NONLINEAR AND PHOTOREFRACTIVE OPTICS -...

P d A d R d T d 5

NONLINEAR AND PHOTOREFRACTIVE OPTICS

CHAPTER 38 NONLINEAR OPTICS

Chung L . Tang School of Electrical Engineering Cornell Uni y ersity Ithaca , New York

3 8 . 1 GLOSSARY

c velocity of light in free space

D displacement vector

d m n Kleinman’s d -coef ficient

E electric field in lightwave

E ̃ complex amplitude of electric field

e electronic charge

f oscillator strength

" Planck’s constant

I intensity of lightwave

k propagation vector

m mass of electron

N number of equivalent harmonic or anharmonic oscillators per volume

n 1 , 2 index of refraction at the fundamental and second-harmonic frequencies , respectively

P macroscopic polarization

P ( n ) n th-order macroscopic polarization

P 0 , 2 , 1 , 2 power of lightwave at the fundamental , second-harmonic , sum- , and dif ference-frequencies , respectively

P ̃ complex amplitude of macroscopic polarization

Q amplitude of vibrational wave or optic phonons

S strain of acoustic wave or acoustic phonons

38 .3

38 .4 NONLINEAR AND PHOTOREFRACTIVE OPTICS

T m n relaxation time of the density matrix element r m n

G j damping constant of j th optical transition mode

d Miller’s coef ficient

« ( E ) field-dependent optical dielectric tensor

« n n th-order optic dielectric tensor

» 0 optical dielectric constant of free space

h amplitude of plasma wave or plasmons

l wavelength

r m n density matrix element

x ( E ) field-dependent optic susceptibility tensor

x 1 or x (1) linear optic susceptibility tensor

x n or x ( n ) n th-order optic susceptibility tensor

v p plasma frequency

k a u p u b l dipole moment between states a and b

3 8 . 2 INTRODUCTION

For linear optical materials , the macroscopic polarization induced by light propagating in the medium is proportional to the electric field :

P 5 » 0 x 1 ? E (1)

where the linear optical susceptibility x 1 and the corresponding linear dielectric constant « 1 5 » 0 (1 1 x 1 ) are field-independent constants of the medium .

With the advent of the laser , light intensities orders of magnitude brighter than what could be produced by any conventional sources are now possible . When the corresponding field strength reaches a level on the order of , say , 100 KV / m or more , materials that are normally ‘‘linear’’ at lower light-intensity levels may become ‘‘nonlinear’’ in the sense that the optical ‘‘constants’’ are no longer ‘‘constants’’ independent of the light intensity . As a consequence , when the field is not weak , the optical susceptibility x and the corresponding dielectric constant » of the medium can become functions of the electric field x ( E ) and « ( E ) , respectively . Such a field-dependence in the optical parameters of the material can lead to a wide range of nonlinear optical phenomena and can be made use of for a great variety of new applications .

Since the first experimental observation of optical second-harmonic generation by Franken 1 and the formulation of the basic principles of nonlinear optics by Bloembergen and coworkers 2 shortly afterward , the field of nonlinear optics has blossomed into a wide-ranging and rapidly developing branch of optics . There is now a vast literature on this subject including numerous review articles and books . 3 , 4 , 5 , 6 It is not possible to give a full review of such a rich subject in a short introductory chapter in this Handbook ; only the basic principles underlying the lowest order , the second-order , nonlinear optical processes and some illustrative examples of related applications will be discussed here . The reader is

NONLINEAR OPTICS 38 .5

referred to the original literature for a more complete account of the full scope of this field .

If the light intensity is not so weak that the field dependence can be neglected and yet not too strong , the optical susceptibility and the corresponding dielectric constant can be expanded in a Taylor series :

x ( E ) 5 x 1 1 x 2 ? E 1 x 3 : EE 1 ? ? ? (2)

or

« ( E ) 5 « 1 1 « 2 ? E 1 « 3 : EE 1 ? ? ? (3)

where

« 1 5 » 0 (1 1 x 1 ) (4)

« n 5 » 0 x n for n $ 2 (5)

and » 0 is the dielectric constant of free space . When these field-dependent terms in the optical susceptibility are not negligible , the induced macroscopic polarization in the medium contains terms that are proportional nonlinearly to the field :

P 5 » 0 x 1 ? E 1 » 0 x 2 : EE 1 » 0 x 3 : EEE 1 ? ? ?

5 P ( 1 ) 1 P ( 2 ) 1 P ( 3 ) 1 ? ? ? (6)

As the field intensity increases , these nonlinear polarization terms P ( n . 1) become more and more important , and will lead to a large variety of nonlinear optical ef fects .

The more widely studied of these nonlinear optical ef fects are , of course , those associated with the lower-order terms in Eq . (6) . The second-order nonlinear ef fects will be discussed in some detail in this chapter . Many of the higher-order nonlinear terms have been observed and are the bases of a variety of useful nonlinear optical devices . Examples of the third-order ef fects are : third-harmonic generation 7 , 8 associated with u x (3) (3 v 5 v 1 v 1 v u 2 , two-photon absorption 9 associated with Im x (3) ( v 1 5 v 1 1 v 2 2 v 2 ) , self- focusing 10 , 11 and light-induced index-of-refraction 1 2 change associated with Re x (3) ( v 5 v 1 v 2 v ) , four-wave mixing 1 3 u x (3) ( v 4 5 v 1 1 v 2 2 v 3 ) u 2 , degenerate four-wave mixing or phase-conjugation 14 , 15 u x (3) ( v 5 v 1 v 2 v ) u 2 , optical Kerr ef fect 1 6 Re x (3) ( v 5 0 1 0 1 v ) , and many others .

There is also a large variety of dynamic nonlinear optical ef fects such as photon echo , 1 7

optical nutation 1 8 (or optical Rabi ef fect 1 9 ) , self-induced transparency , 2 0 picosecond 2 1 and femtosecond 2 2 quantum beats , and others .

In addition to the nonlinear optical processes involving only photons that are related to the nonlinear dependence on the E -field as shown in Eq . (6) , the medium can become nonlinear indirectly through other types of excitations as well . For example , the optical susceptibility can be a function of the molecular vibrational amplitude Q in the medium , or the stress associated with an acoustic wave S in the medium , or the amplitude h of any space-charge or plasma wave , or even a combination of these excitations as in a polariton , in the medium :

P 5 » 0 [ x 1 1 x 2 : E 1 x 3 : EE 1 ? ? ? ] E

1 » 0 [ x q : Q 1 x a : S 1 x h : h 1 ? ? ? ] E (7)

giving rise to the interaction of optical and molecular vibrational waves , or optical and acoustic phonons , etc . Nonlinear optical processes involving interaction of laser light and


molecular vibrations in gases or liquids or optical phonon in solids can lead to stimulated Raman 23 , 24 , 25 processes . Those involving laser light and acoustic waves or acoustic phonons lead to stimulated Brillouin 26 , 27 , 28 processes . Those involving laser light and mixed excitations of photons and phonons lead to stimulated polariton 2 9 processes . Again , there is a great variety of such general nonlinear optical processes in which excitations other than photons in the medium may play a role . It is not possible to include all such nonlinear optical processes in the discussions here . Extensive reviews of the subject can be found in the literature . 3–5

3 8 . 3 BASIC CONCEPTS

Microscopic Origin of Optical Nonlinearity

Classical Harmonic Oscillator Model of Linear Optical Media . The linear optical properties , including dispersion and single-photon absorption , of optical materials can be understood phenomenologically on the basis of the classical harmonic oscillator model (or Drude model) . In this simple model , the optical medium is represented by a collection of independent identical harmonic oscillators embedded in a host medium . The harmonic oscillator is characterized by four parameters : a spring constant k , a damping constant G , a mass m , and a charge 2 e 4 f as shown schematically in Fig . 1 . f is also known as the oscillator-strength and 2 e is the charge of an electron . The resonance frequency v 0 of the oscillator is then equal to [ k / m ] 1 / 2 .

