i.e., for the potential V (r) ∝ r –4 . These eigenfunctions play important role in the presentconsiderations, so they will be discussed in more detail later.

Apart from these two simple models it is also possible to construct other models. Forexample, a kernel for hard spheres or the difference kernel are also sometimes used.

When a model analytical kernel is employed in practical computations, then, thereimmediately arises the question whether the adopted model can be found from physicallyjustified cross section. That is, whether a given model can be derived from relation (22)which relates the kernel and the interatomic potential (via a corresponding cross section).The answer to such a question is usually either negative, or very difficult to be given. Bermanet al [45] addressed this question by relating the suitable moments of the collision kernel tothe collisional integrals Ω (l,s) known from kinetic theory of gases and closely connected withtransport coefficients [37] . These authors analyzed the KS kernel, the hard spheres one andthe difference kernel. They have found that the KS kernel gives the results closest to theexpectations following from the kinetic theory of gases. Thus, they have concluded that theKS kernel may be treated as the model yielding the results more reliable physically thanthe other models. This is so, regardless of the fact that it is not derivable from interatomicpotentials.

The introduced collision kernels and rates (with suitable generalizations for coherences)are ideal candidates for theoretical description of VCC. Simply, the microscopic equation ofmotion for the atomic density operator elements is written as

(32)

162

where the radiative part corresponds to the right-hand sides of the optical Bloch equations(18) and collisional one to the expressions of the type of the right-hand side of Eq.(21). Thenone arrives at the set of equations which constitute the operator L0 describing the radiativeand collisional processes and it is possible to proceed to investigate LIKE in gases.

Concluding this section, we note that any modelling method is in fact arbitrary, becauseany adopted analytical model for description of VCC is, to say the least, loosely if at allconnected with realistic interatomic potentials. Moreover, the coherence collision kernels area problem in the analysis via linear Boltzmann equation. Therefore, we propose a differentapproach which will allow a consistent theoretical method of modelling VCC. The proposedmethod will be shown to avoid at least some of the questions raised in this section.

Collision operators and their properties

To facilitate the presentation and discussion of our modelling technique it is convenientto introduce the concept of the collision operator [46, 47] which stems directly from the linearBoltzmann equation. If is an arbitrary velocity distribution, the collision operatoron is defined as

(33)

where, for sake of clarity, we have temporarily neglected the indexation. The collision kerneland rate are combined into a single entity and the corresponding equation of motion

(34)

together with the given initial distribution is the kinetic equation which gives thetime evolution of the velocity distribution of active atoms due to the influence of theVCC with perturber particles. It is worth stressing that the introduced collision operator is,by definition, neither time nor position dependent.

The properties of collision operators were discussed previously [47] . It seems however,useful to review briefly the main concepts. It is convenient to introduce a vector space F ofvelocity distributions

For ƒ, g ∈ F, we define the scalar product

The particle number conservation given by Eq.(24) when reexpressed in terms of acollision operator reads

(35)

(36)

(37)

Furthermore, as a consequence of the detailed balance condition (25) for collision operatorwe find that for any two distribution functions ƒ, g ∈ F we have

(38)

which means that the collision operator is Hermitian in space F. Moreover, the rate γand kernel satisfy relation (28) which yields [47] the inequality:

(39)

163

which holds for arbitrary velocity distribution ƒ . This relation may be called the non-positivity property of collision operator.

Since the collision operator is Hermitian in space F, there exist left and right eigen-vectors of which are equal and the eigenvalues are real. Denoting the eigenfunctions by

and the real eigenvalues by λα we write

(40)

(41)

The eigenfunctions form a complete set of orthonormal vectors

The general property (39) of the collision operator implies that the eigenvalues are non-positive Moreover, as it was discussed after Eq.(27), the MaxwellianW is a unique stationary solution to the kinetic equation and it corresponds to the non-degenerate zero eigenvalue. So, there must exist an index α, henceforward denoted by 0,such that λ 0 = 0 and Since it is a unique eigenvalue, we have λα < 0 for allα ≠ 0. Expanding the solution of the kinetic equation (34) we see that all initially presentdeviations from Maxwellian (i.e., equilibrium) distributions behave as exp[–λα ( t–t 0)]. Hence,requirement λ 0 = 0 ensures that the Maxwellian is the stationary solution corresponding tothermal equilibrium, as it should be.

Within the presented formalism the kinetic equation (34) can be written as

(42)

(43)

with the kernel expanded in terms of the eigenfunctions as

It is the basis of the modelling method introduced and discussed in [47]. Namely, if we knowa certain set of functions to be the eigenfunctions, then any collision operator can beconstructed according to Eq.(43), provided the eigenvalues are known. Thus, we can modela whole class of collision operators by a suitable choice of the eigenvalues. In the paper [47]

we treated the eigenvalues as free parameters, and we have also shown how to reconstructthe Keilson–Storer and strong collision models by the proper choice of the eigenvalues.

On coherence collision operators. Introducing the concept of collisionoperators we focused attention on the velocity distribution functions which have a simpleprobabilistic interpretation. Hence, all discussion given above is certainly valid for popu-lations. Coherences, however, are in general complex and as such, have no probabilisticsignificance. Corresponding collision operators do not need to satisfy the dicussed require-ments. We will argue, however, that it is possible to generalize the introduced concepts tocoherences.

As it was discussed in connection with optical Bloch equations the atom suffers not onlyvelocity changing collisions but also the dephasing ones. We propose following procedure forcoherence collision operators. The dephasing collisions are accounted for as it was done in theBloch equations (18) via the homogeneous linewidth ΓC . In other words, dephasing is alwaysincluded in the corresponding equations of motion. Since the coherences are conjugates ofeach other the coherence collision operators must also have this property. Hence,effects due to VCC, but other than dephasing, are assumed to be incorporated in a complexcollision operators with zeroth eigenvalue λ0

(c ) = 0 (as for populations). Other

164

eigenvalues λ (c)α are complex, with Re λ (c)

α < 0 which ensures the correct time dependenceof coherences. In such a manner we can easily extend our modelling technique to the case ofcoherences. However, operators do not have to satisfy number conservation, or detailedbalance conditions.

On the other hand, when collisions are isotropic it can be argued [3] that only thediagonal (i.e., population) operators play significant role and those for coherences are negli-gible. Then, only dephasing contributes to evolution of coherences. This statement is a kindof a postulate and its validity (i.e., the necessity of including coherence collision operatorsother than simple dephasing) has not been investigated, although it is frequently employedin practical calculations. Nevertheless, in the view of the above given remarks, our modellingmethod can be easily adjusted to the case when the coherence operators must be included.

Eigenfunctions and eigenvalues of collision operators

The collision operator was shown to be Hermitian and therefore it possesses a completeand orthonormal set of eigenfunctions. Solving the eigenproblem and finding exact eigenfunc-tions and eigenvalues for the collision operator requires the knowledge of a collision kerneland rate from interatomic potential. So it is still more difficult than computation of thekernel and rate directly from the potentials. In order to avoid this difficulty we propose a dif-ferent approach which was introduced earlier [46, 47]. Namely, we adopt a set of orthonormalfunctions as the eigenfunctions of the collision operator and define them as follows

(44)

(45)

where φα might be taken in the Cartesian coordinates, as

where vi are Cartesian components of the velocity and Hn (.) are the Hermite polynomials (allspecial functions in our work are taken according to Ref.[48]). Equivalently, we may choosespherical coordinates and then the eigenfunctions φα become

(46)

oscillator ones allows to see that all the requirements imposed on are indeed satisfied.The adopted eigenfunctions of the collision operator are the eigenfunctions of the Keilson–Storer kernel, as demonstrated by Snider . The eigenfunctions of the KS model correspond[43]

to eigenvalues (superscripts (c) and (s) distinguish Cartesian and spherical cases)

with Ln( l +1/2) being the associated Laguerre polynomials and Ylm the spherical harmonics of

the angles determined by the spatial orientation of the vector The two sets of eigenfunctionsfor either of the coordinate systems are distinguished by different subscripts and the contextshould make it clear which eigenfunctions are considered.

It is worth noting that the slightly modified functions

(47)

are the usual eigenfunctions of the standard quantum-mechanical harmonic oscillator [49] ofmass m osc and frequency ω osc , such that the factor is replaced by u ²0 .

The connection between the adopted eigenfunctions of collision operator and

(48)

165

We should, however, stress that adopting the given functions, in either of the coordinateframes, as eigenfunctions of the collision operator is a postulate. This is a postulate, becauseour choice is in fact arbitrary, although well-justified. We assume that the chosen eigenfunc-tions are good approximations to the true eigenfunctions of any physically reasonable collisionoperator. The fact that the selected eigenfunctions are the ones corresponding to theKS model is one of the arguments supporting our choice. In our earlier paper [47] we havegiven a detailed discussion and justification that such an approach is reasonable and thatthe functions can be considered as good approximations to the true eigenfunctions.On the other hand, in we have assumed [47] that the eigenvalues are unknown, and therebytreated as free parameters of the theory. In particular, the eigenvalues can be different fromthe ones corresponding to the KS kernel. In this manner, using expansion (43) and allowingthe eigenvalues to be free parameters we are able to model a whole class of collision operators.These assumptions allowed us to propose a method for modelling various physical phenomenain gaseous mixtures subjected to electromagnetic irradiation. The examples of applicationsof our modelling approach are given in our previous papers [47, 50] .

We can use either Cartesian form (45), or the spherical one (46) of the eigenfunctionsof the collision operator. The latter form of the eigenfunctions is sometimes more convenientand it is usually used in the literature [43, 45] . However, in practical calculations the choicebetween the forms (45) and (46) is, in fact, a matter of convenience. In the papers devoted tomodelling the physical phenomena eigenfunctions as given in (45) were mainly used.One of the advantages of the eigenfunctions taken in the Cartesian form (45) is that they arefactorized in a way which facilitates calculations of various integrals. Moreover, the case withaxial symmetry (the symmetry axis being determined by the incident radiation) is essentiallya one-dimensional one. Then, the eigenfunctions (45) are especially convenient since all thephysically interesting quantities are expressed by integrals over a single component of velocity.

The choice of either of the possible forms of the adopted eigenfunctions determines thenumbering of the eigenvalues. The situation is similar as in the case of a harmonic oscillator.It is perhaps worth noting that eigenvalues (or energies) of the harmonic oscillator are stronglydegenerate. In the spherical case the energy corresponds to the principal quantum numberN (s) = 2n + l , while in the Cartesian case N (c) = n 1 + n 2+ n 3. The energy levels areg ( N ) = ( N + 1) ( N + 2) times degenerate (with N being the principal quantum number foreither of the cases). However, in this work we consider the collision operator which differsconsiderably from the oscillator Hamiltonian. Therefore, it is reasonable to expect that thedegeneracy specific to the harmonic oscillator will be, at least partially, lifted.

The eigenvalues of the collision operator can be found from the relationSubstituting the collision operator according to its definition (33) into scalar product (36),then using the property (24) of the collision rate, we arrive at the expression

(49)

and either Cartesian eigenfunctions (45) or spherical ones (46) can be used.In a modelling approach, eigenvalues can be considered as free parameters, and as such

can be used to fit theoretical results to the experimental data. The closed theory, however,should not contain free parameters, but only the quantities with well-defined physical sig-nificance. Computation of the eigenvalues according to Eq.(49) requires the knowledge ofthe collision kernel. For populations, K aa is specified in Eq.(22) via the cross sec-tion. The cross section is usually unknown since the potentials for A–P interaction are alsounknown. It is, however, possible to circumvent this difficulty [51] . The idea is to expressthe eigenvalues of the collision operators via the transport coefficients which are directlymeasurable physical quantities. This is done in several steps. Transport coefficients can becomputed in the kinetic theory of gases within Chapman-Enskog approach which gives a

166

transport coefficients via collision integrals ΩNP , as assumed in this work) this method gives

. For example, for active-atom molecules( l,s ) [37]

method to find successive approximations to various physically significant quantities. Withinthe first order (for the case when N A <<

in the perturber gas, diffusion coefficient, viscosity and heat conductivity are given as

The collision integrals Ω(l,s ) for integers l and s are defined as

(50)

(51)

where W R is a Maxwellian with u2 = 2kR B T/µ = u 2P + u 2

0 being the square of the mostprobable speed of the relative motion of A and P particles. Hence, WR corresponds to thedistribution of the velocities of relative motion of the collision partners. The quantity Q(l) (v)which appears in (51) is expressed by the cross section via the following integral

(52)

χ s is the scattering angle between and relative velocities before and after the collision.On the other hand, the eigenvalues (49) of collision operators are given as integrals, over

velocities of an active atom before and after the collisions, of the integrand which includesthe cross section via the collision kernel [see Eq.(22)]. This suggests that eigenvalues areclosely related to the collision integrals (51). The derivation of this relation consists inconstructing the generating function which, by differentiation, allows one to express theeigenvalues directly via a combination of the collision integrals. The employed mathematicalmethods are technical and fairly complicated [51]. Therefore, we present here only the finalresults. Although it is possible to obtain the discussed relationship both for the sphericalcase (for eigenfunctions (46)) and for Cartesian ones (45), we concentrate on the Cartesiancase, since it is more useful for the system with axial symmetry.

The eigenvalues associated with the Cartesian eigenfunctions (45) are labeled by threenonnegative integers. As the general property of collision operators requires, we have λ000 = 0,while for the next several eigenvalues we have

(53a)

(53b)

(53c)

Eigenvalues differing by permutation of indices are degenerate, because no direction is priv-ileged. This is a general property of the Cartesian case [51] . Calculation of next Cartesianeigenvalues is certainly possible, but results are more and more complicated.

Eigenfunctions of the harmonic oscillator in spherical or Cartesian coordinates are con-nected by a unitary transformation [49] . Since eigenfunctions of the collision operator areproportional to the oscillator ones, the mentioned transformation can be employed to obtaina connection between the eigenvalues corresponding to two coordinate systems. One easilyobtains that the eigenvalues for Cartesian (c) and spherical (s) cases are related as

(54)

167

Transport coefficients (50) can easily be expressed by the eigenvalues given in Eqs.(53).Conversely, eigenvalues can be rewritten in terms of combinations of transport coefficients.This fact puts the eigenvalues on the firm physical ground. They are no longer free parame-ters, but are fully expressed by physically understandable transport coefficients. For example,for the first eigenvalues (53a) from (50) we obtain

0

(56a)

(55)

Expressing some of the first eigenvalues by the collision integrals or transport coefficientswe can construct an approximate (because of finite number of eigenvalues used) collisionoperator (43). Thus, it is written in terms of the quantities which are directly measureable.In this way we are able to construct the collision operator without invoking any particularanalytical model. Therefore, we avoid all the difficult questions arising when adopting anyspecific model. Interatomic potentials determine differential cross sections which appear inthe collision integrals, and thus determine the eigenvalues. Thereby the right potential isautomatically accounted for in our (although approximate) collision operator constructedaccording to Eq.(43). Circumventing the problems connected with any particular model ofcollision kernel we provide a method to construct a whole class of collision operators.

The obtained relationship between the eigenvalues of the collision operators and collisionintegrals Ω (l,s ) or transport coefficients clarifies the previously employed modelling method.We have shown that only several eigenvalues of the collision operator usually suffice to describephysical phenomena occurring in the gaseous mixtures [47, 50] . Hence, it would be of interestto reexamine the results of modelling in the view of the present ones. It would be interestingto use explicit expressions for the eigenvalues in the equations for the quantities modelled byeigenvalues – previously taken as free parameters. This may provide some new informationon the light–induced kinetic effects in gases. Moreover, it will be easier to compare the theorywith potential experiments, since the physically significant quantities will be expressed byother ones, which are measurable and have well known physical meaning. This may also bean interesting subject for further investigations.

Bloch–Boltzmann equations and modelling method

In previous sections we have presented a theoretical method allowing the detailed de-scription of the velocity-changing collisions. As we have discussed it earlier, we are interestedin the interplay between collisions and radiative effects which is incorporated in operator L .We follow the lines established in Eq.(32). The radiative part of L 0 is thus given by theoptical Bloch equations, while the collisional operators introduced above provide the neces-sary collisional terms. In this manner we arrive at the set of equations for local, microscopicevolution of the density matrix elements of active atoms. Such an approach may be calledphenomenological, but it seems quite successful and is used by virtually all authors dealingwith similar problems. The obtained equations of motion corresponding to operator L0 canbe called Bloch-Boltzmann equations and are as follows

(56b)

(56c)

with ρ12 = ρ*21 . The notation in Eqs.(56) is the same as in (18), but for simplicity, we haveomitted the arguments indicating the dependence of ρab on velocity, position and time. We

168

recall that Rabi frequency may be implicitly depend on the position. Thereby, the matrixelements of the density operator are also parametrically dependent on position We alsonote that, in general, these equations are subject to initial conditions ρab t = 0) = ρ0

abThe stationary solution , however, is expected to be independent of the initial conditions.

Collision operators (a = 1, 2), incorporate the rates γaa and kernels Kaa

and represent the influence of VCC on the atomic populations. The coherence collisionoperator are assumed to have the properties introduced earlier. Dephasing contribution isincluded in the homogeneous linewidth ΓC . The operator in (56c) describes collisionaleffects other than dephasing. The term containing can frequently be neglected andinfluence of collisions on coherences reduces to dephasing [3, 47] .

Bloch-Boltzmann equations together with the definitions of the quantities appearing intheir right-hand sides constitute the main theoretical framework sufficient to investigate theinterplay between the collisional and radiative effects. When using this model we will assumethat:

1 . All collision operators in the above equations correspond to the same set of eigenfunc-tions ϕα given by Eq.(45) or (46). This assumption is thoroughly discussed andjustified in Ref. [47].

2. Each population collision operator is parameterized by a different set of eigenvalues.That is, a set λ(a)

α corresponds to for a = 1,2. The choice of two different setsof eigenvalues for ground- and excited-state atoms reflects the fact that, in general,the A–P interaction depends on the internal state of the active atom. Different colli-sional cross sections imply different transport properties and hence two different setsof eigenvalues. On the other hand, these eigenvalues can be expressed via the collisionintegrals or transport coefficients characterizing active atoms in either of the atomicstates. Therefore, λ(1)

α and λ(2)α are not free parameters, but have specific physical

significance, as it is implied by above given considerations.

3. If the effect of collisions other than dephasing must be included, then the eigenvaluesof collision operators are treated as complex parameters. Otherwise, these collisionoperators can be neglected.

Since collision operators satisfy the requirement of particle number conservation(37) the density operator is normalized

(57)

as required by Eq.(4). This requirement must be also satisfied by any initial condition ρ0

Before proceeding further, we add some remarks on the calculational procedure. Thecomputation of the light-modified diffusion tensor and the gradient velocity is most easilyperformed when the time integrals in Eqs.(8) and (10) are calculated with a limiting proce-dure. The integrand is multiplied by the factor e–st and, simultaneously, the limit s → 0 +is taken. Performing the integration over time, after simple manipulations we obtain forlight-modified diffusion tensor

( 5 8 )

and for the gradient velocity

( 5 9 )

169

The operator L0 has a zero eigenvalue (since it possesses a stationary solution, as in Eq.(3)),but in these equations its inverse operates on quantities which yield zero after taking thetrace over the atomic states and integrating over velocity. This ensures the existence of thelimit in the above equations. In order to evaluate the integrands in (58) and (59) we need tocalculate the quantities of the type of

where σ0 is a velocity dependent matrix satisfying the requirement

(60)

(61)

The right-hand side of Eq.(60) may be viewed as the Laplace transform of the time-dependentmatrix the time evolution of which is governed by the Bloch-Boltzmann equations (56), i.e.by the operator L0 . Then, the quantity (60) can be evaluated by solving the Bloch-Boltzmannequations in the Laplace domain with σ0 being the initial condition. Hence, we need

(62)

to obtain (58) and (59), respectively.This discussion clearly illustrates a twofold meaning of the Bloch-Boltzmann equations.

The solution with initial conditions satisfying the requirement (57) would yield thetime evolution of the local density operator determined by collisional and radiative processes.Stationary solution is then the limit for t → ∞. On the other hand, solutions withthe special initial conditions (62) specify the integrands in expression (58) for light-modifieddiffusion tensor and in (59) for gradient velocity. Therefore, the explicit solution to theproblem of LIKE may be obtained via the detailed analysis of Bloch-Boltzmann equationswith initial conditions chosen according to the current needs.

The general solutions to the Bloch-Boltzmann equations are found in the Laplace do-main [47] . Since such solutions are rather complicated, they are briefly summarized in theAppendix. We note that the obtained expressions contain collision operators or their inversesacting on the initial functions of velocity. This difficulty can be avoided, because the physicalquantities of interest such as drift velocity (7), light-modified diffusion tensor (8) or gradientvelocity (10) are expressed as integrals over velocity of simple functions of velocity, multipliedby the solutions of Bloch-Boltzmann equations, which via the initial conditions are also func-tions of velocity. This suggests a simple procedure. The functions of velocity are expandedin terms of the eigenfunctions of the collision operators. The solutions to Bloch-Boltzmannequations depend on the combinations of the eigenvalues and eigenfunctions. The arisingintegrals over velocity can then be easily computed. It appears, that due to orthogonality ofthe eigenfunctions only a very few expansion terms are of importance. We will illustrate thisprocedure in more detail in the next section taking drift velocity as an example.

Absorption equation

In our discussion we have noted that the local steady-state density operator mayparametrically depend on the position within the sample due to the spatial variations oflight intensity. Therefore, to close the system of equations we need an equation governingthe dependence of the light intensity within the sample. Analysis of Maxwell equations for alight wave propagating in the gaseous medium [30] yields the equation

(63)

170

where is the direction of light propagation. α denotes the saturated absorptionrate which, in general, depends on the velocity distributions of active atoms in both states.Hence, it is a functional of the local light intensity I It can be shown [30] that

(66)

171

(64)

In the optically thick systems, due to strong absorption, the light intensity varies in spaceand Rabi frequency χ changes accordingly. This is so, because Rabi frequency and the lightintensity are connected as

(65)

with B being the Einstein’s absorption coefficient and c the speed of light. As a result,elements become dependent. Since atomic populations depend on intensity via Rabifrequency the resulting set of equations is strongly nonlinear. In absence of saturation theabsorption rate is directly proportional to the intensity I which considerably simplifies theproblem. In the case of optically thin systems further simplification may be obtained, if oneassumes that the intensity variations across the sample are negligible. In any case, Eq.(63)closes our physical model, because without it we would be unable to determine the variationof light intensity within the sample.

APPLICATIONS AND DISCUSSION

The derived theoretical model seems to be quite elastic to enable the description ofa variety of physical situations. We will, however, restrict our attention and describe onlysome applications directly connected with LIKE in gases, and focus especially on the driftvelocity. Our purpose is to illustrate how the developed formalism may be used to analyzephysical phenomena. We will not dwell upon mathematical details but discuss the method,the basic results and the possibility of approximations useful in some particular cases. We willalso discuss the generalized correlations of the functions of velocity which seem to be quiteimportant, since they may be viewed as Green-Kubo relations for Navier-Stokes transportcoefficients for light irradiated systems. These correlation functions allow us to put light-modified diffusion tensor (8) and gradient velocity (10) into a single theoretical scheme.

Other applications are certainly also feasible. We have employed Bloch-Boltzmannequations (56) to the analysis of radiation distribution (i.e., of the spectra of resonancefluorescence) [50] in presence of VCC. The narrowing of spectral lines, similar as in the Dickeeffect, was then predicted. Also, using a decorrelation approximation, Bloch-Boltzmannequations can be reduced to the rate equations which are especially useful for the discussionof the case with broadband irradiation. The drift velocity and light-modified diffusion tensorwere investigated for such a case .[17]

Drift velocity

Physical concepts underlying LID effect were already discussed above. We can, there-fore, concentrate on theoretical investigation of drift velocity defined by Eq.(7) which werewrite as

where the trace of the local steady-state density operator is replaced by a total velocitydistribution function found in (A.14). The components of velocity are proportional to

the eigenfunctions (45) with one index being a unity and two other ones zeroes. Sinceis expressed by the upper state population (see Eq.(A.13)) we expand into the sameset of eigenfunctions, and we write

(71)

(67)

When using (A.13) in (66) we see that the first term is odd in velocity, so it does notcontribute. Using the orthonormality of the eigenfunctions we obtain

(68)

When the incoming light propagates along the z-axis of the system then Bloch-Boltzmannequations (and thereby the solutions) depend only the z component of velocity. This is thecase with axial symmetry and since it corresponds to a typical experimental situation, inthe forthcoming, we will concentrate on this case only. Generalizations to three dimensionsdo not pose any problem. Thus, all velocity distributions are axially symmetric and theexpansion coefficients as in (67) with first two indices equal to zero are the only nonvanishingones. In such a case, drift velocity given by Eq.(68) reduces to a single term, which is

(69)

The final step in calculating consists in finding the expansion coefficient forthe local, stationary upper state population. These coefficients are given in the Appendix(Eq.(A.17)). For axial symmetry, when first two indices are zeroes, the drift velocity becomes

(70)

with operators and following from Eqs.(A.6) and (A.8) in the Appendix. Deriving (70)we have accounted for the fact that collision operators acting on the Maxwellian WA givezeroes. Note, that the operators depend solely on the z component of velocity and,therefore, all integrals are one-dimensional. The integrand in Eq.(70) is not an eigenfunctionof collision operator, so further calculations and results are fairly complicated [47] . Thus, werestrict ourselves to discussing the general result.

Due to relation (55) the eigenvalues appearing above can be identified with the diffusionconstants for excited and ground state atoms. This conclusion coincides with the experimen-tally confirmed results [8] . Condition equivalent to requirement D 1 ≠ D 2, isnecessary to observe the drift. This condition summarizes the physical requirement that thecollisions must be state dependent and reflect the dependence of the A–P interaction on theinternal state of the active atoms. The same conclusion stems directly from Eq.(66) when(A.16) is taken into account. Moreover, (as discussed previously) we expect that D1 > D 2,which implies that eigenvalues satisfy the inequality

All eigenvalues must be nonpositive, so we see that the first factor in (70) is positive. Foraxial symmetry ∆ = (ωL – ω 21 ) – kvz . therefore when light field is tuned to the blue sideof resonance (i.e., when ω L > ω 21) the contribution of v z > 0 dominates in the integrand of(70) and the integral is positive. Drift velocity becomes positive – atoms are pulled by thelight. Conversely, for irradiation at the red wing of the resonance the contribution of vz < 0

172

dominates and becomes positive. Atoms are pushed by light. It is straightforward tosee that drift velocity vanishes on resonance, because on resonance the integrand in (70) isan odd function of velocity (everything apart from the factor vz is even in velocity).

Frequent collisions quench the drift effect since appreciable thermalization occurs duringatomic lifetime. As a result we expect the drift velocity to decrease when the buffer gaspressure increases. The eigenvalues of collision operators are proportional to perturber densityas it follows from Eqs.(53) and can be proved in a general case [51]. Thus, drift velocitydecreases when N P increases. This is evident in Eq.(70).

The dependence of drift velocity on Rabi frequency is also evident. It vanishes whenχ → 0, because, then, there are no excited atoms. Conversely, when χ is very large, theoperator factor in (70) tends to and yields zero when acting on the Maxwellian W A

so that → 0 for χ → ∞. This conclusion corresponds to the fact that power broadeningsmoothens out the differences in velocity distributions for atoms in both states. The decreaseof velocity selectivity entails, and decreases.

We conclude the general discussion of drift velocity by noting that all the featuresof LID are properly accounted for in the presented formal approach. Our method beingindependent of any particular collision models is, therefore, quite effective in the descriptionof light-induced drift.

In some physical situations it is possible to obtain simpler expressions. In order topresent such a case, let us assume that coherence collision operators may be neglected (asis frequently done by many authors). Then, operator becomes an ordinary function pro-portional to the Lorentzian profile with width determined by the homogeneous linewidth ΓC .The inhomogeneously broadened Doppler absorption profile is usually much broader. Thenthe Doppler approximation is possible, and it consists in approximating the relatively muchnarrower homogeneous profile by suitably normalized delta function. If this approximationis applicable, the integrals in (70) can be computed explicitly and we obtain

(74)

where we have denoted

(72)

(73)

The derived expression (72) for drift velocity within Doppler approximation is still indepen-dent of any collision model. The eigenvalues can be expressed via the transport coefficients,giving drift velocity in terms of measureable quantities. The described approach illustratesthe possibilities inherent in our modelling technique.

Drift velocity given by relation (72) has the properties similar to ones discussed above.The dependence of on Rabi frequency is the only difference. In the present, approximatecase, does not vanish when χ gets very large. This is so, because Doppler limit is notvalid when there is an appreciable power broadening. This restricts the Doppler limit to weakor modest light intensities.

The presented Doppler limit corresponds to a restricted, but well-defined class of ex-periments and is mathematically much simpler than the general one. The Doppler-limitexpressions, exhibiting the major features of LID may be, therefore, quite useful in manypractical applications also ones different from LIKE. For example, calculation of fluorescencespectra [50] can be done within this approximation.

173

Generalized correlation functions

(80)

The correlations of various functions of velocity play an important role in the transporttheory. We briefly show how these correlations can be dealt with in our approach.

For illustration purposes, we first give a simple one-dimensional example. We consideran active-atom-perturber mixture in equilibrium, without any radiation field. For integers pand q, we define a time dependent correlation function of powers of velocity [47] as

(75)

where the double angular brackets denote averaging over the equilibrium velocity distribution,which in this simple case, is just a Maxwellian WA(v). Thus, we can write

(76)

Collision operator (for A–P collisions) appears here, since it is the one which governs theevolution of A atoms in equilibrium. Using the one-dimensional eigenfunctions ϕ n(v) ofcollision operator to expand the above expression and the scalar product (36) we transformthe correlation function into

(77)

where we also employed the completeness and orthonormality of eigenfunctions. Taking ϕn(v)in the Cartesian coordinates (45), we can easily compute the scalar products (integrals). Usingthe generating function of Hermite polynomials [48] we obtain

with the integrals Jn (p) given as

(78)

(79)

Due to the properties of Hermite polynomials integrals Jn (p) vanish when p and n are ofdifferent parity. What is more important, integrals Jn(p) vanish for n > p. Therefore, forgiven integers p and q, the sum in (78) contains only a finite number of terms. We concludethat the considered one-dimensional correlation function is expressed by a finite number ofthe eigenvalues of collision operator.

The quantities of physical interest are then expressed as the time integrals over thefunction G eq , that is, via

and its a trivial matter to find such integrals from (78). Instead of considering a general case,let us focus on a particular example. The one-dimensional diffusion coefficient for A atomsin the perturber bath may be defined [14] as

(81)

174

Substituting expression (78) we note that it contains only one term (since p, q = 1), and weget

(86)

(82)

which is positive since the eigenvalue λ1 must be negative. This is a one-dimensional equiv-alent of relation (55), thus it proves the self-consistency of our approach. The generaliza-tion to three dimensions poses no difficulties. Other functions of velocity can also be easilyanalysed. Problems similar to our example are, however, not very interesting because theyconcern the equilibrium, when there are no excited-state atoms and the velocity distributionis Maxwellian.

The situation changes when the A–P mixture is light irradiated. The system attainsthe stationary, but nonequilibrium state. Its time evolution is much more complicated, sinceit is governed by the operator L0. For example, instead of a simple diffusion coefficient (82)we need to deal with the light-modified diffusion tensor (8). We will briefly discuss how themethod outlined in the above simple case can be generalized for systems excited by incidentlight beam.

Let and be two velocity dependent quantities. They may also have the opera-tor character (as it is in case of gradient velocity), hence, their sequence may be of importance.By analogy to Eqs.(80) and (76) we define [52] the generalized correlation function as a doubleintegral

(83)

The generalization consists in three points. First, we use evolution operator L corresponding0

to the Bloch-Boltzmann equations.stationary density operator . Finally, there is an additional term ⟨b⟩∞ denoting theaverage

Next, the averaging is now performed over the local

(84)

where is the stationary state total velocity distribution of the active atoms as in (A.14).This average is necessary in Eq.(83) in order to ensure the convergence of the time integral,and it is due to the nonequilibrium features of the light-induced stationary state.

Calculating g(a, b) we first apply the limiting procedure, as that leading to Eqs.(58) and(59). This yields the expression

The last terms have an obvious property

(85)

which follows directly from Eq.(84) and from normalization (4) of the density operator. Dueto requirement (86) the limit in Eq. (85) is finite. Expression (85) can be considered asequivalent definition of the generalized correlation functions. Comparison of the definition(85) with previous results allows us to put the light-modified diffusion tensor and the gradientvelocity into a single general scheme, so that they can be expressed as

(87)

175

Returning to generalized correlation function (85) we note that the expression in curlybrackets can again be viewed as a Laplace domain solution to Bloch-Boltzmann equationswith being the initial condition. Thus we are back to the already discussedcomputational procedure. The essential role is played again by Bloch-Boltzmann equations.The formal solutions (A.10) given in the Appendix with suitably chosen initial conditions canbe used in (85). Then, expanding the functions of velocity in the eigenfunctions of collisionoperators we express the correlation function g(a, b) entirely via the physical parameterscharacterizing the irradiated system and via the eigenvalues. Since the latter are related totransport coefficients we obtain closed expressions for quantities od interest.

Finally, let us discuss behavior of generalized velocity correlation functions in the casewhen the light field is absent. The limiting procedure is rather lengthy, so we state the finalresult, best expressed in a form expanded into the eigenfunctions

(88)

(89)

which reduces to a finite sum, when functions and have polynomial character. Italso seems interesting that when the collisions do not distinguish between atomic internalstates, that is when (or, equivalently when for all multiindices α), thegeneral expression (85) reduces to

where the right–hand side is given in (88). This means that the A–P interaction ”doesnot see” excited atoms which collide with perturbers exactly as the ground–state ones do.Although due to irradiation we have for collisional processes all atoms behave asthose in the ground state. For , the A–P collisional interaction is simply insensitiveto the presence of excitation. Hence, in the stationary state the total velocity distributionmust correspond to the equilibrium one, that is to the Maxwellian as in (A.16).

The specific calculations based on Eqs.(87) for light-modified diffusion tensor (8) orgradient velocity (10) are fairly complicated, thus we give only a discussion of the results.

The light-modified diffusion tensor consists of two contributions, one being the coeffi-cient D 1 corresponding to the no-light-field case, and the second one displaying the interplayof collisional and radiative processes. When the irradiation specifies an axis of symmetry, theelements of diffusion matrix exhibit axial symmetry

D11 = D 22 = D t, D33 = Dp. (90)

with D t and D p each being of the form D1 term plus complicated functions of system param-eters and eigenvalues corresponding to both atomic levels. The axial symmetry is preservedon resonance (when drift velocity = 0). In absence of irradiation, or for excitation insen-sitive collisions the second term vanishes (as in (89)) and the elements of the diffusion tensorreduce to D 1, as expected. This confirms the notion of light as a thermodynamic force.

Very similar comments apply also to gradient velocity, for which we obtainThis is an expected result, which fully agrees with the discussed idea of diffusive pulling ofatoms towards the regions with higher light intensity. The components perpendicular to

the incident light are different from which is parallel to the light beam, as it is requiredby the axial symmetry. Gradient velocity also does not vanish on resonance, but it tends tozero when χ → 0. The obtained formulas [52] for gradient velocity are quite involved. Theirmathematical form does not bring anything new to the understanding of the physical effects,thus we do not present them here.

176

Final remarks

We have constructed a closed theoretical model allowing the description of physicalphenomena occurring in the system consisting of a dilute active-atom vapor immersed in themuch denser perturber gas and submitted to the external radiation field. The fact that radia-tive and collisional processes are much faster than the macroscopic light-induced flows allowsthe separation of the evolution of the considered system into two distinct parts. The firstpart, corresponding to macroscopic flows, is the slow one. The flows enter the generalizeddiffusion equation (5) and (6) which depends on the light-modified coefficients. These coef-ficients, in turn, are determined by the interplay between radiative and collisional processesoccurring on a rapid time scale.

Radiative processes are accounted for by means of the optical Bloch equations for asimple two-level atom. The employed model allows us to include all essential effects. It posesno problems to generalize Bloch equations either to an atom with more levels, or to a modelwith spatially degenerate levels specific to realistic atoms. More complex models, however,entail much more involved mathematics or can be dealt with only by numerical calculations.Comparison of theoretical results with experiments usually requires more elaborate models,but it seems that a two-level one is sufficient to give insight into the physics of the problem.

Optical Bloch equations must be augmented by terms describing the collisions. Wetreat the dephasing collisions (influencing the internal atomic state) within the impact ap-proximation by introduction of a homogeneous linewidth ΓC to the equations of motion forcoherences, while velocity changing collisions (VCC) are treated by means of the classicallinear Boltzmann equation which is valid when the perturber gas is much denser than theactive-atom one. The arising terms are combined into a single concept – the collision oper-ator. Collision operators are shown to be Hermitian in the suitably chosen space of velocitydistributions. As such, they possess real eigenvalues and a complete, orthonormal set ofeigenfunctions.

We have discussed the theoretical approach towards determination of collision opera-tors via the eigenvalues which were expressed by the transport coefficients. On the otherhand, some experimental measurements were also done to determine the collision operators(or kernels) (see Refs.[53, 54], and the references given therein). The basic idea of these ex-periments is to excite a resonance transition in a velocity selective manner, and then to probeit by a second (usually weak) laser field by excitation to a higher third level. The spectraof fluorescence from this third level provide information about velocity redistribution in thefirst excited level. Haverkort et al [53] applied such a method to obtain the best fit to theKeilson–Storer kernels. Gibble and Gallagher obtained the necessary spectra and then, bymeans of an appropriate deconvolution procedure found the corresponding (one-dimensional)collision kernels. It is very interesting to relate the experimental fluorescence spectra directlyto collision operators. An attempt in this direction, however with only one exciting laser field,-was presented in [50] . It may be expected that a general and possibly rigorous treatment ofthe fluorescence spectra corresponding to the experimental cases will be quite complex andprobably will require a suitable approximation scheme. It is an interesting subject for furtherstudies, and it seems, at least for the weak probe case, that it will be possible to express thespectra via the eigenvalues of the collision operators. Then, comparison of experimental spec-tra to the theoretical ones should directly yield the eigenvalues and hence the (approximate)collision operators according to Eq.(43).

The collision operators corresponding to each element of the density operator of activeatoms are then added to optical Bloch equations. The obtained set of equations of motion iscalled Bloch-Boltzmann equations (56) and it constitutes the essential theoretical framework

177

of our approach since it gives the sought joint description of rapid radiative and collisionalprocesses. They attain a local stationary, but nonequilibrium state, which is found as thesteady-state solution to Eqs.(56). This solution, denoted by depends parametricallyon local light intensity (or, equivalently, on local Rabi frequency). Hence, it is necessaryto close the system of equations by the absorption equation (63) and (64). In such a waythe description of the system is complete. Let us note, that even for simple two-level atomthis system of equations is strongly nonlinear, mainly due to complicated behavior of lightintensity within the sample.

The formal solutions determining the quantities of physical interest are expanded interms of the eigenvalues and eigenfunctions of collision operators. This results in relativelysimple expressions depending only on a few eigenvalues of collisions operators. The eigenval-ues, in turn, are connected with the transport coefficients following from the kinetic theory ofgases. Therefore, the obtained results contain no free parameters, perhaps except the com-plex eigenvalues of the coherence collision operators. The latter, however, can frequently beneglected and left out of consideration. In such a case our approach gives a slightly simplerdescription of the considered physical system.

Introducing our model we have discussed several underlying assumptions. They mostlyconcern the validity of the linear Boltzmann equation and the postulated form of the adoptedeigenfunctions of the collision operators. On the other hand, our approach is not restrictedby the field strength, detuning or other parameters. This allows easy and well controlledapproximations, such as the Doppler one employed for the analysis of the drift velocity.Moreover, we are not restricted by any specific choice of an analytical model for VCC. We areable to avoid difficult questions, because our results involve the collision-operator eigenvalueswhich are directly expressible by the transport coefficients. The latter automatically includethe interatomic potentials. This seems to be an advantage of our method since it providesa bridge between quantum optics and kinetic theory of gases. The usefulness of our methodis illustrated by two simple examples. The number of effects studied within our formalismis much larger and it may be successfully used for investigations of other effects or differentphysical situations.

We have described a method allowing investigation of various parameters and phenom-ena due to the interplay of radiative and collisional effects. We have argued that light maybe viewed as a thermodynamic force driving macroscopic flows. We also stress the essentialrole played by VCC. Most of the effects disappear when collisions do not diuscriminate be-tween atomic states. Within our model we have shown that local stationary total velocitydistribution function reduces to the Maxwellian when the collisionsdo not depend on atomic internal state, that is when collision operators are the same for anatom in either of its states.

One of the still unresolved problems concerns the coherence collision operators. Theyare frequently left out of the picture, that is, simply neglected. There is no known criterionindicating precise conditions which would determine the situation when omission of theseoperators is indeed justified. It seems, that finding such a criterion will require a complexfully quantum-mechanical investigations. Classical linear Boltzmann equation is insufficientbecause coherences are complex and do not classical analogues. Coherences are responsible forphase relationships between quantum states and thus must be treated quantum-mechanically.Our method allows modelling VCC for coherences via complex eigenvalues with negative realparts, but does not provide any tools for resolving the discussed problem. These are stillopen questions.

Recently, some anomalous features of light-induced drift were reported [55, 56] . Thesimple analysis in previous sections shows that drift-velocity as the function of detuning∆ is expected to have just one zero (see Eq.(72)). Anomalies consist in observation of morethan one zero. The experiment [55] is performed in a perturber gas which is a mixture of two

178

noble gases, and as the authors state, still requires some enhanced resolution, but indicatesthe strong dependence of anomalies on the composition of the buffer gas. The calculationsof Gel’mukhanov [56] connect anomalies with the high temperature effects. It seems that ourformalism may be extremely useful for investigations of such anomalies. The connection withNavier-Stokes formalism via the transport coefficients can be especially useful in studies oftemperature effects. The detailed study of the generalized correlation functions as Green-Kubo relations may also be useful in better understanding of light-induced macroscopic flows.Our approach seems to be helpful in establishing connections between quantum-optical andstatistical points of view on light-induced phenomena.

One more interesting problem is connected with nonlinear effects. The Bloch-Boltzmannequations determine the local stationary state due to radiative and collisional effects butdescription of the long-range, macroscopic effects requires both the diffusion equation (5)and the absorption one (63). When no simplifying assumptions are made, the equationsare strongly nonlinear. It is of interest how the light intensity variations are combined withother phenomena. To our knowledge, the only one work devoted to this subject is due toGel’mukhanov et al [57]. These authors consider the evolution of light intensity inside aninfinitely long sample, while describing the VCC in the simplest strong-collision model (seeEq.(29)). They study both low and high field intensity. For the high field case they obtainand discuss the nonlinear Burgers which leads to soliton-like behavior of light-induced drift.The study of LIKE in gases in the complete manner is thus a separate and quite difficultproblem which, to our knowledge, is almost unexplored. Hence, we conclude that the full (i.e.,including absorption equation) description of LIKE is largely an open problem. Applicationof our approach due to its strong relations to kinetic theory may also prove quite useful infurther, more detailed analysis of nonlinear effects occurring in the propagation of light acrossa gaseous mixture. This is, however, a possible subject for further studies.

One more factor deserves attention. All the considerations presented here are restrictedto bulk effects. That is, we have left out all surface, or boundary effects. Phenomenaoccuring at the walls of the container may strongly influence LIKE in gases. For example,accomodation coefficients different for an atom in either of its states may affect the light-induced drift, or even lead to a surface LID [8] . However, all kinds of wall or boundary effectsare entirely beyond the scope of this work.

Concluding, we feel justified to say that despite the discussed limitations, the approachvia Bloch-Boltzmann equations, eigenfunctions and eigenvalues of the collision operatorsseems to be quite elastic and applicable to the description of a large collection of experimentalsituations. Results presented in this review together with previously published ones seem tohave considerable potential and a consistent, well justified physical basis.

(A.1)

APPENDIX

Formal solutions to Bloch–Boltzmann equations. The BlochBoltzmann equations (56) play an essential role in our approach. Their solutions determinevaroius quantities of interest. Therefore, we briefly discuss the formal solutions and theirproperties.

We consider here a general case in which a 2 × 2 matrix with a, b = 1,2,is assumed to satisfy Eqs.(56) with general initial conditions given by . We do notspecify the meaning of the matrix R ab, it can be adjusted according to current needs. First,we transform the system of equations to Laplace domain by

179

where the tilde denotes Laplace transforms. After the transformation, the set of equationsbecomes algebraic (but includes initial conditions explicitly) and may be solved formally.We omit lengthy, but otherwise straightforward calculations in the Laplace domain, we onlypresent the final results. The element is

(A.2)

The velocity distribution is expressed via the already known element as

(A.3)

Similarly, the off-diagonal matrix element is

(A.4)

The auxiliary quantities depending on initial conditions are defined as

(A.5)

We have also introduced several auxiliary operators. They are defined as follows

(A.6)

(A.7)

(A.8)

(A.9)

Selecting appropriate initial conditions, we can assign specific meaning to the obtained for-mal solutions and we can use them to construct the needed physical quantities such as, forexample, the generalized correlation functions. The corresponding expression is

(A.10)

where [b] denotes the dependence of the matrix, which is a solutions of Bloch-Boltzmannequations, on initial conditions as discussed in the main text.

The stationary solutions to Bloch-Boltzmann equations are obtained by multiplyingEqs.(A.2-A.4) by a factor s and taking the limit s → 0+ . It is easy to notice that

(A.11)

so that the term does not contribute to stationary solutions. The term requiressome care. When the initial condition F0 is expanded into the eigenfunctions, the zerothexpansion term contains ϕ 0 = W A which belongs to eigenvalue zero of the operator Sothe zeroth term in is of the form When the stationary limit is taken, thisterm is the only one which survives, and we have

(A.12)

and it gives the nonzero contribution to the stationary solutions.

180

With the above remarks it is a simple matter to construct the velocity distributionscorresponding to the local steady-state density operator They are given as

(A.13)

(A.14)

(A.15)

where the operators with subscript zero follow from (A.6–A.9) for s = 0.When the A–P collisions do not discriminate between the states of active atoms, then

As a consequence, equation (A.14) yields

(A.16)

although there is nonzero field, and therefore Finally, we note that expansioncoefficients as defined in Eq.(67) can be found from (A.13) and using the scalar product(36) we obtain

(A.17)

The eigenfunction taken in the Cartesian frame, according to (45). We can certainlyalso use spherical eigenfunctions, but for the case with axial symmetry the Cartesian onesseems to be more convenient due to their factorization.

Acknowledgment

Partial support by Gda sk University through grant BW/5400-5-0305-7 is gratefullyacknowledged.

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182

TEMPORAL AND SPATIAL SOLITONS: AN OVERVIEW

A D BOARDMAN, P BONTEMPS, T KOUTOUPES and K XIE

Photonics and Nonlinear Science GroupJoule LaboratoryDepartment of PhysicsUniversity of SalfordSalford, M5 4WTUnited Kingdom

Abstract

This chapter contains a fundamental review of envelope temporal and spatial solitons. Asubstantial effort has been made to give an account of both the historical background andthe physical concepts. Mathematical detail is given to justify the generic nonlinearequations and guide to the inverse scattering method is presented.

1. INTRODUCTION

Long ago, in August 1834 [1] John Scott Russell, a naval architect, was workingfor the Scottish Canal companies to establish the possibility of rapid steamboat transit oncanals. As part of this investigation, he was observing a boat being pulled along, rapidly,by a pair of horses. For some reason, the horses must have stopped the boat rathersuddenly. What happened next was to change science in the most dramatic way. Thestopping of the boat caused a very strong wave to be generated. This wave, in fact, asignificant hump of water stretching across the rather narrow canal, rose up at the front ofthe boat and proceeded to travel, quite rapidly, down the canal. Russell, immediately,realised that the wave was something very special. It was 'alone', in the sense that it sat onthe canal with no disturbance to the front or the rear; nor did it die away until he hadfollowed it for quite a long way. The word 'alone' is synonymous with 'solitary' and,Russell soon referred to his observation as the Great Solitary Wave.The word solitary is now routinely used, indeed even the word 'solitary' tends to bereplaced by the more generic word 'soliton'. Once the physics behind Russell's wave isunderstood, however, solitons, of one kind or another, appear to be everywhere but it isinteresting that the underlying causes of soliton generation were not understood by Russell,and only partially by his contemporaries.

Beam Shaping and Control with Nonlinear Optics183Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

Russell, after graduating in 1824, at the age of sixteen, was a brilliant observer, abeautiful writer and had a great gift for lecturing. It is not surprising, therefore, that healso carried out very careful laboratory experiments on shallow water containers, longenough to see the generated waves evolve. Incidentally, it is incredibly simple to do this;all that is necessary is to strike the water at one end of the water tank with moderate force.What Russell saw was that the solitary waves have a speed proportional to their amplitudeand that they pass through each other [2] without destruction, or change.

These results did not agree with the work of Airy [3] who asserted that largeamplitude waves would self-steepen and break up. Solitary waves were not predicted sosomething was missing from the theory. In the meantime, in 1847, Stokes [4] worked outthat, for deep water, CW (periodic) waves (not localised) with finite amplitude can existwithout breaking up i.e. in mathematical language they have permanent form. What thenwas missing from the earlier by Airy with respect to localised waves? The answer isstrikingly simple, once the point is realised. The explanation of the observations of Russellneeded a study of the delicate balance between dispersion and nonlinearity. This point wasappreciated by both Boussinesq (1871) [5] and Rayleigh (1876) [6] but even they neverarrived at what we now term a nonlinear partial differential equation, for which Russell’swave is a solution. It took until 1895 for two Dutchmen, Korteweg and de Vries [7] toproduce this differential equation. One of the solutions of the KdV equation is the solitarywave observed by Russell. Normally, a hump of water like Russell’s solitary wave isthought of as a packet of waves, all travelling with different speeds. This way of looking atthings comes from Fourier, but it is a linear viewpoint with the end result being thedestruction of the hump due to dispersion. Large amplitudes mean lots of power, so thewaves in the packet are now forced to interact with one another. The forces trying torestore equilibrium are no longer just proportional to the height of the wave and the speednow depends on displacement: this is a nonlinear system. In a balanced situation, the powerof the wave acts against dispersion. The wave then remains intact and a soliton is born!

As is sometimes the way of science, this nonlinear work was then ignored as beingof marginal interest. Indeed, the natural philosophers of the time seemed blissfullyunaware that our world is really rather a nonlinear one. Russell, on the other hand, wasconvinced of the importance of his wave, and its curiosity, right up to his death in 1882.But it took much later research on waves in crystal lattices, by Zabusky and Kruskal in1965, to make plain [8] the elegant generic nature of Russll’s observation to a wider public,and to emphasise that nonlinearity is important to the propagation of such waves; Zabuskyand Kruskal realized that the form of their equations was exactly like that of Kortweg andde Vries.

A really striking feature of the 1965 work, however, was in demonstrating thatsolitary waves retain their shape (this time in a crystal), even after colliding with each other.It is fascinating that Russell had quietly observed this 130 years earlier. The 1965 authorsinvented the name soliton for these waves, just to emphasize that although a soliton is asolitary wave it retains its identity, even after a collision. The change of ‘ary’ to ‘on’ is notsurprising, incidentally - physicists call things ‘ons’ at almost every opportunity (electron,meson and photon, for instance) because the Greek word ‘on’ means solitary, and in eachinstance the ‘on’ suffix signals the principle that the entity concerned retains a characteristicparticle nature.

The fundamental characteristic, for us, is that a soliton can exist that retains itsshape. The fundamental shape of Russell’s soliton happens to be a sech2 function but thatit is not important, at the moment. The real point is that Zabusky and Kruskal ended up,also, with the Korteweg-de Vries equation, even though they were working on discretecrystal lattices. In other words, they got what Russell saw on shallow water. They gotwhat is now known as a Korteweg-de Vries soliton.

184

In broad, physical terms, nonlinear water waves can easily be appreciated, byglancing at figure 1, which is a sketch of what can happen to water waves approaching ashore. Under certain conditions, to be discussed, properly, later on, dispersion andnonlinearity will balance and then solitons on the water surface will appear. Otherwise, avariety of unbalanced conditions occur.

SOLITONS IN WATER

• Waves approaching a shore (beach)

• Wave speed has small ‘nonlinear’dependence on height socrest travel faster

• Water waves also exhibit dispersion- as the water behind it disperses‘rounding’ the edge

• In the right conditions dispersion& nonlinearity balance

Figure 1.sea.

Sketch illustrate some water wave behaviour. Note that solitons are readily generated on theto

Achieving this balance can be thought of in a crude sort of way but it is really quite asophisticated step by Nature, as will be shown later. In fact, it is a chirp balance that isoperating. Chirp, although it is a concept that arouses interest, immediately, will not bedefined, at this stage, however. It is better to introduce it later on.

Earlier on, in this chapter, the Russell observation was referred to as a solitary waveand the word soliton was, confidently, introduced. Some sort of qualification of thislanguage is now necessary, before passing on to greater detail. Basically, it is quite simple.If solitary waves pass through each other, experiencing nothing other than a phase shift,they are elevated to the status of solitons. Of course, linear beams and pulses passthrough each other without any change but this is to be expected because linear physicspermits simple superposition. The big surprise comes when nonlinear beams and pulses donot destroy each other when they interact. In the interaction region, nonlinear forcesoperate forbidding the process of simple superposition. Self-destruction is expected butsolitons survive this. Solitary waves, or solitons, also come in various forms, some ofwhich are shown in figure 2. Some of the most famous members of this large, extended,family are

• Korteweg-de Vries:first (and easily) observed on shallow water by Russell and has avelocity that is proportional to its amplitude

• Envelope: easily observed on deep water and is a solution of the nonlinearSchrödinger equation and has a velocity independent of amplitude.The most famous application is in optical fibres but they are the onesthat will be the centre of attention here for both pulses and beams

• Sine-Gordon: dislocations in solids are described by these solitons and are kinks oranti-kinks with velocities that are independent of amplitude.

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Solitary Waves or Solitons?

Figure 2. Illustrations of solitary waves and some useful comments.

Nonlinear equations are classed as integrable or non-integrable [2] and the integrable oneshave soliton solutions that are preserved during collisions. Non-integrable equations canhave, for certain parameters, solitary wave solutions but these are not always preservedduring collisions. The point about solitons is that they are among Nature’s generic entities,i.e. although their first observation was on shallow water they also turn up in optical fibres,in the same form as those on deep water, and should be looked for wherever dispersiveexcitations, or waves, can be created. In order to be selective, this chapter concentrates onbright envelope solitons, which come in the form of temporal pulses or spatially distributedbeams. These are not Russell’s solitons but another member of the family. Dark solitons(black holes!) are also possible but they will be introduced only briefly here. A typical,undamped, evolution of what is called a findamental bright envelope soliton is shown infigure 3 in which a pulse with arbitrary (normalised) intensity is selected. It varies withtime across its intensity distribution and the plot shows the shape of the pulse with distancedue to propagation. Figure 3, deliberately, for the benefit of this discussion, has no unitsattached to it. If a specific system is in mind, however, and that system is in the form, forexample, of a thin film, then the units, i.e. the scales will, of course, matter. This plot willapply, equally, to a beam. All that is necessary to make it applicable to a beam is to replacetime with distance across the beam.

Figure 3. The fundamental [lowest-order] bright envelope soliton, in the absence of damping.

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2. DISPERSION AND AND TEMPORAL SOLITONS

A temporal pulse is thought of, in a linear regime, as a superposition of waves withdifferent frequencies so that a spectrum of frequencies (its bandwidth) exists around ω0 .If this bandwidth is ∆ω, then the condition ∆ω << ω 0 is necessary, if the concept of

carrier frequency is to mean anything. In other words, although, in principle, a wavepacket [pulse] can be decomposed into Fourier components, their amplitudes are small[negligible] outside ∆ω . Hence, as shown in figure 4, ω(k) is represented rather well by aTaylor expansion about ω0 . In fact, figure 4 shows that the linear dispersion for a vacuumcontrasts rather nicely with the typical (ω,k) variation that can occur for a material. It is

the presence of a finite value of in the Taylor expansion that causes pulse

dispersion, i.e. spreading with local time as the evolution progresses. It is important toobserve the following points [10]

•

• the arrival times of the frequency components spread around ωz, are

0 , at a particular point

the group velocity, Vg (ω), depends upon the frequency

• for a bandwidth ∆ω, the spread in arrival times of all the ω values is equal to thepulse width ∆τ, where

Figure 4. Comparison of pulse behaviour in a dispersionless (vacuum) medium and a medium withmaterial dispersion.

In figure 4 local time is measured across the pulse in a frame of reference moving with a

speed equal to the material group velocity, vg, hence, local time is where t is

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real time. We will, in fact, make the transformation → t and use this new t as

a definition of local time, unless specified otherwise. It is always important to check in anyproblem whether the local time or the laboratory frame time is being used. This local timemodel is a standard and convenient representation. If a laboratory frame is used then allthat will happen to figure 4 is that the pulse cross-sections will be centred over a line drawnat an angle to the z-axis. This is, quite simply, inconvenient! Finally, the local time isplotted from left to right, taking t = 0 at the pulse centre.

Figure 5. Pulse spreading in the time-domain due to material dispersion in a linear medium. There is nota corresponding spreading in the frequency-domain.

Figure 5 proves, rather dramatically, that it is the existence of a finite value of

and hence that causes dispersion. In optics, is called the group-velocity

dispersion. This is so-called because Vg(ω) is a function of ω and this alone causesarrival time spreading of signals.

Dispersion is a material property so a pulse in a vacuum [11] can also be said to be asolitary wave also! While this is strictly correct we can choose any pulse shape to travel ina vacuum. Hence, the vacuum ‘solitary wave’ is not unique, in the sense that the stationarysolution of the KdV equation, or the nonlinear Schrödinger equation, is unique. Alsopulses propagating in a vacuum can pass through each other but no nonlinearity is involvedso it is pointless to consider them as solitons. To take this discussion further, Figure 6gives a few examples of what can happen to a temporal pulse, or spatial beam, because ofthe presence, or absence, of dispersion, or nonlinearity.

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F’igure 6. Solitary waves: to be or not to be! That is the question! [with apologies to William Shakespeare]

3 . CHIRPING AND BRIGHT PULSE (TEMPORAL) SOLITON FORMATION

At this stage of the discussion, it is useful to return to the concept of chirp. Thediscussion begins with figure 7, which summarises the idea of phase.

Figure 7. Summary of the concepts of phase and chirp parameter.

Suppose that the time (local, or otherwise) is t and that p = ωt. The definition that phase

is p = ω t, is a very familiar one. The definition that seems, at first sight, to be

unnecessary, for such a simple relationship. Why not just use p/t? Such a step leads,immediately, into error, if p is not as simple as ωt. The problem is that it is perfectlypossible for the frequency to deviate from the carrier frequency, as time is swept across apulse i.e. the instantaneous frequency can be lower (higher) than the centre (carrier)frequency as the front, or back, end of the pulse is reached. It is perfectly possible for asmooth change to be established through the existence of a so-called chirp parameter Cthat can be positive or negative. For interest, the historical background of the concept ofchirp is summarised in figure 8. In a Bell Laboratory report in 1951, B M Oliver [12],

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referring to work on frequency-modulated radar, introduced the use of chirping. The aimwas to produce compressed radar signals and a famous remark, attributed to him, is "notwith a bang but a chirp". The word chirp is, perhaps, more obviously attributable to birds,because they emit chirped (frequency-modulated) pulses as a matter of course. Torecognise that what birds, and bats [13] do, naturally, lies at the heart of our particularsoliton work can be said to be an example [12] of Pasteur’s view that " fortune favours theprepared mind". The original idea in the B M Oliver report is, first, to generate a squarepulse envelope, in contrast to a continuous wave, which is uninterrupted, and has afrequency ω0, for example. A rectangular or square pulse envelope simply ‘chops’ off thewave, front and rear, encapsulating a carrier, which has a frequency ω0, all the way acrossthe pulse. In other words, in the time-domain ω = ω 0 is not a function of time across thepulse i.e. it is unchirped. This situation can be changed by making the frequency vary withtime across the pulse, A linear 'frequency ramp' is shown in the second segment of figure 8and the frequency variation (chirping) that is established across the pulse is shown in thelast segment of figure 8.

Figure 8. Basic idea on chirped radar put forward, long ago, by Oliver.

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Unchirped input pulse:

same colour through the pulse

same proportion of frequency everywhere

Anomalous-Dispersion Regime: higherfrequency travels faster than lower frequency

leading edge moves towards the blue

trailing edge moves towards the red

Figure 9. Illustration of pulse broadening in a dispersive medium. Time increases from left to right; theenvelope is the bell-shape that encloses the pulse oscillations; it is the locus of the extreme values of theamplitude.

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Nonlinearity changes the refractive Index

results in a phase profile across the pulse

results in a frequency shift across the pulse

In presence of Anomalous-Dispersion

leading edge has concentration of lowerfrequencies (red): slows down

tail has concentration of higher frequencies(blue): speeds up

Figure 10. Illustration of nonlinear pulse propagation in a dispersive medium.

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Figure 11. Diagram designed to illustrate the competing roles of dispersion and nonlinearity in preservingthe shape of a fundamental (lowest-order) envelope temporal soliton. Time is measured left to right.

Since, in this section, the emphasis is on dispersion as a material characteristic, thediscussion of chirp is conveniently centred upon temporal pulses. If dispersion wasunimportant, however, the emphasis would switch to beams which invoke diffraction. It isdiffraction as a beam characteristic that leads, through the introduction of a spatialfrequency (a wavenumber), to an identical argument concerning spatial chirp. A simpleanalysis shows that spatial frequency chirps are introduced into the spatial beams as theyare transmitted. Having made this point clear about the spatial analogue, the argumentswill now return to the temporal case.

The optical fiber (fibre) is the usual transmission vehicle [14] for temporal envelopesolitons, but other waveguides can perform this task as well. It all depends on the state ofthe dispersion and the magnitude and type of the nonlinearity. More details will be givenlater but the ability to transmit a temporal soliton (usually at optical frequencies) depends

upon the sign of [the group velocity dispersion] and γ [the nonlinear coefficient].

γ has not been mentioned before but, as the pulse power increases, nonlinearity affects thewavenumber (or frequency)by shifting it in value by an amount proportional to the power.γ is simply the constant of proportionality. For the time being, all materials to be

considered here have γ > 0, so it is the sign of that counts, Indeed, it was

Lighthill who discovered that, for a carrier frequency is a necessary

condition for soliton existence. If the material is said to have normal

dispersion and if the material is said to have anomalous dispersion. Figures

9, 10 and 11 show the effect of dispersion and nonlinearity on pulse propagation, together

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with a diagram that shows how a balance is obtained. Figure 9 shows that in an unchirpedpulse, in a linear medium, the ‘proportion of frequency’ is the same everywhere in thepulse. In an anomalous dispersion regime, the higher frequencies move faster than theothers. Hence, considering the bandwidth of the pulse, arising from the envelope shape,higher frequencies will arrive first, causing a blue → red distribution i.e. chirping. In apurely nonlinear medium the opposite occurs. If there is nonlinearity and dispersion,chirping, because of the nonlinearity, causes a red → blue distribution, at the beginning ofthe passage of a pulse, so the tail of the pulse speeds up. In an unbalanced state, therefore,a nonlinear pulse in a dispersive medium could compress if the power is high enough.Figure 11 shows how a balance can be interpreted by reading ‘frequency effects’upwards/downwards and ‘time effects’ forwards/backwards. The figure can be used in thefollowing way. Chirping arises from both nonlinearity and dispersion and the balanceachieved in the fundamental soliton keeps the frequency distribution (proportion offrequency) evenly distributed across the pulse. Suppose that anomalous dispersion disturbsthe frequency distribution. In an anomalous dispersive medium, high frequencies (blue)travel faster [14] and so arrive earlier. This is shown in the sketch as an arrow pointing toearlier times. Nonlinearity, on the other hand, lowers lead frequencies and this is shown bythe downward frequency arrow at the lead end of the pulse. Once lowered, thesefrequency components will travel slower and arrive later, as required by an anomalousdispersive medium, Upon being redistributed to the tail of the pulse the nonlinearity thenproceeds to increase the frequencies, once again. The bottom line is that going around thisloop results in the balanced state that is the envelope soliton.

Both material dispersion and power level (nonlinearity) add a chirp to an evolvingpulse without external intervention, In the original radar example, the technique was usedwith a delay line to compress pulses i.e. because different frequencies have differentpropagation times through the delay line, delaying the first arrivals causes ‘a traffic jam’and, hence, a compression.

As an illustration of the behaviour of pulses in a material, figures 12-14 containnumerical simulations that show in detail what happens as a pulse evolves.

Dispersion of a Sech Inputin a Linear Medium

Figure 12. Dispersion of a pulse in a linear medium, The figures refer to beams if time is replaced with a

spatial coordinate.

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Figure 12 shows that, in the absence of any nonlinearity, i.e. no power dependence of thefrequencies, a pulse will disperse. As can be seen the phase p, which is a constant acrossthe pulse at the beginning of the propagation develops a local time dependence as the

evolution proceeds. The derivative where t is the local time, is the chirp and across

the centre of the pulse a very clear linear chirp develops. The slope is negative so the typeof dispersion selected is called anomalous and the chirp parameter is said to be negative. Ifthe medium could be purely nonlinear then a significant power level is possible, withoutdispersion! Of course, this cannot ever be completely true [dispersion is usually present]but it is, nevertheless, more or less possible, especially at the start of a propagation. It isillustrated in figure 13 to make an important point.

Behaviour of Sech Input InA Purely Nonlinear Medium

Figure 13. Behaviour of a pulse, in the time-domain, in a purely nonlinear medium. Replacing time withspace makes the figures appropriate to beams.

This point is that the action of nonlinearity alone does not compress the pulse in the time-domain. Indeed, in the time-domain the pulse progresses without change of shape. Thephase behaviour, on the other hand, is very interesting. A positive chirp develops as theevolution progresses, What then happens in a nonlinear dispersive medium? The answer isthat it is possible for the chirps from dispersion and nonlinearity to cancel, precisely. Whenthis happens, a sech-shaped eigensolution of the system is formed and it this which we willrefer to as an envelope soliton. The solitons are nonlinear states of the system and canexist in various forms, dependent upon the energy available and it is the lowest-order onethat is shown in figure 14. Clearly the chirps cannot cancel if hardly any energy enters thematerial so it is to be expected that some power threshold, for a sech(t) pulse, should bereached before a chirp cancellation, and hence a soliton, will appear. That this is, indeed,the case is shown in figure 15, where a 0.2 sech(t) input disperses, completely, and a 0.8sech (t) input does not. The 0.8 sech(t) input sheds (disperses) energy it does not requirebut, because it is above the threshold energy for soliton formation, then goes on to form a

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stable, lowest-order soliton. Another important point is that, if the threshold energy level isexceeded, the precise input shape does not matter. A dramatic example is given in figure16 in which a rectangular input pulse, above the threshold energy, "gives birth to a soliton".

Figure 14. How the lowest-order, envelope, soliton arises.

Figure 15. A numerical check on the power needed to create the lowest-order, envelope soliton. Note thatlocal time need only be replaced by distance for these figures to relate to beams.

It is interesting that the emerging bright envelope soliton has the sech shape it is requiredto have, as a solution of what we have been calling the nonlinear Schrödinger equation.

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This equation will be shown below, to govern the behaviour of pulses and beams. It is anequation that has rather special properties because an Nsech(t) input generates afundamental soliton if N = 1 and higher-order solitons if N > 1. Technically, fundamentalsolitons are of great interest because they do not change their shape as they propagate i.e.they remain sech-shaped. Higher-order solitons N = 2, N = 3,… change shape [theybreathe but, fortunately, so does the chirp!] as they propagate but keep returning toNsech(t), periodically. The period is known as the soliton period and is a useful lengthscale of the system. Apart from that, higher-order solitons are not expected to be useful ina switching device or communication system.

.Figure 16. A Square input pulse gives birth to an envelope soliton The transverse direction is time (pulse)or distance (beam).

This is because a higher-order, bright, envelope soliton is a bound state of N =1, sech-shaped, fundamental solitons, with zero binding energy, so it is extremely vulnerable toperturbations. Hence an N = 2 soliton, for example, could degenerate quickly to twoN = 1 solitons. This means that N is the soliton content or soliton number for a giveninput pulse. Figure 17 contains a diagramatic explanation of what happens when either aBsech(t) input pulse is used for a nonlinear system described by the nonlinear Schrödingerequation, or a rectangular input pulse B rect(t). The soliton "content" is shown as afunction of B. For example, if a pulse Bsech(t) is entered into a dispersive material then,provided 0.5 < B < 1.5, a fundamental soliton is created. On the other hand, if B rect(t)is entered then a fundamental envelope soliton pulse is created only if 0.5π < B < 1.5π.These conclusions come from an exact mathematical treatment of the nonlinearSchrödinger equation, using inverse-scattering theory [15] [IST] that is to be discussedlater on. The conclusions are easily confirmed out by numerical simulations.

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Figure 17(a). How solitons are created from Bsech(t) input pulses or Brect(t) input pulses.

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Figure 18 Simple idea of diffraction.

4. DIFFRACTION, SELF-FOCUSING AND BRIGHT SPATIAL SOLITONFORMATION

We are well aware that a beam of light, even if it is emerging from the now familiarlaser pointer used by lecturers, will spread out and change shape as it propagates (see

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figure 18). In air, this is all that will happen. But what about light propagation in nonlinearmedia? In this case light always heads for the region of highest refractive index (think ofhow a lens works), and a nonlinear material, such as a quantum well AlGaAs structure,forces the index up, precisely, and only, where the beam is travelling. This is just what isneeded to stop light rays from inside the beam from bending outwards and causing achange in beam shape. When beam spreading is exactly stopped by the nonlinearity, aspatial soliton [16] is formed and the beam can now be used as a stable optical guide, justlike the familiar optical fibre.

The spatial soliton is, however, better that the fixed, ‘hard’ waveguide such a fibreconstitutes. The reason is that it is a ‘soft’ guide that is easily tunable by changing thebeam size. Many applications can be imagined for these novel waveguides, such as usingthem for optical wiring, to steer one beam with another in addressable arrays, and to act asswitching devices and as logic units for a whole range of information processing andcomputing.

Spatial solitons [16,17,18-30] are a significant outcome of modern soliton andmaterials science because of the ease with which they can be manipulated. Although thereare still worries about how fast many materials can react to light, or changes of beamdirection, work on spatial solitons should herald an era of new, all-optical processingdevices that are cheap and easy to implement. Russell’s observation did not change physicsor technology in his own century nor most of our own. But with the real possibility ofoptical soliton communication systems and all-optical processors now on the horizon, itmay well transform them next century.

It is many years now since the basic physics, concerning the self-trapping [30-32],of a powerful beam of light in a nonlinear medium was thoroughly discussed. Some of theimagery evoked in those early, pioneering, discussions will be produced here, through theconsideration of a beam of light with a rectangular distribution of energy, a radius r and awavelength λ, propagating through a dielectric, non-magnetic, medium that has a linearrefractive index n0. First, as stated above, the beam will want to spread out, due todiffraction. Indeed, at a given radius r the light (laser) beam can be imagined to act as itsown aperture and rays of light will diffract through it, with an approximate diffraction angleθd = λ/2nr. If the material becomes nonlinear, in a Kerr-like manner, [i.e. third-order only,

refractive index of the linear medium will change, by the small amount α|E|2 , where E isthe electric field carried by the beam. If the medium has a positive α then the refractive

the tendency of the beam to self-diffract with a nonlinear tendency to self-focus. In thesefigures can be seen a simplistic, use of diffraction through a circular aperture of radius r,but the example is designed to illustrate what nonlinearity can do [30,31] to achievediffraction-free beams.

In figure 19, rays setting out from the beam axis will either escape, or be totallyinternally reflected. This approximate model uses an aperture to create a beam of radius r.Since diffraction does not permit a geometrical transmission, rays making an angle θc withthe axis are totally internally reflected at the beam boundary but other rays can escape thegeometrically defined boundary. In principle, therefore, beam broadening occurs. Becausethe electric field E of the beam causes a nonlinear change in refractive index, the criticalangle depends upon E and, hence, upon power. As figure 20 shows, most of the beampower is enclosed [30-32] within ray angles making an

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with nonlinearity proportional to intensity], with a nonlinear coefficient α , then the

index increases to n + (α|E|2) and, as shown in figures 19 and 20, it is possible to balance

Figure 19 Showing that the critical angle depends upon electric field or power.

