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Series II: Mathematics, Physics and Chemistry - Vol. 31
Soliton-driven Photonics
edited by
A. D. Boardman Joule Laboratory, Department of Physics, University of Salford, Salford, United Kingdom
and
Springer-Science+Business Media, B.V.
Proceedings of the NATO Advanced Study Institute on Soliton-driven Photonics Swinoujscie, Poland 24 September-4 October 2000
A C.I. P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-0-7923-7131-1 ISBN 978-94-010-0682-8 (eBook) DOI 10.1007/978-94-010-0682-8
Printed an acid-free paper
AII Rights Reserved © 2001 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2001 Softcover reprint ofthe hardcover Ist edition 2001 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.
Contents
Preface
IX
Spatial Solitons in Liquid Crystals 41 M.A. Karpierz*
Magnetic Solitons 59 N.V.Ostrovskaia
Evolution of ConcentratedSolutionof Nonlinear Schrodinger Equations in RegularNon-Uniform Medium 69
Yu. N. Cherkashin and V.A. Eremenko
Observation Nonlinear Effects of a Laser Beam Interaction with Waveguide Photosensitive AgCI-Ag Films 73
E.1. Larionova, L.A. Ageev and V.K. Miloslavsky
Featuresand Applications of i 2 ) Vector Spatial Solitons 77
G. Leo and G. Assanto
SolitonTransmission through a Single-Mode Fiber 87 M. Aksoy and M.S. Kilickaya
Nonparaxial Propagation of Parametric Spatial Solitons 91 R. Petruskevicius
Spatial Solitary-Wave Beams in Kerr-Type PlanarOptical Waveguides: Nonparaxial Vector Approach 95
K. Marinov, D.1. Pushkarov and A. Shivarova
Non-Recurrent Periodic Arrays of Spatial Solitons in a Planar Kerr Waveguide 99
C. Camboumac, M. Chauvet, J.M. Dudley, E. Lantz and H. Maillotte
vi
A. Szyrnariska and T.R. Wolinski
Interactions of Solitary Waves in a Photorefractive, Second-Harmonic Generating Medium 107
A.D. Boardman, W. I1ecki,Y. Liu and A.A. Zharov
Analytical Description of Quadratic Parametric Solitons 111 A.A. Sukhorukov
Spatial Solitons in Saturating Nonlinear Optical Materials 115 B. Luther-Davies"
Nonparaxial Solitons 141 A.I. Smirnov and A.A. Zharov*
Spatial Solitons on Nonlinear Resonators 169 C.O. Weiss*, V.B. Taranenko, M. Vaupel, K. Staliunas, G. Siekys and M.F.H. Tarroja
Two-Color MUltistep Cascading - Second-Order Cascading with Two Second-Harmonic Generation Processes 211
S. Saltie1, K. Koynov, Y. Deyanova and Y. Kivshar
The Davey-Stewartson Model in Quadratic Media: A Way to Control Pulses 215 H. Leblond
Experiments on Seeded and Noise Initiated Modulational Instability in LiNb03 Slab Waveguides 219
R.R. Malendevich, H. Fang, R. Schiek and G.I. Stegeman
Soliton Signal in the Magnetic Chain at the External Magnetic Field near to Critical Value 223
LA. Molotkov
Observation of Dipole-Mode Vector Solitons 229 C. Weilnau, C. Denz, W. Krolikowski, M. Geisser, G. McCarthy, B. Luther-Davies, E.A. Ostrovskaya and Y.S. Kivshar
Spatial Self-Focusing and Intensity Dependent Shift in Lil03
using Tilted Pulses 235 B. Yellampalle and K.H. Wagner
vii
Round-Trip Model of Quadratic Cavity Soliton Trapping 239 O.A. Egorov, A.P. Sukhorukov and LG. Zakharova
SpatialSolitary Wavesand Nonlinear k-Space 245 S.M. Blair
Propagation of Short Optical Pulses in Nonlinear Planar Waveguides - Pulse Compression and Soliton-Like Solutions 251
M.E. Pietrzyk
Parametric Emission of Radiation at Spatial Solitons Interaction 257 LV. Shadrivovand A.A. Zharov
Observation of InducedModulation Instability of an Incoherent Optical Beam 261 Z. Chen, L. Klinger and H. Martin
Quadratic Bragg Solitons 267 G. Assanto*, C. Conti and S. Trillo
Effects of Nonlinearly Induced Inhomogeneity on Solitary Wave Formation 293 K. Marinov, D.1. Pushkarov and A. Shivarova*
Instability of Fast Kerr Solitons in AIGaAs Waveguides at 1.55 Microns 317 L. Friedrich, R.R. Malendevich, G.I. Stegeman, J.M. Soto-Crespo, N.N. Akhmediev and 1.S. Aitchison
Extremely Narrow QuadraticSpatial Solitons 321 A.V. Pimenov and A.P. Sukhorukov
SolitonPropagation in Inhomogeneous Media with Sharp Boundaries 325 V.A. Eremenkoand Yu. N. Cherkashin
Photorefractive Photovoltaic Spatial Solitons in Slab LiNb03 Waveguides 329 M. Chauvet, C. Camboumac, S. Chauvin and H. Maillotte
Theory ofCW Light Propagation in Three-CoreNonlinear Directional Couplers 333 P. Khadzhi, O. Tatarinskaya and O. Orlov
Two Approaches for Investigation of Soliton Pulse in a Nonlinear Medium 339 LA. Molotkov and N.L Manaenkova
Photorefractive Solitons through Second-Harmonic Generation 343 A.D. Boardman, Y. Liu and W. Ilecki
viii
Shifted Beam Interaction for Quadratic Soliton Control ' 347 D.A. Chuprakov, X. Lu and A.P. Sukhorukov
Bright Solitary-Wave Beams in Bulk Kerr-Type Nonlinear Media 351 K. Marinov, DJ. Pushkarov and A. Shivarova
Generation of Light Bullets 355 I.G. Koprinkov, A. Suda , P. Wang and K. Midorikawa
Application of Nonlinear Reorientation in Nematic Liquid Crystals 359 W.K. Bajdecki and M.A. Karpierz
Two-Dimensional Bragg-Ewald's Dynamical Diffraction and Spontaneous Gratings 363
V.1. Lymar
Solitons in Optical Switching Devices 397 E. Weinert-Raczka"
Quadratic Solitons: Theory 423 A.P. Sukhorukov*
Non-Adiabatic Dressed States for a Quantum System Interacting with Light Pulses 445
I.G. Koprinkov
Rotating Propeller Soliton 449 T. Carmon , R. Uzdin , C. Pigier , Z.H. Musslimani, M. Segev* and A. Nepomnyashchy
Theory of Cavity Solitons 459 W.J. Firth*
Discrete Spatial Solitons in Photonic Crystals and Waveguides 487 S.P. Mingaleev, Y.S. Kivshar and R.A. Sammut
Generalized Hamiltonian Formalism in Nonlinear Optics 505 VE. Zakharov
Index 519
Preface
It is ironic that the ideas ofNewton, which described a beam of light as a stream of particles made it difficult for him to explain things like thin film interference. Yet these particles, called 'photons', have caused the adjective 'photonic' to gain common usage, when referring to optical phenomena. The purist might argue that only when we are confronted by the particle nature of light should we use the word photonics. Equally , the argument goes on, only when we are face-to­ face with an integrable system , i.e. one that possesses an infinite number of conserved quantities, should we say soliton rather than solitary wave. Scientists and engineers are pragmatic, however, and they are happy to use the word 'soliton' to describe what appears to be an excitation that is humped, multi­ humped, or localised long enough for some use to be made of it. The fact that such 'solitons' may stick to each other (fuse) upon collision is often something to celebrate for an application, rather than just evidence that, after all, these are not really solitons, in the classic sense. 'Soliton' , therefore, is a widely used term with the qualification that we are constantly looking out for deviant behaviour that draws our attention to its solitary wave character.
In the same spirit, 'photonics' is a useful generic cover-all noun, even when 'electromagnetic theory ' or 'optics' would suffice. Indeed, remarkably few photons are needed to permit Maxwell's equations to be used, allowing us to focus upon continuum electric and magnetic field behaviour. Nevertheless, we are always using real materials and any nonlinearity owes its form and roots to the details of the photonic processes .
There is now considerable current momentum in soliton-driven photonics research and it embraces a very broad set of objectives. These can include understanding how materials influence outcomes through , for example, photorefractive, or magnetooptic behaviour. In addition , new mathematical results , or simulation outcomes, or strange results from using higher dimensions, or cavities, or vortices, are all contributing to the excitement level. It was felt, therefore, that a NATO ASI under the title of this book would provide a forum for some global leaders to give overviews that emphasised the common features. The aim was to unite an audience of doctoral and post­ doctoral workers into a common frame of mind, regardless of the material being discussed.
These intentions were realised and the Directors are extremely grateful for the beautiful set of lectures delivered on such a catholic range of topics. Each delivery transmitted the hands-on experience of the lecturer. The material was mainly focused upon spatial solitons because this is an area of growth and high activity, stimulated by the desire to use 'chip-level' photonics for information processing. Although the mathematics is often the same as in the temporal area,
ix
x
the physical descriptions and the applications are diffraction-based, which sets spatial solitons aside from temporal ones.
As always, the blend of topics emphasised the need not only to accumulate some knowledge of basic theory but also a working knowledge of materials and what they are capable of. Again the vision of all-optical switching, or some other kind of signal processing application was ever present in our thinking. To achieve all of this, the ASI was conducted as a School and what a great School it was. With so much expertise from all over the world being concentrated in one place, this NATO ASI was hugely successful.
No set of Directors can undertake an enterprise like this alone, however. They need help, and a lot of it! To be honest, without the fantastic work of Lynn Clarke in Salford and the great care put into the local arrangements by Ewa Weinert-Raczka, in Poland, the School would have been a logistical failure. On the Committee side, we are, as always, particularly indebted to George Stegeman for his help and guidance at every stage of the event. The Directors are extremely grateful to the NATO Science Committee for the substantial financial award that made it possible to attract 75 participants from so many countries.