In the presence of , for example , a monochromatic wave :

E 5 1 – 2 [ E ̃ e 2 i v t 1 E ̃ * e i v t ] (8)

FIGURE 1 Harmonic oscillator model of linear optical media .


the response of the medium is determined by the equation of motion of the oscillator in the presence of the field :

2 X ( 1 ) ( t ) t 2 1 G

X ( 1 ) ( t ) t

1 v 2 0 X ( 1 ) ( t ) 5

2 e 4 f

2 m [ E ̃ e 2 i v t 1 c . c . ] ? x (9)

where X ( 1 ) ( t ) is the deviation of the harmonic oscillator from its equilibrium position in the absence of the field . The corresponding linear polarization in the steady state and linear complex susceptibility are from Eqs . (8) and (9) :

P ( 1 ) 5 2 NeX ( 1 ) ( t ) x 5 1 – 2 [ P ̃ ( 1 ) e 2 i v t 1 P ̃ ( 1 ) * e i v t ]

5 Ne 2 f E ̃

2 mD ( v ) e 2 i v t 1 c . c . (10)

and

» 0 χ ( 1 ) 5 u P ̃ u u E ̃ u

5 Ne 2 f

mD ( v ) (11)

where N is the volume density of the oscillators and D ( v ) 5 v 2 0 2 v 2 2 i v G . The

corresponding real and imaginary parts of the corresponding linear complex dielectric constant of the medium Re » 1 and Im » 1 , respectively , describe then the dispersion and absorption properties of the linear optical medium . To represent a real medium , the results must be summed over all the ef fective oscillators ( j ) :

Re » 1 5 » 0 1 O j

v 2 pj f j ( v 2

0 j 2 v 2 ) ( v 2

0 j 2 v 2 ) 2 1 v 2 G 2 j

(12)

and

Im » 1 5 O j

v 2 pj f j v G j

( v 2 0 j 2 v 2 ) 2 1 v 2 G 2

j 5

v 2 pj f j

2 v

G j / 2 ( v 2 v 0 j )

2 2 ( G j / 2) 2 for v < v 0 j (13)

where v 2 pj 5 4 π N j e

2 / m is the plasma frequency for the j th specie of oscillators . Each specie of oscillators is characterized by four parameters : the plasma frequency v p j , the oscillator-strength f j , the resonance frequency v 0 j , and the damping constant G j . These results show the well-known anomalous dispersion and lorentzian absorption lineshape near the transition or resonance frequencies .

The dif ference between the results derived using the classic harmonic oscillator or the Drude model and those derived quantum mechanically from first principles is that , in the latter case , the oscillator strengths and the resonance frequencies can be obtained directly from the transition frequencies and induced dipole moments of the transitions between the relevant quantum states in the medium . For an understanding of the macroscopic linear optical properties of the medium , extended versions of Eqs . (12) and (13) , including the tensor nature of the complex linear susceptibility , are quite adequate .

Anharmonic Oscillator Model of the Second - order Nonlinear Optical Susceptibility . An extension of the Drude model with the inclusion of suitable anharmonicities in the oscillator serves as a useful starting point in understanding the microscopic origin of the optical nonlinearity classically . Suppose the spring constant of the oscillator representing the optical medium is not quite linear in the sense that the potential energy of the oscillator


FIGURE 2 Anharmonic oscillator model of nonlinear optical media .

is not quite a quadratic function of the deviation from the equilibrium position , as shown schematically in Fig . 2 . In this case , the response of the oscillator to a harmonic force is asymmetric . The deviation (solid line) from the equilibrium position is larger and smaller on alternate half-cycles than that in the case of the harmonic oscillator . This means that there must be a second-harmonic component (dark shaded curve) in the response of the oscillator as shown schematically in Fig . 3 . It is clear , then , that the larger the

FIGURE 3 Response [ x ( t )] of anharmonic oscillator to sinusoidal driving field [ E ( t )] .


anharmonicity and the corresponding asymmetry in the oscillator potential , the larger the second-harmonic in the response . Extending this kind of consideration to a three- dimensional model , it implies that to have second-harmonic generation , the material must not have inversion symmetry and , therefore , must be crystalline . It is also clear that for the third and higher odd harmonics , the anharmonicity in the oscillator potential should be symmetric . Even harmonics will always require the absence of inversion symmetry . Beyond that , obviously , the larger the anharmonicities , the larger the nonlinear ef fects .

Consider first the second-harnomic case . The corresponding anharmonic oscillator equation is :

2 X ( t ) t 2 1 G

X ( t ) t

1 v 2 0 X ( t ) 1 y X ( t ) 2 5

2 e 4 f

2 m [ E ̃ e 2 i v t 1 c . c . ] ? x (14)

Solving this equation by perturbation expansion in powers of the E -field :

X ( t ) 5 X ( 1 ) ( t ) 1 X ( 2 ) ( t ) 1 X ( 3 ) ( t ) 1 ? ? ? (15)

leads to the second-order nonlinear optical susceptibility

» 0 χ ( 2 ) 5 u P ̃ u u E ̃ 2 u

5 Ne 3 f y

2 m 2 D 2 ( v ) D (2 v ) (16)

Unlike in the linear case , a more exact expression of the nonlinear susceptibility derived quantum mechanically will , in general , have a more complicated form and will involve the excitation energies of , and dipole matrix elements between , all the states . Nevertheless , an expression like Eq . (16) obtained on the basis of the classical anharmonic oscillator model is very useful in discussing qualitatively the second-order nonlinear optical properties of materials . Expression (16) is particularly useful in understanding the dispersion properties of the second nonlinearity .

It is also the basis for understanding the so-called Miller’s rule 3 0 which gives a very rough estimate of the order of magnitude of the nonlinear coef ficient . We note that the strong frequency-dependence in the denominator of the χ ( 2 ) involves factors that are of the same form as those that appeared in χ ( 1 ) . Suppose we divide out these factors and define a parameter which is called Miller’s coef ficient :

d 5 χ ( 2 ) (2 v ) / [ χ ( 1 ) ( v )] 2 χ ( 1 ) (2 v ) » 2 0 5 m y / 2 e 3 f 1/2 N 2 » 2

0 (17)

from Eqs . (16) and (11) . For many inorganic second-order nonlinear optical crystals , it was first suggested by R . C . Miller that d was approximately a constant for all materials , and its value was found empirically to be on the order of 2 – 3 3 10 2 6 esu . If this were true , to find materials with large nonlinear coef ficients , one should simply look for materials with large values of χ (1) ( v ) and χ (1) (2 v ) . This empirical rule was known as Miller’s rule . It played an important historical role in the search for new nonlinear optical crystals and in explaining the order of magnitude of nonlinear coef ficients for many classes of nonlinear optical materials including such well-known materials as the ADP-isomorphs—for example , KH 2 PO 4 (KDP) , NH 4 PO 4 (ADP) , etc . —and the ABO 3 type of ferroelectrics—for example , LiIO 3 , LiNbO 3 , etc . —or III-V and II-VI compound semiconductors in the early days of nonlinear optics .

On a very crude basis , a value of d can be estimated from Eq . (17) by assuming that the anharmonic potential term in Eq . (14) becomes comparable to the harmonic term when the deviation X is on the order of one lattice spacing in a typical solid , or on the order of an Angstrom . Thus , using standard numbers , Eq . (17) predicts that , in a typical solid , d is on the order of 4 3 10 2 6 esu in the visible . It is now known that there are many classes of materials that do not fit this rule at all . For example , there are organic crystals with Miller’s coef ficients thousands of times larger than this value .


A more rigorous theory for the nonlinear optical susceptibility will clearly have to come from appropriate calculations based upon the principles of quantum mechanics .

Quantum Theory of Nonlinear Optical Susceptibility . Quantum mechanically , the nonlinearities in the optical susceptibility originate from the higher-order terms in the perturbation solutions of the appropriate Schro ̈ dinger’s equation or the density-matrix equation .

According to the density-matrix formalism , the induced macroscopic polarization P of the medium is specified completely in terms of the density matrix :

P 5 N Trace [ p r ] (18)

where p is the dipole moment operator of the essentially noninteracting individual polarizable units , or ‘‘atoms’’ or molecules or unit cells in a solid , as the case may be , and N is the volume density of such units .