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Figure 20 Illustration of circular hole/single slit diffraction plus introduction of the concept of criticalpower.

angle to the beam axis. Figure 19 is not completely true (it lacks rigour) [33]

but we ought to be able to learn quite a lot from it! The main idea being introduced is thatif the aperture is circular then 2θd is the direction of the first minimum in the (far field)Fraunhofer diffraction pattern. This means that the bulk of the energy in a diffracting beamis, more or less, associated with rays making an angle of θd, or less, to the horizontal axis.For a single slit the first minimum in the pattern should be at θd but, of course, theintensity will not be zero on the scale of figure 20 because the distance between the slit andthe screen is not infinite, as it should be for (Fraunhöfer diffraction: far field), when thesource is at infinity. For figure 20, the observation plane is rather close to the circular hole,or slit, so the actual geometrical shadow if it would also be wider than that which is shown.The greater the distance between hole/slit and screen the more the Fresnel (: near field)pattern goes over to a Fraunhöfer pattern. These points need not worry us in this kind of

what we call a ‘hand waving’ or ‘ball park’ exercise, so we will set

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(approximately) and use it to get an estimate of some distance [30] LD at which a beamcan double in width, due to diffraction. Since this is the case, then θc = θ d is the criticalcondition for the beam to have its diffraction killed by the nonlinearity. This leads to acritical power Pc , which, rather importantly, does not depend upon r. Another way tolook at this problem is to introducc the concept of nonlinear length LNL, as demonstratedin figure 21. At balance, the diffraction balances the self-focusing and diffraction-freebeams are the result. Such diffraction-free beams are called self-trapped and are spatialsolitons. Troublesome questions now arise about Pc. Can it be controlled to producestable spatial solitons?

Figure 21. Introducing diffraction length (LD) and nonlinear length (LN) .

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The important feature is that, if P > P cr, rays are forced towards the axis of thebeam, in a self-focusing action. If P < Pc r, then diffraction takes over. These propertiesof the medium occur because Pcr is fixed, i.e. ε0, n, c, α and λ are fixed parameters ofthe system and it is necessary for any launched input power P to be precisely equal to Pcr

for stability to occur [21,23,30,32]. Inevitably, in any real experiment the power launchedinto a waveguide will be either P < Pcr or P > P cr, even if the difference is onlyinfinitesimal. In either case, such a fluctuation will lead to instability taking the form of a'runaway' to diffraction or to self-focusing. This, then, is the problem with self-trappedbeams in which diffraction and self-focusing is finely balanced in a bulk (infinite with noboundaries) medium. Mathematically, it means that such beams are, by their nature,unstable in a multi-dimensional system and this has been established, for a long time, in theliterature.

The situation described above, although interesting, in principle, at first sight appearto have very little experimental promise because of its instability. Some forms ofstabilisation have been proposed, however, that are proving to be rather successful. Thefirst stabilisation method involves controlling the perturbations that can lead to instabilityby a method based upon a modulation due to interference fringes [34]. This is, relatively,complicated so it will not be discussed here because the aim of this chapter is to studystraightforward spatial solitons, which can be created within [16] a nonlinear planarwaveguide. This procedure reduces the propagation to a one-dimensional diffraction.Given the propagation distance as the other dimension this is then called (1+1)propagation.

The instability, referred to above, that arises in a three-dimensional medium has aPc r that is fixed, in the sense that self-adjustment of the beam such as its radius, whilefocusing, cannot change it. On the other hand, if a beam is launched into a planarwaveguide that has interfaces parallel to the x-z plane, then guiding confinement will occurin the y-direction. Suppose then that the guide is weakly nonlinear, so that the modal fieldintensity is unaffected by nonlinearity, to first order. In this case, the power needed fordiffraction and self-focusing to balance becomes [21]

r changes, due to attempts to self-diffract or self-focus, the system is always forced backto stability. This is the elegant feature of a waveguide being able to induce (1+1)propagation and created some excitement in the field [16] when it was announced.Naturally, since open waveguides, made of transparent dielectric materials, are used, realbeams carry some part of their energy outside the waveguide. This is because tangential

where 2d is the thickness of a guiding layer with linear refractive index n0 . r is now thelargest beam dimension and πrd is, approximately, the area of a beam with most of itsenergy confined within an elliptic cross-section, and within the guide. This is shown inFigure 22, which also contains a schematic explanation of the stability. This new value of

Pcr does not now arise from the cancellation of r2 but has, left within it, a factor A s

field components are continuous at

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Figure 22 Stable beam propagation in a planar waveguide .Launched powers P < Pcr or P > Pc r returntowards P = Pc r. Diffraction/self-focusing balance in the x-direction; linear guiding in y-direction;propagation in z-direction.

the boundaries but, as shown in figure 23, most of the energy is carried, inside thewaveguide, as sketched in figure 22. Hence the 'balancing' arguments are correct, even forrealistic cases.

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Figure 23 Contour plot of the intensity distribution of a stabilised beam in a planar waveguide. Note thatthe contours are further apart as the field weakens into the cladding regions.

In figures 24 and 25 an outline of what self-focusing means, physically, is given. Basically,if an unbounded crystal is used and beam of power P is entered into it with P > Pc , orP >> P then phenomenon of self-focusing occurs. This means that because the power ofc

the beam is more than is needed to achieve the balance power (Pc) needed to killdiffraction, the beam will continue to focus. In figure 24 pure self-focusing is convenientlyillustrated in terms of an equivalent focusing [30,32] lens of focal length LN. In fact, whileLN is (equivalently) only associated with a focusing lens, diffraction is associated with adefocusing lens of focal length LD. The combined effect is obtained in this, geometrical,ray picture by obtaining a combined focal length, in the usual way. Of course, self-trapping(spatial soliton creation) is always going to be possible so the lens model is only useful uptoa point. According to figure 24 as P → Pc , LND → ∞, yet a self-trapped filament can beformed. A thorough investigation of focusing region is needed, therefore, but this is farbeyond the scope of this article. It has been addressed, however, and some answers can befound in the literature that abounds on this topic [32]. Figure 25, sketches out the casewhen beams go supercritical and hints that several filaments (self-trapped) beams can becreated, each with P = Pc. The figure finishes with a final sketch of a spatial solitonlabelled as a self-trapped beam.

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Figure 24 An illustration of self-focusing. Only the equivalent positive (focusing) lens is shown, todemonstrate the dramatic consequences of self-focusing. The focusing region is not accounted for by thissimple theory. Self-trapped filaments can form, for example, rather than the beam becoming smaller andsmaller.

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Figure 25 Paraxial geometrical optics will not accurately describe the focusing region. Many filaments(spatial solitons) could be created, for example.

5. GENERATING THE NONLINEAR SCHRÖDINGER EQUATION [NLS]

The nonlinear Schrödinger equation [NLS] is a familiar sight in the literature andone of the preferred generic forms is

(5.1)

[the literature often shows ½ multiplying the second term but a simple re-scaling of Zplaces the 2, as in (5. l)], where U is an envelope function that is slowly varying, withrespect to Z. This envelope function has been seen pictorially in the previous sections andis literally the “envelope” that “holds”, or “encloses” the rapidly oscillating wave. In theform (5.1) the equation has been stripped of dimensions and any particular physicalapplication. As shown in figure 25, it is entirely generic in character.

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Figure 26 pulse/beam solitons are solutions of the same equation.

Even though it is generic in form, equation (5.1) has preferred forms, as outlined in figure27, so that applications to beams of pulses can be emphasised.

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Figure 27. Preffered form of the NLS.

Having stated what the NLS looks like, how does the equation arise? There are severalanswers to this question to be found in the literature, all leading to the generic equation. Atone extreme, a complicated first principles calculation can be undertaken. At the other, asimple, straightforward, examination of the dispersive nature of the system also reveals thenonlinear Schrödinger equation. It is this latter approach that will be adopted here.Suppose that the dispersion equation is written in terms of kz, the z-component of thewavenumber in the manner shown in figure 28.

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Figure 28 Summary of expansion of dispersion equation about an operating point ( ω0, k0 ). Thisprocedure is like Taylor expanding the characteristics of an electronic device about an operating (bias)point.

Dispersion in a system simply means that the wavenumber (k) has a frequency (ω )dependence that is not a straight line. This means that phase velocity vp = ω/k and group

velocity varies, according to which point on the dispersion curve is selected.

Temporal pulses and spatial beams are modulations of carrier waves [9,14]. If onlytemporal pulses are being considered, then it is clear that the act of producing an envelopeintroduces a frequency spread ∆ ω about the carrier frequency ω0. Provided that weconsider only a small bandwidth around ω all frequency components in a linear system0,lie fairly close to the centre (carrier) frequency ω0 . If, in addition, there is a small spread ofwavenumbers about k then the wavenumber of the complete beam-pulse-like disturbance0can be Taylor expanded [9,15,35] around (ω0,k0) and the series can be safely truncated atthe second-order or third-order terms. The number of terms to be kept will depend uponthe application we have in mind. Figure 28 shows propagation down the z-axis of such a

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pulse/beam envelope, associated with some envelope function u(x,z,t) that varies muchmore slowly than the rapidly varying exp i[ω t-k z], as the propagation proceeds. Note0 0also that y is ‘frozen’ so no variation with y is allowed. Because of the slowly varying

assumption only will eventually appear in the NLS, amounting to a neglect of

In the expansion given in figure 28, since the system is confined in the y-direction, i.e.guiding along z occurs and the x- and z-directions are each allowed to reach ± ∞. Theguiding confinement means that the wavenumber being used here is already the guidedwavenumber. The expansion looks at the deviation, from the guided wavenumber kz0,because of the introduction of wavenumber and frequency bandwidths [∆kx , ∆k ] and ∆ω,zrespectively. Since the whole point of the exercise is to look at beam and pulsepropagation in a nonlinear medium, the deviation, away from k , originating from thez0

power is also added in, intuitively, as a term |u|² term, where u is the

(complex) envelope amplitude. Because both spatial and temporal effects are included, thefinal envelope equation accounts, simultaneously, for diffraction and dispersion. Somedetails will now be given that may help with the derivation. These are the relationships

• k x , ω, |u| ² are to be treated as independent variables

( 5 . 2 a )

• k is a dependent variable; k depends upon ω and |u|²z

( 5 . 2 b )

( 5 . 2 c )

[kx , ω are independent variables in this theory]

( 5 . 2 d )

( 5 . 2 e )

[k x |u|² are independent variables]

( 5 . 2 f )

At the operating point kz = k z0 = k, kx = 0, ω = ω0 so that

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(5.3a)

(5.3b)

Figures 29, 30 and 31 show further developments that lead to the familiar generic equation.Figure 29 contains the fundamental step that ∆ ω , ∆k x and ∆kz are in the space-time

domain, and exist as operators on u(x,z,t). In other words, the NLS is

the application of the inversion to the time-space domain of the operator

to u(x,z,t).

Figure 29(a) Using the operator derived in figure 27 upon u to get the NLS. The result of leaving out

diffraction is given.

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Figure 29(b) NLS appropriate to diffraction cases.

Figure 30 Illustration of length scales in dispersion and diffraction limited propagation in a nonlinearmedium.

214

Figure 31 How to reach the generic form of the NLS

Bright (B) or dark (D) solitons can be generated from the temporal, or spatial, equations byconsidering the sign of γ and/or the sign of β2. This is the application of the famousLighthill criterion, which tests whether β2γ < 0, or β2γ > 0. Diffraction always has thesame sign and so is always there, even if the material is not; hence a laser beam in a vacuumwill always spread out because only material-induced self-focusing can stop it.Nevertheless, in a self-defocusing material dark (D) soliton beams are possible. Thesituation is summarised in figure 32.

Dark solitons will not be discussed any more here, except to point out that they are,strictly speaking, ‘holes’, or ‘absence of light’ in a continuous infinitely extendingbackground. In practice, finite backgrounds can be used such as a ps hole ‘dug out’ froman ns pulse. These ‘holes’ propagate like the bright pulses or beams. A dark soliton has atanh cross-section.

For spatial solitons the preferred notation is

(5.4)

and the general (fundamental) solution for this equation is displayed and demonstrated infigure 33.

215

216

Figure 32 Summary of bright/dark conditions.

Figure 33(a) First steps in checking the general, fundmental solution of the NLS.

Check

Figure 33(b) Final check on the fundamental (first-order) solution of the NLS, formulated for a spatialsoliton beam.

In this beam case, ξ is interpreted as the angle at which the beam centre propagates,relative to the z-axis, and θ0 is just an arbitrary fixed phase that can be set to zero, or anyother value, without any loss of generality.

There are other options, and interpretations, depending on the application in mindand figure 33 contains an appropriate form for beams. For pulses, it is common to use thefollowing notation and form

(5. 5)

in which , even though it is dimensionless, t is regarded as the local time, the ½ is in its‘traditional position’.

The general solution to equation (5.5) can be written down like the general solutionto (5.4) except for a different interpretation. The first point to make is that equation (5.5)describes the motion of a pulse, relative to a frame of reference moving with a speed equalto the group velocity vg [see figure 31 for the relevant transformation]. The question ofwhether a laboratory frame, or a moving frame, is used is really relevant to the temporal(pulse) case. Figure 34 illustrates the frames of reference and emphasises that in themoving frame the measurement of time is local to the pulse i.e. it is not absolute, or global,

time. Hence, we tend to put the pulse centre at and let T range from

217

negative to positive values as the pulse is transversed in time. In the vast literature on thistopic, the symbol t is usually used, as shown in equation (5.5), but it is really the local T,as emphasised in figure 34. The general solution is also shown in figure 34 and relatesimmediately to that shown in figure 33. If x0 = 0 in figure 33, then AΩ z of figure 34 isto be compared to ξz in figure 33. For spatial solitons ξ is the angle the beam axismakes to the z-axis. For temporal solitons Ω is a frequency. A little consideration willshow, noting the change in the position of the ½ and the 2 in equations (5.5) and (5.4), thatthe solutions given in figures 33 and 34 are identical.

Figure 34 Frames of reference.

218

In dimensionless units [36], writing u = Asech where q = AΩZ ,

where A is the amplitude and now Ω is a shift in

reciprocal dimensionless velocity. The phase displacement is is, in

fact, a frequency and the soliton velocity, relative to the already moving frame isSo when Ω is zero, the soliton is at rest in the frame moving at the

group velocity. This ½, versus 2, position accounts for the z being measured as z → 2 zin the spatial beam formula and leads to [note the plus sign etc] as opposed to

[note the minus sign etc]. To recap, the interpretation is (1) ξ is the angle of

beam propagation, where AΩ is the relative soliton pulse displacement (2) Ω is afrequency. In this formulation, for pulses, 2A is the mean soliton energy, Ω is a meanfrequency and the mean time is q/A. Hence, [A, Ω, q, φ] characterise the temporal solitonat any point during its propagation. It cannot be overemphasised that, since we are dealingwith the general formula it is important we are already in the moving frame of the groupvelocity, so quantities like Ω (a frequency displacement) are in addition to this anddescribe a soliton moving relative to this frame. In the spatial soliton case, ξ ≠ 0 meansthat the beam is pointing in a certain direction and that is easy to understand. For pulses,however, usually something has to be happening to the pulse environment for pulses todrift relative to group velocity frame. Such events may be the presence of amplifiers,causing perturbations, or self-frequency shifts acquired when pulses get very short.

6. INVERSE SCATTERING TRANSFORM (IST) METHOD

The generic equation for pulse and beam envelope soliton propagation is

(6.1)

Although Korteweg and de Vries managed to solve their particular nonlinear equation,attempts to formalise exact solutions immediately encountered the problem that the usualFourier transform method does not work on such an equation. This is because of thepresence of the nonlinear term 2|U|2U. For most of this century this mathematicaldifficulty remained unresolved. Thirty years ago, however, Gardner, Greene, Kruskal andMuira [37] invented what is known as the inverse scattering method (IST) to extractsoliton solutions from the Korteweg-de Vries shallow water problem. A beautifulextension by Lax to a wide class of nonlinear problems soon followed. The honours,however, go to Zakharov and Shabat for their discovery of how to deal with (6.1), so it iswidely known as the Zakharov and Shabat problem [38].

At first sight, the IST method appears to be very abstract, especially to physicists.It even seems mysterious but, once mastered, the technique is very appealing and reallydoes have a lot of practical uses. The idea at the heart of the method is to break up thenonlinear equation (6.1) into two ordinary linear differential equations that look like ascattering problem so dear to the hearts of physicists. This move takes us into familiarquantum mechanics territory, in so far as we have become accustomed to generating botheigenvalues and eigenfunctions from some kind of potential function that is acting as ascattering agent. Furthermore, if the number of eigenvalues reveals just how many solitonsare present in an evolving beam, or pulse, this would be a very strong bonus, indeed. If

219

these same eigenvalues told us the amplitude and velocity of the solitons present, then thiswould be even better. Finally, if the scattering potential function is actually the initialvalue, U(0,S) then this would be an excellent outcome! It turns out that this is preciselythe case. To achieve such a remarkable mathematical coup was always going to be adifficult task and, indeed, when the first case applied to the Korteweg-de Vries (KdV)equation, was presented, it was regarded as possibly just luck! Zakharov and Shabatchanged that view and now the IST is applicable to quite a broad class of evolutionequations. Possibly more than 100 equations.

The main problem [2,9,39] is to find linear operators that permit the creation of anauxiliary eigenvalue problem to replace the nonlinear partial differential equation. There isstill a strong measure of intuition needed here, even today. The auxiliary problem uses thepulse or beam shape U(Z = 0,S) as a potential. For this choice of linear operator theeigenvalue must remain fixed as U(Z = 0,S) evolves to U(Z,S), provided that U(Z,S)satisfies (6.1). In fact, given U(Z = 0,S), the eigenvalue problem becomes an ordinary(direct) scattering problem so the evolution of the eigenfunctions, with Z, becomes trivial.Given the eigenfunctions at Z ≠ 0 the function U(Z,S) can be found by inverting theproblem; just like taking an inverse Fourier transform, only more complicated. It isinteresting that Gelfand, Levitan and Marchenko [2,9,39] had provided a method for doingsuch as inverse, quite a few years before it was needed for soliton problems. As will beseen below, we will not need to go all the way to the inverse because even at theeigenvalue equation stage, very powerful quantitative statements about the nonlinearsystem can be extracted.

Having stated in words what the idea is, let us now be more specific. Let us beclear, first of all, that Fourier analysis does not apply to a nonlinear problem but it isimportant to appreciate how beautifully simple Fourier analysis is in the linear case.Suppose that the nonlinear term 2|U| 2 U is absent, then (6.1) becomes

For a plane wavedispersion relation is k = ω 2 and, from the Fourier transform,

where k is wavenumber and ω is angular frequency, the

(6.2)

(6.3)

(6.4)

where (0,ω) is the Fourier transform of the INITIAL (input) function at Z = 0. Herethen is a major clue to generalising the well-known Fourier method to nonlinear problems,because the evolution of U(0,S) to U(Z,S) is trivially asserted by equation (6.3). Inother words, given U(0,S) (0,ω) can easily be found and once (0,ω ) is well-known(6.3) evolves the information to yield U(Z,S). In practice, (6.3) is, of course, the familiarINVERSE TRANSFORM i.e. in mathematical language, the inverse mapping of (0,ω)onto U(Z,S). This little example is very helpful, however, in enabling the task to be seen.If only we could use just U(0,S), coupled to a simple evolution like (6.3) then even if theinverse is more complicated than (6.4) the nonlinear task would be complete. How is itpossible then to proceed?

The first point is that using a traditional Fourier decomposition is out, but if thenonlinear differential equation could be reduced to linear calculations, using only

220

U(Z = 0,S) then the calculation could proceed. The result of such nonl ineardecomposition will be, as stated earlier, to make the problem like a traditional, linear,scattering problem of the type that appears in undergraduate quantum mechanics texts.Indeed, it is well known that if we could collect enough scattering data, such as reflectionand transmission coefficients for any wave scattering off a potential well or bump then areconstruction of what is actually doing the scattering is possible, in principle. This isworking backwards [INVERSELY] from the scattered field to a knowledge of thescattering object. It is like hearing the sounds from a distant drum and using these soundsto reconstruct the shape of the drum. In our case the “drum”, i.e. the potential, is theinitial value U(0,S). From U(0,S) we ought to be able to construct U(Z,S) by some formof inversion. We will not perform the inversion, however, because it turns out that theinitial break down into linear equations yields a pair of equations that permits a verypowerful and straightforward assessment of soliton presence. The problem, then, is eveneasier, in principle, than it looked at first sight. Unfortunately, breaking down equationslike (6.1), in the first place, is not easy and required a burst of inspiration from Zakharovand Shabat, through a consideration of the following. Zakharov and Shabat recast

(6.5)

equation (6.1) by introducing operators and , with the properties

where and are linear operators. Zakharov and Shabat showed that

where 0 < p < 1. At this stage, things do not look too bad! Furthermore

(6. 6)

(6.7)

where ψ1, ψ2 are eigenfunctions and ξ is an eigenvalue. In other words (6.7) is solvedjust like a scattering problem in quantum mechanics but note that ξ remains constant, withrespect to Z, i.e. if the eigenvalue can be found for U(Z = 0,S) it remains the same for allU(Z,S).

Equations (6.7) yield the linear coupled equations [37]

(6.8a)

(6.8b)

where Zi is the initial value Z = Zi , and the eigenvalue is, in general, complex.The beautiful thing about ξ, however, is that (for pulses) its real part ξr gives the

velocity of the soliton, relative to the frame of reference moving at the group velocity andξi gives the amplitude i.e. [40]

221

The interpretation of (6.8a, 6.8b) is as follows. U(Z,S) is the starting (initial ) value of the

(6.13)

(6.14)

According to equations (6.10), and writing Λ = -iξ, the solutions ψ1, ψ2 are

(6.12)

where k2 = N2 - Λ2 and g is some constant.Note that U(0,S) = 0 for S < - ½ and S > ½ and that equations (6.10) reveal that

either ψ 1 or ψ 2 is zero in these regions. In other words, to get functions that do notdiverge [‘blow up’], ψ ≠ 0, ψ2 = 0 in the region S < - ½ and ψ 1 = 0, ψ 2 ≠ 0 in the

1

unity, the coefficient of ψ2 is equal to some constant value, which we call g. In theregion - ½ < S < ½, ψ 1 ≠ 0, ψ 2 ≠ 0 but ψ 1 = 0, precisely, at S = ½ and ψ 2 = 0,precisely, at S = ½, to avoid divergences.

(6.9)

nonlinear solution of (6.1) before any evolution to further points along Z is achieved. Weusually set out at Z = 0, so that

(6.10a)

(6.10b)

An example, which will serve to illustrate the power of equations (6.10) will now be given.It shows how to determine the ‘soliton content’ of an input pulse or beam that has arectangular shape i.e.

(6.11)

1

region S > ½. Because the coefficient of ψ in the region S < - ½ is normalised to

222

Figure 35 Dispersion relation for scattering off the rectangular distribution (6.11), using the coupledlinear equations (6.8).

The boundary conditions at S ± ½ give

From equations (6.15a) and (6.15c)

(6.15a)

(6.15b)

(6.15c)

(6.16)

223

and substitution of equation (6.16) into (6.15b) gives the dispersion equation

(6.17)

and

Equation (6.17) can be solved graphically as shown in figure 35. The ψ1 , ψ 2

solutions for the three eigenstates, predicted by assuming that are shown in

figure 36.

Figure 36 Eigenstates for N = 2π (1.7)

224

Figure 37 Deducing the soliton content of Nrect(S)

The crossing points on the Λ = 0 axis of ( Λ ,k) plots are found from

and these occur at

(6.18)

(6.19)

On the Λ = 0 axis, however, k = N so equation (6.18) tells us the critical values of Nneeded to get n = 1, n = 2, n = 3,.… crossing points of the curves i.e. it reveals thevalues of N needed to generate 1,2,3,… solitons from a given Nrect(S) input condition.Figure 37 illustrates this answer for two crossing points and sets out the soliton existenceconditions.

The procedure described above permits the determination of whether a given inputwill lead to solitons, as the input evolves, during propagation. It also determines how manysolitons lie “buried” in the input condition. The eigenvalue equations (6.10), on workingbackwards yield only the unmodified nonlinear Schrödinger equation yet this can still beused to determine soliton content [40,41] when extra terms, needed to account for

225

damping for instance need to be added to modify the original nonlinear Schrödingerequation. All we need to do is to use the exact numerically determined solution of themodified NLS at any point Z = Z1 and use that in place of the function U(0,S) inequations (6.10). The value of the eigenvalues ξ will then reveal, given the startingfunction U(Z1,S), a certain soliton content at that point of the evolution. What this meansis that if any point during the evolution is used as an input to an unmodified nonlinearSchrödinger equation then the ξ = ξ r + ξi values show how many solitons could emergeand what they will be like in terms of amplitude and velocity. In other words, at everypoint along the propagation direction the soliton content of a pulse, or beam, can easily befound. Indeed, it is rather easy to program (6.10) with modern mathcad software. Plots ofξ, as a function of Z, show just when the system is capable of supporting solitons and aprecise way of finding out when the pulse or beam finally becomes devoid of solitoncontent. The power to determine this comes from the direct scattering equations (6.10).There is no need to worry about taking the inverse. A great deal can be learned about thesystem at this crucial, eigenvalue, stage. The algorithm is illustrated in figure 38.

Figure 38 How to deduce if an evolving pulse has any soliton content.

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1.2.

3.4.5.

6.7.

8.

9.10.

11.

12.

13.14.15.

16.

17.

18.

19.

20.

21.

22.

23.

REFERENCES

J.S. Russell, Report on waves, Proc. Roy. Soc. Edinburgh, 319-320, (1844).P.G. Drazin and R.S. Johnson, Solitons: an introduction, Cambridge University Press,Cambridge (1983).G.B. Airy, Tides and Waves, Encyc. Metrop., Fellowes, London (1845).G.G. Stokes, On the theory of oscillatory waves, Camb. Trans. 8, 441-473, (1847).J. Boussinesq, Théorie de l’intumescence liquid appelée onde solitaire ou detranslation, ce propageant dans un canal rectangulaire, Comptes Rendus Acad Sci(Paris), 72, 755-778, (1871).Lord Rayleigh, On waves, Phil. Mag. 1, 257-279, (1876).D.J. Korteweg and G. de Vries, On the change of form of long waves advancing in arectangular canal and on a new type of long stationary waves, Phil. Mag. 39, 422-443,(1895).N. Zubusky, and M. Kruskal, Interaction of solitons in a collisionless plasma and therecurrence of initial states, Phys Rev Lett 15, 240-243, (1965).M. Remoissenet, Waves Called Solitons, Springer-Verlag, Berlin (1995).D.L. Lee, Electromagnetic Properties of Integrated Optics, John Wiley & Sons, NewYork (1986).A.C. Scott, F.Y.F. Chu and D.W. McLoughlin, The soliton: a new concept in appliedscience, Proc. IEEE 61, 1443-1483, (1973).(a) B.M. Oliver, Bell Telephone Laboratories Technical Memorandum MM-51-150-10, Case 33089, March 8, (1951). (b) S.C. Bloch, Introduction to chirp concepts witha cheap chirp radar, Am. J. Phys. 41, 857-864, (1973).R. Dawkins, The Blind Watchmaker, Penguin, London (1986).G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, San Diego (1995).A.K. Zvezdin and A.F. Popkov, Contribution to the nonlinear theory of magnetostaticspin waves, Sov. Phys. JETP 57, 350-355, (1983).J. S. Aitchison, Y. Silberberg,, A.M. Weiner, D.E. Leaird, M.K. Oliver, J.L. Jackel,E.M. Vogel and P.W.E. Smith, Spatial optical solitons in planar glass waveguides, J.Opt. Soc. Am. B. Opt. Phys., 8(6), 1290-1297 (1991).J.S. Aitchison, K. Al-Hemyari, C.N. Ironside, R.S. Grant and W. Sibbett, Observationof spatial solitons in AlGaAs waveguides, Electron Lett., 28, 1879-1880, (1992).(a) Y. Silberberg, Spatial optical solitons in Optical Solitons, Ed. J Satsuma, Springer-Verlag, Berlin (1992). (b) P.V. Mamyshev, A. Villeneuve, G.I. Stegeman and J.S.Aitchison, Steerable optical waveguides formed by bright spatial solitons, ElectronicsLetters 30, 726-727, (1994).A. Villeneuve, J.S. Aitchison, J.U. Kang, P.G. Wigley and G.I. Stegeman, OpticsLetters 19, 761-763, (1994).G.I. Stegeman, A. Villeneuve, J.S. Aitchison and C.N. Ironside, Nonlinear integratedoptics and all-optical waveguide switching in semiconductors, Fabrication, Propertiesand Applications of Low-Dimensional Semiconductors, Ed M Balkanski and IYanchev, Kluwer Academic Publishers, Netherlands (1995).A.D. Boardman and K. Xie, Theory of spatial solitons, Radio Science, 28, 891-899,(1993).A.D. Boardman, K. Kie and A.A. Zharov, Polarisation interaction of spatial solitons inoptical planar waveguides, Phys. Rev. A., 51, 692-705, (1995).A.D. Boardman and K. Xie, Dynamics of spatial soliton coupling. Studies inClassical and Quantum Nonlinear Optics. Ed Ole Keller, 2-30, Nova Press, NewYork (1995).

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24. A.D. Boardman, K. Xie and A. Sangarpaul, Stability of scalar spatial solitons incascadable nonlinear media, Phys. Rev. A 52, 4099-4106, (1995).

25. A.D. Boardman and K. Xie, Magnetic control of optical spatial solitons, Phys. Rev.Letters, 75, 4591-4594, (1995).

26. A.D. Boardman and K. Xie, Waveguide-based devices: linear and nonlinear coupling,Low-Dimensional Semiconductor Devices, Ed. M. Balkanski, Kluwer Publishers,Amsterdam, (1996).

27. J. Boyle, S.A. Nikitov, A.D. Boardman, J.G. Booth and K.M. Booth, Nonlinear self-channelling and beam shaping of magnetostatic waves in ferromagnetic films, Phys.Rev. B, 53, 1-9, (1996).

28. A.D. Boardman and K. Xie, Spatial solitons in χ (2) and χ(3) dielectrics and control bymagneto-optic materials. Proceedings of Minnesota International MathematicsWorkshop, Springer-Verlag (1997).

29. Boardman, A.D. and Xie, K. Soliton-based switches, logic gates and transmissionsystems. Ed. M. Balkanski, Kluwer Publishers, Amsterdam (1997).

30. S.A. Akhmanov, A.P. Sukhorukov and R.V. Khokhlov, Self-focusing and diffractionof light in a nonlinear medium, Sov. Phys. Usp., Engl. Transl., 93, 609-636, (1968).

31. M. S. Sodha, A.K. Ghatak and V.K. Tripath, Self-focusing of laser beams, TataMcGraw-Hill, New Delhi (1974).

32. O. Svelto, Self-focusing, self-trapping and self-phase modulation of laser beams,Progress in Optics, 11, 1-51, (1974).

33. F.A. Jenkins and H.E. White, Fundamentals of Optics, McGraw-Hill, New York(1950).

34. A. Barthelemy, S. Maneuf and F. Froehly, Propagation et autoconfinement defaisceaux laser par non-linearite de Kerr, Opt. Comm. 55, 201-206, (1985).

35. A.D. Boardman, S.A. Nikitov, K. Xie and H.M. Mehta, Bright magnetostatic spin-wave envelope solitons in ferromagnetic films, JMMM, 145, 357-378, (1995).

36. J.P. Gordon and H.A. Haus, Random walk of coherently amplified solitons in opticalfiber transmission, Optics Letters 11, 665-667, (1986).

37. C.S. Gardner, J.M. Green, M.D. Kruskal and R.M. Miura, Method for solving theKorteweg de Vries equation, Phys. Rev. Lett. 19, 1095-1097, (1967).

38. V.E. Zakharov and A.B. Shabat, Exact theory of two-dimensional self-focusing andone-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP 34,62-69,(1972).

39. G.L. Lamb, Elements of Soliton Theory, John Wiley & Sons, New York (1980).40. A.N. Satsuma and G.M. Dudko, Initial value problems of one-dimensional self-

modulation of nonlinear waves in dispersive media, Prog. Theor. Phys. Suppl. 55, 284-306, (1974).

41. V.V. Afanasjev, J.S. Aitchison and Y.S. Kivshar, Splitting of high-order spatialsolitons under the action of two-photon absorption, Optics Comm, 116, 331-338,(1995).

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S P A T I A L S O L I T O N S I N Q U A D R A T I C N O N L I N E A R M E D I A

Laboratory of PhotonicsDepartment of Signal Theory and CommunicationsUniversitat Politècnica de CatalunyaGran Capità UPC-D3, Barcelona, ES 08034, Spain

Lluís Torner

I n t r o d u c t i o n1

Self-focusing effects in quadratic nonlinear processes were long known to be pos-sible under specific conditions, namely when the parametric interaction is very weakand yields an effective third-order effect for the pump wave.13,14 By and large, however,the full extent of the self-focusing and its implications were not fully appreciated untilrecently. 1 5 The remarkable exception is the theoretical work of Karamzin and Sukho-rukov more than twenty years ago,12 who investigated the mutual focusing of beams inparametric processes and identified its implications for the formation of solitons. In the

last few years, spatial soliton propagation has been observed experimentally in second-harmonic generation settings by Torruellas and co-workers in bulk potassium titanyl

Self-focusing and self-trapping of light have been investigated since the early days ofnonlinear optics. 1,2 Interest in this field has been maintained by the fascinating rangeof new phenomena encountered and their potential applications, such as soliton propa-gation, all-optical switching, and logic for ultrafast signal processing devices. For manyyears such effects have been pursued using the optical Kerr effect in cubic nonlinearmedia, 3 – 1 0 and since more recently using the photorefractive effect.11 However, self-induced trapping of light also occurs in quadratic nonlinear media (hereafter referredto as χ (2) media). 12 In this case, both spatial and temporal solitons form through themutual focusing and trapping of the waves parametrically interacting in the nonlinearmedium. In these lectures we focus on spatial solitons that form with cw light signalsin planar waveguides and in bulk crystals, but most of the mathematical results holdalso in the case of temporal solitons. Also, following the usual convention, throughoutthese lectures we make no distinction between the solitary waves that we study andmathematical solitons, referring to both of them as solitons.

Beam Shaping and Control with Nonlinear Optics229Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

phosphate (KTP),16 and by Schiek et. al., in planar waveguides made of lithium niobate(LiNbO 3).17 Generation of strings of solitons through the modulational instability ofwide intense beams has been observed experimentally by Fuerst and co-workers.18

Solitons in quadratic nonlinear media form in planar waveguides and in bulk media,and the lowest-order bright solitons are stable on propagation in both cases. Stable lightbullets are also theoretically possible. Those solitons might have important applicationsto different optical devices, including switching and routing devices and optical cavitiescontaining χ(2) crystals. The aim of these lectures is to present the background of thetopic in an unified point of view. We shall only discuss the basic properties of thesolitons in the simplest configuration, namely second-harmonic generation, and we shallconcentrate in Type I phase-matching. Naturally, the families of solitons are richer in thecase of Type II phase-matching geometries because of the additional degree of freedomthat this phase-matching offers, and that is important regarding potential applicationsof the solitons, but Type I geometries are simpler and capture the main features ofthe solitons. Hence, except for the last part of the lectures where we discuss a fewapplications of the solitons, we shall focus on Type I phase-matching. Also, we shallonly study bright solitons.

The lectures are organized as follows. In Section 2 we shall describe the physicalsetting considered and the evolution equations used to model the light propagation inquadratic nonlinear crystals. Section 3 is devoted to the basic properties, includingstability, robustness and excitation, of the simplest families of bright solitons that existin the absence of Poynting vector walk-off. In Section 4 we shall meet the solitonsthat exist in the presence of Poynting vector walk-off and discuss briefly their salientproperties. Section 5 is devoted to the dynamics on evolution of beams with topologicalphase dislocations, or optical vortices. In this part we shall meet the modulationalinstabilities existing in quadratic nonlinear media and we will discuss how differentcombinations of topological charges of the input light signals produce certain patternsof bright solitons. To end the lectures, in Section 6 we shall discuss briefly discretesolitons and in Section 7 we shall summarize our main conclusions.

2 Phys ica l Set t ing

2.1 Evolution Equations

We consider cw light beams travelling in a medium with a large, non-resonant χ(2) non-linearity under conditions of second-harmonic generation. We focus on spatial solitonsin Type I phase-matching settings. The electric field of each of the waves is writtenin the form E (r, t) = A (r) exp[ikz – iωt], and we consider experimental conditions forwhich both the scalar and the slowly varying envelope approximations for the fieldshold. These conditions are expected to be fulfilled under most relevant experimental

situations, but they might fail for narrow beams, very high input powers, and in highlyanisotropic crystals.

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We shall study the formation of solitons in planar waveguides and in bulk media.Most of the generic expressions shall be written for the bulk case, but it is assumedthat they hold in both cases. Throughout the lectures we use the subscripts 1 and 2 forquantities associated with the fundamental and second-harmonic waves, respectively.The appropriate χ (2) nonlinear coefficients involved in the two-wave mixing process areincluded in the normalization of the fields. Under such conditions, the beam evolutioncan be described by the reduced equations 19

(1)

where a1 and a2 are the normalized envelopes of the fundamental and second-harmonicwaves, ξ is the normalized propagation coordinate, r = –1, and α = – k1 / k 2 . Here k 1,2

are the linear wavenumbers at both frequencies. In all realistic situations α – 0.5. Theparameter β is a measure of the linear phase mismatch between the fundamental andsecond-harmonic waves and is given by β = k 1 η2 ∆k, where ∆k = 2 k 1 – k2 , is the wavevector mismatch and η is a characteristic beam width. The parameter δ and the unitvector characterize the magnitude and direction of the Poynting vector walk-off dueto the fact that the energy and wave fronts generally propagate in different directionsin a birefringent medium. One has, δ = k1 ηρw , with ρw being the walk-off angle, andwe note that walk-off is absent for propagation along the principal optical axes of thecrystal. Finally, the transverse coordinate is given in units of η, and the propagationcoordinate is normalized in such a way that z/ld = 2ξ, with ld = k 1 η2 /2, being thediffraction length of the fundamental beam.

In the one-dimensional case, equations (1) also hold for pulsed light. Then, diffrac-tion is replaced by dispersion, Poynting vector walk-off is replaced by group velocitymismatch, and r and α are given by the group velocity dispersions.

2.2 Limit of Large Wave vector Mismatch

The self-focusing nature of the wave-mixing process at the regime of large phase-mismatchbetween the waves (β >> 1) and small conversion to the second-harmonic, is simply ex-posed by noticing that in such conditions the governing equations approximately reduceto the nonlinear Schrödinger equation (NLSE)20–22

(2)

which in one-dimensional geometries is known to have stable soliton solutions. Theproperties of the χ (2) solitons are impacted by this fact. 19 However, the NLSE does notallow stable solitons for bulk geometries and a question might arise about the impli-cations of this fact to χ (2) trapping . Actually, the beam evolution in the χ (2) mediumquickly violates the approximations required to derive the NLSE, and stable solutionsdo exist in bulk quadratic nonlinear media.

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However, an important point must be emphasized. Most experimentally relevantsolitons in quadratic media form for small and negative wave vector mismatches, atexact phase-matching, and in general under other conditions where (2) does not hold.Therefore, such solitons exhibit new properties, dynamics, and features. Thus, theyought to be treated accordingly.

( 3 )

2.3 Experimental Values

We shall shortly be giving some values of β and δ that are relevant to experimentalconditions, but the impact of these parameters on the formation of solitons is bestelucidated by recalling their relation to the linear lengths that characterize the low-power beam evolution in the χ (2) medium, namely the diffraction length ld , the coherencelength lc = π/ |∆k|, and the walk-off length lw = η /ρ . One has

Non-critical or temperature-tuned phase-matching configurations that yield a negligiblewalk-off correspond to δ 0; by contrast, δ ≠ 0 occurs when angle-tuning phase-matching is used and walk-off is present. When the diffraction and walk-off lengthsare comparable one has δ ~ 1. Most crystals with large χ(2) nonlinear coefficientsare also highly birefringent, as is the case for most organic materials, and for that casewalk-off would dominate diffraction and large values of δ would result for angle-tunedconfigurations. Under such conditions, equations (1) may not be valid.

The role of the parameter β is exposed by recalling that for focused beams thevalue of ∆k gives only partial information about whether or not it corresponds to nearphase-matching. This is because focused beams contain a broad spectrum of transversewave vectors, and each spectral component of the beam experiences a different wavevector mismatch. The effective phase mismatch, as measured by the second-harmonicgeneration efficiency, depends on the diffraction properties of the beam, and for the casewe are studying here it is given by the value β . Outside exact phase-matching, whichcorresponds to β = 0, it is useful to write β = (η/η0 ) 2 , with η0 = (k1 ∆k )–1/2 beinga characteristic width. Thus, for a given input beam width η , a wave vector mismatchsuch that η ≈ η , effectively corresponds to near phase-matching, whereas a larger ∆k ,0

with |β| >> 1, corresponds to a large phase mismatch.The mutual trapping of beams in the χ(2) crystal is governed by the interplay between

the linear lengths contained in β and δ, in addition to the nonlinear length determined by

the light intensity and by the input conditions. Regarding the linear lengths, favourableconditions for self-trapping occur when all lengths are comparable, so that β ~ ±3, andδ ~ ±1. Typical experimental conditions that yield these values, and which are repre-sentative of the actual parameters involved in the experimental observation of solitons

in KTP cut for Type II phase-matching, 16 are η ~ 15 µm, and ρw ~ 0.1° – 0.5°. Forsuch parameters, the value of δ falls in the range 0.3 – 1.5, and one needs a coherencelength of some lc ~ 2.5 cm, to have β ~ ±3. For typical materials and wavelengths, say

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λ ~ 1 µm, this coherence length corresponds to a refractive index difference ∆ n ~ 10–4

between the fundamental and second-harmonic waves. The above parameters yield adiffraction length of ld ~ 1 mm, so that ξ in the range 0 – 20, corresponds to a few cm.

Large effective wave vector mismatches, in the sense mentioned above, correspondto β ~ 30, whereas exact phase-matching corresponds to β = 0. Large values of δ, sayδ ~ 5, which for the above parameters is obtained in configurations with a walk-off angleof some ρw ~ 1.5°, correspond to a moderate Poynting walk-off. They are representativeof the experimental conditions encountered, e.g., in organic materials with very largequadratic nonlinearities, at appropriate wavelengths close to those where non-criticalphase-matching occurs.

23,24

To grasp the order of magnitude of the optical intensities associated to the normalizedquantities used along the lectures and that are required to form solitons in quadraticnonlinear media, notice that for a beam width η ~ 20 µm and the typical nonlinearcoefficients of KTP, a normalized power (defined below) of some I ~ 50, leads to anactual peak intensity of the order of ~ 10 GW/cm2 . Existing inorganic materials (e.g.,potassium niobate), semiconductors (e.g., gallium arsenide) and organic materials (e.g.,N-4-nitrophenil-(L)-prolinol, NPP, or dimethyl amino stilbazolium tosylate, DAST),with larger quadratic nonlinear coefficients would require lower power requirements.23,24

No need to say that, this is so provided that the appropriate nonlinear coefficients areaccessible at suitable wavelengths and phase-matching geometries with a reasonablewalk-off and low absorption, and provided that long enough, mechanically stable, highquality samples can be made. Quasi-phase-matching of the largest nonlinear coefficientsof suitable materials, e.g., LiNbO3, also leads to reduced power requirements and tosuitable operating conditions, so that it holds promise for future use. 25–27

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2.4 Conserved Quantities

Central to the beam evolution described by equations (1) is the fact that they constitutean infinite-dimensional, Hamiltonian dynamical system.1 0 , 28 To expose this fact, it isconvenient to rewrite the equations in terms of the new fields A 1 = a1 , and A2 =a2 exp(–i βξ) . T h e n , the conserved Hamiltonian writes

(4)

The governing equations can now be written in the canonical form

(5)

where the symbol δF indicates a Fréchet or variational derivative and Ã2 =We shall also make use of two additional conserved quantities: the total power or

energy flow given by the Manley-Rowe relation

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and the total transverse beam momentum

(7)

In Section 5 we shall consider light beams without azimuthal symmetry, and in suchcases it is useful to monitor the evolution of the longitudinal component of the totalangular momentum of the light beams given by

(8)

with being the transverse beam momentum density in expression (7). In the absenceof Poynting vector walk-off, L is also a conserved quantity of the beam evolution. Whenwalk-off is present, one readily finds that

(9)

For the right-hand-side of this equation to vanish in the presence of walk-off, the trans-verse momentum of the second-harmonic beam has to be parallel to Otherwise, thetotal angular momentum is not conserved in the presence of Poynting vector walk-off.

It is also important to examine the rate of energy exchange between the fundamentaland second-harmonic waves. Writing the fields in the form a

1,2= R1,2 exp( iφ

1,2), where

R1,2 and φ 1 , 2 are real quantities, one arrives at

(10)

Therefore, to cancel the energy exchange between the fundamental and second-harmonicbeams, their wave fronts, including the nonlinear wave vector shifts induced by the waveinteraction, ought to verify φ2 ( ξ , r ⊥ ) = 2φ1(ξ, r ⊥ ) + βξ. This is what happens when asoliton is formed: the transverse complex shapes of the two interacting beams inducethe appropriate wave front distortions to cancel diffraction and Poynting vector walk-off,while the energy exchange between the waves is also cancelled. It is worth emphasizingan obvious but important fact: such process occurs dynamically, and it takes an infinitedistance to form a true stationary soliton out of input conditions that do not coincidewith such a soliton, as it is always the case in practice.

3 The Simplest Bright Solitons

3.1 Families of Stationary Solitons

We first consider stationary soliton solutions in the absence of Poynting vector walk-off(i.e., δ = 0). The simplest solutions have the form

(11)

with κ v being the wave number shifts induced by the nonlinear wave interact ion. Forthe solutions to be stationary one needs κ 2 = 2κ1 + β. According to (10) this relation

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Figure 1: Typical shape of (1+1) solitons for different values of the wave vector mis-match. The solitons correspond to I = 30.

32,33

ensures that there is no power exchange between the waves. The coupled equationsobeyed by U1 and U2 are given by

(12)

By and large, these equations have a rich variety of solutions, including solutions withdifferent dark shapes, solutions with multiple peaks, and solutions with exotic shapes.In these lectures we are only interested in the lowest-order solutions. In the case of(2+1) solitons in bulk media (i.e., solitons with two transverse coordinates plus onelongitudinal coordinate), those correspond to nodeless, radially symmetric solutionswith no azimuthal angular dependence. In equations (12), r, α and β contain materialand linear wave parameters, while the nonlinear wave number shift κ1 parametrizes thefamilies of solutions. In all plots presented in these lectures r = –1, and α = –0.5.

In the case of (1+1) solitons, one analytical solution with a bright shape is known.Namely, 12,19,29–31

(13)

where s is the transverse coordinate. This solution occurs at β = –2(α – 2r), with thespecial value κ1 = –2 r. The whole family of solutions for different values of κ1 and atother values of β can be found by solving equations (12) numerically using a shooting ora relaxation algorithm. Figure 1 shows the typical shape of a few representative solitons.In the case of (2+1) solitons no analytical solutions are known, but whole families ofsolitons are readily obtained by solving the equations numerically.32–35

3.2 Similarity Rules

Even though the families of stationary soliton solutions are only known numerically,

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important information about their properties can be obtained analytically by examiningthe scaling properties of eqs. (12). One readily finds that such equations are invariantunder the similarity transformations

(16)

(15)

(14)

with µ ≠ 0 being an arbitrary parameter. Thus, the soliton solutions can be transformedinto each other following such rules.

This self-similarity has important consequences. For example, when doing the nu-merical calculations it can be used to solve equations (12) in an efficient way. Moreimportant, it has direct physical consequences. In particular, it shows that the gen-eral features of the solutions at either side of phase-matching are similar. Similarly,it implies that at phase-matching (β = 0), all the solitons in the absence of Poyntingvector walk-off are self-similar. Therefore, at phase-matching the relation between theamplitude and the width of the family of solitons is

amplitude × (width)² = constant.

Outside phase-matching, the stationary solutions exhibit different amplitude-width re-lations. We shall examine them below.

3.3 Stability

One crucial property of the families of solitons is their stability under propagation. Herewe refer to the stability of the solitons when propagated in the system with the samenumber of “transverse dimensions” than the system where they have been found. Inother words, modulational instabilities against higher-dimensional perturbations are notconsidered. Otherwise, solitons in quadratic nonlinear media are known to be modula-tionally unstable in both (1+1) and (2+1) geometries if the corresponding perturbationsare allowed to grow in the experimental setting considered.36,37 We shall discuss this issue

in Section 5.The stability of the families of stationary solutions can be elucidated by using dif-

ferent methods.38—42 In these lectures we shall only discuss a geometrical approach.42

Such geometrical derivation of the stability criterion is useful by itself, because in manycases it gives direct insight into the stability of the solitons, and also because it showsthe universality of the stability criterion for similar systems.

One first finds that the stationary solutions with the form (11) occur at the extremaof the Hamiltonian for a given energy flow, i.e., they correspond to

Now, the stability of the stationary solutions can be elucidated by noticing that theglobal minimum of H gives stable solutions, whereas local maxima yield unstable solu-

tions. This is a consequence of Lyapunov theorem about dynamical systems applied tothis case.10,28 Therefore, to elucidate the stability of the families of solitons one has toplot the curve H = H (I) and identify its lower and upper branches. The condition of

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Figure 2: Hamiltonian and wave number shift versus energy for the families of (1+1)solitons. (a) and (c): phase-matching and positive β; (b) and (d): negative β. Solidlines: stable solitons; dashed lines: unstable solutions. 42

marginal stability, that separates stable from unstable solutions, is given by the pointthat separates the lower from the upper branches of the curve H = H (I ). There aredifferent ways to determine that point. For example, examination of the correspond-ing curves, shown in Figure 2 for the case of (1+1) solitons, leads immediately to theso-called Vakhitov-Kolokolov criterion, 43 given by

(17)

The mathematical proof is given by Whitney’s theorem about two-dimensional mapsapplied to this case. 44 In the case of spatial solitons in quadratic nonlinear media, thelower branches of the curves in Figure 2 are found to correspond to the global minimumof H , and the upper branches to local extrema. Therefore, criterion (17) holds.

Physically, the main conclusion to be raised from Figure 2 in the case of (1+1)solitons, and from the similar plot but for (2+1) solitons,34,35 is that at positive phase-mismatch and at phase-matching, all the lowest-order stationary solutions in the absenceof walk-off are stable. Very narrow regions of solutions that would be unstable exist atnegative phase-mismatch near the cut-off condition for the soliton existence, but thosehave a very limited physical relevance to the experimental formation of solitons. Thisis so because of several reasons. First, because solutions near cut-off correspond toincreasingly broader beams; second, because in reality the input beams never matchexactly the shape of the unstable stationary solutions; and third, because above thethreshold light intensity for the existence of solitons there is always a stable soliton.Therefore, the excitation of the stable solitons that exist reasonably far from cut-off iswhat dominates the dynamics of the beam evolution.

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3.4 Two Useful Properties

The families of stationary soliton solutions contain important information. In theselectures, we shall only discuss two of them, that have direct experimental implications.

One important property of the families of solitons is their amplitude-width relations.At phase-matching the amplitude-width relation is given by (15), and otherwise it hasto be calculated numerically. Figures 3(a)-(b) show the outcome in the case of (2+1)solitons at β = ±3.35 One important consequence of the existence of soliton families witha given amplitude-width relation is that under the influence of any small perturbationswhich lead to adiabatic changes of the energy flow along the propagation direction, likesmall material or radiative losses or gain, the shape of the solitons tends to adiabaticallyevolve following the corresponding relation. Figures 3(a)-(b) show that in the case ofsolitons in quadratic nonlinear media, at reasonably large amplitudes the width increasesrather slowly as the amplitude of the solitons decreases. Therefore, such solitons retaintheir shape a great deal while they propagate under the influence of the non-conservativeperturbations, provided that they are small. It is worth noticing that the behavior shownin Figs. 3(a)-(b) is consistent with the behavior of the soliton width for different solitonenergies that were observed experimentally.16

Another important feature of the family of solitons is the fraction of the total energythat is carried by each of the waves, fundamental and second-harmonic, that form thesoliton. Figure 3(c) shows how this fraction depends on the wave vector mismatch fortwo representative values of the total energy flow of the soliton. 33 This particular plotcorresponds to (1+1) solitons, and it shows that at positive phase-mismatch most of theenergy is carried by the fundamental wave, whereas at phase-matching the energies arecomparable and at negative β the largest amount of energy is carried by the second-harmonic. Naturally, all this has important implications when it comes to the excitationof solitons with different input light conditions.

Figure 3: (a) and (b): width of the (2+1) solitons as a function of their amplitude.35

(c): fraction of energy carried by each wave forming the (1+1) solitons as a function ofthe wave vector mismatch. Solid line: I = 18; dashed line: I = 30.33

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Finally, notice that expression (16) provides the starting point of a variational ap-proach to obtain approximate analytical expressions of the families of solitons. Oneneeds to calculate I and H for given beam shapes and then optimize their parametersto minimize H. See ref. 45.

3.5 Excitation, or “Oscillating Solitons”

In practice, the input light conditions do not coincide with the shape of the stationarysolitons. Therefore, the excitation of solitons with arbitrary input light beams is acentral issue, that has to be investigated in detail. This is particularly true in the caseof solitons that form in the presence of walk-off, because walk-off poses severe restrictionsto the formation of solitons in realistic experimental conditions. Also, notice that atpresent, the longest χ(2) crystals available are a few centimeters long, which correspondsto a few tens of linear diffraction lengths, so that in single pass experiments only thesoliton evolution during a relatively short distance is relevant.

In general, one has to elucidate both the soliton content of the input conditionsconsidered, and the dynamics of the evolution of such input conditions. In the caseof mathematical solitons, namely soliton solutions of so-called completely integrableevolution equations, a great deal of this information can be obtained using the toolsprovided by the Inverse Scattering Transform; in particular, the soliton content of theinput conditions can be determined a priori. However, this is not so for solitons of non-integrable evolution equations, as it is the case of the system (1). Therefore, the studyof the dynamics of the soliton excitation relies heavily on numerical experiments. Thoseare performed by solving the evolution equations (1), typically with a split-step Fourieror a finite-difference standard scheme, for given arbitrary input conditions. Motivatedby its experimental relevance, one might consider sech-like, or Gaussian input beamswith the form

(18)

with A and B being the amplitudes of the fundamental and second-harmonic beams,respectively.

The excitation of bright solitons has been examined for a wide variety of input con-ditions, both in the case of (1+1) and (2+1) geometries. The numerical experimentsshow that solitons emerge from the input beams in a wide variety of input conditions,in terms of wave vector mismatches and input light shapes and intensities, not nec-essarily close to those given by the stationary soliton solutions. Solitons also emergewith inputs which fall very far from those solutions indeed, and in particular when onlythe fundamental beam, or only the second-harmonic plus a small fundamental seed, issupplied at the input face of the χ(2) crystal. In such a case, the input beams reshape,exchange power and adjust themselves through radiation of dispersive waves, and then

they mutually trap. See refs. 46 and 47 for details.Here we shall only recall one point that is found: the excitation of solitons with ar-

bitrary inputs, that hence contain one stationary soliton plus some amount of radiation,

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Figure 4: Excitation of solitons with the input conditions (18). Left: Peak amplitude ofthe fundamental beam, scaled to the input value, as a function of propagation distance.The plot is for (2+1) propagation. Right: Detail of the beam evolution in the caseA = 20, B = 0.47

produces “oscillating states”. The amplitude of the oscillations decreases as the beamsshed dispersive waves away, but after the initial reshaping process the leak is extremellysmall. Figure 4 shows the typical beam evolution during the first tens of diffractionlengths. The plot corresponds to propagation in bulk media.

The oscillations might have several experimental consequences. For example, whenonly fundamental beams are initially launched with energy well above the threshold forthe existence of stationary solitons, the numerical simulations predict fast oscillationsthat produce sharp intermediate stages. 47 Under those conditions the peak power atthe centre of the beams reaches very large values. Such values might be high enoughto damage the crystal. Also, at such values equations (1) might not be valid, and theactual beam evolution inside the crystal might uncover new important features thatought to be investigated.

3.6 Solitons in Quasi-Phase-Matched Samples

By and large, quasi-phase-matching (QPM) ffo ers an attractive approach to produce

highly-efficient parametric wave-mixing interactions in quadratic nonlinear optical me-dia (for a comprehensive review, see refs. 25-27). Conventional phase-matching tech-niques used to compensate for the wave vector mismatch between the waves of differencefrequencies that interact in the nonlinear medium are based on the birefringence andthermal properties of the materials involved. As a consequence, the overall efficiencyof the interaction can be limited by a variety of effects, mainly Poynting vector spatialbeam walk-off due to the different propagation directions of energy and phase fronts

in anisotropic media, non-convenient operation temperatures or crystalline optical axesorientations, and the value of the quadratic nonlinear coefficients accessible through thepolarization of the waves involved in the interaction.

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With QPM, the coupling between the fundamental and second-harmonic waves canbe chosen so that non-critical phase-matching can be achieved at a suitable temperature,without Poynting vector spatial walk-off between the interacting beams, by using thelargest second-order coefficients of the material employed, and with an optimized overlapbetween the guided modes involved in waveguide devices. Such advantages hold verypromise for many applications, in both waveguide and bulk settings, and in particularto the formation of solitons.

QPM can be potentially used for a variety of materials and operating frequencies.Typical experimental conditions give coherence lengths lc = π /|∆ k| in the range 2 – 20µ m, with the actual value depending on the material involved and the wavelength used.In the normalized units of equations (1), one has β ~ 500 – 1000. For the typical valuesof the other various involved parameters, given in Section 2, β ~ 500 corresponds to acoherence length of some 7 µm. These are the conditions encountered, e.g., in the QPMof the diagonal d

33nonlinear coefficient of LiNbO3 at λ ~ 1.5 µm.

QPM relies on the periodic inversion of sign of the nonlinear χ( 2 ) coefficient at givenmultiples of the coherence length lc , and the so-called m -th order QPM correspondsto a periodic domain inversion with period m = 2mlc (see refs. 25-27 and referencestherein, and Marty Fejer’s lectures). In general, the biggest nonlinearity is obtained for1-st order QPM. In such a case, the corresponding evolution equations at leading-order,after averaging over the periodicity of the QPM structure, are analogous to equations (1),but with effective nonlinear coefficients 2/π smaller than the actual material coefficientsand a global phase shift between the fundamental and second-harmonic waves. Hence,so are the solitons.

48Higher-order corrections have been considered in ref. 49.

Here we only wish to discuss the robustness of the soliton formation and evolutionagainst random deviations of the domain length. Such random deviations occur as aconsequence of the fabrication tolerances of the QPM domains. The physical nature andstatistical properties of the random deviations from the nominal lengths depend a greatdeal on the specific fabrication technique used to implement the QPM sample. Herewe only discuss the so-called duty-cycle errors, 25 that occur when the periodicity of thedomains is very well controlled, but the positions where the domain walls actually formdiffer from the nominal ones. In the resulting structure, the ending wall of the n -thdomain is located at the position where is the nominal lengthof each domain and Rn is the random shift. Such errors are short-range correlatedalong the longitudinal coordinate. Therefore, they have a small, adiabatic impact onthe solitons, because when the correlation length is much shorter than the characteristicsoliton length the stochastic effects can be averaged out to a large extent over everycharacteristic soliton length. On the contrary, long-range correlations might be far frombeing averaged out over a soliton period, hence they impact more strongly the solitonevolution. This is the case of the so-called random-walk errors that occur when theending wall of the n -th domain is located at

Figure 5 shows the typical outcome of the excitation of a soliton in a QPM samplewith duty-cycle domain length errors. The plots show the fundamental beams propa-

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Figure 5: Excitation of a soliton in QPM samples with duty-cycle domain-length errors.Dashed lines: case of an ideal structure; solid lines: case of structures with differentrandom domain errors, with typical deviation: in (a) 10 % ; in (b) 20 %.48

gated some 20 diffraction lengths. The dashed lines show the soliton in the case of anideal structure and the solid lines correspond to structures with random errors with aGaussian distribution around the nominal domain length. On average the duty-cyclerandom errors induce small radiative losses in the solitons. Because losses are small, oncethe soliton is formed after the input beams reshape slightly, they evolve adiabaticallyfollowing the amplitude-width relation of the family of solitons. 48,50

4 Walking Solitons

4 . 1 Motivation

By their very nature, solitons in quadratic nonlinear media are made out of the mu-tual trapping of several waves. Here we consider the formation of spatial solitons underconditions for second-harmonic generation, therefore the solitons exist due to the mu-tual trapping of the fundamental and second-harmonic beams. In general, except undersuitable conditions, in the low-power regime the beams propagate along different direc-tions due to the Poynting vector walk-off present in anisotropic media and this fact hasimportant experimental implications when it comes to the choice of suitable materials,input light wavelength and general conditions suitable to the formation of solitons.

However, when a soliton is formed the interacting waves mutually trap and in thepresence of Poynting vector walk-off the beams drag each other and propagate stuck,or locked together. Such a beam locking opens the possibility to specific applications ofthe solitons, 51–56 and it also poses new challenges to the understanding of the solitonformation. This is so because the “walking” solitons existing in the presence of walk-off exhibit new features in comparison to the non-walking solitons. Investigation ofthese new features is important regarding their potential applications, but also from afundamental viewpoint because the approach and outcome have implications to walkingsolitons existing in other analogous but different physical settings.

2 4 2

4.2 Families of Solitons

(20)

Next, we examine stationary solutions of equations (1), with δ ≠ 0, describing mutuallytrapped beams walking off the ξ axis. Those have the form

(19)

where U and φ are real functions, η = s – v ξ is the transverse coordinate moving withthe soliton, and φ v (ξ,s) = κ

vξ + ƒv (η). Here v is the soliton velocity, and ƒv (η) are the

wave fronts of the solitons. According to (10), to avoid energy exchange between thewaves one needs κ2 = 2κ1 +β, as for non-walking solitons. However, one also needs thatthe wave fronts verify ƒ 2(η) = 2ƒ1( η) everywhere, or alternatively Uv (η) and ƒv (η) oughtto be symmetric and anti-symmetric functions of η, respectively. The solitons that existin the absence of walk-off fulfil the former condition, whereas with this exception all thewalking solitons fulfil the latter.

Substitution of (19) into (1) yields the system of coupled nonlinear ordinary differen-tial equations fulfilled by the functions U v (η) and ƒv (η). They can be solved numericallyto obtain the families of walking solitons.57–60 Recall that κ1 and v parametrize thefamilies of walking solitons. Experimentally, such parameters correspond to the lightintensity and to the angular deviation of the solitons from the longitudinal propaga-tion axis. Walking soliton solutions exist for values of κ 1 and v such that solitons arenot in resonance with linear dispersive waves. The resonance condition can be readilycalculated to obtain57

Figure 6 shows the typical amplitude and wave front shape of a walking soliton. Asshown in the plot, the walking solitons have curved wave fronts. Such curvature de-pends on all the parameters involved, including the wave vector mismatch, the walk-offmeasure, the soliton energy and the soliton velocity.57–60

As it is the case of the non-walking solitons studied previously, important informationabout the families of walking solitons can be obtained from the conserved quantities ofthe wave evolution, as follows. One first finds that the stationary walking solitons withthe form given by (19) occur at the extrema of the Hamiltonian for a given energy flowand a given transverse momentum, i.e., they occur at

(21)

This is an important result that has important implications to the soliton stability. Onecan also use it to find approximate analytical expressions of the shape of the walkingsolitons using a variational approach.

The walking solitons have curved wave fronts and therefore their transverse mo-mentum is not simply proportional to their velocity. The actual relation between thevelocity and the momentum for the walking solitons can be elucidated by examining theevolution of the energy centroid of the bound state constituted by the fundamental andsecond-harmonic beams propagating stuck together. One finds 51

243

(24)

In contrast to this result, the travelling-wave solutions that occur in the absence ofPoynting vector walk-off have a flat wave front and a tilt given by the soliton velocity.In such conditions, one finds the particle-like results

(23)

where Hv = 0 is the Hamiltonian of the zero-velocity solitons and the term Iv 2 /2 representsthe kinetic part of the Hamiltonian of the solitons walking with velocity v. However,such is not the case of the solitons that exist in the presence of Poynting vector walk-off.

4 .3 S tab i l i t y

The stability of the families of walking solitons can be elucidated by different ways. Herewe shall use the geometrical approach discussed previously for the case of non-walkingsolitons. Once again, our starting point is the variational expression (21). Becausethe families of stationary walking solitons realize the extrema of H for given I and J,one concludes that solutions that realize the global minimum of H are stable, whereasthose that realize a local maxima are unstable on propagation. Therefore, we ought toexamine the surface H = H(I, J) and identify its lower and upper sheets (Fig. 7).

In the case of “smooth” H = H (I, J) surfaces, the curve that separates the lower andupper sheets of the surface can be determined by noticing that over the curve the vectornormal to the surface is contained on the horizontal plane. A similar procedure holds inthe case of sharp surface foldings. Sheet-crossings ought to be treated separately. Thevector normal to the two-parametric surface i sgiven by the expression

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Figure 6: Typical shape of a (1+1) walking soliton. The plots is for β = –3, δ = 1,and I = 30. In (a)-(b): v = –0.5; in (c)-(d): v = 0.5. 57

(22)

Figure 7: Sketch of the procedure to determine geometrically the condition of marginalstability of the walking solitons.

where the symbol ∂ (a,b ) stands for a Jacobian matrix. The vector( F, G ) / ∂ is containedon the horizontal plane when the last term in the RHS of (24) vanishes, so that

(25)

This is the condition of marginal stability for the families of walking solitons that isderived using a linear stability approach.59

To identify the stable and the linearly unstable walking solitons one has to evaluatethe condition (25) for the families of walking solitons. Actually, only the families thatexhibit multi-valued surfaces H = H (I, J) need to be studied in detail. See ref. 60 forthe details. The main conclusion is that, similarly to the families of solitons that existin the absence of walk-off, narrow regions of unstable solutions exist near cut-off, butwith such exceptions all the walking solitons are stable on propagation.

4 . 4 E x c i t a t i o n

The excitation of walking solitons with different input beams is governed to a largeextent by Eq. (22). or our present purposes it is better to write it as F

(26)

This expression has to be used with caution because it holds for the families of stationarywalking solitons, but not for the input light conditions. The difference is that unlessthe input conditions exactly match the shape of the walking soliton solutions, the beamdynamics and reshaping towards the formation of a walking soliton always producessome radiation that takes energy and momentum away. However, when the radiationproduced is small Eq. (26) provides a direct estimation of the velocity of the walkingsoliton that eventually gets excited. The ratio I2 /I depends strongly on the linear wave

vector mismatch between the waves and also on the total energy flow. In particular, atpositive β the solitons have small I 2 /I, and they walk slowly. At phase-matching andat negative β the solitons have larger I

2/I, thus they walk faster.

245

5 Vortices

5.1 Motivation

Vortices, or topological wave front dislocations, are ubiquitous entities that appear inmany branches of physics. 61 Regardless the physical setting considered, the vorticesdisplay fascinating properties and a strikingly rich dynamical behavior. Optical vorticesare not an exception. They are spiral, or screw dislocations of the wave front thathas a helical phase-ramp around a phase singularity. They appear spontaneously inspeckle-fields, in the doughnut laser modes, and in optical cavities, and otherwise theycan be generated with appropriate phase masks or by the transformation of laser modeswith optical components. Optical vortices have been investigated in linear media, cubicnonlinear media, and photorefractive crystals.62–70

In this section we shall examine the phenomena generated by intense beams con-taining optical vortices in bulk quadratic nonlinear media under conditions for second-harmonic generation. With moderate input powers and wide beams, when only the fun-damental pump beam is initially launched light undergoes frequency doubling togetherwith the generation of a phase dislocation nested in the second-harmonic beam. How-ever, with intense, tightly focused beams soliton effects become crucial and a whole newrange of phenomena appears. In particular, the composite states of mutually trappedbeams containing the phase dislocations self-break inside the quadratic crystal into sep-arate beams, that then form sets of spatial solitons.71–73

Such a behavior defines the principle of operation of a new class of devices thatcan process information by mixing topological wave front charges and producing certainpatterns of spatial solitons. The number of output solitons can be controlled by thevalue of the “array” of topological charges of the input light signals. Next, we shallpresent the basic ingredients needed to understand the device behavior. Namely, theexistence of solitary-wave vortices and their instability, and the dynamics generated bythe mixing of beams with different topological charges.

5.2 Bright Vortex Solitary Waves

We examine families of stationary solutions of equations (1), with δ = 0, that correspondto solitary-wave vortices. Specifically, we consider topological phase dislocations nestedin the centre of beams with a bright, Gaussian-like shape. Such solitary waves appearas higher-order solutions of the governing equations. They have the generic form

(27)

where ρ is the radial cylindrical coordinate, and ϕ is the azimuthal angle. In the case ofsolutions with a phase dislocation nested in the centre of the beams, mv are the topolog-ical charges of the dislocations and sgn(mv ) their chirality. The transverse profiles Uv

are assumed to be real, radially symmetric functions. To avoid power exchange betweenthe fundamental and second-harmonic waves one needs κ2 = 2κ1 + β, and m2 = 2m1.

246

Figure 8: Typical shape of bright vortex solitary waves. Solid lines: fundamental beam;dashed lines: second-harmonic. Conditions: β = 3, κ1 = 3.72

The coupled equations obeyed by U1 and U2 are

(28)

These equations have a rich variety of solutions. In particular, there are families ofbright, higher-order solutions of two types. Namely, higher-order solutions withoutvorticity, that are characterized by the number of nodes, or field zeroes of the beams,and solutions without nodes, but with phase dislocations of increasingly higher charge.Here we only consider the latter. Figure 8 shows the typical shape of a solitary-wavevortex with charge m1 = 1 and with charge m1 = 2. The whole families of solutionsin the case of Type I geometries can be found in ref. 72. Type II geometries produceanalogous results.73 However, in the case of Type II the problem is richer, because itinvolves three waves (see appendix A). In particular, the families of solitary-wave vorticesoccur for the combinations of topological charges that verify

m 2ω = m oω + me ω , (29)

where moω and meω correspond to the ordinary-polarized and the extraordinary-polarizedfundamental beams.

The important point for our present purposes is that the bright vortex solitary wavesare unstable. In general, such instability produces the self-breaking of the beams alongthe azymuthal direction. Such self-breaking is due to the modulational instability ofthe top of the ring-shaped beams, somehow analogous to other spatial and tempo-ral modulational instabilities arising in χ (2) media,36,37,109–111 and similar to azimuthal

modulational instabilities that occur in χ( 3 ) media.6 3 , 6 5 , 6 7 , 7 4

247

Figure 9: (a) and (b): Growth rate of perturbations with different azimuthal indicesfor vortices with m1 = 1. (c): Typical decay of the solitary-wave vortex in Fig. 8(a). 72

5.3 Azimuthal Modulational Instability (AMI)

To examine the stability of the solitary-wave vortices against azimuthal perturbationsone can seek solutions of the form

(30)Inserting (30) into (1), and linearizing around the perturbations, one obtains the set offour coupled linear partial differential equations obeyed by ƒv (ρ,ξ) and gv ( ρ,ξ). Suchequations have many possible types of solutions, but now we are only interested inthose that display exponential growth along ξ. To obtain them one can use the methoddescribed in refs. 74 and 75. Figure 9 shows the typical outcome of such an stabilityanalysis. See refs. 72 and 73 for the details.

The main result predicted by the plots is that the larger the parameter κ1 , hencethe light intensity, the stronger the instability. For vortices with charge m1 = 1 , t h eperturbations with n = 3, together with and n = 2 exhibit the largest growth rate.

For m1 = 2, the perturbations that tend to dominate are n = 5, together with n = 4.Therefore, under ideal conditions when all the perturbations are excited with equalstrength, the vortex solitary waves tend to split into the corresponding number of beams.This is so at the initial states of the evolution, where (30) is justified. By and large,numerical simulations confirm such predictions. Figure 9 (c) shows the result of arepresentative simulation.71

5 . 4 From Topological Charge Information to Sets of Solitons

By now we have introduced the ingredients needed to describe the principle of operationof a class of devices that mix wave front topological charge dislocations nested in focused

light beams and produce certain patterns of bright spatial solitons. The devices canoperate in different regimes, as follows. Let us consider Type II geometries.

248

Figure 10: Sets of solitons obtained with different combinations of the topologicalcharges of the input light.

In a first regime only the fundamental beams are input in the crystal, with a highenough intensity to form spatial solitons. Then, a second-harmonic is generated with thetopological charge m2ω = moω + meω, and the beams self-split into a certain number ofsolitons because of the AMI of the solitary-wave vortices. Ideally, the number of outputsolitons is given by the index of the azimuthal perturbations with the highest growthrate, or by the interplay between different perturbations when there are some withsimilar growth rates. In practice, the several existing asymmetries in the experimentalset-up, including Poynting vector walk-off, can seed a dominant perturbation.

In a second regime, a coherent second-harmonic seed is present at the input togetherwith the two fundamental beams. When m2ω = moω + me ω the beam evolution is similarto that of the regime mentioned above. A totally new situation is encountered whenthe topological charges of the input light are chosen to verify m2ω ≠ moω + meω . Then,the number of output solitons is dictated by the different dynamics experienced by thedifferent azimuthal portions of the beams.76

Down-conversion schemes, where an intense second-harmonic light beam is input inthe crystal together with noise at the fundamental frequency, are another possibility.