The School was located in Swinoujscie, a magnificent place on the Baltic coast, and the ambience of the hotel, which housed the School, was superb. We were looked after extremely well and everybody enjoyed both the work and the relaxation periods. We believe that we have a classic collection of material in this book, which we hope will stand the test of time. We wish everybody health and happiness and we have a real desire to maintain contact with everybody we met in Poland.
Allan Boardman Anatoly Sukhorukov
A.D. BOARDMAN and M. XIE Department ofPhysics, School ofSciences, University ofSalford, Salford, M54WT, UK
1. Introduction
The word gyrotropy turns up quite often in physics and it comes from the Greek word gyros, meaning circle [I]. It is used not only in science but in engineering too, as a generic description of an event involving some rotation of the plane of linear polarisation of light. In fact, following Fresnel's proposition that linearly polarised light is a superposition of two forms of light called left and right-circularly polarised light it is clear that a gyrotropic material is associated with the appearance of elliptically, or circularly, polarised light. It is a very important area that embraces many complex materials which display a wealth of fascinating properties, like optical activity. In general then, a complex relationship exists between the field vectors E and H and the induction vectors D and B, where these quantities have their usual meanings. This relationship can be adjusted to take into account that gyrotropy can be free , natural or forced [I]. Free gyrotropy and forced gyrotropy are in the same category, because 'forced' means that it is created by an external magnetic field, for example, and 'free' is associated with internal fields. The best known example of natural gyrotropy is optical activity that is exhibited by sugar solutions and this is immediately distinguishable from the forced case by the following signature. Suppose a plane linearly polarised light wave passes once through a natural gyrotropic material causing the plane of polarisation to be rotated. If the same beam is reflected back through the material then, because it has natural gyrotropy, the rotation on the first pass is undone and no final rotation results i.e, no reversal of handedness occurs in this case [2]. This is a very important distinction from forced gyrotropy, which is the property of magnetooptic materials where the rotation of the plane of polarisation would have been doubled. Faraday discovered this and the Faraday effe ct, as it is now called, involves propagation parallel or antiparall el to an applied magnetic field. Other well-known magnetooptic effects are Voigt and Cotton-Mouton after their discoverers, which occur when the wave propagation is perpendicular to an applied field. Either name can be used but, historically, Voigt dealt with vapours while the second name-pair used liquids. Voigt will be the term adopted here to denote this type of birefringence, which is also revealed by uniaxial crystals, when a wave propagates perpendicular to the optic axis. As will be shown later, for bulk media, the Faraday effect is a non-reciprocal phenomenon and the bulk Voigt [Cotton-Mouton] [1,2] effect is reciprocal. The really interesting outcome, however, is that even the Voigt effect is non-reciprocal in an asymm etric waveguide.
A.D. Boardman and A.P. Sukhorukov {eds.), Soliton-driven Photonics, 1-20. © 2001 Kluwer Academic Publishers.
2
This is the principal source of interest in this chapter because the 'perpendicular field format' permits the use of transverse magnetic (TM) waves, which makes the systems easier to design [3-5].
The reversal of handedness in the Faraday effect and the Voigt-asymmetric guide effect reminds us of chirality in what are now termed complex media. The latter can be constructed with embedded elements and, in a very real sense, embedding is what will be done here. The huge advances in material science in recent years makes many integrated formats possible, so the issue of nonreciprocity is a control one for many applications . Its full realisation may have to depend heavily on waveguide design, or some other ideas, to overcome weaknesses, created by fabrication tolerances , showing up experimentally, but the global trends show that integrated magnetooptic devices are now set to become rather important.
The integration of magnetooptic [6] devices with semiconducting substrates that include active devices, such as lasers, detectors , and amplifiers is now possible . It would appear , then, that there has never been a better time to pursue the combination of nonlinearity and guided-wave magnetooptics. A start in this direction has shown that an external magnetic field, applied to a waveguide containing third-order optically nonlinear material and magnetooptical elements, can force bright solitons from a state of attraction into isolation from each other [7,8].
Magnetooptic material are available in static form through the application of an external magnetic field to a film of material such as yttrium iron garnet (YIG). The YIG is magnetised by an external magnetic field, to create off-diagonal elements in its permittivity tensor that are proportional to the induced magnetisation. Such elements
are usually defined as n~Q, where [9] nm is the linear refractive index of the
magnetic material and Q is the magnetooptic parameter. Q is, in general, a complex quantity and is a function of the magnetisation of the material. It will vanish in the absence of any applied magnetic field, or permanent magnetisation . Any original reservation that both the length required for significant phase shifting (in TE-TM interactions , for example) and the attenuation coefficient, for a YIG film, are too large for practical integrated-optical circuit devices has given way to the use of materials such as (LuNiBiMFeAl)sO I2 as guiding layers deposited upon Gd3GasO l 2 substrates [9]. Also, the substitution by cerium of the rare-earth garnet yields large magnetooptic coefficients . This fact, coupled with the use of bismuth and aluminium doping, to reduce attenuation to acceptable integrated-optics applications levels, shows that magnetooptic devices are promising candidates for switching and routing.
A dynamical way to introduce magnetooptic behaviour into a waveguide structure is to use magnetostatic waves, propagating in the same waveguide structure . Even though, the Russian literature [10] makes it abundantly clear that material in which both electric polarisation and magnetisation are present presents a potential conceptual problem . This is because polarisation and magnetisation are not uniquely separable but a good phenomenological model, at optical frequencies , is to assume that the materials have a relative permeability of unity i.e. they can be described in terms of a permittivity tensor. This second-rank tensor has, in principle, a complete set of nonzero elements. Indeed, the relative permittivity tensor reflects the fact that the optical refractive-index changes are a function of the magnetisation tensor vector.
3
To use known materials to take maximum advantage of magnetooptic effects, one needs to design and optimise a waveguide structure. In the static case, unfortunately, the nonlinear optical coefficient ofYIG is not yet available, but it is expected to be large because of the semi-empirical Miller rule, which states that a large refractive index implies the existence of large nonlinear coefficients. The nature of optical nonlinearity in YIG, e.g., whether it is thermal, resonant, or nonresonant, is also not known.
In the dynamic case [6], that uses magnetostatic waves it is far better to use a heterostructure. For example, in this structure, optimisation for the microwave propagation (magnetostatic waves) in one layer and optical guiding in another structure can be achieved. Hence, for a linear TE-TM conversion structure, the requirements of optical guidingand microwave guidingdo not have to be achieved in a single layer. As far as the optical waves are concerned, they propagate in a layer that has its dielectric behaviour significantly altered by the presence of a magnetostatic wave optimised in a neighbouring layer.
Clearly, the outcome of usingmagnetooptics [7,8,II] is to introduce a newdegreeof freedom into the kind of guided-wave processes now used in optics. Indeed, exploiting magnetooptical properties will produce very impressive integrated units, when compared to those basedpurelyupon GaAs or LiNb03 technologies.
To achieve this goal, the key issues promoting success are the selection of magnetooptic materials, creation of magnetooptic waveguide structures and choosing the nonlinear process. The production of diffraction-free beams, called spatial solitons, is an excellent choice using the soliton dynamics to engage in soliton switching or other forms of solitonchannel control.
Material and thin film technology is already in an advanced state becauseof the very strong interest in magnetooptic recording media. This is implemented through a wide variety of materials and structures. The latter includes periodic structures and Co/Pt and Co/Pd ultra-thin films but the general aim is to exploit modem controllable magnetic properties and try to achieve resistance to oxidation. For waveguides, low loss propagation must be achieved and the classic material is an expitaxially grown YIG film, with the possibility that YIG-semiconductor structures could be used to achieve amplification during propagation. The loss associated with propagation in YIG films is substantially lowered by bismuth substitution, so YIG epitaxially grown onto single crystal substrates, has excellent optical quality. Indeed, it is transparent in the 1.1 11m range and is saturable by small, easily generated, magnetic fields. Specifically, bismuth-substituted yttrium-iron-garnet films such as Y3_xBixFesOI2, where o::; x::; 1.42 are excellent candidates for optical waveguide applications. They are very responsive and transparent in the photon energy range 0.7 ev to 4.8 ev and exhibit a large magnetooptical Faraday effect from the visible to the ultraviolet. Lutetium and lanthanum and gallium-substituted [9] yttrium-iran-garnets - (LuNdBi)3 (FeAI)3 0\2 ­ are also optically transparent. All these types of materials can be deposited upon gadolinium-gallium-garnet (GGG) substrates.
The common magnetooptic configurations [12] are shown in Figure I which shows the saturation magnetisation M of a magnetooptic material, relative to the propagation direction. A magnetooptic material, in addition to possessing a linear refractive index nm, must have the role of the magnetisation ofthe material specified rather carefully.
4
Longitudinal
-- propagation
Polar
-- propagation
Transverse
Figure1. Common magnetooptic configurations M is the magnetisation.
For example, suppose that z is selected, as the propagation direction and that the transverse configuration is used, in which M is perpendicular to this propagation direction. A light beam can be entered on the [x,y,z = 0] plane of a waveguide with diffraction only allowed in the x-direction. For these applications no diffraction will be permitted in the y-direction. Instead, the beam is trapped in the x-z plane and the guide is of finite width in the y-direction. This guiding confinement "freezes" the y-direction so that no beam spreading, due to diffraction, can occur along this axis. As a consequence, the field is uniform - approximately - in the y-direction, and a so-called (1+1) solitary wave can be formed [13,14].