The density-matrix satisfies the quantum mechanical Boltzmann equation or the density-matrix equation :

r m n

t 1 i v m n r m n 1

r m n 2 r # m n

T m n 5

i " O

k [ r m k V k n 2 V m k r k n ] (19)

where r # m n is the equilibrium density matrix in the absence of the perturbation V m n and T m n is the relaxation time of the density-matrix element r m n . The n th-order perturbation solution of Eq . (19) in the steady state is :

r ( n ) mn ( t ) 5

i " O

k E t

2 ̀

[ r ( n 2 1) mk ( t 9 ) V k n ( t 9 ) 2 V m k ( t 9 ) r ( n 2 1)

kn ( t 9 )] exp F S i v m n 1 1

T m n D ( t 9 2 t ) G dt 9 (20)

The zeroth-order solution is clearly that in the absence of any perturbation or :

r (0) mn 5 r # m n (21)

In principle , once the zeroth-order solution is known , one can generate the solution to any order corresponding to all the nonlinear optical processes . While such solutions are formally complete and correct , they are generally not very useful , because it is dif ficult to know all the excitation energies and transition moments of all the states needed to calculate χ ( n $ 2) . For numerical evaluations of χ ( n ) , various simplifying approximations must be made .

To gain some qualitative insight into the microscopic origin of the nonlinearity , it can be shown on the basis of a simple two-level system that the second-order solution of Eq . (20) leads to the approximate result :

χ ( 2 ) ~ [( v g e 2 v )( v g e 2 2 v )] 2 1 3 k g u p u e l 3 2 [ k e u p u e l 2 k g u p u g l ] (22)

It shows that for such a two-level system at least , there are three important factors : the resonance denominator , the transition-moment squared , and the change in the dipole moment of the molecule going from the ground state to the excited state . Thus , to get a large second-order optical nonlinearity , it is preferable to be near a transition with a large oscillator strength and there should be a large change in the dipole moment in going from the ground state to that particular excited state . It is known , for example , that substituted benzenes with a donor and an acceptor group have strong charge-transfer bands where the transfer of charges from the donor to the acceptor leads to a large change in the dipole moment in going from the ground state to the excited state . The transfer of the charges is mediated by the delocalized π electrons along the benzene ring . Thus , there was a great deal of interest in organic crystals of benzene derivatives . This led to the discovery of


many organic nonlinear materials . In fact , it was the analogy between the benzene ring structure and the boroxal ring structure that led to the discovery of some of the best known recently discovered inorganic nonlinear crystals such as b -BaB 2 O 4 (BBO) 3 1 and LiB 3 O 5 (LBO) . 3 2

In general , however , there are few rules that can guide the search for new nonlinear optical crystals . It must be emphasized , however , that the usefulness of a material is not determined by its nonlinearity alone . Many other equally important criteria must be satisfied for the nonlinear material to be useful , for example , the transparency , the phase-matching property , the optical damage threshold , the mechanical strength , chemical stability , etc . Most important is that it must be possible to grow single crystals of this material of good optical quality for second-order nonlinear optical applications in bulk crystals . In fact , optical nonlinearity is often the easiest property to come by . It is these other equally important properties that are often harder to predict and control .

Form of the Second-order Nonlinear Optical Susceptibility Tensor

The simple anharmonic oscillator model shows that to have second-order optical nonlinearity , there must be asymmetry in the crystal potential in some direction . Thus , the crystal must not have inversion symmetry . This is just a special example of how the spatial symmetry of the crystal af fects the form of the optical susceptibility . In this case , if the crystal contains inversion symmetry , all the elements of the susceptibility tensor must be zero . In a more general way , the form of the optical susceptibility tensor is dictated by the spatial symmetry of the crystal structure . 3 3

For second-order nonlinear susceptibilities in the cartesian coordinate system :

P (2) i 5 O

j ,k » 0 χ (2)

ijk E j E k (23)

χ (2) ijk in general has 27 independent coef ficients before any symmetry conditions are taken

into account . Taking into account the permutation symmetry condition , namely , the order E j and E k appearing in Eq . (23) is not important , or

χ (2) ijk 5 χ (2)

ikj (24)

the number of independent coef ficients reduces down to 18 . With 18 coef ficients , it is sometimes more convenient to define a two-dimensional 3 3 6 tensor , commonly known as the Kleinman d -tensor : 3 4

(25) 1 P x

P y

P z 2 5 » 0 1 d 1 1 d 1 2 d 1 3 d 1 4 d 1 5 d 1 6

d 2 1 d 2 2 d 2 3 d 2 4 d 2 5 d 2 6

d 3 1 d 3 2 d 3 3 d 3 4 d 3 5 d 3 6 2

E 2 x

E 2 y

E 2 z

2 E y E z

2 E x E z

2 E x E y

A B rather than the three-dimensional tensor χ (2)

ijk 5 χ (2) ikj . One obvious advantage of the

d i m 5 tensor form is that the full tensor can be written in the two-dimensional matrix form , whereas it would be dif ficult to exhibit on paper any three-dimensional matrix .

An additional important point about the d -tensor is that it is defined in terms of the complex amplitudes of the E -field and the induced polarization with the 1 / 2 factor explicitly separated out in the front as shown in Eq . (8) . In contrast , the definition of χ i j k may be ambiguous in the literature because not all the authors define the complex amplitude with a 1 / 2 factor in the front . For linear processes , it makes no dif ference , because the 1 / 2 factors in the induced polarization and the E -field cancel out . In nonlinear


processes , the 1 / 2 factors do not cancel and the numerical value of the complex susceptibility will depend on how the complex amplitudes of the E -field and polarization are defined .

For crystalline materials , the remaining 18 coef ficients are , in general , not all independent of each other . Spatial symmetry requires , in addition , that they must satisfy the characteristic equation :

χ (2) ijk 5 O

a b g

χ (2) a b g R a i R b j R g k (26)

where R a i , etc ., represent the symmetry operations contained in the space group for the particular crystal structure and Eq . (26) must be satisfied for all the R s in the group . For example , if a crystal has inversion symmetry , or R a i , b j , g k 5 ( 2 1) d a i , b j , g k , Eq . (26) implies that χ (2)

ijk 5 ( 2 1) χ (2) ikj 5 0 as expected . From the known symmetry elements of all 32

crystallographic point groups , the forms of the corresponding second-order nonlinear susceptibility tensors can be worked out and are tabulated . Equation (26) can in fact be generalized 3 3 to an arbitrarily high order n :

χ ( n ) ijk ? ? ? 5 O

a b g ? ? ?

χ ( n ) a b g ? ? ? R a i R b j R g k ? ? ? (27)

for all the R s in the group . Thus , the forms of any nonlinear optical susceptibility tensors can in principle be worked out once the symmetry group of the optical medium is known .

The d -tensors for the second-order nonlinear optical process for all thirty-two point groups derived from Eq . (26) are shown in , for example , Ref . 34 . Similar tensors can in principle be derived from Eq . (27) for the nonlinear optical susceptibilities to any order for any point group .

Phase-matching Condition (or Conservation of Linear Photon Momentum) in Second-order Nonlinear Optical Processes

On a microscopic scale , the nonlinear optical ef fect is usually rather small even at relatively high light-intensity levels . In the case of the second-order ef fects , the ratio of the second-order term to the first-order term in Eq . (2) , for example , is very roughly the ratio of the applied E -field strength to the ‘‘atomic E -field’’ in the material or :

χ 2 E

χ 1 <

E

E atomic

(28)

which is on the order of 10 2 4 even at an intensity level of 1 MW / cm 2 . The same ratio holds very roughly in each successively higher order . To see such a small ef fect , it is important that the waves generated through the nonlinear optical process add coherently on a macroscopic scale . That is , the new waves generated over dif ferent parts of the optical medium add coherently on a macroscopic scale . This requires that the phase velocities of the generated wave and the incident fundamental wave be ‘‘matched . ’’ 3 5

Because of the inevitable material dispersion , in general the phases are not matched because the freely propagating second-harmonic wave will propagate at the phase velocity corresponding to the second-harmonic while the source polarization at the second- harmonic will propagate at the phase velocity of the fundamental . Phase matching requires that the propagation constant of the source polarization 2 k 1 be equal to the propagation constant k 2 of the second-harmonic or :

2 k 1 5 k 2 (29)

Multiplying Eq . (29) by " implies that the linear momentum of the photons must be


FIGURE 4 Phase-matching requirement and the ef fect of materials dispersion on momentum mismatch in second-harmonic process .

conserved . As shown in the schematic diagram in Fig . 4 , in a normally dispersive region of an optical medium , k 2 is always too long and must be reduced to achieve proper phase matching .