All such processes are driven by the azimuthal dependence of (10), that in the caseof Type II phase-matching writes

(31)

The central result is that the “information” coded in the value of the input array[moω, meω , m2ω ] is transformed into a certain number of output soliton spots. Figure

10 shows the outcome of typical numerical experiments with different combinations oftopological charges and energies of the input light beams.

249

6 Discrete Solitons

6.1 Motivat ion

Discrete solitons on nonlinear lattices are a subject of intense research and continuouslyrenewed interest, as they appear in many models of energy transport in a variety ofphysical, chemical and biological scenarios (see, e.g., refs. 77-82, and references therein).Discrete solitons form as a consequence of the balance between linear coupling thattends to spread the excitation across the lattice, and nonlinearity. Different types ofnonlinearities, and among them cubic nonlinearities that yield evolution equations be-longing to the family of the NLSE, have been investigated for many years. However,discrete solitons also exist in quadratic lattices.

Besides its potential application to arrays of optical waveguides, discrete solitons onquadratic lattices might have applications to mathematically analogous, but physicallydifferent systems. As a matter of fact, parametric interaction of modes in general is auniversal phenomenon, hence similar discrete solitons might also exist in other branchesof nonlinear science. Naturally, the same is true for the continuous solitons.

6.2 A Quadratic Lattice

Consider the differential-difference normalized evolution equations

(32)

which come from the standard discretization of equations (1). In general, these equationsmight hold as a model for the parametric interaction of two generic modes in differentscenarios, not only in Optics. In any case, β is the phase-mismatch between the modes,and n the position on the lattice. The parameters α1 and α2 give the strength of thelinear spreading effects. In the numerics we set α1 = 0.5 and α2 = 0.25, to allowcomparison with the spatial solitons discussed previously in the continuous case.

For later use it is convenient to introduce the quantities Qn = An , Pn = Bn exp(–iβξ) .We shall make use of two conserved quantities of the corresponding evolution equations,namely the norm

(33)

and the Hamiltonian

(34)

Those are the discrete versions of the continuous quantities (4)-(6). However, noticethat one crucial difference between the continuous and the discrete systems is that noanalog to the transverse momentum (7) is known in the discrete case.

250

Figure 11: Typical shape of two discrete solitons (β = 0). (a): κ1 = 1; (b): κ1 = 5.

6.3 The Simplest Discrete Solitons

Let us consider solutions of (32) with the form An = q n exp(iκ 1 ξ), B n = pn exp( iκ2ξ) ,where for simplicity here the numbers qn , pn are assumed to be real. As always, A n

and B n ought to be phase-locked, so that κ 2 = 2κ 1 + β. One readily finds that suchstationary solutions realize the extrema of the Hamiltonian for a given norm, with κ1

being the corresponding Lagrange multiplier. The difference equations obeyed by theseries of amplitudes qn and p n are

(35)

Similarly to other discrete systems, eqs. (35) have several different types of localizedsolutions. Here we only consider the simplest ones, namely those that have a brightshape and a constant phase across the lattice (i.e., “unstaggered”), and that have asingle maximum that is located in one lattice site (i.e., “on-site” solitons). To elucidatethe existence of such solutions one might solve numerically (35) searching for convergingseries q n, pn , with the symmetry conditions qo > qn for n > 0, and qn = q – n , andsimilarly for pn . Other families of solutions, including “inter-site” (i.e., q )n = q– n + 1

and “staggered” solitons, also exist. Figure 11 shows the typical shapes of two discretesolitons. The most interesting is the strongly localized soliton in Fig. 11(b).

By and large, the properties and features of discrete solitons can be drasticallydifferent from those of their continuous counterparts. Solitons of the continuous andthe discrete NLSE provide a paradigmatic example. Discreteness modifies the shapeand confinement features of the solitons, and modifies the existence conditions andproperties of walking solitons that move across the lattice. In particular, this includesthe amplitude-width relations of the families of solitons, and in the case of solitonssupported by quadratic nonlinearities, the fraction of energy (or norm) in each modeforming the soliton, that we discussed in 3.4 for the continuous solitons.

As an illustrative example, let us examine the ratio of the peak amplitudes of thetwo modes forming the discrete soliton, i.e. q0/p0 . For the continuous families of solitons

251

such ratio is given by32,33

(36)

According to the similarity rules (14), at phase-matching (β = 0) all the continuoussolitons are self-similar. Thus, the ratio q(0)/p (0) is identical for all members of thesoliton family. Numerically, q (0)/ p(0) 1.2069. The discreteness of (35) breaks suchsimilariry rules, so that in particular q0 /p0 is no longer constant at β = 0. For example,in the case of the weakly-localized soliton of Fig. 11(a) one has q0 / p0 1.20, whereasfor the strongly localized soliton of Fig. 11(b) one finds q0 / p0 1.33.

Discreteness introduces a variety of other differences. Other discretizations of (1)different from (32), and the corresponding discrete solitons, might be also of interest.

7 Concluding Remarks

In these lectures we have only examined the basics of solitons in quadratic nonlinearmedia. Specifically, we only considered the simplest bright solitons, mostly in TypeI phase-matching geometries. However, much more is known.8 3 Important investiga-tions have been reported by many authors about Type II geometries,8 4 – 8 6 dark-like

and symbiotic solitons, different types of temporal solitons,19–21 ,29–32 ,87–91 includingThirring and gap solitons, 9 2 – 9 5 higher-order soliton solutions,96–99 soliton interactionsand collisions, 1 0 0 – 1 0 3 the effects of higher-order nonlinearities that may be present inaddition to the quadratic nonlinearity,1 0 4 – 1 0 9 magneto-optic effects, 1 1 0 modulational in-stabilities in various geometries and dimensionalities,36 ,37 ,71–73 ,111–113 some of which havebeen observed experimentally,1 8 and so forth.1 1 4 – 1 1 6 A few applications of the solitons,mainly those potentially relevant to all-optical switching schemes but also other appli-cations, have been examined and some experimentally observed. 52,53,117

The field is growing vigorously, theoretically and experimentally, and new areaswhere spatial, temporal and spatio-temporal solitons in quadratic nonlinear media mighthave important applications are emerging already. Optical cavities containing quadraticnonlinear crystals and quantum optical devices are fascinating and promising examples.In principle, new and existing materials with very high quadratic nonlinearities holdvery promise for the future to reduce the power requirements to form solitons and hencerender them closer to practical applications.

From a broader viewpoint, it is worth stressing that solitons in quadratic nonlinearmedia can potentially have important implications not only to various parts of nonlinearoptics, but also to other branches of nonlinear science. That is so because parametricwave mixing is a universal phenomenon. 118 Therefore, the formation of stable multidi-mensional soliton entities in mathematically similar, but physically different settings isa potentially important possibility that has to be explored.

252

Acknowledgements

This work has been supported by the Spanish Government under grant PB95-0768. Myown work in the topic addressed in these lectures has been done in close collaborationwith many colleagues, including George I. Stegeman, Curtis R. Menyuk, Dumitru Mi-halache, Dumitru Mazilu, Juan P. Torres, Nail N. Akhmediev, William E. Torruellas,Ewan M. Wright, Dmitrii V. Petrov, José-Maria Soto-Crespo and Maria C. Santos. Iam most grateful to all of them. The numerical work has been carried out at the C 4-Centre de Computació i Comunicacions de Catalunya. Support by the European Unionthrough the Human Capital and Mobility Programme is gratefully acknowledged.

A Type II

The normalized evolution equations for the slowly-varying field envelopes in Type IIphase-matching geometries for second-harmonic generation can be written as

(37)

where a 1, a2 and a3 are the normalized envelopes of the ordinary polarized fundamentalbeam, the extraordinary polarized fundamental beam and the second-harmonic beam,respectively. In the case of spatial solitons α 1 = –1, α 2 –1, and α3 –0.5. Theequations for Type I are obtained from these equations by setting a 2 = a 1 = aω , α1 =

α 2 = r, α 3 = α, δ 2 = 0, δ3 = δ, and a 3 = a2 ω . Equations (37) have several conservedquantities, including the corresponding Hamiltonian. For our present purposes, we onlyneed the energy flow given by the Manley-Rowe relation

(38)

the unbalancing between the energies of the two fundamental waves Iu = I1 – I2 , andthe transverse beam momentum

(39)Non-walking solitons of (37) are a two-parameter family, whereas walking solitons con-stitute a three-parameter family. Physically, such parameters are related to the totalenergy flow, to the unbalancing energy and to the soliton velocity. The extra degreeof freedom relative to Type I geometries, namely the unbalancing energy, has impor-tant implications. 84 In particular, it can be used to control the velocity of the walking

253

solitons. The analogous of Eq. (26) but for Type II geometries writes

254

(40)

Changing the polarization of the input light at the fundamental frequency modifies theratios I2 /I and I3 /I, hence it changes the velocity of the excited walking soliton.53,55

The properties of the families of solitons in Type II geometries can be found in refs. 84and 85. Walking solitons have been also studied. 86

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258

PHOTOTOREFRACTIVE SPATIAL SOLITONS

Mordechai Segev¹ , Bruno Crosignani² , Paolo. Di Porto²,Ming-feng Shih¹ , Zhigang Chen¹ , Matthew Mitchell¹ and Greg Salamo¹

¹Electrical Engineering Department, PrincetonUniversity, Princeton, New Jersey 08544²Dipartimento di Fisica, Universita' dell'Aquila, 67010L'Aquila, Italy and Fondazione Ugo Bordoni, Roma, Italy³Physics Department, University of Arkansas,Fayetteville, Arkansas 72701

INTRODUCTION

The advent of Nonlinear Optics, mainly due to the work of Bloembergen and coworkersat the beginning of the '60, has opened the way to a number of fundamental discoveries andapplications [1]. No matter how sophisticated the microscopic or phenomenologicalapproach adopted to deal with this topic, one of the central problems to the completeunderstanding of nonlinear processes is always the solution of the associated wavepropagation equation, which is, of course, intrinsically nonlinear. This circumstance can beconsidered in many cases as an obstacle to a full comprehension of the problem, due to thewell-known mathematical difficulties usually encountered when trying to solve this type ofequations. However, it is precisely the nonlinear nature of the wave equation which givesrise to a wealth of solutions, and thus of possible behaviors of the propagating field, andmakes nonlinear optics much more interesting that its linear counterpart.

A typical situation is the one associated with the so-called optical Kerr effect. In thiscase, the third-order polarizability takes a particularly simple form and the nonlinearity ischaracterized by a contribution to the refractive index n proportional to the intensity I ofthe propagating field, that is n=n +n I. Inserting this expression into the wave propagation1 2

equation allows one to deduce the equation of evolution of the field amplitude in the formof a spatio-temporal partial differential equation, usually referred to as nonlinearSchrödinger equation (NLSE), whose solutions have been investigated in great detailstarting with the pioneering work of Chiao et al. [2] and of Zakharov and Shabat [3]. In

Beam Shaping and Control with Nonlinear Optics259Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

particular, this analysis has shown the existence of a peculiar kind of solutions (solitons),able to propagate without distortion (or periodically recovering their initial shape), and hasgiven rise to an entirely new field of applied research. [4] [5]. It is now well established thatsolitons are a general phenomenon in nature: they appear in many fields in which wavespropagate and can exhibit both dispersion and nonlinearities, including surface waves influids (where they were initially discovered), volume fluid waves (deep sea waves), chargedensity waves in plasma and in solid state, etc. [6]. In Optics, two distinct types ofsolitons are known: temporal solitons [4] and spatial solitons [5]. Temporal solitons arepulses of very short duration that maintain their temporal shape while propagating overlong distances. They are now routinely generated [7] and are the backbone of future high-speed telecommunication links. Conversely, it is much more difficult to generate andobserve spatial solitons, in particular those that resulted from the Kerr effect (so-calledKerr-type solitons), which are optical beams that propagate without diffraction (stationarysolutions of the NLSE), for several reasons. First, because the nonlinear change in therefractive index scales with the optical intensity and since the values assumed by n2 a r echaracteristically very small (e.g., in silica glasses n2 is of the order of 10

- 1 6 cm² /Watt),

very large optical power densities are required [8]. Second, Kerr-type spatial solitons arestable only in two-dimensional systems, i.e., one longitudinal dimension along which thebeam propagates and one transverse dimension in which the beam diffracts or self-traps(this configuration is often referred to as a (1+1) D system). Full 3D optical beams undergocatastrophic self-focusing [9], and (1+1) D beams become transversely unstable [10] whenthey propagate in self-focusing Kerr-like nonlinear media. This means that Kerr-type spatialoptical solitons can be observed in single-mode waveguides, as demonstrated in [8]. In thisrespect, there is an apparent asymmetry between temporal and spatial solitons,notwithstanding the intrinsic spatio-temporal symmetry of the wave equation, due to theadditional dimensionality in the latter case.

Optical spatial solitons have undergone a sharp and dramatic conceptual change duringthe last few years. It has been driven by the discovery of three distinct nonlinearmechanisms that have been shown to support three dimensional [i.e., (2+1) D] solitons[11], that is, beams that are self-trapped in both transverse dimensions: the photorefractivenonlinearity [12,13], the quadratic nonlinearity [14], and the vicinity of an electronicresonance in atoms (or molecules) [15]. This article is dedicated to solitons inphotorefractive media.

The photorefractive nonlinearity occurs in high-quality lightly-doped electro-opticcrystals, such as BaTiO 3 (barium titanate), LiNbO3 (lithium niobate), SBN (strontiumbarium niobate). The photorefractive effect [16] was originally interpreted as an "opticaldamage" of the crystal provoked by the optical beam. It exhibits a reversible variation ofthe refractive index induced by the spatial variation of the optical intensity. Thismechanism, which is, by comparison to standard nonlinear optical effects, rather slow but iseffective at extremely small optical powers, possesses the capability of recording in realtime the information encoded in the spatial modulation of the beam and, because of thiscapability, has been widely used in the frame of real-time holography and optical phase-conjugation. [17], [18],[19]. These properties suggested to look for propagation of spatialsolitons in photorefractive materials, hoping that, in the nonlinear regime where the inducedrefractive index variation affects the very beam which has produced it, the beam diffractioncould be compensated by its self-focusing and self-trapped (non-diffracting) propagationwould result [12]. In fact, this hope has been more than justified by the great deal oftheoretical and experimental results which have been obtained in the last five years andwhich have shown the existence of one and two-dimensional photorefractive spatialsolitons, bright and dark, of vortex solitons, of photovoltaic solitons, together with a

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number of interesting applications (like, e.g., soliton-induced waveguides in bulkphotorefractive media) and of self-trapping of incoherent light beams.

In this review, our scope is to introduce the general topic of nonlinear propagation inphotorefractive media and focus more specifically on photorefractive solitons. We will startwith the basic band transport model of the photorefractive effect [20,21], which suffices fortreating most of the situations we will be dealing with. As a first application, this will allowus to consider the concept of photorefractive gratings and to introduce the formalism oftwo-wave mixing as the simplest example of nonlinear propagation. We will then considerthe propagation of a generic beam in a photorefractive crystal and introduce the formalismfor describing this kind of propagation. Following that we will examine the conditionsunder which it should be possible to find self-trapped solutions, that is, particular transversebeam profiles which propagate without distortion (photorefractive solitons). Afterreviewing the main theoretical results which have been established up to now, we willpresent some of the impressive experimental evidence which confirms the greatpossibilities of application of the PR effect and discuss interactions between (or among)solitons.

THE PHOTOREFRACTIVE EFFECT

The photorefractive nonlinearity gives rise to light-induced changes in the refractiveindex of certain types of (typically non-centrosymmetric) crystals. More specifically, thespatial ditribution of the intensity of the optical beam (or beams) gives rise to aninhomogeneous excitation of charge carriers, which in turn produces, by redistributingthemselves through diffusion and drift, a space charge separation whose associated electricfield modifies the refractive index of the crystal via the Pockels effect. This modification,the photorefractive effect, can be qualitatively described by means of the simple bandtransport model introduced in [21]. More precisely, let us refer to Fig. 3.1 of Ref. [18], inwhich the energy diagram of a typical photorefractive material is shown. A largeconcentration (1018 -10 19 cm- 3) of identical, uniformly distributed, donor impurities, whoseenergy state lies somewhere in the middle of the bandgap, can be ionized by absorbingphotons. Correspondingly, the generated electrons are excited in the conduction bandwhere they are free to diffuse or to drift under the combined influence of self-generated andexternal field (if any). During this process, some of the electrons are captured by ionizeddonors neutralizing them, the successive ionization rate being proportional to the localillumination. In this way, the ensemble of the non-mobile donors acquires aninhomogeneous charge distribution which tends to be positive in the illuminated regionsand to vanish in the dark regions. The combined presence of this charge distribution, of theelectrons in the conduction band and of a number of ionized acceptors present in the crystal[22], gives rise to a low-frequency electric field (space-charge field, E). It is this field that,through the standard mechanism of Pockels' effect (δn α r E, where r is some electro-opticcoefficient of the crystal), produces the refractive index variation responsible for thephotorefractive (PR) effect.

In order to translate in quantitative terms the above considerations, we need to writedown the equations necessary to determine the space charge field E. The first one is therate equation describing the donor ionization rate as a result of the competition betweenthermal and light induced ionization and recombination with free electron charges, that is

(1)

261

where I is the optical intensity, ND the donor density , the ionized donor density, Ne theelectron density, β the thermal generation rate, s the photoionization cross-section and γthe recombination rate coefficient. The second equation is the Gauss Law

, (2)

where is the low-frequency dielectric constant and the charge density ρ is given by

, (3)

-q and NA representing the electron charge and the acceptor density, respectively. The lastrelevant equation is the continuity equation

(4)

where the current density

(5)

, (6)

where is the distance between the two crystal faces to which the external bias staticpotential V is applied.

choosing x,y,z along the principal dielectric axes of the crystal, for the typical valuesassumed by the space-charge field, it is possible to write [17]

We will first consider the set of Eqs.(1)-(6) as if the optical light intensity,associated with the field Eop propagating at optical frequencies were a prescribed function.The main task of determining Eop will then be faced by solving in a self-consistent way thepertinent wave equation, written in the presence of the induced tensorial refractive indexcontribution associated with the linear electrooptic (Pockels) effect . More precisely, by

(7)

where rijk is the linear electrooptic tensor (of rank three).

THE SPACE-CHARGE FIELD

We start dealing with the steady-state situation in which the time derivatives in Eqs.(1)and (4) can be assumed to vanish (actually, as we shall see in Sect.4, this corresponds toconsider time larger than the so-called dielectric relaxation time). In this asymptotic

The set of Eqs. (1)-(5) has to be supplemented by the appropriate boundary condition:is the sum of the drift and diffusion contributions, µ representing the electron mobility.

262

regime, it is possible to show the possibility of achieving self-trapped beam propagation(screening solitons). The derivation presented in the next Sections closely follows thatworked out in [23].

Let us first eliminate between Eqs.(1) and (2). With the help of Eq.(3), we obtain,in terms of suitable normalized quantities and assuming Ne << NA ,

(8)

Eq.(8) expresses the relation between the normalized electron density and thenormalized space-charge field, respectively given by

(9)

and Q= (I+Id )/Id , that is

where

(10)

(11)

length and the diffusion field ED = (KBT/q)k D .

Eq. (5), a single equation connecting Y to Q, that isEquation (8) can now be inserted into Eq.(4), which yields, after taking advantage of

(12)

Equation (12) can be cast in a simplified if approximate if the quantity Y isneglected with respect to one (and thus to α which is typically much larger than one), anhypothesis which needs to be verified a posteriori after Y is explicitly found. With thisassumption, Eq. (12) becomes

(13)

In the following, we will consider situations in which the light intensity I is independentfrom z (or, at least, its scale of variation over z is much larger than those over x and y), sothat Y and Q can both depend on the transverse coordinates x and y (2-D case) or on xalone (1-D case).

having introduced α=(ND - NA )/NA (typically a number much larger than one), β1 = β /γ,

s1 = s/ γ , the dark intensity I d =β1 /s1 , the normalized intensity |u|²=I/I d , the D ebye

263

In the 2-D case, the most general solution of Eq.( 13) can be written, with the sameboundary conditions as in the 1-D case, in the form

In the 1-D case, Eq.(13), together with the proper boundary conditions Ey=Ez=0 for x=

± /2 and Ex(x= /2)=Ex(x=- /2)=EDY0 , admits of the simple solution

where

(14)

(15)

where are unit vectors in the x and y

direction and f(ξ,η) is an arbitrary function which can be determined by imposing thecondition (the electric field has to be conservative)

(16)

By doing that, we obtain an equation for f(ξ,η) which reads

(17)

which has to be solved with the appropriate boundary conditions at zero and infinity.

NONLINEAR OPTICAL BEAM PROPAGATION IN PHOTOREFRACTIVEMEDIA

In the previous derivations, |u|2 has been assumed to be a prescribed function. Ingeneral, once in possession of the expression of Y, i.e., of E, the next step is to solve in aself-consistent way Helmholtz’s equation for the propagating optical (high-frequency) fieldin the presence of the refractive index determined by Eq.(7). Actually, in most cases, Eq.(7)can be reduced to a scalar form by noting that very often one of the diagonal components

of the electro-optic tensor ( say r ≡ r xx x = r 3 33 ) is much larger than all other components,

so that if both the applied bias field E and the optical field Eop are directed along the x-axis(parallel to the crystalline c-axis), the problem becomes a scalar one. This, in turn, allows todescribe optical propagation by using the scalar Helmholtz equation for the x-componentof Eop . Furthermore, if we write

(18)

264

where r⊥ =(x,y) and k=(ω/c)n1, and assume the validity of the slowly-varyingapproximation for the field amplitude A, Helmholtz’s equation can be approximated by theparabolic (Fock-Leontovich) wave equation

(19)

where It is convenient at this point to introduce suitableadimensional variables defined through the relations

(20)

In these new variables, and recalling Eq.(15), Helmholtz’s equation takes the form

(21)

where f and we have neglected

the diffusive term which is responsible for self-bending effects [24].Equation (21) has to be solved together with Eq.(17), which can be rewritten in the form

(22)

The set of coupled equations (21) and (22) provides the analytical description ofnonlinear propagation in the general 2-D case.[23]

SELF-TRAPPED BEAM PROPAGATION

In the present Section, we examine the possibility of propagating self-trapped beamsinside the PR crystal by exploiting an exact compensation between diffraction and nonlinearself-focusing induced by the PR effect. The results of the previous Section, showing thatthe diffusion term in the space-charge field is responsible for an energy transfer betweentwo different plane waves while the nondiffusive one induces a phase change but no energytransfer, indicates that the presence of the bias field E0 is fundamental for achieving self-trapped propagation. In fact, diffraction involves accumulation of phases that are linearwith the propagation distance to each individual plane-wave component of the beam whichcannot be compensated by power exchange but, eventually, by a strong phase coupling. Itis therefore desirable to eliminate the power-exchange term in the space charge fieldaltogether. Although this cannot be accomplished in full (diffusion always exists), in allpractical situations of self-trapped propagation of a single beam, the diffusion field (the firstterm on the RHS of Eq. (14)) is very small as compared to the "screening field" (the secondterm on the RHS in Eq. (14)), which is proportional to E0 and gives rise to solitons. Thisapproximation is valid for all types of PR solitons found thus far and occurs whenever thebeam diameter is larger than ~5 µm [the diffusion field is ~ 50 V/cm whereas the trappingfield E0 is typically larger than 1 kV/cm]. Accordingly, we shall consider in the following

265

situations where the first term on the RHS of Eq.(14), which is independent from E0, canbe neglected. [24]

Figure 1. Formation of a bright photorefractive screening soliton.

266

Spatial screening solitons in one transverse dimension

Screening solitons were first predicted in [25],[26] and independently in [27]. Atpresent, most of the experimental studies with PR solitons employ the (1+1) D screeningsolitons and

they are considered best understood among all other types of PR solitons. An extensivereview of the theoretical derivation of screening solitons (including those at very highoptical intensities) can be found in Ref. [26], along with all a discussion on the physicalmaterials parameters and proposed applications. Here we summarize the main steps in thetheoretical formulation of screening solitons.

Intuitively, one may view the formation of bright (low intensity) screening solitons in thefollowing manner. A narrow light beam propagates in the center of a biased dielectricmedium. As a result of the illumination, in the illuminated region the conductivity increasesand the resistivity decreases. Therefore, the voltage drops primarily in the dark regions (asimple voltage divider) and this leads to a large space charge field there. The refractiveindex changes in proportion with the space charge field.If δn is positive, this process results in an antiguide (a large positive index change in thedark regions) which would strongly defocus the beam. However, if δn is negative, thelarge negative index change in the dark regions creates a "graded index waveguide" thatguides the beam that generated it, in a self-consistent manner as illustrated in Fig. 1.

For a rigorous treatment of the underlying theory, we recall the results of Sections 3 and4. The wave equation for the field amplitude u reads

We look for self-trapped (stationary) solutions, which correspond to propagationwithout change in the beam transverse profile, of the form

(23)

(24)

where β is real, and we now defined having introduced thebackground illumination Ib which represents the intensity of a background optical beamused to illluminate the crystal uniformly (thus artificially increasing the dark current Id). By

assuming to be a real function (which means that we do not analyze here greysolitons, which were discussed in Ref. [27], or higher-order solitons), and indicating with a

prime the derivative with respect to the argument , Eq.(23) becomes

(25)

Equation (25) respectively describes the case of a 1D beam which goes asymptotically(actually, on a transverse spatial scale much shorter than the distance between theelectrodes) to zero (u0=0, bright soliton) or to a constant value u0 different from zero(dark solitons). Note that in both cases it reduces, for u<<1, to the well-known nonlinear

267

Schrödinger equation describing propagation in the presence of the optical Kerr effect[26]. It can be integrated by quadrature, thus transforming it in a first-order equation, bymeans of the change of variable

(26)

which implies

(27)

so that it becomes

(28)

For bright solitons one requires the boundary condition u(±∞)= 0, while for dark solitons

Let us first consider dark solitons. Since the structure of Eq.(25) implies an asymptoticbehavior of the kind

Figure 2. Width (FWHM of the intensity, u2, profile) of a bright low-intensity soliton (upper curve) as afunction of u(0) and of a dark low-intensity soliton (lower curve) as a function of u(∞).

268

the only possibility consistent with the boundary condition is β+s=0. Furthermore, sincethe dark soliton condition requires u(0)=0, Eq.(25) yields u"(0)=0,which is consistent withan odd

solution , corresponding to in the neighborhood of = 0. On the otherhand, in the same zone the second term of Eq.(25) is negligible with respect to the third

one (in fact, ), so that s has to be negative, i.e., s=-1, and, consequently, β=1.Finally, by imposing in Eq.(28) the condition p(u=±u0)=0, it is possible to find p(u=0) sothat Eq.(28) can be rewritten as

(29)

(30)

Let us now consider bright solitons. In this case, the asymptotic behavior furnished byEq.(29) implies β+s>0 . Besides, if we put p(0)=0 in Eq.(28), its asymptotic behavior for

yields β = –sln[1+ u2 (0)]/ u2(0). Thus, the requirement β+s > 0 is possible

only if s=1. If we now specializes Eq.(25) around = 0 , we obtain

which is consistent with an even solution. By imposingin Eq.(28) the condition p[u=u(0)]=0, it is possible to find p(u=0) and rewrite it as

(31)

We are now in the position to check the validity of the approximation

underlying the derivation of Eq.(25). To this end, we use Eq. (14) (neglecting the diffusionfield) in order to write ∇ . Y in the form

(32)

If we now introduce Eqs.(30) or (31) into Eq.(32), it is possible to check in a self-consistent way the validity of the approximation leading to Eq.(25), that consisted inneglecting dYX/dξ with respect to unity. The integration of Eq. (31) can be accomplishedonly numerically. The problem has been extensively treated in [25], [26] and [27] and thesoliton waveforms are shown there. Here, we show only the corresponding solitonexistence curves, that is the relation which has to be satisfied between the full width at half-maximum (FWHM) and u(0) (bright) or u0 (dark) in order to assure the existence of thesoliton. The existence curves of both bright and dark solitons are shown in Fig. 2.

Spatial screening solitons in two transverse dimensions

We look for self-trapped solutions of Eq. (21), that is, for solutions whose transverseprofile remains the same during propagation (spatial solitons), of the kind

(33)

269

With an argument similar to that adopted for the 1-D solitons, it is possible to show thats=1 , β = p 2 – 1 and s=-1 , β=1 for bright and dark solitons, respectively. [23] If we now

introduce polar coordinates ρ and θ thorough the relations and

write, respectively for bright and dark vortex solitons, o r= w(ρ ) exp (im θ ), where w is a real function and m is the so called topological

charge (which can assume only integer values, [12]), Eq.(21) takes in the two cases theform

(34)

(35)

and

where W 0 is the boundary value of w at large distance. They have to be solved togetherwith Eq.(22), that is

where Q = 1 + w2 and g has to go to zero for large ρ.The problem of finding cylindrically symmetric solutions of this set of equations has

been extensively treated in [23] and the interested reader is referred to it..

(36)

DIFFERENT KINDS OF SPATIAL SOLITONS : QUASI-STEADY-STATE ANDPHOTOVOLTAIC SOLITONS

In the previous sections, we have investigated the possibility of generating spatialsolitons in the asymptotic temporal regime in which the space-charge field has reachedsteady-state (see Sect. 4). On its way toward equilibrium, the space-charge field canassume , in a quasi-steady-state time-window, values and spatial distributions such that it ispossible to propagate spatial solitons. These possess properties different from those ofscreening solitons, like, e.g., their independence on the absolute light intensity and its ratioto dark irradiance. While a large amount of experimental evidence shows the existence andallows to determine the properties of these quasi-steady-state spatial solitons [13],[28]-[34], their analytical description is still based on a phenomenological approach[12],[35],[36] and a recent study of a numerical nature [37] in the frame of (1+1) D theory.We refer the interested reader to a recent review [38] in which quasi-steady-statephotorefractive solitons have been described in detail.

In some photorefractive crystals, such as LiNbO3 , BaTiO3 and LiTaO3 , uniformillumination can generate (without any external field) a bias field driven by the bulkphotovoltaic (or photogalvanic ) effect [39],[40]. The bulk photovoltaic effect arises from

270

free charge carriers that are optically-excited from deep donor levels that reside in non-centrosymmetric potential (i.e., dopants that substitute a particular atom in the crystallinematrix of a non-centrosymmetric crystal). The photo-excited carriers possess excess kineticenergy in a preferential crystalline direction (the direction of the lower potential) and,before they "thermalize" (scatter from the lattice and/or from other electrons, impurities,etc.), give rise to a polar current. The photovoltaic current Jp v depends on the optical

polarization and intensity I through a third rank tensor κ and gives rise to an additionalterm in the current density of Eq.(5) of the form

(37)

In ferroelectric photorefractive crystals, the photovoltaic current is predominantly oriented

along the ferroelectric axis (c) and simplifies to (note thatphotovoltaic currents driven by circular polarization also exist, as described in Ref. [40]).The bulk photovoltaic effect gives rise to a new kind of soliton, namely the photovoltaicsoliton [41], both in the open and closed circuit configurations [42],[43]. Photovoltaicdark solitons and vortex solitons have been recently observed experimentally [44],[45] anda recent report [46] has presented a z-scan study of the photovoltaic self-defocusingnonlinearity.

EXPERIMENTS WITH SCREENING SOLITONS

Photorefractive screening solitons have been extensively investigated, both theoreticallyand experimentally, during the last three years. At present, the existence of both (1+1) Dand of (2+1) D is well established, in both bright and dark realizations. In the last Sections,we provide an overview of the most significant experimental work on screening solitons,along with more advanced topics such as soliton interactions ("collisions"), and self-trapping of incoherent light beams.

Observations of bright screening solitons

In Sects. 5.1 and 5.2 we have analytically investigated the possibility of propagating self-trapped beams (both in 1 and 2 dimensions), in the asymptotic time regime where thespace-charge field has reached its asymptotic value. We recall that the main feature of thesePR screening solitons is the existence of a unique relation among the soliton width, thetrapping voltage and the intensity ratio (the ratio of the soliton peak irradiance to the sumof the equivalent dark irradiance and uniform background irradiance), a large deviationfrom this relation (which we call existence curve , see Fig. 2) proving unsuitable for solitonpropagation. In particular, the most favorable operation point (i.e., the point on theexistence curve at which the narrowest soliton can be trapped with a given nonlinearity

) is at the minimum of the existence curve, that is, it is obtained atintensity ratio roughly equal 2.4 [26], which means that the solitons peak intensity shouldbe roughly 2.4 times the sum of the dark plus background intensities. The dark irradiance isvery small in all dielectric photorefractive media (on the order of mWatt/cm² -µWatts/cm² ), so at the absence of background illumination solitons can, in principle, beobserved with nWatt (or less) optical powers. However, operation at ultra-low opticalpowers poses a real experimental challenge, because the soliton formation time is of theorder of the dielectric relaxation time, which in turn is inversely proportional to the solitonintensity. For very long (hours) formation times, small vibrations can eliminate any

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observation by distributing the space charge field over a region (in space) much broaderthan the optical beam. A second alternative for generating bright screening solitonscorresponds to operating at very large intensity ratios while keeping the soliton intensitywithin a reasonable range. But, from the existence curve it is apparent that this optionrequires very large applied fields. For example, a ~ 1 µWatt soliton is more intense thanthe dark irradiance by at least 106 times, which implies a required trapping voltage of morethan 10 6 Volt, to be applied across 5-mm electrodes - also unphysical (leads to surfacearcing and at higher field to dielectric breakdown). The third choice is to generate"artificial" dark irradiance, using uniform background illumination, and controlling itsmagnitude relative to the soliton peak intensity. The background beam is typically a laserbeam polarized orthogonal to the soliton beam and co-propagates with it. In this manner,both beams experience almost the same absorption (thus the soliton peak amplitude, u 0 ,remains unchanged throughout propagation despite the absorption), nonetheless, thebeams can be easily separated from each other after the crystal by use of a simple polarizer.This configuration also compensates for absorption, that must be fundamentally present inall PR media, because both the soliton and background beams experience roughly the sameabsorption so that the ratio between them is maintained, enabling soliton propagation overlarge distances (many absorption lengths are possible; in fact, the limiting factor is possibledichroism, that gives rise to a small difference between the absorption constants of bothpolarizations). This is the commonly used method for generating bright screening solitons.It was first used by Iturbe-Castillo et al. [47] for making the first observation of steady-state self-focusing effects in biased photorefractive crystals. An identical configuration wasused by Shih et al. to observe the first screening solitons [48], and has been used since thenfor all experiments with bright screening solitons.

At present, both (2+1) D [48],[49] and (1+1) D [50] bright screening solitons havebeen observed. A typical top view photograph on a (2+1) D screening soliton in SBN isshown in Fig. 3 (top), where, for comparison, we also show the normally-diffracting beamwhen the applied voltage is set to zero (bottom). It is well established now that both(2+1) D and (1+1) D cases exhibit stable self-trapped propagation over many (>10)diffraction lengths and robustness against localized perturbations (of a transverse scalemuch smaller than the soliton width) that could have fairy large amplitudes [49],[50]. Thesolitons are also stable against considerable (~10%) deviations in their initial conditions,i.e., the input beam profile, width, and phase profile [50]. In characterizing the solitonstability properties, it turns out that the existence curve plays a far more important role insoliton formation than initially thought. As evident from a series of papers on (1+1) Dscreening solitons (the first one being Ref. [50]), all these observations were made in 3Dbulk crystals. This is despite the fact that any (1+1) D beams propagating in a three-dimensional (bulk) Kerr-like nonlinear medium should be transversely unstable [10].Obviously, (1+1) D photorefractive solitons behave in a manner different than their Kerrcounterparts. From many experiments with photorefractive screening solitons (in fact, withany photorefractive solitons), it is apparent that, as long as the parameters of the beam andnonlinearity are "on", or close to, the existence curve, and as long as the intensity ratio isgreater than ~ 0.1 (i.e., the screening solitons are not in the Kerr regime but rather thenonlinearity is saturated, at least in the center of the beam [26]), the transverse modulationinstability is "arrested" to a degree that it is not observed even within a propagationlength of > 15 diffraction lengths. (Many examples for this "instability arrest" can be foundin the references in the next sections on soliton interactions and collisions). However, if theparameters deviate considerably from the existence curve, transverse instabilities becomedominant and the one-dimensional beam becomes distorted and tends to break up into

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multiple filaments (see experiments in Ref. [50]). In this respect, it is instructive to mentionexperiments and numerical work that has reported just the opposite and has claimed thatthe (1+1) D screening solitons (bright and dark) always suffer from transverse instabilities[51]. It appears that the results presented in that paper were in error, as it contradicts alarge amount of experimental data obtained by several groups. The mistake in [51] hasbeen recently found by Infeld et al. [52], who have shown that bright (1+1) D screeningsolitons propagating in a 3D medium are transversely stable, in the conditions discussedabove are indeed satisfied.

At present (2+1) D screening solitons have been observed experimentally by severalgroups and their existence is well established (see, e.g., Refs. [48],[49] and many otherreferences in the next sections). Experimental work has shown that they are robust andstable against small perturbations (such as material inhomogeneities) as long their intensityratio is roughly larger than unity. Furthermore, as shown in [49], a Gaussian beam evolvesinto a soliton even if its initial parameters deviate from those that give rise to a soliton by~10%. In a manner similar to the (1+1) D screening solitons, it is evident that theparameters (2+1) D solitons must obey an existence curve. As shown in [49] and [48], 10-20% deviations from this curve (typically in the

Figure 3. A top view photograph of a 10 µm wide spatial soliton in a 5 mm long photorefractive crystal(top)), and, for comparison, the same beam diffracting naturally when the nonlinearity is « turned off »bottom.

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applied in the voltage) lead to elliptical self-trapped beams, whereas large enoughdeviations lead to instability and beam breakup [49]. However, at present the full theory forsuch solitons is not available yet, so a direct comparison (via the existence curve) betweentheory and experiments has not been demonstrated. In fact, early numerical work on (2+1)D beam propagation employing the photorefractive screening nonlinearity has steered muchcontroversy, as it failed to find (2+1) D solitons [53]. That paper states explicitly that theevolution of a Gaussian-type beam in a biased photorefractive crystal is characterized byoscillations of its diameters in both transverse axes and by spreading, uncharacteristic ofsolitons. The simulations in [53] have predicted oscillations in the beam diameter from 35µm to 17.5 µm every 2 mm and that these oscillating-beams cannot be axially symmetric.The experimental results(e.g., [49]), on the other hand, prove that the soliton beammaintains an axially-symmetric constant diameter and no oscillations in its diameter wereobserved. It is obvious that the experiments contradict those numerical simulations. Thereason for this contradiction is that solitons exist for a specific set of parameters: those thatare "on" the soliton existence curve. This means that, similar to the (1+1) D case, in a givencrystal, a soliton of a specific diameter (at a given wavelength) and at a given value ofintensity ratio, exists at a single value of the external field. For example, in the 1-D case"optical beams that significantly differ from soliton solutions tend to experience cycles ofcompression and expansion" (see [24]). The parameters used in the numerics of [53] weresimply too far from those that can support a soliton. Furthermore, similar numericalcalculation with (1+1) D screening solitons (for which the theory is well established) havefailed to find solitons [54], for the very same reason: the parameters did not correspond topa point "on" the existence curve. Altogether, this is not surprising given the extensiveliterature on solitons in Kerr media: Burak and Nasaski [55] have shown that a Gaussianbeam propagating in Kerr media is also characterized by oscillations in its diameter andthat the beam converges a soliton only if the parameters are close enough to the solitonparameters. This means that finding the set of parameters that supports a soliton istherefore the critical issue: knowing them, one can simulate numerically the evolution of aGaussian beam into a soliton, as done for the (1+1) D case. The soliton parameters,however, require an analytic treatment (it is an eigen-value problem) and are very hard tofind using multiple numerical simulations. A first analytic treatment was recently given inRef. [23], but, as explained there, the existence curve has been thus far found explicitlyonly for (2+1) D dark (vortex) solitons for intensity ratios < 2.

Finally, it is worth noting that both (1+1) D and (2+1) D screening solitons self-bendtowards a preferential direction (typically the c-axis in uniaxial PR crystals). This effect isdriven by the first correction to the space charge field, which is the first term on the RHS ofEq.(14). Self-bending was first predicted in Ref. [24] and first observed in [49].

Observations of dark screening solitons

Experiments with dark screening solitons fundamentally differ from those with brightsolitons inseveral important items (apart from the sign of the applied field, which controls the natureof the nonlinearity: self-focusing or self-defocusing). First, the existence curve of (1+1) Ddark screening solitons is a monotonically decreasing function of intensity ratio andconverges asymptotically to a constant value. This implies that the "most favorableobservation point" (the point on the existence curve at which the narrowest soliton can betrapped with a given nonlinearity, is obtained at the largest intensity ratio. The theoretical

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upper limit on this ratio is set by the validity of the approximation for the field at theelectrodes (see Appendix of [26]), whereas the experimental limit is obtained in the absenceof background illumination. Of course, this implies that, unlike their bright counterparts,dark screening solitons can be observed without additional background illumination. Thisis true only if the dark soliton is borne on an infinite beam, i.e., the notch-bearing soliton-forming beam covers the entire face of the crystal . This means that to observe dark PRscreening solitons one needs to launch a "true" dark soliton [56], rather than launching adark notch borne on a finite beam (which will form an "approximate" dark solitonwhenever the beam diameter is much larger than the width of the notch), as commonlydone with Kerr-like dark solitons [57],[58]. The second item is that fundamental darksolitons require a π phase shift in their center. Combing these two poses an experimentalchallenge: to obtain a narrow dark notch of a few µm width, in which the transverse phasejumps by π at its center, on the background of a uniform beam that covers the entire inputface of the photorefractive crystal. The first attempt to generate dark screening solitons[59] employed the "conventional" method of using a tilted glass plate [57],[58], in whichthe "aspect ratio" between the width of the notch and the diameter of the beam is restrictedby the thickness of the glass. Accordingly, this attempt has indeed demonstrated self-defocusing but self-trapping of the notch was not conclusive [59]. Later on, Chen et al.have used a different technique to obtained the necessary waveform [56]. They have used aλ /4 step mirror made of an InP wafer, of which one half is etched to a λ/4 depth. This step-mirror was illuminated by a collimated beam and its reflection provided a dark notch (dueto the π phase jump) in a broad beam of 1 cm in diameter. When this notch-bearing beampassed through a properly biased crystal, the formation of a fundamental dark soliton in

Figure 4. Typical experimental results of photographs and intensity profiles taken at the exit of a SBNcrystal The figure shows an-odd -number sequence of dark (1+1)D soilitons that have all evolved from asingle notch borne on an otherwise uniform beam. The applied field (and thus the nonlinearity) increasewith increasing number of dark solitons. This figure is taken from Ref. 61.

steady-state was observed. The shape-preserving behavior of the dark soliton wasconfirmed in our earlier experiments by measuring the beam profile as it propagates

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throughout a specially-cut (wedged) crystal and by guiding a beam of a differentwavelength [56]. Later experiments have demonstrated “higher-order” or “multiple” darksolitons, all evolving from a single input notch. In general, dark spatial solitons can begenerated by launching an optical beam with two different initial conditions. One is the“odd” initial condition, which provides a π phase shift into one half of the beam (known asphase discontinuity or phase jump). The other is the “even” initial condition, whichprovides an amplitude depression at the center and an even-symmetry in the phase of thebeam (known as amplitude discontinuity or amplitude jump). For PR screening solitons, ifthe initial width of the dark stripe is small, only a fundamental soliton [56] or a Y-junctionsoliton [60] is generated, corresponding to the lowest order in the odd- or even-numbersoliton sequence. As the initial width and the bias

field are increased, a progressive transition from a lower-order soliton to a sequence ofhigher-order multiple solitons is observed. Odd initial conditions always generates an oddnumber of dark solitons, whereas an even initial condition gives rise a an even number ofdark solitons (in this respect, the PR screening solitons are similar to Kerr-like darksolitons). A detailed review of multiple dark screening solitons can be found in Ref. [61].Typical experimental results of photographs and intensity profiles taken at the exit face of aSBN crystal are shown in Fig. 4. The figure shows an odd-number sequence of dark (1+1)D solitons that have all evolved from a single notch borne on an otherwise uniform beam.The applied field (and thus the nonlinearity) increase with increasing number of darksolitons. This figure is taken from Ref. [61]. For fundamental dark screening solitons, theanalytic theory (presented in the previous sections and in [25],[26],[27]) has predicted theexistence curve given in Fig. 2. A detailed experimental study of this curve [56] has showngood agreement with the theory. For higher-order dark solitons, the comparison betweentheory and experiments was based more on numerical methods, as an analytic theory forhigher order dark screening solitons is not available yet (in fact, such theory will be, mostprobably very challenging as this saturable nonlinearity is not integrable). Nevertheless, thecomparisons between experiments and numerical simulations on high-order dark screeningsolitons were shown to be in good agreement [61].

As for observation of (2+1) D dark screening solitons (namely, vortex solitons), thesame group that has tried to find bright screening solitons using numerical methods only(and did not succeed, see Ref. [53]), has attempted to find numerically vortex solitons, withno success either. This has lead them to believe that singly-charged vortex screeningsolitons cannot exist [62]. However, a recent paper by Chen et al. [63] has demonstratedjust that: a self-trapped singly charged vortex screening soliton. They key idea that hasenabled this observation was, once again, to launch a vortex nested on a broad beam thatcovers the entire PR crystal. In fact, the vortex soliton can be generated only when it isborne on an “infinite” beam, and it breaks up when it is nested on a finite donut-shapedbeam. When the vortex beam is a donut-shaped narrow beam, it breaks up into twoelongated “slices” (with a self-defocusing nonlinearity) or into two focused “filaments”(with a self-focusing nonlinearity) [63]. Typical experimental results showing the input(left), diffraction output (middle) and vortex screening soliton output (right), intensitydistributions in a SBN crystal are shown in Fig. 5.

WAVEGUIDES INDUCED BY PHOTOREFRACTIVE SCREENING SOLITONS

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Optical solitons are created when diffraction is balanced by self-focusing, that is, theoptical beam modifies the refractive index of the medium in such a way that diffraction iseliminated. In essence, a soliton forms when the beam induces a waveguide (via thenonlinearity) and, at the same time, is guided in the waveguide it has induced. This meansthat the soliton is (must be) a guided mode of the waveguide it induces. Although this “self-consistency” idea is fairly old [64], it has gained momentum in the last few years as Snyderand co-workers were able to use it for making predictions [65],[66],[67] on solitoninteractions, collisions, stability and other properties in a general nonlinear medium (apartfrom Kerr medium, for which analytic results are available [3]). Experimentally,waveguides induced by (1+1) D Kerr solitons [68],[69],[70] and by (2+1) D vortex (dark)Kerr solitons [71] have been demonstrated. Following the first observation ofphotorefractive solitons [13] it was clear that such solitons should also exhibit waveguidingproperties. Indeed, soon thereafter waveguides induced by quasi-steady-statephotorefractive dark solitons have been demonstrated [33].

Figure 5.Typical experimental results showing the input (left), diffraction output (middle) and vortexscreening soliton output (right).

Since the properties of all photorefractive spatial solitons greatly differ from thoseof Kerr-type solitons, one expects that the waveguides they induce will exhibit new featuresas well. First, photorefractive spatial solitons are stable when trapped in either one or twotransverse dimensions. This implies that they can induce two-dimensional waveguides,

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which indeed has been demonstrated by Shih et al. [72] with (2+1) D photorefractivescreening solitons. Second, the required optical power for formation of photorefractivesolitons is very low (as low as 1 micro Watt, compared to 100 kWatt for optical Kerrsolitons). This means that the waveguides can be induced by ~1 µWatt solitons [72].Finally, since the response of photorefractive media is wavelength-dependent, one cangenerate a soliton with a very weak beam and guide in it a much more intense beam of awavelength at which the material is less photosensitive [33]. This enables steering andcontrolling intense beams by weak (soliton) beams. Therefore, photorefractive spatialsolitons seem very promising for various applications, such as light controlling light, opticalwiring, near-field multi-channel to multi-channel interconnects, and frequency conversion inthe soliton-induced waveguides (see discussion on potential applications in Ref. [26]).

A recent paper [73] has presented a detailed experimental and theoretical study ofthe waveguiding properties of (1+1) D photorefractive screening solitons. It has shown thatthe number of possible guided modes in a waveguide induced by a bright screening solitondepends on the intensity ratio of the soliton. In particular, the number of guided modesincreases monotonically with increasing intensity ratio from a single mode. When intensityratio is much smaller than unity (the Kerr-limit [26]), the waveguide can support only onemode for the guided beams of the wavelengths equal to, or longer than, that of the soliton.At large intensity ratios the number of possible guided modes can be large. One can adjustthe numerical aperture of the soliton-induced waveguide while keeping the soliton beam ata fixed shape by simply “walking along the existence curve”, that is, varying the intensityratio and maintaining a soliton by adjusting the applied voltage according to that curve. Onthe other hand, waveguides induced by dark screening solitons can support only a singleguided mode for all intensity ratios.

For two-dimensional waveguides induced by (2+1) D screening solitons, it has beenfound [72] that the same principle holds, although no accurate comparison could be madebetween theory and experiments since the existence curve is available only for a limitedregime. It was also shown [72] that the 2D induced waveguide is roughly isotropic as longas the (2+1) D soliton is circular. On the other hand, when the self-trapped beam has anelliptical profile, the induced waveguide is anisotropic in its modal properties [74].

Finally, these studies of the properties of the soliton-induced waveguide haveproven to be useful for understanding and predicting interactions between solitons(collisions),. as explained in the next section.

SOLITON COLLISIONS

Amongst all soliton properties, interactions (commonly referred to as “collisions”)between solitons are perhaps their most fascinating feature, since, in many aspects, solitonsinteract like particles. As early as 1965, Zabusky and Kruskal have shown [75] that twointeracting (colliding) Kerr solitons always conserve their individual energies and linearmomentum, just like particles do. For this reason, Kruskal has named these nonlinearcreature “solitons” to emphasize the similarity between these localized wave-packets andparticles. Although it may seem rather logical that solitons possess this kind of behavior, itis by no means trivial. Keeping in mind that solitons result from nonlinearities, there is noseemingly straightforward reason as to why two such colliding solitons would not simplytake out of balance the entire nonlinear system that supports them and disintegrate intosmall fragments that have no particular order. For many years soliton interactions were

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solitons.

understood only in the framework of sophisticated numerical or analytical methods (e.g.,inverse-scattering transform [3]). The more intuitive insight suggested recently by Snyder[66], which employs the idea that solitons must be guided modes of the waveguides theythemselves induce, simplifies the understanding of soliton interactions by considering lightpropagation in waveguides at close proximity. Using this method, Snyder’s group wascapable of predicting new phenomena, such as fusion and fission of solitons upon collision[76], and soliton spiraling [77]. In any case, before 1990, all soliton interaction studieswere limited to collisions in a single plane, since practically all of the observed stable brightsolitons before that time were one dimensional [(1+1) D]. The discovery of stable (2+1) Dbright solitons (including photorefractive solitons which are described in detail in section7.1, quadratic solitons [14], and solitons in resonant atomic vapor [15]), has initiated theexploration of full 3D interactions between bright solitons. During the last 2-3 years theideas of soliton fusion, fission and spiraling upon collision were all demonstrated withphotorefractive screening solitons (see Refs. [78], [79] and [80], respectively). Similarobservations in other nonlinear media (e.g., [15]) confirm that these intriguing phenomenaare not restricted to photorefractive solitons but are rather general features of interacting

Figure 6. Illustration of coherent and incoherent interactions between solitons.

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To intuitively understand soliton collisions, one needs to consider two opticalwaveguides that are brought to close proximity. The optical beams guided in thesewaveguides (in the form of guided modes) overlap, primarily in the "center" regionbetween the waveguide where the evanescent "tails" of the modes coexist. Now, there aretwo possible scenarios for these self-trapped beams to interact: coherent versusincoherent interactions. Both types of interactions are illustrated in Fig. 6.

Coherent interactions occur when the nonlinear medium can respond to interferenceeffects between the overlapping beams. They occur in all nonlinearities with aninstantaneous (or extremely fast) time response (such as the optical Kerr effect and thequadratic nonlinearity). For all other nonlinearities that have a fairly long response time(e.g., photorefractive and thermal), the relative phase between the interacting beams mustbe kept stationary on a time scale much longer than the response time of the medium.When this occurs, the material responds to interference between the overlapping beams.When the beams have a zero relative phase ("in-phase"), they interfere coherently and theintensity in the center region between the induced waveguides is increased. In a self-focusing medium, this leads to an increase in the refractive index in that region, which inturn, attracts more light to the center, moving the centroid of the solitons towards it andhence the solitons appear to attract each other. When the interacting beams are π out ofphase from each other, they interfere destructively and the index in the center region islower that it would have been if the beams were far away from each other, As a result, thesolitons appear to repel each other. In fact, both "attraction" and "repulsion" betweensolitons are actually due to asymmetries in their induced waveguides that is caused (via thenonlinearity) by the close proximity of the beams.

Incoherent interactions occur when the relative phase between the (soliton) beams variesmuch faster than the response time of the medium. In this case, the medium cannot respondto inteference effects but responds only to the time-averaged intensity (average taken overa time longer than material response time), which is identical to a simple superposition ofthe intensities. Therefore, irrespective of their relative phase, the intensities of the beamsadd up and the intensity in the "center" region between the solitons is increased (ascompared to a single isolated beam). Since these solitons propagate in a self-focusingmedium, this leads to an increase in the refractive index in that region. As a result, morelight is "attracted" towards the center region and the solitons appear to attract each other.Such an incoherent "interaction force" is always attractive (for bright solitons), since theintensity in the center region cannot decrease by merely the coexistence of two solitonbeams at close proximity.

Collisions in Kerr media. The soliton interaction forces are, in principle, the same forall nonlinear media that can support solitons. There are, however, several very importantdifferences in the outcome of collision processes between Kerr-type and saturablenonlinear media. First, in Kerr media all solitons are (1+1) D and the collisions are boundto occur in one single plane. In addition, in Kerr media, all collisions are fully elastic, whichimplies that the number of solitons is always conserved. Furthermore, the system isintegrable, and therefore no energy is lost (to radiation waves) but rather conserved in eachsoliton. In addition, the trajectories and "propagation velocities" of the solitons recover totheir initial values after each collision (whether attractive or repulsive). This equivalencebetween solitons and particles is the reason for the term "soliton". In an attractive coherentcollision in Kerr media in which the solitons' trajectories are separated by a large angle, thesolitons simply go through each other and remain virtually unaffected by the collisionprocess (apart from a tiny displacement and a small change in absolute phase). When the

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attractive collision occurs at small angular separation between the solitons, they movetowards each other, combine and separate periodically. On the other hand, in a repulsiveKerr collision the solitons simply move away from each other. Coherent collisions in Kerrmedia were demonstrated in Refs. [82] and [83].

Collisions in photorefractive and in other saturable nonlinear media are, in manyaspects, much richer than those in Kerr media and therefore more interesting. First,saturable nonlinear media can support (2+1) D solitons and therefore collisions can occurin full 3D, giving rise to new effects that simply cannot exist in Kerr media. Second, asexplained above, the self-induced waveguides in saturable nonlinear media can guide morethan one mode. This gives rise to new phenomena, including soliton fusion, fission, andannihilation. In 1992 Gatz and Herrmann have found [84] numerically that solitons insaturable nonlinear media which undergo a coherent collision at shallow relative angles canfuse to each other. One year later,. Snyder and Sheppard have shown theoretically thatcolliding solitons can undergo "fission", that is, generate additional soliton states uponcollision, or, in other cases, annihilate each other.[85] Their explanation was simple andelegant: since both solitons induce waveguides, one needs to compare the collision angle tothe critical angle for guidance in these waveguide (that is, to the angle above which totalinternal reflection occurs and a beam is guided in the waveguide). If the collision occurs atan angle larger than the critical angle, the solitons simply go through each other unaffected(the beams refract twice while going through each other's induced waveguide but cannotcouple light into it). If the collision occurs at "shallow" angles, the beams can couple lightinto each other's induced waveguide. Now if the waveguide can guide only a single-mode(a single bound state), the collision outcome will be identical to that of a similar collision inKerr media. However, if the waveguide can guide more than one mode, and if the collisionis attractive, higher modes are excited in each waveguide and, in some cases, thewaveguides merge and the solitons fuse to form one soliton beam. Such a fusion process isalways followed by some small energy loss to radiation waves, much like inelastic collisionsbetween real particles. Experimentally, fusion of solitons was observed in all kinds ofsaturable nonlinear media: atomic vapor, photorefractives, and quadratic. Specifically withphotorefractive screening solitons, fusion of two colliding soliutons was observed inincohertent [78],[81] as well as in coherent [79],[86],[87] interactions. Fission ofphotorefractive screening solitons was observed by Krolikowski and Holmstrom.[79] Toillustrate these processes, we show the experimental results of collisions betweenphotorefractive solitons in Figs. 7 and 8. Figure 7 shows a top-view photograph of anattractive incoherent collision in which the solitons pass through each at a large angle. Anexample of fusion in Fig. 8 shows the intensity distribution a long distance after a collisionin which the same solitons collide at shallow angles and fuse to form a single beam.Soliton “collisions” for phase differences intermediate between 0 and π lead to energyexchange between the solitons. The interference pattern formed by the overlap of the“tails” of the solitons is intermediate in phase to that of both of the solitons so that poweris scattered from one soliton into the other. The higher intensity soliton narrows in space,and the weaker one broadens. These effects have been observed with screening solitons, asdescibed in Ref. [86].Since saturable nonlinear media can support (2+1) D solitons, one can also look atcollisions of solitons with trajectories that do not form a single plane. When the solitons areindividually launched, they move in their initial trajectories. When they are launchedsimultaneously, they interact (attract or repel each other) via the nonlinearity and theirtrajectories bend. If the soliton attraction exactly balances the “centrifugal force” due torotation, the solitons can "capture" each other into orbit and spiral about each other, much

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like two celestial objects or two moving charged particles do. This idea was suggested byMitchell, Snyder and Polodian [77] in the context of coherent collision. Recently, Shih,Segev and Salamo have demonstrated such spiraling-orbiting interaction employing anincoherent collision between photorefractive screening solitons. [80] Under the properinitial conditions of separation and trajectories, the solitons capture each other into anelliptic orbit. This is shown in Fig. 9. If the initial distance between the solitons is increased,the solitons' trajectories slightly bend toward each other but their "velocity" is larger thanthe escape velocity and they do not form a "bound pair". On the other hand, if theirseparation is too small, they spiral on a "converging orbit" and eventually fuse. An effectsimilar to the spiraling-fusion part of the experiment was also observed in Ref. [88] withatomic vapor.

Figure 7. Top - view photograph of an (attractive) incoherent collision between two photorefractivescreening solitons in which the solitons pass through each other at a large angle.

Figure 8. Fusion between the same solitons when the collision occurs at a shallow angle. Shown are theintensity profiles and photographs of beams A and B at (A) the entrance plane, (B) each individual solitonat the exit plane when the other is absent, and © the fused beam at the exit plane.

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The observation of spiraling brings about an interesting question: Do interactingspatial solitons also conserve angular momentum? Indeed, recent calculations by Y.Kivshar's group in Australia in collaboration with Segev's group confirm [89]that spiralingsolitons conserve angular momentum whenever there is no power exchange between theinteracting (coherently or incoherently) beams, i.e., the beams do not fuse and do notundergo fission [82]. The reason for these exceptions is intuitive: fusion and fission arealways followed by "loss" of energy which is radiated away from the solitons (via radiationmodes of the respective induced waveguides). Therefore, if energy (or power) is notconserved, angular momentum does not have to be conserved either. In the true spiralingprocess however, (i.e., excluding the fusion part of Ref. [80]), preliminary results indicatethat all the power is conserved and therefore so is the angular momentum. It seems thatthis property is general, and should exist in any 3D nonlinear system of self-trapped wave-packets in nature.

SELF-TRAPPING OF INCOHERENT LIGHT BEAMS

As the final section of our review on photorefractive solitons, we wish to introducea new, perhaps somewhat futuristic, direction into which solitons seem to be evolving:incoherent solitons. Until 1995, all soliton experiments and theories in nature employed acoherent "pulse", either in space, time, or both. In other words, given the phase at a givenlocation on the pulse (space or time) one can predict the phase anywhere on that self-trapped pulse. However, pulses or wave-packets do not necessarily need be coherent. Forexample, one can focus into a narrow spot a light beam from a natural source such as thesun or an incandescent light bulb. Can such beam self-trap in a nonlinear medium?

Figure 9. Spiraling of two colliding photorefractive screening solitons with initial trajectories that do notlie in the same plane. Shown are the photographs of the optical beams. (a) Beams A and B at the inputplane, (b) the spiraling soliton pair after 6.5 mm of propagation and (c) the spiraling pair after 13 mm ofpropagation. Note that occurs in elliptical beams. The triangles indicate the conters of the correspondingdiffracting beams. After 6.5 mm the solitons have spiraled about each other by 270° and after 13 mm thespiraling angle doubles to 540°.

Last year, Segev's group at Princeton has demonstrated self-trapping of beams (spatial"pulses") upon which the phase varied randomly in time/space across any plane.[90] In the

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first experiment, a quasi-monochromatic partially spatially-incoherent light beam wasemployed. The beam originated from a laser and passed through a rotating diffuser thatintroduced a new (random) phase pattern every 1 microsecond. The beam was launchedinto a slowly-responding photorefractive crystal and, under appropriate conditions, theenvelope of this beam self-trapped into a single non-diffracting narrow filament. Earlier thisyear, in a subsequent experiment Mitchell and Segev have demonstrated that an incoherentwhite light beam, i.e., a “pulse” that is both temporally and spatially incoherent, self-traps. [91] In this experiment the self-trapped beam originated from a simple incandescentlight bulb which emitted light between 380-780 nm wavelength [see Fig. 10].

To understand the new ideas involved, we need first to explain some aspects ofincoherent light. A spatially-incoherent beam is nothing but a multi-mode (so-called"speckled" beam) whose structure varies randomly with time. The beam consists of brightand dark "patches" (thus the notion "multi-mode") that are caused by a random phasedistribution, which varies randomly with time. The envelope of this beam is defined by thetime-averaged intensity. To illustrate this, consider a detector array (e.g., human eye)which monitors the beam. When this detector responds much slower than the characteristic

Figure 10. Self-trapping of an incoherent white light beam.

284

phase fluctuation time, all it will "see" is the time-averaged envelope. Typically,such anincoherent beam diffracts much more than a coherent beam, since every small bright"patch" (speckle) contributes to the diffraction of the envelope. In the limiting case of thespeckles being much smaller than the beam size, diffraction is dominated by the degree ofcoherence, i.e., the size of the speckle, rather than the diameter of the beam's envelope.Instantaneous nonlinearities cannot self-trap such a beam. If an incoherent beam islaunched into a self-focusing nonlinear medium that responds instantaneously (e.g., theOptical Kerr effect), each small speckle forms a small "positive lens" and captures a smallfraction of the beam. These bright-dark features on the beam change very fast throughoutpropagation and these tiny induced-waveguides intersect and cross each other in a randommanner. The net effect is beam breakup into small fragments and self-trapping of thebeam’s envelope will not occur.

For self-trapping of an incoherent beam (an incoherent soliton) to exist, severalconditions must be satisfied. First, the nonlinearity must be non-instantaneous with aresponse time that is much longer than the phase fluctuation time across the incoherentbeam. Such a nonlinearity responds to the time-averaged envelope and not to theinstantaneous "speckles" that constitute the incoherent beam. Second, the multi-mode(speckled) beam should be able to induce a multi-mode waveguide via the nonlinearity.This is achievable in any saturable nonlinear media. Third, as with all solitons, self-trappingrequires self-consistency: the multi-mode beam must be able to guide itself in its owninduced waveguide. Theory of incoherent solitons was presented in two recent papers byChristodoulides and Segev groups.[92],[93] From the theory it is apparent that the self-trapping process re-shapes the statistics of the incoherent beam. For example, incoherentsources (e.g., the sun) have non-localized statistics: the correlation length (loosely definedas the distance beyond which two points are no longer phase-correlated) does not dependon the absolute location. In the incoherent soliton, however, the correlation length has adifferent value at the center of the beam and at its margins. Furthermore, it is possible to"engineer" (at least to some extent) the coherence properties of an incoherent beam by theself-trapping process. The rapid progress in this direction brings about many interestingfundamental ideas (such as coherence control) and possible applications (e.g., using self-trapped beams from incoherent sources, such as, Light Emitting Diodes) for reconfigurableoptical interconnects and beam steering.

CONCLUSIONS

The last few years have witnessed a great renewal of interest in the photorefractive effectthanks to the realization of the possibility of propagating stable 2D spatial solitons at verylow optical powers in PR crystals. This result, besides possessing a great scientificrelevance in itself, associated with the first experimental demonstration of a non-diffractingtwo-dimensional pencil of light, has also an interesting applicative potential thanks to theguidance properties of the waveguide associated with the soliton, which survives in thedark also after the soliton has been turned off. In this paper, we have tried to present a self-contained approach to the theory of self-trapped nonlinear propagation, which is far frombeing definitive, together with the most important experimental demonstrations obtained tillnow. The novelty of the field, and the fact that it is still undergoing a rapid growth, hasmade our task not an easy one and we apologize to the readers for the many inevitableomissions both in the subjects we have chosen to emphasize and in the references.

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We cannot conclude this review without mentioning several important developmentswhich started to be investigated during the completion of this paper. The first concernsself-trapping of optical beams in biased photorefractive semiconductors, such as InP:Fe, inwhich both electrons and holes participate in the formation of space charge field [94],[95].Interestingly enough, the self-focusing effects undergo a large enhancement when the rateof optical excitation of holes is close to (but smaller than) the thermal excitation rate ofelectrons. When the optical excitation of holes exceeds the thermal excitation rate ofelectrons, self-focusing turns into self-defocusing, i.e., the sign of the optical nonlinearitycan be reversed by all optical means. These type of solitons seem important for applicationsbecause they are sensitive in the near-infrared wavelengths range and because they respondmuch faster than all other photorefractive materials (with the same intensities). The secondprogress has to do with the existence of solitons in centro-symmetric photorefractive media[96], which fundamentally do not possess quadratic nonlinearities. The change in therefractive index that gives rise to these solitons is driven by the dc Kerr effect, which issimilar to Pockels' effect but δn is now proportional to (E s c)

² and thus to 1/(I+Id )² . Recentexperiments performed by Del Re, Crosignani and collaborators at the Burdoni Instituteand University of L'Aquila (Italy) have demonstrated these solitons. Finally, in the last twoyears several groups have predicted the existence of photorefractive vector solitons [97]and two-component solitons [98],[99],[100]. Of particular interest are the incoherentsoliton pair (which form a system that resembles Manakov's solitons. These were predictedby Christodoulides et al. [101] and subsequently observed by Chen et al. at Princeton[102],[103],[104].

Acknowledgments

We would like to dedicate this review to our friends and collaborators who have gone withus a long way over the last few years, exploring these new and exciting field of physics:Demetri Christodoulides, George Valley, Yuri Kivshar, Marty Fejer, Matt Bashaw, MinoruTaya, Galen Duree, Mark Garrett, Toni Degasperis, Stefano Trillo, Matthew Chauvet,Tamer Coskun, and Amnon Yariv.

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290

SUB-CYCLE PULSES AND FIELD SOLITONS:NEAR- AND SUB-FEMTOSECOND EM-BUBBLES

A. E. Kaplan, S. F. Straub * and P. L. Shkolnikov

Electrical and Computer Engineering DepartmentThe Johns Hopkins University

Baltimore, MD 21218

ABSTRACT

We demonstrate the feasibility of strong (up to atomic fields) and super-short (few- or evensub-femtosecond) sub-cycle (non-oscillating) electromagnetic solitons -- EM bubbles(EMBs) in a gas of two-level atoms, as well as EMBs and pre-ionization shock waves inclassically nonlinear atoms. We show that EMBs can be generated by existing sources ofradiation, including sub-picosecond half-cycle pulses and very short laser pulses. Weinvestigate how EMB characteristics are controlled by those of originating pulses. Ourmost recent results are focused on the related transient phenomena, including EMB forma-tion length, multi-bubble generation and shock-like waves. We also develop the theory ofthe diffraction-induced transformation of sub-cycle pulses in linear media.

* Also with Abteilung für Quantenphysik, Ulm University, Ulm, Germany

1. INTRODUCTION

Contemporary optics usually operates with almost-harmonic, multi-cycle oscillationsmodulated by an envelope much longer than a single cycle of the oscillations. In fact, anynarrow-line radiation is an envelope signal, be it a coherent radiation of a laser, or anincoherent light filtered through a spectroanalyzer. This is also true for any optical pulse,including self-induced transparency (SIT) solitons in two-level systems (TLS) [1],described by Maxwell-Bloch or sine-Gordon equations; mode-locked laser pulses [2] dueto multi-mode cavity interaction with laser medium; and optical-fiber solitons [3] due toKerr-nonlinearity, described by a nonlinear Schrödinger equation [4],etc. To describe anyof those pulses, slow-varying envelope approximations are used in both the propagation(by reducing Maxwell equations to a parabolic partial differential equation) and thematerial response (rotating-wave approximation in constitutive equations). Due to the

Beam Shaping and Control with Nonlinear Optics291Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

availability of very short laser pulses (down to ~ 6ƒs length [5a] and even below 5ƒs [5b]),with just a few laser cycles, the efforts are made to improve the envelope approximation atleast for linear propagation (see e. g. [6]).

A lot of new experimental techniques and applications, however, such as time-domainspectroscopy [7] of dielectrics, semiconductors and flames [8], and of transient chemicalprocesses, e. g. dissociation and autoionization [9], new principles of imaging [10], andatomic physics by means of photoionization [11], would greatly benefit from the availabil-ity of short and intense electromagnetic pulses of non–oscillating nature, i. e. sub-cycle(almost unipolar) "half-cycle" pulses (HCPs). The spectra of currently available HCPsgenerated in semiconductors via optical rectification, reach into terahertz domain; theseHCPs are ~400 – 500 ƒs long, with the peak field of 150 – 200 KV/cm [9].

In our recent work [12-14], we have proposed two new different principles of generat-ing much shorter (down to 0.1 ƒs = 10 –16 s) and stronger (up to ~ 1016 W/cm 2 ) pulses. Oneof these principles is based on stimulated cascade Raman scattering and would result in thegeneration of an almost periodic train of powerful sub-femtosecond pulses [12], while theother relies on the generation of powerful "EM-bubbles" (EMBs) [13,14], sub-cycle soli-tary pulses of EM radiation propagating in a gas of two-level or classically nonlinearatoms. The latter effect would allow one to generate a single EMB, or a few EMBs withcontrollable parameters, each of EMBs propagating with different velocity such that onecan easily separate them into individual pulses. In this paper we review our recent researchon EM-bubbles and present new related results on transient processes, in particular, on theformation length of EMBs, the generation of multi-EMBs, and the formation of shock andshock-like waves.

Such super-short and intense sub-cycle pulses might be of great interest for the host ofapplications (see below). Especially significant are non-oscillating solitary waves that areable to propagate over substantial distances with unchanged shape and length. The exactsoliton-like solutions for the nonlinear propagation of unipolar pulses in the strongly-driven two-level system (TLS), described by full Maxwell + full Bloch equations, werefound quite a while ago [15]. The solutions have a familiar, 1/cosh, profile, with its dura-tion and velocity related to its amplitude. At that time the authors of Refs. [15] did notbelieve that these nonlinear pulses were feasible; the main stumbling point they saw wasthat the pulse intensities would exceed ~ 1014 W/cm 2 , the level unaccessible then. Nowoptical fields a few orders of magnitude larger are available; however, one of the majorproblems in the generation of such short (and intense) pulses lies in that the TLS modelused in the theory [15] (and in some more recent research [16,17]) will be stretched farbeyond its limitations, since intensities above ~ 1014 W/cm2 cause very fast over-the-barrier ionization. What are the largest intensities (and thus the shortest lengths) of thesepulses that can still be supported by atomic gasses? What are new properties of thesepulses beyond the TLS approximation? Fortunately enough, these and other questionsabout high-intensity super-short pulses, can be addressed using the very fact that the atomis so strongly excited that one can use again its classical (as opposed to quantum) descrip-tion [13]. In the intermediate domain, a multi-level quantum approach has to be used.

We show here that EMBs are not only feasible but natural for many nonlinear system,both quantum and classical. Their length may range from picoseconds to sub-femtoseconds, depending on their intensity. We call them EM-bubbles to stress their non-envelope nature. We demonstrate that field ionization, a fundamental factor not consideredpreviously, imposes an upper limit on the EMB amplitude and a lower limit on its length;after an EMB reaches its shortest length at some peak amplitude, further increase of theamplitude results in EMB broadening. At some threshold amplitude, the EMB degeneratesinto a shock wave that is a precursor of a dc ionizing field -- a new feature which is notpresent in TLS model. Furthermore, we show that even at much lower peak intensities,

292

when TLS model may still be valid, the initially smooth HCP may drastically steepen toform a shock-like wave which then breaks in a multi-EMB solution [14]. Unlike a dc-ionization precursor shock wave, this shock-like wave can appear far below the ionization.