This is most interesting because a whole range of magnetooptic signal processing devices such as filters, correlators, spectrum analysers, switches , modulators, frequency shifters and tunable filters are either in use, or appear to be on the horizon. Such devices will be even more useful if power can be added in as another degree of flexibility . The main theme of the magnetooptic advantages is nonreciprocity. In other words, the study of nonlinear magnetooptical interactions is of prime marketplace importance. It is a realistic prospect because modern film production technology is so much better than it used to be, and, literally, bears no comparison with what was possible in previous decades. It is now easy to make integrated optical building blocks and it is possible to imagine, in the near future, integration of magnetooptic devices with semiconducting substrates containing active devices [6].
2. Some Linear Magnetooptics of Bulk (Infinite) Media
(A)THEDIELECTRIC TENSOR
Magnetooptics, being a form of forced gyrotropy, introduces the role of an external magnetic field. The consequence is that off-diagonal terms appear in the permittivity tensor, which are proportional to the magnitude of this applied field. That this is the case can be readily appreciated, from the behaviour of free electrons in the presence of a constant magnetic field H, = (O,O ,Ho). Suppose that an electron has a mass m and
5
charge q and that its velocity is v. If an electromagnetic plane wave carrying an electric field E has the form expi(oot-kz), where 00 is angular frequency and k is a wavenumber, propagates through a free-electron gas, then the equationof motion is
dv m- = qE+qvxBodt
(2.1)
where Bo = 110"0 and 110 is the permeability of free space. Deploying Maxwell's equationsin the form
curlE =- aB curlB =~ oE + lloj (2.2a)at ' c2 at j = Nqv, B, = 1l0Ho (2.2b)
where c is the velocity of light, j is the current density, is time, B is the total magnetic field and N is the carrier density, leads to
curlH =ioo£;o [I+~]. E =ioo£;o£; ' E (2.3) 100£;0
Here £;0 is the permittivityof free space, £; is the relative dielectric tensor and 1 is a unit matrix tensor. The conductivity tensor is, therefore,
(2.4)
where OOc = qBo is called a cyclotron frequency. The conclusion is that the role of a m
magnetic field can be expressed through a dielectric tensor that has the same symmetry as cr. An overall, generic conclusion is that all materials, exposed to "0 have a general form with off-diagonal elements that are proportional to the applied field. In fact, the common magnetooptic configurations have all the diagonal elements that are almost equal, in practice, and off-diagonal elements involving Q, which is simply proportional to the applied magnetic field.
(
.'QO 2]-I nm 2nm
(
iQsinSsin~ J -iQSi~Bcos ~ (2.5)
where the angles are defined in Figure2, n is the refractive index of the material, and Q is the magnetooptic coefficient induced by Ho.
z
y
Figure2: Coordinate systemfor the applicationof an external magnetic field 8 0•
(B) WAVE PROPAGATION
Consider a planewave propagating alongthez-direction withthe form
E(x,Y,z) =E oexp(ioo - i~z), H(x,y,z) =ho exp(ioo( - i~z) (2.6)
where 00 is the usual angular frequency and ~ is the propagation constant (wavenumber), which has to be found.
The firstof Maxwell's equationsyields
i~hoy = iOH'.on;, (Eox- iQcosSEoy+iQsin Ssin ~Eoz)
i~hox =-i» Eon;, (iQ cos SEOx+ Eoy - iQsin Scos ~Eoz )
0= iooEon;, (-iQsin Bsin ~Eox +iQsin Scos~Eoy +Eoz)
Finally, div D = 0 yields
(-iQsin Bsin ~Eox + iQsin Bsin ~Eoy + Eoz) =0
and div B = 0 gives
7
(2.8a)
(2.8b)
(2.8c)
(2.9)
(2.10)
Not all of these equations are independent, and it is important to note that the propagating wave, although it still has a transverse magnetic field, does not necessarily have a transverse electric field.
Hence,
00 2 n; e;
2 n;, -iQsin8sin~ iQsin 8cos ~ Eoz
which implies that
(2.12)
The two solutions of ~ refer to waves whose polarisation does not change in the course of the propagation. Also note that ~ is not dependent upon ~. This is not a surprise because the bulk medium under consideration has rotational symmetry about the z-axis, The direction of Ho, the applied de magnetic, field can, without loss of generality, be placed in the xz plane. In which case, there is no problem in setting ~ = O. There are two particular geometries which have attracted great experimental interest. They are the Faraday configuration (8 = 0) and the Voigt (Cotton-Mouton)
configuration (S = ~ ) .
8
The Faraday configuration Faraday geometry refers to propagation parallel to the applied de magnetic field Ho,
so substituting e= 0 into (2.12) gives
(2.13)
These waves are circularly polarised in the form
E+ =1]I(x+iy)exP(irot-i~+Z)
E_ = 1] I ( x - iy ) exP ( iro t - i~_ Z )
(2.14)
(2.15)
(2.16)
(2.17)
where 1Eo1= ~E;x + E;y. The waves are completely transverse, because both s, and
Hz are zero. A linearly polarised wave can be decomposed into the sum of the two circularly
polarised waves. E = A+E+ + A_E_, where A± are the corresponding complex amplitudes of E±. Again, without loss of generality, it can be assumed that the wave is linearly polarised in the x-direction. Hence, at Z =0, E(z =0) =1EoIxexp(irot) = A+E+(z =0) + A_E_(z =0) i.e.
(2.18)
=1 ~o I(X+iY)exP(ioot-iP.z)+1 ~o '(X-iy)exp(ioot-iP_z)
= 1 ~o lexP(ioot-ij3z)[(X+iy)exp(-i P, ;P- z)+(X-iy)exp(-i P- ;P. z)]
=1 Eo IeXP(ioot-iPZ)[xcos(P+;P- z J + YSin ( P, ;P- zJ]
9
where p= P. +P- . 2
This is a linearly polarised wave with the electric field vector set at an angle
8F = P. -P- z , with respect to the x axis. It should be noted that the polarisation 2
angle is proportional to the propagation distance z, which means the direction of polarisation is rotating in the course of propagation. This phenomenon is the well­ known Faraday rotation (1,2,6,12] .
For the commonly used magnetooptic material (GdBiCa)3Fes0I2, when magnetised to saturation, Q is about 7.5 x 10-\ so the rotation angle, per unit length, is
d8F=P.-P_=180 0
nQ . For example, if A = 1.152 urn, n = 2.398 , dz 2 A
d8F = 2800° / cm. The required length Lr , to obtain a 1800 rotation, is therefore, dz
0.64 mm. Loss can be included in the analysis by assuming that the refractive index n is complete, i.e. n = n, - in; A decay length can be defined as the propagation distance
over which the amplitude of the wave is reduced to ~ of its original value i.e. e
Ld =~. For a (GdBiCahFesOI2 film , n, = 0.19 x 10-4 , at A = 1.1.52 urn, so that
2nn j
Ld =9.65 mm. Since Ld » L" magnetooptic film can be regarded as transparent at this frequency and the effect of loss can be safely neglected in the Faraday configuration.
The Voigt Configuration This is the name usually given to the case when a wave propagates perpendicularly to
the field i.e. Ho is applied in the x-direction and 8 =2: . This gives the solutions 2
10
EOY
Ell is a transverse wave, linearlypolarised parallel to Ho (hence the subscript II). E.L
is elliptically polarised in the plane perpendicular to Ho (hence the .L subscript that identifies this mode). Forthe E.L wave, Dx==Dz=O, so D.L==DoyYexp(iO)t-i~.Lz)
is linearly polarised in the y-direction. Note, further, that the polarisation of E.L does depend on the actual value of Q and thus on the medium properties. For some commonmaterials Q - 10-4
, so E.L is almost linearlypolarised, in they-direction.
The difference ~ .L - ~II is called the magnetically-induced birefringence, which has
a beat length Li; defined as the propagation distance over which a 21t phase difference is introduced into the E.L wave, with respect to the Ell wave
Now, for (GdBiCa)3FeS012 and "A = 1.15 11m, Q == 7.8 X 10-4 so Lb == 1.71m. This means that the magnetically-induced birefringence is much smaller than the Faraday effect in a magnetooptic medium, whenever Q « I, because the birefringence is proportional to r;j, while Faraday rotation is proportional to Q. This is an interesting summaryof some rather importantmagnetooptic effects but it is clear that, for the Voigt effect, ~II is independent of Q and ~.L is proportional to (f. In other words,
changing the sign of Q does not change the signs ~II or ~.L ' Waves in the bulk
experience reciprocity and reversing the magnetisation does not change the dispersion. What is needed is an asymmetric waveguidestructure, an example of which consists of a magnetooptic substrate, supporting a non-magnetic nonlinear film bounded by an air cladding. For such a guide O(Q) effects are introduced, and the propagation of TM waves becomes non-reciprocal [3-5]. This is the type of transverse configuration that will be investigated next.
3. The Transverse Configuration Envelope Equation
This analysis is to be applied to the type of asymmetric waveguide, described earlier. For a fast time variation exp(ioot), where 00 is the usual frequency and t is time, the wave equation, ignoring losses, is
11
(3.1)
where c is the velocity of light in vacuo, E =n; is the dielectric constant distribution
for the given structure that exists in the absence of any magnetic effects. PM and PNL
represent the polarisations introduced by the magnetooptic and nonlinear properties, respectively . TM waves will be studied here and these are defined as
TM : E =A(X,Z)[~y(y)y+~,(y)zJexp(-i~PZ )exP(iwt) (3.2)
Guiding is assumed in the y-direction, propagation is along the z-direction and diffraction and/or self-focusing takes place in the x-direction.
x,y,z are unit vectors , P is a dimensionless wavenumber [effective refractive
index] and ~(y) introduces the waveguide as a modal field shape. A(x,z) is an amplitude that is permitted to vary slowly as the propagation proceeds along z but is also contains a transverse x-dependence as well.