In bulk crystals , the most ef fective and commonly used method is to use birefringence to compensate for material dispersion , as shown schematically in Fig . 5 . In this scheme , the k -vector of the extraordinary wave in the anisotropic crystal is used to shorten k 2 or lengthen 2 k 1 as needed . For example , in a negative uniaxial crystal , the fundamental wave is sent into the crystal as an ordinary wave and the second-harmonic wave is generated as an extraordinary wave in a so-called Type I phase-matching condition :

2 k ( o ) 1 5 k ( e )

2 (30)

or the fundamental wave is sent in both as an ordinary wave and an extraordinary wave while the second-harmonic is generated as an extraordinary wave in the so-called Type II phase-matching condition :

k ( o ) 1 1 k ( e )

1 5 k ( e ) 2 (31)

In a positive uniaxial crystal , k ( e ) 2 in Eqs . (30) and (31) should be replaced by k ( o )

2 and k ( o ) 1 in

Eq . (30) should be replaced by k ( e ) 1 . Crystals with isotropic linear optical properties clearly

lack birefringence and cannot use this scheme for phase matching . Semiconductors of zinc-blende structure , such as the III-V and some of the II-VI compounds , have very large second-order optical nonlinearity but are nevertheless not very useful in the bulk crystal


FIGURE 5 Phase matching using birefringence to compensate material dispersion in second-harmonic generation .

form for second-order nonlinear optical processes because they are cubic and lack birefringence and , hence , dif ficult to phase match . Phase matching can also be achieved by using waveguide dispersion to compensate for material dispersion . This scheme is often used in the case of III-V and II-VI compounds of zinc-blende structure . Other phase-matching schemes include the use of the dispersion of the spatial harmonics of artificial period structures to compensate for material dispersion .

These phase-matching conditions for the second-harmonic processes can clearly be generalized to other second-order nonlinear optical processes such as the sum- and dif ference-frequency processes in which two photons of dif ferent frequencies and momenta k 1 and k 2 either add or subtract to create a third photon of momentum k 3 . The corresponding phase-matching conditions are :

k 1 Ú k 2 5 k 3 (32)

The practical phase-matching schemes for these processes are completely analogous to those for the second-harmonic process . For example , one can use the birefringence in a bulk optical crystal or the waveguide dispersion to compensate for the material dispersion in a sum- or dif ference-frequency process .

Conversion Ef ficiencies for the Second-harmonic and Sum- and Dif ference-frequency Processes

With phase matching , the waves generated through the nonlinear optical process can coherently accumulate spatially . The spatial variation of the complex amplitude of the generated wave follows from the wave equation :

2

z 2 E i ( z , t ) 2 1 c 2

2

t 2 E i ( z , t ) 5 1

c 2 » 0

2

t 2 P i ( z , t ) (33)

where E i ( z , t ) 5 1 – 2 [ E ̃ 0 ,i e

ik 0 z 2 i v 0 t 1 c . c . ] 1 1 – 2 [ E ̃ 2 , i ( z ) e ik 2 z 2 i v 2 t 1 c . c . ] (34)


and

P i ( z , t ) 5 P ( v 0 ) i ( z , t ) 1 [ P (2 v 0 )

source , i ( z , t ) 1 P (2 v 0 ) i ( z , t )] (35)

P (2 v 0 ) i ( z , t ) 5 1 – 2 F O

j » 0 χ (1)

ij (2 v 0 ) E ̃ 2 , j ( z ) e ik 2 z 2 i v 2 t 1 c . c . G (36)

P (2 v 0 ) source , i ( z , t ) 5 1 – 2 [ P ̃ s , i ( z ) e i 2 k 0 z 2 i v 2 t 1 c . c] (37)

P ̃ (2 v 0 ) s ,i ( z , t ) 5 1 – 2 O

jk » 0 χ (2)

ijk (2 v 0 ) E ̃ 0 , j E ̃ 0 , k (38)

The spatial variation in the complex amplitude of the fundamental wave E ̃ 0 , i in Eq . (34) is assumed negligible and , in fact , we assume it to be that of the incident wave in the absence of any nonlinear conversion in the medium . It is , therefore , implied that the nonlinear conversion ef ficiency is not so large that the fundamental intensity is appreciably depleted . In other words , the small-signal approximation is implied . Solving Eq . (33) with the boundary conditions that there is no second-harmonic at the input and no reflection at the output end of the crystal , one finds the second-harmonic at the output end of the crystal z 5 L to be :

I 2 v 0 ( z 5 L ) 5 2 d 2 I 2

0

c » 0 n 2 ( n 1 2 n 2 ) 2 sin 2 S L v 0

c D ( n 1 2 n 2 ) (39)

where d is the appropriate Kleinman d -coef ficient and the intensities refer to those inside the medium . When the phases of the fundamental and second-harmonic waves are not matched , or n 1 ? n 2 , it is clear from Eq . (39) that the second-harmonic intensity is an oscillating function of the crystal length . The maximum intensity is reached at a crystal length of :

L max 5 l

4 u n 2 2 n 1 u (40)

which is also known as the coherence length for the second-harmonic process . The maximum intensity that can be reached is :

I 2 v 0 max ( z 5 L m a x ) 5

2 d 2

c » 0 n 2 ( n 1 2 n 2 ) 2 I 2

0 (41)

regardless of the crystal length as long as it is greater than the coherence length . The coherence length for many nonlinear optical materials could be on the order of a few microns . Therefore , without phase matching , the second-harmonic intensity in such crystals corresponds to what is generated within a few microns of the output surface of the nonlinear crystal . A much more interesting or important case is clearly when there is phase matching or n 1 5 n 2 .

The second-harmonic intensity under the phase-matched condition is , from Eq . (39) :

I 2 5 S 8 π 2

c » 0 n 2 1 n 2

D S L l 1 D 2

d 2 ef f I

2 0 (42)

where d ef f is the ef fective d -coef ficient which takes into account the projections of the


E -field and the second-harmonic polarization along the crystallographic axes and the form of the proper d -tensor for the particular crystal structure . The intensities in this equation refer to the intensities inside the nonlinear medium and the wavelength refers to the free-space wavelength . Equation (42) shows that the second-harmonic intensity under phase-matched conditions is proportional to the square of the length of the crystal measured in the wavelength , as expected for coherent processes . The second-harmonic intensity is also proportional to the ef fective d -coef ficient squared and the fundamental intensity squared , as expected .

One might be tempted to think that , to increase the second-harmonic power conversion ef ficiency indefinitely , all one has to do is to focus the beam very tight since the left-hand side is inversely proportional to the beam cross section while the right-hand side is inversely proportional to the cross section squared . Because of dif fraction , however , as the fundamental beam is focused tighter and tighter , the ef fective focal region becomes shorter and shorter . Optimum focusing is achieved when the Rayleigh range of the focal region becomes the limiting interaction length rather than the crystal length . A rough estimate assumes that a beam of square cross section doubles in width ( w ) due to dif fraction in an ‘‘optimum focusing length , ’’ L o p t , w 2 / l 2 , and that this optimum focusing length is equal to the crystal length L . Under such a nominally optimum focusing condition , the maximum second-harmonic power that can be generated in practice is , therefore , approximately :

P (opt) 2 5 S 2 π 2

c » 0 n 2 1 n 2

D S L l 3

2 D d 2

ef f P 2 0 (43)

Note that this maximum power is linearly proportional to the crystal length . It must be emphasized , however , that this linear dependence is not an indication of incoherent optical process . It is because the beam spot size (area) under the optimum focusing condition is linearly proportional to the crystal length . Numerically , for example , approximately 3 W of second-harmonic power could be generated under optimum focusing in a 1-cm-long LiIO 3 crystal with 30 W of incident fundamental power at 1 m m .

Equation (43) can , in fact , be generalized to other three-photon processes such as the sum-frequency and dif ference-frequency processes :

P (opt) 1 5 S 2 π 2

c » 0 n 1 n 2 n 1 D S L

l 3 1 D d 2

ef f ( v 1 5 v 1 1 v 2 ) P 1 P 2 (44)

and

P (opt) 2 5 S 2 π 2

c » 0 n 1 n 2 n 2 D S L

l 3 2 D d 2

ef f ( v 2 5 v 1 2 v 2 ) P 1 P 2 (45)

In using Eqs . (44) and (45) , one must be especially careful in relating the numerical values of the d ef f coef ficients for the sum- and dif ference-frequency processes to that measured in the second-harmonic process because the two low-frequency photons are degenerate in frequency in the latter process .