(1.1)

EMBs can potentially be as short as 10 – 0.1ƒs, with the amplitudes approaching theatomic field. These super-short and intense sub-cycle pulses might be of great interest forthe host of applications. They can be used for a "global" spectroscopic technique based ona shock-like excitation across the entire atomic spectrum (to the extent similar to passingatoms through a foil), including normally prohibited transitions. The ionization by a pulseshorter than the orbital period may bridge a gap between conventional photoionization andcollisional ionization by a particle [11], with the important difference being that EM pulsesoffer a control of the quantum state of the atom during the entire process, and hence a con-trol of its final state. This, in turn, has far-reaching implications for applications in time-resolved spectroscopy of transient chemical processes occurring on a femtosecond timescale, e. g. dissociation and autoionization (see e. g. [9]), especially for quantum control ofchemical transformations (see e. g. [18]). These new pulses may expand time-domainspectroscopy of dielectrics, semiconductors, and flames [8] from presently available THzdomain [7-11] to optical frequencies. One can also envision their applications to probinghigh-density plasmas, testing the speed of light, imaging molecules and atoms at surfaces,and for an order of magnitude frequency up-conversion due to the large Doppler shift of acounter-propagating coherent light backscattered by EMB, etc. A train of sub-femtosecondpulses with very high repetition rate (~ 125 THz, or with the spacing ~ 8 ƒs), feasible incascade stimulated Raman scattering [12], can be used for the stroboscopy of atomicmotion in a molecule (e. g. during its dissociation).

Another property of EMBs, which may be greatly instrumental in their applications, istheir extremely broad spectrum, which ranges ideally from radio-frequencies to visible oreven ultra-violet domains. A single pulse of such nature would have a continuous powerspectrum from zero frequency to the highest (cutoff) frequency of the pulse,

where tp is the pulse duration (evaluated at half-intensity). For example, with τ=0.2 ƒs, thecutoff wavelength, λcut =2πc/ωcut ~ 2.4 c tp , is ~1440°A , i. e. in the far UV. It would beseen by a human eye as an extremely short and powerful burst of white light. Even thespectrum of a much longer, 1ƒs pulse, with λ cut ~7200°A, would still cover the infrared,millimeter, microwave, and rƒ domains. Thus the propagation of EMB would be greatlysensitive to a material in which they propagate. The EMB spectrum will be affectedstrongest by metallic particles or any other good conductors (the part of the spectrumbelow the respective plasma frequency will be absorbed), or by the presence of water orother substance having strong absorption bands, especially in infrared. Designating theEMB radiation here "S-rays" (where "S" stands for "sub-cycle" or "sub-femtosecond") inanalogy to recently demonstrated "T-rays" [10] (THz pulses, see below), we note that thefact that different materials have different transparency for S-rays, suggests a great numberof possible applications utilizing EMBs to emulate X-rays without X-ray-induced ioniza-tion damage. These S-rays can be used e. g. to monitor processing of high-density com-puter chips, screening food products at the food-processing facilities, luggage and con-cealed weapons in the airports, etc. One can also envisage applications of S-rays, similarto T-rays, but in the new frequency domain and with orders of magnitude broader spectra,for medical imaging, in particular, for a new kind of tomography, "S-tomography", with anadditional possible advantage of positioning an S-ray source inside a human body. S-rayscan also be a useful tool for the diagnostic of high-density fusion plasmas.

This paper is structured as follows. Section 2 discusses a general relationshipbetween the field and polarization, which results in a solitary wave as a solution of full

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Maxwell equation + arbitrary constitutive equations. Section 3 addresses an exact EM-bubble solution of full Maxwell+Bloch equation for a two-level system. Section 4 is onEM-bubbles and shock waves which are due to a classical anharmonic potential with ioni-zation. Section 5 discusses various approximate approaches to the transition processes innonlinear EMB propagation. Section 6 concentrates on EMB generation by half-cyclepulses. Section 7 elaborates on multi-EMB solution; Section 8 discusses shock-like wavefronts due to multi-EMB formation, and Section 9 gives an example of EMB formation byshort laser pulse. In Section 10, we develop the theory of diffraction-induced transforma-tion of sub-cycle pulses. In conclusion, we briefly discuss future research on and physicalramifications of EMBs.

2. MAXWELL EQUATIONS AND GENERAL SOLITARY WAVE CONDITION

Maxwell equation for the electric field of a plane EM wave propagating along the z-axis,is:

(2.1)

where is polarization density. Considering a pulse that propagates with a constant velo-city, c β EMB , introducing retarded variables, and imposing a steady-state condition,

(2.2)

(2.3)

we reduce Eq. (2.1) to the "solitary wave (EMB) Maxwell equation":

where

(2.4)

is an EMB’s normalized relativistic "momentum". Stipulating now that an EMB car-ries finite energy per unity area of cross-section, i. e. that (a so calledbright soliton condition), and integrating

andEq. (2.3) twice, we obtain a universal "EMB-

replication" relationship between

(2.5)

Note that Eq. (2.5) is valid regardless of constitutive relationship between and Forour further calculations, we assume the field is linearly polarized, so that the wave equationcan be reduced to scalar equation, and introduce dimensionless variables: field,ƒ, polariza-tion, p, time, where ω0 is a characteristic frequency of the system, and distance,

as well as dimensionless particle density, Q. All these variables and parametersare defined below for quantum and classical models separately; using them, we writeMaxwell equation as

and EMB-replication relationship (2.5) as

(2.6)

(2.7)

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3. EM-BUBBLES IN TWO-LEVEL SYSTEM

Consider first the pulse propagation in a medium of quantum TLS characterized bythe dipole moment, and resonant frequency, ω0. We introduce normalized variables: thefield

(3.1)

where ΩR is Rabi frequency; the polarization per atom, p = ρence per atom, η = ρ

12 + ρ 21 ; the population differ-11 – ρ22 , where ρ jk ( j, k =1, 2) are density matrix elements of a TLS

(with ρ11 +ρ22 =1 and and time τ= (t – z/βEMB c) ω0 , to write full Bloch equa-tions as:

(3.2)

where the overdot designates ∂/∂τ; we use the notation of [19], which addressed high har-monics generation in a super-dressed TLS. Note that (3.2) is not based on rotating waveapproximation. Relaxation is not included in (3.2) since we consider pulses much shorterthan TLS relaxation times. The first integral of Eq. (3.2) is square of the Rabi sphereradius,

(3.3)

The polarization density here is where N is the density of particles; therefore, theparameter Q in (2.6) is as:

(3.4)

where e is the electron charge, λ0 =2πc/ω0 and is the fine structure con-stant.

To find an EMB solution for TLS, we substitute the condition (2.7) (with unknown atthis point M or βEMB ) into (3.2). Having in mind the invariant (3.3) for atoms being ini-tially at equilibrium, η → 1 at | τ| → ∞,η(τ) = 1 –ƒ2/(2QM),

such that the first of Eqs. (3.2) gives uswe obtain from the second of Eqs. (3.2) a nonlinear equation for the

EMB field, ƒ( τ ), as:

(3.5)

which is a so called Duffing equations. Its first integral is

(3.6)

(the integration constant C = 0 under the bright soliton condition), which determines aseparatrix in the phase plane, and ƒ, starting and ending at the point = ƒ = 0. The nextintegration gives us finally an EM bubble, a solitary, non-oscillating wave:

(3.7)

(3.8)

(3.9)

the polarization and population are then:

In Eq. (3.7), the amplitude of EMB and its length are respectively:

Dimensional EMB amplitude, EEMB , by the definition of ƒ, Eq. (3.1), is

(3.10)

(For EMB length, T, defined at a half–peak field, i. e. T ≈ tEMB /1.32 = τEMB /1.32ω0, wehave E EMB

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Instead of having ƒEMB as function of M or β E M B , we can express β EMB in terms ofƒ

EMB :

or, if Q << 1 (see below, Section 5),

(3.11)

(3.12)

Shorter EMBs have higher amplitudes and move faster, approaching the vacuum speed oflight. The lowest allowed speed of a bubble is

(3.13)

which corresponds to a linear propagation of an adiabatically slow pulse.

The Fourier spectrum of EMB,

(3.14)

spreads from zero to the cutoff frequency,

(3.15)

Phase-portrait considerations show that with ƒ =p=1–η=0 a t |τ | → ∞, t h enon–oscillating EMB (3.7) is the only soliton supported by the system. Therefore, surpris-ingly, regular SIT envelope solitons [1], which have been obtained in the rotating-waveapproximation, are inconsistent with the exact solution (3.7) based on full Bloch equations(3.2). This indicates that higher-order approximations may render SIT solitons unstable atlong enough distances. EMB (3.7) may be regarded as a "full-Bloch" 2ππ -soliton; by intro-ducing phase (or area)

(3.16)

we get φ R(∞) =2ππ, which points to a "full-Bloch" area theorem.

A similar EMB solution, (3,7), holds also for amplifying TLS media with theinversed population, ηη (|ττ | → ∞→ ∞) = – 1 . In this case, however [13],

(3.17)

Since a TLS with ηη ∞ = –1 is a non-equilibrium system storing pumping energy, β E M Bhere is not the speed of energy propagation, so that the fact that (i. e. the EMBmoves faster than light) is not incompatible with special relativity. More intense EMBshere move slower, approaching the speed of light from above as their amplitude increases.

4. EM-BUBBLES AND SHOCK WAVES IN A CLASSICAL POTENTIAL

The solution (3.7) is valid within the limitations of our TLS model. In particular, theEMB duration, must be shorter that all the atomic relaxation times, whichstill allows for EMBs as long as ~ 10– 9 s, with longer EMBs having lower peak amplitude,Eq. (3.8), and moving slower, Eq. (3.9). It is instructive to consider an example of Xe,with ωω0 – 8.44eV, effective dipole size, d/e ~ 7°A (based on the "super-dressed TLS" datafor high-harmonic generation in Xe [19]), and N ~ 1019 cm –3 (Q ~10–2). For a 10ps longEMB, we have E p k ~103 V/cm. Longer pulses can be considered within the TLS modelwith relaxation. Of a particular interest, however, are the shortest and most intense pulses.When the EMB field approaches the atomic field (~108 –109 V/cm), the EMB formation isaffected mainly by the atomic ionization potential, which limits EMB length and

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on EMB within a classical 1-D model of an atom, with a strongly nonlinear potential, U(x) ,limited at |x| → ∞, to allow for ionization; here x is the electron displacement. ThenBloch equations (3.2) are replaced by a classical normalized equation for the electronmotion:

(4.1)

Thus, the maximal field strength, E

=20eV and x

2 9 7

with u – 1 ≈ p –1 , at |p | → ∞when

with the dimensionless variables and parameters of the system defined as

(4.2)

where x0 is an atomic characteristic size, and U0 is a characteristic energy (e. g., the ioni-zation potential) [20]; me is the mass of electron. The polarization density here isP=Nxe=Nex 0 p. Note that for EMB, TLS Bloch equations (3.2) reduce to a simpleDuffing equation for, e. g. p,

(4.3)

with A = QM – 1 and B = (QM) 2/ 2, which is equivalent to Eq. (4.1) (with ƒ =pMQ ) for thesimplest classical anharmonic potential,

with a = const > 0. (4.4)

Hence, the potential (4.4) can give rise to the same EMB, Eq. (3.7). For an arbitrary poten-tial u(p), the family of EMB solutions, p(ττ), is found from Eq. (4.1) through the quadrature[13]:

(4.5)

A "bright" solitary solution to Eq. (4.5) exists, however, only for particular nonlinearities.For example, for (4.4) the nonlinearity must be "positive", a > 0 [21]. In general, if u is asmooth, monotonically increasing function of p2 , the "bright" solitary solution exists onlyif near p = 0

(4.6)

This requires the atomic potential to have sufficiently "hard walls", which holds for somemodel potentials [22] (but not for a "soft" potential as e. g. u = – (1 +p2 ) –1/2 [22]). Anexample of a model potential that allows for an explicit analytic solution of Eq. (4.5) is:

with b = const >0 . (4.7)

To illustrate the limitations imposed by over-the-barrier ionization, consider first a classical"box" potential, u(p) = 0 for |p| < 1, and u(p) = 1 otherwise, in which case the EMB field is

with ƒEMB ≤ 2, (4.8)

and M =ƒ EMB /Q. (We presume here that an electron always starts its motion at p = 0.)max , and shortest EMB length, tmin , are:

(4.9)

where U 0 is an ionization limit, and 2x0 is the total box width. Emax is of the same natureas an atomic field, Ea t =Emax / 2, i. e. the atom is ionized (in classical terms) by a pulse of acertain shape [here, Eq. (4.8)], if its peak amplitude exceeds Emax ; t min is the time requiredfor such a field to pull an electron out of the potential well. (With U0 0 =1°A,this results in E

m a x ≈ 2 109 V/cm, and t m i n ~ 0.4 10– 1 6 s.) To make a connection to atoms

with Coulomb long-range attraction, consider now a potential

(4.10)

It has a single well and satisfies hard-wall condition onlyFor a given U0 and atomic number, Z, we have

(4.12)

here r e = e 2/m ec2 is the classic electron radius. As an illustration, consider a limiting casewith b = 0. Small-amplitude EMBs are governed again by a Duffing equation,

its solitary solution being (Fig. 1, curve 1).

Fig. 1. Normalized field amplitude, ƒ, vs time, τ, τ, for steady-state EMB (curves 1-3) and a shock wave (curve

4) due to ionization potential. Curves: 1 -- MQ = 0.12, 2 -- MQ = 0.187, 3 -- MQ = (MQ)ion – 1 0 –5 ; 4

-- MQ = (MQ)ion ≈ 0.3403.

Here and therefore, β cr = 0, i. e. small-amplitude EMBs here can move veryslowly, a typical feature of any potential with du(0) /d(p2) = 0. The EMB peak amplitude is

Hence, as its amplitude increases, an EMB moves faster, and shortens.However, at pp k ≈ (8/45) 1/4 ≈ 0.65, ƒpk ≈ 0.122, EMB length (at the half-peak amplitude)reaches its minimum, τA min ≈ 5.3 (at the half-peak amplitude, Fig. 1, curve 2) or I min ≈ 2(at the half-peak intensity). Assuming U0 ≈ 2 4eV and Z = 2, as in He, one obtains the shor-test EMB length:

(4.13)

(Significantly shorter EMBs can be attained with ionized atoms, e. g. ion beams, whichmay have ionization potential, U0 , orders of magnitude larger.) As the field amplitudecontinues to rise, EMB begins to broaden, becoming a flat-top pulse (Fig. 1, curve 3).Finally, at a threshold amplitude, pp k ≈ 1.245, ƒp k ≈ 0.42, it becomes a shock (anti-shock)wave whose single leading (trailing) edge is a front of an ionizing (de-ionizing) cw field(Fig. 1, curve 4) [23]. The amplitude front rises (falls) as e x p (ττ/ ττion ), w i t hττ ion ≈ (MQ) –1/2 ≈ 1.7. This shock wave is typical to any hard-wall potential with ionization.Our preliminary results indicate, though, that a single-front shock wave becomes unstable,producing a short precursor that travels as a pilot EMB at a faster speed ahead of the groupof other, longer and closely spaced EMBs, which merge into a cd field far behind the pre-cursor. This pattern persists if one accounts for the plasma due to ionization behind thepilot group of EMBs. In a more detailed picture of a shock wave, the classical over-the-barrier ionization near the threshold must be modified by quantum tunneling.

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τ

5. VARIOUS APPROXIMATE APPROACHES TO EMB PROPAGATION

To demonstrate the existence of EMBs (in both quantum and classical eases) mostrigorously, we have used so far a "double-full" approach: full Maxwell equation (2.1) + fullconstitutive equations (3.2) or (4.1) (i. e. without rotating wave approximation). The prob-lem with this "double-full" approach is that at this point we do not have a mathematicaltheory which would allow us to handle a general solution of the problem (including thecase of an arbitrary initial/boundary conditions) with the same degree of confidence andinsightfulness as the inverse scattering theory provides for the so called fully integrablepartial differential equations, like Kortdeweg-de-Vries, nonlinear Schroedinger, and someother equations. There are no results regarding the full integrability of "double-full" equa-tions, nor even about them being of the same class of equations that can lend themselves tothe inverse scattering theory. In physical terms, the very fact that the full nonlinearMaxwell equation allows for the coupling between forward and backward propagatingwaves, creates a significant complicating factor. Hence, in theoretical consideration of thepropagation, as well as in numerical simulations, in particular for all kind of transientproblems, we need to look for connections to some better understood equations, at least incertain meaningful limits. Closer consideration shows, fortunately, that "double-full"equations can often be reduced to much simpler equations (with some of them being fullyintegrable), while keeping them free from a rotating wave approximation and hence opento broad-spectrum solutions.

Our computer simulations have shown that at low density, Q << 1 (e. g. in gasses,where typically, Q ~ 10 –4 -10 –1), Maxwell equation can be reduced to approximate first-order equation without losing any significant feature of nonlinear propagation. In particu-lar, the EMB solution have the same form, as for full Maxwell equation. This is explainedby the fact that when Q << 1, the propagation velocity approaches the speed of light,1 – β = O(Q) Q << 1. and any retroreflection can be neglected. Assuming now that the wavepropagates only in one direction (e. g. positive ), using retarded variable τ = τ – /β

LN,

and keeping in mind definition (3.13), we transform Maxwell equation (2.6) to the equa-tion:

(5.1)

Neglecting in it the term (which is small since the pulse changes relatively slow asit propagates along the axis), and eliminating one derivative, ∂/∂τ, by integrating theresulting equation over ∂τ, we can write now:

(5.2)

By rescaling the propagation coordinate, we finally obtain:

(5.3)

The physical implication here is that nonlinear retroreflection is neglected; the counter-propagating waves are decoupled. The validity of the reduced Maxwell equation can beverified by e. g. using it instead of Eq. (2.6) to obtain EMBs in either quantum and classicallimits, as well as by numerical simulations of the transient propagation [14,13]. We havefound also that Eq. (5.3) can still be used even if Q is not small, if the field spectrum doesnot stretch beyond ω0 .

In order to describe the studied process by even simpler equations, and especially byfully integrable ones, one can work now on the simplification of constitutive equations. Amajor step in this direction is based on the observation that for the most of nonlinear gassesof interest, in particular, for noble gasses, the TLS frequency of the first transition, ω0 , is

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extremely high, so that even near-femtosecond pulses and laser oscillations are relativelyslow compared with a cycle of that frequency. How slow is "slow" in this case? Xe hasthe lowest energy of the first excited level among the noble atoms; with

~ 8.5 ev, one cycle of ω 0 is 2 ππ /ωω 0 ~ 0.5fs. For He, with ~ 20 ev, one cycle is~ 0.2 fs. For any available half-cycle pulses (HCP), the HCP length, t0 , is is three orders ofmagnitude longer; even the full cycle of e. g. Ti:Spph laser oscillation is ~ 2.7 fs -- stillmuch longer than 2 ππ/ω/ω0 in these gasses. Another important parameter is the dimensionlessamplitude of the incident field ƒ0 , (3.1), which is also related to the time scale of the non-linear motion in TLS, ƒ 0 = O(ττ EMB ). Thus, introducing a parameter, wecan see that it is very small for HCPs available now or in the foreseeable future. For noblegasses, for example, and with E 0 ~ 2 MV/cm, which is an order of magnitude higher thanpresently available HCP, we have ƒ0 ~ 10–2

; with t 0 ~ 400 ƒs, we also haveHence, if εε << 1, we can use a "slow motion" (but no envelopes!) approximation, wherebyƒ(ττ), p(ττ) and ηη (ττ) have their Fourier frequencies much smaller than ω0 . As a first step,"instantaneous weak response", we neglect in the second of equations (3.2), so thatp1 =ƒη, η, and substitute it in the first of equations (3.2). By integrating it and having inmind the invariant (3.3) (i. e. η η = 1 at we obtain

(5.4)

Writing , and neglecting again in (3.2), by assuming now thatwe obtain in the next approximation: This, after

evaluating from the latter equation and substituting it into the former one, results inintegration of which yields ∆η ∆η ≈

For ƒ – p we have now:

(5.5)

Two last terms in the rhs of (5.5) reflect the Rabi dynamics, without which EM bubbleswould not exist. Rhs of Eq. (5.5) is O( εε3 ); the next approximation correction is O(εε5 ). Inthe limit ε → ε → 0, Eq. (5.5) can be further simplified by noticing that since

as well as andsion O(εε5 ):

we can write, still with the preci-

(5.6)

Eqs. (5.3) and (5.6) yield a single self-contained abridged Maxwell-Bloch equation:

(5.7)

It can be readily shown that solution (3.7) of the full Maxwell-Bloch equations are alsosolutions of Eq. (5.7). (5.7) is one of the so called Modified Kortdeweg-de Vries (MKdV)equations. The MKdV solutions could be associated [24] with a regular KdV equation,where in the second term, instead of ƒ2 , one has ƒ. MKdV is fully integrable by usinginverse scattering method and has an infinite number of invariants [24].

Similar equation can be obtained for a classical anharmonic oscillator (4.1) if theamplitude is not large, i. e. when approximation (4.4) with the coefficient of first-ordernonlinearity, a, can be used. In this case, similarly to (5.6), we can write

(5.8)

and the self-contained wave equation, similar to (5.7), will be again MKdV:

(5.9)

If the incident field is due to a laser, and is, therefore, oscillating and strong, we mayhave ε ε >> 1. Presuming that TLS model and Bloch equations (3.2) are still valid, theMaxwell-Bloch equations can be reduced to another well-explored equation. Since we

300

expect the driven polarization, p, to vary rapidly, we can drop p from the second of equa-tions (3.2), assuming that Solving (3.2) in this approximation, we readily obtain:

(5.10)

where φR is Rabi phase (3.16); in this approximation, The reduced Maxwellequation (5.3) can then be written as or

(5.11)

With a proper choice of retarded coordinate, τ1, it can be reduced to even simpler equation,

(5.12)

These are different forms of so called sine–Gordon equation, which is fully integrable. Ithas again the same soliton solution, Eq. (3.7), as general Maxwell+Bloch equations do.While using Eqs. (5.10)-(5.12), one has to be cautious about choosing boundary conditions;only those functions ƒ(τ) at ζ = 0 that satisfy a condition on the area of the pulse (seebelow), S ≡ φR(∞) = 2πn, where n is integer, are applicable for simulations. (The bestchoice would be S = 0, since it would allow one to vary the amplitude of the incident fieldonce the shape of the field is chosen). If S is not integer of 2π, this approximation will beinconsistent with the physics in the sense that it may happen that neither area S(ζ )of thepropagating pulse nor its energy W are invariants. Fortunately, the condition S = 0 caneasily be satisfied for an oscillating field, which is exactly the area of the intended applica-bility of (5.10)-(5.12). However, even more stringent conditions may be imposed by thefact that TLS model is invalid when the Rabi frequency, ΩR, exceeds the TLS frequency,ω0, i. e. when ƒ >> 1. Some other approximations that result in fully integrable equationscan be found in [15-17].

We have to note that at this point no mathematical proof exists that in the general,"double-full" formulation, EMBs are real solitons in the sense of full integrability of thefull Maxwell + full constitutive equations, and that, therefore, EMBs are absolutely stable.Our numerical simulations for both TLS and nonlinear classical potentials show that smallEMB due to reduced Maxwell equation (5.3) are stable against both small and large (e. g.collision with another EMB) perturbations, which is consistent with the results of Ref. [15]for TLS. Large EMBs (approaching the ionization threshold) may become unstable andbreak down into smaller EMBs. In a related simulation, we have discovered thatsignificantly below the ionization threshold the EMBs are remarkably stable upon temporalor spatial changes of medium parameters. In particular, when the gas density, N, waschanged by two orders of magnitude along the path of propagation, the EMB profile and itslength remained stable; only its velocity, βEMB , was adjusting to a varying density, suchthat

N(z ) M(βEMB ) = inv . (5.13)

An EMB generated e. g. in a gas jet can therefore "slide" into vacuum without distortion.

Finally, it is worth noting that a very interesting recent work [25] suggested genera-tion of non-oscillating or unipolar EM solitons and shock-like waves in nonlinear dielec-trics due to collective effect (phonons) in a crystal lattice. The time scale of these solutionsis much larger than those discussed here, with the soliton length t being much longer thanpa cycle of the transverse optic lattice resonant frequency, ωOT, i. e. would beno shorter than a few picoseconds for the best of materials. The nonlinearity in [25] scalesas E 2 , with the single soliton having the profile of cosh–2; the important fact is that, asshown in [25], the full nonlinear Maxwell equation for the case in consideration (based onsimplified constitutive equation that uses the assumption of low frequencies and relativelyweak field) can be reduced to a fully integrable Boussinesq-like equation.

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6. EM-BUBBLE GENERATION BY HALF-CYCLE PULSES

One of the major avenues of EMB-generation [13] is to use existing half-cycle pulses(HCPs) [7-11] to launch much shorter EMB pulses in a nonlinear medium via a transientpropagation process. At first, it is important to obtain 50 ƒs to ~ 5 ƒs long EMBs, thusattaining one to two orders of magnitude enhancement over available HCPs. In our com-puter simulations [13,14], we found that distinct individual EMBs could be obtained with aHCPs. In these simulations, we used incident HCPs (i. e. the solution ƒ( ζ , τ ) at ζ = 0 ) o fvarious profiles resembling a typical experimental profile, in particular, Gaussian,

(6.1)

and 1/cosh profile, i. e. the same as EMB, (3.7), but with its amplitude, ƒ0, unrelated to itslength, τ0 :

(6.2)

All of them show very similar behavior in the formation of EMBs; in this paper we willaddress 1/cosh profile (6.2) as the only one that can bring up exact analytical results relatedto the formation of multiple EMBs; however, most of our results here on the parameters ofthe leading (i. e. largest and fastest) EMB, "EMB-precursor", in particular its amplitude,length, and formation distance, will be valid for any incident HCP. The TLS approxima-tion is valid with great margin, if the instantaneous Rabi frequency is relatively small,ƒ << 1, which, as was shown in the previous section, is the case for the available HCPs; forinstance, in noble gasses, even with still unavailable E = 2 MV/cm, ƒ ~ 10 –2.

For a given length of the incident HCP, τ0, HCP’s threshold (minimal) amplituderequired to attain a single EMB, provided the HCP has a profile (6.2), according to (3.9)and (3.10), is:

(6.3)

In most of our runs, we used τ0 = 4000, which corresponds to t0 ~ 313 ƒs (or 413 ƒs atpulse’s half-amplitude), for Xe in this case, ƒthr = 1 0 –3

and Ethr ≈ ≈ 60 KV/cm.Typical patterns of EMB formation are shown in Figs. 2 and 3.

Fig. 2. Double-EMB formation by HCP with ƒ0 = 2ƒthr .

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Fig. 2 depicts a double-EMB formation for ƒ = 2ƒ0 thr ; the larger EMB here is 3ƒthr = (3/2)ƒ0→ 180 KV/cm, and of the weaker one, ƒthr ; they are respectively 104 ƒs and 313 ƒs long.Fig. 2 can also be seen as a collision of two EMBs, with each of them coming out unaf-fected by the collision (aside from slight shift of their center lines of propagation); this canbe shown by retracting the plot back in ζ < 0. For larger ƒ0 , more EMBs are formed andthe strongest EMB moves faster than the rest of the pack, leading the train as a precursor.The "density plot" showing the linear trails of individual EMBs moving with differentvelocities, with the front trail being due to EMB-precursor, is depicted in Fig. 4. As ƒ0increases, the precursor is growing stronger and shorter, and the distance, ζEMB , for it tobreak away from the mother-HCP, is decreasing. Fig. 3 depicts multi-EMB formation forE 0 =2 MV/cm (ƒ0 ≈ 3.3 × 10–2 = 33ƒ thr), expected to be available in the near future. In thiscase, z EMB is estimated (see end of Section 8) from ζ EMB ~ 1.23 × 105 ; for Xe at 10 atm(Q ~ 0.57 ), it translates into z EMB ~ 12.5 cm. The precursor here is 4.8 ƒs long, two ordersof magnitude shorter than available HCPs.

Fig. 3. The formation of multi-EMBs and a shock-like wave front for as the wave propagates inζ; inset: superimposition of the field profiles at different ζ illustrating the front formation.

Experiment- and application-wise, it is important to know how the properties of EMBs arecontrolled by the incident HCP. In this respect, one has to answer a few important ques-tions: given the amplitude, E0 , and length, t0 , of the incident HCP, (i) what are the leadingEMB’s amplitude and (ii) its length? (iii) how many EMBs (per one HCP) can be

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generated? (iv) what are amplitudes and lengths of these EMBs? and (v) what is the for-mation distance for the first EMB? Although the equations governing the wave propaga-tion are fully integrable in certain approximations (see the preceding section), the abovequestions cannot in general be answered analytically. Our combined numerical and analyt-ical efforts [14] allowed us, however, to obtain remarkably simple results, which could besummarized as follows: the amplitude of EMB, EEMB, is proportional to (and larger than)the amplitude E0 of an incident HCP; the EMB’s length is inversely proportional to EEMB ;the number of the EMBs is proportional to the area of the incident HCP. We have alsodiscovered that when multiple EMBs are generated, they form at some point a shock-likewave front, its its formation being is proportional to We have shown that the dis-tance of the first EMB formation is proportional to if E 0 is relatively small; forsufficiently large E0 , this process coincides with the shock-like wave formation. The maingood news is that very short EMBs can be generated by a long HCP with sufficiently largeamplitude.

Fig. 4. Density plot for the propagation shown in Fig. 3; note that the "trails" of individual EMBs make

straight lines, i. e. each one of them propagate with its individual constant velocity, βEMB , (3.11).

In our computer simulations [14], using HCPs of various profiles [in particular, Gaus-sian (6.1), and cosh–1 (6.2)], we found that an EMB-precursor shows a linear dependenceof its amplitude, ƒEMB , on the incident amplitude, ƒ0 , regardless of the profile:

with a = const ~ 2.

For the profile (6.2), Eq. (6.4) becomes exact with a = 2, so that

ƒEMB =2ƒ0 – ƒthr ,

see Fig. (5). This also gives the precursor’s length:

(6.4)

(6.5)

(6.6)

Due to (6.5) and (6.6), the amplitude and length of the largest EMB tend to constants as theHCP’s length τ0 increases (Fig. 6). Hence, to attain large and short EMB, there is no needto use a short incident pulse; the only prerequisite is a sufficiently high amplitude.

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To explain these results and to find other characteristics of the EMBs, in particulartheir formation distance, we use here the approach reminiscent of that developed in thetheory of modulation instability in self-focusing and in propagation of pulses in nonlinearoptical fibers. Approximating an initially long and smooth HCP by an almost dc wave withthe amplitude of the incident slow pulse, and evaluating the propagation characteristics ofthis wave, we analyze the behavior of small perturbations of this wave.

Fig. 5. The EMB’s amplitude, ƒEMB , vs the amplitude, ƒ0 , of the incident HCP (both normalized to ƒthr ) .Curves: solid -- EMB-precursor; broken -- higher order EMBs.

Fig. 6. The EMB’s amplitude, ƒEMB , vs the length, τ 0 , of the incident HCP (normalized to ƒthr and τ thr ,respectively). Curves: solid -- EMB-precursor; broken -- higher order EMBs.

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Linearizing the original equations with respect to these perturbations, and deriving adispersion equation for their spectral components, we find then the spectral componentwith the fastest phase change. The speed of propagation of this unique component, its fre-quency, as well as a spatial scale (the shortest of all the components) at which a sufficientphase accumulation occurs, -- all this point to an EMB-precursor that will develop fromthis component.

Note that this approach allows us to still work with full Maxwell+constitutive equa-tions; to demonstrate it, we show here how the meaningful results can be obtained for fullMaxwell+Bloch equations (2.6) and (3.2); we will also keep track of simplifications stem-ming from reduced equations (5.3) and (5.6), (5.7). Assuming a field with the amplitudeƒ0 = const in Eqs. (2.6) and (3.2), one obtains that the population difference, polarizationper atom, "momentum" parameter, and the speed of this "dc" field are respectively:

(6.7)

where

(6.8)

is the Stark-shifted frequency of TLS due to the field effect. Solving now (2.6) and (3.2)for small perturbations of this solution, and representing these perturbations in terms ofspectral components, we obtain the dispersion relationship between thewave number of the perturbation component q and its frequency Ω::

(6.9)

In the linear (ƒ0 → 0), low frequency (Ω → 0) limit we have:

q LN = Ω (1+ Q) 1/2 . (6.10)

The part of q which is due to both the nonlinearity and dispersion if Q << 1 is thus:

(6.11)

The lowest ∆q(Ω) < 0 corresponds to the fastest perturbation. Looking for the minimum of∆q, we obtain the frequency, Ω = Ωfast , of this component as:

(6.12)

hence if we have

(6.13)

Substituting Ω = Ω fast into (6.9), we evaluate qfast and the phase velocity β fast ≡ Ω fast /qfast ,of this component in the case Q << 1 as:

(6.14)

Comparison with (3.11) shows that a matching EMB, βEMB = β fast , has an amplitude:

(6.15)

or, for

ƒfast ≈ 2ƒ0 (6.16)

Eq. (6.16) confirms the linear dependence between ƒ0 and ƒfast in (6.4) and fits perfectly thecoefficient a = 2 in (6.5). Note that in an ideal dc field, τ0 → ∞ and thus ƒthr → 0, whichexplains the difference between (6.5) and (6.16).

The same approach can be used to estimate the precursor formation distance,but only in the limited range of the parameters, since in general depends on total areaof HCP (see below). To still use perturbation approach, we substitute again Ω = Ωfast i n t o

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(6.9) and estimate ζEMB as a distance at which a certain change of phase, φ= 0(2π), isaccumulated (the best fit is provided by In particular, if ƒ0 << 1, ∆qfast ≈

and [14]

(6.17)

(curve 1 in Fig. 7). Thus, ζEMB , estimated by a phase change, scales as This com-pares well with the distance of first appearance of a saddle point, (dotsin Fig. 7), in the numerically obtained field profile up to (ƒ0 /ƒthr )cr = Ncr ~ 4.

Fig. 7. The normalized formation distance,

Curves: 1 -- (6.17),

of EMB-precursor vs normalized incident amplitude,

(6.18); dots -- first saddle point appearance in a

field profile.

For larger ƒ0 , when multiple EMBs are generated (see below), right before the EMB-precursor breaks away, the initially smooth HCP drastically steepens to form a shock-likewave (Fig. 3), which, unlike a dc-ionization shock wave (section 4), can appear now farbelow ionization. Its formation distance, ζ sh (curve 2 in Fig. 7), that can be analyticallycalculated based on the theory of shock-like wave, section 8 below, is as [14]:

(6.18)

which scales as now.

7. MULTI-BUBBLE SOLUTION

When the incident amplitude of HCP, ƒ0 , sufficiently exceeds the threshold of EMBformation, ƒthr , more than one EMB will be generated, as one can see from Figs. 2 and 3.In the limit ƒ0 << 1, when the propagation is described by modified KdV (5.7), one candevelop the analytical theory on N-bubble solutions for the profile (6.2). The results of ourtheory, to be published elsewhere, are based on invariants of MKdV [24], and are brieflysummarized here. Total number of EMBs, NEMB , is:

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(7.1)

where L(x) the largest integer not greater than x. For N0 >> 1, NEMB is proportional to theincident HCP area, ƒ0τ0 , With the EMB-precursor designated by number 1, the amplitude,ƒn of the n -th EMB is given by an amazingly simple formula:

(7.2)

such that the decrement,

(7.3)

is independent of n. Since each bubble with the amplitude ƒEMB carries an energy:

(7.4)

which is proportional to its amplitude, ƒEMB , a unique quality of the function (6.2) is thatthe energies of the bubbles generated by it, are equidistant, or quantized, in the way remin-iscent of the energy spectrum of a linear oscillator, with the "quantum" (or "quton") of theirenergy spectrum being ∆Wq = 2 Wthr . It is worth noticing that when N0 is an integer, thelowest energy of a bubble is exactly Wthr = ∆ Wq /2, which again is reminiscent of theenergy of the ground level of a linear oscillator, being equal to half the quantum of excita-tion.

For the EMB-precursor, n = 1, Eq. (7.2) coincides with Eq. (6.5), as expected. If ƒ0 i san integer of ƒthr (6.3), the incident HCP gives rise to an exact N-bubble solution. Other-wise, a part, ∆Wrad , of its incident energy, W0 , is radiated away into non-trapped modes;their relative impact decreases rapidly as the total number of EMBs increases:

(7.5)

Thus ∆Wrad is always smaller than the critical (smallest) energy, Wthr of a bubble (for thefixed τ0). Furthermore, as the incident amplitude increases, the relative maximum energyof un-trapped radiation greatly decreases:

(7.6)

8. SHOCK-LIKE WAVE FRONTS

When the incident HC-pulse is sufficiently strong to generate many EMBs (N0 >> 1) ,right before the EMB-precursor is formed, the initially smooth HCP drastically steepensand forms a shock wave at the front of the pulse. The formation of the EMB-precursorcoincides with the point in space at which the shock wave is steepest; this front is aboutτEMB long. After this point, the shock wave breaks into the train of EMBs. To investigatethis shock wave formation and estimate the location of the breaking point, we make furtherapproximation, which may be called "instantaneous reaction", by dropping the higherderivative terms in the constitutive equations of the system. In the limit Q << 1, Eq. (5.3) isreplaced in the case of TLS by

(8.1)

and in the case of the anharmonic classical oscillator (5.3) -- by

(8.2)

If the HCP amplitude is small, ƒ << 1, Eqs. (8.1) and (8.2) are further simplified to

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(8.3)

with a being the same as in Eq. (4.4) for classical anharmonic potential; a = 1/8 for TLS.Any equation in the form

where F(ƒ) is some smooth function, has a general solution whereby each point of the solu-tion, ƒ = ƒ1 , moves with the fixed velocity determined by ƒ1 :

(8.5)

where in the case of Eq. (8.1), F(ƒ) = 1 – (1 +f2)–3/2 , and of Eq. (8.3), -- F( ) = 12aƒ2 .Evaluating now the derivative ∂f/∂τ at a point ζ, we find

(8.4)

ƒ

(8.6)

(8.7)

such that for any nonzero there will be a point ζ, at which ∂ƒ/∂τ → ∞,which signifies the formation of shock wave. Since the formation of the shock wave willbe arrested at the amplitude ƒ1 , at which is maximal, we find that for the profile(6.2), such a point is at c osh or at with

Hence the distance of formation of the shock wave is

which in the case of TLS (i. e. when a =1/8 and τ0 = 4/ƒthr ) gives Eq. (6.18). At the pointof shock wave formation, the full constitutive equation will prevent the discontinuity of theexact solution, and break the shock wave into the train of solitons; the length of thesteepest rise of the shock wave is thus determined by the EMB-precursor time, (3.9).

Consider an example, E0 = 2 MV/cm with t0 ~ 313 ƒs (or 413 ƒs at pulse’s half-amplitude), in Xe. In this case ƒthr = 10–3 and Ethr ≈ 60 KV/cm (ƒ0 ≈ 3.3 × 10 –2 = 33ƒthr ) ,and the formation distance is estimated, Eq. (6.18), as ζEMB ~ 1.54 × 106 , which under10 atm pressure (Q ~ 0.57 ) translates into zsh ~ 12.5 cm. The EMB-precursor here is 4.8 ƒslong, two orders of magnitude shorter than available HCPs. Note that in all these exampleswith HCPs, the field ƒ << 1 (ΩR << ω0) is much below the super-dressed regime of TLS, andtherefore far from the ionization. The distance of the shock wave (and first EMB) forma-tion can be shortened, if its leading front is sharpened (e. g. by a shatter), such thatτlead < τ0. The distance ζ sh can then be evaluated by multiplying (8.7) by a factor τlead/ τ0;in the above example, if the HCP leading front is shortened down to ~ 40 ƒs, the shock for-mation distance reduces to ~ 1.25 cm.

9. EM-bubbles generation by a short laser pulse

Even the highest realistically expected fields of HCPs are still much lower that theamplitudes readily attainable in lasers. The possibility of the EMB formation in each lasercycle, therefore, increases tremendously, although the ensuing picture becomes more com-plicated due to the multiple EMB interactions, when the regular laser radiation with manyoscillations in the envelope is used instead of HCPs. Indeed, since the laser cycle is muchshorter (e. g., the cycle duration for the radiation with λ=0.9 µm is ~ 3 ƒs ), with the laserintensities of ~ 1014 W/cm2 (which corresponds to the field ~ 2.7 × 108 V/cm), the EMB for-mation distance reduces to less than 1 mm, and the EMB becomes an order of magnitudeshorter than the optical cycle. Fig. 8 shows a group of EMBs developing from a very short(6 ƒs) laser pulse with the relatively low peak intensity 6.8 × 1012 W/cm2 . One can see that

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the EMB formation length is about 0.1 – 0.2 mm, and the length of the EMB-precursor isabout 0.5 ƒs. There is a distinct possibility that the very high-order harmonics generation[26] in noble gasses might be to a substantial degree attributed to the multiple EMB forma-tion, which would explain many major features of the HHG phenomenon, such as its puz-zling insensitivity to the phase mismatch at different high harmonics, broadening and shiftof harmonic spectra, etc.

Fig. 8. EMB formation from an oscillating laser pulse (tp ≈ 6 ƒs). Inset -- a final cross-section magnified to

show EMBs.

10. DIFFRACTION-INDUCED TRANSFORMATION OF SUB-CYCLE PULSES [27]

So far we were focusing on nonlinear propagation of sub-cycle pulses, being mostlyinterested in formation of solitons and EM-bubbles in nonlinear media. >From the applica-tion point of view it is important also to know what would happen with a sub-cycle pulsewhen it is emitted into a linear medium (e. g. air or vacuum). It is clear that the diffractionwill immediately affect not only spatial profile of the pulse, but also its temporary profile,since different Fourier components of such a broad-band pulse diffract differently. Low-frequency components diffract most drastically, almost as the radiation of a point source,while very high-frequency components may propagate almost without diffraction likegeometro-optical rays. Thus, on-axis radiation will be loosing the low-frequency part of its

310

spectrum, while the off-axis radiation will be loosing its high-frequency part, which willresult in a peculiar transformation of both of them. In particular, we show below that theon-axis pulse in the far field area will be mimicking the time-derivative of the originalpulse (thus formating "full-cycle" pulse out of half-cycle pulse), while in addition to that,the off-axis propagation also results in the lengthening of this pulse.

In this Section, following our recent work [27], we develop an analytic approach tothe theory of linear diffraction transformation of pulses with super-broad spectra and arbi-trary time dependence, in particular half-cycle (unipolar) pulses. Since this theory hasmuch broader applications than nonlinear processes, we will develop it for spatially 3Dcase (or precisely speaking, 2+1+1D case, i. e. with the spatial cross-section of the beamhaving two dimensions, and other two dimensions formed by the axis of propagation andtime). We found close-form solutions for pulses with initially Gaussian spatial profileshaving either cosh– 1 -like or Gaussian time dependence. The far-field propagation demon-strates time-derivative behavior regardless of initial spatio-temporal profile.

Diffraction is one of the fundamental manifestations of the wave nature of light. Thediffraction theory of monochromatic light has been developed in great details (see e. g.[28]). In general this theory is heavily loaded with various special functions, but thedevelopment of lasers advanced the use of Gaussian beams, which are auto-model solu-tions of a so called paraxial approximation (PA) and allow one to handle the diffraction ofspatially-smooth optical beams in a very simple way (see e. g. [29]).

Recent developments [7-11,18] in optics resulted in the generation of short andintense EM pulses of non–oscillatory nature, or almost unipolar "half-cycle" pulses(HCPs), with extremely broad spectra that start at zero frequency; even existing pulseshave many exciting applications (see introductory section and Refs. [7-11,18]). The spec-tra of currently available HCPs generated in semiconductors via optical rectification, reachinto terahertz domain; they are ~400–500 ƒs wide, with the peak field up to 150–200 KV/cm. In our recent work part of which is described in this paper (see also Refs.[12-14]), we proposed new different principles of generating much shorter (down to0.1 ƒs = 10–16 s) and stronger (up to ~ 10 16 W/cm2 ) HCPs. As it was pointed out above,different frequency components in HCPs diffract differently, far away from the sourceHCPs propagate with significant dispersion and distortion even in free space [7-11,18,30,31]. This phenomenon calls for the diffraction/transformation theory of pulseswith super-broad spectra, preferably comparable in its simplicity and insights with that ofGaussian beam diffraction of monochromatic light. Such a theory could also apply to otherfields of wave physics: acoustics, solid-state physics and quantum mechanics.

The work [31] analyzed asymptotic field behavior in far-field area of a beam with an(initially) Gaussian spatial profile and found time-derivative behavior in that area. Ref.[31] considered only the fields with an also (initially) Gaussian temporal profile. However,even for that profile, no global analytic solution (even for on-axis field) for the fieldbehavior along the entire propagation path was found, leaving the theory without the majoradvantage of a standard theory of monochromatic Gaussian beams. The ability of theory toglobally address the propagation, in particular, in between near- and far-field areas, isessential, since in practice, that intermediate area could be of most significance, with thediffraction distance for the highest spectral frequency, (where 2t0 is a pulsetime-width, and 2r0 a pulse transverse size), being considerably large. For 2t0 ~ 400 ƒs and2r0 ~ 1 cm [3], one has zd ~40cm, the same as e. g. for 2t0 ~ 4 ƒs and 2r0 ~ 1 mm.

We derive here a simple equation for on-axis field (with a Gaussian initial spatialprofile) valid for an arbitrary temporary profile and for any distance from the source.Using that equation, we obtain close-form solutions for the field transformation due to dif-fraction for some temporal profiles, in particular cosh– 1 -like and Gaussian profiles, andshow that in far-field area pulses demonstrate time-derivative behavior regardless of their

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frequency limit, we find a general solution valid for any spatia1 and temporal profiles of thefield, which also explains in simple antenna terms the nature of time-derivative behavior;this solution is also valid in far-field area for any frequency.

Consider now a pulse propagating along the z axis and having an arbitrary timedependence and a known transverse profile, at the point z =0. We assume at this point ahigh-frequency limit, meaning that the shortest temporal scale of the pulse, t0 (in theextreme case of non-oscillating, half-cycle pulse, it is its initial half-timewidth, see below),and its respective longitudinal scale, ct0 , are much shorter than its transverse radius, r0 ,

(10.1)

The frequency components of the largest part of its spectrum, will pro-pagate with relatively small diffraction, so that one can apply a standard paraxial approxi-mation (PA) to each one of them. Within PA, the diffraction of a monochromatic field,Eω exp[– iω(t – z/c )], in a free space, is described using a PA wave equation similar to aSchrödinger equation for a free electron:

(10.2)

where is a transverse Laplacian; note that PA allows one to neglect polarization of thefield and reduce the problem to a scalar one. We will also assume the field cylindricallysymmetric in its cross-section, so that where r is the radial distancefrom the axis z in the cross-section. With the most of the available or to be availablesources of HCPs, one can assume that the field at the source has a plane phase front for allthe spectral components, so that their waists are located at the source. The spatially-Gaussian field at the source, z=0, can then be written as where r0 is theradius of the spatial field profile at the level exp(– 1/2) of peak amplitude, Writing thesolution of PA equation (10.2) as a Fourier transform:

(10.3)

where is a retarded time, we have the field spectrum S(ω,r,z ) for a Gaussian modeas:

(10.4)

(10.5)

is the spectrum of original pulse, and

is a diffraction factor due to PA. For the on-axis field, r=0, we haveBy substituting this into (10.3), and introducing a dimensionless

retardation time, and propagation distance, we derive asimple equation for the temporal dynamics of an on-axis field at any point ζ:

(10.6)

If the full energy of the field at source is finite, the solution for the on-axis field is:

(10.7)

where s(x ) = –1 if x < 0, and s(x) = 1 otherwise. One can see that, as expected, in a near-field area, ζ << 1, the original temporal pulse profile is almost conserved, Eon ≈ E0(τ). T h emost interesting and universal (see below) pulse transformation occurs in far-field area,ζ >> 1. In this case, E on can be expanded as

(10.8)

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Here

so that as ζ → ∞, the on-axis far-field replicates time derivative of the original pulse:

(10.9)

All of the results (10.6)-(10.9) are true for an arbitrary initial temporal profile, E0(τ). I nparticular, any HCP is transformed in the far-field area into a single-cycle pulse.

Writing Eq. (10.6) in real time, where we noticethat it coincides with an equation for the voltage UR ∝ Eon at a resistor R in a series RCcircuit (high-pass filter) driven by a source US ∝ E0 so that the circuit relaxation time isT=RC. Since the only parameter with the dimensionality of resistance in a free-space pro-pagation is the wave impedance of vacuum (R = 120 π ohm), the circuit capacitance is then

which is consistent with a capacitor formed by electrodes having the pulse waistarea ∝ and spaced by z and thus provides an interesting and simple interpretation of thenature of pulse transformation in free space.

A simple example of the field evolution along the the entire path of propagation (i. e.for an arbitrary ζ ), is given by a smooth bell-shaped initial profile

with exponential tails at |τ | → ∞. Eq. (10.7) yields then:

(10.10)

at ζ = 1 and τ > 0, the rhs of (10.10) is e x p ( – τ )( 1 – 2τ2)/4. A familiar profileE 0 = ξ 0 /cosh(τ) does not behave as nicely; in this case, a solution (10.7) in elementaryfunctions exists if ζ is any rational number, but its form is different for different ζ’s. Atζ =1, one has

The solution (10.7) for an Gaussian initial temporal profile(here t0 is the pulse half-width at exp(–1/2) peak

amplitude), having the spectrum is handled analytically for any ζ:

(10.11)

where As the pulse propagates, its total on-axis energy

per unity area,

(10.12)

which in the limit ζ → ∞ yields w(ζ ) → (2 ζ2)–1 , as expected. The evolution of the profileand spectrum of the on-axis Gaussian HCP as it propagates away from the source, is shownin Fig. 9; one can clearly see that the pulse sheds off lower frequencies to finally formalmost exact mimic of the time-derivative of the original Gaussian pulse. The zero point(i. e. the moment τz where E = 0), is moving closer to τ = 0 as the distance ζ >> 1 increases.Using first two terms in the expansion (10.8), with the zero point found from

, we have τz ≈ ζ –1 .

The lower-frequency radiation diffracts stronger and hence is found mostly off-axis,where, by the same token, the higher frequencies are weaker. all of which results in thelengthening of the pulse. The smaller the diffraction angle at the cut-off frequency,θ d = ct0 /r0 (<<1 due to (10. 1)), the stronger this effect is pronounced. In far-field area,ζ >> 1, we introduce the angle of observation, θ = r/z, and an angular factor due to diffrac-tion, and approximate the spectrum (10.4) of the pulse as: Soƒƒ (ω, θ, ζ) ≈

which in the case of Gaussian initial tem-poral profile, yields an off-axis pulse in far-filed area:

(10.13)

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(10.13) resembles (10.9) with the main difference being that off-axis pulse stretches in timeby the factor Θ and its spectrum is respectively "squeezed" by the same factor.

Fig. 9. The evolution of the on-axis temporal profile (normalized field, Eon 0 vs normalized time, τ, and

the normalized amplitude spectrum, | S | 0 t0 , vs normalized frequency, v ≡ ωt0 (inset), of the initially

Gaussian half-cycle pulse, as it propagates along the axis Curves: 1, ζ = 0 ; 2, ζ = 0.25; 3,

ζ = 0.5; 4, ζ = 1; 5, ζ = 2 ; 6, ζ = 4. For the sake of comparison, each curve in the main Fig. is scaled up by

the factor w –1/2 (ζ).

In the so called low-frequency limit r0 << ct0 , opposite to (10.1), with the source sizebeing much smaller that the wavelength λ =2πc/ω of any frequency component, all thecomponents have the same dependence on the angle of propagation; also, the initial spatialprofile of the field becomes unimportant. The radiation pattern at each frequency is thendetermined by an elementary (i. e. point-like) dipole formed by the field distribution,

(t,x,y). At the distance from this point-like source (i. e. awayfrom the very small near-field area ρnear << λ), and assuming that the field is linearlypolarized, the spectrum of radiative waves is:

(10.14)

where θ is now the angle between the axis z and the direction of propagation,in the plane of the vector of polarization, and the observation point.

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The Fourier transform of (10.14) produces the same time-derivative profile everywhere,

(10.15)

where is the polarization vector of radiative field, = cosθ. E q .(10.15) explains pulse transformation in simple terms of elementary dipole antennae drivenby a current which is induced by the dynamics of one of the dipole electrical"charges" q0 originated by the source field, E0 ; hence time-derivative temporal profile.Bearing in mind that for a Gaussian beam, Eq. (10.15) at θ=0 is consistentwith the Gaussian on-axis far-field (10.9), indicating that the results (10.6-10.12) for theon–axis field are valid regardless of the condition (10.1). Furthermore, in far-field area,Eq. (10.15) describes an on-axis field for any distribution, regardless of whether it is Gaus-sian or not.

The dispersion and transformation of the pulse due to the propagation and diffractioncan to great degree be reversed. The feasibility of that is related to the time/space recipro-city manifested here by Eq. (10.6) being invariant to the simultaneous sign reversal of timeτ and distance ζ. (The same is true for the solution (10.7), if E0 ( τ ) is a symmetric function,see Eqs. (10.10)-(10.12).) In practice, the diffracted HCP can be transformed back almostinto its original temporal profile (except for its cw component) by reflecting its diffractedwave front e. g. from a spherical concave mirror, if the angular aperture of the mirror issignificantly larger than the diffraction angle, θd . If such a mirror has the radius of curva-ture Rm and is situated sufficiently far from the source, with the distance between thembeing ƒ1 >> zd , the pulse is focused again into a tight spot at the distance ƒ2 , determined by astandard optical mirror formula, If ƒ2 <ƒ 1 , the area of this spot is smallerthat that of the original spot, and the amplitude of the focused pulse is larger by the factorƒ1 /ƒ2 . The residual distortion of the pulse (in particular, slight bipolarity of initially unipo-lar HCP) will be due to lower-frequency diffraction losses at the mirror; the larger the mir-ror size, the smaller this effect. In the case of point-like source, pulse restoration can beachieved by using a full ellipsoid of revolution, with the source and observation pointssituated at the foci of the ellipsoid.

11. CONCLUSION

In conclusion, we have theoretically demonstrated feasibility of powerful, near- andsub-femtosecond sub-cycle EM pulses and solitary waves, EM bubbles, supported by bothquantum and classical nonlinear media. We have shown how their maximum amplitudeand minimum length are limited by the atomic ionization. It follows from our theory that10 – 0.1 ƒs EMBs can be generated by the available half-cycle pulses and short laser pulses;the peak EMB intensity can reach ~ 1014 – 1016 W/cm 2. Those results represent only thevery first steps in the exploration of the new time domain. Our hope is that EMBs will beexperimentally observed in the near future. This will pose new set of problems, such asEMB detection and characterization, separation, gating, control, focusing and guiding, andexploring various EMB applications. In a transverse-limited EM field, a zero-frequencyspectral component of the incident HCP will not propagate beyond the near-field area, andin the far-field area, EMB will assume a modified profile. Using analytic approach to thediffraction-induced transformation of pulses with arbitrary temporal profiles, includinghalf-cycle pulses, we found close-form solutions for the propagation of most commonlyused initial spatio-temporal profiles, and explained the nature of time-derivative transfor-mation in far-field area for arbitrary pulses.

315

The new time domain, being largely an uncharted territory, holds a lot of promises forthe physics of field-matter interaction. The most familiar nonlinear effects and parametersassociated with coherent light-matter interactions (harmonic generation, self-induced tran-sparency, photon echoes, soliton generation and propagation, saturation of all kinds, n2 ,χ (3) , etc.) are likely to take on entirely different forms, or may even cease to exist. One ofthe most fundamental and intriguing phenomenon is the field ionization of atoms,molecules, and semiconductor quantum wells by a super-short pulse with the amplitudecomparable to or larger than the ionization threshold. Such pulses could cause a substan-tial "shake-up" excitation or ionization of an atomic system within the time much shorterthan any characteristic time of the system. In our most recent research [33] we showed thata few ƒs long and unipolar EMB acting upon a semiconductor quantum well, can causeboth forward and backward field ionization, with the photoelectrons emitted in both direc-tions (i. e. not only in the direction of the ionizing unipolar field) with comparable intensi-ties. Even more fundamental and exciting results are obtained for the hydrogen atom hitby a sub-cycle pulse with an sub-atomic unit amplitude. We also observed that the ioniza-tion response of the atom consists of a sequence of well-separated peaks resulting in strongspatio-temporal inhomogeneity of the photoelectron cloud, and found an explanation ofsuch a behavior.

This work is supported by AFOSR. The work by SFS is in part supported by theDeutsche Forschungsgemeinschaft. AEK is a recipient of the Alexander von HumboldtAward for Senior US Scientists of AvH Foundation of Germany.

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For a harmonic oscillator with a frequency ω0 , it is natural to choose with, where is the Compton wavelength.

If nonlinearity is negative, a < 0, one can expect formation of "dark" EMB (a solitary"hole" propagating on a cw field background): ƒ(τ)∝ƒ0 tanh(τ ƒ0), ƒ 0 =const.

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317

NONLINEAR WAVEGUIDING OPTICS

R. REINISCH

Laboratoire d'Electromagnétisme, Microondes et Optoélectronique, UnitéMixte de Recherche 5530 du Centre National de la Recherche Scientifique,Institut National Polytechnique de Grenoble - Université Jopseph Fourier,ENSERG, 23 ave des Martyrs, BP 257, 38016 Grenoble Cedex, France. e-mail : name@enserg.fr

I. INTRODUCTION

Nonlinear waveguiding optics is concerned with nonlinear optics in devices whichsupport guided waves or surface plasmons (or any combinations of them)[1]. These wavesmay be resonantly excited using prism or grating couplers. The theories which allow thestudy of such nonlinear optical interactions are the total field analysis[2] and the coupled-mode approach[3,4]. In the total field analysis no hypothesis is made regarding the transversefield map (i. e. perpendicular to the direction of propagation of the guided waves). Thus thisis a general method which also applies to non-guiding structures. But its generality does notallow an easy insight in the underlying physics. In the coupled-mode approach theelectromagnetic (EM) field is expanded on the guided modes and on the radiation fields of theguiding device. This method is convenient when guided modes are excited. However this isnot always the case : when considering prism or grating couplers or Fabry-Perot's no guidedmodes are present because these devices behave as "open" resonators. Then one is left withthe radiation fields that is to say with an integral representation of the EM field which doesnot lead to an easy analysis of "open" resonators. But it is known that there exists a close linkbetween EM resonances and poles[5,6] arising from the solution of what is called thehomogeneous problem. Once the importance of poles in some suitable complex planes isrecognized, it is tempting to solve the homogeneous problem in order to use these poles forthe study of nonlinear optical interactions in "open" resonators. It is at this level that thedifficulty appears : due to the "open" feature, the EM resonances involves leaky modes[5,7].These modes, when derived from the solution of the homogeneous problem, are found toexist by themselves and consequently, due to their leaky feature, to diverge at infinity. Thisdifficulty comes from the fact that the solution of the homogeneous problem yields the polesbut tells nothing about the way they contribute to the EM field. In other words the solution ofthe homogeneous problem, although useful, is not sufficient to get a complete solution of thiskind of problems. When looking for the expression of the EM field due to a source (incidentbeam(s) and/or nonlinear polarization), it is found that leaky modes have no individualexistence : they always constitute a portion of the EM field i. e. they only exist in a finiteregion of space preventing in this way any divergence at infinity[8]. As already mentioned,the coupled-mode formalism[3,4] only involves the radiation fields but does not exhibitexplicitely the leaky modes although these modes constitute a powerful tool for the study of"open" resonators in nonlinear optics. However no coupled-mode formalism has beendeveloped bringing into play leaky modes.

In this chapter we show how leaky modes can be simply used for the study of nonlinearoptical interactions in "open" resonators. This requires to go somewhat in the detail of the

Beam Shaping and Control with Nonlinear Optics319Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

theory of leaky modes, to show that they arise from poles which are called "improper"(guided modes are associated to "proper" poles), to explain that several expansions of the EMfield are possible. Passing from one of these expansions (the transverse one) to another one(the leaky mode one) leads to a relation between leaky and radiation modes. This relationserves as a basis to derive the equation of evolution of a leaky mode amplitude. Cerenkovsecond harmonic generation (SHG) is chosen as an example to illustrate the interest of leakymodes in guided wave nonlinear optics.

II. LEAKY MODES

Various representations of the EM field will be considered : the first one is thelongitudinal, or Fourier, representation from which two other expansions are derived : thetransverse representation which displays the guided modes and the radiation field(s)supported by the structure and the leaky mode representation which is appropriate whenconsidering "open" or leaky resonators. Section 2 is devoted to the derivation of the equationof evolution of a leaky mode amplitude.

1. Different representations of the EM field in linear planar structures

The structure of interest is a linear planar multilayer configuration illuminated by a lightbeam (time dependence e –iω t ) under incidence θi . Figure 1 shows an example corresponding

to a system involving three media 1,2,3 with relative permittivity ε1 , ε 2 , and ε3 with ε2 >( ε 1,

ε3 ) in order to allow for guided modes.

Figure 1. The structure of interest.

For simplicity, we assume :i) lossless media,

ii) a two-dimensional situation where Thus the EM field which

depends on x and y (i. e. is either TE polarized (specified by the z-component

of the electric field or TM polarized (specified by the z-component of the

magnetic fieldThe analysis closely follows refs. [3,8-10].

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1.1 The longitudinal representation. The solution (x, y) is expressed as aFourier integral :

(1)

Let α q denote the y-component of the wavevector in the outside media q (q=1,3). The

quantities αq and γ are related by :

(2a)

or

(2b)

with:

(c : speed of light in vacuum) (3)

In eq. 1 the integration is performed along the path P corresponding to the real γ-axis.Hence the name of longitudinal representation given to this expansion which includesimaginary values of αq since γ varies from −∞ to +∞. Equation 1 represents Φz (x,y) as acontinuous infinite spectrum of waves : each elementary spectral component is described by

ϕz (γ ,y)e iγ x and corresponds to a plane wave solution with longitudinal wavevector

component γ and transverse field map ϕz (γ ,y). It is the quantity ϕz (γ ,y) which depends onthe particular structure of interest : multilayer configuration, gradient index device…

The advantage of eq. 1 is its relative simplicity. The drawback comes from the fact thatthis expression does not exhibit explicitely possible guided modes which, as is known, maybe present in the structure fig. 1.

1.2 The transverse representation : guided modes and radiation fields.This representation is obtained from eq. 1 by performing a contour deformation where thepath P of integration along the real γ-axis is transformed into a path Pγ in the complex γ-plane.Such a procedure is valid provided the function ϕz (γ,y)eiγ x is analytical[11] in region C

bounded by P and Pγ (Fig.2). This requirement implies a search of the singularities ofϕ z(γ,y)e i γ x. These singularities are of two types :* poles which contribute to the integral in the complex γ-plane through the residue theorem,

* branch points, due to the square root in eq. 2b, which require cuts in the complex γ-plane inorder to avoid multivalued functions.

Concerning the branch points, some care must be exerted. From equation 2b, it is seenthat there are branch points at ± kq (q=1,3). In medium 2 passing from one determination ofthe square root of the transverse components of the wavevectors eq. 2b to the other one doesnot give a different EM field[9]. Thus the determination chosen for the square roots in thismedium is not of importance. For q=l,3 the choice of the branch cuts is rather arbitrary.However convenient cuts are those for which the radiation condition at infinity is fulfilled onthe entire top sheet of the four-sheeted Riemann γ -plane.

The radiation condition writes :

Im(αq) > 0 (q= 1,3) (4a)

321

and the cuts correspond to :

Im(αq ) = 0 (q=1,3) (4b)

With these cuts the entire top sheet of the four-sheeted γ-plane is mapped on the upper

half of the complex αq -planes (q=1,3). The image of the cuts eq. 4b in the complex γ-planecorresponds to the hyperbolas :

γ 'γ" = k'qk"q (q= 1,3) (4c)

with :

γ = γ' + iγ" (4d)

kq =k'q +ik"q (4e)

In eqs. 4c-e and in fig. 2, the outside media exhibit infinitesimal losses in order to avoidcuts located on the γ" and γ' axis. The resulting contour of integration Pγ is illustrated in fig.2 .

Figure 2. Top sheet of the complex γ-plane. The bold lines represent the branch cuts.

When γ is swept around the cut, αq remains real and varies from +∞ to -∞ . In theregion x>0, the semi-circle at infinity does not contribute to the integral. For x<0, this semi-circle is in the lower half of the top sheet of the γ-plane.

All these mathematical considerations lead to the following transverse representation ofΦ z (x,y) which results from a deformation of the contour from P to Pγ and from a change ofvariable of integration:

322

(5)

Since the entire integration is carried out in the top sheet of the complex γ-plane, all the poles

which are captured when deforming P into Pγ are located only in the upper half of the αq-planes (q=1,3) and consequently comply with the radiation condition eq. 4a. These poles,which are called "proper", are associated to non-diverging EM fields obtained solving thehomogeneous problem, i. e. without excitation. Thus the corresponding residues account forguided modes described by the summation over m in eq. 5. The two continuous spectra(Q=I,II) come from the paths around the cuts and describe two types of radiation fields whichinclude imaginary values of γ : the class I radiation field arises from the branch point k

q=1

whereas class Q=II is due to the branch point kq=3 . The term α Q denotes α1 for class Q=I and

α3 for class Q=II. These so-called radiation modes correspond to the solution obtained whenilluminating the system by a plane wave of unit amplitude incident on the interfaces y=-t andy=0 for Q=I and Q=II respectively. Figure 3 is a possible representation of the radiationmodes of class Q=I,II[3].

Figure 3. Class I and class II radiation fields : ρI,II and τI,II

and transmitted fields for the two classes of radiation fields.denote respectively the amplitude of the reflected

As compared to the longitudinal representation eq. 1, expression 5 of Φz(x,y) exhibitsthe guided modes and the radiation fields. In expansion 5, a coefficient cm corresponds to theamplitude of the mt h guided mode and cQ(αQ) to the amplitude of a spectral component of the

radiation field of class Q; ϕm,z(y) and ϕQ,z(αQ,y) represent the associated transverse field mapwhich depends on the guiding structure. It is seen that the transverse representationcorresponds to the expansion of the EM field used in the coupled-mode formalism[4].

1.3 The transverse representation : orthogonality relation[3]. The modes ofthe transverse representation (i. e. guided or radiated) have orthogonality properties which area consequence of the Lorentz reciprocity theorem[12]. It is assumed that the permittivitytensor [ε] and the permeability tensor [µ] are symmetrical tensors of rank 2. Let

and

and n (guided or radiated). Use of the Lorentz reciprocity principle leads to the followingorthogonality relation wheni) at least one of the modes is a guided one :

be the EM fields for two different modes m

323

(6a)

In eq. 6a δm,n denotes the Kronecker symbol with the following meaning :

(6b)

Therefore notice that δm,n = 0 when γn = γ m .

ii) two radiation modes α and β are involved, the orthogonality relation writes :

(7a)

In eq. 7a δ[γ(α) + γ (β)] is the Dirac-δ function : two radiation modes α and β for which

γ(α) + γ(β) ≠ 0 (7b)

are orthogonal.In eq. 6a and 7a, ⟨....⟩ means an integration in the cross-section plane :

and = (1,0,0).Equations 6a and 7a show that only the transverse components of the EM field enter theorthogonality relations.

The interest of the orthogonality relations eqs. 6a and 7a is that they allow to determinethe unknown coefficients occuring in the transverse representation of the EM field :

(8)

According to eqs. 6 and 7, it is worth noting that in order to derive the amplitude of agiven mode one has to use the orthogonality relations with a mode having the same absolutevalue of the longitudinal wavevector component but propagating in the opposite direction.Stated differently, one has to associate two counterpropagating modes.

Proceeding along these lines, the following equations yield the amplitude (thesuperscript ± stands for forward (+) and backward (-) modes)* of a guided mode :

(9a)

with

(9b)

32 4

* of a radiation mode :

(9c)

with

(9d)

(10a)where

1.4 The transverse representation : equation of evolution of a modeamplitude[3]. Let us consider now the situation where a nonlinear polarization is present.This leads to x-dependent modes amplitudes in eq. 8. The aim is to derive the equationobeyed by these coefficients.

(10b)

(11)

The starting point is the Lorentz reciprocity relation used under the following conditionsNL

: one of the EM solution is generated by a nonlinear polarization (x, y) whereas the otherone corresponds to an eigenmode n (guided or radiated). Thus :

In eq. 11 is the EM fields arising from the nonlinear polarization

NL(x,y).

Since the field tends to zero at infinity, eq. 11 leads to :

Equations 9a and 9b show that the following equations are obeyed by* the amplitude of a guided mode :

(12)

o r

(13a)

(13b)

325

* the amplitude of a radiation mode :

(14a)

or

(14b)

It is worth noting that eqs. 13, 14 are first order differential equations although no slowlyvarying envelope approximation has been made.

The important consequence of the orthogonality relations eqs. 6,7 is that guided modescannot be excited by a beam incident on the device from the substrate or the superstrate. Theexcitation of guided modes is possible either using the but-coupling technique or "opening"the structure. This leads us to the third representation.

1.5 The leaky mode representation. The transverse expansion eq. 5 is of greatinterest when considering situations where guided modes exist. But "open" resonators do notexhibit proper poles. Thus eq. 5 only includes the radiation fields which often describe anEM field traveling along an oblique direction with respect to the x-direction. For thesestructures eq. 5 is not more helpful than eq. 1 is. Such situations are conveniently handledintroducing another complex plane which we call the complex w-plane. Let us consider theEM field Φz(x,y) in medium q=3. The integration in eq. 1 is carried out in the complex w-plane :

w = u + iv, (15a)

through the following change of variables :

γ = k3 sin w

α3 = k3 cos w

together with the use of polar coordinates :

(15b)

(15c)

(15d)

(15e)

x = r sin θ

y = r cos θ

where r and θ are shown in fig. 4a.

Figure 4a. The change of variables eqs. 15d,e. A leaky mode only contributes to the raditation field withinthe shaded region.

326

It should be noted that the transformation eqs. 15 b,c maps the sheets of the γ-plane into

a strip of width 2π in the complex w-plane avoiding, in this way, the branch cut associated to

the determination of α3. This mapping plots the four quadrants of the top and bottom sheets

of the γ-plane into the regions denoted Ti and Bi (i=1,2,3,4) respectively (fig. 4b). The pathPw in fig. 4b is the image of the original contour of integration P, used in the integration ofeq. 1, through the transformation eqs. 15b, c.

Figure 4b. The complex w-plane. Solid curve : SDP (θ=0), dashed curve : SDP (θ= π /4), dotted-dashedcurve : SDP (θ= π/2).

In view of an asymptotic evaluation of Φ z(x,y) in medium 3, Φ z(x,y) is written(according to eqs. 15) :

(16a)

The leaky mode representation involves a contour deformation from path Pw to the steepestdescent path (SDP). The calculation shows that there is a saddle point at w=θ and that theSDP obeys the equation :

cos(u - θ)cosh v = 1 (16b)

Before going further on some comments are in order.i) According to eq. 16b, the position of SDP in the complex w-plane depends on the directionof observation θ i. e. SDP=SDP(θ ). The SDP curve crosses the v=0 axis at u= θ.ii) Only those poles located between paths Pw and SDP contribute to the integral eq. 1. SinceSDP generally passes through strips B1 or B3, poles located on the bottom sheet of the γ-plane may be captured. These poles, which do not fulfill the radiation condition eq. 4a, arecalled "improper", and correspond to leaky modes whose amplitude diverges at infinity. Butfig. 4b shows that an improper pole contributes provided the angle of observation θ is greater

than a critical angle, θc,l (the index l labels a pole), the value of which depends on the

327

location of the pole in the w-plane (in the example fig. 4b θc,l = π /4). Stated differently, theimproper poles contribution only occurs within a finite angular region corresponding toθ>θc,l (shaded region of fig. 4a) preventing, as y → +∞, any divergence in the EM fieldrepresentation.iii) These leaky modes are characteristic of open resonators : the improper feature of pole γ l

comes from the EM coupling with the outside. These modes are obtained solving thehomogeneous problem.iv) It was pointed out in section 1.2 that improper poles cannot be captured within thetransverse expansion. Thus leaky modes are part of the radiation field.

Finally it is seen that the EM field eq. 1 is the sum of the space wave Φsw,z,3 resultingfrom the integration along SDP and of the residue contributions coming from the poleslocated between Pw and SDP :

(17a)

The discrete summation in eq. 17a is carried out considering all the poles located between Pwand SDP : the leaky ones and the "proper" ones within the angular regions where theycontribute.The asymptotic integration of eq. 16a along SDP leads to :

(17b)

In eq. 17b asymptotic means k3 r >>1. The far-field corresponds to a cylindrical wave whichdecreases as r-1/2 together with an angular dependence ϕz(θ) cos θ independent of r. This is aclassical result of far-field radiated by antennaes[13].