The global equation for A, obtained after averaging across the waveguide structure, to exploit the character of the linear modal fields, is [15]
where
_ 3 f2x,-"y(I~J +1~Jr dy+ fx xwr(~~ +~:)((2 +~ndy
i: =4 pf(l ~y 1
2 + I~ z 1
(3.3)
(3.4)
The integrations , over the whole waveguide structure, produce Evz the effective
magnetooptic coefficient and Xm the effective nonlinear coefficient. Diffraction is
82
accounted for through the - term and averaging offers the possibility of optimising Ox 2
the magnetooptic influence upon spatial soliton propagation. Only the simplest, physically acceptable, nonlinearity will be used here, and is the
Kerr third-order type and for which the polarisation has the x-component
12
(3.5)
where Ex, Ey and E, are electric field components and Eo, the permittivity of free­ space, has been absorbed into the nonlinear material susceptibility components Xu»,
XxY.Yx.
4. Discontinuity Created by an Interface and Split-Field Method
In Figure 3 optical beams encounter a discontinuity in the form of an interface located at x = O. They can pass from a magnetooptic material to a nonlinear medium, for example . The linear guiding structure, before the interface , in the form of a step, magnetooptic and nonlinear effects are introduced, is defined as
y<O O<y<d
y>d (3.6)
d' " "
Figure 3. Creating an interface. An optical beamcanbe launchedeitherin the magnetooptical material or inthe nonlinear material. Themagnetooptic material is magnetised bya current stripdeposited anywhere on the upper surfaceof the waveguide structure. Notethat the cladding is air andthe
substrate is a compatible non-magnetic material likeGGG.
The main effect , because the magnetooptic and nonlinear effects are small, is that s, is perturbed, because of the deliberately introduced discontinuity. Such a discontinuity can be created by a guide made up of a ridge in which a magnetooptic material has a
13
thickness d to the left of the origin x = 0 and a nonlinear material with a thickness d'
exists to the right of x = O. An effective index E, is created , therefore, where
J - J ' J
(l-n;,) J(I~J + 1~J)dy+(n~L -n;,) J(l~ yI2 +1~J)dy o a-a: (3.7)
In this case, the cladding is air, with n. = I, and the linear refractive index in the core of the guide is nz = nm to the left of the discontinuity and n2 = nNL to the right of it. Both nm and nNL are the linear values of the, respective, refractive indices. Note that the substrate refract ive index value does not enter, explicitly, into the definition of the effective index E" but its influence comes from the shape of the modal field.
The advantage of this configuration is that E, can be designed to be a very small
quantity by choosing the materials carefully and also, the modal fields fall off very rapidly. Only the modal field components in the x < 0 region are needed . Furthermore, although the integration is over the whole waveguide structure , the only contributions are those shown in (3.7), The contributions from this step discontinuity are of opposite
sign so E, -+ 0 in the region of d' =:!.. This means that E,. can readily be made the 3
O( B,,)' The basic global TM magneto optic envelope equation for this discontinuous
situation is
8 2
A 2- A 2- A - A I A 12 1--=-2---2+ eY' + e,. +Xm
ro 8z roP 8x (3.8)
where EyZ ' E" Xm are all functions of the transverse coordinate x. The model assumes
that in is zero for x < 0 and that BY' =E, =0 for x > O.
If the beam width is Do, then the measure of the 'diffraction' length of the system,
over which linear beam spreading occurs, is Lo =2PD~ ~. These typical scales are c
useful in performing the transformations x =Dox', z =Loz' and A =_c_ /1-1V. roDo ",PXm
ol 2- - Hence, after the definition VI = 2P 2 Do eY' + e, the TM envelope equation simplifies,
C
V I x<o V -
L - 0 x> 0' (3.11)
In order to proceed with a detailed analysis of the role of the interface, it is useful to split [13] IJI into two parts IJIl and 1J12, which are solutions of
(3.I2a)
(3.12b)
where
(3.13)
(3.14)
The homogeneous equations (F I = 0, F2 = 0) have solutions IJI:O), IJI ~O) , with the
interpretation that IJI ~O ) is a self-focused beam in the nonlinear medium and IJI :O) is a
radiative field. The general solution of(3 .12b) for F2 = 0 is
Equation (3.15) describes a nonlinear beam [soliton] centred at x - x= (xo- e;z) in
which S is the angle the beam makes to the interface, in the xz plane , Xo is the position
of the beam centre at z = 0, T) is the amplitude and e=(T)2 +~} is the phase . To
zeroth order, IJII is interpreted as the radiative part of IJI and 1J12 is the soliton part of IJI. Since a soliton in the x > 0 region will only have a weak exponential tail in the linear (x < 0) half-space, F1 is very small and is a driving term that causes only a
perturbation to the homogeneous form IJI:O) . Indeed, if there is no incident radiation
15
upon the interface arriving from x = - 00 then \jJ I , driven by F" is always at a low
level. F2 will also be small, relative to the homogeneous soliton form \jJ ~O) . It, too,
can be treated as a perturbation. Hence, an iterative approach to solving equations (3.12) is very efficient.
The first iteration, obtained by substituting the homogeneous solutions \jJ ~ O), \jJ ~0)
into(3 .13)and(3.14),gives F; ( ' )=_VI_21\jJ~O) 12\jJ~O ) , for x <O, F;(') =0, for x >O
and F;(') = O. This iteration, therefore, causes no change to \jJ ~O ) i.e. after the first
iteration \jJ ~ 0 ) ~ \jJ:1) and \jJ ~O ) ~ \jJ ~I ) . A second iteration yields F 2 (2) = 0 [x < 0]
and F2(2) *' 0 [x > 0]. The main results, at this stage, are some equations for the
evolution of the amplitude of the spatial soliton and the position of its centre i.e,
(3.16)
(3.17)
Equation (3.16) shows that a strong beam (1']2 > VI) keeps its identity when it interacts
with an interface but a weak beam (1']2 < VI) loses its power because of the interaction.
For a weak beam, all the power is lost to radiation and ends up in the linear medium. Equation (3.17) shows that for 1'] 2> VI > 0 the soliton is radiation-free and is a/ways attracted to the interface. For 0.5 VI < 1'] 2 < VI the soliton is initially attracted to the
interface but its amplitude 1'] will eventually reduce to below JO .5v l , because of
radiation, at which point the interface becomes repulsive. For 1'] 2 < 0.5 V" the soliton will a/ways be repulsed by the interface and is radiative. For VI < 0, the soliton is radiation-free and is repulsed by the interface. For 1'] 2 > VI > 0, 1'] is a constant, because there is no radiation. The distance, along the z-axis, covered by a soliton, before it crosses an interface, is
I
[ ~]2ex x 1'] + V1'] - VI Z = p(1'] 0) arctan exp(21']x ) - J2
4 2 ~ 0 1'] 1']-v1'] -VI
(3.18)
For VI < 0 the force is repulsive and the final position of the soliton is at the x = 00,
with a final velocity
(dX ) =4l'JeXP(-l'Jxo)(~-~]~ (3.20) dz a-eec l'J -VI +l'J
All of these conclusions can be tested numerically by direct simulation and in all of them VI depends upon the magnetisation of the magnetooptic material. The magnetisation can be created easily by a simple, or a complex electrode wire structure. It is a matter of choice and application. In all cases , VI is a function of x. The electrode structure can be deposited upon the upper surface as thin (-I urn) current­ carrying metal strips (-25Jlm) wide. The parameter controlling the behaviour of beams
2
near to an interface is VI =2PO)z D; (EF+E,)=VF+v, so that there are two c .
contributions to VI : EF is from the magnetooptic part of the waveguide and E, is
from the waveguide structure. In fact, E, can be designed out by adjusting the film
thickness at different parts of the waveguide, but it is fixed once the waveguide is fabricated. Because EF is due to a magnetooptic effect, it is controlled by an applied
static magnetic field i.e. the electric current in the deposited strip electrode structure . Since the magnetooptic material is controlled by the placement of an electrode
structure on the surface of the guide the magnetic field drops rapidly to zero in the (±x )-directions. In this way, a magnetic field creates a magnetisation that is either saturated, or otherwise, over a limited region of the magnetic substrate .
The fact that a magnetooptic material experiences a saturation of its magnetisation at a particular value of magnetic field means that increasing the magnetic field beyond a value of about 300 Oe, in many cases, will not increase the magnetisation (M) any further. Since Vj(x) is proportional to the magnetisation it means that v)(x) has a maximum value associated with the saturation phenomenon. The magnetic field (H) distribution provided by an electrode structure is readily calculated from Maxwell's equations, but its relationship with v)(x) is not so straightforward. Nevertheless, a simple hyperbolic tangent function is a model that is very close to observed magnetisation vs magnetic field behaviour, for many magnetic materials . Accordingly, the model V = A tanh (KHIH.) is assumed here, where H is the magnetic field and A and K are just empirical constants, selected to make tanh ~ I in the region of high
17
magnetic field and H., is the saturation magnetic field, causing (x) to acquire a saturation value v (x) = A. For smaller H, the tanh function falls away rapidly, making v (x) -+ 0 as x -+ zco.
It should be emphasised that electrode structures are used, routinely, to create linear couplers which consist of silver electrodes 20llm wide, carrying as little as IrnA of electric current. For simplicity, it is assumed here that the magnetic field that they produce can be approximated to that produced by an infinitely long thin wire, or indeed, a set of such wires. Assuming that I = 60 rnA and d = OJ urn the maximum field directlyunder a single wire is H = 1/2nd = 400 Oe. Supposethat the magnetic substrate becomes saturated at 300 Oe = 23.87 x 103 Am-I, then the x-position at which the magnetic field is sufficient enough to saturate the magneticmaterial is, for example, x = 0.27Ilm: v = A tanh(KH/H,) ;:::: A = 1.6.
In the calculations reported here the current (wire) strip is assumed to be infinitely thin but, of course, it has width, in practice. It should be expected, therefore, that v(x) is somewhat wider than the theoretical result. This is not a very significant effect, however. Note that a potential barrier/wall U is created that is opposite in sign to v. This can be established using a Lagrangian variational analysis to be reported elsewhere. Note also that as the beam width Do changes so does v and the power P. In fact, as Do increases, v increases and P decreases and vice-versa. In practice, what will happen is that high power solitonscross the finite v(x) region but the smaller powerbeams will break up.