The Optical Parametric Process

A somewhat dif ferent , but rather important , second-order nonlinear optical process is the optical parametric process . 36 , 37 Optical parametric amplifiers and oscillators powerful solid-state sources of broadly tunable coherent radiation capable of covering the entire spectral range from the near-UV to the mid-IR and can operate down to the femtosecond time domain . The basic principles of optical parametric process were known even before the invention of the laser , dating back to the says of the masers . The practical development


of the optical parametric oscillator had been impeded , however , due to the lack of suitable nonlinear optical materials . As a result of recent advances 3 8 in nonlinear optical materials research , these oscillators are now practical devices with broad potential applications in research and industry . The basic physics of the optical parametric process and recent developments in practical optical parametric oscillators are reviewed in this section as an example of wavelength-shifting nonlinear optical devices .

Studies of the optical parameters of materials clearly have always been a powerful tool to gain access to the atomic and molecular structures of optical materials and have played a key role in the formulation of the basic principles of quantum mechanics and , indeed , modern physics . Much of the information obtained through linear optics and linear optical spectroscopy came basically from just the first term in the expansion of the complex susceptibility , Eq . (2) . The possibility of studying the higher-order terms in the complex susceptibility through nonlinear optical techniques greatly expands the power of such studies to gain access to the basic building blocks of materials on the atomic or molecular level . Of equal importance , however , are the numerous practical applications of nonlinear optics . Although there are now thousands of known laser transitions in all kinds of laser media , the practically useful ones are still relatively few compared to the needs . Thus , there is always a need to shift the laser wavelengths from where they are available to where they are needed . Nonlinear optical processes are the way to accomplish this . Until recently , the most commonly used wavelength-shifting processes were harmonic generation , sum- , and dif ference-frequency generation processes . In all these processes , the generated frequencies are always uniquely related to the frequencies of the incident waves . The parametric process is dif ferent . In this process , there is the possibility of generating a continuous range of frequencies from a single-frequency input .

For harmonic , sum- , and dif ference-frequency generation , the basic devices are nothing more than suitably chosen nonlinear optical crystals that are oriented and cut according to the basic principles already discussed in the previous sections and there is a vast literature on all aspects of such devices . The spontaneous optical parametric process can be viewed as the inverse of the sum-frequency process and the stimulated parametric process , or the parametric amplification process , can be viewed as a repeated dif ference-frequency process .

Spontaneous Parametric Process . The spontaneous parametric process , also known as the parametric luminescence or parametric fluorescence process , is described by a simple Feynman diagram as shown in Fig . 6 . It describes the process in which an incident photon , called a pump photon , propagating in a nonlinear optical medium breaks down spontaneously into two photons of lower frequencies , called signal and idler photons using

FIGURE 6 Spontaneous breakdown of a pump photon into a signal and an idler photon .


a terminology borrowed from earlier microwave parametric amplifier work , with the energy and momentum conserved :

v p 5 v s 1 v i (46)

k p 5 k s 1 k i (47)

The important point about this second-order nonlinear optical process is that the frequency condition Eq . (46) does not predict a unique pair of signal and idler frequencies for each fixed pump frequency v p . Neglecting the dispersion in the optical material , there is a continuous range of frequencies that can satisfy this condition . Taking into account the dispersion in real optical materials , the frequency and momentum matching conditions Eqs . (46) and (47) , in general , cannot be satisfied simultaneously . In analogy with the second-harmonic or the sum- or dif ference-frequency processes , one can use the birefringence in the material to compensate for the material dispersion for a set of photons propagating in the nonlinear crystal . By rotating the crystals , the birefringence in the direction of propagation can be tuned , thereby leading to tuning of the signal and idler frequencies . This tunability gives rise to the possibility of generating photons over a continuous range of frequencies from incident pump photons at one particular frequency , which means the possibility of constructing a continuously tunable amplifier or oscillator by making use of the parametric process .

A complete theory for the spontaneous parametric emission is beyond the scope of this introductory chapter because , as all spontaneous processes , it requires the quantization of the electromagnetic waves . Detailed descriptions of the process can be found in the literature . 4

Stimulated Parametric Process , or the Parametric Amplification Process . With only the pump photons present in the initial state , spontaneous emission occurs at the signal and idler frequencies under phase-matched conditions . With signal and pump photons present in the initial state , stimulated parametric emission occurs in the same way as in a laser medium , except here the pump photons are converted directly into the signal and the corresponding idler photons through the second-order nonlinear optical process and no exchange of energy with the medium is involved . The stimulated parametric process can also be viewed as a repeated dif ference-frequency process in which the signal and idler photons repeatedly mix with the pump photons in the medium , generating more and more signal and idler photons under the phase-matched condition .

The spatial dependencies of the signal and idler waves can be found from the appropriate coupled-wave equations under the condition when the pump depletion can be neglected . The corresponding complex amplitude of the signal wave at the output E s ( L ) is proportional to that at the input E s (0) , as in any amplification process : 3 9

E s ( L ) 5 E s (0) cosh gL (48) where

g 5 d ef f u E p u 4 k s k i

2 n s n i (49)

is the spatial gain coef ficient of the parametric amplification process . d ef f is the ef fective Kleinman d -coef ficient for the parametric process . k s and k i are the phase-matched propagation constants of the signal and idler waves , respectively ; n s and n i are the corresponding indices of refraction .

Optical Parametric Oscillator . Given the parametric amplification process , a parametric oscillator can be constructed by simply adding a pair of Fabry-Perot mirrors , as in a laser , to provide the needed optical feedback of the stimulated emission . The optical parametric oscillator has the unique characteristic of being continuously tunable over a very broad


FIGURE 7 Schematic of singly resonant optical parametric oscillator .

spectral range . This is perhaps one of the most important applications of second-order nonlinear optics .

The basic configuration of an optical parametric oscillator (OP) is extremely simple . It is shown schematically in Fig . 7 . Typically , it consists of a suitable nonlinear optical crystal in a Fabry-Perot cavity with dichroic cavity mirrors which transmit at the pump frequency and reflect at the signal frequency or at the signal and idler frequencies . In the former case , the OPO is a singly resonant OPO (SRO) and , in the latter case , it is a doubly resonant OPO (DRO) . The threshold for the SRO is much higher than that for the DRO . The tradeof f is that the DRO tends to be highly unstable and , thus , not as useful .

Tuning of the oscillator can be achieved by simply rotating the crystal relative to the direction of propagation of the pump beam or the axis of the Fabry-Perot cavity . As an example of the spectral range that can be covered by the OPO , Fig . 8 shows the tuning

FIGURE 8 Tuning characteristics of BBO spontaneous parametric emission ( 3 and 1 ) and OPO (circles) pumped at the third-(355 nm) and fourth- (266 nm) harmonics of Nd-YAG laser output . Solid curves are calculated .


curve of a b -barium borate OPO pumped by the third-harmonic output at 355 m m and the fourth-harmonic at 266 m m of a Nd : YAG laser . Also shown are the corresponding spontaneous parametric emissions . The symbols correspond to the experimental data and the solid curves are calculated . 3 8 With a single set of mirrors to resonate the signal wave in the visible , the entire spectral range from about 400 nm to the IR absorption edge of the b -barium borate crystal can be covered . With KTiO 2 PO 4 (KTP) or the more recently developed KTiO 2 AsO 4 (KTA) crystals , the tuning range can be extended well into the mid-IR range to the 3- to 5- m m range . With AgGaSe 2 , the potential tuning range could be extended to the 18- to 20- m m range .

The ef ficiency of the SRO that can be achieved in practice is relatively high , typically over 30 percent on a pulsed basis . Since the OPO is scalable , the output energy is only limited by the pump energy available and can be in the multijoule range .

A serious limitation at the early stage of development is the oscillator linewidth that can be achieved . Without rather complicated and special arrangements , the oscillator linewidth is typically a few Angstroms or more , which is not useful for high-resolution spectroscopic applications . The linewidth problem is , however , not a basic limitation inherent in the parametric process . It is primarily due to the finite pulse length of the pump sources , which limits the cavity length that can be used so that the number of passes by the signal through the nonlinear crystal is not too small . As more suitable pump sources are developed , various line-narrowing schemes 4 0 typically used in tunable lasers can be adapted for use in OPOs as well .

The OPO holds promise to become a truly continuously tunable powerful solid-state source of coherent radiation with broad applications as a research tool and in industry .