To summarize for large y (x constant), leaky modes do not contribute and the spacewave eq. 17b constitutes a good approximation of the radiation field. In, or very close to, theguiding layer (medium 2), it can be shown[3,9] that the leaky modes provide a gooddescription of the radiation field. Besides it is known[5] that improper poles are intimatelylinked to EM resonances. Therefore in the resonance domain it is possible to replace theradiation fields involved in the EM resonance process by the associated leaky mode. Thisimportant result greatly simplifies the study of optical resonators in nonlinear optics as weshow in the next section.

2. Equation of evolution of a leaky mode amplitude

The starting point is constituted by the transverse and leaky mode expansions (eqs. 5and 17a respectively). In this section the following hypothesis are assumed which correspondto usual situations when interested in "open" resonators :a) we look for the EM field in, or in the vicinity, of the guiding layer (medium 2),b) their exists isolated improper polesc) the working point remains in the vicinity of such a pole i. e. of the order of the distancefrom the pole to the real γ-axis.When points a-c apply, eqs. 5 and 17a yield :

(18)

According to equation 18, a leaky mode constitutes a good approximation of theradiation field of class Q involved in the resonant process.Use of eqs. 14b and 18 leads to :

328

(19)

In eq. 19 the first term arises from the existence of the nonlinear polarisation NL whereasthe second one describes the in-coupling of possible incident beam(s) on the system.

We now show that, thanks to hypothesis a-c, eq. 19 can be greatly simplified.

2.1 Simplification of equation 19. It is worth noting that and

have different expressions. Indeed is the solution of the homogeneous problem

whereas is the solution corresponding to an incident plane wave. This means that

includes a resonant denominator contrary to which does not. Besides the

comparison of and is possible only in the vicinity of an isolated pole i. e.close to resonance.

Let be the transverse field map at resonance defined as follows :

(20a)

In order to exhibit the resonant term in we notice that it is possible to express

in terms of and of the transmission withand (fig. 3) :

(20b)

In the following Ai =1, hence :

(20c)

Since the functions and are defined within a multiplicative constant, thesimplest choice, which will be done throughout this chapter, is :

(20d)

and

Besides :

Provided hypothesis a-c apply, the constant K is close to 1 :

(20e)

(20f)

K ≈ 1 (20g)

Let us consider now successively the two terms of eq. 19.

329

2.2 Simplification of eq. 19 : the influence of the nonlinear polarization.Equation 19 becomes :

(21a)

Hence from eq. 20f :

(21b)

To avoid unecessary complicated notations, we pursue considering forward propagatingmodes and we drop the superscript +.It is convenient to rewrite eq. 21b under the form :

with

The knowledge of NQ,rad requires the calculation of :

(22a)

(22b)

(23)

(24a)

The term is calculated in ref. [3] :When dealing with nondegenerated radiation modes :

where :denotes the reflectivity for a radiation mode of class Q=I,II,

(24b)

ηQ is the constant value of η (y) in the outside medium 1 for class I and medium 3 for classII.

For symetrical structures :

(25a)

where + and - stand for even and odd modes respectively.If in addition the system is lossless then :

330

Moreover close to an improper pole γl [5] :

(25b)

(26a)

(26b)

In eq. 26b γ z,Q denotes the zero[5] of the reflectivity for class Q radiation modes.According to eqs. 26 and 25b, eq. 25a can be rewritten as :

(27a)

(27b)

with

In eqs. 26 and 27 t Q, rQ et nQ do not depend on αQ. Equation 26a shows that is a

lorentzian centered at γ 'l , with a width 2 γ" l and of peak value |tQ |² . This lorentzian is

associated to the EM resonance giving rise to the improper pole γl . Thus the link betweenimproper poles and EM resonance appears markedly.

The integral eq. 23 only depends on αQ and can always be calculated numerically. Infact, according to the expressions of τQ(αQ ) and NQ(αQ) on the one hand, assuming thatpoints a-c apply on the other hand, IQ can be easily calculated using the theorem of theresidues[11]. Keeping in mind points b) and c), a tedious but straightforward calculationyields* for lossless symetrical structures :

* in the general case :

with

(28a)

(28b)

(28c)

(28d)

In eqs. 28 αQ ,l denotes α1, l for Q=I and α 3, l for Q=II.

331

2.3 Simplification of eq. 19 : the in-coupling process. One has to considerthe second term of eq. 19. Because no source term is assumed in this section, a coefficientcQ does not depend on x :

(29)

(30)

Equations 20b and 20f show that :

where A i(αQ) is the spectral component of the incident beam in the transverserepresentation.Use of eq. 30 shows that eq. 29 writes :

(31)

Let ai(x) be the expression of the incident field at y=0 :

(32a)

(32b)

and

equations 26a and 32 lead to :

(33)

2.4 General equation of evolution of a leaky mode amplitude. When anonlinear polarization is present together with incident beams, the amplitude of a leaky modeobeys the following equation derived from eqs. 22a and 33 :

According to eq. 20g, eq. 34a writes :

In eqs. 34 γi is the longitudinal wavevector component of the excitation (nonlinearpolarization and/or incident field(s)).

2.5 Alternative derivations of Nprocedure. Other methods allow to derive N

Q,rad : total field method, "closing"

“closing” procedure. In both cases the validity of eqs. 34 is assumed. The problem whichQ,rad : these are the total field method and the

remains to be solved is the determination of γl, tQand NQ,rad : the solution of the homogeneous

(34a)

(34b)

332

curve eq. 26a, concerning NIn the first one it is noticed that N

maps. This suggests to derive the expression of Nobtained from the total field method[2] using the undepleted pump approximation (UPA) and

derived by “closing” the leaky resonator. This means that for a Fabry-Pérot the reflectivity of

corresponding value obtained with the closing method the agreement is

(35)

"

“closing” procedure. The latter yields accurate results. The theory developed in section 2,

it suffices to know the improper pole α. Finally the method of ref. [19] allows to relate the expression of the EM field outside

the guiding layer to the known one inside the guiding layer.

SHG has been recently reconsidered [20,21].

III. AN EXAMPLE : CERENKOV SHG

2 γ l "measures" the aperture of the nonlinear optical resonator.Thus leaky modes appear as well suited candidates for the study of nonlinear

interactions in nonlinear optical resonators. It is seen that the difficulty of the divergence ofthe leaky modes is an apparent one : in fact, leaky modes have no individual existence, theycontribute to the EM field only in a finite region of space. Use of eqs. 34 requires thedetermination of NQ,rad . There are three possibilities : eqs. 28a,b, the total field method or the

which allows the demonstration of eq. 34, can be considered as a justification of the“closing” procedure used in previous studies with no demonstration. The main interest of the“closing” procedure and of eqs. 28 yielding NQ,rad is their simplicity since they rely on acalculation of linear optics. For NQ,rad l , the zero γz,Qand tQ

Having demonstrated the equation of evolution of a leaky mode amplitude and explainedhow NQ,rad can be calculated, let us choose, as an example, Cerenkov SHG. This type of

problem yields γl , tQ is easily obtained from the knowledge of the peak value of the resonance

Q,rad two methods have been employed.Q,rad only involves the transverse (i. e. along y) field

Q,rad through a comparison with the result

a simple x-dependence of the nonlinear polarisation of the form eiγ x . This was done in refs.[14,15]. The drawback is the necessity of solving a linearized nonlinear problem.

Another method, which is an approximate one, has also been used. It relies on the factthat high finesse resonators are considered for which the EM leakage is small. Thus NQ,rad i s

the mirrors is 100%, for a prism coupler the prism is removed, for a grating coupler theleakage due to the radiated diffracted orders is neglected. Figures 3 and 4 of ref. [16] and fig.2 of ref. [17] show that the “closing” method yields results in good agreement with thosederived from the total field analysis. This “closing” procedure has also been used in ref. [18].The main advantage of such a method, as compared to the total field calculation, is itssimplicity since it only involves quantities which can be obtained through a linear calculation.

When comparing NQ,rad derived from the theory developed in section 2.2 with its

* excellent :of the order of 5 10-2 for prism couplers (guided waves or surface plasmons),* good :of the order of 10-1 for dielectric Fabry-Perot’s.

The fact that the agreement is less good for dielectric Fabry-Perot’s than for prismcouplers is presumably due to the low finesse of such Fabry-Perot’s. In that case thetransmission can no longer be approximated by a lorentzian, eq. 20g does not hold and onehas to calculate K in eq. 20f.

2.6 Brief discussion. The comparison of eqs. 19 and 34 clearly shows the interestof leaky modes : a leaky mode replaces the “packet” of radiation modes involved in theresonant process i. e. for which :

The Cerenkov regime is interesting because it gives the opportunity to use differentmethods to derive the EM field : the total fied analysis, the transverse representation (orcoupled-mode formalism), the leaky mode representation. Figure 5 represents a typical

333

Cerenkov configuration : the fundamental frequency ω field is guided and the SH field isradiated.

Figure 5. A typical Cerenkov configuration

In this section we only consider the total fied method and the leaky mode representation.The transverse representation involves the radiation field and therefore leads to tediouscalculations.

1. The total field method

This method has been used in refs. [22-24]. The interest is in the EM field at the SHfrequency 2ω. The starting point is Maxwell equations at 2ω :

(36a)

(36b)

Solving Maxwell equations in media 1,2 and 3 yields :

(37a)

(37b)

(37c)

In eq. 37b the terms with amplitudes A2 and B 2 correspond to the free solution with

longitudinal wavevector component γ2ω and ℑ NL(y)e+i2γωx

describes the driven EM field

with longitudinal wavevector component 2γ ω . The expression of ℑ NL (y) is not of importancehere.

The boundary conditions at y=0,ei) require that the longitudinal wavevector components of the free and driven waves at 2 ω b eequal yielding :

334

γ 2ω = 2 γω (38a)

ii) lead to :

(38b)

Equation 38a is nothing but than the nonlinear Snell’s law[25]. In eq. 38b, is a

column matrix whose elements are the unknown amplitudes A1, A2, B2, A3 of the free SH

fields in media 1, 2, 3 and is the known column matrix associated to the driven

solution. The solution of eq. 38b shows that the amplitudes of the free waves are inversely

proportional to a quantity which is the determinant ∆ of matrix . The important point

is that ∆ has zeros in the complex γ .2ω -plane. These zeros of ∆ are poles of A1, A2, B2 and A3

In the vicinity of such a pole, assumed to be isolated, a very good approximation of 1/provided by the first term of the Laurent expansion[11]:

∆ is

(39a)

Since in Cerenkov SHG the SH field is radiated, the poles of 1/∆ are improper poles (see

section 1.5). Equation 39a shows that the peak value of |1/∆|2 is reached when :

(39b)

Phase matching is achieved when γω fulfills eq. 39b. It is worth noting that eq. 38a doesnot imply eq. 39b. In other words eq. 38a does not correspond to phase matching with aneigenmode at 2ω. The Cerenkov phase-matching condition is intimately linked with a

resonance effect associated to the complex zeros γl, 2ω

of ∆ : resonance occurs when eq.

39b holds. The curve |1/∆ |2 is a Lorentzian provided the improper poles i) are spread out in

the complex γ2ω -plane without clustering and ii) are not too close from cut-off (not too closemeans that the distance to cut-off is large as compared to the width at half-maximum of|1/∆|2).

In order to vizualise the Cerenkov phase matching process, it is convenient to plot

|1/∆ |2 in the complex neff2-plane This is done in figs. 6 with

the numerical values of ref. [20] :

e=4.90784 µm

n2(ω)= 1.6992

n1 (ω)=n3 (ω)=1.6875

n2(2ω) = 1.69772

n1(2ω)=n3(2 ω)= 1.71.

335

Figure 6. The Cerenkov phase-matching surface.

The peaks correspond to the improper poles of 1/∆. In an SH experiment thelongitudinal wavevector of the fundamental frequency guided mode is real. Thus only the realaxis (n"eff2 = 0) is accessible : when sweeping neff1 , phase matching takes place for the values

neff1,res of ne f f 1 fulfilling eq. 39b. Clearly the smaller n"e f f 2 at the peak position, the more

efficient the associated EM resonance. Figures 6, which display the complex poles of 1 /∆ ,represents the Cerenkov phase-matching surface : this surface clearly demonstrates theexistence of phase-matching in Cerenkov SHG.

2. Leaky mode analysis

Since the Cerenkov regime involves radiation fields, we know from section 1. 5 that theassociated poles at the SH frequency are improper poles describing leaky modes at 2 ω. ThusCerenkov SHG can be consideredi) either as a guided mode at ω - radiation fields at 2ω interactionorii) as a guided mode at ω - leaky mode at 2ω interaction.

In this section scheme ii) is considered[21].

336

The starting point is eq. 34b at frequencies ω and 2ω. The guided mode amplitude c1(x)

at ω and the leaky mode amplitude c2 (x) at 2ω obey the following set of equations (onlyforward modes are considered) :

with :

( 4 0 c )

(40d)

The " - " denotes the complex conjugate.Let us pursue in the framework of the UPA. Equation 40b yields :

(40a)

(40b)

( 4 1 a )

Hence from eq. 41a, the EM field Φ z (2ω, x, y) in the waveguide is given by (see eq. 17a) :

(41b)

Equation 41b shows that Cerenkov SHG involves two types of waves : a leaky wave,with complex longitudinal wavevector , which decays on a length characterized by the

out-coupling length c :

(41c)

and a wave with real longitudinal wavevector component 2γω. For x>> c, the SH intensity isconstant and its value at y=y

0is given by :

(42a)

Equation 42a describes the same resonant process than that corresponding to eq. 39a.The SH intensity is a Lorentzian centered at with width and peak value :

(42b)

337

The width of the Lorentzian may be interpreted as a measure of the amount of radiationmodes involved in the resonant excitation of the leaky mode , . The smaller the EM

leakage, the longer c and the larger the enhancement of the SH efficiency resulting from thisresonance process.According to eq. 42a the peak value of I2is obtained when :

Equation 43 is identical to eq. 39b and expresses the resonant excitation of a leaky modeat 2ω. Consequently this equation constitutes the phase matching relation for Cerenkov SHG.Clearly eq. 43 is not automatically satisfied: a phase matching condition exists with one ofthe leaky modes of the structure.

Figure 7 is a plot, with the numerical values of ref. [20], of I2 as a function of the

effective index neff1 at ω ( neff1 The intensity at 2ω has been derived in two ways:

from eq. 42a with ξ 2 calculated using the "closing" method (dashed line) and from the totalfield analysis (solid curve).

(43)

Figure 7. Plot of I 2 as a function of the effective index ne f f 1 at ω.

The resonance effect, and as a result the existence of a phase-matching condition,clearly appears. The agreement with the total field calculation can be considered as excellent.This shows the interest of the "closing" procedure: the knowledge of the SH intensity

requires the determination of and ξ2 . The values of these

parameters are obtained solving the homogeneous problem. With the numerical values usedin fig. 7 : n eff1 = 1.69666, neff2, = 1.69654+ i5.071610–4, ξ2 = 7 .4610–21 + i8.3810–20.

Leaky modes play an important role in Cerenkov SHG, It is seen that the situation isvery similar to SHG at prism or grating couplers which, as is known, behave as leakyresonators : in Cerenkov SHG the leaky waveguide is equivalent to a resonator with a highfinesse because the SH field is at grazing incidence. The description of guided wave SHG

338

corresponding to the guided mode at ω - guided mode at 2ω interaction is obtained fromCerenkov SHG letting = 0. Thus guided wave SHG and Cerenkov SHG are similar.The only difference is that for the former the waveguide is equivalent to a closed opticalresonator whereas for the latter the SH interaction takes place in an open, i. e. leaky,resonator. From this point of view there is no difference between Cerenkov SHG and SHGat distributed couplers and the phase-matching condition which exists for the latter is alsopresent for the former.

IV. CONCLUSION

The equations of evolution of a leaky mode amplitude (eqs. 34) apply to a wide class ofnonlinear optical effects at leaky resonators : χ2 , χ3 . . at prism or grating couplers, Fabry-Perot's, multilayer devices…. It is not the geometry and the specificities of the resonatorwhich are important but the set of parameters characterizing the EM resonance brought intoplay: γ l or α l , γ z ,Q , tQ. In this sense the theory presented here constitutes a unified approachof leaky resonators in nonlinear optics.

The development of the leaky mode formalism is an opportunity to present and discussseveral expansions of the EM field together with the link between them. The longitudinalrepresentation constitutes the starting point of two types of coupled-mode analysis : thetransverse expansion which corresponds to the "usual" coupled-mode theory[3,4] and theleaky mode representation whose main interest comes from the fact that a leaky mode replacesthe "packet" of resonantly excited radiation modes. The resulting simplification allows aneasy insight in the physics of nonlinear optics in "open" resonators. Thus leaky modesprovide a convenient framework and a powerful tool for the study of nonlinear opticalinteractions in such devices.

Acknowledgment

I wish to thank Dr. G. Vitrant, Prof. M. Nevière and Dr. E. Popov for helpfuldiscussions.

REFERENCES

1. See, for instance, G. I. Stegeman, Introduction to nonlinear guided wave optics in :Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, Eds., NATO ASI Series,Kluwer Publ., Dordrecht (1992).2. V. C. Y. So, R. Normandin, and G. I. Stegeman, Field analysis of harmonic generationin thin film integrated optics, J. Opt. Soc. Am. 69: 1166 1979).3. C. Vassalo, Théorie des Guides d'Ondes Electromagnétiques, Tomes 1 and 2, CNET-ENST, Eyrolles, Paris (1985).4. H. Kogelnik, Theory of dielectric waveguides in : Integrated Optics, ed. T. Tamir,Springer-Verlag, New-York (1975).5. M. Neviere, The homogeneous problem in : Electromagnetic Theory of Gratings, ed. R.Petit, Springer-Verlag, New-York (1980).6. M. Nevière, E. Popov, R. Reinisch, Electromagnetic resonances in linear and nonlinearoptics : phenomenological study of grating behavior through the poles and zeros of thescattering operator, J. Opt. Soc. Am. A12:513 (1995).7. T. Tamir, Leaky waves in planar optical waveguides, Nouv. Rev. d'Optique 6:273(1975).8. T. Tamir, A. A. Oliner, Guided complex waves, fields at an interface, Proc. IEE 110:310(1963).9. T. Tamir, L. B. Felsen, On lateral waves in slab configurations and their relation to otherwave types, IEEE Trans. Antennas and Propag. 13:410 (1965).10. V. V. Shevchenko, Continuous Transitions in Open Waveguides, The Golem Press,Boulder, Colorado (1971).11. G. Arfken, Mathematical Methods for Physicists, Third Edition, Academic Press (1985).

339

12. R. E. Collins, Field Theory of Guided Waves, IEEE Press (1991).13. J. D. Kraus, Antennas, Mc Graw-Hill, New-York (1950).14. G. Vitrant , M. Haelterman and R. Reinisch, Transverse effects in nonlinear planarresonators II. Modal analysis for normal and oblique incidence, J. Opt. Soc. Am B7: 1319(1990).15. R. Reinisch, G. Vitrant and M. Haelterman, Coupled-mode theory of diffraction-induced transverse effects in nonlinear optical resonators, Phys. Rev. B44:7870 (1991).16. R. Reinisch, M. Neviere, E. Popov, and H. Akhouayri, Coupled-mode formalism andlinear theory of diffraction for a simplified analysis of second harmonic generation at gratingcouplers, Opt. Comm. 112:339 (1994).17. R. Reinisch, G. Vitrant, J. L. Coutaz, P. Vincent, M. Neviere, and H. Akhouayri,Modal analysis of grating couplers for nonlinear waveguides, Nonlinear Optics 5: 111 (1993).18. M. Haelterman, G. Vitrant and R. Reinisch, Transverse effects in nonlinear planarresonators I. Modal theory, J. Opt. Soc. Am B7: 1309 (1990).19. R. Reinisch, M. Neviere, P. Vincent, G. Vitrant, Radiated diffracted orders in Kerr typegrating couplers, Opt. Comm. 91:51 (1992).20. G. J. M. Krijnen, W. Torruellas, G. I. Stegeman, H. J. W. Hoekstra, and P. V.Lambeck, Optimization of second harmonic generation and nonlinear phase-shifts in theCerenkov regime, IEEE Journ. of Quant. Elect. 32:729 (1996).21. R. Reinisch, G. Vitrant, Phase-matching in Cerenkov second harmonic generation : aleaky mode analysis, Optics Letters 22:760 (1997).22. M. J. Li, M. de Micheli, Q. He, and D. B. Ostrowsky, Cerenkov configuration secondharmonic generation in proton-exchanged lithium niobate guides, IEEE Journ. of Quant.Elect. 26: 1384 (1990).23. N. Hashizume, T. Kondo, T. Onda, N. Ogasawara, S. Umegaki, and R. Ito, Theoreticalanalysis of Cerenkov-type optical second harmonic generation in slab waveguides, IEEEJourn. of Quant. Elect. 28:1798 (1992).24. Y. Azumai, I. Seo, and H. Sato, Enhanced second harmonic generation with Cerenkovradiation scheme in organic film slab guide at IR lines, IEEE Journ. of Quant. Elect. 28:231(1992).25. N. Bloembergen, Nonlinear Optics, Benjamin Inc., New-York (1965), pp. 74-83.

340

QUADRATIC CASCADING: EFFECTS AND APPLICATIONS

Gaetano Assanto

Department of Electronic EngineeringTerza University of RomeVia della Vasca Navale, 84Rome, ITALY 00146

INTRODUCTION

In the search for a viable ultrafast, low power, lossless approach to all-opticalprocessing of signals, quadratic cascading, i.e. the aimed sequence of two second-order nonlinear processes, has emerged in the past few years as one of thesolutions better suited for such a broadband task. Despite its foundation lies inthe nonlinear optics (NLO) knowledge of the late 60's,1-6 quadratic cascading hasstimulated a vast scientific effort in fundamental NLO and its applications onlyafter extensive research in cubic materials with an intensity-dependent refractiveindex and the parallel development of material science in noncentrosymmetricmaterials for electrooptic and parametr ic effects . Fol lowing numeroustheoretical 1,3,5,6,9,10,13,15-17 and experimental 4,7,8,11,12,14 reports on phase-effectsthrough quadratic nonlinearities, the second-harmonic generation (SHG)experiment performed by DeSalvo et al. in a KTP crystal clearly demonstrated theimplications of a cascading-induced nonlinear phase shift for all-opticalprocessing. 18

Several steps forward have been accomplished since then, both theoreticallyand experimentally, exploring and/or discovering various areas in whichcascading does or could play a leading role, from guided-wave all-opticalswitching devices to all-optical transistors or isolators, from self-guiding beams inspace or time to gap solitons, from wavelength shifters to nonlinear competition.In this Chapter, for the sake of conciseness although at the expense of coverage,we will discuss illustrative features of cascading in bulk and in waveguideconfigurations, with specific reference to quasi-cw cases and excluding theimportant situations involving laser cavities and competing nonlinearities.

The Chapter is organized in Sections dealing with plane waves, guided wavesand self-confined beams, respectively: Section I introduces the model equationsfor cw propagation in both Type I and Type II SHG, discussing features ofcascading phase-shift and amplitude modulation of a fundamental frequencywave. Section II illustrates various all-optical transistors/modulators, whereas

Beam Shaping and Control with Nonlinear Optics341Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

Section III examines several nonlinear integrated devices for all-optical switching.Finally, Section IV is devoted to some applicative features of spatial solitary wavesthrough Type II SHG in bulk media.

I. MODEL EQUATIONS

Quadratic cascading is based on the consecutive application of two nonlinearoperators, namely second-order susceptibility tensors, to the electromagneticfield(s). Denoting by subscripts 1, 2 and 3 the three waves involved in theinteraction, the simplest case of cascading is described by an upconversion processwith ω 3= ω1+ ω2 followed by downconversion with o r ap r o c e s s c o n v e n i e n t l y d e s c r i b e d b y t h e p r o d u c t o f s u s c e p t i b i l i t i e s

. Its simplest implementation is in Type ISHG, with and the two fundamental frequency (FF) fields at ωoscillating with the same polarization. The other frequency degenerate caseinvolves optical rectification through , i.e. the electro-optic effect. 3,22 The latter relies on the generation of a DC field and will not bediscussed further.

Using the standard slowly-varying-envelope approximation, in the presenceof a quadratic nonlinearity and material dispersion such that a single three-waveinteraction is nearly phase matched (i.e. the three waves have nearly the samephase velocities) and is therefore dominant, the evolution of the plane-waveamplitudes Ej (z) of the copropagating electric fields along z is described by:

(1)

with the wavevector or phase mismatch. In writing eqns. (1) theinteract ion has been reduced to a scalar one through the project ions

and similar ones, with e 1 , e2 , e3

the unit polarization vectors for the three fields. Taking a further step in order tosimplify the notation, assuming all the wavelengths to be far from materialresonances such that the frequency dependence in d(2) can be neglected (i.e.invoking Kleinman symmetry), and scaling the amplitudes according to

such that (j=1,2,3) are the field intensities (W/m2 ), eqns. (1)reduce to:

(2)

342

with ζ =z/L and ∆ =∆kL the normalized propagation distance and mismatch,respectively, and the nonlinear coefficient.

It is instructive to analyze separately the cases of Type I and Type IIinteractions, the first referring to complete degeneracy between the fields withsubscripts 1 and 2, and the second to a "true" three-wave process.

Type I Second Harmonic Generation

When subscripts 1 and 2 effectively refer to identical parameters, i.e. the fieldsE 1 and E2 are indistinguishable, taking into account the degeneracy factorseqns. (2) reduce to two coupled nonlinear ordinary differential equationsdescribing the interaction between two waves of frequencies ω and 2ω:

(3)

with

For SHG in the low depletion limit, i.e. assuming a negligible secondharmonic (SH) and , the second of (3) can be integrated from 0to ζ and substituted in the first of (3), to yield:

(4)

with n2,eff (ζ) an equivalent z-dependent (focusing or defocusing) Kerr coefficient,and β eff (ζ) an equivalent two-photon absorption. The intensity dependence in theFF field evolution establishes a parallelism with the effects of a cubic nonlinearity,provided the z-variant feature is averaged out in the limit of large mismatches, i.e.when the low-depletion approximation holds. In this limit, the nonlinearabsorption is zero and the Kerr coefficient takes the value:

(5)

This equivalent Kerr response, obtained through a quadratic (SHG)interaction in the limit of low conversion efficiency, encompasses a lineardependence on the usual figure of merit for quadratic materials, namely (d(2))2/n3,and an inverse dependence on the phase mismatch. In addition, the factor L/λ canbe regarded as an enhancement term the role of which, in geometries involving atransverse resonance at the generated frequency, is played by the quality factor ofthe cavity.23 Moreover, the effective change in index or, more appropriately, thenonlinear phase shift changes sign according to ∆k. This implies that a focusing ordefocusing behaviour depends on the material arrangement with respect to theincident field to be doubled, for instance orientation, temperature, periodic poling,

343

applied voltage, etc. Some of these important characteristics were verified in bulkby DeSalvo et al. in a 1mm KTP crystal at 1.064µm via Z-scan,18 and later on byDanielius et al. in BBO via self-diffraction, 24 by Ou in potassium niobate inside acavity,25 by Fazio et al. in KDP, 26 by Wang et al.2 7 in the organic NPP and byVidakovic et al. in periodically poled lithium niobate (PPLN).28

The shown similarity between a phase mismatched SHG process and a cubicfrequency degenerate interaction is, however, of limited use. Although it allows arough estimate of the potential of a cascaded process in mimicking the Kerr effectin quadratic media,29 it disregards some important features of cascading which, ina measurable sense, does not induce any actual change in refractive index. It isworthwhile, therefore, to examine the induced phase shift directly, dropping thelow-depletion approximation and resorting to a simple integration of eqns. (3).Such integration can be performed exactly30-31 or numerically 19,21 , with the aim ofinvestigating the evolution of amplitude (or intensity) and phase shift of the FFfield component. Figs.1a-b show the evolution of phase shift and normalizedintensity versus propagation distance for various mismatches and a fixedexcitation Γ |Φ 1(0)|= 20. The FF intensity undergoes oscillations, recovering itsinitial amplitude with periodicity decreasing for increasing mismatch, as expectedbased on coherence length considerations. Similarly, the degree of conversion tosecond harmonic becomes higher when approaching the optimum condition, i.e.phase matching. It is rather more interesting the evolution of the FF phase,graphed in Fig.1a. The phase grows stepwise, with jumps corresponding toenergy flowing back to the FF through downconversion, and plateaus separatedby π /2 in the case of complete FF depletion. It is apparent that the linear

Fig. 1 a) FF nonlinear phase shift ϕNL in units of π and b) FF throughput T vs. propagation forvarious mismatches (solid lines). The dashed lines refer to an input SH seed of 0.001 in intensityand relative phase of π (discussed later in the text). Here

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behaviour discussed above in the framework of an undepleted FF is only adequateto describe the evolution of the phase in its initial stages or for high mismatches,when the corresponding FF amplitude remains nearly constant. In another limit,for intense excitations and/or large phase shifts, the dependence becomes linear ininput amplitude, rather than in intensity. 19 An intuitive explanation of such acomplex behaviour can be provided in terms of photons (or energy) which,partially converted from FF to SH, propagate at different velocities and acquire aphase difference before eventually recombining at the FF after a coherencelength. 21,32 A phasorial representation of the field amplitudes in snapshots duringpropagation, as shown in Fig. 2, helps the intuitive understanding. The linearphase rotation caused by a nonzero mismatch on the SH phasor allows thedownconverted photons to tilt the progressively more depleted FF phasor, until a≈π/2 phase change is completed after one cycle.

Fig. 2 Phasorial representation of the cascaded FF phase rotation during the first cycle (seeFig. 1). At various propagation distances, the FF and SH components are graphed, the additionalvectors being the previous- and present-step contributions to the FF field in each picture. Here ∆=πand

The evolution of the FF throughput versus input excitation is displayed (inphase and transmission) in Fig. 3, for a fixed mismatch. For increasing input, thephase plateaus become progressively extended, while the recursive FF depletionbecomes more and more pronounced, according to a nonlinear improvement ofwavevector matching. The latter effect can be better appreciated when plottingthe FF throughput versus ∆, as in Fig. 4. A stronger excitation tends to depletemore the FF, narrowing the center lobe and making the sidelobes grow deeper.The occurrence of flat phase-plateaus when the energy flows back into the FFallows the realization of phase and polarization interferometers, with reducedpower requirements when a folded geometry or two crystals are employed. 33-34

345

Fig. 3 FF throughput (left axis) and phase (right axis) vs. excitation

for ∆ =π, and no SH seed (solid lines), SH seed 1% in intensity of the FF

input and with relative phase of π (dashed lines).

Fig. 4 FF throughput vs. mismatch for excitation for low excitation

Γ |Φ1(0)|=1 and high excitation Γ |Φ1(0)|=4 without (solid line) and with

(dashed line) an SH seed of intensity 1% than the FF and phase π.

The features illustrated so far refer to SHG, i.e. to a process with a single FFinput. The interaction is, however, inherently coherent, because it does notdepend exclusively on the field moduli. 21 This translates in the possibility ofseeding the process with another input at the second harmonic, controlling theoverall evolution by acting on its amplitude or on the relative phase between Φ 1(0)and Φ 3(0). In order to illustrate this concept, various cases in the presence of aweak SH seed have been added to Figs. 1,3 and 4. Since the phase plays a majorrole even though the added input might be rather small, this opens up the way tovarious possibi l i t ies for phase- to-ampli tude t ransducers or a l l -opt icalmodulators/amplifiers, as discussed in Section II. Several schemes, based onreflecting configurations where the coherently generated second harmonic and theleftover fundamental propagate back into the crystal, can also be envisaged.35

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Type II Second Harmonic Generation and Non-Degenerate Frequency Mixing

When more (than one) inputs are required to generate the new frequency,extra degrees of freedom are available and can be exploited for cascading. This isthe case of Type II phase-matching in SHG and, in general, of all three-wavemixing processes with distinguishable field components.20 Eqns. (2) can bemanipulated to give:

(6)

for Φ 1 , and similar for the others. Eqn. 6 shows that the evolution of each wavedepends on the intensity of the others. Proceeding in analogy to the previoussubsection (Type I SHG), let us assume that the third wave is entirely generatedthrough the nonlinear interaction, i.e. Φ3(0)=0. If we let the two inputs to beunbalanced, with in the low conversion approximation for Φ3eqn. (6) reduces to:

(7)

which shows how the more intense wave induces a phase shift on the weakercomponent, even for a vanishing phase mismatch (i.e. |∆|→0). This observation,implicit in the non-degenerate analysis,20 was outlined in the context of SHG byBelostotsky et al.36 Experimental verification of cascaded phase-shifts throughfrequency non-degenerate interactions have been reported by Nitti et al. in theorganic crystal MBA-NP 37 and by Asobe et al. in PPLN.38 Related effects havebeen discussed in [39]. For simplicity and without lack of generality, we will

Fig. 5 FF phase shift vs. mismatch for Type I SHG with Γ2 |Φ1 (0)|2 =150(solid line), and for Type II SHG (long and short dashes) withΓ

2[|Φ1(0)|2 + |Φ (0)|2 ]=150 and |Φ2 1 (0)|

2 / |Φ2 (0)| 2 = 0.5. The short dashes

represent the phase of Φ 2 .

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henceforth refer to SHG with orthogonal FF inputs, as this is the case of a fewapplications discussed later. Fig. 5 is a plot of nonlinear phase shift in the twounbalanced FF components versus mismatch, calculated from numericalintegration of (2). A nonlinear phase-shift is induced on the weaker componenteven in perfect matching, whereas the phase of the stronger wave goes throughzero in analogy to the Type I SHG case. As visible in Fig. 6, however, for a givenintensity ratio ≠ 1, in contrast to Type I SHG the maximum phase jump is π, with astaircase evolution sharper for better wavevector matching and/or higher inputimbalance (not shown) . 40 The corresponding FF throughput versus imbalance isgraphed in Fig. 7, where a sharp transition can be observed as an input ratio ≠1 isintroduced. This abrupt change in transmission, further discussed in theframework of all-optical transistors, can be related to a dramatic shortening of thecharacteristic length for energy exchange between the three waves near phasematching, 41-42 leading to FF modulation at the end of a finite length medium. Forperfectly balanced inputs at FF, conversely, cascading via Type II and via Type ISHG present identical phenomenologies.

Fig. 6 Phase shift of Φ 1 v s . exc i t a t i on fo r Type I I SHG wi th|Φ 1 (0)|2/ | Φ2(0)| 2 = 0.1 and various mismatches.

Fig. 7 Throughput T2 of the strong polarization component Φ 2 v s .ratio |Φ1(0)|

2/ |Φ2(0)| 2 for various mismatches.

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In the next Sections, the phase shift and related amplitude/throughputmodulation obtainable through cascading will be discussed in the framework of afew specific but paradigmatic applications. Phase conjugation, 43 four-wavemixing,44 laser mode-locking and pulse conditioning29 v i a c a s c a d i n g a r eimportant effects and applications among those left out of this review. Thefollowing section is devoted to all-optical transistors, and the next deals withguided-wave geometries involving cascading.

II. ALL-OPTICAL MODULATORS VIA SHG CASCADING

Fig. 8 shows a few basic schemes of all-optical transistors based on quadraticcascading via second-harmonic generation. Many of them have already beendemonstrated experimentally in bulk crystals of KTP, and we will try to outlinemajor advantages and drawbacks of each.

Type I SHG transistor with SH input signal

This implementation (Fig.8a) of an all-optical transistor is based on theintrinsic coherence of the quadratic SHG interaction, such that a weak input at theSH can alter the evolution of the two waves (FF and SH) along propagation,resulting in phase and/or amplitude modulation of the FF component.21,45-46 Fora fixed input FF intensity, a variation in the relative phase between the SH seedand the FF pump induces a shift in the overall phase matching response, as seen inFig. 4. If the operating ∆ -point is chosen in order to maximize this phase-to-throughput modulation (i.e. along the slope of the main lobe), a large change intransmission can result from a small variation in input phase. Conversely,keeping the phase constant, the presence of an SH input determines the responseof the device, i.e. results in an intensity controlled transistor.

Fig. 8 All-optical transistors based on SHG cascading: a) Type I SHG with SH signal; b) Type ISHG with SH pump; c) double Type I SHG with orthogonal FF inputs coupled to one SH wave; d)Type II SHG with orthogonal FF inputs; e) Type II SHG with SH pump; f) double Type II SHGwith coupler and amplifier.

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Fig. 9 shows the results of the experiment performed by Hagan et al. in KTP,together with the numerical simulation.46 The FF throughput, despite the use oftemporal pulses with a Gaussian beam distribution, clearly exhibits a modulationversus input relative phase, with a contrast as high as 4.6:1 when employing SHpulses with energy 1.2% of the pump at 1.064µm. Notice that, although KTP isType II phase-matched for SHG at this wavelength, the FF input was injected at45° with respect to ordinary and extraordinary crystal directions, thereby resultingin standard Type I cascading.

The major drawbacks of this scheme are the use of a signal of frequency twicethe output, and the need for an interferometrically stable setup in order toproperly control the phase. The first problem can be solved by working with anSH pump (Fig. 8b), i.e. in downconversion with input and output signals at the FF.The use of a short-wavelength pump, however, may be undesirable. The secondproblem can be solved by employing a Type II configuration, as discussed later.

Fig. 9 FF transmission vs. relative phase between SH seed and FFpump. The peak FF intensity is 20GW/cm 2 , and the seed energy fractionis 1.2%. (After Ref. 46)

Double Type I SHG transistor

Crystal symmetries can, sometimes, a l low more than one quadrat icinteraction to be nearly phase matched. Considering the additional possibilitiesoffered by Quasi-Phase-Matching in ferroelectric crystals, this opportunity can beexploited in coupling two orthogonal FF waves (Φ1 and Φ2 ) through the nonlinearsusceptibility by means of two distinct Type I SHG interactions (labelled "a" and"b"), i.e. via a single SH wave Φ3.47 The resulting equations:

( 8 )

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encompass cross-phase modulation between Φ1 and Φ2 , and the possibility ofnonlinear polarization rotation, FF modulation and small-signal amplificationwhen varying either FF input with respect to the other, either in phase or inamplitude. This scheme (Fig.8c), potentially feasible in a temperature tunedperiodically poled lithium niobate crystal, 47 has not yet been demonstrated.

Type II SHG transistor

As outlined above, the use of a Type II phase-matching geometry allows tointroduce more degrees of freedom in SHG cascading, with the extra advantage ofphase insensitivity when inputting the FF only. This latter feature, on the basis ofthe input-output response shown in Fig. 7, stimulated the study of a frequencydegenerate, phase insensitive, all-optical transistor (Fig.8d).4 0 - 4 1 , 4 8 - 5 0 Thetransmission at FF at the end of the crystal, in either or both polarizations, willabruptly change when a small imbalance is introduced between the input FFcomponents, provided the interaction is nearly phase-matched. This process doesnot depend on the relative phase between Φ1 and Φ 2 , and can be exploited aroundan appropriate bias point in order to obtain small-signal amplification, as sketchedin Fig. 10. In addition, through the phase shift experienced by the weakest input, arotation of the polarization state by nearly π /2 at the output can be exploited forall-optical switching through an output analyzer. 50-51 Analytical expressions forthe obtainable gain can be derived under plane wave and cwapproximat ions, 31,34,41-42 although the use of gaussian beams and pulses inexperiments is bound to degrade the performance to some extent .4 8 - 5 0

Experiments performed at 1.064µm have demonstrated amplifications as high as21.6dB in KTP crystals, 48 and a typical set of data together with a numericalsimulation (from eqn. (2) including beam and pulse shapes) is shown in Fig. 11.49

This transistor scheme, although phase insensitive and with inputs andoutput at the same frequency, relies on an input signal which, for best operation,

Fig. 10 FF transmission of a Type II SHG transistor vs. inputimbalance. The two marked regions (dashes) identify the pre-amlifier orcoup le r ( |Φ1(0) |

2<< |Φ2(0)|2) and the amplifier (|Φ1(0)|2 ≈ |Φ2 (0)|2 ) ,

respectively. Notice that ∆Tc <<∆Ta for comparable changes in imbalance.Itot is the total FF input intensity.

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should have amplitude comparable with the other FF component.40-41 This limitsits usefulness in any realistic application, because a weak signal to be amplifiedshould be coherently superimposed to a large background in order to satisfy thesmall-imbalance requirement. A Type II downconversion (half-harmonicgeneration) process (Fig. 8e) is in principle able to solve this problem at the

Fig. 11 Experimental FF transmission for the extraordinary (squares)and ordinary (triangles) components and the total (filled circles) vs inputimbalance through a 2mm-long KTP crystal. The solid line is calculatedusing eqns. (2) including beam and pulse profiles. Here Itot=12GW/cm2

(peak value). (After Ref. 49)

expense of a strong pump at the second harmonic, because in this case a smallinput at FF in either polarization would trigger the parametric downconversionfrom SH and result in large gain at FF. An alternative way to employ cascadingfor the amplification of a small signal incoherent with the pump is described in thefollowing paragraph. Finally, the injection of a weak coherent seed can, even forthe Type II transistor, widen the range of possibilities available to control theinteraction. 31

Double Type II SHG transistor

This scheme (Figs. 8f and 12) employs two nonlinear stages pumped by thesame strong FF wave: the first couples a weak FF signal to a large FF pump in theorthogonal polarization through Type II SHG, while the second operates withinputs of comparable intensities as described in the previous paragraph.52-53

Effectively, the coupler or preamplifier works in the initial (large imbalance)region of the characteristic of Fig. 10, in order to generate the coherentsuperposition required by the amplifier. The latter, in fact, is best suited to workin the central region (small imbalance) of Fig. 10. Although the experimentalimplementation is rather complicated due to the use of pulsed beams and thepolarization constraints, preliminary results utilizing KTP crystals havedemonstrated its feasibility with amplifications of the order of 7dB at 1.064µm.54

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F i g . 1 2 Schemat ic of a two-s tage Type I I SHG t rans is tor .PBS=polarizing beam-splitter, Coupler and Amplifier are Type II crystalsfor SHG. (After Ref. 52)

III. QUADRATIC CASCADING IN GUIDED-WAVE GEOMETRIES

Dielectric waveguides for transverse confinement of the electromagneticfields over propagation lengths not limited by diffraction are a convenient settingfor the evaluation of nonlinear effects and the implementation of nonlinearintegrated optics devices. 55 In the framework of SHG cascading, experimentalmeasurements have been reported in KTP,56-57 in lithium niobate 58-63 and i nDAN [4-(N,N-dimethylamino)-3-acetaminonitrobenzene], 6 4 and a number ofguided-wave geometries and approximations have been investigated theoreticallyor numerically. Here, with specific reference to the simple case of cw excitation inlossless structures, we will briefly review the basic concepts involved inwaveguide nonlinear optics through quadratic cascading and describe several all-optical devices demonstrated and/or proposed to date.

The equations describing quadratic cascading in waveguides are substantiallysimilar to those relative to plane waves, provided the overlap of the modal fieldsis properly taken into account and the mismatch calculated with the guided-wavewavevectors. Introducing pairs of (single) superscripts "l,k" identifying thechannel (planar) waveguide eigenvalues and eigenmodes, normalizing the modalfield transverse distributions (taken real) such that areguided powers (in units of W for channels, in W/m for planar guides), for Type ISHG eqns. (3) hold with

(9)

(10)

β 3(p,q) and β(m,n)

1 being the wavevectors of the eigenmodes at 2ω and ω ,respectively. A straightforward extension is valid for Type II SHG. The overlap

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integral in (9) alters the bulk nonlinearity, and proper parity of the involvedmodes has to be selected in order to maximize the conversion efficiency.

Single channels

A small phase-mismatch is required for efficient cascading to take place, atleast in type I SHG. In waveguides this requirement translates in properorientation of the medium, polarization of the excited eigenmodes, temperature orelectro-optical adjustements of the effective indices (i.e. wavevectors), variationsof the nonlinearity along the propagation direction (QPM), etc.65-69 In manyapplications a single-mode waveguide at the FF is desirable, together with alimited number of modes at the higher frequency. When more modes are presentat the SH, competition between them can take place provided they are nearlyphase-matched to the same FF wave, as demonstrated by Treviño-Palacios et al. ina lithium niobate QPM waveguide.6 2

More complicated geometries, with Cherenkov emission at the SH, have alsobeen investigated both experimentally (in DAN) and theoretically, showing thatlarge nonlinear phase shifts of the FF can be achieved in crystal cored fibers withthe SH field trapped in the fiber cladding. 64,70-71 Lee et al. have investigated self-phase modulation in SH surface emitting geometries with counterpropagating FFwaves, 72 and Ueno et al. in semiconductor waveguides subject to large walk-offand losses. 73

Modulators. Single channels can be operated as amplitude or phasemodulators controlled by the intensity of the incoming FF wave(s). This is indeedthe signature of quadratic cascading, as it has been verified experimentally bySundheimer et al. in KTP channels,56-57 and by Schiek et al. in lithium niobatechannels. 58-59 The experiments performed in KTP were carried out by measuringthe spectral broadening of laser pulses through self-phase modulation due tocascading, whereas the lithium niobate waveguides were evaluated with aninterferometric setup, and phase matched through temperature tuning with anonuniform longi tudinal profi le (due to the oven geometry) . The la t terconfiguration, with lower temperature at the crystal ends, allowed efficient SHGconversion internally to the sample, with a resulting conspicuous phase shift (ashigh as 1.5π ) but low depletion of the FF wave at the output of the channels. Thisis a desirable situation for all-optical devices based on frequency conversion.Fig.13 shows the FF transmission and the corresponding nonlinear phase shiftversus temperature. Notice that the phase matching temperature TPM for Type ISHG between the TM00 (ω) and TE00(2ω) modes was 336.6°C. Temperatures higher(lower) than TPM induce a negative (positive) phase contribution, resulting in anoverall large positive phase shift for T<TPM.58

The introduction of a nonuniform distribution of wavevector mismatch ornonlinearity along the propagation direction gives origin, through Maxwell’sequations and coupled mode theory, to a set of equations of the form:

(11)

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Fig. 13 Measured and calculated a) FF transmission and b) nonlinear phase shift vs. temperaturein a 15µm-wide lithium niobate channel. Peak power is 60W and wavelength 1.319µm. (AfterRef. 58)

with Γ(ζ) the longitudinally varying nonlinearity, andthe propagation-dependent mismatch. The ζ-variant

linear contributions are linked to variations in the linear parameters, with

(12)

and similarly for δβ3 ( ζ ). The ζ-variant nonlinearity includes quasi-phase-matching, and can also be taylored to specific bandwidth-enhanced high-efficiency applications. 7 4 - 7 5 Defining the new amplitudes and

and the phase

eqns. (11) can be recast in the form:

(13)

(14)

Solving the second equation in (14) in the undepleted FF approximation, theamplitude a3 (1) results proportional to the Fourier transform of a function whichincorporates the perturbation, i.e.:

(15)

with

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(16)

This approach, although rigorously valid only for low conversions, allows toinvestigate and design tentative profiles for various operations. A linear variationwas induced through the oven temperature profile in Ref. [58-59], and a nonlinearone through ion-exchange in Ref. [56-57]. Other interesting possibilitiesencompassed by (11) or (14) will be illustrated below. Finally, it is worthmentioning the possibilities offered by Type II and SH-seeded guided-waveinteractions, in analogy to the plane-wave cases discussed above.21,40-42,45-46

Wavelength Shifter. If one of the two processes involved in cascading ismade non-degenerate in frequency, a small frequency shift can be obtained at theexpense of a FF input pump through the cascaded interaction:

(17)

as demonstrated in bulk BBO by Tan et al ., 76 and proposed in a MQW waveguidewith a vertical SH resonator by Gorbounova et al.77 and in a QPM lithium niobatechannel by Gallo et al. 78 This configuration for wavelength shifting offers a large3dB bandwith and amplification at the expense of the input pump, which can bewithin the useful spectral range of the Er-doped fiber amplifiers used intelecommunications. Fig. 14 shows the calculated response in the latter case for aQPM channel in lithium niobate, with an additional reflector at SH in order tomaximize the useful interaction length.78 Preliminary results in a 9.5mm-longwaveguide (without reflector) are in qualitative agreement with the simulations.79

Fig. 14 a) Sketch and b) calculated conversion efficiency (ω –δ → ω+δ) vs. signal wavelength λ2 f o ra λ -shifter in a QPM lithium niobate channel of length L and with a 2 ω-reflector to maximize thedifference-frequency-generation.

All-optical diode. If a ζ non-uniform perturbat ion, e i ther l inear ornonlinear, is also non-symmetric with respect to the terminations of a channel, thequadratic interaction will induce a nonreciprocal response of the device uponexcitation from different sides. This corresponds to a purely dielectric isolator orall-optical diode, which is - in principle - able to transmit the FF component only

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for increasing (or decreasing) ζ. A simple implementation of this concept has beenreported by Treviño-Palacios et al. in a QPM channel partially overcoated with athin photoresist film in order to introduce a change in wavevector mismatch. 6 3

The results - reproduced in Fig. 15 - demonstrate the expected nonreciprocity. Forthe case of nonlinear tayloring through the QPM periodicity, Fig. 16 shows thecalculated response of a diode with a flat + linearly chirped mismatch profile(Fig.16a): the isolation effect is apparent at various input power levels, with a largeextinction ratio when propagating in opposite directions (Fig.16b)..L

Fig. 15 SHG wavelength scan of a 7 µm-wide 1cm-long QPM lithiumn i o b a t e c h a n n e l w i t h o u t ( d a s h e d l i n e ) a n d w i t h ( s o l i d l i n e s ) asuperimposed thin film to realize a non-uniform mismatch distribution.Open and filled circles refer to forward and backward propagation,respectively. (After Ref. 63)

Fig. 16 a) Chirped mismatch profile vs. length and b) FF transmittance vs. excitation for an all-optical diode with a flat+linear mismatch induced by varying the QPM periodicity. Forward(solid line) and backward (dashed line) FF transmissions show a good degree of isolation at givenpowers.

Integrated Mach-Zehnder Interferometer

The integrated interferometer is, among various devices, the most obviousimplementation of an all-optical switch based on a nonlinearly-induced phase-

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shift.5 5 When employing cascading, a simple configuration with geometricallyidentical waveguides (Fig. 17a) characterized by opposite signs of phase-mismatchallows to minimize the power requirements, because the nonlinear phase shifts inthe two arms will effectively combine to produce throughput switching.8 0 - 8 1

Moreover, the phase plateaus (versus excitation, see Fig. 3) allow this device towork with good switching contrast even in the presence of pulses with acontinuous distribution of instantaneous power. An experimental demonstrationhas been reported by Baek et al. in a lithium niobate channel with a non-uniformtemperature profile for phase matching,60 and a typical set of experimental datawith a numerical simulation is reproduced in Fig. 17b. Excellent performance hasalso been predicted for the Mach-Zehnder device with soliton-like pulses injectedat FF.82

Fig. 17 Guided-wave Mach-Zehnder interferometer. a) sketch of the device; b) measured (solidline) and calculated FF transmittance vs. input power at 1.319µm. (After Ref. 60)

Directional Coupler

The directional coupler, widely studied in the context of Kerr nonlinearities, 55

has also been investigated in the case of cascading nonlinearities. 80,83-88 Thepertinent equations describing a coupler with negligible overlap between thefields Φ 3 and Ψ 3 at the SH, and linear coupling strength κ between the fields Φ 1

and Ψ 1 at FF, are:

(18)

with the linear detuning.Despite the fact that, due to the periodic character of the FF amplitude in

propagation (Fig.1b), a "clean" switching in a uniform coupler with identical arms(ΓΦ = Γ Ψ , δ =0) is predicted only in the "cross" channel (Fig.18a), experimental

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results for a non-uniform phase-matching profile in lithium niobate channels haveshown the feasibility of such a switch,55 as apparent in Fig. 18b. Conversely, anonlinearly asymmetric but otherwise uniform coupler, with one arm substantiallylinear, is expected to exhibit good switching performances.8 5 Directional couplersbased on Type II SHG can be operated as logic AND/NAND gates or spatialdemultiplexers. Notice that eqns. (18) can also describe the Mach Zehnderinterferometer by setting κ to zero, or a counterpropagating geometry withdistributed feedback coupling by changing the sign of ζ in the bottom equations.Lee et al. have investigated a geometry involving surface emitted second-harmonic, 8 7 whereas soliton-based directional couplers have been analyzed bySchiek, 83 Karpierz,

86and De Angelis et al.88

Fig. 18 Half-beat length symmetric directional coupler. a) Calculated FF transmission through bar("=") and cross ("X") arms vs. excitation (∆ =8π); b) measured and calculated FF transmission in anon-uniformly temperature-tuned (Τ=343.5° C) lithium niobate device operated at 1.319µm. (Ref. 61)

Distributed Couplers

Prisms or gratings for input coupling a radiation field into a planar waveguideneed to be properly tuned to a geometric resonance in order to satisfy momentumconservation. In the presence of a quadratic process such as SHG in thewaveguide, in-coupled power at FF can be converted to SH and cause a cascadingphase shift, thereby altering this linear tuning and affecting the overall couplingefficiency through interference effects in addition to local energy conversion. Thisphenomenon, in the form of a travelling wave interaction, for a prism coupler(Fig. 19a) exciting the FF guided mode Φ 1 of eigenvalue can be described by:

(19)

with t the coupling strength, δ = (ki n sinθ in −

β1( m )

)L the linear detuning and 1ω , 1 2ωreirradiation lengths into the prism. Here Φ in f in ( ζ) represents the input field withits distribution, typically gaussian with size dz along z. A similar set of equationscan be written for a grating coupler. Numerical solutions of (19), with optimizedlinear coupling conditions at FF (L = 0.87 d z, lω = 0.74 dz) have shown that a) a

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β1( m )

detuning δ translates into a nonlinear phase shift of the FF mode throughcascading, even under perfect SHG matching (∆=0); b) an SHG mismatch canpartially compensate the input detuning, maximizing the FF coupling efficiency ata given power level; c) a power/detuning scan of coupled FF power can provideinformation about nonlinearity and phase mismatch of the nonlinear structure.Fig. 19b shows the calculated response of a prism-waveguide system with a zero(dotted line, FF) or a small SHG phase mismatch (solid line FF, dashed line SH)and the other parameters optimized for best input coupling at low powers:compared to the perfectly matched case, cascading induces a secondary maximumin efficiency at high powers.8 9

Fig. 19 Distributed prism coupler: a) sketch of the geometry; b) calculated coupling efficiency atFF (solid and dotted lines) and transfer efficiency to SH (dashed line). The dotted line refers to acoupler in linear and SHG matching (δ=∆ =0), whereas the other lines are calculated for δ=0, ∆ = 0 . 9 .

Based on a coupled-mode formalism and linear theory of diffraction,90

cascading through second-harmonic generation in distributed couplers has alsobeen shown to give rise to optical bistability.91

Mode Mixer

Mode mixing devices allow the spatial routing of information in bimodalwaveguides where interference between two FF modes ( Φ1 a n d Ψ1) takes placedepending on input power or on the presence of a coherent seed (see sketch in Fig.20a). The existence of two modes at FF implies, however, several modes at SH,with various Type I and Type II interactions potentially contributing to thecascading phase shift. To this extent, a channel waveguide in z-cut lithiumniobate has been numerically investigated by De Rossi et al. for operation atλ =1.55µ m, and the results show good switching contrast with low FF depletion.9 2

A typical set of calculated results is shown in Fig. 20b. Preliminary experimentalresults at 1.319µ m in a temperature-tuned lithium niobate channel confirm thetheoretical expectations.

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Fig. 20 Mode mixing device. a) Sketch of the waveguide with the two FF modes involved; b)calculated FF throughputs in the two channel halves (Right: dashed line, Left: solid line) vs. FFinput excitation (equally distributed between the two modes). The dotted line is the total FFtransmission.

Distributed Feedback Grating and Gap Solitons

A distributed feedback grating (Fig. 21a) of period Λ is able to couple forwardand backward waves at a given wavelength λ≅2 Λ in the medium, i.e. it opens as top-gap in the dispers ion diagram of the wave. When employed in aquadratically nonlinear structure for SHG, the grating can either induce a Braggreflection at one or both frequencies, depending on dispersion, polarization,index/depth profi le . Due to the coupling between counterpropagat ingcomponents and the energy exchange between fields at different frequencies, adeformation of the stop-gap(s) is expected with input excitation, potentiallyleading to induced transparency and optical bistability. While in the Kerr limit( | ∆| → ∞) this is reminiscent of Kerr-induced energy localization in Braggcoupling structures,9 3 the situation becomes more complicated when FF depletionand stop-gaps at both frequencies are involved.

In the limit of a single stop-gap at FF [see eqns. (18) with Ψ and Ψ back-1 3propagating waves along –ζ ], Picciau et al. have identified a range of parametersfor which localization actually takes place and is able to induce optical bistabilitywith or without the injection of a weak coherent SH seed.9 4 - 9 5 Fig. 21b showsb i s t a b l e l o o p s f o r a n u n s e e d e d c a s e w i t h a z e r o B r a g g d e t u n i n g ,

and various SHG mismatches, whereasFig. 21c displays a loop obtained with an in-phase coherent seed input at SH withδ=∆=0 (linear and SHG matching).

When two stop-bands are considered (i.e. also Φ a n d Ψ3 are Bragg-3coupled), it is necessary to define the detunings δ 1 a n d δ 3 with respect to thebo t tom o f t he s t op -bands a t FF and SH, r e spec t i ve ly , t he mi sma tch

and the curvatures ω"1 and ω"3 (in wavevector space) of the closestBloch eigenvalues to FF and to SH. In this case the equations describing theFF (u ) and SH (u ) envelopes in the structure can be cast in the form:1 3

361

(20)

with ρ1 a n d ρ 3 overlap integrals between the pertinent Bloch eigenfunctions.96

Equations (20) are formally equivalent to those describing dispersive SHG afterinterchanging t and z and, therefore, can exhibit solitary-wave solutions, i.e. theylead to the existence of gap solitons in quadratic media.9 7 - 9 8 Such form of z-localization, or mutual trapping of the FF and SH fields, is obtained through thecounterbalance of grating dispersion and parametric mixing, provided the parityof the Bloch solutions is such that ρ1 and ρ 3 do not vanish, i.e. the nonlinearity iseffective. This condition corresponds to require that the SH is not close to theupper edge of its stop-band. Stable and stationary bright-bright and twin-holedark solitons have been identified for the two envelopes, with several non-stationary and unstable solutions as well.97 The non-stationary solutions are ableto travel at reduced speeds inside the structure, making ideal candidates forpower-dependent delay lines or optical buffers. Since the dispersion normallyavailable in uniform media is not large enough to permit the observability oftemporal solitons of the quadratic nonlinearity (simultons), propagation in periodicstructures, including the cases of out-gap frequencies,9 9 - 1 0 0 might actually openthe way to their use in ultrafast all-optical devices for processing and switching.In this framework, for a singly FF resonant grating, Fig. 22 describes the excitationof a two-color solitary wave: an incident sech-shaped FF pulse reaches the

Fig. 21 Distributed feedback grating. a) Sketch of the doubly-resonant Bragg reflector operating atboth FF and SH; b) calculated FF transmission vs. excitation for various SHG mismatches and exactBragg resonance at FF only; c) FF transmission vs. in-phase SH seed in a coherently controlleddevice with no Bragg resonance at the second harmonic.

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nonlinear periodic medium (z>0), where the required SH is generated and lockswith the FF to originate a slowly moving simulton (propagating at 30% of thenatural FF group velocity). In Fig. 23 the FF reflectivity is calculated versus inputpeak intensity for two sets of detunings, only one of which allows the formation ofa travelling simulton. In the latter case the reflectivity drops below 50% at highexcitations.1 0 0

Fig. 22 Snapshot of the z-distributions of the reflected (z<0) andtransmitted (z>0) intensities at FF (thick solid lines) and SH (thin solidlines) after the nonlinear interaction has taken place. The dashed linerepresents the incident pulse, 100ps in durat ion. Intensi t ies aren o r m a l i z e d s u c h t h a t a p e a k v a l u e p = 1 0 c o r r e s p o n d s t o a b o u t500MW/cm² in KNbO 3, whereas distances are expressed in gratingcoupling lengths.

Fig. 23 FF reflectivity vs. incident peak intensity in z=0, for a largepositive mismatch (∆ =22 π) and for FF close to either the upper (dashedline) or the lower band edge (solid line). In the latter case, a simulton isgenerated at the interface and the reflectivity drops.

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IV. APPLICATIONS OF VECTORIAL SPATIAL SOLITARY WAVES VIA SHG

Self-guided beams or spatial solitary waves (SSWs) in quadratic media are, inthe framework of cascading nonlinearities, a major field of investigation.

2 9 F o r

this reason, such mixed states of mutually trapped interacting fields areextensively discussed elsewhere in this Book. After their first obser vationin acrystal for Type II SHG,101 however, they have gained considerable importancedue to their potential applications. In this framework, and as a natural extensionof some concepts introduced in Sections I and II, in this Section we will discuss afew applications of a specific class of SSWs emerging from Type II SHG. TheseSSWs contain an SH and two orthogonally-polarized FF components and, for thelatter reason, are often referred to as vectorial SSWs.

Self-guided beams in space originate in quadratic media through the balanceof natural spreading and parametric gain in the presence of a nonlinear phasefrontdistortion due to cascading, provided the nonlinear length is shorter then theinvolved diffraction and walk-off lengths. 101-102 Their evolution, assumingstandard slowly-varying envelope and paraxial approximations, is described bycoupled equations which, for an "eoe" interaction and in the presence ofbirefringent walk-off, take the form:

(21)

with and ρ²ω the walk-off angles in the x-z plane and

0 ethe wavevectors of Φ1 Φ2 and Φ

3, respectively. Subscripts "e" and

"o" refer to "extraordinary" and "ordinary", respectively. These equations,integrated numerically with a split-step propagator, can describe the formation ofSSWs at large enough intensities and for a finite FF beam excitation. Due to thepresence of walk-off and the extra degree of freedom introduced by unbalancingthe FF inputs, however, they encompass a rich variety of phenomena rangingfrom angular steering to switching, from imbalance-controlled collisions to phase-and polarization-insensitive down-conversion SSWs, etc. We will briefly discusssome of these applications in the following.

All-Optical Steering and Switching

Fig. 24 shows the evolution in the x-z plane of an FF beam which, launched atz=0 without any SH and with , propagates in a KTP-like medium.The chosen intensities are large enough to induce self-guidance, and it is apparentthat, once the injected FF waves have generated enough SH to sustain parametricgain and overcome linear diffraction and walk-off, the SSW propagates along adirection which depends on the prevailing input polarization component. When

, the SSW travels along a direction comprised between the "e" axes

364

Fig. 24 FF propagation maps of a spatial solitary wave in the x-z planeof a 1cm-long KTP-like crystal. Here ∆ =0.1 π and the input intensities in a20µm-waist beam are (units are GW/cm²):

at FF and SH, whereas for the SSW evolves along the "o" axis.Finally,when if the power is sufficient to support two SSWs, theinput beam feeds two diverging solitary waves (a slight asymmetry with respectto perferct balance is due to the presence of two "e" out of three interactingwaves).103 This behaviour, inherently associated with a three-wave process andwalk-off in Type II SHG, indicates the feasibility of all-optical angular steering, thelatter controlled by the relative input intensities rather than the phase of acoherent SH seed. 104 Fig. 25 shows the results of experiments performed at1.064µm in a 1cm-long KTP crystal: a single FF gaussian beam was launched atθ≈45° with respect to "e" and "o" axes, and a variation in θ allowed to vary theimbalance. 103

The demonstrated steering action can occur over a substantial range ofimbalances spanning transverse distances much larger than the input waist, ass h o w n i n F i g . 2 6 . M o r e o v e r , t h e t r a n s v e r s e d i s p l a c e m e n t c a n b eenhanced/reduced by the introduction of an input tilt in the x-z plane, leading toamplified/compensated steering depending on the relative signs of tilt andbirefringent walk-off. 102 Finally, an aperture (mirror) placed at the crystal outputcan convert the induced s teer ing into switching (rout ing) , making thisconfiguration an ideal candidate for digital all-optical operations in bulk. Inaddition, in analogy to Type II SHG in the plane- or guided-wave approximations,the phase shift experienced by the weakest FF component over the strongest oneleads to polarization rotation of the outcoming SSW beam, as investigatedtheoretically in [105-106] in the absence of walk-off.

365

Fig. 25 Measured x-profiles of the SSW emerging a 1cm-long KTPcrystal for various rotations of the λ /2 waveplate from theposition. The input beam was delivered at 1.064µm in 35ps pulses, with a12µm waist. (After Ref. 103)

Fig. 26 Sketch of an all-optical beam steering device based on vectorial solitary waves in KTP. Theoutput position after 1cm propagation depends on the FF input imbalance. FF input waist is 20µm.

Collisional Interactions

Parametric solitary waves tend to propagate without diffraction through gainand phasefront distortion. It is worth emphasizing that, unlike their Kerrcounterparts, they are not associated to an actual change in refractive index. Thischaracteristic leads to some peculiar effects when two (or more) SSWs collide,because the corresponding field distributions Φ1 , Φ2 and Φ3, both in amplitudeand in phase, can locally interact through interference depending on allparameters defining the SSW at any given point in space. l 0 7 - 1 1 4 Collisional

366

interactions of vectorial quadratic solitary waves can, therefore, be controlled bytheir overall intensity/power, phase, polarization content, SH seed, etc.113-114

After colliding, the quadratic SSWs can either cross, or coalesce, or repel, or form abound state. The important feature of vectorial SSWs being the possibility offeredby the FF input imbalance to control their evolution, it is rather appealing that thesame parameter can also condition the outcome of their interaction, as shown inFig. 27 for a few cases.113 Relying on relative intensity, rather than phase, in order

Fig. 27 Propagation of vectorial SSW pairs launched at an angle inθvarious cases: a) | φ 1 (0)|² = 21.5GW/cm², |φ 2 (0)|² = 18.5GW/cm², θ=4ρω ; b)|φ1 (0) | ²=21GW/cm², | φ 2 (0) | ² = 19 GWcm², θ=4ρω ; c) | φ 1 (0)|² = 20GW/cm²,|φ 2 (0) | ²=20G W/cm ² ,θ = 4 ρ ω ; d) | φ 1 ( 0 ) | ² = 2 0 G W / c m ² = 2 0 G W / c m ² ,θ=8ρω . x and z axes are in µm and in diffraction lengths, respectively,∆=0.2π and the input beams are identically gaussian with waist=20µm.

to regulate the outcome of SSWs’ collisions is definitely a step towards theirapplications to all-optical switching and processing. Finally, taking advantage ofthe asymmetry introduced by walk-off and imbalance, even pairs of SSWslaunched parallel to each other can be made to collide or diverge depending ontheir input polarization state, as demonstrated in Fig. 28.

Down-conversion switching

A Type II phase-matched frequency-degenerate interaction can also beutilized in down-conversion, i.e. feeding energy through an SH input andconverting it to the FF. Specifically, if the pump is associated to the Φ3 beam, asmall input in either Φ 1 or Φ 2 will promote the growth of the FF at the expenseof the pump, allowing the formation of a self-guided beam. An example of suchbehaviour, calculated for a KTP-like medium, is shown in Fig. 29. While the pumpnormally diffracts in the absence of an FF seed (Fig. 29a), the latter will originatean SSW carrying a substantial portion of the input power (Fig. 29b-c). The solitary

367

Propagation of SSWs launched parallel to each other and 60µmapart, but with different imbalances: a) Left and right beams:

b) Leftand right beam same as in a). Units are as in Fig. 27.

Fig. 28

wave is, then, switched-on by a small FF input polarized along either the "o" or the"e" axes, irrespective of the input relative phase. In addition, for FF seedssubstantially weaker than the pump, the undesired situation of both FF inputpolarizations present (which, based on a Type II three-wave interaction, would beexpected to become phase-sensitive) does not alter the phenomenology except forthe initial stages of the parametric interplay.115 Finally, Fig. 30 shows the total FFpower carried by the formed SSW versus a wide range of input FF signals and forvarious mismatches. Clearly the process appears rather "robust" with respect tovariations in either parameters. Such results are in agreement with preliminaryexperimental demonstrations performed at 1.064µm in a 1cm-long KTP crystal. 116

Fig. 29 Evolution in the x-z plane of and (d) fordown-conversion SSWs. In (a) and Φ3 diffracts; in (b)thru (d) a weak FF seed has been introduced (either Φ1 or Φ2 ), with peakamplitude five orders of magnitude down with respect to the SH pump.Distances along x and z are in µm and in diffraction lengths, respectively.

368

Fig. 30 Total FF power vs. input seed power normalized to the SHpump, after propagation in a 2cm KTP-like material (no walk-off), forvarious mismatches: ∆=0 (solid line), ∆=0.2 π (short dashes), ∆=-0.2π (longdashes), ∆ = 2π (dash-dotted line), ∆ =-2π (dotted line).

CONCLUSIONS

Today quadratic cascading is to be considered a well-established approach toall-optical processing based on nonlinear optics. Effects and applications of thephase shift intrinsic to sequential up- and down-conversion processes arenumerous and diversified, ranging from bulk to guided-wave configurations forboth analog and digital operations. The interplay between phase distortion andparametric gain can support the propagation of diffractionless beams, which canalso be employed towards applications. The experimental results in knownquadratic crystals have demonstrated the practical relevance of these classes ofphenomena, paving the way to more efficient and low-power implementations inadvanced materials with larger nonlinearities, low absorptions and limited walk-offs. The field, in its infancy from a historical perspective but rather mature interms of accomplishments and understanding, is still rich of potentials in theframework of both fundamental science and optical engineering.

ACKNOWLEDGMENTS

I am indebted to several colleagues and collaborators: G.I. Stegeman and E.W.Van Stryland (CREOL-UCF, Orlando), K. Gallo, G. Leo, and C. Conti (Terza Univ.Rome). Partial support was provided by the Italian Ministry for Research (MURST40% "Photonic Technologies...") and the National Research Council (grants96.01844.CT11 and 96.02238.CT07).

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374

NONLINEAR OPTICAL FREQUENCY CONVERSION:

MATERIAL REQUIREMENTS, ENGINEERED

MATERIALS, AND QUASI-PHASEMATCHING

M. M. Fejer

Edward L. Ginzton LaboratoryStanford UniversityStanford, CA 94305fejer@leland.stanford.edu

1. INTRODUCTION

Since the first demonstration in 1961, quadratic nonlinear optical interactions have

developed into widely used tools for the generation of coherent radiation. While more than

three decades old, this field is still rapidly developing today. New generations of solid-state

pump lasers and nonlinear optical materials have led to a renaissance in quadratic nonlinear

optics over the past decade, leading both to practical sources of coherent radiation at

wavelengths inaccessible to convenient laser transitions and to applications in areas such as

signal processing, quantum optics, and cascade nonlinearities.

The interactions considered here are those due to the lowest order nonlinear

susceptibility, the quadratic polarization response to applied fields, given by P = εε0d : E 2. In

the presence of applied fields at frequencies ω1 and ω2, the polarization response, and hence

the generated field, contains components at the sum (ω1 + ω2 ), difference (ω1 – ω2 ), and

harmonic (2ω1 and 2ω2 ) frequencies, referred to as sum frequency generation (SFG),

difference frequency generation (DFG), and second harmonic generation (SHG),

respectively. Other important interactions include optical parametric amplification (OPA), in

which a long wavelength signal field is amplified at the expense of a higher frequency pump

Beam Shaping and Control with Nonlinear Optics375Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

field, and optical parametric oscillation (OPO), where an OPA in a cavity with losses smaller

the parametric gain generates outputs built up from the noise in the presence of only a high

frequency pump wave.

The first observation of a quadratic nonlinear interaction was by Franken in 1961

(Franken 1961), who generated the 347 nm second harmonic of a 694 nm ruby laser in a

piece of crystal quartz. Despite a 3 J input pulse energy, the harmonic output was only about

10 nJ, a result of the small magnitude of the nonlinear susceptibility and the lack of phase

velocity matching between the fundamental and harmonic waves, as was already understood

in this first paper. Franken pointed out that the ratio of the nonlinear to the linear polarization

should be on the order of the ratio of the applied optical electric field to the interatomic field in

the solid, and hence be orders of magnitude smaller for reasonable applied fields. He further

considered that the intensity of the generated field should be proportional to the square of the

“volume of coherence”, and hence proportional to [λ /(n2 – n 1)] 2 , where λ is the fundamental

wavelength and n 2 and n1 are the refractive indices at the second harmonic and fundamental

wavelengths, respectively. He added that “the lateral extent of the coherence volume ... [is]

determined by the coherence characteristics of the pump laser” and that “a maser of the gas

discharge type is clearly more favorable in this respect than the ruby device.” While the

evolution of laser technology has reversed the roles of solid state vs gas discharge lasers in

this respect, it is remarkable that the major challenges of the following thirty years of research

in nonlinear optical devices were already described in this two page letter: materials with

larger nonlinear susceptibilities, materials allowing phase velocity matching, and pump lasers

of high power and high spatial coherence.