For transverse magnetisation, 8yz < 0 for forward propagation and 8yz > 0 for
backward propagation. Furthermore, 8yz is the order of 10-4, so that for 'A. = 1.551lm
and Do = l Oum, vy: is the order of ±I. Suppose that the waveguide is designed to make Vs = +0.5 then VI = -0.5 for a forward wave and VI = 1.5 for a backward wave. If the soliton is in a layer and hence interacts with two surfaces, the force acting on the forward soliton (assuming that 11 = I)) is repulsive from each interface so the soliton will stay at the centre of the layer, where the repulsive force from each interface comes to a balance. This soliton is also radiationless so it will pass through the guide without any loss of power. On the other hand, the force on the backward soliton (with 11 = I (initial amplitude)) is, initially attractive from both interfaces. The consequence is that if the soliton deviates from the centre of the layer it will become unbalanced and move towards an interface where it is radiated away into the linear medium. If the power loss is so quick that 11 reduces below a critical value of 11 = 0.87 before it crosses one of the interfaces, the force acting on it from the interface will become repulsive. It may return to the centre of the layer and stay there. The decay process continues, however, because the soliton keeps on radiating energy into the linear medium. If the waveguide is made long enough, there will virtually be no output energy at the end of the guide. Effectively, the backward soliton can not pass through this device: an isolator action is thereforeachieved.
The above design can also be used as an optical switch. When the applied current is on, the soliton can pass in a forward direction but will be radiated away in the backward direction. If the current is switched off, the value of v, will be 0.5 for both directions. In this situation, a soliton with v = I will be radationless but will be attracted towards
18
one of the interfaces and be lost to the linear medium, so there is no possibility of an output in either direction. Hence, switching off the applied current virtually switches off the waveguide, while switching on the applied current only causes the forward soliton to pass, with the backward solitonstill beingswitched off.
5. Simulations
The examples selectedare exact numerical solutions of equation (9) but each figure has an artificial line, drawn parallel to the z-axis to show where an interface is situated. This is only done for convenience and has no othermeaning. In eachcase, a single wire electrode is placed on top of the waveguide structure. This wire has a radius a = lOum and carries a current of 200 rnA. The centreof the wire is burn from the interface and the arrangement creates an x-dependent function v)(x) , because of the current flow. This x-dependence enables vyz to be manipulated, which combines with the fIXed Vs
that arises once the waveguide structure has been created. Vs could be created by a dislocation line, for example, thus slightly disordering the waveguide and the magnetic effect could be confined to the x < 0 region by the wire electrode structure. On the other hand, if necessary, Vs can be designed out by addinga phaseshift to beams in the x < 0 region, thus leaving vyix) as the control. The v)(x) displayed in Figures 4a and 3b show, for a single interface, what happens to a soliton-shaped light beam that sets out at x = -10, when the current wire has its centreat -5 [in arbitrary units]. In this example, however, IHsl = 23.87 Oe, b = Sum, I = 200 rnA, Yl = 1.8 and Vs = 0.5 and because the current is switchedit cannotcross the v,(x) region and a solitoncannot be created in the x > 0 region. Figure 4c shows that when the current is switchedoff a solitonis formed in x > 0 even though someradiation of "unwanted" energyoccurs.
Figure 4 (a) position of beamentry point intothe linearmedium to the leftof the currentwirecreating VI(X) (b) currentswitchedon: the lightbeamfails to reachthe interface and radiates back intothe linearhalf-space
(c) currentswitchedoff: light not only reaches the interface but alsocreatesa solitonchannel in the x > 0 nonlinear half-space.
b = 51lm, x = -lOllm, 1= 200 rnA, 11 = 1.8, v; = 0.5IH,j= 23.87Oe
19
Figures 5(a,b,c) show the b = I cases for both ±200 rnA. For a current of 200 rnA, the soliton is stable and propagates close to the interface. For -200mA, however, the soliton is fatally attracted into the interface and is eventually destroyed. So in one case an output is achieved and in the other no output is obtained : on this basis, switching can be claimed.
6. Conclusions
This chapter addresses the combination of linear magnetooptics and intrinsic optical nonlinearity, to make nonlinear waveguide systems that permit solitary wave control and a possible new range of devices . Such applications are simple in concept - as, indeed, are all schemes ever proposed for realistic optical switching - yet they are within the currently available material technology.
-,.' • -I
·1 ,
(a)
10
(b) (c)
Figure 5 (a) Vl(X) for a positive currentof200 mAcarriedby a wire positioned at b =-ll!m, close to the interface
(b) positivecurrentswitched on: soliton beamis stableand is heldawayfromthe interface (c) currentswitched to the oppositedirection : the solitonlosesenergyto radiation; eventually
the soliton'dies'.All otherdata is the same as for Fig.3.
A discussion is presented of a magneto optic configuration in a standard planar format. The latter is the building. block of planar technology photonics because the objective is an all-optics 'chip-level' format that will participate in, and control all­ optical processing operations in the future . The way forward is to use spatial soliton beams in which the diffraction length is the operative length scale . The reason for this is that diffraction operates over only the order of mm and so it fits the 'chip' design very well. Once solitons are created , controlling their dynamics becomes an important issue so magneto optics is put forward here as a very attractive option . Building a planar structure upon a magnetooptic substrate leads to a number of possibilities, but for the moment, we choose TM waves, within what is called the transverse magnetooptic configuration. The new idea is to use a transversely varying magnetooptic parameter, created by deploying electrode structures. These electrode structures, will, in practice, be narrow strips of metal , which can be created in any desired pattern. Even the simplest of them gives an impressive degree of control over the soliton dynamics. An
20
interesting example is presented to illustrate the capability of this area but added or buried , electrode structures will become a feature of the all-optical chip technology of the future . There is a lot of work to be done! It will give a real possibility of manipulating the solitons in any way that is desired. The realisation of this aspiration is assured by the availability of magneto optic materials through the global technology that is driving the linear magnetooptic field.
References
I. Petykiewicz,1. (1992) WaveOptics, Kluwer Academic Publishers, Dordrecht. 2. Hecht, E. and Zajac, A (1974)Optics, Addison-Wesley, Reading. 3. Mizumoto, T. and Naito, Y. (1982)IEEE Trans. Microwave Theoryand Techniques 30, 922. 4. Bahlmann, N., Lohmeyer, M., Zhuromskyy, 0 ., Dotsch, H. and Hertel, P. (1999)Opt. Comm. 161,330. 5. Shintaku, T. and Uno, T. (1994) J App. Phys. 76,8155.
6. Stancill, D.S. (1991)IEEEJ QuantumElectronics 27, 61. 7. Boardman, AD . and Xie, 1. (1997)Phys. Rev.E 55,1899. 8. Boardman, AD. amd Xie, K. (1997) J. Opt. Soc. Am. B. 15,3102. 9. Zvezdin, AK . and Kotov, VA (1997) Modern Magnetooptics and Magnetooptical Materials (loP,
Bristol). 10. Prokhorov, AM ., Smolenskii, GA and Ageev, A.N. (1984)Sov. Phys. Usp. 27,339. II. Boardman, AD. and Xie, K. (1995)Phys. Rev. Lett., 76,4591. 12. Sokolov, A.V. (1967)OpticalProperties ofMetals, Blackie, London. 13. Aliev, Yu M., Boardman, AD ., Smimov, AI., Xie, K. andZharov, AA (1996) Phys. Rev. E. 53,5409 . 14. Boardman, AD . and Xie, K. (1994)Phys. Rev. A SO, 1851.
EXPERIMENTS ON QUADRATIC SOLITONS
GEORGE I. STEGEMAN School ofOptics and CREOL University ojCentral Florida 4000 Central Florida Blvd. Orlando, FL 32816-2700, USA
I. Introduction
Spatial solitons are beams that do not diffract by virtue of a strong nonlinear interaction with the medium in which they propagate,[ I] Quadratic solitons are a rather special member of the spatial soliton family because their existence is not linked to a self­ induced refractive index change and subsequent guiding of the beam by the induced waveguide. Instead they exist by virtue of the strong coupling via the second order nonlinearity X(2 ) between beams of different frequencies.[2] For example, for Type I second harmonic generation (SHG) in which there is a single fundamental and harmonic beam, the beams exchange photons with propagation distance leading to mutual self­ trapping. Although such solitons were predicted back in the 1970s, they were not observed experimentally until the mid 1990s.[2-4] Over the last five years there has been a great deal of experimental progress in this field and the purpose of this chapter is to review some aspects of this work. (A separate chapter by Sukhorukov in this book deals with the details of the theory of quadratic solitons and related effects.)
2. Properties of Quadratic Solitons
The importance of this particular type of soliton is that it demonstrated experimentally that, in nonlinear optics, nonlinear wave mixing can lead to soliton formation. Here we will use very simple examples to illustrate the beam focusing that takes place in wave mixing. The key concept is that photons are exchanged between beams of different frequency when they are coupled by the X(2) nonlinear susceptibility. Although all that is necessary to predict quadratic solitons is to look for appropriate solutions to the coupled mode equations, it is useful to understand on a physical level how solitons can be formed. Far off phase-matching, the concept of "cascading" gives a clear picture, and near phase­ matching the beam narrowing can be understood in terms of the structure of the nonlinear polarization fields.[5]
Consider first how "cascading" leads to solitons. It is based on how the exchange of energy, for example between a fundamental and harmonic beam, leads to a nonlinear phase shift. This is illustrated in Figure I. For cases where the fundamental is the
21
A.D. Boardman and A.P. Sukhorukov (eds.) , Soliton-driven Photonics, 21-39. © 2001 Kluwer Academic Publishers.