3 8 . 4 MATERIAL CONSIDERATIONS

The second-order nonlinearity is the lowest-order nonlinearity and the first to be observed as the intensity increases . As the discussion following Eq . (26) indicates , only materials without inversion symmetry can have second-order nonlinearity , which means that these must be crystalline materials . The lowest-order nonlinearity in a centrosymmetric system is the third-order nonlinearity .

To observe and to make use of the second-order nonlinear optical ef fects in a nonlinear crystal , an ef fective d -coef ficient on the order of 10 2 13 m / V or larger is typically needed . In the case of the third-order nonlinearity , the ef fect becomes nonnegligible or useful in most applications when it is on the order of 10 2 21 MKS units or more .

Ever since the first observation of the nonlinear optical ef fect 1 shortly after the advent of the laser , there has been a constant search for new ef ficient nonlinear materials . To be useful , a large nonlinearity is , however , hardly enough . Minimum requirements in other properties must also be satisfied , such as transparency window , phase-matching condition , optical damage threshold , mechanical hardness , thermal and chemical stability , etc . Above all , it must be possible to grow large single crystals of good optical quality for second-order ef fects . The perfection of the growth technology for each crystal can , however , be a time-consuming process . All these dif ficulties tend to conspire to make good nonlinear optical materials dif ficult to come by .

The most commonly used second-order nonlinear optical crystals in the bulk form tend to be inorganic crystals such as the ADP-isomorphs NH 4 H 2 PO 4 (ADP) , KH 2 PO 4 (KDP) , NH 4 H 2 AsO 4 (ADA) , CsH 2 AsO 4 (CDA) , etc . and the corresponding deuterated version ; the ABO 3 type of ferroelectrics such as LiIO 3 , LiNbO 3 , KNbO 3 , etc . ; and the borates such as b -BaB 2 O 4 , LiB 3 O 5 , etc . Although the III-V and II-VI compounds such GaAs , InSb , GaP , ZeTe , etc . generally have large d -coef ficients , because their structures are cubic , there is no birefringence that can be used to compensate for material dispersion .


Therefore , they cannot be phase-matched in the bulk and are useful only in waveguide forms . Organic crystals hold promise because of the large variety of such materials and the potential to synthesize molecules according to some design principles . As a result , there have been extensive ef forts at developing such materials for applications in nonlinear optics , but very few useful second-order organic crystals have been identified so far . Nevertheless , organic materials , especially for third-order processes , continue to attract a great deal of interest and remain a promising class of nonlinear materials .

To illustrate the important points in considering materials for nonlinear optical applications , a few examples of second-order nonlinear crystals with their key properties are tabulated in Tables 1 through 3 . It must be emphasized , however , that because some

TABLE 1 Properties of Some Nonlinear Optical Crystals*

Crystal LiB 3 O 5 b -BaB 2 O 4 f

Point group mm 2 a 3 m

Birefringence n x 5 a 5 1 . 5656 b

n y 5 c 5 1 . 5905 n z 5 b 5 1 . 6055

n e 5 1 . 54254 n o 5 1 . 65510

Nonlinearity [pm / V] d 3 2 5 1 . 16 b d 2 2 5 16 d 3 1 5 0 . 08

Transparency [ m m] 0 . 16 – 2 . 6 c 0 . 19 – 2 . 5

G m a x [GW / cm 2 ] , 25 b , 5 g

SHG cutof f [nm] 555 d 411

, D T [ 8 C ? cm] 3 . 9 e 55

, D Θ [mrad ? cm] , CPM 31 . 3 e 0 . 52

, 1 / 2 D Θ [mrad (cm) 1/2 ] 71 . 9 e NCPM ê 148 . 0 8 C Not available

, D l [ Å ? cm] Not available 21 . 1 D y 2 1

g ê 630 nm [fs / mm]

OPO tuning range [nm]

240 d

, 415 – 2500 d

( l p 5 355)

360

, 410 – 2500 ( l p 5 355)

Boule size 20 3 20 3 15 mm 3 e [ 84 mm 3 18 mm

Growth TSSG e ê , 810 8 C TSSG from Na 2 O ê , 900 8 C

Predominant growth defects Flux e inclusions Flux and bubble inclusions

Chemical properties Nonhygroscopic e (m . p . , 834 8 C) Slightly hygroscopic ( b 5 a , 925 8 C)

* Data shown is at 1 . 064 m m unless otherwise indicated . G m a x —surface damage threshold ; , D T —temperature-tuning bandwidth ; , D Θ , CPM—critical phase-matching acceptance angle ; ,

1 / 2 D Θ —noncritical phase-matching acceptance angle ;

, D l —SHG bandwidth ; D y 2 1 g —group-velocity dispersion for SHG at 630 nm .

a Von H . Konig and A . Hoppe , Z . Anorg . Allg . Chem . 439 : 71 (1978) ; M . Ihara , M . Yuge , and J . Krogh-Moe , Yogyo - Kyokai - Shi 88 : 179 (1980) ; Z . Shuquing , H . Chaoen , and Z . Hongwu , J . Cryst . Growth 99 : 805 (1990) .

b C . Chen , Y . Wu , A . Jiang , B . Wu , G . You , R . Li , and S . Lin , J . Opt . Soc . Am . B6 : 616 (1989) ; S . Liu , Z . Sun , B . Wu , and C . Chen , J . App . Phys . 67 : 634 (1989) . On the basis of d 3 2 5 2 . 69 3 d 3 6 (KDP) and using the value d 3 6 (KDP) 5 0 . 39 pm / V according to R . C . Ekart et al ., J . Quan . Elec . 26 : 922 (May 1990) .

c 0 . 16 – 2 . 6 m m : C . Chen , Y . Wu , A . Jiang , B . Wu , G . You , R . Li , and S . Lin , J . Opt . Soc . Am . B6 : 616 (1989) . 0 . 165 – 3 . 2 m m ; S . Zhao , C . Huang , and H . Zhang , J . Cryst . Growth . 99 : 805 (1990) .

d Calculated by using Sellmeier equations reported in reference ; B . Wu , N . Chen , C . Chen , D . Deng , and Z . Xu , Opt . Lett . 14 : 1080 (1989) .

e T . Ukachi and R . J . Lane , measurements carried out on Cornell LBO crystals grown by self-flux method . f Reference sources given in : ‘‘Growth and Characterization of Nonlinear Optical Crystals Suitable for Frequency

Conversion , ’’ by L . K . Cheng , W . R . Bosenberg , and C . L . Tang , review article in Progress in Crystal Growth and Characterization 20 : 9 – 57 (Pergamon Press , 1990) , unless indicated otherwise .

g Estimated surface damage threshold scaled from detailed bulk damage results reported by H . Nakatani et al ., Appl . Phys . Lett . 53 : 2587 (26 December , 1988) .


TABLE 2 Properties of Several Visible Near-IR Nonlinear Optical Crystals*

Characteristics KNbO 3 † LiNbO 3 ‡ Ba 2 NaNb 5 O 1 5

Point group mm 2 3 m mm 2

Transparency [ m m] 0 ? 4 – 5 . 5 0 . 4 – 5 . 0 0 . 37 – 5 . 0 Birefringence negative biaxial

n x 5 c 5 2 . 2574 n y 5 a 5 2 . 2200 n z 5 b 5 2 . 1196

negative uniaxial n 0 5 2 . 2325 n e 5 2 . 1560

negative biaxial n x 5 b 5 2 . 2580 n y 5 a 5 2 . 2567 n z 5 c 5 2 . 1700

Second-order nonlinearity [pm / V]

d 3 2 5 12 . 9 , d 3 1 5 2 11 . 3 d 2 4 5 11 . 9 , d 1 5 5 2 12 . 4

d 3 3 5 2 19 . 6

d 3 3 5 2 29 . 7 d 3 1 5 2 4 . 8 d 2 2 5 2 . 3

d 3 2 5 2 12 . 8 , d 3 1 5 2 12 . 8 d 2 4 5 12 . 8 , d 1 5 5 2 12 . 8

d 3 3 5 2 17 . 6

( n v 2 n 2 v ) / T [ 8 C 2 1 ] 1 . 6 3 10 2 4 2 5 . 9 3 10 2 5 1 . 05 3 10 2 4

T p m [ 8 C] 181 , d 3 2 2 8 , d 3 1 89 , d 3 2 101 , d 3 1

, D T [ 8 C-cm] 0 . 3 0 . 8 0 . 5 l S H G (cutof f)[ m m] ê 25 8 C 0 . 860 , 1 . 08 1 . 01