Subsequent progress was rapid. In 1962 the seminal paper of Bloembergen and

coworkers presented a perturbation expansion for the nonlinear susceptibilities, quantitative

analysis of the propagation effects associated with phase-matched and phase-mismatched

interactions, and presented several schemes for accomplishing phase-velocity matching,

including the quasi-phasematching scheme that will be the subject of the second section of

this chaper. (Armstrong 1962) The most widely used phasematching scheme, based on

orthogonally polarized waves in birefringent crystals, was not given in Bloembergen’s paper,

but also appeared in 1962, independently in (Giordmaine 1962) and (Maker 1962).

All the basic quadratic nonlinear phenomena, including SHG, SFG, DFG, OPA, and

OPO were demonstrated by 1965. Since that time, progress in the field has been driven

largely by improved nonlinear materials and pump lasers. Notable among current

developments in pump lasers are high power diode lasers (1 W in a single transverse mode),

diode-pumped solid-state lasers (1 - 10 W single-mode power available commercially at

several near-IR wavelengths), rare-earth-doped fiber lasers (compact 30 W 1 µm lasers in

Yb:glass, compact fs sources in Er:glass), and tunable/ultrafast lasers such as Ti:sapphire and

Cr::LiSAF (fs pulses, TW peak powers, 400 nm tuning ranges). A good source of

information on the current state of solid state laser technology is the series of Ref. (Pollock

1997).

376

Developments in nonlinear optical materials and their applications are the subject of the

remainder of this chapter, which is divided into two main components. The first deals with

the basics of quadratic nonlinear optical interactions, with emphasis on issues relating to the

necessary properties of nonlinear optical materials. The second addresses two closely related

topics of growing practical importance, engineered nonlinear materials and quasi-

phasematching.

2. BASICS OF QUADRATIC NONLINEAR FREQUENCY CONVERSION

2.1 Plane Wave SHG

In order to discuss the important properties for practical nonlinear materials, it is useful to

first review the basics of nonlinear frequency conversion.(Shen 1984) Let us consider

second harmonic generation as a prototypical interaction. Defining the envelope fields Ej (r)

b y , where E j( r, t)is the total space and time dependent

field at frequency ω = ωj , and k j j nj /c, the evolution of the second harmonic and

fundamental fields is given by

(1)

(2)

where the k-vector mismatch is ∆k = k 2 – 2k1, and d eff is the effective nonlinear coefficient, a

combination of components of the nonlinear susceptibility tensor discussed in section 2.4.

In the low conversion limit, where depletion of the pump can be neglected, the solution

for the second harmonic field in a crystal of length L is

(3)

Note that the integral takes the form of a Fourier transform of the nonlinear coefficient, a

point that will be important in later discussions. If we further assume that the nonlinear

coefficient is uniform throughout the crystal, we arrive at an expression for the efficiency η,

(4)

where I is the intensity, the nonlinear drive η0 = C2L2 I1 is expresssed in terms of the material

constant and the dephasing δ = ∆kL/2 .

In the "phasematched" case, where δ = 0, the efficiency reduces to

(5)

so we see that the efficiency scales quadratically with the length of the crystal and the

nonlinear susceptibility, and linearly with the input intensity. If we relax the assumption of

377

low conversion, but retain the assumption of phasematching, it can be shown that the

efficiency obtained from the exact solution to Eqs. (1) and (2) is given by

(6)

In both cases, the efficiency depends only on the nonlinear drive; the properties of the

material enter only through the parameter C2 . The simplest figure of merit for a nonlinear

material is then d 2/n3 . This is also one of the most widely used figures, but is not always the

most relevant for practical applications. The tight grouping of materials used in commercial

products in the low (< 1 pm/V), e.g. KDP, LBO, to moderate (< 5 pm/V), e.g. KTP, BBO,

LiNbO3, range of effective nonlinear susceptibilities, suggests that other considerations may

dominate the actual utility of a nonlinear material. Elucidating these issues, which are

dominated by the linear optical properties of the nonlinear materials, is the subject of section

3.

For non-phasematched (∆k ≠ 0) SHG in the undepleted pump limit, the factor sinc 2(δ) in

Eq. (4) shows that the efficiency is sharply peaked around ∆k = 0, falling to half of its peak

value for δ = ±0.443π or equivalently ∆ k = ±0.886π /L. This behavior is a result of the

periodic alternation of the direction of energy flow between the fundamental and second

harmonic waves as their relative phase drifts due to the difference between their phase

velocities. The distance for a half cycle of this oscillation is known as the coherence length,

(7)

For typical materials, the dispersion of the refractive index leads to coherence lengths in the

range of several microns to several tens of microns for visble SHG. In order to obtain

efficient SHG, the coherence length must exceed the length of the crystal. For 1 µm SHG in

a 1 cm long crystal, this condition corresponds to matching the refractive indexes at the

fundamental and the harmonic to better than 2x10-5 . Satisfying this strict criterion dominates

much of the practical aspects of nonlinear frequency conversion.

Before discussing the methods used to accomplish phasematching, it is worth discussing

the solutions to Eqs. (1) and (2) in the presence of simultaneous high conversion and

phasemismatch. The exact solutions can be stated in terms of Jacobian elliptic

functions,(Armstrong 1962) but this level of detail is not necessary for the present

discussion. One important point for subsequent analyses is that the central lobe of the

phasematching curve narrows dramatically with increasing nonlinear drive. An approximate

expression (valid for η0 >> 1) for the first zero in the conversion efficiency (which occurs at

δ = π in the low conversion limit) is (Eimerl 1987)

(8)

This narrowing of the phasematching peak, one manifestation of cascade nonlinearity, has a

profound effect on the efficiency of high peak power SHG. Defining an effective

phasemismatch an approximate expression (valid for η0 >> 1)

for the efficiency within the first peak of the phasematching curve is

(9)

378

Note that because of the increase in δeff with increasing η0 at fixed δ, the efficiency is not a

monotonically increasing increasing function of η0 unless δ = 0. For any finite value of δ,

the efficiency increases with η0 (e.g. with increasing laser power) up to a maximum (always

smaller than 1), and then decreases with further increase in the laser power. This point will

be essential in the discussion of high peak power SHG in section 3.1.

2.2 Birefringent Media

Phase velocity matching requires compensation for dispersion in the refractive index,

which for typical materials leads to a difference of 0.01 – 0.1 between n2 and n 1 . The

common method for meeting this requirement takes advantage of the difference in the

refractive indexes for orthogonally polarized waves in birefringent crystals. As the effects of

this birefringent phasematching are crucial to the understanding of mechanisms that

practically limit the efficiency of nonlinear interactions, we briefly summarze the key points

of propagation in birefringent media.

In lossless nongyrotropic media, the dielectric tensor can be diagonalized with respect to

an orthogonal coordinate system; in this principal axis representation, the three independent

components of the dieletric tensor, εx x, ε y y , and εzz , are real. In the simplest “uniaxial

birefringence” case, i.e. for crystals of symmetry higher than orthorhombic, two of the

principal values are degenerate (conventionally εxx = ε yy ) and are known as the ordinary

component, ε o . The third component, εzz, is known as the extraordinary component, εe, and

the z axis is known as the optical axis. For εe < εo , the crystal is called negatively

birefringent, for εe > εo positively birefringent.

In a uniaxially birefringent crystal, light polarized perpendicular to optical axis propagates

with a phase velocity c/n o , independent of propagation direction, where the ordinary

refractive index no = ε1/2o . For light polarized in a plane containing the optical axis, the phase

velocity is c/n e(θ), and the extraordinary index n e (θ ) depends on the direction of

propagation. For a given crystal, ne is a function only of the angle θ between the k-vector

and the optical axis. The extraordinary refractive index varies continuously from no for

θ = 0°, to n o = ε1/2 e for θ = 90°, (Yariv 1984)

(10)

where the mean index = (ne + no )/2, and δ n = ( no – ne )/2. The approximate form,

accurate to first order in the birefringence, is convenient for estimating bandwidths, but is not

sufficiently accurate for the sensitive calculation of the phasematching angle itself.

The propagation of an ordinary wave is essentially identical to that of a wave propagating

in an isotropic medium, but the extraordinary wave differs in several ways. For our

purposes, the most important of these is that the phase and group velocities are not parallel to

each other. The “walkoff angle” ρ between the phase velocity, parallel to the wave vector k,

and the group velocity, parallel to the Poynting vector S, also depends on θ. The exact

dependence is straightforward (Yariv 1984); a useful approximate form is

379

ρ ≈ (2 δn / ) sin(2θ). This expression, exact at θ = 0°, 45°, and 90°, and correct to second

order in δn/ elsewhere, shows that the walkoff angle vanishes for propagation along a

principal axis, and takes its maximum value at θ = 45°. Note that this is also the direction for

which the extraordinary refractive index varies most rapidly with angle. For crystals of

moderate birefringence δn / ≈ 0.02, the maximum value of ρ is typically 4°, which will be

seen to significantly affect the SHG efficiency.

For crystals of orthorhombic and lower symmetry, all three principal values of the

dielectric tensor are different, and the propagation effects for a general direction become

substantially more complicated than in the uniaxial case. For propagation normal to one of the

principal axes (in practice the situation in most cases of interest), the behavior is similar to the

uniaxial case, with the wave polarized along the principal axis taking the role of the ordinary

wave, and the wave polarized in the plane normal to the principal axis taking the role of the

extraordinary wave.

2.3 Birefringent Phasematching

In the simplest birefringent phasematching scheme, the pump is polarized as either an

ordinary or an extraordinary wave, and the propagation direction is chosen to satisfy

2 k1 = k2 . For this "Type I" interaction, the phasematching condition can be restated as

n 2 = n1. For a negatively birefringent crystal, the second harmonic wave is polarized for the

lower (extraordinary) index, and we have ne2(θ) = no1. For a positvely birefringent crystal,

the roles of the ordinary and extraordinary waves are of course reversed. In the slightly more

complex “Type II” scheme, the pump is polarized to excite both an ordinary and an

extraordinary wave. For example, in a neatively birefringent crystal, or,

equivalently,

It is clear that if Type II phasematching is possible in a given crystal, Type I is also. The

reasons that Type II may be of interest are better use of the crystal propeties, e.g. a larger de f f

or larger acceptance bandwidths, or technical advantages such as, for optical parametric

oscillators, easier separation of the orthogonally polarized output beams, or better controlled

tuning behavior near degeneracy.

2.4 Effective Nonlinear Coefficient

The nonlinear susceptibility is a third rank tensor, whose form depends on the point

group symmetry of the crystal.(Yariv 1984) Which combination of its components are

pertinent to evaluating the effective nonlinear deff appearing in Eqs.(1) and (2) depends on

the choice choice of polarization and propagation directions. Calculation of deff involves

projecting the third rank tensor onto the unit vectors parallel to the three electric fields

involved in the interaction. The details are straightforward, but tedious. Tabulations for the

different possible phasematching schemes in each point group can be found, for example, in

(Kurtz 1975). Note that since the k-vector mismatch depends on the even-rank dielectric

tensor, it is even in θ ( ∆k(–θ) = ∆k(θ )), while the effective nonlinear coefficient, which

380

depends on the odd-rank nonlinear susceptibility tensor, is not, (de f f (–θ) ≠ de f f

(θ)). Thus,

care must be exercised in choosing the quadrant of propagation to ensure not only

phasematching, but also the largest possible de f f .

2.5 Phasematching Bandwidths

The phasematching requirement, |∆k | ≤ 0.886 π / L sets several acceptance bandwidths

for parameters such as propagation angle, temperature, and spectral bandwidth. If ∆k

depends on a parameter ξ, and we define then the first term of a Taylor

expansion around the phasematched point ξ = ξ0 yields ∆k(ξ) = β ξ δξ, where δξ ≡ ξ – ξ 0 .

The FWHM acceptance bandwidth is then

(11)

Note that the bandwidth is inversely proportional to the length of the crystal, and depends on

the material properties through β ξ . In the special case where the first derivative vanishes, the

second term in the Taylor series yields ∆k (ξ ) = (βξ ξ /2)δξ2

, where β ξξ ≡ ∂2

∆k/∂ ξ2

. The

FWHM accepatance bandwidth is then

(12)

For such cases the bandwidth decreases only with the square root of the crystal length,

substantially increasing the bandwidth compared to the critical case, a condition often termed

“noncritical phasematching”. In the remainder of the discussion of bandwidths, we will

round 0.886 ≈ 1, and 1.776 ≈ 2 to simplify the resulting formulae.

The most important of the acceptance bandwidths is the angular acceptance. From the

approximate form of Eq. (11), and Eq. (10) we can evaluate βθ

. We find

(13)

Eq. (13) applies to Type I phasematching in a negative uniaxial crystal. Similar relations can

be found for other phasematching cases. We see that the bandwidth is inversely proportional

to crystal length, and is smallest for θ = 45°. For typical cases, the acceptance bandwidths,

normalized to the length of the crystal, are in the range 0.1 – 1 mrad-cm. The divergence in

the bandwidth for θ → 0 reflects the approach to noncritical phasematching. With Eqs. (10)

and (12), we find the bandwidth for θ = 0

(14)

Typical bandwidths in this case, normalized to the square root of crystal length, are 10 – 30

mrad-cm1/2, a significant practical advantage over the critically phasematched case.

In order to take advantage of the attractive features of noncritical phasematching, a

continuously tunable parameter other than angle is necessary to meet the phasematching

condition. In general, this parameter is temperature. Following the same method as for

angular acceptance, we find

381

(15)

For typical thermooptic coefficients, the bandwidth is in the range of 0.1 – 10 K-cm. The

temperature acceptance bandwidth can be especially important in high average power devices

where absorbed optical power results in significant self-heating of the crystal.

For tunable, pulsed, or spectrally noisy lasers, the wavelength acceptance bandwidth

becomes an important parameter. Again following the same method as for the spectral

acceptance, we find

(16)

where the second form uses the definition of the group index Typical

wavelength bandwidths are 0.1 – 10 nm-cm. It is interesting to note that while ordinary

phasematching requires only matching of the refractive indices, if, in addition, the group

indices are matched at the fundamental and second harmonic frequencies, the denominator in

Eq. (16) vanishes, and the interaction is wavelength noncritical. Many materials have at least

one such wavelength pair. We will see the importance of this condition in section 3.2 on

ultrafast nonlinear interactions.

2.6 Focused Interactions

According to Eq. (4), the SHG efficiency is proportional to the pump intensity,

suggesting that for a given available input power, focusing the input beam to a small spot will

increase the efficiency. While this is true up to a point, the efficiency does not increase

monotonically with reducing spot size; there exists an optimum spot size. This limit can be

understood by again referring to Eq. (4). For a sufficiently tightly focused beam, the

diffraction length will be smaller than the length of the crystal, and the L2

scaling of the

efficiency will no longer apply. The optimum spot size comes from this tradeoff between a

tight focus for for high intensity and a loose focus for a long effective interaction length.

For our discussion of focused SHG, it is necessary to first establish some basic

properties of gaussian beam propagation. For a beam with a minimum waist w0 , the beam

waist propagates according to where the characteristic diffraction

length, known as the Rayleigh length is In the near

field, the beam radius is nearly constant, while in the far field, the radius

grows essentially linearly with distance. The far field can thus be characterized by a

diffraction angle, θ D = λ /πnw 0 . Note that the Rayleigh length decreases quadratically, and

the diffraction angle increases linearly, with the spot size.

For zR >> L, diffraction can be neglected, and the efficiency, defined here as the ratio of

SH power out to fundamental power in, can be obtained by averaging Eq. (4) across the

profile of the gaussian beam. We find an effective area for the beam of πw , and so for the02

near field efficiency,

382

(17)

Detailed calculations for the efficiency of focused SHG (Boyd 1968) are beyond the

scope of this review. Here we confine ourselves to motivating some basic results necessary

for the remainder of our discussion. Consider first the case of noncritical phasematching. A

reasonable guess for the optimal spot size is to choose it such that the entire crystal just fits in

the near field of the gaussian beam, i.e. 2zR = L, or equivalently, πw0

2 = Lλ/2n, leading to

an expression for the “confocal” efficiency

(18)

where the coefficient has units [%/W-cm]. Note that the pertinent

material figure of merit is now de f f2 / n

2 , and the efficiency scales with the inverse cube of the

wavelength, rather than inverse square as in the plane wave case. In a material with

deff = 5pm/V, and n=2, γ n c = 0.4 %/W - cm for 1 µm SHG.

For critical phasematching, an additional consideration comes into play, the Poynting

vector walkoff that leads to non-parallel propagation of the fundamental and SH beams. A

characterisitc distance for the beams to walk off each other is the aperture length, la = w 0 / ρ .The optimum focusing remains close to confocal, but for L/ la > 1 the efficiency rapidly falls

below the result in Eq. (17). The decrease in efficiency is conveniently described in terms of

a parameter B ≡ ρ(πnL/ 2λ)1/2

, which is the ratio L/la evaluated for a confocally focused

beam. For B = 1, the efficiency is 0.58 ηnc ; for B > 1.5 the efficiency for critical

phasematching is accurately approximated by In this latter limit, the efficiency for

noncritically phasematched SHG is

(19)

where the coefficient has units [%/W-cm1/2 ]. Note

that the pertinent material figure of merit for this case is d 2e f f / n 5/2 ρ, and that the efficiency

scales with λ- 5 / 2

. For a material with the same parameters as the example for noncritical

phasematching, but with a walkoff angle ρ = 2°, γcr = 0.04 %/W-cm1/2

, an order of

magnitude lower than in the noncritical case.

Quantitative calculation of the efficiency for the general case of focused SHG can be

accomplished by calculating the nonlinear polarization distribution for a given pump field,

and treating it as a source term in an appropriate Green’s function for the generated field in an

anisotropic medium. The results of this calculation are available in a readily used general

form in Ref. (Boyd 1968). Their results for the general case are conveniently summarized as

(20)

where the dimensionless function h contains all the information on focusing and

birefringence. Among the key results are that for noncritical phasematching and confocal

focusing ( L/ zR= 2, B = 0) h = 0.8, and that the actual optimum efficiency occurs for tighter

focusing ( L/ z R = 5.7, B = 0) h = 1.07, though this tighter focusing is often not used for

practical reasons such as crystal damage effects.

383

3. MATERIALS ISSUES IN NONLINEAR FREQUENCY CONVERSION

The basic results of the previous sections can be combined to elucidate the combinations

of material parameters of particular importance in particular types of nonlinear interactions.

The following sections several of these cases. It should be clear that these categories are not

orthogonal, e.g. a device can operate simultaneously at high peak and high average power,

so that more than one set of these criteria may apply, and in fact one may compound the

difficulties presented by the other.

3.1 High Peak Power SHG

Q-switched lasers are attractive pump sources for nonlinear frequency conversion,

because, for a given average power, the increase in peak power leads to higher conversion

efficiency. However, optimization of SHG for high peak power pulses involves subtleties,

often overlooked, which lead to counterintuitive design rules. The origin of the complexity is

in the narrowing of the phase matching acceptance bandwidths that accompanies high

conversion efficiency SHG.

The important quantity for pulsed SHG is the energy conversion efficiency, ηE,defined

as the ratio of the energy in the second harmonic output pulse to the energy input in the

fundamental pulse. With nanosecond and longer pulses, for which dispersion is generally

negligible, ηE can be calculated by averaging the instantaneous conversion over the entire

temporal pulse shape. For a pump that is Gaussian in space, one must also average over the

radial intensity distribution. For a pulse that is gaussian in space and time, the ratio of ηE to

the conversion efficiency at the peak of the pulse ηp, is 1/2 in the undepleted pump limit.

For example, for an energy conversion efficiency of 5%, the peak conversion must be 14%.

This difference does not have serious implications in the low conversion limit, but as the

conversion increases, the peak of the pulse is driven deeply into the saturated regime before

the wings of the pulse reach significant conversion. Consider the following table showing for

several values of the energy conversion efficiency, the corresponding peak conversion

efficiency and the nonlinear drive (defined in Eq. (4)):

η E 5% 25% 50% 75% 85%

ηP 14% 56% 87% 99.0% 99.9%

η0 0.15 0.94 2.8 9 0. 17.2

Note that for relatively modest energy conversion of 75%, the necessary peak conversion is

already 99%, and the nonlinear drive is 9. With Eq. (8) we see that the phasematching peak

has narrowed by approximately a factor of 3 in this case; for 85% energy efficiency, the peak

is narrowed by a factor of 6.

To understand the limits imposed by this intensity dependent narrowing of the angular

acceptance, we must establish one further result. For a given input power, the nonlinear

drive, according to Eq. (4), is η 0 = C2 L

2 P / πw2 , where w is the radius of the beam, while

the dephasing due to the angular divergence of the beam is (Eq. (13))

δ = (L/2)β θ δθ = (L/2) β θλ / πnw, where the second form follows from the usual result for

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the diffraction angle of a finite beam. Thus, the ratio η0 /δ2 is independent of both the length

of the crystal and the focusing of the pump beam. This ratio is often termed the normalized

brightness, Φ = η 0 / δ2. The importance of the normalized brightness is its connection to the

maximum possible SHG conversion efficiency. From the solution for depleted pump SHG,

(the exact Jacobian elliptic function solution or the approximate Eq. (9)) it can be shown that

the maximum possible efficiency is a function only of Φ, given by (Eimerl 1987) (Beausoleil

1992)

(21)

and that this maximum efficiency occurs for an optimum value of the nonlinear drive

(22)

where K is the complete elliptic integral, and the approximate form holds for ηm < 0.95. It is

important to note that for a given laser and a given nonlinear material, i.e. for a given Φ, η m

from Eq. (22) is the highest efficiency possible. If this value is inadequate, then either a more

powerful laser or a better nonlinear material is required.

Given the optimum drive, we can calculate the optimum aspect ratio, for the beam,

(23)

Note that focusing more tightly than this condition for a given crystal length, or using a

longer crystal than this for a given spot size will actually reduce the efficiency, even if the

corresponding beam divergence does not exceed the conventional angular acceptance

criterion. This is a nonlinear result, depending on the narrowing of the acceptance bandwidth

at high conversion.

It is conventional to present Φ in the form Φ = 4πP/Pt h , where the “threshold power”

Pt h is a material property defined by

(24)

The brightness must exceed 2, or equivalently the laser power must exceed Pt h / 2π in order to

obtain 50% efficiency. Thus, the pertinent material figure of merit for high peak power SHG

is d 2 /n2 β θ . This figure of merit diverges for a noncritically phasematched interaction; a

similar analysis shows that in this case, any desired conversion efficiency can be obtained if

sufficiently long crystals are available.(Eimerl 1987)

While this analysis was based on pulses uniform in space and time, it has been found that

they accurately predict the performance of real gaussian pulses if 1/e2

pulse lengths and beam

radii are used to relate energy to power and power to intensity, respectively.

3.2 Ultrafast SHG

For SHG with ultrafast pulses, the essential quantities are the energy conversion

efficiency η E , defined as the ratio of the SH pulse energy out to the fundamental pulse energy

in, and the distortion of the pulse shape/duration in the SHG process. It is convenient to

385

describe the efficiency (in the undepleted pump limit) by modifying Eq. (20) to (Arbore

1997)

(25)

where e 1 is the energy of the fundamental pulse, τ is the FWHM wifth of the pulse, g is a

dimensionless function that accounts for the time dependence of the conversion efficiency

and for spectral bandwidth effects. This decoupling of the spatial (h) and temporal (g)

dependences is appropriate if the interaction is in the near-field limit or in the “quasi-static”

(SHG acceptance bandwidth much greater than pulse bandwidth) limit.

In the quasi-static regime, the required extension of the CW analysis is straightforward.

At each instant in time the efficiency can be calculated according to the CW expression

evaluated at the instantaneous intensity. In this case,

(26)

where the normalized time is = t/τ. For a gaussian pump pulse, g = 0.664; for a sech²

pulse, g = 0.589.

If the pulse bandwidth approaches the acceptance bandwidth, the analysis becomes more

complex. If we choose the crystal length so that the SHG acceptance bandwidth just equals

the pulse bandwidth, we have with Eq. (16)

(27)

where ∆λ is the spectral FWHM of the pulse, and the group index mismatch is

∆ ng = ng2 – ng 2 . We can approximate the energy conversion efficiency in various cases by

setting the crystal length L = l B W , and averaging the resulting instantaneous power conversion

over the pulse shape.

If the pulse is transform limited, and hence ∆λ is directly related to the pulse length τ,some general conclusions may be reached. We can write l BW in terms of τ

( 2 8 )

where q is a dimensionless factor dependent on the pulse shape, taking the value 1 for a

gaussian and 1.4 for sech² . For typical materials, lτ is several mm for 100 fs near-IR pulses.

It is worth noting that this length lτ can also be interpreted, in the time domain, as the length

over which fundamental and SH pulses separate by a time equal to the pulse duration, due to

their unequal group velocities. With L in Eq. (25) replaced by lτ from Eq.(28), the quasi-

static approximation for the energy efficiency becomes

(29)

Note that the quantity in parentheses in Eq. (29) has dimensions of [%/nJ], i.e. the efficiency

is a function only of the pulse energy, and not the pulse length. Typical values in the near-IR

386

are several %/nJ, unless a near degeneracy exists for the group velocities at the fundamental

and SH. The material figure of merit for ultrafast SHG is seen to be . Evaluation

of g without making the quasi-static assumption shows that the simple estimate of ηE for

L = l τ from Eq. (29) overestimates the true value by a factor of 2 for sech² pulses, an

indication that bandwidth limitations are coming into effect in this regime.

(30)

These conclusions hold as long as the length is limited by bandwidth limitations, rather

than by spatial walkoff effects, i.e. as long as lτ < l a . This condition always holds for

noncritical phasematching, but may be violated for critical phasematching, especially for long

pulses. In this case, the energy efficiency scales as

and is inversely proportional to the square root of the pulse length. Details are beyond the

scope of this discussion.(Arbore 1997)

It may be apparent from the previous discussion that a close analogy exists between time

and space domain propagation effects in SHG, i.e. between group velocity mismatch and

Poynting vector walkoff, and between diffraction and dispersion. This analogy can be made

quantitative, so that analyses made in one domain are immediately transferrable to the

other.(Akhmanov 1975)

3.3 SHG in Lossy Materials

To this point the materials effects that we have been considering are connected to

limitations imposed by phasematching. There are also adverse effects associated with optical

losses. We consider here only linear loss mechanisms, which can be described by

I (z) = I 0 exp(–κz), where the extinction coefficient κ is independent of the intensity I. The

extinction generally is composed of two components, κ = α + σ, where α is the absorption,

and σ is the scattering. The solution for plane-wave SHG in the presence of loss is

straightforward,(Chemla 1987) and shows that the efficiency is significantly reduced

compared to the lossless case when κL > 1. For a crystal of the optimum length,

, the efficiency is given, to a good approximation, by replacing L in Eq.

(5) by When one of the extinction coefficients (κ > ) is much larger

than the other, the optimum length approaches infinity, but the asymptotic limit of he

efficiency is not significantly larger than that obtained for a crystal whose length is 1/κ > . The

combination of material properties relevant for SHG in lossy media is then

Similarly, for focused SHG, one can to a good approximation replace L

in Eq. (18) by lk, in which case the pertinent combination of material properties is

The importance of this ratio of nonlinear susceptibility to loss is the

reason why the huge enhancements associated with resonant nonlinearities have not been

used to produce efficient nonlinear devices.

It is important to note that the analysis presented in this section applies only if it is

possible to fabricate a crystal whose length is on the order of the extinction length. While this

can be only several microns for resonant media, for low-loss dielectrics this length can easily

387

be several meters, impossible to obtain in any practical sense. However, it is possible by

embedding the nonlinear crystal in a Fabry-Perot cavity to achieve efficiencies approaching

those of a crystal of length lκ.

3.4 Resonator SHG Devices

The single-pass SHG conversion efficiencies in the range of several tenths of one percent

per watt available in typical nonlinear crystals are too low for use with most CW lasers. The

high circulating fields available either inside a laser cavity or in a low-loss Fabry-Perot cavity

can be used to significantly enhance the SH conversion efficiency. Because the field

enhancement is inversely proportional to the losses in the crystal, these resonator applications

depend on the availability of low-loss nonlinear materials.

Consider externally-resonant SHG,(Kozlovsky 1988) (Schiller 1993) with a nonlinear

crystal inside a ring Fabry-Perot cavity resonant at the fundamental frequency, with input

mirror transmission T, and all other mirrors high reflectors for the fundamental and

transmitting at the SH. We take the total cavity losses, exclusive of extinction in the nonlinear

crystal, to be Aƒ. The matched resonator condition, choosing the input mirror transmission to

be equal to the total cavity losses, i.e. T = κ1 + Aƒ eliminates any reflection from the cavity

so that all the input power is coupled into the cavity. For this condition, the ratio of the

circulating power in the cavity to the input power is Pc / P1 = 1/( κ1L + Aƒ ), and the efficiency

is enhanced by the square of this factor. The optimum length for the crystal comes about

from the tradeoff between increasing the single-pass efficiency with L at the expense of

increased loss and decreased circulating power. Inserting Pc into Eq. (18) for noncritically

phasematched SHG, and maximizing with respect to L, we find the optimum length is

Lopt = Aƒ / κ1 . For this choice, the efficiency is the same as a single pass device of length

1 /4κ A1 ƒ , i.e.

(31)

and the material figure of merit is For a crystal with κ1= 0.3%/cm and a cavity

with Aƒ = 0.3%, the optimum length is 1 cm and the efficiency enhancement over a single

pass device is 3 x 104. This analysis has neglected the loss that conversion to the SH

represents to the circulating fundamental. When this conversion becomes comparable to the

static losses in the cavity, the input coupling must be chosen equal to the total loss (linear and

SH conversion) in order to achieve impedance matching. The analysis is more complicated,

but the final result is simple.(Schiller 1993) To achieve 50% conversion to the SH, the

necessary input power is

(32)

A similar analysis for a critically phasematched crystal (Bordui 1993) leads to

(33)

and hence a material figure of merit

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For intracavity SHG, with the nonlinear crystal placed inside a laser cavity, the analysis

is more complex, but similar ratios of nonliner susceptibility to loss emerge as the pertinent

material parameters.

3.5 High Average Power SHG

For SHG at high average power, the absorbed power deposited in the crystal as heat can

significantly raise the temperature. Through the thermooptic effect, this temperature rise can

spoil the phasematching as well as create a thermal lens, both of which effects can reduce the

efficiency and distort the beam shape. For simplicity, consider a flat-top beam of radius w

and average power Pav passing through a crystal of radius R with absorption coefficient αand thermal conductivity k th , whose surface is held at a fixed temperature T0 . Solving the heat

equation with a volume heat source αI(r), the temperature field in the crystal as a function of

the radial coordinate r is found to be

(34)

The temperature drop from center-to-edge of the illuminated region is ∆T = αPav / 4πkth ,

independent of w and R, and is parabolic in form. The temperature drop from the center-to-

edge of the crystal is a factor 2 ln( R / w) + 1 larger, but is less problematic as it can usually be

compensated by a change in the heat sink temperature T0 .

The parabolic temperature profile causes a radially varying phase mismatch across the

beam, whose amplitude depends on the thermooptic coefficients of the crystal.(Okada 1971)

(Gettemy 1988) While the average of this temperature rise can again be compenstaed by a

shift in T0, the radial variation cannot be corrected by simple means. This thermal phase

mismatch is given by

(35)

The maximum allowable phase mismatch, ≈ π / 2, combined with Eq. (35), sets a limit on the

allowable power in the crystal. Note that this limit scales inversely with the length of the

crystal, and is proportional to

Comparing Eq. (18) for the SHG efficiency in a noncritically phasematched interaction,

and Eq. (35) for the thermal phase mismatch, we see that if absorption of the fundamental

dominates the thermal loading, both are proportional to the product PavL, and independent of

spot size w, so that their ratio is a function only of material parameters

(36)

The material figure of merit in this case is If for a given material

this ratio is not adequate for a desired application, the only alternative, other than complex

cooling geometries,(Eimerl 1987) is to use a different material. The commonly held notion

that increasing the spot size can ameliorate the problem is counterproductive; the ratio η nc /δ t h

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decreases monotonically with increasing spot size. A comparable expression can be

developed for the case where SH absorption dominates, though the axial variation in the SH

power, which leads to an axial variation in the temperature drop from center-to-edge of the

crystal, must be taken into account.

In addition to the thermal dephasing, absorbed power also leads to thermal lensing. The

thermal phase shift across the beam takes a similar form to the thermal mismatch

(37)

This phase shift is distributed parabolically across the illuminated region (Eq. (34)), so to

first order it simply produces pure lensing with focal length fth . The focal power 1/ f th is

given by

(38)

which again scales with the product Pav L. While the focal power is proportional to w ² , and

hence can be reduced by increasing the spot size w, the thermal phase shift across the beam,

often a better measure of the effect on beam propagation, cannot. It is worth noting that

proper optical design can correct for the effect of a pure thermal lens, but for real (gaussian)

beams the parabolic thermal field characteristic of idealized flat-top beams holds only near the

axis. Higher terms in the power series representation of the exponential integral solution in

the gaussian beam case lead to an aberrating lens that cannot be corrected without aspheric

elements.(Stein 1974)

3.6 Optical Parametric Oscillators

In the presence of a strong pump at frequency ωp , an input signal at frequency ωs both

generates an “idler” wave at frequency ω i = ωp –ω s and experiences parametric gain. The

small signal parametric gain g, defined as , near degneracy (ω s ≈ ωi) is

equal to the second harmonic conversion efficiency. For focused noncritically phasematched

interactions, the parametric gain is again equal to SH conversion efficiency, so that with Eq.

(20) we have . For typical near-IR materials, the gain, like the SH

efficiency is in the range of 0.1%/W-cm. The Poynting vector walkoff associated with

critically phasematched interactions affects parametric gain more than SH efficiency. In the

limit of large walkoff angle ρ, the parametric gain reaches a limiting value given by

independent of length for optimal focusing, where the effective length

. For typical critically phasematched crystals, le f f can be less than 1 mm,

imposing a severe limit on the gain available in such materials.(Byer 1975)

If the parametric amplifier is placed in a cavity that resonates the signal wave, and the

gain exceeds the loss, the output will build from the noise, creating an optical parametric

oscillator. The oscillation condition is g = 2As , where As is the round trip power loss in the

cavity at the signal frequency. For a gain of 0.2%/W and cavity losses of 1%, the threshold

pump power is 10 W. If the cavity losses are dominated by mirror and interface losses, the

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material figure of merit for OPO applications is d2 / n2 for noncritical phasematching, and

d 2 / n3ρ2 for critical phasematching.

At high intensities, the gain goes over to an exponential dependence, and

I s ( L)/ Is (0) = For pulse lengths shorter than the build up time of the OPO

cavity, the threshold intensity increases above the CW value, approaching a condition where

the threshold intensity is inversely proportional to pulse length. In this limit, it is more useful

to quote a threshold fluence (energy/area) than a threshold intensity. (Brosnan 1977) Since

the allowable fluence is limited by surface damage effects, there exists a minimum crystal

length for which threshold can be reached before damage occurs. In these cases, the critical

parameter of the material is d2 Idam / n3, where Idam is the surface damage threshold.

Detailed discussion of optical parametric oscillators can be found in (Byer 1975) and

(Byer 1977). A more recent review is in Ref. (Tang 1992) Two special issues on OPOs

(Byer 1993) contain collections of current information, which can also be found in Ref.

(Pollock 1997).

3 . 7 Waveguide devices

Nonlinear frequency conversion in waveguides can have much higher normalized

conversion efficiency than interactions in bulk media, because the diffractionless propagation

available in channel waveguides eliminates the tradeoff between tight focusing for high

intensity and loose focusing for long interaction lengths characteristic of confocally focused

interactions.

It can be shown that the efficiency of a waveguide SHG device takes the same form as

for a plane wave interactions

(39)

where the normalized efficiency ηnor, with dimensions [%/W-cm2], is given by

ηnor = C 2 / Aeff , where the effective area is, (Stegeman 1985) (Bortz 1994)

(40)

The modal fields with propagation constants βj are normalized according to

where <ƒ> is defined as the integral over an infinite plane A∞ normal to z, and we have

allowed for the possibility that the nonlinear susceptibility is depth-dependent by defining a

normalized (|g| ≤ 1) spatial dependence of deff by g (x , y) = d eff ( x, y )/ d . For a weaklymax

guiding waveguide, it can be shown that Aeff ∝ λ2 / n (nco – ncl ), where nco and ncl are the

refractive indexes of the core and the cladding, respectively, so that overall the efficiency of

waveguide SHG scales as d 2 / neff (nco – ncl )λ4 For visible SHG in LiNbO3 waveguides,

typical values of ηnor are 100 – 1000 %/W-cm2.

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3 . 8 Materials Requirement Summary

As should be clear from the previous discussion, different applications put different

demands on nonlinear materials; a number of material parameters other than the nonlinear

susceptibility appear in the figures of merit applicable in these different cases. Note that these

figures of merit address only the readily quantifiable properties of the nonlinear material.

There are several other properties, less amenable to quantitative description, that are also

important for determining the utility of a material. A number of these have to with fabrication

issues. It is not uncommon for difficulties in the growth of crystals of adequate size and

homogeneity to delay for many years, or in some cases prevent, otherwise attractive materials

from moving from the laboratory into practical use. Similarly, problems with polishing (e.g.

soft, easily cleaved, or hygroscopic materials) and coating (e.g. materials with strong

anisotropy of thermal expansion) can complicate utilization of a material. Many nonlinear

materials exhibit long-term aging phenomena, often due to photochromic or photorefractive

responses, or sensitivity to environmental factors such as moisture, that degrade the

properties of the material over time scales of tens to hundreds of hours. Such problems are

often not apparent in laboratory device demonstrations, but can be prohibitive in practical

applications. Aspects of these issues are addressed in Ref. (Bordui 1993).

Given the variety and complexity of the requirements placed on nonlinear materials, it is

not surprising that a large number of them have been explored over the years, and that no one

material is suitable for all applications. The following section discusses some of the more

commonly used nonlinear crystals in the context of these issues.

4. NONLINEAR OPTICAL MATERIALS

The first NLO experiment used crystal quartz as the nonlinear material, a choice which

severely limited the SHG efficiency due to the lack of adequate birefringence for

phasematching. The first phasematched interactions used potasium dihydrogen phosphate

(KDP) and ammonium dihydrogen phosphate (ADP), two materials produced in large

quantities for piezoelectric applications, which fortuitously had adequate birefringence for

phasematching SHG of 1 µm lasers. Extensive materials research during the first decade of

nonlinear optics research produced a number of new materials, including some still in use

today. Two examples are the ferroelectric niobates, LiNbO3 and Ba2NaNb5O15, which both

have nonlinear susceptibilities an order of magnitude larger than those of KDP, and can

noncritically phasematch SHG of 1 µm radiation. Lithium iodate, whose large birefringence

enabled phasematching to wavelengths as short as 380 nm, also emerged in this era. Ref.

(Zernike 1973) contains an interesting discussion of the early nonlinear materials research.

One of the factors stimulating the renewed activity in nonlinear frequency over the past

decade has been the emergence of nonlinear materials with improved properties over those of

the first generation materials. Here we briefly review the properties of some of these

materials. More extensive discussion and references can be found in the review in Ref.

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(Bordui 1993), and in Ref. (Dmitriev 1997). Ref. (Bordui 1993) also contains a tabular

summary of the properties of several commonly used materials, and their figures of merit for

various applications.

4 . 1 Currently Available Materials

KDP remains the most widely used nonlinear material, though it is undistinguished

except for the large size and high quality of crystals that can be grown from low temperature

aqueous solution, and its high surface damage threshold. This observation reinforces the

importance of crystal growth and control of extrinsic properties in a successful nonlinear

material, and the inadequacy of simple figures of merit like d2/n3 for predicting a material’s

ultimate practical utility.

LiNbO3 is transparent from the mid-IR (5 µm) to the near UV (350 nm), has moderately

large nonlinear susceptibility (5 pm/V), can be noncritically phasematched for SHG of 1 µm

lasers, and can be grown in large, high quality boules by the Czochralski method. Its primary

disadvantages are its relatively low surface damage threshold and its susceptibility to

photinduced refractive index changes (photorefractive damage) for visible radiation. The

5%MgO:LiNbO 3 variant largely eliminates this latter limitation.

K(TiO)PO4 (KTP) is transparent from the mid-IR (4.5 µm) to the near UV (350 nm), has

moderate nonlinear coefficients (3 pm/V), and birefringence adequate for nearly noncritical

phasematching for SHG of 1 µm lasers. In these properties, it is similar to LiNbO3, over

which its primary advantages are resistance to photorefractive damage, high surface damage

thresholds, and large temperature acceptance bandwidths. Its primary disadvantages are

moderately difficult flux or hydrothermal crystal growth, high absorption in the mid-IR, and

susceptibility to photo-induced absorption (grey-track damage).

KNbO3 is transparent from the near-UV (400 nm) to the mid-IR (5.5 µm), has a large

nonlinear coefficient (12 pm/V), can be noncritically phasematched into the blue (430 nm),

and has good resistance to laser induced damage. Its primary disadvantages are difficult

crystal growth and processing, and small temperature acceptance bandwidths.

The borate family of materials, including BaB2 O4 (BBO) and LiB3 O5 (LBO) are

characterized by small (0.8 pm/V, LBO) to moderate (2 pm/V, BBO) nonlinear

susceptibilities, broad phasematching in the UV (fifth harmonic of 1 µm lasers in BBO, third

harmonic generation of 1 µm lasers in LBO) and transparency to wavelengths below 200 nm,

low optical loss and excellent resistance to laser damage (>l0 GW/cm2 for 10 ns pulses).

Their major limitations are due to difficult crystal growth, large walkoff angles (BBO) and

optical coating problems (LBO).

The chalcopyrite materials, such as AgGaS2, AgGaSe2, and ZnGeP2, are the primary

materials for mid-IR nonlinear optics, with transparency extending to beyond 10 µm, and

large nonlinear susceptibilities (18 pm/V AgGaS2, 33 pm/V AgGaSe2, and 70 pm/V

ZnGeP2). They have various limitations, including low thermal conductivity (AgGaSe2) and

moderately large IR losses (AgGaSe2, ZnGeP2). In all three the growth and processing are

difficult, and serious surface damage limitations exist, especially for AgGaS2 and AgGaSe2.

393

4 . 2 Approaches to Finding Improved Nonlinear Materials

There remain needs for better nonlinear materials in all spectral ranges. In the UV,

materials combining non-critical phasematching, ease of crystal growth, and robustness with

respect to aging effects are lacking. In the visible, materials combining adequately large

nonlinear susceptibility and noncritical phasematching to enable efficient single-pass doubling

of high-power CW lasers or CW optical parametric oscillators, with the thermophysical

properties to allow support of multiwatt visible beams, are unavailable. In the mid-IR,

current materials suffer from combinations of difficult growth, low surface damage theshold,

limited non-critical phasematching, and high optical losses. For nontraditional applications

like cascade nonlinearities or mixers and parametric amplifiers for communications and

quantum optics, a generation of materials with an order of magnitude higher efficiency than

those currently available are needed.

There are several possible approaches to developing improved nonlinear materials. The

most obvious, though often overlooked, of these is suggested by the observation that many

of the problems of current materials, such as absorption, photorefractive and photochromic

effects, limited homogeneity and size of available crystals are usually not intrinsic material

properties. Thus, improving an old material is often the most direct approach to finding a

“new” material. It is not uncommon for 10 – 20 years to elapse between the first laboratory

demonstration of a new material, and its emergence in practical applications. Examples

include the reduction of photochromic effects in KTP and photorefractive effects in LiNbO 3

through the addition of suitable dopants, and the application of improved crystal growth

methods to make available large crystals of the ZnGeP2 , a material first described more than

25 years ago.

Finding wholly new materials can proceed by several paths. The first approach followed

historically was to take advantage of materials that were already well-developed for other

applications, primarily piezoelectricity, as it demands the same noncentrosymmetry

requirement as nonlinear optics. This approach, which led to the early use of KDP and ADP,

is attractive, as it takes advantage of years of preexisting materials development, but limited

as only a small number of materials fall in this category. We will see the power of a variant of

this approach in the discussion of engineered materials in the following section. The second

approach followed historically was to search the mineralogical tables, guided by Miller’s

rule, for crystals of high refractive index and large birefringence. This approach, followed in

the 1960’s, led, for example, to the use of cinnabar (HgS) and proustite (Ag3AsS3), but

appears to offer no current prospects. An important method is to search compounds

isostructural to a known useful material. The KTP family is one such example, where

K(TiO)AsO4 (KTA) was found to have similar properties to KTP, but with substantially

lower mid-IR absorptivity. Similarly, a large number of crystals in the KDP family have been

explored for advantageous phasematching properties. All these approaches have yielded

useful results, but do not offer a systematic approach to designing a material suited to a

particular interaction. Such design methods are the subject of the following section.

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4 . 3 Engineered and Microstructured Materials

In this context, the term engineered materials is applied to materials which can be tailored

in a systematic fashion to suit specific applications, through, for example, control over their

transparency range, nonlinear coefficient, or phasematching. The commonest approach to

this goal has historically been through molecular engineering techniques. These have been

widely applied to organic molecules, for which accurate techniques for design and synthesis

of molecules with desired absorption edge and nonlinear susceptibility have been developed.

Numerous examples of molecules with enormous nonlinear polarizabilities exist, some of

which are discussed in other chapters of this volume. These techniques have been applied on

a more limited scale to inorganic materials, especially the borate family, and again good

results for design of the absorption edge and nonlinear susceptibility have emerged.(Chen,

1985) Far less successful have been efforts to design the birefringence for phasematching,

which, as the small difference of large numbers is inherently more difficult, and the

“growability” of these materials, which has precluded the use of otherwise attractive

compounds.

Techniques for achieving continuously tunable non-critical phasematching would be very

powerful for optimizing materials for a variety of applications. One such approach is based

on solid solutions such as AgGa1-xInxSe2 and K(TiO)P1-xAsxO4 , which allow continuous

interpolation of properties between those of one end compound and the other. If one of these

compounds has too much birefringence for a given application and the other too little, in most

cases a composition exists in between with the desired birefringence. While these techniques

have been successfully demonstrated, they have not been widely exploited, because mixed

crystals are almost always difficult to grow homogeneously, compounding an already

challenging growth problem.

An alternative approach, which doesn’t require solution of a new set of growth problems

for each application, would be a very powerful simplification of the process of engineering

nonlinear materials to suit particular interactions. An increasingly important family of these

techniques are based on microstructured materials.

The microelectronics industry provides a good model for the application of

microstructured materials to nonlinear optics. In microelectronics, a broad range of devices,

from high-current switches to microprocessors to RF amplifiers, are fabricated in one or two

very well understood materials (Si and GaAs) by applying a small set of well-controlled

processes steps, often lithography based. One such technique currently developing rapidly in

the nonlinear optics field is quasi-phasematching, in which a periodic spatial variation

imposed on the properties of a material allow a nonlinear interaction to proceed efficiently in

the absence of phase-velocity matching. Other techniques are based on a variety of thin-film

media, especially polymers and III-V semiconductors, in which, for example, waveguides

with tailored dispersion and modal overlap for modal phasematching,(Wirges 1997) and

multilayers with controllable form birefringence for birefringent phasematching (Fiore 1996)

are being developed. The discussion in this chapter will focus on the quasi-phasematching

technique.

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5 . QUASI-PHASEMATCHING

In a quasi-phasematched interaction, the phase error that accumulates between waves

propagating with different phase velocities is periodically reset by a periodic variation in the

properties of the nonlinear material. Taking SHG as an example, the relative phase of the the

second harmonic field and the nonlinear polarization (proportional to deff E ²1 ) drifts by π

every coherence length lc (Eq. (7)). Because the direction of energy flow between the

fundamental and the second harmonic depends on this relative phase, in a non-phasematched

interaction the power that flowed into the SH in the first coherence length flows back to the

fundamental over the next coherence length, so that there is no average growth of the second

harmonic field. In a phasematched interaction, the relative phase is constant, so the second

harmonic field grows linearly with distance. In the ideal quasi-phasematched interaction, the

sign of the nonlinear polarization is changed every coherence length, so that the sign of the

nonlinear polarization is reversed every coherence length, effectively adding the π shift

necessary to properly rephase the nonlinear polarization and the SH field. The SH field then

grows monotonically with distance, “quasi-phasematching” (QPM) the interaction.

It is also useful to view this interaction in momentum space, rather than the real space

picture presented above. In a phasematched interaction, 2k 1 = k 2 , i.e. the sum of the k-

vectors of the two fundamental photons destroyed equals the k-vector of the SH photon

generated, conserving photon momentum. In a phase mismatched interaction, 2k 1 + ∆ k = k 2 ,

momentum is not conserved, and the efficiency is very low. If an additional structure with a

spatial frequency Kg = ∆k exists in the medium, a sort of “quasi-momentum” conservation

exists, and the interaction can proceed efficiently. This form of the analysis is clear from the

integral form of the solution for the SH harmonic field, Eq. (3), which shows that a periodic

variation in d(z) with spatial frequency equal to ∆k will cancel the oscillatory term in the

integrand, and lead to monotonic growth of the SH.

5.1 Basic Theory of QPM

For quantitative analysis of QPM,(Fejer 1992) it is useful to write the spatially varying

nonlinear susceptibility in the form

(41)

where deff is the maximum value of the nonlinear coefficient, and g(z) is a variable normalized

to g(z) < 1 containing all the information about the spatial distribution of d(z). Substituting

Eq. (41) for d(z) into Eq. (3), we can write the generated field as

(42)

where

(43)

396

Eq. (43) makes clear that the generated field depends on the Fourier amplitude of g( z) at the

spatial frequency ∆k. Note that if g = constant Eq. (43) simply reproduces the usual sinc²

tuning for conventional SHG. If g(z) is periodic with period Λ, we can write g(z) as

(44)

where the mt h spatial harmonic of the grating is

(45)

Assuming that the mth spatial harmonic is much closer to ∆k than any of the others, that term

will dominate in the integral in Eq. (43), and the generated SH field will be, neglecting an

unimportant phase factor,

(46)

which is identical to that obtained for a conventionally phasematched interaction with a

nonlinear coefficient dm = G m deff and a phase mismatch shifted by an amount Km . If the

grating is a rectangular wave of unit amplitude and period Λ, with positive sections of length

l, the required Fourier coefficient Gm can be expressed in terms of the duty cycle D = l/Λ; the

resulting value of dm is

(47)

For an optimum choice of D, e.g. 50% for m odd, we find

(48)

For example, in periodically-poled lithium niobate (PPLN), with a nonlinear coefficient of

d 33 = 27 pm/V,(Roberts 1992) the nonlinear coefficient for first order QPM is d 1 = 1 8

pm/V, while second and third order QPM have nonlinear coefficients of 9 pm/V and 6 pm/V,

respectively, compared to the coefficient used for birefringent phasematching, d 31 = 4.3

pm/V. In the case of a quasi-phasematched waveguide interaction, the overlap integral

includes the depth-dependent Fourier component of the nonlinear coefficient, i.e. the function

g(x,y) in Eq. (40) is replaced in the overlap integral by Gm (x,y) of Eq. (44), which can have

a major impact on the efficiency for common waveguide materials.(Bortz 1994).

Summarizing, if we can construct a grating in the nonlinear susceptibility whose period is

an integer multiple (m) of 2lc , a quasi-phasematched interaction proceeds essentially as a

conventionally phasematched interaction with a nonlinear susceptibility reduced by a factor

2/mπ. This discussion has focused on pure modulation of the nonlinear susceptibility,

though QPM based on modulation of the linear properties is possible as well, generally at a

significant reduction in efficiency.(Fejer 1992)

397

5.2 Advantages of QPM

By eliminating any dependence on birefringence for phasematching, QPM offers several

major advantages compared to conventional phasematching techniques. QPM allows a

material to be used for any interaction within its transparency range, and can even be applied

in non-birefringent materials like the zincblende semiconductors. Noncritical phasematching

is possible, eliminating the deleterious effects of Poynting vector walkoff, especially

important for applications in optical parametric oscillators. Any combination of polarizations

can be used, allowing, for example, parallel polarization of all the fields to take advantage of

the often-large diagonal components of the nonlinear susceptibility tensor, inaccessible to

birefringent phasematching. In LiNbO3 , the ratio d33 / d 31 = 7, a factor that enters as the

square in the calculation of the conversion efficiency. Perhaps most important in the long run

is the ability QPM affords to systematically tailor one material for many applications. Despite

these several advantages, QPM has not been widely used until the past several years, due to

the difficulty in fabricating the necessary fine pitch (typically microns to tens of microns)

gratings in the nonlinear susceptibility.

5.3 Early History of QPM

QPM was first discussed in the seminal 1962 paper by Bloembergen,(Armstrong 1992)

and a patent issued to him in 1968. Despite its invention prior to that of birefringent

phasematching, the latter has become the basis for almost all practical nonlinear frequency

conversion devices. Early demonstrations of QPM were based on stacks of rotated plates of

III-V and II-VI semiconductors. References to this early work can be found in (Fejer 1992).

Development in this era was slow, due to the difficulty in fabricating the necessary stacks of

thin plates, and losses associated with the many air-to-semiconductor interfaces. Practical

QPM approaches awaited the advent of monolithically patternable media.

A number of materials have been explored for monolithic patterning of the nonlinear

susceptibility, including ferroelectrics, poled polymers, poled glass, patterned orientation

growth of semiconductor films, and asymmetric quantum wells. By far the best developed of

these are the periodically-poled semiconductors, which will be the subject of the remainder of

this chapter.

5.4 Ferroelectrics for QPM

Ferroelectrics are materials in which each unit cell develops a spontaneous electric dipole

moment at temperatures below a critical “Curie” temperature.(Lines 1977) These dipoles

form into macroscopic regions of aligned polarization, known as “domains”, separated by

domain walls. The domains can be preferentially oriented by application of an electric field

exceeding a value characteristic of the material, the coercive field. In many of these respects,

ferroelectrics and electric fields are analogous to ferromagnets and magnetic fields.

For a large class of ferroelectrics, there are two allowed orientations of the spontaneous

polarization, 180° rotated with respect to each other. For the purposes of QPM, the important

398

point is that the sign of odd-rank material tensors, including the nonlinear susceptibility,

changes from one domain orientation to the other. Thus, a periodic array of ferroelectric

domains produces the periodic reversal in the sign of the nonlinear susceptibility necessary

for QPM. The problem of creating a monolithic QPM structure is thus reduced to the problem

of creating a micron-scale array of ferroelectric domains.

This possibility was recognized early in the history of nonlinear materials development.

As early as 1964, the effects of random arrays of domains in as-grown crystals was

investigated.(Miller 1964) Subsequent to that, during the 1980’s, periodic arrays of domains

were created by periodically perturbing the growth of bulk and fiber ferroelectric crystals.

While interesting QPM SHG results were obtained, proving the potential of the method,(Xue

1984) (Jundt 1991) development was limited by the difficulty in creating adequately periodic

structures over interaction lengths longer than about 1 mm. Note that the effective interaction

length is limited to the distance over which accumulated drift in the domain location becomes

on the order of one coherence length.(Fejer 1992) Widespread use of QPM techniques did

not begin until lithographically controlled methods for patterning domains were devised. An

important practical point is that at least two of the ferroelectrics widely used for optical

applications, LiNbO3 and LiTaO3 , are also used for surface acoustic wave filters. This

industry uses several tons (>106 wafers) per year, so the growth of these materials is very

well developed, with high quality 3-inch diameter wafers available for prices in the range of

$100.

5.5 Lithographic Patterning of Ferroelectric Domains

The first lithographic domain patterning methods developed were based on patterned

indiffusion of dopants into ferroelectric substrates.(Lim 1989) (Webjorn 1989) Initially,

indiffusion of a Ti grating into LiNbO3 was used, though techniques based on indiffusion of

protons in LiTaO3 , (Mizuuchi 1991) and Ba and Rb into KTP (van der Poe1 1990) soon

followed. The microscopic mechanisms causing the domain reversal in these methods are not

well understood. These methods all produced shallow gratings, whose depth was on the

order of the grating period (several microns), so that this material could be used only in

waveguide interactions, in which the radiation is trapped in the waveguide region, of

comparable depth to the domain pattern. Progress in these waveguide devices was rapid;

SHG of visible light with efficiencies with normalized efficiencies in the range of 100%/W-

cm² and several milliwatts of output power were demonstrated by 1992. Early work in

patterned dopant QPM methods is reviewed in Ref. (Fejer 1992b).

The next major step in the development of periodically-poled ferroelectrics was poling

using electric fields applied through periodic electrodes patterned on the surface of the

wafer.(Yamada 1993) This approach, surprisingly, created domains that propagated all the

way through 0.5 mm thick substrates, allowing use as a bulk material as well as for

waveguide applications. Initially developed in LiNbO3 , this technique was again rapidly

applied to other ferroelectrics, including LiTaO3 , (Zhu 1995) and KTP (Chen 1994). Typical

conditions for the poling of LiNbO3 are the application of voltages around 10.5 kV across a

399

0.5 mm thick wafer (fields of 21 kV/mm) for durations on the order of 100 ms, with an

insulating film covering a periodic electrode on one face of the crystal, and a uniform ground

electrode on the other face. The dynamics of the domain formation process are complex, but

a model that explains important aspects of the poling behavior has been developed.(Miller

1996) A key point in explaining the high fidelity of the transfer of the pattern from the

periodic electrode to the domain pattern is the strong dependence of the lateral velocity of the

domain walls on the applied electric field. Measurements showed that the velocity changed by

five orders of magnitude for a several percent change in the field in the vicinity of 21 kV/mm.

When domains that nucleate under the electrodes begin to spread beyond the electrode and

into the region covered by the insulator, the polarization charge deposited on the surface is

uncompensated by free charge, reducing the average field across the wafer. Because of the

steep dependence of the domain wall velocity on the electric field, these small changes are

enough to stop the domains’ penetration under the insulating regions; The upshot is that it is

possible to pole entire 3 inch (75 mm) diameter wafers with high quality volume domain

gratings, with periods from the 30 µm range suitable for QPM of mid-IR generation, down

to the 6 µm periods necessary for SHG of green light. Shorter period gratings are a topic of

current research, with material suitable for operation into the blue spectral region

fabricated,(Pruneri 1995) (Goldberg 1995) but over smaller areas than the longer pitch

gratings. Shorter period gratings appear to be easier to pole in LiTaO3 and KTP than in

LiNbO3 , with usable material fabricated with periods around 2 µm, and applied to the

generation of UV radiation. (Mizuuchi 1997) (Meyn 1997)

6. CURRENT RESULTS FOR QPM DEVICES

Fabrication and device application of quasi-phasematched materials is a very rapidly

expanding field of research, with hundreds of papers published over the past several years. It

is not possible to comprehensively review the work in this area in the space available. Here

we focus on some key results in two areas, visible light generation and mid-IR OPOs.

6.1 Visible Light Generation

Waveguide devices for visible SHG were the first to take advantage of electric-field-poled

substrates. The major advantage over the chemically poled substrates was eliminating the

difficult tolerances associated with overlapping the waveguide modes with the shallow,

depth-dependent periodically-poled layer.(Bortz 1994) Efficiencies increased rapidly, again

in all three common ferroelectrics, LiNbO3 , LiTaO3, and KTP. Normalized efficiencies for

SHG of blue light of 1600%/W-cm² have been reported in LiNbO3 waveguides,(Kintaka

1996), 1500%/W-cm² in LiTaO3 ,(Yi 1996), both using annealed proton exchanged

waveguides, and over >400%/W-cm² in KTP ion-exchanged waveguides.(Chen 1996).

Relatively high powers have been generated in the waveguide devices, 25 mW of 490 nm

output for 120 mW of 980 nm pump power in a LiNbO3 waveguide,(Webjorn 1997), 23

400

mW output for 121 mW IR pump in an LiTaO3 waveguide,(Yamamoto 1992) and 12 mW

output for 146 mW IR pump power in KTP.(Chen 1996) While these output levels appear to

be stable, higher outputs (up to 100 mW) have been observed in LiNbO3

waveguides,(Webjorn 1997b) though with evidence of photorefractive instabilities. An

important step toward control of photorefractive damage has been the recent progress in

poling MgO:LiNbO3 substrates, which are much more resistant to photorefractive damage

than conventional LiNbO3 and LiTaO3. An efficiency of 1200%/W-cm2 has been reported,

with 5.5 mW of 434 nm light generated,(Mizuuchi 1997b). This device included a high-

index top cladding layer of Nb2O5, which improved the efficiency by creating a more

symmetrical waveguide, bringing the peaks of the fundamental and SH modes closer

together, thereby increasing the overlap integral. Another device on MgO:LiNbO3 , generating

19 mW of 434 nm radiation at 600%/W efficiency using buried domains on an x-cut

substrate, was reported.(Mizuuchi 1997c) Efforts to efficiently package QPM waveguide

SHG devices have also been succesful, with cm3 devices including pump diode, coupling

hardware, and the SHG chip, generating as much 15 mW blue output.(Webjorn 1997),

(Kitaoka 1995)

Scaling to higher powers than is practical in waveguides requires the use of bulk crystals.

Ideally, these would have adequate conversion efficiency to operate in a single pass, thus

avoiding the complexities associated with intracavity or externally resonant devices. In an

ideal PPLN crystal, the normalized efficiency for SHG of green light, given the effective

nonlinear coefficient of 18 pm/V, is γnc = 4%/W-cm. For a 5 cm long crystal, practical with

current technology, a theoretical efficiency of 20%/W is obtained. For 1 µm pump laser with

several watt output power, rather moderate by current standards, efficiencies in the saturated

regime can be expected. The reduced susceptibility to photorefractive damage associated with

the periodic variation in electrooptic and optogalvanic effects in periodically-poled crystals is

an important practical advantage.(Taya 1996) (Sturman 1997)

The highest CW output power reported to date was in a 5.3 cm long PPLN sample,

whose 6.5 µm period domain grating was of sufficient quality to provide a nonlinear

susceptibility 78% of the ideal 18 pm/V. Temperature tuning showed a nearly ideal sinc2

response, indicating that the sample was uniform over its entire length, and a normalized

efficiency of 10.5%/W. The maximum 532 nm output power, 2.7 W, was obtained at an

internal 1064 nm pump power of 6.4 W, or 42% overall conversion efficiency. This is

approaching the performance necessary for an ideal solid state replacement for the argon laser

in many applications. At powers exceeding 2 W, significant thermal focusing effects were

observed. Subsequent studies suggest that these are a result of a green-induced IR absorption

phenomenon that must be reduced before higher single-pass powers are obtained in these

long crystals.(Miller 1997) 330 mW of average green power was generated by SHG of a

mode-locked 1 µm laser in PPLN, with an average conversion efficiency of 52%.(Pruneri

1996)

Extension into the blue is challenging due to the difficulty in poling the necessary fine

pitch (3 - 4 µm) period gratings, but recent results have been promising, with 49 mW of CW

401

473 nm radiation generated in a 6 mm long PPLN sample with near ideal 19 pm/V nonlinear

susceptibility, and 3 K temperature tuning bandwidth.(Pruneri 1996b) Doubling of a 980 nm

diode laser to produce 6.7 mW in bulk MgO:LiNbO 3 corona poled with a 5.3 µm period, for

a normalized efficiency of 2%/W-cm has also been reported.(Harada 1996)

6.2 Optical Parametric Oscillators

The noncritical phasematching available in PPLN, along with the large d33

coefficient of

LiNbO3, are particularly advantageous for use in OPOs, which have been perhaps the most

impressive demonstrations of QPM materials to date. The first QPM OPO, based on an

annealed proton exchanged waveguide in a dopant-diffusion-poled LiNbO3 substrate, was

demonstrated in 1995.(Bortz 1995) Shortly thereafter, the first QPM OPO in bulk material

was demonstrated in electric-field poled LiNbO3 (Myers 1995) Progress since that time has

been rapid.

The gain of a bulk PPLN OPA at degeneracy, the same as the normalized SHG

efficiency, is 4%/W-cm for 0.5 µm pumping, and, scaling as λ–3 , (Eq. (18)), is 0.5%/W-cm

for 1 µm pumping. These high gains permit pumping with much lower power lasers than are

possible with conventional materials. In the first bulk PPLN OPO, pumped with 7-ns pulses

from a Q-switched diode-pumped Nd:YAG laser, the threshold for a 5 mm long PPLN

crystal was 135 µJ, ten times lower than has been obtained with birefringently phasematched

LiNbO3, and more than 20 times below the surface damage fluence of 3 J/cm 2 , an important

consideration for practical applications.(Myers 1995) In the same apparatus, a 15 mm long

crystal and tighter pump focusing led to threshold of 12 µJ, well-matched to pumping with

high-repetition-rate Q-switched diode-pumped lasers.(Myers 1995b) By use of multiple QPM

gratings patterned on a single chip, it was possible to tune a 1-µm-pumped OPO from 1.36 to

4.8 µm, simply by translating the crystal to bring different gratings into the pump

beam.(Myers 1996) (Myers 1995b) compares QPM OPOs vs birefringently phasematched

OPOs, provides tabular data on the phasematching properties of PPLN for IR OPOs, and

reviews pulsed singly-resonant and CW doubly-resonant OPOs in PPLN.

An important application of PPLN, taking advantage of the high gain and low loss, has

been in CW singly resonant OPOs. The first CW 1 µm pumped OPO, using a 5-cm-long

PPLN crystal, demonstrated a threshold pump power of 3.5W, an output power at 3.25 µm

of 3.5 W, and a pump depletion of 94% at the maximum pump power of 13.5 W, consistent

with the high gain and low (<0.1 %/cm) mid-IR losses of PPLN.(Bosenberg 1996) Electric-

field-poled PPLN has also been used in a waveguide OPO, with threshold of 1 W; design

approaches to sub-100-mW thresholds seem realistic.(Arbore 1997b)

Operation of PPLN OPOs has been extended to 0.5 µm pumps, including Q-switched

(Pruneri 1995b) and CW (Batchko 1997) operation. The latter had a threshold of 0.9 W.

Picosecond (Butterworth 1996) and femtosecond (Burr 1997) OPOs with IR pumps are

another important current direction, taking advantage of the large ultrafast figure of merit

(d 2/n2∆ng).

402

7. FUTURE DIRECTIONS

The field of QPM nonlinear optics continues to grow rapidly. In addition to the well-

established applications in mid-IR OPOs, applications are now growing in ultrafast nonlinear

optics, especially for low pulse energy systems such as fiber lasers where the high figure of

merit is of critical importance. Ultrafast SHG,(Arbore 1997) OPG,(Galvanauskas 1996) and

OPOs (Butterworth 1996) (Burr 1997) have all been demonstrated. The use of Fourier

synthesis techniques to design QPM gratings with desired spectral amplitude and phase

response is just beginning to be investigated,(Arbore 1997c) and offers many exciting

opportunities. As the extrinsic loss is reduced in the visible spectral region, and the

techniques for reliably patterning short-pitch domain gratings are improved, applications of

long periodically-poled crystals to high power visible light generation will expand.

Applications beyond sources of coherent radiation are also appearing. Wavelength

convertors for WDM communications systems implemented by near-degenerate difference

frequency generation in QPM waveguides are characteristic of the optical signal processing

functions made possible by the very high mixing efficiencies available in waveguide QPM

devices.(Xu 1995) (Yoo 1996) There appear to be many other opportunities in this area,

e.g., quantum optics experiments using waveguide (Anderson 1995) (Serkland 1995) and

bulk (Lovering 1996) OPAs.

Other materials systems are being developed that address limitations of PPLN.

Periodically-poled lithium tantalate extends the UV edge to 280 nm.(Meyn 1997)

Periodically-poled KTP isomorphs hold the promise of larger apertures, higher damage

thresholds, and more resistance to photorefractive effects.(Reid 1997) III-V semiconductors

offer large nonlinear susceptibilities and transparency beyond the 5 µm multiphonon edge

characteristic of oxide ferroelectrics. Diffusion-bonded stacks of plates (Zheng 1977) and

growth of films controllably twinned by a template substrate (Angell 1994) (Yoo 1996) are

both progressing towards practical device implementations. The surprisingly large nonlinear

susceptibilities created by electric field poling of glass fibers has been used in QPM

interactions, offering interesting opportunities for long devices in convenient waveguide

forms.(Kazansky 1997)

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Miller, G.D., Batchko, R.G., Tulloch, W.M., Weise, D.R., Fejer, M.M., and Byer, R.L., 1997, 42%efficient single-pass second harmonic generation of a continuous-wave Nd:YAG laser output in a 5.3cm length periodically-poled lithium niobate crystal, presented at the OSA Topical Meeting onAdvanced Solid State Lasers, Orlando, FL, Jan. 1997.

Miller, R.C. 1964, Optical harmonic generation in BaTiO3 single crystal, Phys. Rev. 134:A1313.Mizuuchi, M., Yamamoto, K., and Taniuchi, T., 1991. Second harmonic generation of blue light in LiTaO3

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Myers, L.E., Eckardt, R.C., Fejer, M.M., Byer, R.L., and Bosenberg, W.R., 1996, Multigrating quasi-phase-matched optical parametric oscillator in periodically poled LiNbO3 , Opt. Lett. 214:591.

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LOW-POWER SHORT WAVELENGTH COHERENT SOURCES:TECHNOLOGIES AND APPLICATIONS

D.B. Ostrowsky

Laboratoire de Physique de la Matière CondenséeUniversité de Nice-Sophia-AntipolisParc Valrose06108 Nice Cedex 2, France

INTRODUCTION

The object of this course is to attempt to compare three techniques that currentlyappear to be the most promising for the realization of relatively short wavelength visiblecoherent light sources furnishing a few tens of milliwatts: parametric conversion viasum-frequency generation, rare-earth doped fiber upconversion lasers, and, shortwavelength diode lasers. The course will essentially concentrate on the two latter typesof devices since the parametric devices have been extensively described in other coursesat this school. In any case, it should be emphasized that all three techniques will nodoubt prove to be viable for certain applications. The comparison will not result in a“winner take all" situation. In order to emphasize this we will begin by mentionningsome of the motivations for the development of such sources. This will enable us tounderstand the different constraints various applications will impose. We will thendescribe each technique in turn and determine the type of performance each can beexpected to provide. We will conclude with a description of some interestingapplications using the techniques and short-wavelength sources that have beendescribed.

APPLICATIONS OF SHORT-WAVELENGTH SOURCES

In this course we shall define "short-wavelength" as wavelengths below thosecurrently produced by continuous wave diode lasers, i.e. essentially wavelengths in thegreen to violet portions of the visible spectrum. The applications for coherent sources inthis range can be, somewhat arbitrarily, grouped into three major classes: data storageand printing, display, and scientific instrumentation. Among these applications, data

Beam Shaping and Control with Nonlinear Optics407Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

storage is currently the most important economic motor for device development and wewill, therefore, examine it in somewhat more detail than the others.

It is clear that data storage will be the main consumer application for shortwavelength lasers in the forseeable future. Currently, on the order of 150 million diodelasers are sold each year for this purpose - at an average price on the order of 1$ perlaser! With the red diode lasers already being used in commercial Digital Video Disks(DVD's) one attains a 4 GB per face capacity, and a demonstrated possibility of usingfouoverall 16 GB capacity. This improvment over the current 650 MB CD-ROM diskwe know and love is attained through a reduction by 2.2 in the interline spacing - nowdown to around 0.8 microns, close to the finest lines used in microprocessors, and afactor of 2 in spot size due to better quality optics and a slightly shorter laser wavlength.However, only a small advantage has been gained by the passage from the 780 to 640nanometer diode lasers, the rest has been achieved by improved coding. So, the race ison to fabricate diode lasers in the 400 nm range, with excellent beam quality, that wouldpermit tripling or quadrupling the capacity to the 15 GB/face range. For this applicationit is difficult to imagine other sources being cost effective but, since the laser currentlyonly represents a few per cent of the entire device fabrication cost, an enablingtechnology, such as fiber lasers, or the frequency doubling of existant high power near-IR diode lasers, could probably compete if they offer superior performance, and attainfabrication costs of under 10 $. However, whether this will be possible remains to beseen.

Displays are currently an essentially non-existant market for coherent sources, butone that has enormous potential for mass dvelopment if the enabling technologies aredevelopped. These technologies include both the three color laser sources and adequatescanning technology, or high brightness LED arrays. On a much more pedestrian level,GaN LED's are being proposed for many more immediate applications, the paradigmbeing traffic lights, where good energy efficiency and long life are primordial.However, this is an application field we shall not address.

The other field of applications we will address is scientific instrumentation. Theseare often based on fluorescence techniques, which, using short wavelength sources, hasalways had many scientific and technological applications. In this course we shalldescribe three such applcations, quite arbitrarily chosen, as examples of the use ofvarious phenomena and devices that shall be described in the paper. These are geneticidentification, a fiber optic temperature sensor, and, waveguide examination.