22
X(2)(-200; 00, (0)
Figure I. Schematic of the cascading process in whichthe fundamental is thedominant field. The fundamental is up-converted to the harmonic which travels at a different phasevelocity from the fundamental. On phase-match the down-converted fundamental would be II out of phasefrom theoriginal fundamental. Off phasematch, there is an additional phaseshift, leading to a net (nonlinear) phaseshiftof the total fundamental.
dominant field and Akl, = (2k1-k2)L > 0, V2 > VI where V is the phase velocity. As discussed in the caption, there is a nonlinear phase shift on the fundamental due to the exchange of energy between the two waves. The higher the fundamental intensity, the larger the phase shift. This leads to an effective self focusing nonlinearityof the form
81r[x;if n ----=--
2,eff - 41 Ak Gocn /I..U
which leads to solitons with a small harmonic component. In this limit the quadratic solitons closely resemble Kerr solitons (although that small harmonic field is absolutely necessary for their existence).
Because the "cascaded" beam narrowing always occurs for the dominant beam in a parametric interaction, it also occurs for a dominant harmonic when the fundamental is small. Thus quadratic solitons also exist in the limit where the harmonic that undergoes a nonlinear "cascaded" phase shift and self-focusing of the harmonic occurs, but with ilk defined as ilk = k, - 2k,. The diagram equivalent to Figure I is shown in Figure 2.
Beam narrowing can also be understood in terms of the nonlinear polarization generated by the wave mixing.[5] Consider the case of a waveguide in which the field distributions along the x-axis are locked in by the waveguide. From the usual expansion of the nonlinear polarization in terms of the products of the interacting fields, the nonlinear polarization term driving the harmonic is of the form P2NL(y,) o: a,2(y) and the one driving the regenerated fundamental is ptL(y) «: a2(y)a,'(y) where a,(y) and a2(y) are the fundamental and harmonic field respectively, and y is the co-ordinate along which diffraction (or self-trapping) occurs. Again assume for simplicity that the initial
(I)
23
Figure 2. Cascaded nonlinear phase shift tor a dominantinputharmonic tield. Heret1k = k2- 2k,.
P2NL(y) a: exp[-2//wo 2] and thus the polarization source and hence the harmonic
generated b{ it are both narrower in space than the fundamental. Similarly PINL(y) o:
exp[-3/ /wo-], i.e. the polarization source which regenerates the fundamental is narrower than the original fundamental. Although the actual field distributions for the fields are more complicated than Gaussian, these arguments are valid for all beams of finite width. Therefore this parametric interaction leads to beam narrowing for both beams. This counteracts diffraction and results in stable solitons. Note that any process involving the product of finite beams will lead to mutual self-focusing, and presumably spatial solitons.
(} ~ ,
,:)exp[-it1kz1 -- 0: oy --
where t1k = 2kl-k2and focX(2)(-2w; oi,m). For bulk media a term (&/&2) would be added for diffractionalong the x-axis also. and the field would be a function of (x.y.z).
The detailed field solutions are discussed in the chapter by Sukhorukov. Here only some simple. physically intuitiveaspects of the fields are discussed. The soliton fields are stationary, i.e. lad and la21 are independent of (x.y,z). The fundamental and harmonic are in phase, and they rotate together on propagation . This is in sharp contrast to the case of classical second harmonic generation in which only the fundamental is input, shown in Figure 2 for the phase-matched case t1k =0. From equations (I) and (2) and neglecting diffraction, the increment in the harmonic field t1a2 (from oa2/8z) for classical SHG is
24
always orthogonal to a, so that these two fields are always orthogonal to one another and the fundamental depletes as the harmonic grows. This is in contrast to the soliton case in which both the increments ~a , and ~a2 are orthogonal to a, and a2 respectively (because the fields are in phase). (From this argument alone it is clear that the steady state fields must be in phase for the amplitudes to be a constantl) Thus both a, and a2rotate together and the rotation angles are ~<I> 1 ex: z and ~<I>2 ex: z which effectively modifies the wavevectors of the waves (without changing the refractive index). The stationarity condition that the two fields must stay in phase (i.e. not to change their amplitudes) fixes their relative amplitudes. As indicated in the inset of Figure 1, this results in two waves, mutually self-confined along the y (diffraction) direction with different field widths. If the geometry is not phase-matched, i.e. ~k :1= 0, the phase rotation rates and relative amplitudes are those required to keep the fields rotating together. In general, the rotation rate is proportional to the total power associated with the soliton fields, the higher the power, the faster the rotation rate.
Quadratic Soliton Second Harmonic Generation
[j(mf ~·;(2m I l~ I~E ....~.I ••••
/.~ / ... r· -+.,.
Figure3. Evolution of the fundamental and harmonic fields with propagation distance z for a quadratic soliton and for classical second harmonic generation.
Clearly the field amplitudes and relative phases at the input, i.e. the boundary conditions, should playa role in how the fields evolve with distance. On the other hand, quadratic solitons are the high power eigenmodes associated with X(2) interacting waves, i.e. the stable solutions. This means that if there is sufficient intensityat some point in the two fields, the two waves will evolve into solitons by re-adjusting their relative amplitudes and phases via the couple mode equations, with potentially the concurrent emission of excess energy into radiation fields. The limiting case occurs when only the fundamental is input, and indeed it has been found experimentally that this leads to quadratic solitons, albeit inefficiently. An example is shown in Figure 4 of a soliton field obtained with a fundamental only input, photographed (visible because of crystal scattering sites) for propagation over 5 diffraction lengths.
25
- Diffra Ii n L .ngth:
Figure 4. Photographof a 20 urn wide quadratic soliton propagatingover 5 diffraction lengths in bulk KNb0 3 at 982 nm forType I non-critical phase-matching.
The requirements for quadratic soliton generation can be understood in terms of the characteristic lengths in an experiment, i.e. lengths fixed by the experimental parameters and geometry. For 6k = 0, these are:
2 Diffraction length: Ld = mvo n / A
Parametric gain length: L pg = ~~ 20JX efj IG\ I
For a spatial soliton to exist, Ld > Lpg. Physically this means that the photon exchange between the fields must occur over a distance (characterized by Lpg) smaller than the diffractiondistance. If the group velocitiesof the fundamental and harmonic fields do not propagate in the same direction, there is also a walk-off length Lwo so that Lwo > Lpg is also needed for soliton generation.
3. Family of Quadratic Spatial Solitons
The first experimental reports of quadratic spatial solitons in ID (slab waveguides) and 20 (bulk media) utilized Type I and Type II birefringent phase-matching respectively.[3 ,4] This work, just as all subsequent experiments, relied on inputting a single frequency (typically the fundamental) and relying on field evolution on propagation to lead to other frequency components needed for forming solitons. An example of the beam narrowing that takes place and the stabilizationof the beam width in the steady state is shown below in Figure 5 for a fundamental only excitation case.[3] Note how the self-focusing cascading reduces the threshold, where-as self-defocusing increases it.
Most of the experiments, with the sole exception of the work on LiNb0 3 slab waveguides, have been focused on bulk media. One of the key measurements is the solitons as a function of the wavevector mismatch. The results of such experiments are
26
100
100
,~~75
'!--I- --I------I - -" -l -I I - AkL =0 - - -----. J
o
Figure5. The evolution of the outputbeamwidthfor the fundamental wavein KTPat 1064 nm fordifferent initialphase-mismatches. The vertical dashed line is identifies thethreshold intensity.
variation of the soliton threshold intensity (energy for pulsed lasers) required to generate a shown in Figure 6 for KTP in the complicated situation where the energy propagation directions are different for the three interacting waves for Type II SHG.[3] Note that the shapes are similar for solitons obtained both on up-conversion (fundamental input) and down-conversion (harmonic input with fundamental seed), as expected from the cascading picture discussed previously.[6] The input threshold pulse energy is minimum at Akl, = 0 and increases for increasing I~L I.
15 T
t· t·
8 10 12
Figure6. Thresholdpulseenergy(30 ps pulses) required to formquadratic solitonsat 1064 nm in TypeII KTP. The righthandside is for up-conversion (fundamental input, Ak= 2k1 - k2) and the
left hand side for down-conversion (strong harmonic, weakseedfundamental, Ll.k = k2 - 2kl) .
27
To date spatial solitons have been observed in bulk samples of KTP, KNb03, LBO, LiNb03 and LiIOd3,4,6-9] The lowest soliton thresholds were obtained in LiNb03
which used quasi-phase-matching for SHG and hence accesseda nonlinearity of about 17 pm/V.[8] The work on LBO was noteworthy because the soliton generation occurred in a OPG which was triggered by noise.[7]
One of the most exciting developments has been the extension of quadratic solitons into the time domain. The same interaction mechanism which leads to beam narrowing in space with the consequent balancing of diffraction also leads to pulse compression in time. This was demonstrated proposed in BBO.[10] For a temporal soliton, the temporal pulse spreading is caused by group velocity dispersion (GVO) and a temporal soliton requires that the dispersion length, LuVQ, be larger than Lpg. Here To is the lie half-width for a gaussian-like pulse, vg is the group velocity and dvldOJ is the GVO. In 20 a
T,2v2o gLevD = ---- dVg / dOJ
temporal soliton requires that LevD> Lpg be satisfied. For a spatio-temporal soliton, i.e. an "optical bullet", the condition l-ev» "" Lel> Lpg must be satisfied for both transverse beam dimensions and at all the frequencies which constitute the soliton. But dvldOJ is different at all the frequencies making up a quadratic SHG soliton. Using pulse tilting techniques involving a grating that was introduced in the pulse compression work, it is possible to control the GVO at one frequency, but not independently at all ofthem .[1I] However, by working far from phase matching, Akl, » 0, the harmonic component is very small and only the GVO of the fundamental is important. This pulse-tilting approach also lends itself to only one transverse dimension, i.e. to waveguide systems. Wise and coworkers applied it to a bulk medium by using a very wide beam in one dimension (so that it wouldn't diffract) and produced a quasi-spatio-temporal soliton, i.e. a quasi-optical bullet, in LI0 3 by satisfying L(il'f) "" LJ > Lpg at the fundamental frequency and for one transverse dimension. Their key results are shown in Figure 7.[9]
160
400 ..-.,
'';;; >-
~ -a Q) Q)
0 0 20 40 60 80 20 40 60 80
Intensity (GW/cm2 ) Intensity (GW/cm2)
Figure 7. l3eam narrowing in both space and timeshowing steady state beam width in one dimension and steady state pulse width, i.e. a quasi-opticalbullet. .