G m a x [MW / cm 2 ] Not available , 120 40

Phase transition temperature ( 8 C)

225 and 435 , 1000 300

Growth technique TSSG from K 2 O ê , 1050 8 C

Czochralski ê , 1200 8 C

Czochralski ê , 1440 8 C

Predominant growth problems

Cracks , blue coloration , multidomains

Temp . induced compositional striations

Striations , microtwinning , multidomains

Postgrowth processing Poling Poling Poling & detwinning

Crystal size 20 3 20 3 20 mm 3

(single domain) [ 100 mm 3 200 mm

(as grown boule) [ 20 mm 3 50 mm (with striations)

* Unless otherwise specified , data are for l 5 1 . 064 m m . (Data taken from : a , e – i ; a , b – c ; and a , d , respectively . † There is a disagreement on the sign of the nonlinear coef ficients of KNbO 3 in the literature . Data used here are taken from Ref . e

with the appropriate correction for the IRE convention . a

‡ Data are for congruent melting LiNbO 3 . b Five-percent MgO doped crystals gives photorefractive damage threshold about 10 – 100

times higher . k , l The phase-matching properties for these crystals may dif fer due to the resulting changes in the lattice constants . j a S . Singh in CRC Handbook of Laser Science and Technology , vol . 4 , Optical Materials , part I , M . J . Weber (ed . ) , CRC Press , 1986 ,

pp . 3 – 228 . b R . L . Byer , J . F . Young , and R . S . Feigelson , J . Appl . Phys . 41 : 2320 (1970) . c R . L . Byer in Quantum Electronics : A Treatise , H . Rabin and C . L . Tang (eds) , vol . 1 , part A , Academic Press , 1975 . d S . Singh , D . A . Draegert , and J . E . Geusic , Phys . Re y . B 2 : 2709 (1970) . e Y . Uematsu , Jap . J . Appl . Phys . 13 : 1362 (1974) . f P . Gunter , Appl . Phys . Lett . 34 : 650 (1979) . g W . Xing , H . Looser , H . Wuest , and H . Arend , J . Crystal Growth 78 : 431 (1986) . h D . Shen , Mat . Res . Bull . 21 : 1375 (1986) . i T . Fukuda and Y . Uematsu , Jap . J . Appl . Phys . 11 : 163 (1972) . j B . C . Grabmaier and F . Otto , J . Crystal Growth 79 : 682 (1986) . k D . A . Bryan , R . Gerson , and H . E . Tomaschke , Appl . Phys . Lett . 44 : 847 (1984) . l G . Zhong , J . Jian , and Z . Wu , 1 1 th International Quantum Electronics Conference , IEEE Cat . No . 80 CH 1561-0 , June 1980 , p . 631 .

of the materials are relatively new , some of the numbers listed are subject to confirmation and possibly revision . Discussions of other inorganic and organic nonlinear optical crystals can be found in the literature . 4 1

As nonlinear crystals and devices become more commercialized , the issues of standardization of nomenclature and conventions and quantitative accuracy are becoming increasingly important . Some of these issues are being addressed 4 2 but much work remains to be done .


TABLE 3 Properties of Several UV , Visible , and Near-IR Crystals*

Crystal KDP KTP (II) †

Point group 42 m mm 2

Birefringence n e 5 1 . 4599 n o 5 1 . 4938

n x 5 a 5 1 . 7367 n y 5 b 5 1 . 7395 n z 5 c 5 1 . 8305

Nonlinearity [pm / V]

d 3 6 5 0 . 39 d 3 2 5 5 . 0 , d 3 1 5 6 . 5 d 2 4 5 7 . 6 , d 1 5 5 6 . 1

d 3 3 5 13 . 7

Transparency [ m m] 0 . 2 – 1 . 4 0 . 35 – 4 . 4 G m a x [GW / cm 2 ] , 3 . 5 , 15 . 0 SHG cutof f [nm] 487 , 990

, D T [ 8 C-cm] 7 22

, D Θ [mrad-cm] 1 . 2 15 . 7 , D l [ Å -cm] 208 ‡ 4 . 5 D y 2 1

g ê 630 nm [fs / mm] 185 Not applicable

OPO tuning range [nm] [nm]

, 430 – 700 ( l p 5 266)

, 610 – 4200 ( l p 5 532)

D T F [ 8 C] 12 Not available

Boule size 40 3 40 3 100 cm 3 , 20 3 20 3 20 mm 3

Growth technique Solution growth from H 2 O TSSG from 2KPO 3 -K 4 P 2 O 7 ê , 1000 8 C

Predominant growth defects Organic impurities Flux inclusions

Chemical properties Hygroscopic (m . p . , 253 8 C) Nonhygroscopic (m . p . , 1172 8 C)

* Unless otherwise stated , all data for 1064 nm . (Data taken from c , e ; a , b , f , m ; and d , g – i , respectively . ) † KTP Type I interaction gives d ef f , d 3 6 (KDP) or less for most processes . m The d i j values d are for crystals grown by the

hydrothermal technique . j – l Significantly lower damage thresholds were reported for hydrothermally grown crystals . d

‡ The anomalously large spectral bandwidth is a manifestation of the l -noncritical phase matching . n This is equivalent to a very good group-velocity matching ( D y

2 1 g , 8 fs / mm) for this interaction in KDP .

a D . Eimerl , J . Quant . Elect . QE-23 : 575 (1987) . b D . Eimerl , L . Davis , S . Velsko , E . K . Graham , and A . Zalkin , J . Appl . Phys . 62 : 1968 (1987) . c D . Eimerl , Ferroelectrics 72 : 95 (1987) . d Y . S . Liu , L . Drafall , D . Dentz , and R . Belt , G . E . Technical Information Series Report , 82CRD016 , Feb . 1982 . e Y . Nishida , A . Yokotani , T . Sasaki , K . Yoshida , T . Yamanaka , and C . Yamanaka , Appl . Phys . Lett . 52 : 420 (1988) . f A . Jiang , F . Cheng , Q . Lin , Z . Cheng , and Y . Zheng , J . Crystal Growth 79 : 963 (1986) . g P . Bordui , in Crystal Growth of KTiOPO 4 from High Temperature Solution , Ph . D . thesis , Massachusetts Institute of

Technology , 1987 . h Information Sheet on KTiOPO 4 , Ferroxcube , Division of Amperex Electronic Corp ., Saugerties , New York , 1987 . i P . Bordui , J . C . Jacco , G . M . Loiacono , R . A . Stolzenberger , and J . J . Zola , J . Crystal Growth 84 : 403 (1987) . j F . C . Zumsteg , J . D . Bierlein , and T . E . Gier , J . Appl . Phys . 47 : 4980 (1976) . k R . A . Laudis , R . J . Cava , and A . J . Caporaso , J . Crystal Growth 74 : 275 (1986) . l S . Jia , P . Jiang , H . Niu , D . Li , and X . Fan , J . Crystal Growth 79 : 970 (1986) . m L . K . Cheng , unpublished . n 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 , 1987 , pp . 146 – 159 .

3 8 . 5 APPENDIX

The results in this article are given in the rationalized MKS systems . Unfortunately , many of the pioneering papers on nonlinear optics were written in the cgs gaussian system . In addition , dif ferent conventions and definitions of the nonlinear optical coef ficients are


used in the literature by dif ferent authors . These choices have led to a great deal of confusion . In this Appendix , we give a few key results to facilitate comparison of the results using dif ferent definitions and units .

First , in the MKS system , the displacement vector D is related to the E -field and the induced polarization P in the medium as follows :

D 5 » 0 E 1 P

5 » 0 E 1 P ( 1 ) 1 P ( 2 ) 1 ? ? ? (A-1)

The corresponding wave equation is given in Eq . (33) . For the second-order polarization and the corresponding Kleinman d -coef ficients , two definitions are in use . A more popular definition in the current literature is as follows :

P ( 2 ) 5 » 0 d 2 : EE (A-2)

In an earlier widely used reference , 3 4 Yariv defined his d -coef ficient as follows :

P ( 2 ) 5 d (Yariv) 2 : EE (A-3)

The numerical values of d (Yariv) 2 in this reference (e . g ., Table 16 . 2) 3 4 are given in

(1 / 9) 3 10 2 22 MKS units . The numerical value of » 0 in the MKS system is 10 7 3 (1 / 4 π c 2 ) in MKS units . Thus , for example , a tabulated value of d (Yariv)

2 5 0 . 5 3 (1 / 9) 3 10 2 22 MKS units in Ref . 34 converts to a numerical value of d 2 5 0 . 628 pm / V in MKS units .