Having briefly outlined some potential applications of interest, we will now go onto describe the three most promising types of low-power short wavelength coherentsources: upconversion fiber lasers, short wavelength diode lasers, and frequencydoubled near IR diode lasers.

UPCONVERSION FIBER LASERS

Introduction to fiber lasers

Actually, the advantages of a doped fiber structure for the realization af amplifiersand lasers was recognized and demonstrated by Snitzer as early as 19611 in work thatprobably began before the demonstration of the first laser by Maiman in 1960. With thematuring of laser technology, resulting in the availability of a variety of "classical" lasersources for pumps, optical fibers became a remarkable medium for the fabrication of a

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new class of lasers: the laser pumped fiber lasers. The essential advantage of fiber lasersis that of "plumbing" i.e. the fiber allows the confinement of a high density of pump andlaser powers over distances impossible to attain with bulk optics. The advantages ofthis configuration, highly schematically shown in the following figure, are numerous.

Figure 1. Grossly simplified schematic of a fiber laser.

The essential point to note is that this configuration, in contrast to the bulk laserconfiguration, allows an independant choice of the pump spot size and the interactionlength. This allows several orders of magnitude higher pump power concentration withcorrespondingly lower thresholds. Furthermore, since essentially no optical power islost over the relatively short fiber lengths ( a few centimeters to a few 10's of meters)usually used in fiber lasers, the doping concentration can be optimized for each desiredphenomenon by properly adjusting the length. It is all of these degrees of freedom thathave permitted the development of a variety of "standard" fiber lasers and amplifiers inboth three-level and four level atomic systems2. By standard, we mean lasers that arepumped by wavelengths that are shorter than their lasing wavelength. In the followingwe shall be concerned with upconversion lasers, i.e. those that are pumped withwavelengths longer than their lasing wavelength. All of the previous advantages citedcontinue to apply but the phenomena involved in the excitation process are somewhatmore complicated. We will outline these processses in the following sections.

Introduction to up conversion

In the early 1960's, before the advent of visible Light Emitting Diodes (LED's) andefficient semiconductor photodetectors in the infrared, there was a major interest in theincoherent upconversion of IR light for both displays and IR detection.

In part due to a suggestion by Bloembergen3 for an infrared quantum counter(IRQC) based on what we would now call Excited State Absorption (ESA), aconsiderable amount of work was carried out in this direction over the followingdecades.

A number of upconversion processes identified at that time and their respectivepump power density normalized efficiencies 4 are shown on the following figure.

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Figure 2 Upconversion processes and a rough estimate of their power density normalized efficiencies perpump power density following Kushida and Tamatani, and Auzel.

These normalized efficiencies were defined as where η = Pe / Pp and Pe

is the high frequency power, and S the interaction surface, were essentially estimatedfrom experimental data on bulk samples, available around 1970 by the authors ofreference 10, without taking into account the sample lengths. What was surprising tomany people was the extremely high relative efficiency of the double energy transferprocesses, as first pointed out by Auzel5. Being French, he called this process Additionde Photons par Transfert d'Energie and both the acronym APTE, and the name Auzeleffect, were extensively used in the literature over the following years. Today, energytransfer is the commonly used name for this process.

The atomic, as opposed to dielectric, phenomena most highly investigated wereESA and energy transfer processes in which several ions were involved. In both casesthe active materials of prediliction were Rare Earth (RE) ions in various hosts. Whilethe original motivations for upconversion of that ancient era have been adequatelyaddressed by other techniques, the scientific work has proven to be invaluable with theadvent of guided wave optics, leading to the the fiber amplifiers, lasers, and sensors thatare of current active interest.

Since the performance of many of these devices is based on the particularities ofthe RE ions we will summarize the essential spectroscopic aspects of these ions beforeoutlining some pertinent examples of the actual upconversion processes . .

Rare earth ion spectraThe rare earths play an extremely important role as active elements in the optical

region of the spectra. Since they are the essential actors in the atomic upconversionscenarios we will develop, we present here a brief outline of their key properties.

The essential characteristics of these ions in various hosts consist of a multitude ofrelatively fine absorption and emision lines. This is due to the fact that the optically

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active electrons in the partially filled 4f shell are inside the filled 5s 2

indicated heuristically on the following figure.and 5p 6 shells as

They are, therefore, partially fielded from the fields in the their immediateenvironment and their emission and absorption lines are an order of magnitude finerthan those of transition metal ions, for example as can be seen on the following figure.

Figure 3. Highly imaginative heuristic schematization of the outer orbitales of Cr3 + and Nd .3 +

These emission and absorption lines arise from transitions between the levels in theground states of the elements in their stable +3 ionisation state. These levels areconventionally labelled as 2 S + 1 L J where S, L, and J represent the spin quantumnumber, the orbital angular momentum quantum number, and, the total angularmomentum quantum number respectively. The orbital angular momentum quantumnumber L is historically designated by the letters S,P,D,F,G,H,I,...which correspond toL= 0,1,2,3,4,5,6,7.... respectively. Within a given LSJ level the local field raises the2J+ 1 degeneracy leading to a number of Stark levels, which depends on the symmetryproperties of the field as well as J. Even more important is the fact that the field breaksthe inversion symmetry of the ion’s environment which permits transitions between

Figure 4. Fluorescence bands of transition metals (Ti, Cr) and rare-earth (Nd, Er) ions.

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Figure 5. Energy level for two LSJ manifolds of a rare-earth ion in a crystal field. Each manifold is splitinto various Stark levels

It is important to note that the splitting of the individual manifolds is on the order ofkT at 300°K, which corresponds to 210 cm- 1 . This means that these levels will rapidlythermalize among themselves allowing us to consider the manifold as a singlehomogeneous band. It is this rapid thermalization that depopulates the upper Stark

levels of the 4I 15/2 level of Er permitting the 1.48 µ m pumping of the ubiquitous 1.54

µ m amplifier. This rapid thermalization process will also serve as the basis for atemperature sensor we shall describe later.

A last preliminary, but essential, subject to be treated before entering the discussionof the excitation processes themselves, are the phonon spectra associated with thepossible host materials. This will be outlined in the following section.

Role of the host material phonon spectraWhen rare earths are used to dope a given material, the excitation process, and

hence the useful lasing levels, will depend on an essential material parameter: thematerial’s phonon spectrum. In order to demonstrate this we will begin by discussingthe system that currently appears to be the most promising for the realisation of a bluefiber laser: thullium doped ZBLAN fiber. ZBLAN is an acronym for ZrF4 - BaF 2 -LaF 3 - A1F3 - NaF, a glass almost accidentally discovered at the University of Rennesin 1974 6 . While silica fibers have led to very performant lasers in the infrared and nearinfrared, it is the ZBLAN fibers that have permitted the realization of most of theupconversion based visible lasers. The reason for this lies in the phonon spectrum ofthe two materials. ZBLAN, a so-called “soft” glass, has a much lower average phonon

energy than silica. As a result, if a rare-earth LSJ manifold is less than about 4500 cm -1

above the next lower energy, manifold, in a silica host, non-radiative phonon emissionprocesses can depopulate the upper manifold. For the ZBLAN host the the limitingenergy separation neeeded to avoid this is only about 3000 cm- 1 . This is evident fromthe multiphoton decay rate versus energy gap figure shown in the following figure7. .

metastabilty of many of the levels.

levels that would have been parity forbidden in a symmetric field. The relativeweakness of the oscillator strengths for such transitions partially accounts for the

In a typical rare earth the levels will consist of LSJ manifolds having a total width

on the order of several hundred cm-1 sparated from one another by several thousand

cm -1 , with a typical pair being shown in the following figure.

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Figure 6. Phonon induced decay rate as a function of the energy gap between levels in silica and ZBLANhosts.

As a result of this, rare-earths in ZBLAN hosts exhibit many more metastable statesthan in silica hosts. These states will permit both lasing and the possibility of ESA orenergy transfer to higher level. This is the essential reason for the widespread use of theZBLAN host for upconversion lasers. Having introduced the necessary backgroundmaterial, we will go on to discuss the main excitation and de-excitation processes in thedoped fiber materials.

Excited State Absorption (ESA) and Energy Transfer (ET) processesWe will now outline the basic excitation and de-excitation processes that will

determine the population dynamics of the systems of interest for upconversion: ExcitedState Absorption (ESA) and Energy Transfer (ET).

For ESA the basic effect is schematized on the following figure.

Figure 7. Schematic of a typical ESA upconversion process.

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process an ion absorbs a pump photon and transfers the energy to a similar level in aFigure 8. In this ETneighbouring ion

In this processs a pump photon has been absorbed exciting the ET ion to the level1. The ET ion then transfers it’s excitation to the neighbouring ion’s level 3,simultaneously relaxing to it’s ground state. The lasing ion relaxes non-radiatively to theupper lasing level, 2. If there is a second excited ET ion in the vicinity of the lasing ion,and a corresponding excited state in the lasing ion having a level whose energy isaround the transfer energy above the state 2, a double energy transfer (see fig. 2a) cantake place leading, eventually to upconversion.

Another multiple ion phenomena which occurs is that of quenching. This process isschematized in the following figure.

Energy transfer processes can occur between identical ions, or other ions havingsimilar energy levels, which are in close proximity to one another. In the followingfigure we schematize an ET process which enables an ion to non-radiatively transferabsorbed energy to a neighbouring ion.

As indicated on the figure, a first pump photon is absorbed leading to an excitationof level 2 which non-radiatively decays rapidly to level 1, a metastable level. The atomin level 1 then absorbs a second pump photon to attain level 4, from which it rapidlydecays non-radiatively to level 3. Upconversion occurs when level 3 is radiativelycoupled to the ground state.since, despite the energy losses due to the non-radiativeprocesses, the 3-1 fluorescence has a higher frequency than the pump photons. This isthe simplest possible example of ESA. Clearly the process does not have to begin orterminate on the ground state, and higher order processes are possible, but figure 7expresses the essential idea. It is also the process which leads to green fluorescence withEr doped silica fibers when pumping at 800 nm, and which we shall describe in moredetail later.

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Figure 9. Quenching of an excited state due to ET to a neighbouring ion.

In this example, typical for a four-level laser ion such as Nd , the pump photon isabsorbed to excite level 3 which rapidly relaxes non-radiatively to the upper lasinglevel, 2. In the presence of a quenching ion the excitation can be trasferred to that ionthereby reducing the population of the excited state and lowering the pumpingefficiency. Such transfers do not always have deleterious effects. They can be used toadvantage, for example to depopulate a relatively long lived terminal laser transitionlevel, or, as we shall show, to realize useful technological devices such as the molecularbeacon we shall describe in the application section.

Having identified and described most of the essential phenomena involved in theexcitation and de-excitation of rare-earth ions we are, finally, in a posi tion to describesome fiber upconversion lasers, which we shall do in the following section.

Examples of upconversion lasers

ternperature versions previously , the paper of Allain et alWhile upconversion lasers had been demonstrated in pulsed and CW low-

8 on the first CW roomtemperature upconversion fiber laser was a major breakthrough. This laser was based ona holmium doped (1200 ppm by weight) ZBLAN monomode fiber of around 1 mlength. The fiber laser was pumped by a krypton laser operating at 647 nm and had athreshold of about 150 mw. It produced up to 10 mw of power at 550 nm and exhibiteda slope efficiency of approximately 20%. While this was a very impressiveaccomplishment, more recent work has centered on the use of thulium doped ZBLANand has led to efficient upconversion fiber lasers operating at 480 nm and we shalldescribe these lasers in more detail

The first CW room temperature blue upconversion laser was reported by Grubb etal9. The fiber used was a 1000 ppm Tm 3+ doped 2 m long ZBLAN fibre with anumerical aperure of 0.21 and an LP11 cut-off wavelength of 800 nm. The pump was aNd:YAG laser operating on three closely spaced lines at 1112, 1116, and 1123 nm. Thethree step ESA upconversion pumping scheme used is shown on the following figure.

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Figure 10. Three photon ESA pumping scheme for 650 and 480 nm laser action in Tm3+ doped ZBLANfibers.

The first version of this laser had a threshold of 46 mw of coupled pump power anda 32% slope effiicency with respect to absorbed pump power. The maximum outputwas 56 mw.

More recently, output powers of up to 230 mW of power were demonstrated with a1.6 W Nd:YAG pump10 and 106 mw of 480 nm light has been obtained using two diodelasers as a pump for a nearly identical fiber 11 . The threshold for the latter device (anincident power threshold of 80 mw was cited) and slope efficiency were very close tothose of the configuration in reference 9. We believe this to be the most interestingfrequency conversion schemes realized to date with diode laser pumping and it certainlygives a glimpse of the type of devices we might see appearing commercially in the nearfuture.

Before leaving the discussion of such lasers it is worth noting that in view of thecomplexity of the rare-earth spectra and the various means of obtaining upconversion, itis still possible that other combinations of ESA and/or ET will lead to viable lasers. Oneextensively studied possible material is Pr doped and co-doped ZBLAN which has ledto dual wavelength pumped lasers 12 based on ESA, and single wavelength pumping ofNd:Pr doped fiber 13 , which could also prove to be useful. Nevertheless, since theperformance of these devices is not equal to the Tm doped systems we shall not discussthem any further. They are simply cited as examples of the multiplicity of posible pathsto upconversion fiber lasers in general.

Nevertheless, while these lasers allow quite acceptable nm tolerances for the pumpwavelength they suffer from the extremely high current prices of the ZBLAN fibers. Anorder of magnitude reduction will probably be necessary to make such laserscommercially viable.

Having concluded this discussion of upconversion fiber lasers, we shall now go onto describe the state of the art of short wavelength diode lasers.

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SHORT WAVELENGTH DIODE LASERS

In view of the applications for short-wavelength lasers, it might seem surprisingthat the first demonstration of a blue-green injection laser came nearly 30 years after thefirst such lasers realised in the GaAs system. While lasing had been demonstrated in thewide-gap II-VI semiconductors using optical and e-beam pumping the report of Haaseet al 14 in 1991 of injection pumped lasing at 490 nm in ZnCdSe-Zn(S)Se waspractically a surprise. Research workers had clearly identified the most significantproblem as the inability to realize highly p-doped materials. This was overcome throughthe development of nitrogen doping using an RF (radio-frequency) plasma source

which permitted attaining the 1018 cm -3 range of dopant concentration. This allowedoperation of a gain guided laser having the structure shown in the following figure.

Figure 11. Highly simplified schematic of a ZnSe injection laser structure.

The laser functionned in a pulsed mode with a threshold of around 70 mA at atemperature of 77 K.

This report led to a considerable amount of research activity throughout the world,leading to a demonstration of CW operation at room temperature by a group at Sony in1993 15 . Currently, CW operation of such lasers has attained 100 hour levels at roomtemperature 16 . This limitation is apparently imposed by the formation of nonradiativecenters during current injection. The degradation mechanism apparently originates atpre-existing defects which are created when the first layers of ZnS nucleate on theGaAs substrate. Incorrectly aligned atoms form stacking faults which expand duringsubsequent crystal growth to become so-called dark-line defects which then propagateto in or near the active layer. In the active quantum well region these faults present sitesfor non-radiative recombination of electrons and holes which leads to a release ofaround 2.5 eV of energy, a level capable of creating further defects, and hence, rapiddegradation. A major part of the research on this system consists of characterizing, inthe hope of eventually eliminating, the defect nucleation sites.

However progress at this time seems to be more rapid using another class ofmateials based on GaN and it's alloys. It was only in 1994 that the extensive research on

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the Al-Ga-In-N alloy family led to dramatic results- the commercialization by Nichia, aJapanese company, of extremely bright LED's emitting in the green to ultraviolet range.This was followed by the announcement by the same group of injection laser operationunder pulsed conditions in late 199517 and a CW room temperature version emitting1.5 mw per facet around 420 nm and lasting 35 hours, with operating currents between100 to 280 mA, in 1996 18.

The type of structure used is highly schematically shown in the following figure.

Figure 12. Highly simplified schematic of a GaN injection laser.

Obvious differences with other laser structures include the fact that the substrate isan insulating crystal. This leads to the necessity of depositing the n electrode at the sideof the structure and adapting the buffer layer to the sapphire structure.

While these lasers still have threshold current densities on the order of 3 kA/cm2,an order of magnitude greater than GaAs lasers, the defect structures, which are provingdifficult to eliminate in the II-VI lasers, appear to be far less deleterious in the nitrides.The technology is continuing to develop, with a strong push coming from the displayapplications for which GaN LED arrays are already commercially available. Thiscombination of an existing market (display) as well as the promise of blue lasers seemsto give a strong advantage to the GaN family in the ongoing race towards reliable shortwavelength diode sources.

The parametric process that has been most developped for the practical generationof CW blue light is Second Harmonic Generation (SHG). For the power levels we areconsidering, on the order of 10 mW, the Quasi-Phase-Matched (QPM) waveguideconfiguration, realized in lithium niobate or tantalate, described in the lectures ofProfessors Stegeman and Fejer at this institute, appears to be the most promising. Sincethe underlying phenomena and theory have been developped in those courses we shallonly present the briefest of outlines here before proceeding to a comparison with theother techniques. The basic QPM waveguide configuration is shown on the followingfigure.

PARAMETRIC SOURCES

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Figure 13. Basic waveguide QPM configuration for SHG.

The essential idea is that the periodically inversed domains, and hence sign-

inverted nonlinear coefficients, correct the π dephasing developped between thenonlinear polarization and the propagating wave at the harmonic frequency. This cleveridea, first proposed by the group of Bloembergen19 was demonstrated in waveguideform in 1989 by groups at Stanford 20 and the Institute of Optical Resarch inStockholm 21. Extensive analyses of the tolerances on periodicity, waveguide regularity,pump wavelength spectrum, and temperature variations have been carried out by the

two groups 22,23 . In reference 23 the authors presented a very useful formula giving thefollowing first-order Taylor series expansion approximation of the full width at halfmaximum (FWHM) acceptance bandwith of the SHG efficiency for an arbitrary

parameters variation:

where ξ is the arbitrary parameter and ∆k = k 2 – 2k1 – K m with K m the inversion

period for the mth order QPM being used For the frequency doubling of 860 nm lightthis indicates that the FWHM tolerances for a 1 cm long sample in lithium niobate areapproximately those given in the following table.

parameter δ λ (nm) δT(°K)

.07 1.4

δυz (mrad)

11

δυo(mrad)10

Table 1 FWHM acceptance bandwidth for doubling of aan 850 nm source with a 1 cm long QPM lithiumniobate saample. δυ z and δυo represent angular deviations of the sample cut around the z and ordinaryaxes respectively.

While such tolerances appear rather draconian, a recent development in the field ofdiode laser pumps should allow satisfying these conditions. The device in question is anelectrically tunable, high power, single frequency diode laser fabricated by SDL24 . Thestucture consists of a Distributed Bragg Reflector (DBR) configuration with separateelectrodes allowing independant current injection in the gain and DBR regions.Injection of current in the DBR region locally heats the grating leading to an increase ofthe emittted wavelength. The device can be tuned over a 10 nm range in 0.08 nm steps,the steps being due to the longitudinal mode separation in the cavity. Such devices have

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already demonstrated over 20,000 hour lifetimes, in ongoing tests, while emitting 200mW continuously.

That such a device, in conjunction with a lithium niobate QPM doubler, is a viablesource for optical disk recording is underscored by the recent (June 9, 1997)announcement by Panasonic, of the realization of a complete recording head based onthis technology. The source, using an X-cut MgO doped periodically poled lithiumniobate crystal 25 with an overlay of Nb2O5 provides 15 mW of power at 425 nmwithout demonstrating photorefractive effects. The direct modulation permitted by thesource allowed the demonstration of the reading and writing of optical discscorresponding to a 15 GB/face capacity, roughly a 4-fold improvement over existingtechnology.

In conclusion, in view of the performance of the system reported by Panasonic, itlooks like near IR diode laser pumped QPM parametric generators could become thefirst example of a nonlinear optical mass consumer application. If this should come topass, the repercussions would be enormous for the field of nonlinear optics as themarket would generate the means for an enormous expansion of related research anddevelopment.

OTHER EXAMPLES OF APPLICATIONS

In this section we shall describe three examples of techniques and devices based onthe various excitation and de-excitation phenomena we have presented. To underlinethe sort of pleasant surprises science can afford us we will begin with two examplesbased on what are usually considered to be nefast phenomena. The first is a fiber optictemperature sensor, the second is a molecular beacon technique used for geneticidentification. We shall conclude with a description of a fluorescent waveguideexamination technique.

An ESA based auto-referenced silica fiber optical temperature sensorIn this section we will outline, as an example, a highly performant sensor

prototype 26 , based on ESA in Er doped silica fiber, that will illustrate some of theparticularities one can encounter in rare-earth spectroscopy. The ESA in question is thegreen fluorescence observed when pumping erbium doped fibers with 800 nm light, andis the essential reason that such pumps cannot be used for Er doped fiber amplifiers.The ESA phenomena used is schematically shown in fig 14.

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Figure 14. The 550 nm green fluorescence due to ESA of 800 nm light

A closer examination shows that the upper level of the green fluorescence is in facttwo levels that are thermally quasi-isolated from the rest of the atomic system. Thesetwo levels, which thermalize rapidly between themselves, give rise to two overlappingcomponents of green fluorescence as shown in the following figure.

Figure 15. Two component green fluorescence due to Er ESA

A measurement of the intensity ratio of the two lines allows determining thetemperature. An important point to note is that the measurement is based on an intensityratio eliminating the need for calibration. Another important point to note, as shown inthe figure inset, is that the intensity coming from the small (3%) population of the upperexcited state level at ambiant temperature is compensated by its 20 times largeremission cross section, resulting in roughly equal total fluorescence for the twocomponents. It is this gift of nature that allows the temperature sensor to have an

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extremely large dynamic range.A schematic of the experiment carried out to verify this,and the experimental results, are shown in the following figures.

Figure 16. Experimental schematic of the ESA based temperature sensor.

In this set-up the doped fiber, typically 50cm long, was spliced to a 20 meter lengthof standard telecommunication fiber. Power levels in the 50 to 200 mW range, at 850nm were supplied by the Ti:saphire laser, which could be replaced by commerciallyavailable diode lasers.

Figure 17. Experimental results for the ESA based fiber temperature sensor.

This sensor can also be excited by an energy transfer process using Yb-Er codopedfibers and a 980 nm pump and quasi-distributed, rather than point sensors, can berealized by analyzing the ratio of 1130 and 1240 nm lines which also appear whenpumping at 800 nm27 , but a discussion of this configuration would take us too far afieldfrom the subject of upconversion.

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Molecular beaconsThe molecular beacon is another application of blue light that is based on an

apparently nefast phenomenon: fluorescence quenching by a neighbouring ion. Thisphenomenon is one in which an excited atom, capable of fluorescing, transfers it’senergy non-radiatively, to another ion, which then decays, also nonradiatively, to it’sground state. This sort of quenching is common in the rare-earths and is the sort ofprocess that limits the useful doping levels to a few per cent. However, such aphenomenon has been put to good use in a "device" for genetic identification: themolecular beacon28 . The structure of such a beacon is shown in the following figure.

Figure 18. A molecular beacon consisting of a single strand of DNA folded back upon itself with afluorescent (F) and quencher (Q) molecule attached at the extremity.

The letters A,C,G, and T represent the adenine, cytosine, guanine,and thyminemolecules attached to the DNA strand. Adenine links to thymine and guanine tocytosine to form the famous double helix. In the beacon extremity the complementarymolecules are used to close the structure and fluorescent and quencher molecules areattached to opposite sides of the strand. Commonly used molecules are fluorescin forthe fluorophore and DABCYL for the quenching molecule. As long as the extremity ofthe structure is closed, the proximity of the quenching molecule will inhibitfluorescence when the beacon is illuminated with blue light. When such a structure isplaced in a solution containing single strands of DNA it will move about due to thermalfluctuations, and attach itself to the normal, much longer DNA strands if it finds acomplementarily coded segment. Since the longer segment is much more rigid,attachment will force the closed end of the beacon open, separating the F and Qmolecules, allowing fluorescence, and hence localisation of the researched segment.Such beacons allow the discrimination between segments that differ from one anotherby only a single nucleotide29 . This technique is expected to play a major role in futureresearch on gene sequence identification and manipulation.

Having concluded our two example of good uses of usually nefast phenomena, wewill go on to describe our last example of blue light or upconversion processes,waveguide examination by fluorescence.

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Waveguide examination by fluorescence

While calculations give beautiful graphics concerning the evolution of fields inintegrated optical devices it is rather difficult to directly observe such fields. What oneusually observes when looking at a waveguide are scattering centers which do not yielda reliable image of the fields. A technique that has been demonstrated for overcomingthis is to coat the guide with a low index plastic film doped with a fluorescent materialand observe the emitted fluorescence induced by the evanescent wave. Thisfluorescence, which is directly proportional to the local guided wave power, gives anexcellent map of the field30. A schematic of the experimental setup is shown on thefollowing figure.

Figure 19. Schematic of the waveguide observation by fluorescence set-up.

In the first experiments demonstrating this effect the guide was realized usingpolyphenylsiloxane (PPS), an electron resist material that allowed direct e-beam writingof guides 31. In these experiments a structure consisted of an 800 nm thick PPS guidewith an index of 1.565 at 632 nm, covered with a 300 nm polymethylmethacrylate(PMMA) film with an index of 1.49, doped with 6% by weight of Rhodamine B. Thismolecule exhibits an interesting anti-Stokes fluorescence line (an upconversion processwe have not discussed) at 595 nm when pumped with a He-Ne laser. In the structuredescribed approximately 10% of the guided light propagates in the evanescent fieldwithin the doped film and 3 mW of guided light leads to a fluorescence power on the

order of 10 -10 W/µm 2 emitted from the guide surface which is a detectable powerlevel. This allowed, therefore, field maps to be made in varying structures such ascouplers, Y-junctions and curved guides. In order to observe the fields propagating inlithium niobate and tantalate guides, which, due to their higher indices, only have on theorder of 1% of the guided light in the evanescent field, it was necessary to dope withRhodamine 6G and pump with a blue line of an argon laser. This yielded equivalentfluorescent power and allowed the observation of fields propagating in these materials.A chosen few have even seen a film made with this technique that demonstrates theobservation of switching in electrically controlled lithium niobate directionalcouplers 32 .

Since this direct observation of switching is as close to the main thrust area of thisinstitiute as we shall get in this contribution, we choose to draw the curtain here.

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CONCLUSION

In this course we have described the current state of the art concerning threetechnologies being used to realize low-power coherent short wavelength sources:upconversion fiber lasers, diode lasers, and second harmonic generation. We have alsogone at some length into describing three rather original applications for such sourcesand their technology: a fiber optic temperature sensor based on upconversion, a DNArecognition technique, and a waveguide examination technique, both of thelatter usingshort wavelength induced fluoresence. We hope to have shown that this is a rapidlyevolving field, involving pluridisciplinary interactions, that has led to important results,and will, in the future continue to provide, as it has in the past, pleasant surprises.

The author wishes to acknowledge fruitful discussions with P. Gibart concerningGaN and ZnS lasers, Jean-Pierre Lehureau, concerning compact disc technology, AlbertLibchaber and Gregoire Bonnet, concerning molecular beacons, Jean-Paul Pocholle,Gerard Monnom and Bernard Dussardier concerning rare-earth doped fibers, andArnaud Grisard, Marc De Micheli and Pascal Baldi concerning parametric sources. Theauthor alone is totally responsable for any errors introduced as well as the opinionsoffered.

1 E. Snitzer, "Optical maser action of Nd3+ in barium crown glass", Phys. Rev. Letts. 7, p 444, (1961)2 J. R. Armitrage, Optical Fiber Lasers and Amplifiers, ed P.W. France, p14- 49, CRC Press (1991)3 N . B l o e m b e r g e n , P h y s . R e v . L e t t s . 2 , 8 4 ( 1 9 5 9 )

4. T. Kushida and M. Tamatani, "Conversion of infrared into visible light", supplement to J. Japan Soc.Appl. Phys. 39, pp 241-247, (1970), F. Auzel, "Materials and devices using double-pumpedphosphors with energy transfer", Proc. I.E.E.E. 61, pp 758-786, (1973)

5. F. Auzel, "Compteur quantique par transfert d'énergie enttre deux ions de terres rares dans un tungstatemixte et dans un verre", C.R. Acad. Sci.262, pp 1016-1019, (1966)

6 M. Poulai, M. Poulain, and J. Lucas, Mat. Res. Bull. 10, 243 (1975)7 Zheng H. and Gan F. Chinese Phys. 6,978 (1986)8 J.Y. Allain, M. Monerie, and H. Poignant, " Room temperature CW tunable green upconversion

holmium fibre laser", Electron. Lett. 26,261 (1990)9 S.G. Grubb, K.W. Bennett, R.S. Cannon and W.F. Humer, CW room-temperature blue upconversion

fibre laser", Electron. Lett. 28, 1243 (1992)10 R. Paschetta, N. Moore, W.A. Clarkson, A.C. Tropper, D.C. Hanna, G. Mazé, 230 mWof blue light

from thulium:ZBLAN upconversion fiber laser, Proceedings CEO 97 paper CTuG3 p 80 (1997).11 S. Sanders, R.G. Waarts, D.G. Mehuys, and F.D. Welch, "Laser diode pumped 106 mW blue

upconversion laser", Apl. Phys. Lett. 67, p 1815, (1995).12 R.G. Smart, D.C. Hanna, A.C. Tropper, S.T. Davey, S.F. Carter, and D. Szebesta, Electron. Lett. 27,

13

14. M.A. Haase, J. Qui, J.M. Depuyt, and H. Cheng, Appl. Phys. Lett. 59, p1272, 1991151617

18

1307 (1991).S.C. Goh, R. Pattie, C. Byrne, and D. Coulson, "Blue and red laser action in Nd3+: Pr3+ co-doped

fluorozirconate glass", Appl. Phys. Lett., 67, 768 (1995).

N. Nakayama, et al, Electron. Lett. 2, p. 2194, 1993S. Tanaguchi, etal, Electron. Lett. 32, 552, 1996. Nakamura, S., et al , InGaN multi-quantum-well structure laser diodes, Jpn. J. Appl. Phys. 35, L74

(1996)Nakamura, S. Characteristics of InGaN multi-quantum-well structure laser diodes, Mater. Res. Sec.

Soc. 449, : 1135 (1996)

19 J.A. Arrmstrong, N. Bloembergen, J. Ducuing, and P.S. Pershan, Interactins between light waves in anonlinear dielectric, Phys. Rev. 127, 1918 (1962)

20 E.J. Lim, M.M. Fejer, R.L. Byer, and W.J. Koslovsky, Blue light generation by frequency doubling ina periodically poled lithium niobate channel waveguide, Electron. Lett.25 pp 731-732 (1989).

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21 J. Webjorn, F. Laurell, and G. Arvidsson, Blue light generated by frequency doubling of laser diodelight in a lithium niobate channel waveguide, IEEE Photon. Technol. Lett 1 pp 316-318 (1989)

22 S. Helmfrid and G. Arvidsson, Influence of randomly varying domain lengths and nonuniformeffective index on second-harmonic-generation in quasi-phase-matched waveguides, J. Opt. Soc.Am. B. 8,797 (1991)

23 M.M. Fejer, G.A. Magel, D.H. Jundt, and R.L. Byer, Quasi-phase-matched second harmonicgeneration tuning and tolerances, IEEE J. Quantum Electron. 28,2631 (1992)

24 V.N. Gulgazov, H. Zhao, D. Nam, J.S. Major, and T. Koch, Tunable high-power AlGaAs distributedBragg reflector laser diodes, Electron. Lett. 33, 58 (1997)

25 K. Mizuuchi, K. Yamamoto and M. Kato, Harmonic blue light generation in X-cut Mg0:LiNbO3waveguide, Electron. Lett. 33, 806, (1997)

26 E. Maurice, G. Monnom, B. Dussardier, A. Saissy, D.B. Ostrowsky, and G.W. Baxter, Erbium-dopedsilica fibers for intrinsic fiber-optic temperature sensors, Appl. Optics 34, p 8019, (1995)

27 E. Maurice, G. Monnom, D.B. Ostrowsky, and G.W. Baxter, 1.2-µm transitions in erbium-dopedfibers: the possibility of quasi-distributed temperature sensors, Appl. Optics, 34, p 4196, ( 1995)

28 S. Tyagi and F;R; Kramer, Nature Biotechnol, 14,303 (1996)29 G. Bonnet, S. Tyagi, F.R. Kramer, and A. Libchaber, Molecular beacons for probing information in

DNA, to be published30 D.B. Ostrowsky and A.M. Roy, Visualisation de la propagation dans un guide d'onde optique par

fluorescence anti-Stokes, Revue tech. Thomson-CSF, 6,973 (1974)31 D.B. Ostrowsky, M. Papuchon, A.M. Roy, et J. Trotel, Electron beam fabrication using an electron

sensible film, App. Opt. 13 , p, 636 (1974).32 M. Papuchon, B. Puech, C. Puech, and D.B. Ostrowsky, A movie on the visualization by fluorescence

of the electrically controlled directional coupler, Paper Tu A2, Proceedings of the Topical Meetingon Integrated and Guided Wave Optics, Salt Lake City, (1978)

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ARTIFICIAL MESOSCOPIC MATERIALS FOR NONLINEAR OPTICS

C. Flytzanis

Laboratoire d’Optique Quantique,Ecole Polytechnique91128 Palaiseau cédex, France

INTRODUCTION

The mesoscopic materials which have dimensions intermediate between those ofthe bulk material and its constituent molecular units are becoming an important andmultifaceted issue in materials science and in a wide range of important technologicalapplications. Whenever the size of such systems becomes comparable to a characteristicphysical parameter with a dimension of length these mesoscopic particles and theirensembles reveal properties which can be markedly different from those of the bulk aswell as of its constituent molecular units. In certain classes of composites formed withmesoscopic particles of cystalline materials like transparent dielectrics, conjugatedpolymers, semiconductors or metals this is strikingly revealed in their optical andspectral features and a fortiori in their nonlinear optical properties. The characteristiclengths we have in mind here are such as the optical wavelength, the electrondelocalization, the Bohr radius or the electron mean free path.

The observed size dependence of the optical properties of such mesoscopicsystems can be traced to the change of the confinement regime that the photons orelectrons undergo as their size approaches any of these characteristic lengths. This leadsto a modification of the relevant density of states and the appearance of morphologicalresonances that profoundly affect both the magnitude and dynamics of the opticalnonlinearities. It also provides a way of artificially controlling these nonlinearities bycontrolling the size and interface of these mesoscopic systems during the fabricationprocedure.

Here we shall present a succinct discussion of the optical nonlinearities inmesoscopic systems and the way they are affected by the confinement. The emphasiswill be on the so called photoinduced nonlinearities in particular the optical Kerrnonlinearity both because it is bound to play an important role in all-optical beamreshaping and control and also because the confinement has its strongest impact there

Beam Shaping and Control with Nonlinear Optics427Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

since it is multiresonantly enhanced. After reviewing some general aspects of thesenonlinearities and the mesoscopic systems in the following section we present a generaldiscussion of the impact of the electromagnetic or photon confinement on the opticalnonlinearities of the mesoscopic dielectric materials and subsequently that of theelectron confinement which is manifested either as dielectric or quantum confinementand is illustrated with the metal and semiconductor nanocrystals respectively.

OPTICAL NONLINEARITIES AND MESOSCOPIC SYSTEMS.

Efficiency of nonlinear optical effects

The nonlinear optical effects in a medium have their origin in the inducedpolarization sources

(1,2)nonlinear in the field amplitude as they appear in the

development

(1)

and are classified according to the power of the field amplitudes and frequencies,wavevector and polarization vector configurations ; the above development is valid aslong as <Ec and χ (n) ~ 1/En

c where Ec is a cohesive electric field for the outer shellelectrons in the medium. The efficiency of the resulting nonlinear effects is certainlydetermined by the behavior(3) of the effective nonlinear susceptibilities χ(n) at the chosenconfiguration but also by geometrical and temporal features connected with theevolution of the fields amplitudes which in the simplest stationary plane wave regime inthe slow varying envelope approximation are obtained by the nonlinear one-dimensionalpropagation equation(1.2)

(2)

where is the appropriate nonlinear polarization source at a frequency ω which storesand transfers energy to the field of amplitude A = ê ê is the unity polarizationvector, λ the optical wavelength and n0 the refractive index all at frequency ω ;∆k = is the wavevector mismatch (or equivalently ∆kz is the phase mismatch)between the wavevector of the induced nonlinear polarization and the wave vectorthat can propagate in the medium at frequency ω for the chosen configuration namelyk = ω n/c. This wavevector mismatch is a manifestation of nonlocality and introducessevere selectivity in the growth of the nonlinear process since it limits its growth over adistance = π/∆k, the coherence length. If the phase mismatch can be compensatednaturally by exploiting the linear(4) or circular (5) birefringence of the materials orartificially in the so called quasi-phase matched configuration it is quite clear from (2)that the efficiency of the nonlinear optical process will be enhanced either by increasingthe optical nonlinearity of the medium as defined in (1) or by increasing the interactionlength as defined by (2) when ∆k = 0 still keeping the geometrical dimensions of thematerial small.

A key role in all-optical beam reshaping and control is played by the optical Kerrnonlinearity which is related to the cubic nonlinear optical polarization(2,6)

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(3)

where stands for an effective third order susceptibility and the frequencies ω' and ωcan be different, or ω' can be equal to ω or zero. Note that this effective susceptibility

besides the direct contribution also contains indirect(3,7) ones because of retardation(or cascading effects) and may consequently depend on the wavectors as well. Thegeneral case ω ≠ ω' corresponds to the modification of the optical characteristics of abeam at frequency ω by another beam at frequency ω ' while the other two cases ω' = ωand ω' = 0, correspond to the beam self action and static Kerr effect (or Kerrelectromodulation of a beam) respectively. In all these cases one can define an effectivecomplex refractive index

(4)

where I = (n and n20c / 2π) = for the case where ω = ω'and similarly for ω ≠ ω'. The real part of n2 (or χ(3) ) affects the dispersion while theimaginary part of χ(3) can be associated with additional two photon losses or gain in themedium. Along with the magnitude and phase of n2 in the stationary regime a key rolein the envisaged applications of the optical Kerr effect is also played by its temporalevolution(2,6) of n2 which for many purposes can be modeled by the Debye equation

or(5)

(6)

where τ is a decay time related to energy or population relaxation. In certain cases oneintroduces figures of merits to assess the materials as regards the optical Kerr effect andone such figure of merit is

(7)

where α ω is the absorption coefficient at the operating frequency ω. It is evident from itsdefinition that this refers to the nonlinearity per photoexcited electron. Other figures ofmerit can also be defined according to the application one has in mind but in general oneshould be cautious regarding the conclusions that can be drawn from the figures of meritin general. The key issue of nonlinear materials science is clearly to improve theefficiency of optical nonlinearities taking into account several other aspects like opticallosses, photochemical stability, processing, and interfacing or doping, miniaturizationand other important fabrication and stability constraints and these aspects cannot beeasily incorporated in such figures of merit.

Mesoscopic systems. Composites

The efficiency of a large class of nonlinear optical effects can be enhanced byexploiting either the photon or electron confinements. As will become shortly evidentthe two confinements cannot be simultaneously implemented in the same material in theoptical range as they address material aspects incompatible with each other there. The

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first one is achieved by spatially confining(8,9) the interacting optical beams in guides orcavities (resonators) whose minimal dimension is of the order of the optical wavelengthλ so that the electromagnetic modes can be resonantly accommodated in space andthrough multiple reflections effectively increase their interaction path ; this is theelectromagnetic confinement and can be achieved in highly transparent nonlinear opticalmedia where the nonlinearities involving very localized electrons, even for the mostfavorable cases are weak but the absorption losses can be kept minimal and the beamcoherence is preserved over long propagation distances. The second one consists inenhancing the optical nonlinearity of materials with very delocalized valence electrons,like in metals, semiconductors or conjugated polymers, by artificially confining thevalence electrons in regions much shorter than their natural delocalization length in thebulk, which can extend over many units cells or even infinity ; this is the case of theelectron confinement(10) and its most conspicuous feature is the appearance of discreteoptical resonances whose position, oscillator strength and dynamics depend on theextension of the artificial electron confinement and hence can be externally modified tomeet certain requirements. Referring to our previous definition of the cohesive field E c

in connection with (1) this amounts reducing E without substantially altering thec

cohesion of the material.(11)

concerns dimensions of the order of the optical wavelength λ namely a few hundreds(10)

dimensions of the order of the electron Bohr radius or delocalization length namely afew nanometers up to several tens nanometers. These are the dimensions of themesoscopic systems, organic or inorganic, that are increasingly being studied in modernscience and technology. Besides their intrinsic features due to their finite mesoscopicsize their interface with the surrounding media as well as the doping are also of crucialimportance in these considerations and play a very important role in many applications.Accordingly appropriate fabrication techniques have been and are being developed toartificially grow such particles and control their size and form as well as their dopingand interfacing with the surrounding medium ; the role of the later is not simply tosupport the fine mesoscopic particles but also to endow them with certain properties thatare very important for applications.

The particles that exploit the electromagnetic or photon confinement and havesizes of the order of the wavelength λ or smaller are made(11) from high qualitytransparent amorphous dielectrics, polymeric or inorganic ones ; they can have sphericalor ellipsoidal shapes and are uniformly dispersed in a another transparent dielectricamorphous matrix, solid or liquid. Their concentration and spatial arrangement in thematrix depend on the effects one wishes to enhance. Periodic arrays of such particles,the so called colloidal crystals can also be obtained by different techniques in particularoptical ones whereby the particles are displaced and positioned by interfering opticalbeams. The fabrication of such spherical dielectric particles of well calibrated shape andsize is now well controlled and extensively used(12,13) in a wide range of applicationsranging from efficient light scatterers, high quality paints and laser microcavities toname a few. In some of these applications in addition one needs to wrap these particleswith an appropriate film or layer while in others to appropriately dope them withspecific molecular or atomic ions. Whatever the case may be the underlying material isa high optical quality amorphous dielectric with high featureless transparency regionthat can extend up to the near UV to keep absorption losses as low as possible and allowthe electromagnetic waves to undergo many reflections without losing their coherence.

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In the optical range we are interested here the electromagnetic confinement

nanometers (nm) up to micrometer (µm) while the electron confinement concerns

The particles that exploit the electron confinement(14) on the other hand and havesizes up to a few tens of nanometers are made out of crystalline materials with verydelocalized electrons such as covalent semiconductors or metals. The confinement heremodifies the electronic density of states distribution and by the same token that of theoptical oscillator strength distribution and relaxation processes; both play an essentialrole in shaping the linear and nonlinear optical properties of these materials andconsequently determine their potential use in microoptoelectronics and other areas.Another feature of much interest, recurrent to all composite materials obtained byinterfacing materials of different chemical constitutions and more or less mobileelectrons, is the interfacial charge transfer and similar processes that may take placethere and can be exploited to produce new functionality(15) artificial materials. The lateraspect clearly involves quite sophisticated chemical as well as physical considerationsbut the former, namely the confinement, can be addressed and its main consequencesfollowed up to a large extent by physical considerations regarding the electron behaviorin confined geometries. Several approaches exist for preparing such mesoscopicsystems, metal(16) or semiconductor (17) nanocrystals uniformly embedded in a transparentliquid or solid matrix.

For the case of semiconductor nanocrystals the most extensively studied(14,18) onesare the Cd(S,Se)-doped glasses which are used as sharp cut-off filters in the yellow tored part of the spectrum and are available from several glass makers. To make such asemiconductor-doped glass (SDG), one usually starts adding the semiconductorconstituents (or their oxides) to a melt of alkali-silicate glass at very high temperature,typically 1300°C. The melt is then rapidly cooled. At this point, the glass is still almostcolorless, nucleation of Cd(S,Se) particles being still at an early stage. The glass is thenannealed at a temperature lower than 500°C in order to produce a stress-free opticalquality glass. It then undergoes what is known as the striking process: it is furtherheated to a temperature of 500-700°C during which the particles grow. Using such atechnique, CuCl-, CuBr-, CdS-, CdSe- and CdTe-doped glasses have been made andalso GaP but in general this procedure has produced poor results for the III-Vcompounds because of the high volatility of the constituent ions.

The growth of the particles is usually thought to be dominated by the ripeningprocess. In the ripening stage, the volume fraction p occupied by the particles remainsconstant, the bigger particles growing at the expense of the smaller ones by atomicdiffusion through the glass matrix. Particle growth in this ripening stage has beenstudied theoretically by Lifshitz and Slezov(19). Assuming spherical particles, theiraverage radius is given by :

(8)

where s is the surface tension in the interface, D is the diffusion constant, c is a constantthat depends exponentially on the striking temperature and t is the duration of thestriking process. Particle growth being controlled by a diffusion process, this inevitablyleads to a large size dispersion. The expression of the size distribution has also beengiven by Lifshitz and Slezov. One can achieve better size control and eventually verynarrow size dispersion in colloidal solutions where in addition one can control theparticle interface by covering it with organic or inorganic layers.

Metal nanocrystals, (mainly the noble ones like Cu, Ag, Au, Pt) in glasses or othertransparent amorphous dielectrics are produced(16) by somewhat different approaches

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that schematically involve four steps : solution of noble metal salts, reduction, dialysisfor the removal of foreign ions and solidification and eventually coating andsolidification. The coating process is used in order to achieve high volumeconcentrations since otherwise uncoated particles tend to agregate at concentrationshigher than 10- 5. Colloidal solutions are similarly obtained with very narrow sizedistribution.

The most direct way of measuring the particle size and shape and the sizedistribution is electron microscopy(20), but it is a time-consuming and destructivemethod. Inelastic light scattering can also be used(21) to determine the average size of thenanoparticles. X-ray diffraction gives access to the crystalline structure as also does to

as nanocrystalssome extent the EXAFS technique. The main conclusions drawn from these structuralcharacterization studies is that the particles in the glass matrix behave(20)

with the same lattice constant and symmetry as the bulk and show facets compatiblewith their crystalline symmetry like large crystals do. For most purposes regarding theelectron confinement however they can be regarded as almost spherical. The size mayalso be obtained using small angle X-ray or neutron scattering(22) which are nondestructive tools. The absence of scattering at very small angles indicates that eachparticle is surrounded by a depletion zone. There are several indications that thesenanocrystals are defect free and in particular exempt of impurities. The averagecrystallite size can cover a wide range of values, from several micrometers down to afew nanometers, with more or less narrow size distribution depending on the preparationtechnique and the quality of the interface with the surrounding dielectric that cansubstantially differ from case to case. The later has much relevance on the "surface"features but usually for sufficiently large nanocrystals containing thousand or moreatoms corresponding to particle diameters larger than a few nanometers we may simplyassume that the interface with the surrounding dielectric is sharp and of infinite potentialheight. The volume concentration can also vary over a wide range down toconcentrations as low as 10- 6 or up as high as several percent which allows one to studysingle particle features as well as features that arise from interparticle interactions. Incertain cases ordered arrays of such particles have been achieved by physicochemical orartificial techniques, for instance self-organization on semiconductor or polymericsubstrates or artificial nanofabrication techniques involving particle or photon beams ;these are still in an experimental stage and mostly concern two dimensional arrays.

Here we shall be concerned with metal or semiconductor particles uniformly andrandomly dispersed in a glassy or other transparent dielectric with concentrations lowenough that interparticle excitation transfer can be neglected and only the electronconfinement within such a single particle embedded in the glass has to be considered.

ELECTROMAGNETIC CONFINEMENT AND NONLINEAR SCATTERING

A highly transparent amorphous dielectric has very low optical Kerr nonlinearitiesbut because of the low absorption and scattering losses there the cumulative effect of thenonlinear interactions over long distances can give rise(23) to sizable effects when thephase mismatch is not an issue. Classical examples here are those of the temporal andspatial optical solitons which are extensively discussed elsewhere in this volume.Actually it is not necessary for the light to propagate long distances to benefit from thecumulative effect of the optical interactions; indeed multiple passages over a smallspatial region of few optical wavelengths λ in size can effectively provide sufficiently

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long interaction lengths for the nonlinear optical processes to fully develop. In theoptical frequency range this can be achieved(12,24) by electromagnetic confinement in afew micrometer size spherical dielectric particles in air or eventually in anothertransparent dielectric of lower refractive index. The multiple reflections of the radiationinside such a single dielectric particle provide efficient optical feedback to enhance thenonlinear optical processes.

Actually even in an assembly of randomly or periodically distributed dielectricmicroparticles of sizes of the order or less than the optical wavelength λ embedded inanother dielectric one may expect comparable feedback and enhancement of thenonlinear optical processes this time through multiple reflections and scattering off suchsmall nanoparticles.

Such internal or external multiple reflections and nonlinear scattering whichactually preserve a certain coherence can give rise to some quite striking effects that canbe controlled either by the particle size, their concentration or spatial distribution andaverage interparticle distance. We shall briefly discuss below some aspects of theseeffects without going into quantitative considerations which need a detailed descriptionof the electromagnetic propagation in confined space or in random media.

Single dielectric microparticle. Electromagnetic confinement.

A spherical dielectric particle or a liquid droplet with a size parameter κ = 2πa/λ, abeing its radius, and refractive index n0 acts as a high Q optical cavity at specificwavelengths corresponding to the normal electromagnetic modes of a dielectric sphere(or morphology dependent resonances) which can be calculated(25) using the Mie-

Lorentz theory. These occur at discrete size parameters κm ,l where m and l are the modenumber and order respectively ; they possess a (2m+1)- degeneracy. For a spheroid withpolar and equatorial radii ap and ae respectively the (2m+1)-degeneracy of the sphere islifted and the mode frequencies are shifted with respect to those of the sphere of radius

distorsion eby amounts that can be calculated by perturbation theory when the

= |ae - ap| /a is small (e<<1). Their Q can be made very large and theradiation lifetime τ = Q/ω where ω is the mode frequency.

The morphology dependent resonances provide(24) efficient feedback and enhanceseveral nonlinear optical processes such as stimulated Raman and Brillouin scattering,nonlinear Mie scattering, self-phase modulation, third order harmonic generation andothers ; when the particles are doped they provide very efficient stimulated emission andlasing action. Actually these resonances can be modified by processes such aselectrostriction and the nonlinear optical processes being very sensitive to the resonancecharacteristics can be used as very efficient diagnostics to study such distorsions andother features related to the particle form and surface.

In all these cases the new optical waves generated inside the dielectricmicroparticle through the cumulative feedback of the nonlinear optical processes adoptthe spatial distribution of the morphology dependent resonances and eventually leak outfrom the rim of the dielectric in different directions and not in a single well defineddirection as is the case in nonlinear interactions in a homogeneous dielectric. Severalfeatures affect their efficiency that are not present in the one-dimensional wavepropagation (1) usually considered inside a homogeneous dielectric. Thus the processesoccur only when the input laser wavelength coincides with one of the morphologicalresonances and similarly for the emitted frequency in the case of the stimulatedprocesses. The spatial distribution of the modes strongly affect the overlap of the

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nonlinear optical waves and the optical feedback as also does the leakage and the Q ofthe resonance.

The study (24) of the nonlinear optical processes in such microparticles besidesproviding a very elegant demonstration of such processes in a micrometer size dielectricit also provides well controlled means to efficiently enhance these processes and offersthe possibility to exploit them in certain applications.

Two-component dielectric composite

In the previous case the multiple reflections of the radiation and its "trapping"inside the rim of a single dielectric microparticle provided efficient optical feedback toenhance the nonlinear optical processes. Under certain conditions such a feedback andefficient enhancement of nonlinear optical processes can also be achieved in thereversed configuration outside the particles namely through external multiple reflectionsand scattering off these particles in an orderly or disorderly distribution in a solid orliquid dielectric of different refractive index and nonlinear Kerr coefficient.

Photonic band gap composites. Quasi-phase matching. In the case of orderedor periodic distribution of such particles or voids in a transparent dielectric one has aphotonic band gap dielectric or semiconductor(26) and several possibilities of nonlinearprocesses can be envisaged. Any optical coefficient χ here can be assumed periodicallymodulated and written as a Fourier series

(9)

where R and K are vectors in the real and reciprocal space respectively R being theposition vector of the nanoparticles in the periodic array with interparticle distances(lattice constant) of the order of the optical wavelength λ.

Accordingly one has a situation analogous to the one prevailing(27) in X-rayscattering in a crystal or in the electron band states in a crystalline semiconductor. Theoptical waves can now be generated in directions fixed by Bragg von Laue typeconditions in the optical range. The quasi-phase matching configuration(1,28)extensivelydiscussed elsewhere in this volume is actually a one-dimensional realization of such aperiodic array for second order processes and similar ones can be envisaged in two orthree dimensions with quadratic or cubic nonlinearities. Actually in the previouslyconsidered case of multiple reflections inside a single dielectric microparticle one canunfold the nonlinear optical processes to occur in an equivalent periodic array and findseveral similarities between the two cases.

Random composites. Here we shall concentrate our attention on the case of arandom distribution of such dielectric micro or nanoparticles (component A) in atransparent dielectric (component B) of different refractive (nA ≠ nB) and optical Kerr( n 2A ≠ n 2B) indices. Light scattering is the dominant process here and the opticalcoefficients cannot be written as in (9). In the frequency preserving (elastic) scatteringin the Rayleigh and Rayleigh-Gans-Debye regimes, ka << 1 and (εA/εB - 1) ka << 1 withεA ≈ ε respectively, where k = 2B π/λ , and εA ( εB) is the dielectric constant of componentA(B), the transfer of photons from the incident forward propagating mode to thescattering modes of same frequency but different directions is expressed(29) by ascattering loss coefficient

434

(10)

where ∆ε = εA - εB and ∆ n = nA - nB. One also introduces an elastic (coherent) photonmean free path

(11)

which will be assumed much larger than the inelastic mean free path la related to realabsorption losses inclusive the ones involving frequency shifts during the scatteringprocess. In the elastic (frequency preserving) scattering the relative coherence ispreserved to a certain extend and this may have drastic implications on the overallscattering pattern.

We shall briefly concentrate our attention in two cases where the nonlinearbehavior strongly interfers with the scattering pattern because of the modification of therelative coherence in the scattering process. As the intensity of the incident beamincreases the refractive indices of the components A and B, the inclusion and theembedding medium respectively, are modified and become ñA = n A + n 2AI andñ B = nB + n2 BI where I is the beam intensity. Lacking a rigorous theory of nonlinearscattering comparable to the Lorentz-Rayleigh in the linear regime we make the ansatzthat one can introduce an elastic scattering loss coefficient and mean free pathwith a functional dependence on or ∆ñ as in (10) and (11) or

(12)

respectively where

(13)

with ∆n0 = n A - nB and . ∆n2 = n 2A - n 2 B . This ansatz although intuitively acceptable is nottrivial and at presently we lack a theoretical support for its validity; we shall discusssome of its implications.

White self-transparency. It is clear from (12) and (13) that the scattering lossesare suppressed(30,31) for a critical light intensity

(14)

if sign (∆n 0.∆n2) = -1 or otherwise stated the scattering waves are recycled into theincident forward propagating mode without loss of coherence through a suppression ofthe refractive index fluctuation. The net result is that the incident beam propagateswithout attenuation in the random two-component composite for this particular intensityand independently on its frequency as long as this is in the transparency region of thecomposite material. This is the white self-transparency effect(31). The stability conditionsfor this effect are quite involved and will not be discussed here but we mention(30,32)

below a few important implications that follow from such a situation:

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- for optical pulses with peak intensity exceeding the critical one (14) the centerand the wings of the pulse are attenuated and one may recover pulses with rectangulartemporal profile

- a similar process in space allows one to reshape the spatial profile of an intensebeam with peak intensity larger than Ic into rectangular spatial profile

- the reflexion of an intense beam from a mirror located behind such a mediumexhibits bistable behavior as its intensity is varied

- one may shorten pulses- starting with a two component composite with equal linear refractive indices for

the two components or ∆n0 = 0 (index matching) but different optical Kerr coefficients∆ n2 ≠ 0 the scattering losses grow as the intensity grows and this effect can be exploitedin intensity limiting(33,34).

Multiple nonlinear scattering. The previous case effectively concerned thesingle scattering regime which is spatially isotropic. In the multiple scattering regime(34)

interferences along certain scattering paths introduces an axial asymmetry in thescattering radiation pattern along the incident propagation direction. Indeed thescattering pattern exhibits an enhanced scattering in the backward direction within a

cone of solid angle θ ~ λ /lc superposed on the isotropic scattering pattern. This is theweak localization regime a precursor to the strong localization regime the latter

expected to occur when the Iofe-Regel criterion(36) is satisfied, namely l s ~ λ ~ a, whichqualitatively can be visualized as the establishment of stationary interference patterns inall directions following the multiple scattering processes. Within the radiative transporttheory the strong localization regime appears when the effective diffusion constantvanishes and this establishes a threshold condition as also does the Iofe-Regel criterion.We see that within this context and assuming that ansatz (12) is valid in the nonlinearregime both the weak and strong localization behavior can be strongly affected.

Thus in the weak localisation regime, in the case of ∆n0 ≠ 0 (the index mismatchconfiguration) the backward scattered cone will become narrower as the intensityincreases to the critical value I c if sign (∆n0 ∆n2) = -1 and eventually disappears as thisvalue is reached because of the suppression of the refractive index fluctuation in the twocomponent random dielectric ; it angularly broadens in a monotonic way if sign(∆n0 ∆n2) = 1 since the refractive index mismatch cannot be now suppressed. In the caseof ∆n0 = 0 the refractive index matching configuration a cone appears and angularlybroadens as the light intensity increases.

In the strong localization regime when the nonlinear regime becomes operativetwo cases may occur ; depending whether the initial configuration is on or off the strong

localization condition l s ~ λ ~ a, one may switch off switch on the strong localizationand this will in particular affect both the forward as well as the backward propagatingwaves with respect to the incident propagation direction.

These processes besides their fundamental interest can have several importantapplications.

ELECTRON CONFINEMENT

All composite materials formed by uniformly and randomly dispersed metal orsemiconductor crystallites in a transparent dielectric share in common two important

436

features. First, in the metal or semiconductor nanocrystals the otherwise delocalizedvalence electrons in the bulk find themselves confined in regions much smaller thantheir natural delocalization length ; this drastically modifies their quantum motion asprobed by optical beams but also their interaction with other degrees of freedom.Second, because the size of the crystallites is much smaller than the wavelength andtheir dielectric constant is very different from that of the surrounding transparentdielectric, the distribution of the electric field that acts on and polarizes the chargesinside these crystallites can be vastly different from the macroscopic Maxwell field inthe composite.

These two effects, the first quantum mechanical and the second classical, go underthe names of quantum and dielectric confinements respectively and are particularlyconspicuous, in the optical frequency range. The first can be treated within the effectivemass approximation if the particles are large enough so that surface effects can beneglected with respect to volume effects ; the second can be treated within the effectivemedium approach if the particle size and volume concentration are small enough. Themost conspicuous signature of both confinements however is the appearance ofmorphological resonances related to quantum confined dipolar transitions and collectivedipolar modes for the quantum and dielectric confinement respectively. They have adrastic influence on the electron dynamics and optical nonlinearities in these systems.

Dielectric confinement.

For very low volume fraction p of spherical crystallites in the transparentdielectric, or p << 1, and particle size d << λ, where d = 2a is the spherical particlediameter and a its radius, λ is the optical wavelength which is also larger than theinterparticle distance, one can introduce an effective dielectric constant for thecomposite which within the effective medium approach(38) is written

(15)

with εd and εm(ω) being the dielectric constants of the transparent dielectric and theembedded crystallite respectively. This is a straightforward consequence of theClausius-Mosotti approximation for the dipole induced in a spherical polarizableparticle immerged in a dielectric and can also be related to the Mie theory(29) of lightscattering from a diluted gas of spherical particles by imposing the vanishing of theforward scattering amplitude and neglecting all terms of higher order than the dipolarone.

Taking into account the dielectric polarization effect since a << λ the field Ein

inside the particles is related to the Maxwell field in the composite by the relation

where f l (ω) is approximately given by

(16)

To the extent that εd is frequency independent and real while ε m(ω) is frequencydependent and complex, ε ), the absorption coefficient of them(ω) = ε 'm(ω ) + iε"m(ωcomposite is easily obtained from (15) and (16) as

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(17)

where n ≈ εd1/2 is the refractive index of the composite.

It is evident from (15) and also (16) and (17) that these quantities are resonantlyenhanced close to a frequency ω s such that

(18)

which is the condition for the surface plasmon resonance. It arises from the restoringforce of the build-up of surface charge distribution that leads(39) to resonant densityfluctuations of the conduction electrons and accordingly (18) can also be viewed as thecondition for a collective dipolar mode which introduces an antishielding. Its widthwhich also measures the coherent damping of the mode is determined by the imaginarypart ε"m(ω) ; the value of ωs clearly depends on the values of ε'm and εd which can beartificially modified.

An intense light beam can modify the dielectric constant through the optical Kerrnonlinearity by an amount

(19)

(20)

where by extending (40,41) the previous approach to the nonlinear regime one finds

for the effective third order susceptibility of the composite related to the optical Kerreffect, when the contribution of the surrounding dielectric is neglected with respect tothat of the nanoparticles which is resonantly enhanced, being thecorresponding third order susceptibility of the later. The resonances are either thesurface plasmon one or the quantum confined ones and inspection of (19) and of thequantum mechanical expression of clearly shows that in the case of theoptical Kerr effect because of the frequency degeneracy the resonant enhancement ismultiple and the neglect of the nonresonant Kerr nonlinearity of the surroundingmedium is justified.

In the previous discussion we have tacitly assumed that one can replace thesummation over the size distribution by an average value for the optical coefficients ofthe nanoparticles. In a more rigorous approach such aspects can be properly and simplyincorporated and in fact lead to minor deviations from the above sketched derivation.

Our field form factor fl is a macroscopic concept relating the internal field assumed tobe uniform inside the particle and the field in the glass matrix. There is also a

microscopic local field effect but it is built in the response coefficients, ε and χ(3), which

describe the response of the electrons of the particles to the uniform internal field Ein.The dielectric constant ε(ω) of the particle being frequency dependent with

, the factor fl (ω ) may show a resonant behavior whennamely at the surface plasma resonance for spherical particles. Such

resonances are important in the case of metal particles but not for semiconductor ones.Indeed below the absorption edge of bulk semiconductors, the dielectric constant ε isapproximately frequency indépendant, but for semiconductor nanoparticles, we will seethat discrete transitions appear due to size quantization and one could then expect a

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resonant behavior of ε and fl at the corresponding frequencies. It has however beenshown(42) that, for SDGs, even near such transitions, the frequency dependence of ε isweak so that it may safely be replaced by the high frequency dielectric constant ε∞ ineqn (18). The local field factor may then be considered as real and constant. For CdSe-doped glasses for example, one has ε∞ ≈ 6.1 and f ≈ 0.63 which is at least an order ofmagnitude less than the values that prevail in noble metal doped glasses when ω ~ ωs.

Quantum confinement.

For metal and semiconductor nanocrystals containing a few thousand or moreatoms the effective mass approximation (EMA) provides a convenient framework todiscuss the modification of the electron states that results from the confinement. Thisapproach is concerned(43) with the behavior of the delocalized electrons in a crystalperturbed by an aperiodic potential and was initially devised for impurity and electron-hole pair states in crystals and subsequently extended(44) to the case of quantum confinednanostructures. In contrast to the dielectric confinement the quantum confinement has adrastic impact on the optical and spectral features of the semiconductor nanocrystals butmuch less so on those of the metal nanocrystals. We shall discuss to some extent thesemiconductor cases.

In the one-electron picture(43) of a perfect crystal the electrons occupy band stateswhich form energy band continua separated by forbidden energy regions ; theirwavefunctions are of the form

(22)

(21)

where is the pseudo-momentum, n labels the band, has the periodicity of thelattice and is closely connected with the atomic wave-functions that form the basis set ofthe bands and exp(ikr) is the enveloppe wave function. The band states are filled up toan energy level EF , the Fermi level. Within this picture the electronic transitions areviewed as the promotion of an electron from an allowed occupied state below EF to anunoccupied allowed state above EF leaving behind a positively charged hole the twointeracting predominantly through Coulomb forces within the sea of all carriers. Themain distinction between a metal and a semiconductor is related to the Fermi level E F

being an allowed state situated within a band, a half filled band in the case of a metal, orbeing a forbidden state situated within a gap between a filled valence and emptyconduction band in the case of a semiconductor or equivalently that the electron-holespectrum extends down to zero or to a finite energy gap Eg respectively.

We first concentrate our attention on direct-gap semiconductors, such as CdSe, inwhich the bottom of the conduction band and the top of the valence band occur at thesame point of the Brillouin zone, most generally its center G. In the vicinity of suchextrema, the k-dependence of energy may be approximately written in a free-particleform:

m-1 being the inverse effective mass tensor. Neglecting anisotropy, m is a scalar that wewill denote me , the effective mass of the electron, for the bottom of the conduction band

439

and mh, the effective mass of the hole, for the top of the valence band. At the same time,

the k-dependence of un, k

is usually neglected and this lattice periodic part is denoted uc

and uv

for the two bands of interest; referring to the previous remark these are s- and p-

type wave-functions respectively.

Effective mass approximation. We assume the semiconductor to be intrinsic, theground state of a crystallite corresponding to complete filling of the levels of the valenceband, the conduction band being empty. The simplest excited states correspond toexcitation of an electron from the valence to the conduction band, leaving a hole in thevalence band. We assume that the wave-functions vary slowly on the unit cell scale sothat, we may use(43,44) the effective mass approximation which is concerned with thebehavior of electrons in a crystal perturbed by an additional aperiodic potential. Wefurther assume that the bands are isotropic and parabolic in the vicinity of the G point,that the bands are non degenerate and that the potential barrier between thesemiconductor particle and the glass matrix is infinitely high. The electron-hole wave-function φ(re ,rh ) then obeys the equation :

(23)

where ∆ is the Laplacian operator, Wc is the confining potential, assumed to be constantinside the crystallite and infinite outside. The last term in the Hamiltonian correspondsto Coulomb interaction between electron and hole and assumes this simple form onlywhen we neglect the difference between the dielectic constant of the semiconductor andthat of the glass matrix. ε is the low frequency dielectric constant of the semiconductorand r

e-hthe distance between electron and hole. φ (re ,rh) is the envelope wave-function.

There are two length scales that enter the problem: the Bohr radius of the Wannierexciton with µ = memh /(me + mh) and the radius a of the particle if weassume it to be spherical. Two different energies appear(45) in the problem: the kinetic

confinement energy of the carriers of order and the Coulomb interaction energyof order e2/ε a or e2/εaB (whichever is larger). Since these two energies have differentsize dependences, it may easily be shown that, when a >> a

B, the Coulomb interaction

energy is much larger than the confinement energy whereas, when a << aB, the opposite

is true.When a >> a

B, we are in the weak confinement regime(45) , the Wc terms can be

omitted as a first approximation in the Hamiltonian of eqn (23) which then reduces tothe well-known hydrogenic Hamiltonian. The Wannier exciton then exists and movesfreely inside the crystallite. One may then take confinement into account, the resultbeing confinement of the exciton as a whole with its translational mass M = m e + mh

and a small confinement energy of order . On the other hand, when a << aB, we are

in the strong confinement regime(45) the Coulomb interaction term can be omitted as a,first approximation in the Hamiltonian of eqn (23) which is then the sum of twoindependent Hamiltonians. The electron and the hole are confined independently. The

electron envelope wave-function ϕe(r) for example then obeys the equation :

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(24)

(25)

(27)

When the independent electron and hole problems have been solved, one may then takeCoulomb interaction into account using perturbation theory. Finally, when a ~ aB , we arein the intermediate confinement regime and no simple solution exists although anadiabatic decoupling approach has been put forward.

Experiment tells us that the conditions of validity of these limiting cases are not sostringent and the strong confinement regime can be used when a < aB and the particlesthen are denoted quantum dots. Semiconductor quantum dots are thus semiconductorparticles whose size is smaller than that of the Wannier exciton in the bulk. CdSeparticles, for which a

B≈ 5.6 nm, are easily obtained in the strong confinement regime.

In the case of CdSxSe

1-x, aB varies between the CdS value (≈ 3.2 nm) and the CdSe one.

We concentrate on the strong confinement case. As a first approximation, theelectron envelope wave-function is a solution of eqn (24). This is the simple particle in abox problem and for a spherical particle, ϕe(r) assumes(45) the form :

where the Ylm 's are the spherical harmonics, jl(x) is the spherical Bessel function of

order l, ai n

its nth zero and BIn a normalization constant. r, q and j are the sphericalcoordinates of r. Taking the zero of energy at the top of the valence band, thecorresponding eigenenergy is:

(26)

which shows a (2l + 1)-fold degeneracy. Instead of having a quasi-continuous distribu-tion of allowed levels in the conduction band as in the bulk, we now have a discrete setof such levels, the first ones being 1s with l = 0, n = 1 and a

01= π, then 1p with l = 1,

n=1 and a 4.49, then 1d, 2s and so on. We emphasize that the total electron wave-11

=

function is :

in which uc(r

e) has the periodicity of the lattice and is predominantly s- type.

If we assume the valence band to be nondegenerate, the hole envelope wave-function obeys(45) a similar equation whose eigenfunctions are identical to those obtainedfor the electron, the corresponding eigenenergy being given by :

(28)

Here again, the total hole wave-function is :

(29)

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with a different lattice periodic part uv which is predominantly p-type.Several simplifications and approximations(46) are incorporated in the previous

model for the quantum confinement. The spherical particle approximation is one but itshould not be too drastic. For instance the main result which is size quantization of thelevels is not strongly shape dependent and the spherical case correctly gives the order ofmagnitude of the confinement energy. Also the assumption of parabolic and isotropicbands in the vicinity of the G point is not too severe. The main consequence of nonparabolicity is a slight decrease of the confinement energy.

The assumption of an infinite confining potential instead of the ill-defined energygap of the glass matrix ~ 5-6 eV has minor impact on the first levels but may becomemore important for higher excited levels. Finite potential barriers have been considered.The main result here is again a reduction of the confinement energy due to a largerspatial extent of the envelope wave-function.

Because of the difference in dielectric constant between the semiconductor and theglass matrix, the Coulomb interaction between electron and hole is the sum of the lastterm in the Hamiltonian of eqn (23) and of correction terms denoted polarization(43) orsolvation terms. Basically, the electron for example interacts with the hole but also withthe hole image and with its own image. These additional terms have been taken intoaccount using variational or perturbational approaches. They do not change much thevalue of the eigenenergy but they slightly modify the wave-functions, pushing the holetoward the semiconductor-glass interface.

Far more drastic is the approximation of non-degenerate band(48). For theconduction band which originates from s atomic orbitals, this is not a problem. Theconduction band is non degenerate except for the electron spin degeneracy. For thevalence band, the situation is quite different. Since the valence band originates from patomic orbitals, it shows a 3-fold orbital degeneracy and a 6-fold degeneracy when spinis taken into account and this substantially complicates the picture ; a valence bandmixing results(48-50).

Finally, the effective mass approximation (EMA) ceases to be valid for smallsemiconductor particles where the surface layer still makes an important contributionand the only way out of the EMA, which suppresses these surface states is a molecularapproach (51). In such an approach molecular orbitals are constructed, using the trueatomic orbitals as the starting basis set. Projecting the MOs thus obtained on the wave-functions of the inner or outer atoms, these preliminary results allow(52) to visualize theformation of the valence and conduction bands and show the presence of surface states.This MO approach also allows to consider particles the surface of which has been partlyor totally capped or passivated. It is however very demanding in terms of numericalcomputation and has been limited to a total number of a few hundred atoms.

Interband and intraband transitions. The selection rules for transitions betweenthe quantized states are determined(14) by the matrix elements of the momentum operatorp. We consider first interband transitions in which an electron is promoted from an(l,m,n) state in the valence band to an (l ',m',n') state in the conduction band if we neglectthe valence band degeneracy. Since the momentum operator is proportional to thegradient operator and since y

lmn(r h) is the product of a rapidly varying part uv and of a

slowly varying one ϕh, we have :

(30)

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The transition matrix element is approximately the product of what is traditionallydenoted p

cvand of the overlap integral of the electron and hole envelope wave-

functions; its value thus does not depend on the particle radius a.The ϕ

lmns given by eqn (25) form an orthonormal basis set and the overlap integral

is simply δl l δmn'δ

nn'where dij is the Kronecker symbol. The only allowed transitions are

then those which preserve the quantum numbers l, m and n (enveloppe conservation).We then have the 1s-1s, 1p-1p and so on transitions. They occur at frequency ωIn givenby ..

(32)

(31)

in which the last term, corresponding to Coulomb interaction, is obtained using pertur-bation theory. The number β is approximately 1.8 for ns-ns transitions. Taking thevalence band degeneracy into account slightly modifies the selection rules. Transitionsare allowed for example from all the nS

3/2levels to the 1S

eone but the most important is

the 1S3/2

-1Se one.