28
4. Quadratic Soliton Interactions
Quadratic solitons share many properties in common with other coherent solitons in saturating media. One of these is the nature of their interactions where-in in phase solitons attract one another, out of phase solitons repel and for intermediate relative phase angles there is an exchange of energy between them.[12] These conclusions have been drawn essentially from numerical BPM (Beam Propagation Method) simulations of the interactions. Here an approximate but simple coupled mode approach will be used to illustrate the essential physics of these interactions for quadratic solitons.
Figure 8. Schematic of two overlapping soliton fields. The darker region represents the harmonic field and the lighter region the fundamental field.
Solitons interact when their fields overlap . For example in Figure 8, there are two equivalent solitons defined by their fields aj and b, (i = I fundamental field; i = 2 harmonic field) which are functions of y, the in-plane co-ordinate. That is the fields are understood to be of the form a\(y-yo), a2(y-yO), bl(y+yo) and b2(y+yo) where the peak to peak separation of the solitons is 2yo. The nonlinear polarization due to the mixing of the
. . (2)···· (2) 2 2fields tS given by PI = coX [a2al + b2bl + a2bl + b2al ] and P2 = coX [a, + b, + 2a lb.]. If the fields are weak in their overlap region, then coupled mode theory is a reasonable approximation and the changes in the fundamental and harmonic soliton fields are given by
db./dz = if[b2b l' + b2al'+ a2bl' + a2al'] , (2)
In order to complete the calculation, one needs to multiply each equation by the appropriate field envelopes and integrate over y. For example, the first equation would be multiplied by al' , the second by bl' etc. For the first equation this would lead to the RHS
. '2 *2 •terms proportional to Ia2(y-yo)a, (y-yo)dy, Ia, (y-yo)b2CY+Yo)dy, and Ia2(y-yo)al (y- yo)bl·(Y+Yo)dy. The leading term involves the product of the fields associated with the soliton and gives the nonlinear phase rotation discussed previously . The other terms are much smaller if Yo is larger than the soliton's spatial width and these are the corrections
29
due to the interaction. Their value depends on the relative phase between the solitons, as will be discussed next. The same type of terms appear in the three other equations, and have the same interpretation.
In order to interpret the nature of the interactions it is necessary to introduce the relative phases between the solitons. If the solitons are in phase , the peak fields are given by bl(O) = a)(O) and bz(O) = az(O). Also using the fact that the fundamental and harmonic fields are in phase with one another for each soliton, this gives
daj/dz = dazldz = if[C,a,z(O)az(O) + 2CZai \O)alO)] db. /dz = dbj /dz = if[Clalz(O)alO) + 2CZalz(0)aZ(0)]
with C, » Cz. For both solitons the interaction terms (ocCz) increase the nonlinear rotation rate. That is the net result of the attraction is that there is more power associated with each soliton. This results in an attractive force because the solitons are drawn in towards each other.
For out ofphase solitons, bl(O) = -al(O) and bz(O) = -alO) and the resulting coupled equations are
da.zdz= daj/dz = if[Clalz(O)alO) - 2CZalz(0)alO)] db.zdz= dbydz = if[Cla/(O)az(O) - 2Cza,z(0)az(0)]
Now the interaction terms reduce the nonlinear rotation rate, tending to separate the solitons and leading to a repulsive force.
For solitons with a I w2 relative phase, the fields are bl(O) = ±ial(O) and bz(O) = ±i az(O) and the resulting coupled equations are
da.zdz= daj/dz = if[Clalz(O)aZ(O)] K 2f[Cza/(0)az(0)] db./dz = dbydz = ir[Clalz(O)alCO)] ± 2r[CZalz(0)aZ(0)]
In this case the perturbation terms are in or out of phase with the fields so that there is a net exchange of energy between the solitons. Note that when the soliton fields a, deplete, the soliton fields b, grow, and vice-versa. Finally, reversing the relative phase by 7t
reverses the direction of the energy flow between the solitons. In summary, in phase solitons attract, out of phase soliton repel, and for solitons in
phase quadrature energy flows from one soliton to the other. When the relative phases between the solitons take on intermediate values, the result is a combination of both attraction/repulsion and energy exchange.
There have been essentially two experiments reported on quadratic soliton interactions, one in 10 (slab waveguides) with Type I birefringent phase-matching and the second in 20 (bulk media) with Type II phase-matching.[13,14] For the 10 case, a LiNb0 3 Ti:indiffused waveguide was used The results for four different relative phase angles are shown in Figure 9. Note that the results are in excellent agreement with the predictions discussed above.
Similar experiments were performed with bulk quadratic solitons in KTP.[14] In this case, the solitons were launched at small angles relative to each other. As shown in Figure 10, at small enough angles the two solitons fused together and only one soliton appeared at the output facet of the crystal. This type of behavior is typical of collisions in saturating media, a class to which the quadratic solitons belong.[12]
30
400 400 4<1>~ lt/2 200 M=O 2 00
:lOO '5 0 t 50 ~~ =31112
-;
~ !:: ~
0 en -4 00 -2 00 200 400400 -2 00 2 00 4 ,
Ci5 ·200 0 200 ·200 200 Z
Z W 2 50 2 50
~400 400 I-
300 300
• L\~ =1l
L\~ =1112150 t 50
200 200 100 100
100 100 50 50
·200 200 .200 200 -4 0 0 -2 00 200 40().4Q O -2 00 200 4 ,
POSITION (jlm) POSITION (11m)
Righthandside- numericalsimulations.
launched andwhenbothare launched simultaneously. Lefthandsideshows thesimulation.
5. Modulational Instabilities
It is well-known in the nonlinear dynamics field that the same phenomena that lead to soliton formation also lead to periodic instabilit ies under appropriate conditions.[15] This was noted in the early days of nonlinear optics when catastroph ic self-focus ing occurred
31
and led to material damage.[16] This "modulational instability", MI, has been observed in optics in Kerr media, photorefractive media and second harmonic active media.[17-20] In the Kerr case, quasi-cw waves in a fiber broke up into a periodic sequence of temporal pulses.[17] In the spatial domain, MI has been investigated in LiNb03 waveguides in the vicinityof Type 1 phase-matching, in bulk KTP and in photorefractive media, with both coherentand partiallycoherent light.[18-21]
5.1 IDMI
MI occurs because the plane wave eigenmodes in a self-focusing nonlinear medium are unstable, i.e. they break up in the presence of perturbations. The plane wave eigenmodes for a 10 quadratically nonlinear medium are well-known.[22,23] They are the steady­ state solutions to equations(I) for plane waves, i.e. in the absence of the diffraction term. Writingal= P,al(O)exp[-i<p I] , a2= P2al(0)exp[-i<p2] , ~ = oodefp l al(O)/nc, tls = tlk/~ and ~ = uz, the relations between the fields for the two eigenmode solutions are obtained from 4p2 = - tls ± [tlsz+ 4p/]IIZ with <PI = -2pz~ and <PZ = - Pl z~ /pz for the two nonlinear phase rotations. Note that once the intensity of the fundamental is fixed (via a,(O», all of the parameters are fixed. When the + sign is chosen, for tls ~ 00, PzlPI ~ 0 which is associated with the "Kerr limit" of quadratic solitons. Alternatively, if the - sign is chosen, for tls ~ - 00 , P,/p2~ O. Thus in general the nonlinear eigenmodes consist of a combination of fundamental and harmonic fields.
For MI diffraction is now included again in the nonlinear equations because any localized perturbation that occurs will be affected by diffraction.[23] The experiments to be discussed next were performed with a fundamental only input for Akl, > 0 where the closest eigenmode is the one with the + sign. For large enough Akl, > 0, azlal ~ 0 and a fundamental only field is an excellent approximation to this eigenmode. Defining now II = [kl~]' IZy , a, = Aal(O) and b, = Bbl(O), the coupled mode equationsbecome
i dA/dc, + 1/2 dZA/dllz + BA*exp[-itlsc,] = 0 (3)
We now consider a perturbation with periodicity 21t/K, gain coefficienty and amplitudes FI and F2of the from
A = [PI + F1cos(Ky)exp(yz)]exp[ipzc,] B = [pz+ FzCos(Ky)exp(yz)]exp[ iprE,!p/]. (4)
Note that these are just the eigenmode if IF d = IF21 = O. Substituting into equations (3) and assumingthat IF d « PI and IFzl « pzgives
There are solutions to these equations that yield real (physical) values for Kand y over a range of PI (i.e. intensity). These are shown for the case of birefringently phase-matched SHG in a Ti:in-diffused LiNb0 3 slab waveguide in Figure II . When a beam with superimposed random noise is input into the crystal, the periodicity at which the gain is
32
maximum is preferentially amplified and a periodic pattern appears at the output, as indicated in Figure 12. Note that the eigenvectors F1 and F2 are complex which means that both phase and amplitude fluctuations can trigger MI. One can analyse the MI for the other eigenmode, the one in which the harmonic field dominates, in exactly the same way.
Figure II . The MIgain versusMI period for different inputintensity levels. The arrows indicate what the expected maximum gainandperiodshould be for random noiseinput
Figure 12: Typical evolution of a MI induced periodic pattern from a noisyinputbeam.
Figure 13: The experimental geometry fordemonstrating modulational instability in LiNb03Ti:indiffused slabwaveguides nearTypeI phase-matching at 1320 nm.