In the cgs gaussian system , the displacement vector D is related to the E -field and the induced polarization P in the medium as follows :

D 5 » 0 E 1 4 π P

5 » 0 E 1 4 π P ( 1 ) 1 4 π P ( 2 ) 1 ? ? ? (A-4)

The corresponding wave equation is :

2

z 2 E i ( z , t ) 2 1 2

c 2 t 2 E i ( z , t ) 5 4 π 2

c 2 t 2 P i ( z , t ) (A-5)

The conventional definition of d 2 is as follows :

P ( 2 ) 5 d 2 : EE (A-6)

The numerical value of d 2 in cgs gaussian units is , therefore , equal to (3 3 10 4 / 4 π ) times the numerical value of d 2 in rationalized MKS units . Thus , continuing with the numerical example given in the preceding paragraph , d 2 5 0 . 628 pm / V is equal to 1 . 5 3 10 2 9 cm / Stat- Volt or 1 . 5 3 10 2 9 esu .

As a final check , the expression Eq . (42) for the second-harmonic intensity in the MKS system becomes , in the cgs gaussian system :

I 2 5 S 512 π 5

cn 2 1 n 2

D S L

l 1 D 2

d 2 ef f I

2 0 (A-7)


All the intensities refer to those inside the medium , and the wavelength is the free-space wavelength .

3 8 . 6 REFERENCES

1 . P . A . Franken , A . E . Hill , C . W . Peters , and G . Weinreich , Phys . Re y . Lett . 7 : 118 (1961) . 2 . J . A . Armstrong , N . Bloembergen , J . Ducuing , and P . S . Pershan , Phys . Re y . 127 : 1918 (1962) ; N .

Bloembergen and Y . R . Shen , Phys . Re y . 133 : A37 (1964) . 3 . N . Bloembergen , Nonlinear Optics , Benjamin , New York , 1965 . 4 . See , for example , H . Rabin and C . L . Tang (eds . ) , Quantum Electronics : A Treatise , vol . 1A and B

Nonlinear Optics , Academic Press , New York , 1975 , and the references therein . 5 . See , for example , Y . R . Shen , The Principles of Nonlinear Optics , J . W . Wiley Interscience , New

York , 1984 . 6 . M . D . Levenson and S . S . Kano , Introduction to Nonlinear Laser Spectroscopy , Academic Press ,

New York , 1988 , and the references therein . 7 . P . D . Maker and R . W . Terhune , Phys . Re y . A 137 : 801 (1965) . 8 . See , for example , secs . 7 . 3 and 7 . 4 of Ref . 5 . 9 . H . Mahr , ‘‘Two-Photon Absorption Spectroscopy , ’’ in Ref . 4 .

10 . G . A . Askar’yan , So y . Phys . JETP 15 : 1088 , 1161 (1962) ; M . Hercher , J . Opt . Soc . Am . 54 : 563 (1964) ; R . Y . Chiao , E . Garmire , and C . H . Townes , Phys . Re y . Lett . 13 : 479 (1964) [Erratum , 14 : 1056 (1965)] .

11 . See , for example , Y . R . Shen , ‘‘Self-Focusing , ’’ chap . 17 in Ref . 5 . 12 . See , for example , R . W . Boyd , Nonlinear Optics , chap . 4 , Academic Press , 1992 . 13 . See , for example , chap . 15 in Ref . 5 . 14 . Y . B . Zeldovich , V . I . Popoviecher , V . V . Ragul’skii , and F . S . Faizullov , JETP Letters 15 : 109

(1972) . 15 . R . W . Hellwarth , J . Opt . Soc . Am . 68 : 1050 (1978) ; A . Yariv , IEEE J . Quant . Elect . QE-14 : 650

(1978) . 16 . See , for example , A . Yariv and P . Yeh , Optical Wa y es in Crystals , Wiley , New York , 1984 , p . 221 . 17 N . A . Kurnit , I . D . Abella , and S . R . Hartmann , Phys . Re y . Lett . 13 : 567 (1964) ; S . Hartmann , in R .

Glauber (ed . ) , Proc . of the Int . School of Phys . Enrico Fermi Course XLII , Academic Press , New York , 1969 , p . 532 .

18 . C . L . Tang and B . D . Silverman , ‘‘Physics of Quantum Electronics , ’’ P . Kelley , B . Lax , and P . E . Tannenwald (eds . ) , McGraw-Hill , 1966 , p . 280 . G . B . Hocker and C . L . Tang , Phys . Re y . Lett . 21 : 591 (1969) ; Phys . Re y . 184 : 356 (1969) .

19 . R . G . Brewer , Phys . Today , May 1977 . 20 . S . L . McCall and E . L . Hahn , Phys . Re y . Lett . 18 : 908 (1967) ; Phys . Re y . 183 : 457 (1969) . 21 . N . Bloembergen and A . H . Zewail , J . Phys . Chem . 88 : 5459 (1984) . 22 . M . J . Rosker , F . W . Wise , and C . L . Tang , Phys . Re y . Lett . 57 : 321 (1986) ; J . Chem . Phys . 86 : 2827

(1987) . 23 . E . J . Woodbury and W . K . Ng , Proc . IRE 5 0 , p . 2347 (1962) ; R . W . Hellwarth , Phys . Re y .

130 : 1850 (1963) . 24 . E . Garmire , E . Pandarese , and C . H . Townes , Phys . Re y . Lett . 11 : 160 (1963) . 25 . C . S . Wang , ‘‘The Stimulated Raman Process , ’’ chap . 7 in Ref . 4 . 26 . R . Y . Chiao , C . H . Townes , and B . P . Stoichef f , Phys . Re y . Lett . 12 : 592 (1964) ; E . Garmire and C .

H . Townes , App . Phys . Lett . 5 : 84 (1964) . 27 . C . L . Tang , J . App . Phys . 37 : 2945 (1966) . 28 . I . L . Fabellinskii , ‘‘Stimulated Mandelstam-Brillouin Process , ’’ chap . 5 in Ref . 4 .


29 . See , for example , sect . 10 . 7 in Ref . 5 . 30 . R . C . Miller , App . Phys . Lett . 5 : 17 (1964) . 31 . C . Chen , B . Wu , A . Jiang , and G . You , Sci . Sin . Ser . B 28 : 235 (1985) . 32 . C . Chen , Y . Wu , A . Jiang , B . Wu , G . You , R . Li , and S . Lin , J . Opt . Soc . Am . B6 : 616 (1989) . 33 . P . A . Franken and J . F . Ward , Re y . of Mod . Phys . 35 : 23 (1963) . 34 . A . Yariv , Quantum Electronics , John Wiley , New York , 1975 , pp . 410 – 411 . 35 . J . A . Giordmaine , Phys . Re y . Lett . 8 : 19 (1962) ; P . D . Maker , R . W . Terhune , M . Nisenhof f , and

C . M . Savage , Phys . Re y . Lett . 8 : 21 (1962) . 36 . W . H . Louisell , Coupled Mode and Parametric Electronics , John Wiley , New York , 1960 . 37 . N . Kroll , Phys . Re y . 127 : 1207 (1962) . 38 . See , for example , C . L . Tang , Proc . IEEE 80 : 365 (March 1992) . 39 . Ref . 4 , p . 428 . 40 . See , for example , L . F . Mollenauer and J . C . White (eds . ) , Tunable Lasers , Springer-Verlag ,

Berlin , 1987 . 41 . See , for example , S . K . Kurtz , J . Jerphagnon , and M . M . Choy , in Landolt - Boerstein Numerical

Data and Functional Relationships in Science and Technology , New Series , K . H . Wellwege (ed . ) , Group III , vol . 11 , Springer-Verlag , Berlin , 1979 ; Nonlinear Optical Properties of Organic and Polymeric Materials , D . Williams (ed . ) , Am . Chem . Soc ., Wash ., D . C ., 1983 ; Nonlinear Optical Properties of Organic Molecules , D . Chemla and J . Zyss (eds . ) , Academic Press , 1987 .

42 . D . A . Roberts , IEEE J . Quant . Elect . 28 : 2057 (1992) .

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