Eqn (30) as well as EMA assume that ϕ varies slowly on the unit cell scale. Thisis the case for not too small particles when the quantum numbers 1 and n are smallintegers but the assumption is quickly invalidated when 1 and/or n increase. EMA is thenno longer valid. The selection rules are then modified and this may be the reason why,in real spectra, only the first transitions can be identified. At higher photon energy, thespectra have the same continuum-like aspect as in the bulk.

We may also consider intraband transitions in which for example an electron ispromoted from an (l,m,n) state in the conduction band to another (l',m',n') state of thesame conduction band. The situation is quite different here and the matrix element isapproximately given by :

The selection rule is then ∆l = ± 1. We may have for example excitation from the 1Se

state to nPe ones. The transition matrix elements (32) are now ~ a but the transition

probability quickly decreases with increasing n.Above we outlined the description of the electron states in semiconductor

nanoparticles within the effective mans approximation. For metal nanoparticles thesituation at the outset seems to be simpler in the optical frequency range because wemay concentrate our attention solely on the electrons in the half filled conduction band(s-p type). Since the low frequency dielectric constant in metals is very large (inprinciple infinite at ω = 0) the screening of all interchange interactions is complete andone may view the electrons as non-interacting, freely moving quasiparticles in theconfining potential of the metal nanoparticle and their states are then simply describedby an equation of the type (24), with m e ≈ m the free electron mass, and with wavefunctions of type (27). The transitions in the optical range then are either intraband onesbetween such states with intraband transition matrix elements given by (32) or interbandones between the filled d-band states and the quantum confined ones of the half-filleds-p band.

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METAL NANOCRYSTALS IN GLASSES

Using the previous simplified discussion of the dielectric and quantumconfinements we may now proceed to describe the linear and nonlinear properties ofmetal nanoparticles in a glass matix.

Linear properties

We remind(53) that in the case of bulk metals, because of the zero energy gap andlarge density of states at EF, the wave vector dielectric constant is infinite for = 0resulting in a complete screening of the electron-hole interaction potential within adistance rF ≈ 1/k F, the inverse of the Fermi wavevector which is of the order of a fewAngströms or roughly equal to the lattice constant. Thus the electrons and holes oneither side of EF can behave and move as free noninteracting particles over any distancein a perfect metal occasionally interacting and being scattered by temporal and spatialdisorders ; such interruptions of the coherent free electron motion can be expressed interms of a free mean path for the electrons which in an ideal metal we will assume to belimited by the unklapp electron-electron and electron-phonon scattering processeslumped together : one finds there that

(33)

for the dielectric constant of a metal were ωp is the plasma frequency, ω2 = 4π e2n/m, n isp

the electron density, δεinter is the interband contribution and τ0 is a scattering time forthe electrons which for an ideal defect free metal we assume to be related to thecompound effect of electron-electron and to electron-phonon scattering.

Within this context in a metal nanocrystal the main modification then will resultfrom an additional restriction of the free electron motion because of the presence of theinterface with the surrounding dielectric which to a good approximation can bevisualized as a spherical potential well of infinite height. In addition the electron-phonon coupling has to be modified because of the modification of the vibrationalspectrum in the metal nanocrystal with respect to the bulk. Assuming the two processesuncorrelated this only amounts to replacing 1/τ0 in the denominator in (33) by

where vF is the electron velocity on the Fermi level and 1/τ' 0

relaxation rate in the crystallite.This classical picture which however pro

now is the electron-phonon

perly incorporates the Fermi statistics canbe also justified within a quantum mechanical approach(40,54) in terms of the confinedelectronic states in the spherical potential well set up by the interface with thesurrounding dielectric. These are now discrete states which incorporate the boundaryconditions because of the requirement that the wave functions must vanish at theinterface ; as previously discussed more realistic models introduce only minor changes.

(34)

444

Although the enveloppe equation (24) together with the boundary condition lead to

a discrete spectrum with energy spacing as we move up in the energyspectrum this becomes close spaced and in fact one may again introduce a density ofstates

(36)

(38)

(39)

where V is the spherical particle volume, and a Fermi level

(35)

both independent on the nanocrystal size and equal to their corresponding bulk values.Calculating the transition dipole moment between such states and introducing(55) acommon coherence relaxation time T2 related to electron-lattice interactions for allallowed dipole transitions one may proceed to calculate the optical properties. Takinginto account also transitions from the filled d-band which is unaffected by the quantumconfinement one obtains the remarkable result that the dielectric constant takes(40,54,55) theform

(37)

namely the same as for the bulk with the damping time τ in the Drude term given byexpression (34) previously obtained by a classical argument, while the imaginary part ofthe interband term is given by

where P is the transition matrix element of the momentum operator between the d-andthe conduction s-p bands, and J(ω) is the joint density of states for the two bands. Thusone recovers for the metal particles an expression similar to that of the bulk metal (33)with the the damping replaced by expression (34) ; this also justifies the use of the bulkexpression for εm(ω) in équation (15) as anticipated. With this expression for ε

m(ω) we

obtain

for the expression of the surface plasmon resonance in terms of the dielectric constantsεd and εb = 1 + δε inter. We see that ωs can be modified by several ways by acting on thephysical parameters εb, εd or ωp in particular through photoaction.

Optical nonlinearities

The optical Kerr effect results from the photoinduced modification of the complexrefractive index of the medium or δñ = n2I, where I is the light beam intensity and theoptical Kerr coefficient n2 is simply related to the third order susceptibility χ (3)(ω, –ω,ω)

445

or χ(3)(ω – ω' ,ω ') which can be resonantly enhanced whenever ω and/or ω' are close to a,resonance of the medium. In the case of the metal or semiconductor nanocrystals inglasses the relevant resonances are the morphological ones resulting from the dielectricand quantum confinements. A simple inspection of the expressions of the third orderoptical Kerr susceptibilities shows that one has multiple resonant behavior there leadingto strong enhancement close to these resonances and also strong dependence on thefeatures of these resonances which also reflect those of the confinement.

Broadly speaking, in such a resonant regime, the physical mechanismscontributing to χ ( 3 ) can be distinguished in two classes(48,56). The first type involves realpopulation of excited states or excitation of charge carriers there resulting from realtransitions ; it is accompanied by linear absorption and the response speed is limited bythe population relaxation time. The second type of mechanisms contributing to χ(3)

effectively results from light induced shifts on the electronic levels, direct or indirectones, and can be connected with virtual transitions. The most conspicuous directcontribution here is the one related to two-photon resonances while indirect ones mayresult either from interpair electron-hole (biexcitonic) interactions which can be viewedas an optical Stark shift mutually induced between photoinduced e-h pairs, or throughinteraction with the field set up by an interface trapped photocarrier which can beviewed as a static Stark effect ; the three mechanisms can be distinguish through theirdifferent frequency dependence and different response times they imply.

In the case of metal nanocrystal because of the complete screening that isoperative there all contributions of the second class to the Kerr nonlinearity mentionedabove are drastically reduced and only those of the first class contribute there. This hasalso been confirmed experimentally. In addition in metal nanocrystals the effective thirdorder Kerr susceptibility is fourfold resonantly enhanced close to the surface plasmonresonance ωs .

The optical nonlinearities of metal nanocrystals in transparent dielectrics havebeen studied to a certain extent over the last ten years and a good understanding of thesenonlinearities and their relation to the electron dynamics has emerged that confirm themain aspects stated above. Reliable predictions regarding their behavior in noble metalsare indeed feasible. Thus the fourfold resonant enhancement of

(3) of the composite

material close to the surface plasmon resonance (39) as predicted by (19) has beenstrikingly demonstrated(22) in gold and silver colloids first and subsequentlyconfirmed (57,58) also in the case where the gold and silver particles are embedded in glass.Similar behavior has also been evidenced (59) in copper nanocrystals in glass where thesurface plasmon resonance strongly overlaps with the interband transitions. Using theexperimental values of and and relation (18) one easily finds thatfe ∼ has values in the range 10-20 for noble metals resulting inenhancement factors in the range 103-105 for over the surface plasmon resonancewidth (34).

The impact of the quantum confinement is related to the behavior of

which can be extracted from (19) after the factor fl (ω) has been accounted for in termsof the dielectric confinement as stated above using expression (16) and condition (18).The detailed experimental study of ref. 57 using the degenerate four wave interactiontechnique or optical phase conjugation where several parameters were varied, such asthe size of the nanocrystals, the temperature or the polarization state of the input beamand in addition the phase of χ ( 3 ) was determined, allowed to analyse its behavior in thelight of a detailed theoretical modelling and single out the origin of the nonlinearity ; itconfirmed on the one hand the minor role played by the quantum confinement there and

446

on the other the key role played by the photoinduced population rearrangement close theFermi level and its dynamics.

Careful analysis(55,57) of the quantum mechanical expression of χ (3)(ω,–ω,ω)m

indeed shows that the second class of contributions previously termed as photoinducedlevel shifts are irrelevant in metal nanocrystals and only the first class of contributions,termed photoinduced population changes are responsible for the optical nonlinearitiesthere. These can be additionally split into three contributions :

an intraband contribution involving only transitions between two quantumconfined states within the conduction s-p conduction band which can be written

(40)

where κ1 is a constant factor, T1 et T2 are respectively the energy lifetime and dephasingtime and a which is roughly 100-200Å in metal nanocrystals ; thus (3)

0 ≈ T2(2EF /m)1/2 χ intra

is imaginary with Im χ(3) < 0 and strong size dependence characteristic of quantumintra

confinement,an interband contribution involving transitions between a d-band state and a

quantum confined s-p conduction band state which can be written

(41)

is a constant, T 'where κ ' '1 and T are the energy lifetime and dephasing time for the two-2

level system and P and J(ω) were previously defined in equ. (38) ; thus χ (3) isinter

imaginary with Imχ 0 and size independent,<(3)inter

a hot electron contribution which can be viewed as a hybrid term of the previoustwo and results from the photoinduced modification of the occupation of the conductionband states above and below the Fermi level accessible to the d-electrons. Thismodification is provoked by the absorption of photons, close to the surface plasmonresonance frequency, by the conduction electrons which are promoted to initiallyunoccupied states above the Fermi level and the same time liberate states below it.Assuming this leads to a quasithermal distribution in a subpicosecond to picosecond

time scale at an electronic temperature Tone gets

e much higher than the lattice temperature T l

(42)

for this contribution where C is the conduction electrons heat capacity, τ /τ0 e f f measuresthe number of all collisions the electrons suffer before they thermalize, ε''L i s t h eimaginary part of interband term at the point L of the Brillouin zone which makes thedominant contribution close to the surface plasmon resonance frequency ; thus χ (3) ishe

imaginary with Imχ(3) > 0 and size independent.he

The detailed experimental study in ref. 57 showed that in the size range 30-100 Åthe hot electron contribution is the dominant one in χ (3) being by one and two ordersx x x x

of magnitude larger than the inter and intraband ones respectively. In χ (3) this hotxyxy

electron contribution being incoherent does not contribute while the interband one doesand this is confirmed by the measured value and sign on the ratio (3)χ / χ (3)

xyxy . Similarxxxx

conclusions were also reached in the detailled study of the optical nonlinearities in

447

copper nanoparticles in glasses where the interband transitionsintraband ones involved in the surface plasmon resonance.

strongly overlap with the

Electron dynamics

The study of the optical nonlinearities in metal nanocrystals brought into focus theimportant role played there by the electron dynamics in a confined space. The study ofsuch processes was initiated ( 5 3 ) quite early in electron transport in connection with theelectrical conductivity in metallic wires and plates or thin films and it was associatedthere to the modification of the mean free path of the electrons because of theircollisions with the walls. The Landau theory of Fermi liquids provides a rigorous albeitcomplicated framework to study such aspects in terms of the interaction of quasiparticles (electrons and holes) among themselves or with other excitations and with thewalls. Much insight however can also be gained with a simpler description and byproper accounting for the Fermi statistics.

In the past these experimental and theoretical studies were predominantlyconducted in the frequency domain under the assumption that the different interactionsthat cause damping were uncorrelated and the long time regime prevails. Deviationsfrom such a behavior however were clearly noticeable but difficult to analyze infrequency domain. The recent advances in ultrashort laser pulse techniques opened theway to address these problems directly in the time domain, in particular the short timebehavior, and assess the interplay of the different processes there. The most commonlyused time resolved technique is the pump and time delayed probe technique and itsvariants : in essence in all these techniques the spatiotemporal evolution of a quicklyphotoinduced modification of an optical property is interrogated by a short probe pulsewith variable delay and related to the dynamics of the relevant electronic transitions.

The recent application (55,60,61) of time resolved nonlinear optical techniques in thefemtosecond time scale opened the way to study the early stages of ultrafast dynamicsof photoexcited electrons in metal nanoparticles confirmed( 5 5 ) the importance of theinterplay of the e-e and e-p collisions in the process of their thermalization. Numericalsolutions with certain simplifying approximations fully confirm this effect. The case ofthe copper metal particles where some of these studies were performed ( 6 0) presents theinconvenience that the interband transitions in copper strongly overlap with theintraband ones of the surface plasmon resonance. The case of the gold or silver particlesembedded in glass is be exempt of such complications.

Recently studies were performed (55) in silver nanoparticles embedded in a glassmatrix ; here the plasmon resonance and interband transitions are well separatedpermitting a selective probing of the surface plasmon resonance. The preliminary resultsobtained with 100 fs resolution indicate that thermalization does not set in beforeseveral hundred femtoseconds. The study yielded evidence of a time dependent red shift∆ω s and broadening of the surface plasmon resonance which set in instantaneously andfollow the thermalization process. The frequency shift can be related to thephotoinduced modification of ε b in (39) because of the optical Kerr effect while thebroadening is attributed to the increment of the collision rate of the electrons with thesurface as their average velocity increases. Additional studies with shorter timeresolution are needed to unveil the athermal behavior.

Extensions

448

The study of the optical nonlinearities and electron dynamics in metalnanocrystals embedded in glass matrix is connected with some fundamental aspects ofthe behavior of electrons in metals in general and in confined geometries in particular.In this respect two aspects should retain our attention : one is the crucial role played bythe dielectric confinement to enhance the nonlinear optical coefficients close to thesurface plasmon resonance and other morphological resonances ; the other is therestrictions imposed by the Pauli principle and Fermi statistics on the available phasespace for the electrons leading to interference of the scattering processes and eventuallyto the breakdown of the addivity assumption of the damping rates and that of thethermal evolution of the photoexcited electronic distribution. Indeed the impact of thesetwo aspects is such that any specifically quantum mechanical features are effectivelywashed out. This is in striking contrast with the situation in the semiconductornanocrystals discussed below where specific quantum mechanical features prevail in theoptical spectra and electron dynamics.

At the same time the recognition of the important role played by the dielectricconfinement and the Fermi statistics shows the way to some interesting future studies.Thus similar studies in transition metal nanoparticles where the d-band is also half filledor noble metal nanoparticles containing magnetic impurities should show some newfeatures regarding electron dynamics and optical nonlinearities because of the drasticmodification of the electron density distribution at the Fermi level that occur there.Another line of study is that of much higher volume concentrations for the metalnanoparticles where the Maxwell-Garnett effective medium approach breaks down andmust be replaced by more sophisticated ones like the Bruggemann approach( 6 2 ) or morecomplex ones where the topology and fractality are properly introduced. Such studieshave already been initiated uncovering some new aspects(63) Finally there is certainly.interest (6 4 ) in the study of glasses co-doped with metal nanocrystals and rare earth ionswhere the dielectric confinement through the surface plasmon mode it sets up inside thenanocrystal may also show its effect on the rare earth ions outside the metalnanocrystals. In these and other cases the radiative damping which was althogetherdiregarded in the previous discussion may play a certain role for sufficiently largenanocrystals since its contribution grows as the cube of the particle size and must nowbe included. In this respect we wish to point out that the absorption losses in the farinfrared of the composites formed by dispersing metal nanocrystals in dielectrics can notbe accounted for even when all the refinements of the present theory are included. Theseand other studies will be pursued both for their fundamental interest as well as for theirimpact in designing artificial composite materials with specifies functions.

SEMICONDUCTOR NANOCRYSTALS IN GLASSES

The linear and nonlinear optical properties of semiconductor nanocrystals( 1 4 , 1 8 )

have been extensively studied the last two decades in particular in the strong quantumconfinement regime. The most conspicuous spectroscopic feature is the appearance ofthe quantum confinement resonances related to the morphology and size of thenanocrystals. To the extent that the interface with the surrounding dielectric contains asizeable proportion of atoms surface states and traps play an important role anddrastically effect the dynamics of the confined resonances. Below we shall first reviewsome spectroscopic features and then discuss the nonlinear optical mechanisms in suchquantum confined nanocrystals.

449

feature is observed. When discussing below the resonant non-linear optical properties ofthese materials, we will mainly deal with this feature and vaguely denote it 1S3/2 -1S e .

feature only appears as a shoulder to the 1S 3 / 2-1S e peak. At room temperature, only one

From the previous considerations on interband transitions, the linear absorptionspectrum of semiconductor quantum dots is expected to be structured with an absorptionedge shifting to the blue when the particle size is reduced. This is indeed what weobserve in low temperature absorption spectra for the extensively studied CdSe-dopedglasses. The second derivative of the α(ω) spectrum clearly shows substructures whichbear the signature of valence band mixing( 4 6 ) The first two substructures correspond to.the 1S

3 /2-1S

eand 2S

3 /2-1S

etransitions. Then, usually comes the 1P

3/2-1P

eone.

These substructures were observed in several CdSe-doped samples grown in apure silicate matrix. Usually however, in less pure samples, even when working at lowtemperature and taking the second derivative of the absorption spectrum, the 2S 3/2-1S

e

In order to better understand the linear optical properties of SDGs and the levelsof relevance in the vicinity of the absorption edge, luminescence measurements(18) arealso important. When SDGs are excited with a c.w. laser beam, the luminescencespectrum consists of a relatively narrow peak slightly Stokes shifted from the 1S 3 / 2 -1Se

absorption peak and a usually more intense, broader luminescence band showing amuch larger Stokes shift. When excited with a picosecond laser pulse, one observes thatthe narrow luminescence peak is a rather fast component with a lifetime of the order of1 ns whereas the broad band is a slow component with a lifetime of the order ofmicroseconds. In fact, in this broad band, the larger the Stokes shift, the slower theluminescence decay. The narrow peak was therefore interpreted as direct recombinationof an electron in the 1Se level with a hole in the 1S 3 / 2 one. The broad luminescence bandwas interpreted as trapped carrier recombination.

Traps. As previously stated traps play a very important role in these nanocrystalsbut their identification and characterization is very difficult. Until now, we assumed thesemiconductor to be intrinsic and we did not pay attention to the semiconductor-glassinterface. Very little is known about possible inner impurities or defects. In the sameway, little is known about the role of the surface of the nanocrystals. One could imaginethat dangling bonds lead to the presence of surface states. One could imagine that thesedangling bonds are at least partly saturated by surface reconstruction or by bonds withthe Si-O matrix. For commercial Schott filters, the broad luminescence band is muchmore intense than for specially made (denoted experimental) glasses grown from thesame melt. This means that commercial filters contain many more traps thanexperimental samples. And, for experimental samples grown from a same melt, thesmaller the particles are, the larger the number of traps is. The traps are thereforethought to be located at the semiconductor-glass interface. This is further supported bythe observation of drastic changes in the luminescence behavior when the surface ofcolloidal CdSe nanoparticles is capped with organic species.

Another important point was reported in ref. 65. When time resolving the rise ofthe luminescence signal, the authors observed that the narrow peak and the blue edge ofthe broad band start appearing at the same time. One does not observe a progressive riseof the broad band accompanying the temporal decay of the narrow peak as would beexpected if the carriers were progressively trapped. This suggests the existence of two

450

volume of the particle and directly recombine giving rise to the narrow peak andparticles with traps in which the carriers very quickly trap at the interface and thenslowly recombine giving rise to the broad feature.

Photodarkening. When SDGs are exposed to a laser beam at frequency ω suchthat > E g for a long time, they experience a phenomenon known as photodarkening

as first observed independently by Roussignol et al(66) and Mitsunaga et al(67) . This effecthas several consequences, It leads to a drastic change of the luminescence spectrum.After darkening, the magnitude of the narrow luminescence peak is somewhat reducedwhereas the broad band almost completely disappears. Darkening also reduces themagnitude of the recombination time of free carriers, which explains the reduction in theintensity of the narrow luminescence peak. It also reduces the magnitude of the opticalKerr susceptibility in the resonant case. It does not significantly change however theabsorption spectrum. The disappearance of the broad luminescence feature seems to bedue to the opening up of a new and more efficient decay channel in particles with traps.

The degree of darkening which may be quantified in several different ways, forexample by measuring the intensity of the broad luminescence feature, depends only onthe integrated irradiation dose the sample has received. The photochemical processleading to darkening was elucidated by Grabovskis et al(68) using thermally stimulatedluminescence: the absorbed photons create carriers which may be ejected from thenanoparticles and trapped in the glass matrix. This is a quasi permanent phenomenon.At room temperature, it may be considered as permanent but, by heating the samples to~ 370°C for a few hours, the ejected electrons diffuse back to their original particle andthe doped glass recovers its original properties.

Line-broadening. The observed transitions are broad and this is mainly attributedto size dispersion. The transition frequency is a-dependent: smaller particles absorb athigher photon energy. The size distribution leads to inhomogeneous broadening. Thisinhomogeneous broadening hides the intrinsic or homogeneous width of the transitionsas would be given by single particle spectra.

If we are able to excite only one particle size, probing it will give us narrowerfeatures. This may be done using nonlinear techniques. We may fix the excitationfrequency and 1) probe the change in the absorption spectrum : this is saturation or holeburning spectroscopy or 2) measure the luminescence spectrum: this is known asfluorescence line narrowing (FLN) spectroscopy and reveals spectral structures forinstance in the 1S 3/2-1S e peak due to a vibronic or phonon progression. The simplest andclearest FLN spectra are obtained by exciting only the biggest particles. Increasing thephoton energy, one would excite smaller particles in the zero-phonon line and largerones in the one-phonon line.

Symmetrically to case 2) above, one may detect only the blue edge of theluminescence peak and tune the excitation frequency. One then obtains aphotoluminescence excitation (PLE) spectrum. This way, we also observe the phononprogression of the 1S

3/2-1S e transition. FLN or PLE spectroscopies are ideal tools to

observe finer details. Using PLE, Norris and Bawendi (50) have been able to follow thesize dependence of ten interband transitions and could clearly identify the first six ofthem. Using PLE again, Norris et al (50) could also study the very fine structure of thezero phonon line in the 1S3/2- 1S feature. The 1Se 3/2 -1S level is eight-fold degenerate ande

451

the very fine structure is due to lifting of this degeneracy by deviation from sphericity,deviation from the zinc-blende structure and by the confinement-enhanced exchangeinteraction.

Electroabsorption. A static (or very-low-frequency) electric field E leads to thestatic Kerr effect which modifies the optical properties. The change in the absorptionspectrum δα(ω) is usually measured; it is proportional to the imaginary part of

χ(3)

(0,0, ω). The real part of χ(3)

could be obtained using the Kramers-Kronigrelationships. Such an effect may be used (69) in hybrid devices such as the SEEDs madefrom quantum wells. But the static Kerr effect is also interesting per se. The change inthe optical properties is due to the shift of the energy levels and to admixture ofneighboring wave-functions. When dealing with atoms for which the spacing betweenlevels is large, perturbation theory is used and the effect is termed Stark effect. Whendealing with a bulk semiconductor for which the levels are very closely spacedcompared with eaE, the Hamiltonian:

(43)

where H0 is the zero-field Hamiltonian, must be solved. The effect is then termed Franz-Keldysh effect. This is one and the same effect, only the mathematical approach differs.

Electroabsorption measurements were performed(70,71) on CdS 0.5 Se0.5 samples

containing particles of various mean sizes. The electric field with Ein = 2.10 4 V cm- 1

isapplied via indium tin oxide transparent electrodes deposited on the two sides of a thinSDG slab. The applied field oscillates sinusoidally at a frequency of 1 kHz and thechange in absorption δα is measured using lock-in detection at 2 kHz. Oscillations areobserved (70) in the vicinity of the 1S3/2-1S peak with a replica in the vicinity of the split-eoff band edge.

The position of the oscillations is independent of the magnitude E of the applied

field and their amplitude is proportional to E². This is typical of a quantum-confinedStark effect: confinement has led to well-resolved discrete levels. Similar results havebeen obtained by other groups. When the particle size is larger, then Franz-Keldyshbehavior is observed (71). The amplitude of the oscillations decreases when the particlesize is reduced. This is understandable since smaller particles are less polarizable. UsingRayleigh-Schrödinger perturbation theory, these experimental results could be fittedwith good accuracy. Variational calculations have also been performed. We note that

(0, 0, ω) ~ 10 -12 e.s.u., a fairly large value, and that a static field leads to a decrease ofthe absorption coefficient at the posi-tion of the 1S 3/2 - 1S peak and to an increase one

each side of it.

Nonlinear optical properties. The optical Kerr effect

The nonlinear property we will concentrate on and the one that has the largestnumber of applications is the optical Kerr effect. The static Kerr effect is themodification of the susceptibility χ(ω) of a material (or of its index of refraction) by astatic electric field E0 , the modification of c being proportional to the square of E0 . The

452

electroabsorption previously discussed is a special case of the static Kerr effect when ωis close to a resonance. The optical Kerr effect generalizes this to the field of a laserbeam of same (ω) or different (ω') frequency. The first case is a self-action, the second

(3)one is a cross-action. The real and imaginary parts of χ give rise to nonlinearrefraction and nonlinear absorption, respectively.

As previously stated two different physical mechanisms may contribute to thethird-order nonlinearity. The first type involves real population of excited states orexcitation of charge carriers here. It corresponds to resonant nonlinearities when thefrequency w of the field is close to that of an optical transition of the medium. It isaccompanied by linear absorption losses and the response speed is limited by the carrier

recombination time. The second type of mechanisms contributing to χ (3) effectivelyresults from light induced shifts of the electronic levels, direct or indirect, and can beconnected to virtual transitions. It has a femtosecond response time and is then muchfaster, and usually smaller, than the resonant one. The most conspicuous contributionhere is the one related to two photon resonances. In a semiconductor of energy gap Eg ,the third-order nonlinear absorption coefficient becomes significant when the photonenergy of the driving beam is larger than Eg /2 and, in the transparency range, is atwo-photon absorption (TPA) process.

Experimental Techniques. A variety of techniques(14,18) may be used to study theKerr nonlinearity. Nonlinear absorption, which is one of these techniques, may beimplemented in different ways. One may use a single beam of frequency ω and measurethe transmission of the sample as a function of the incident intensity. One may use twolaser beams in a pump-probe scheme. A probe pulse of frequency ω' measures thechange in absorption coefficient δα(ω') induced by a pump pulse of frequency ω.Delaying the probe pulse allows to study the time behavior of this nonlinear response. Ifa white spectrum probe pulse is used, the change in the absorption spectrum can beobtained in a single laser shot. Nonlinear absorption gives access to the imaginary part

o f χ(3)(ω,–ω,ω) in the first case andχ (3) (ω,–ω,ω') in the second one.In optical phase conjugation (OPC), three beams at the same frequency ω are

incident on the sample, two counterpropagating pump beams and a usually weakerprobe beam making a small angle with the forward pump beam. A conjugate beampropagating in the direction opposite that of the probe beam is generated and detected.OPC is also known as degenerate four-wave mixing in the backward geometry. Mostcommon configuration is on the case where the forward pump and probe beams arecopolarized and when the backward pump beam is cross polarized. OPC then has asimple holographic interpretation: the forward pump and probe beams create an indexgrating off which the backward pump beam is diffracted. In the resonant case, apopulation grating is created and again, delaying the backward pump pulse allows to

time resolve the nonlinear res-ponse. OPC gives access to the modulus of χ (3) (ω,–ω,ω).In frequency mixing, two beams of frequency ω and ω' are incident on the sample

and the beam generated at frequency 2ω - ω' is detected. Contrary to the othersituations, we do not have automatic phase-matching in this case. This technique gives

access to the modulus of χ (3)(ω,ω,-ω'). When dispersion is not important, this χ (3)

i s

approximately equal to χ (3) (ω ',–ω',ω' ). This technique is important in the two-photonabsorption regime since, when the difference ω - ω' is much larger than the inversecarrier recombination time, modulation of the population of the excited state at

453

frequency ω - ω ' is negligible: this implies that free carriers do not contribute togeneration of the beam at frequency 2ω - ω'.

If we consider a propagating beam with a gaussian intensity profile, the nonlinearphase follows the intensity distribution: it is maximum at the center and vanishes at theedge of the beam. The nonlinearity then produces a self-focusing (defocusing) of the

beam when the real part of χ(3)is positive (negative). Scanning the sample through the

focal zone of a lens and measuring the intensity transmitted through a partially closedaperture allows to measure the magnitude and, most importantly, the sign of the real

part of χ(3) (ω, –ω,ω): this is the Z-scan technique extensively discussed elsewhere in thisvolume. An open aperture Z-scan allows to measure nonlinear absorption.

The resonant regime. In the resonant case, luminescence may also giveindications on the nonlinear response. When the samples are excited with an intensepicosecond pulse and when concentrating on near-edge luminescence, hot luminescencepeaks appear corresponding to direct recombination from the 2S 3/2-1S e level and evenhigher excited states. This is indicative of state filling in the quantum dots, a behaviorcontrasting with the bulk-like band filling observed in larger particles.

Model and theory. The first prediction regarding the resonant nonlinear responsewas that each particle would behave (72) as an isolated two-level system made of theground state (level 0) and the 1S3/2 -1Se excited state (level 1). Saturation of this two-level system would lead to the nonlinear response. Absorption saturation had indeedbeen observed

(73)in SDGs. It was soon realized(74)

however that, after excitation of onecarrier pair in level 1, a second pair may be created, leading to the presence of a two-pair level 2. The Hamiltonian for such a two-pair state is the sum of the twoHamiltonians for each pair plus terms corresponding to Coulomb interaction betweenthe two pairs. It has been shown that the interaction energy between the two pairsincreases when the particle size is reduced but remains small. The transition frequencyω is smaller than ω10 but not very different from it so that, when ω ≈21 1 0 , we alsohave ω ≈ ω 21. Assuming the same dephasing time T2 for the 0 → 1 and 1 → 2 transitionsand keeping only the triply resonant terms, the degenerate Kerr polarizability is easilyobtained :

(44)

where d 10 and d21 are matrix elements of the electric dipole moment operator, propor-tional to those of the momentum operator, and T1 is the lifetime of level 1. g is the sum

of three terms, the first one corresponding to saturation of the 0 → 1 transition, thesecond one to induced absorption from level 1 to level 2. The third term is a coherentterm involving only off-diagonal elements of the density matrix. Since T2 << T1 , thisthird term is negligible. The resonant nonlinearity mainly originates from population

454

effects. From eqn (44), the susceptibility of the SDG is obtained by multiplying gwith the number density N of quantum dots and with the local field correction factor. Inthe resonant regime, the contribution χd

(3) of the glass matrix is negligible.A nanoparticle should behave as a three-level system if we ignore trapped carriers.

(47)

The important role of trapped carriers had been pointed out by several groups. Toaccount for the trap levels, we add a fourth level denoted 3 whose decay is neglected onthe nanosecond time scale. Carriers are excited to level 1. From there, they can relaxdown to level 0 with rate constant k or be trapped in level 3 with rate constant k'. k' isknown to be large.

The problem has been treated in two cases : one class of (or identical) particlesand two classes of particles, particles with traps and particles without traps. Since thelifetime T1 of level 1 is longer than the laser pulse duration tp , the response functionformalism was used. The response may also be characterized by an effectivesusceptibility which takes the transient nature of the response into account. Theresponse is made of three terms: population of level 1 leads to saturation of the 0 → 1transition and to induced absorption between levels 1 and 2, carriers in level 3 modifythe optical response through the static electric field E0 they create which is of ordere/εa². Assuming rectangular pulses, the response may be calculated analytically. Weonly reproduce here the results pertaining to the case of two classes of particles and theOPC geometry. When t > tp , the amplitude of the nonlinear polarization giving rise tothe conjugate beam is :

(45)

with t the delay of the backward pump pulse and:

(46)

where -i(2a - b) is (within a numerical factor) the contribution of free carriers to the true(steady-state) :

(48)

N w and N w/oare the number densities of particles with and without traps respectively.The two terms in eqn (48) correspond to the first two terms in eqn (44) and generalizethem in that the dephasing time T'2 for the 1 → 2 transition is not necessarily equal toT2 . The second factor in (46) accounts for the transient nature of the response. It is close

to tp/T1 when tp << T1 . -iCf and -iCs are effective susceptibilities. The amplitude of theincident fields are defined as Ai (t) = Ai when 0 < t < t p and 0 otherwise with f, b, pstanding for forward, backward and probe respectively. d' = (ω 21 - ω)T'2 is thenormalized detuning for the 1 → 2 transition. From the expression (45) of Pω

NL (t) , thefluence f (t) of the conjugate pulse may be calculated.

c

455

Regarding the induced absorption contribution, until now level 2 was supposed tobe a two-pair state and we limited ourselves to interband transitions. We may extend theresult to intraband transitions as well in which, for example, the electron is promotedfrom level 1Se to an nPe level. Close to resonance, the susceptibility is expected to bemainly imaginary. Indeed, from what has been said before, a, b and Cs are expected tobe mainly real and positive. For two of the three contributions, numerical estimates can

be obtained. For the trapped carrier term, one gets for Cs a number of order 10-10

e.s.u.

The bleaching contribution to Cf

is estimated to be of order 10-8

e.s.u. The inducedabsorption term is much more difficult to estimate. When the carrier pairs are noninteracting, ω21 = ω

10, T' 2 = T2

and and it is is clear from eqn (48) thatinduced absorption cancels absorption saturation.

Experimental results. Using the nonlinear absorption technique, absorptionsaturation as well as induced absorption have been observed (77,78)

. The nonlinearresponse of SDGs had been observed to be usually comprised of a fast and a slowcomponent. But the exact role of free and trapped carriers could be elucidated onlyrecently by time resolving the nonlinear response of a CdS

0.3Se

0.7sample having

experienced various degrees of darkening. For the fast component, the decay time wasobserved to decrease upon darkening whereas the magnitude of this fast component wasobserved to be indepen-dent of the degree of darkening. The magnitude of the slowcomponent steadily decrea-ses upon darkening. These results support the two classes ofparticles hypothesis in agreement with theory (if we assume one class of particles, C

finvolves a contribution from trapped carriers). Particles without traps give rise to thefast (free carrier) compo-nent, the lifetime T1 decreasing from ~ 1.4 ns to ~ 30 ps upondarkening. Particles with traps in which the carriers are very quickly trapped give rise tothe slow component, darkened particles no longer contributing to the response.

For a non darkened sample, the slow component usually dominates whereas, for adarkened one, the fast component dominates and the slow component eventuallyvanishes. But the dominant mechanism cannot be known a priori: it depends on theorigin of the sample and on its past history. Using a reference sample, the magnitude of

Cf and C can be obtained. For a fresh sample, C S is found to be of order 10-10

e.s.u. ins

agreement with the estimate. Cf is found to be also of order 10-10

e.s.u. when the

numerical estimate for the bleaching term is 10-8 e.s.u. This indicates that, even in

nanoparticles, we still have substantial cancellation between induced absorption andbleaching.

If we think of applications, SDGs in the resonant regime are characterized by thefigure of merit where is the absorption coefficient and τ

rthe carrier

recombination time (also denoted Tl in the previous model). The response time of thefast component can be made very small by darkening (at the same time, the slowcomponent disappears) and a bistable device with a response time of ~ 25 ps has beendemonstrated. Carrier recombination may also become faster at higher laser intensitybecause of Auger processes but such high intensities correspond to a saturated responseand must be avoided.

The size dependence of the nonlinearity and of the corresponding figure of meritare also of interest. In the resonant regime, in a working device, the SDG would rapidlybe darkened. One would then be left with the fast component. It has recently been

456

observed(79) that the figure of merit associated with this fast component decreases whenthe particle size decreases.

(50)

Two-photon absorption and free carrier absorption. We now turn to the casewhere the laser beam is not absorbed. In the bulk, we would say ω < E

g. But we point

out that, for SDGs, Eg is quite often defined as the photon energy at which the

absorption coefficient reaches a certain value (for example 2 cm-1

). This defines aneffective E

g which takes confinement into account. Although when ω < E g we do not

have one-photon absorption, when Eg/2 < ω < E

gwe may have two-photon absorption

(TPA). In the non resonant regime, TPA was the first property to be studied, the mainpoint being again to pinpoint the effect of quantum confinement. When a laser beamexperiences TPA, its intensity is attenuated according to:

(49)

where b, proportional to the imaginary part of (ω,–ω,ω), is the two-photonabsorption coefficient of the SDG. The second term accounts for what is known as freecarrier absorption (absorption due to free carriers created by TPA). N is the numberdensity of free carriers and is given by :

neglecting carrier recombination on the picosecond time scale ; σ is the absorption crosssection of free carriers.

The selection rules for TPA are different from those of linear absorption as hasbeen verified (80) experimentally. Measuring the transmission of a sample as a function ofthe incident intensity and fitting with eqn 49, one may obtain the values of β and σ.When using picosecond laser pulses, free carrier absorption was observed(81,82) to play an

important role, σ being of the order of 10- l 8

cm2 . This role could be minimized by usingfemtosecond pulses which allow a direct measurement of TPA. Since the glass matrix

is real and does not contribute to the imaginary part of (3) , from the measuredvalues of β and of the volume fraction one may deduce the value of the imaginary part

of the nanoparticle χ(3). When plotted as a function of ω /E

g, it was recently

observed (80,82) to have the same value and behavior as in the bulk semiconductoralthough different results had been reported(83) earlier.

The dispersive nonlinearity. There have been some studies of the real part ofin the same non resonant frequency range. When TPA is present, the free carriers

thus created modify the index of refraction. This is a χ(5)mechanism. When working

with a single laser frequency, this free carrier refraction is usually dominant and, inoptical phase conjugation measurements, the conjugate intensity was observed to scaleas the fifth power of the laser intensity. To avoid the problems of free carrier refraction,frequency mixing, also known as non degenerate four wave mixing, was used. When thedifference ω - ω ' is large enough, free carriers do not contribute to generation of thewave at frequency 2ω - ω'. In this way, the real part of (3) could be measured(84,85) . The

457

(3)

General remarks

dominant contribution was found to be due to the glass matrix. Subtracting χd(3) and

taking the volume fraction p into account, the real part of the nanoparticle χ was

obtained and plotted as a function of ω /Eg. The real part of χ(3)

was also found to be thesame as in the bulk and to agree with the results of a two-band model. Here again,different experimental results have been reported.

Contrary to the resonant case in which the nonlinear response is due to populationchanges, in the non resonant case it is due to the optical Stark effect which shifts theenergy levels as was shown in . In the non resonant regime, since the absorption lossesare very small and the response is very fast, a material is characterized by a figure ofmerit different from the one quoted above. The criterion is now that the real part of (3)

be larger than its imaginary part.

The optical nonlinearities of the II-VI semiconductor dots in glass matrices and inparticular their optical Kerr nonlinearity are now reasonably well understood and can berelated to some key spectroscopic features of these nanoparticles. This regards themagnitude, dynamics and frequency behavior of these nonlinearities. Along with thesaturation mechanisms, these are affected by two-photon direct or indirect transitionsand the presence of traps in the semiconductor-glass interface.

The case of CdSe and its CdSxSe1-x

alloys considered here is not unique and resultsand conclusions apply to other II-VI and III-V semiconductor quantum dots althoughthe relative contribution of the different processes may be different. There are also casesof semiconductor quantum dots where additional studies must be performed beforeconclusive predictions can be made regarding the optical nonlinearities.

These materials have potentials in optoelectronics pending certain improvements inthe fabrication and characterization techniques.

EXTENSIONS AND GENERAL REMARKS

The previous discussion was concentrated on the optical Kerr nonlinearities incomposite materials formed by interfacing or embedding artificial mesoscopic materialsin transparent dielectrics. Besides these all optical or photoinduced modifications of theoptical characteristics of these materials a whole class of other or similar effects innonlinear optics can be envisaged there that result from the combined impact of anintense light beam and another external agent such as a static electric or magnetic fieldor even an acoustic wave. Here we have in mind modifications and modulations of thecharacteristics mediated through electro-optic, acousti-optic or magneto-optic coupling.All of them can be strongly affected by the photon or electron confinement along thelines discussed above. Of particular interest are the nonlinear magneto-optic effects asthey exhibit some quite distinctly new features with respect to the other two classes.These effects and in particular the photoinduced Faraday rotation allow a very efficientphotoinduced control of the polarization state of an optical field and the development ofreciprocal optical devices like optical valves and others.

The photoinduced Faraday effect originates(86) from the combined effect of theFaraday rotation and the optical Kerr effect. In an isotropic medium in the presence of astatic magnetic field H0 the two eigenmodes of frequency ω in the direction of H0are theleft and right circularly polarized waves with indices n- and n+ respectively ; through the

458

optical Kerr effect the latter become intensity dependent for high light intensities orI. Accordingly the polarization direction of a linearly polarized input wave

E of frequency ω after propagation through a length L in such a medium collinear withH0 rotates by an angle

where θF = ωL (n- - n+) /2nc is the usual linear Faraday rotation angle and ∆θ NL= θ2 I =ωL (n ) / 2nc is the photoinduced change of the latter and is proportional to the2- - n 2+

difference of the optical Kerr coefficients for the left and right circular polarizations.Accordingly the previous photon and electron confinements here too will lead to

an enhancement of the photoinduced Faraday rotation in mesoscopic materials whenworking close the morphology related resonances and certain provisions are made thatthe difference of the optical Kerr coefficients for left and right polarizations is large.This can be done for instance in II-VI semiconductor nanocrystals like CdS, CdSe orCdTe doped with magnetic impurities such as Mn, the so called semimagnetic or dilutedmagnetic semiconductors. Through the spin exchange interaction of the band andimpurity electrons the Landé factor of the band electrons is enhanced (87) by almost twoorders of magnitude and similarly the magneto-optical coupling and the Zeemansplitting of the electron states. Otherwise stated, the magnetic impurities act as localamplifiers of the static field. Without optimizing the interaction configuration, giantphotoinduced Faraday rotations have been observed in bulk(88) semimagneticsemiconductors like Cd1-x MnxTe and their quantum confined nanostructures(89). In thelatter case in fact these photoinduced Faraday rotations are as large as the linear Faradayrotation when the saturation regime of the quantum confined resonances is reached ;thus in a 1 µm thick multiple quantum well of ten repeat units CdTe/Cd1-x Mn xTephotoinduced Faraday rotation angles as high as 20-30 degrees are achieved(89) formoderate light and magnetic field intensities when the frequency is tuned close to thequantum confined excitonic transitions. Similar effects are expected in Cd1-xMn xTenanocrystals in a glass matrix and much effort is presently concentrated in thefabrication of such materials.

This and other cases clearly justify the present interest and growing effort in theartificial fabrication of composite materials formed with mesoscopic particles embeddedin a transparent dielectric, their interfacing and doping with other constituents as well asthe control of their morphology. This is a rich field for future investigations that opensnew avenues for materials research.

In summary in the previous four sections we discussed the way photon andelectron confinement affect the optical Kerr nonlinearities in artificial compositematerials formed from mesoscopic particles in a transparent dielectric. As it wasstressed in the introductory section the two confinements cannot be simultaneouslyimplemented in the same material in the optical range. The first one leads to anenhancement of the effective interaction path through multiple reflections or scatteringand can be achieved only in mesoscopic particles of highly transparent dielectrics and ofdimension of the order of the optical wavelength embedded in another transparentdielectric of different linear or nonlinear index. The second one, namely the electronconfinement, leads to an enhancement of the optical nonlinearities close to certainmorphological resonances due to either dielectric or to quantum confinement asexemplified in metal and semiconductor nanoparticles respectively embedded in a glass

459

matrix and can be achieved when the size of these nanoparticles is smaller than certain acritical length.

The enhancements achieved in all these cases are substantial but the keyadvantage of these materials relies on the fact that the confinement is artificiallyimplemented and accordingly their properties can be tailored to meet several criteria andrequirements for their use in devices. Indeed besides the ones relevant to their intrinsicnonlinear optical properties these materials also possess most of the other propertiesrequired for device applications such as robustness, photochemical stability,miniaturisation, interfacing etc, in fact to a higher degree than bulk materials, and makethem attractive and competitive with the best bulk materials.

Certainly more work is needed to bring them to the level where their propertiescan be precisely tuned and adapted to a prescribed device application but the essentialfeatures regarding their nonlinear optical behavior are now fairly well understood andcan be used as a basis for developing appropriate growth and fabrication techniques.Several new approaches are presently emerging in this direction. In addition thesemesoscopic materials have several other interesting properties for physicochemical,chemical or even biological applications and the combined effort and interest in theseareas has produced remarkable progress in bringing together these features and forgingnew interdisciplinary research topics.

460

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463

CONTRIBUTORS

V. M. AGRANOVICH, Institute of Spectroscopy, Troitsk, Moscow Obl., Russia

G. ASSANTO, Optoelectronic Laboratory, University « Roma Tre », Rome, Italy

H. BENISTY, Ecole Polytechnique, Palaiseau, France

P. BONTEMPS, Dept. of Pure and Applied Physics, University of Salford, Salford, UK

A; A. BOARDMAN, Dept. of Pure and Applied Physics, University of Salford, SALFORD,UK

S. BRASSELET, France Telecom, Bagneux, 92220-Bagneux, France

Z. CHEN, Department of Electrical Engineering, Princeton University, Princeton, NewJersey, USa

B. CROSIGNANI, Dipartimento di Fisica, Universita dell’Aquila, L’Aquila, Italy

P. DI PORTO, Dipartimento di Fisica, Universita dell’ Aquila, L’Aquila, Italy

M. FEJER, Ginzton Laboratory, Stanford University, Stanford, CA 94305, USA

Ch. FLYTZANIS, Laboratoire d'Optique Quantique, Ecole Polytechnique, Palaiseau, France

F. KAJZAR, CEA, LETI - Technologies Avancées, CE Saclay, Gif sur Yvette, France

A. E. KAPLAN, Dept. of Electr. & Comp. Eng. The Johns Hopkins University, Baltimore,MD, USA

K. KOUTOUPES, Dept. of Pure and Applied Physics, University of Salford, Salford, UnitedKingdom

S. KRYSZEWSKI, Institute of Theoretical Physics and AstrophysicsUniversity of Gdansk, Gdansk, Poland

M. MITCHELL. Department of Electrical Engineering, Engineering Quadrangle,Princeton University, Princeton, NJ, USA

J.- M. NUNZI, CEA, LETI - Technologies Avancées, CE Saclay, Gif sur Yvette, France

465

D. OSTROWSKYLaboratoire de Physique de la Matiere Condensee, Universite de Nice - Sophia Antipolis,NICE, France

R. REINISCH, LEMO-ENSERG, Grenoble, France

G. SALAMO, Department of Electrical Engineering, Engineering Quadrangle, PrincetonUniversity, Princeton, NJ, USA

M. SEGEV, Department of Electrical Engineering, Engineering Quadrangle, PrincetonUniversity, Princeton, NJ, USA

M.-F. SHIH, Department of Electrical Engineering, Engineering Quadrangle, PrincetonUniversity, Princeton, NJ, USA

P. L. SHKOLNIKOV, Dept. of Electr. & Comp. Eng, The Johns Hopkins University,Baltimore,MD, USA

G. I. STEGEMAN, University of Central Florida, CREOL, Orlando, FL, USA

F. STRAUB, Abteilung für Quantenoptik, Ulm University, Ulm, Germany

L. TORNER, Dept. Signal Theory and Communications, Polytechnic University of Catalonia,Barcelona, Spain

E. W. VAN STRYLAND, Physics and Elec.& Comp. Eng., University of Central Florida,Orlando, FL, USA

C. WEISBUCH, Ecole Polytechnique, Palaiseau, France

K. XIE, Dept. of Pure and Applied Physics, University of Salford, Salford, United Kingdom

J. ZYSS, France Telecom, Bagneux, France

466

INDEX

Absorbance, 112, 113Acceptance bandwith, 385–386Active amplifier nonlinearities, 24AgGa1-x In xSe 2, 393AgGaS 2,393AgGaSe 2, 393All-optical amplifiers, 346

devices, 19diode, 350Mach–Zehnder switch, 19modulators, 341, 346poling, 31,95, 99, 104, 107, 121processing, 341,369switching, 46, 342,351,229transistors, 341

Ammonium dihydrogen phosphate (ADP), 392,394

Amplitude modulation, 341of soliton, 229

Analog operations, 161Angular acceptance criterion, 385

momentum, 234steering, 365

Anharmonic oscillator, 300potential, 309

Anomalous-dispersion regime, 193Anticrossing region, 144Approximation of non-degenerate band, 442Area theorem, 140AsGa quantum well, 139Attenuated total reflection (ATR) technique, 119Auger processes, 456Auzel effect, 410,411Axial order, 112Azimuthal modulational instability, 248

Bandfilling, 24Bare cavity, 141

cavity photon, 142Basic envelope soliton solution, 223BaTiO3 (barium titanate), 260,270BBO (BaB2O4), 344,356,378,393Beam locking, 242Biexciton two-photon transition, 133

Birefringence, 110Birefringent media, 379

phasematching, 379, 380walk-off, 365

Bloch 2π-soliton, 296full equations, 295,296optical equation, 158

Bloch–Boltzmann equations, 150, 168, 170, 171,175, 177,297

Bohr radius, 440Bound electron effects, 24

quantum well excitons, 137Boussinesq-like equation, 291,299Bragg coupling, 361

reflection, 361Bright pulse, 266

Carrier frequency, 187, 192heating, 27

Cascading effects, 29phase-shift, 341

Causality, 40Cavity model, 136, 139

cross-section of, 145normal modes, 135polariton modes, 142polaritons, 141

Cd1-xMnxTe, 459CdS, 141,441CdSe, 443,458Cerenkov phase matching, 9

radiation, 320,333–340Charge transfer molecule, 106χ (2) grating in glass fibers, 123χ (2) susceptibility, 29–30, 94, 97χ (3) susceptibility’ 28–29Chirp, 191–192

balance, 185from dispersion, 199from nonlinearity, 199parameter, 190, 197positive, 198spatial, 195

Chirping, 190–l91, 196

467

Chromophore, 105orientation of, 119

Cinnabar (HGS), 394Coherence length, 344, 378Coherent anti-Stokes Raman spectroscopy (CARS),

16frequency degenerate effects, 15interactions, 280

Collision integrals, 167kernel, 159–160, 162operator, 163–166, 168–169, 177–178interactions, 366in Kerr media, 280in photorefractive nonlinear media, 261in saturable nonlinear media, 261

Colloidal crystals, 430Complete elliptic integral, 385Composites, 427Confinement, 214

energy, 440Contact poling, 107Continuity equation, 262Convention factors, 103Conversion efficiency, 378,385–386,401

assymptotic limit of, 387Corona poling, 106–108, 126Correlation function, 174, 175, 179Coulomb long-range attraction, 297Coupled-modes 319, 323, 333, 339, 340Critically phasematched crystal, 388Cross-phase modulation, 21Crosslinking, 107Crystal Violet, 82–83Cubic nonlinearity, 343Cut-off frequency, 313Cyanobiphenyl, 109

DAN, 353–354Dark intensity, 263Dark-line defects, 417DAST, 233Data storage, 407DC poling, 114Debye length, 263Defect nucleation, 417Defocusing, 343Degeneracy factors, 3,101, 103Degenerate four-wave mixing (DFWM), 53

geometry of 18Depletion, 344Dielectric breakdown, 108

composite, 434confinement, 428,437constant, 18, 111photonic band gap, 434susceptibility, 141tensor, 379

Difference frequency generation, 375Diffraction, 202,229

length, 207limited propagation, 219

Diffusion field, 263tensor, 153, 157, 175time, 142

Digital operations, 142(1,5)-Dinitro,(2,6)-di(N,N)-n-butylaminobenzene

(DNDAB), 82–83,94Diode, 356

lasers, 407,408; 425Dipolar χ (2) symmetry, 98

approximation, 101interaction energy, 113moment, 107

Directional coupler, 358Disperse red #1 (DR#1), 82–83,97, 120

in PMMA, 121Dispersion, 144

balance, 185equation, 215law, 141limited propagation, 219

Dispersive nonlinearity, 457Display, 407–408Distributed Bragg reflector (DBR), 419

couplers, 359feedback grating, 361

DNA, 425,425Doppler approximation, 173

limit, 173shift, 293

Down-conversion, 334, 342,369DR1-MMA, 96,98Drift velocity, 171Drude term, 445Duffing effect, 260–261,264,285

equation, 295, 298

Effective length, 390mass approximation, 439–440medium approach, 437nonlinear coefficient, 380phase mismatch, 378

Efficiency of the polar ordering, 116Eigenmodes, 293Eigenvalues, 353Electric field induced second harmonic generation

(EFISH), 77,80Electroabsorption, 452Electrode poling, 107Electromagnetic bubbles (EMB’s), 291–318

amplitude of, 305generation of, 302length of, 298non-oscillating, 296precursor of, 303–308propagation of, 299pulses, 315solution, 294,297steady-state, 298small-amplitude, 298unipolar, 314

468

Electromagnetic confinement, 430, 432solitons, 291

Electron beam poling, 107–108confinement, 428, 430–431, 436

Electrooptic (Pockels) effect, 13, 262coefficient, 311–312materials, 14tensor, 262

Energy efficiency, 386–387flow, 233localization 361regulators, 70transfer (ET), 410, 413, 414, 416, 422

Envelope conservation, 443equation, 215,218function, 143,215, 216soliton, 200,213

Epitaxy, 104Excited State Absorption (ESA), 37, 409, 410,

413–414, 416, 420Exciton bleaching, 24

branches 134donor-bound, 133

Exciton–exciton screening effects, 137External self-action, 53Extraordinary crystal directions, 350

refractive index, 379EZ-scan technique, 53–55

Fabry–Perot cavity, 388quasi-modes, 146

Faraday effect photoinduced, 458Faraday rotation, 458

photoinduced, 458–459Fermi level, 439, 442 ,447 ,449Ferroelectrics, 399–400Ferroelectric crystals, 350Fibers, 354Fiber amplifiers, 356

optic temperature sensor, 420, 425Figure of merit, 378,429First order hyperpolarizability, 87, 89Fluorescence, 423

green, 414line narrowing (FLN), 451

Focal power 390Fock–Leontovich wave equation, 265Focusing, 343Four-wave mixing, 134, 349Fourier components, 187

frequencies, 300Franz–Keldysh behavior, 452Fraunhofer diffraction pattern, 206

far field, 206Free carrier absorption, 59, 457

refraction, 59Frenkel excitons, 133Frequency conversion, 375Fresnel (near field) pattern, 206Functionalized polymers 105

GaAs, 417GaN, 408, 417–418Gap solitons, 341, 361–362Gauss law, 262Gaussian beams, 311, 365

half-cycle pulses (HCP), 311spatial profile, 311temporal profile, 311

Gibbs–Boltzann distribution function, 112–113Glasses, 27Glass transition temperature, 108Gradient velocity, 154, 157Gratings, 359Grating coupler, 359Group index mismatch, 386Group velocity, 138, 144, 188, 216, 379

dispersion, 189, 195,221Guest–host systems, 105Guided modes, 319–326, 336, 337, 339

powers, 353wavenumber, 216waves, 341,369wave all-optical switching devices, 341wave wavevectors, 353

Half-cycle pulses (HCP), 302–315Half-harmonic generation, 352Hamiltonian, 440,452

dynamical system, 233Harmonic generation, 316

light scattering, 80–81, 82polarized, 94, 96

Helmholtz’s equation, 264–265Heteroepitaxy, 104Hole burning spectroscopy, 451Holmium, 415Hot electron contribution, 447

Idler wave, 390Imbalance, 351–352Improper poles, 328, 331, 335, 336Incoherent all-optical effects, 15

interactions, 280Induced polarization, 428

transparency, 361Infrared quantum counter, 409InP:Fe, 286Interband contribution, 447

nonlinearities, 25transitions, 442–443

Interferometer, 357Inverse scattering transform (IST) method, 226Inversion symmetry, 101Isolators, 341

K(TiO)P1-x AsxO4, 395KDP crystal, 97, 378Kerr effect, 40, 259, 260, 262, 268, 286

coefficient, 342–343limit, 278medium, 274, 277, 280

469

Kerr effect (cont.)nonlinearity, 20, 27, 133, 143, 146, 291polarizability degenerate, 454regime, 272solitons, 262, 277suceptibility, 446

Kerr-like dark solitons, 275manner, 203nonlinear medium, 272

Kleinman permutation symmetry, 94KNbO3, 393Kohlrausch–Williams–Watt (KWW) stretched expo-

nential function, 128Kortdeweg–de-Vries (KdV) differential equation,

184–185, 389modified, 115, 300, 307

Kramers–Kronig relationship, 23, 40, 452KTA(K(TiO)AsO 4, 394KTP(KTiO)PO 4, 233, 344, 351–354, 364–368, 378,

393, 399–401

Laboratory frame time, 188–189reference frame, 110

Landau theory of Fermi liquids, 448Lanée factor, 459Langmuir–Blodgett technique, 104

films, 10Laser mode-locking, 349

pumped fiber lasers, 408Leaky mode, 319–320, 326–328, 332–340Legendre polynomials, 112–113LiB2O3, 393Light bullets, 230Light emitting diode (LED), 408, 418

arrays of, 408Light-induced diffusive pulling (LIDP), 156–157

drift (LID), 149, 151, 155, 171, 178kinetic effects (LIKE), 149, 151

Lighthill criterion, 220test, 221

LiNbO3 (lithium niobate), 233, 260, 270, 354, 356,358, 360, 378, 397–403, 418, 420

electrically poled, 402waveguides, 391

Line broadening, 451Linear Boltzmann equation, 160

cavity polaritons, 141coupled equation, 229polarizability, 87, 110

Liquid crystals, 116crystalline polymer, 107nematic, 112smectic, 112

Liquid droplet, 433LiTaO3 , 270, 399–401Local field factor, 85, 103

time, 189–189, 197Localization regime strong, 434

weak, 434Localized wave, 184Lorentz–Lorenz formula, 103

Low conversion limit, 314, 377frequency limit, 314

LSJ manifolds, 412

Macroscopic polarization, 102second order NLO suceptibility χ(2), 102

Magneto-optic coupling, 458Main chain polymers, 105Manakov’s solitons, 286Manley–Rowe relation, 233Material figure merit (FOM), 19, 387; see also Fig-

ure of meritnonlinearities, 20systems, 20

Maxwell equation, 291, 294, 299–300reduced, 301

Maxwell + Bloch equation, 294, 301, 306Maxwell + full Bloch equation, 292Maxwell–Boltzmann distribution, 83, 85Maxwell–Garnett effective medium approach, 449MBA-NP, 347Mesoscopic materials, 427, 431, 444Metal nanoparticles, 448Metastable level, 414Microcavities, 430Mid-Ir OPO, 400Mie theory, 437Miller’s rule, 394Modal dispersion phase matching (MDPM), 7–8Modal fields, 391Mode mixing, 360Model kernel, 162Modified Kortdeweg–de-Vries (KdV) equation, 115,

300, 307NLS, 393spherical Bessel functions, 113

Modulational instabilities, 236Molecular axis, 110

beacon, 415, 420, 423deposition techniques, 134diode, 77engineering, 90epitaxy, 104hyperpolarizabilities, 101nonlinear optics, 99nonlinearities, 22reference frame, 110re-orientation, 120

Morphological resonances, 427Morphology dependent resonances, 433Multiple quantum wells (MQW), 356MSVP model, 116–117Multi-bubble solution, 307Multi-Electromagnetic bubbles, 303Multiphoton absorption, 89Multiple dark sceening solitons, 276

scattering regime, 436Multiplicative factor, 103Multipolar molecules, 79, 89Multipolar symmetry, 97, 99

systems, 78

470

Near-resonant response, 23Nematic liquid crystals, 112; see also Liquid crystalsNLO susceptibility tensors, 102Non-integrable equations, 186Non-resonant response, 23Noncentrosymmetric materials, 341Nondegenerate nonlinearities, 62Nonlinear absorption, 343

atoms, 291coefficient, 377competition, 341drive, 377–378, 385integrable equations, 186integrated devices, 342, 353length, 207materials, 29 392–394, 398–403Maxwell equation, 301mechanisms, 21medium, 219non-integrable equations, 186optical beam propagation, 264partial differential equation, 184phase shift, 341polarization, 2processes, 341scattering, 432

multiple, 436scattering cross-sections, 81Schrödinger equation (NLS), 134, 144, 186, 191,

199, 213–214, 219–220, 231, 259,267–268

fundamental solution of, 222, 223spectroscopy, 16susceptibility, 3, 375, 377

spatially varying, 396Nonlinearity balance, 185Nonlocality, 428Nonresonant Kerr nonlinearity, 439Normal modes, 137Normalized brightness, 385

intensity, 263

Octupolar configuration, 97χ (2) symmetry, 98macroscopic susceptibility, 90molecules, 93

One photon absorption, 20contribution, 93excitation probability, 91–93

One-dimensional diffraction, 208Optical bistability, 34 l–342

density, 112dielectric constant, 111, 115engineering, 369Kerr effect, 448, 452–453, 458–459; see also Kerr

effectnonlinearity, 427–429, 438, 446

limiting, 62parametric amplification (OPA), 375

amplifier, 9generator (OPG), 9, 403

Optical Parametric Oscillator (OPP), 99, 390–391,402, 403

oscillation, 9–10, 376phase conjugation, 258, 453, 455piston, 156poling, 89; see also All optical polingrectification, 292, 342signal idler, 390Stark effect, 458

shift, 446Optimum aspect ratio, 385

drive, 385length, 388

Orbital angular momentum, 411Order parameter, 111, 112Ordinary crystal directions, 350

refractive index, 379Orientation distribution function, 96, 112Orientational averages, 117Overlap integral, 7

Parametric amplifier, 390; see also Optical paramet-ric amplifier

conversion, 407effects, 341interplay, 368

Paraxial approximation, 311–312Periodically poled LiNbO (PPLN), 344, 347, 402,3

420–421LiTaO 3, 403OPO, 402

Phase and polarization interferometers, 345Phase conjugation, 260, 349; see also Optical phase

conjugationconjugation, 260matched crystals, 390matching, 344, 378, 387

birefringent 395condition, 5critical, 387, 391modal, 395noncritical, 387, 391, 394SHG, 5type I, 240,380type II, 253, 342, 347, 380

mismatch, 342, 389shift, 369velocity, 216

Phase-to-amplitude transducers, 346Phase-velocity matching, 376Phonon spectrum, 412

thermal LO, 142Photoassisted poling, 104, 107, 119, 123Photo-crosslinking polymers, 105, 107Photodarkening, 451Photogalvanic effect bulk, 270Photoinduced χ(2) , 96

motion, 94nonlinear response, 97

tensor, 98processes, 78, 83, 86

471

Photoinduced χ (2) (cont .)reorientation mechanism, 99susceptibility, 95

Photoisomerization, 86Photoluminescence excitation (PLE), 451Photon confinement, 428, 430

echoes, 316Photonic engineering, 77, 90Photorefractive crystal, 261, 273

soliton, 274effect, 13, 260–261, 264, 285

precursor of, 303, 306–308in LiNbO3 , 394screening solitons, 271, 279, 281–282; see also

Solitonsspatial solitons bright and dark, 259–260; see also

Solitonsvector solitons, 286; see also Solitons

Photothermal poling, 107–108Photovoltaic (or photogalvanic) effect bulk, 270

current, 271self-defocusing nonlinearity, 271soliton, 260, 270–271

Planar waveguide, 209–210, 353Plane wave, 2, 341Plane waves, 341PMMA-DR #1, 109, 121, 125; see also DR #1-

PMMAPockels effect, 261, 286; see also Electrooptic effectPolar order, 112

growth of 125orientation of chromophores, 117spatial profile of, 125

Polarization of seeded SHG coefficient, 125configuration, 96driven wave function, 4field, 108rotation of, 351

Polaritons linear, 134Polarized HLS, 94, 96Poled polymer, 10, 104, 111, 129Poling, 95, 99, 107, 123

efficiency of, 109, 129field, 110

Polymers thermally crosslinking, 105, 107Potassium dihydrogen phosphate (KDP), 392, 394Potassium niobate, 344; see also KNbO 3Poynting vector, 205

walkoff, 231, 242, 387, 390Prism coupler, 359Proper poles, 320, 323, 326, 328Pulse and beam envelope soliton propagation, 226

conditioning, 349dispersion, 346

in a linear medium, 197in nonlinear medium, 198

propagation in linear media, 193in nonlinear medium, 194

spreading, 189pump-probe spectra, 421pump-probe Z-scan, 59

Quasi phase matching (QPM), 6, 8, 13, 31, 233, 240,350, 375, 377, 395–403, 418, 420

configuration of, 428,434history of, 398interaction, 396materials, 400mid-IR generation of, 400OPO, 402theory of, 396

Quadratic cascading, 341, 343materials, 343nonlinear frequency conversion, 377

Quantum confinement, 428, 446confinement Stark effect, 452well, 133–134, 136, 142–143, 146, 417

asymmetric, 12–13exciton, 137

Quartz, 392Quasi-steady-state photorefractive dark solitons,

277solitons, 270spatial solitons, 270

Quenching, 414, 423

Rabi frequency, 295, 301–302phase, 301splitting, 299splitting dimensionless, 142

Radar signals, 192Radiation fields, 319–321, 323, 326, 328, 334, 336Radiative waves, 122Random composites, 432Rare earth (RE), 410

doped fiber, 407Real-time holography, 260Recombination time, 142Refractive index nonlinear, 30Relaxation, 128

rates, 129times, 129

Repulsive Kerr collision, 281Resonance, 319, 328–329, 331, 333, 335–336,

338–339Resonant enhancement, 3

response, 23Retardation, 429Reverse-saturable absorber, 62

Saturable absorption, 22plasma density, 142

Saturation spectroscopy, 451SBN (strontium barium niobate), 260, 275–276Scattering potential function, 227Schott filters, 450Schrödinger equation, 200, 222, 291

nonlinear solution of, 222nonlinear unmodified, 235solution of, 294, 297

Scientific instrumentation, 407Screening solitons, 271, 279, 281–282, 272–274,

263

472

Second harmonic field, 377Second harmonic generation (SHG), 4, 95, 124; 126,

230, 341, 375, 378–380, 418, 425acceptance bandwidth, 386asymptotic limit of efficiency, 387Band width limitations, 387conversion efficiency, 385–386, 390critically phasematched, 388effective length, 390efficiency, 389energy efficiency, 386–387externally resonant, 388figure of merit, 5, 12, 387, 388focused, 387growth of, 118high average power, 389in lossy materials, 387noncritically phasematched, 388normalized efficiency of, 401–402phase matching in waveguides, 6phase mismatch, 389in quasi phase matched structures, 394–403in silica, 78single pass conversion efficiency, 389in situ, 118spatial walkoff effects, 387temporal decay of, 128thermal phase mismatch, 389Type I, 4, 6Type II, 6ultrafast, 385, 387, 403in waveguide, 12waveguide devices, 391

Second-order hyperpolarizability, 87soliton, 201, 234susceptibility, 342

Seeding, 346, 361power dependence, 127procedure, 125, 129

Self focusing, 40, 276, 286, 202, 204–205, 207–212229

effects, 272Self-confined beams, 341Self-defocusing, 276

nonlinearity of, 271Self-guided beams, 364Self-induced transparency, 133, 145, 291, 316Self-induced transparency (SIT) soliton, 139

pulses, 137Self-phase modulation, 21Self-trapped beams, 208, 210, 212

beam propagation, 263, 265solutions, 261(stationary) solutions, 206, 267

Self-trapping, 221, 229of an incoherent beam, 261, 285

Semiconducting band gap, 25Semiconductor, 37, 47

doped glasses, 431microcavities, 133nanocrystals, 431, 449

Semiconductor (cont .)nonlinearities, 24photonic band gap, 434

Semimagnetic semiconductor, 459Shallow water, 183–184Shock formation distance, 309

waves, 296Shock-like wave fronts, 308Side chain polymers, 105Sidelobes, 345Signal parametric gain, 390

wave, 390Similarity rules, 235Simultons, 362Sine-Gordon equation, 138, 291, 301SIT pulses, 139–141SKS model, 116Slowly-varying-envelope approximation, 342Small-signal amplification, 351Smectic liquid crystals, 112; see also Liquid crys-

talsSolitary wave, 183–186, 190, 292, 315

waves spatial, 342, 364Soliton, 183–286, 291, 301

(1+1), 235(2+1), 235amplitude of, 229, 305annihilation of, 271basic envelope solution, 223beam, 274bright, 234, 267, 268, 279

condition for, 295envelope of, 186, 187, 199photorefractive screening, 266screening, 271–272spatial, 202vortex, 270

buried, 236collisions, 278, 280content, 200critical energy, 140dark, 186, 220–221, 267, 269, 275–276

sceening, 275solutions, 145vortex, 270

discrete, 250excitation of, 239existence conditions, 230

curves, 26first-order, 234fission of, 89, 271formation, 198fundamental, 201fusion of, 89, 271generation of, 183, 316grey, 267high-order, 200, 266–267incoherent, 283, 285

incoherent pair, 286induced waveguides, 261intensity, 271

473

Soliton (cont.)interactions, 271like, 156lowest-order, 198multiple dark, 276numbers, 201optical Kerr, 278

fiber, 291oscillating, 239period, 200photorefractive, 261, 272, 279photovoltaic, 260, 270, 271quadratic, 31in quadratic nonlinear media, 230in quasi-phase-matched samples, 270propagation, 316pulse nonlinear, 134Russel’s, 185–186solution, 144, 146, 186spatial, 183, 206, 208, 210, 212–224, 229, 260,

269–270, 283bright, 202, 260dark, 260screening, 267, 269in a waveguide, 209

spiraling, 199, 283temporal, 183, 187, 195, 213, 218, 221, 224–225,

260,362velocity of, 146, 229, 243in water, 185width of, 272

Space charge field, 125–126, 261, 272, 286wave, 328

Spectral hole burning, 27Spherical Bessel functions, 112

harmonic functions, 79Spin quantum number, 411Split modes, 146, 364Square pulse envelope, 191Surface plasmon resonance, 448Stark effect, 114

levels, 411shifted frequency, 306

Static field poling, 107Kerr effect, 452–453

Steady-state nonlinear susceptibility, 143Stimulated Brillouin scattering, 16–18

cascade Raman scattering, 292Raman scattering, 16–17Rayleigh scattering, 16–17

Stop-gaps, 361Strong confinement regime, 440Sum-frequency generation, 407Surface damage threshold (in SHG), 391Surface plasmon resonance, 438

switching, 46, 229, 341–342Symmetry of the induced χ(2) susceptibility, 126

Taylor expansion, 187–188TE modes, 136Temporal pulse, 195

Thermal focusing, 401lensing, 390mismatch, 390phase mismatch, 389

Thermally crosslinking polymers, 105, 107Thermo-optic effect, 24Thermodynamic force, 154, 162Third-order Kerr nonlinearity, 140

NLO phenomena, 15nonlinearity, 136soliton, 234susceptibility, 37, 47

Three-wave mixing, 134, 347Three-level model, 87Three-wave interaction, 342Threshold intensity, 391

fluence, 391power, 199

Three photon absorption, 20Thulium, 415Ti:Sapphire laser, 300Time-domain, 198TM modes, 136Topological charge, 270

formation of, 248wave front dislocations, 2586

Trans-ci isomerization, 119–120Transport coefficients, 167, 168, 171, 178–179Transverse beam momentum, 234

radius, 312Two-beam coupling, 37

interactions, 37Two-color Z-scan, 56Two-component solitons, 286Two-Level Atom, 45, 138, 267, 291

anharmonicity, 146medium, 137model, 89system, 454

Two-parabolic band model, 47Two photon absorption (TPA), 20, 23, 91–93, 453,

457

Ultrafast nonlinear absorption, 37refraction, 37signal processing, 229

Unbalanced components, 348Up-conversion, 342, 369, 409–410, 422–425

fiber lasers, 408, 416lasers, 407, 409, 415

Vacuum Rabi splitting, 142, 146Vakhitov–Kolokolov criterion, 237Valence band mixing, 442Vectorial quadratic solitary waves, 367

spatial spatial solitary waves, 364Velocity-changing collisions (VCC), 150, 177Visible light generation, 400Vortex (dark), 277

solitons, 246, 260, 276

474

Walking solitons, 241Walkoff angle, 379, 380

spatial, 387Wannier exciton, 441Wave vector mismatch, 231Waveguide devices, 391

examination, 420, 423, 425SHG devices overall efficiency of, 391SHG devices, 391

Waveguides, 353Wavelength shifters, 341, 356Weak confinement regime, 440

Weak-wave retardation, 37, 341, 346White self-transparency effect, 435Whitney’s theorem, 237Wide gap dielectric, 37

Z-scan technique, 53–75, 271, 285, 4542-color, 56

Zakharov and Shabat problem, 226ZBLAN, 412–417ZnCdSe-Zn(S)Se, 417ZnGeP2,393–394

475