The experimental set-up is shown in Figure 13.[19] The beam at the output facet is shown in Figure 14 as a function of increasing power. The beam breaks-up into a sequence of peaks whose spacing decreases with increasing power. Note that this is random noise-induced as shown in the middle panel which was taken with two successive laser shots. Measurement of multiple shots in a given laser power range yielded an
33
31kW
28kW
I ; i
0 ........--+--'---'-+--............+-......"""""'"-1 100
0 ...........-.+-_......-!---~-.--1 100
Figure / 4. The output beam profiles observed with increasing input power. Note the middlepanelwhich shows the pattern obtained on two successive laser shots.
average periodicity for the patterns and this is plotted in Figure 15 as a function of input power . Numerical simulations that included non-ideal experimental conditions like a finite width beam, non-uniform wave-vector mismatch , waveguide losses, lack of SH seeding and pulsed instead of cw input were used to show empirically that the measured gain coefficient is reduced by a factor of about five from the value predicted by the linearized theory. Note that the agreement with theory is excellent.
• Experiment
0..
100
Peak Power [kW]
Figure /5. Measured period of M! versus the peak inputpower.
34
Because of the exponential growth of the periodic patterns, the process moves into a saturation regime very quickly and it is difficult to deduce the growth coefficients from such noise initiated experiments, especially since the noise is not reproducible from shot­ to-shot. Perhaps because of this reason there are no measurements of the gain coefficients available, not just in the optics domain, but also in other physical systems. In order to measure the MI gain coefficients we used the geometry in Figure 16 to seed a periodic ripple onto an incident fundamental beam. As shown in Figure 17, this ripple is amplified
Attenuator
Mirror
............. Low power High power
800
Figure17. Amplification due to MIof the periodically seeded beam.
and the gain coefficient can be estimated from the change in intensity of the periodic pattern. Furthermore, by changing the angle between the seed and main beams, and the phase-mismatch, the parameter space of y versus intensity, periodicity and wave-vector mismatch can be explored. The results shown in Figures 18 exhibit excellent agreement with theory.
5.2 2D MI
MI can also occur in bulk quadraticallynonlinear media. However, in this case there is no analytical theory possible, even for the nonlinear eigenmodes, and the relationship must be obtained numerically.
35
& ~kL= 1451l
! o. c:
200
Figure /8. The measuredgain coefficients versus peak intensityfor various phase mismatches (upper)and modulation periods (lower).
The first experiments were performed on KTP in a Type II phase-matching geometry.[20] As shown in Figure 19, the input is an elliptical fundamental beam which breaks-up into periodic patterns whose period decreases with increasing input power.
As compared to the ID case, the patterns observed are much cleaner for 2D. In fact, detailed measurements have shown that the output beams are essentially circular and, given their intensity, they are for all practical purposes quadratic solitons. So why does MI lead to solitons in this case, yet in the ID case the maxima are still far from solitonic in character. In both cases the beams try to evolve into the nonlinear eigenmodes for finite beams, i.e. solitons. The answer seems to lie in the ability of the periodic pattern to efficiently shed excess energy in the 2D case, i.e. the energy can be radiated away into directions orthogonal to the long axis of the elliptical beam. In the ID case, radiative loss occurs preferentially in the waveguide plane and it is trapped between the principal maxima.
An interesting application of MI to the generation of perfectly formed periodic patterns has been proposed and demonstrated. [24] A mask with a 2D periodic transmission pattern was placed between a broad input laser beam and a doubling crystal. The soliton formation was "seeded" by the grid and the authors were able to get clean patternsdown to a 80 urn center-centerseparation.
36
The soliton formation was "seeded" by the grid and the authors were able to get clean patterns downto a 80 urn center-center separation.
• • • • -•• e e e •
. LowerMiddle - output beamfor an inputintensity of 58 OWlcm2
. Bottom - outputfor an inputintensity of 150OW/cm
There are other instabilities that occur with quadratic solitons. A vortex is a stable soliton of a 2D self-defocusing medium. It is a dark hole in a bright background with a phase integral, integrated around the hole, of mn where m is the "topological charge". It is not a stable soliton for a quadratic medium and destabilizes a bright beam in such a medium.
The experiments were performed in a KTP doubling crystal near Type II phase­ match at A=1064nm.[25] Petrov et al nested a vortex into a bright beam with a suitable mask. As shown in Figure 20, a doubly charged vortex splits into 2 vortices on propagation. With increasing power, this evolves into3 separate brightsolitons.
6. Summary
Quadratic spatial solitons have proven to be a very interesting soliton system. The guidingmechanisms have to do with photon exchange between multiple wavesat two or more frequencies . This results in the locking together of the waves, and mutual self­ trapping, even in the presence of group velocity walk-off of the individual waves. The self-trapping mechanism leads to pulse compression and quasi-optical bullets have been
37
Figure 20. The evolution of a brightbeam witha vortex nestedinto it at the input. Thesequence (a) --+ (e) corresponds to theoutputfor increasing input power.
The last pictureshows a 3D scanof theoutputthreesolitons.
The collision behavior of quadratic solitons is similar to that found in saturating Kerr media. It is possible, however, to use simple approximatecoupled mode theory to predict the nature of the interaction for different relative phase angles between the solitons. The predictionsobtained in this way are in good agreement with the experiments.
Modulational instability, and instabilities in general, are an interesting and fruitful area of quadratic soliton research. In keeping with expectations linking the existence of solitons to modulational instabilities in the general field of solitons, MI exists in quadratically nonlinear media. MI has been investigated in ID (waveguides) and 2D (bulk) media. Of these, it proved possible to do a direct comparison between analytical theory and experiment in the ID case and excellent agreement was obtained. Furthermore, by seeding with an interference pattern, it proved feasible to measure directly the exponential gain coefficients.
Another instability that has been investigated is the break up of a bright beam by nesting a vortex inside it. At high enough powers, the broad beam broke up into discrete solitons.
This research was supported by the US National Science Foundation, and by an AROMURI .
38
7. References
I. Segev, M. and Stegeman, G.1. (1998)Self-Trapping of Optical Beams: Spatial Solitons, Physics Today 51 42-8.
2. Karamzin, Y.N. and Sukhorukov, A.P. (1974) Nonlinear Interaction of Diffracted Light Beams in a Medium with Quadratic Nonlinearity: Mutual Focusing of Beams and Limitation on the Efficiency of Optical Frequency Convertors, JETP Lett, 20, 339-42; Karamzin, Y.N. and Sukhorukov, A.P. (1976) Mutual Focusing of High-Power LightBeams in Mediawith Quadratic Nonlinearity Zh. Eksp. Teor. Phys 68,834-40 (Sov. Phys.-JETP 41, 1420).
3. Torruellas, W.E., Wang, Z., Hagan, OJ. , VanStryland, E.W., Stegeman, G.I., Tomer, L. and Menyuk, C.R. (1995)Observation of Two-Dimensional Spatial Solitary Waves in a Quadratic Medium Phys. Rev. Lett., 74,5036-9.
4. Schiek, R., Baek, Y. and G.1. Stegeman, G.1. (1996) One-Dimensional Spatial Solitons Due to Cascaded Second-Order Nonlinearities in PlanarWaveguides Phys. Rev. E, 53, 1138-41 .
5. Stegeman, G.I., Hagan, OJ . and Tomer, L. (1996) X(2) Cascading Phenomena and Their Applications to All-Optical Signal Processing, Mode-Locking, Pulse Compression and Solitons 1. Optical and Quant. Electron. 28 1691-1740.
6. Canva, M.T.G., Fuerst, R.A., Baboiu, D., Stegeman, G.1. and Assanto, G. (1997) Quadratic Spatial Soliton Generation By SeededDown Conversion ofa StrongPumpBeamOpt. Lett. 221683-5.
7. Di Trapani, P., Valiulis, G., Chianglia, W. and Adreoni, A (1998) Two-Dimensional Spatial Solitary Waves fromTravelingWaveParametric Amplification of the Quantum NoisePhys. Rev. Lett. 80265-9.
8. Bourliaguet, B., Couderc, V., Barthelemy, A., Ross, G.W., Smith, P.G.R., Hanna, D.C. and De Angelis, C. (1999) Observation of Quadratic Spatial Solitons in Periodically Poled Lithium Niobate Opt. Lell. 24, 1410-2.
9. Liu, X., Qian, LJ . and Wise, F.W. (1999)Generation of Optical Spatiotemporal Solitons Phys. Rev. Lett . 824631-4.
10. Di Trapani, P., Caironi, D., Valiulis, G., Dubietis, A., Danielius, R. and Piskarkis, A. (1998) Observation of Temporal Solitons in SecondHarmonic Generation withTiltedPulses Phys. Rev. Lett., 81 570-3 .
II. Dubietis A , Valiulis, G., Danielius, R., and Piskarskas, A (1996) Fundamental-Frequency Pulse Compression Through Cascaded Second-Order Processes in a Type 11 Phase-Matched Second Harmonic Generator Opt. Lett . 21, 1262-1264; Valiulis, G., Dubietis, A , Danelius, R., Caironi, D., Visconti, A and Di Trapani, P. (1999)Temporal Solitons in X (2) Materials withTiltedPulses 1. Opt. Soc. Am. B, 16722-31
12. Stegeman, G.1. and Segev, M. (1999) Optical Spatial Soliton and Their Interactions: Universality and Diversity Science 286 1518-23.
13. Y. Baek, Y., Schiek, R., Stegeman, G.1., Baumann, I. and Sohler, W. (1997) Interactions Between One­ Dimensional Quadratic Solitons Opt.Lett. 22 1550-2; Schiek, R., Baek, Y., Stegeman, G.I., Baumann, I. and Sohler, W. (1998)Interactions Between One-Dimensional Quadratic Soliton-like Beams Opt. Quant. Electron. 30 861-879.
14. Costantini, B., De Angelis, C., Barthelemy, A , Bourliaguet, B., and Kermene, V. (1998) Collisions Between Type 11 Two-Dimensional Quadratic Solitons Opt. Lett. 23 424-426.
15. Enfield, E. and Rowlands, G. (1