Infrared dispersion of third-order susceptibilities in dielectrics: Retardation effects

Prog. Quant. Electr., Vol. 4, pp. 271-300. Pergamon Press, 1976. Printed in Great Britain

INFRARED DISPERSION OF THIRD-ORDER SUSCEPTIBILITIES IN DIELECTRICS: RETARDATION EFFECTS CHR. FLYTZANIS Laboratoire d'Optique Quantique, C.N.R.S.-Ecole Polytechnique, Route de Saclay, 91120 Palaiseau, France

and N.

BLOEMBERGEN

Division of Engineering and AppJied Physics, Pierce Hall, Harvard University, Cambridge, Mass. 02138, U.S.A. Abstract T h e infrared dispersion of the third-order susceptibility describing the mixing of visible light b e a m s in transparent dielectric crystals is discussed. Important contributions arise from local field effects and w a v e s created by lower order nonlinearities in n o n c e n t r o s y m m e t r i c crystals. The dispersion of g °) (-to3, col, toa, - t o o in cubic crystals is considered in detail w h e n to~, to2 and to3 = 2tot-to2 are in the transparent region of the crystal. Strong dispersive r e s o n a n t behavior occurs w h e n ~o~- to2 ~ tOph. . . . and/or 2to, = to.~ .... The interference b e t w e e n the different p r o c e s s e s allows an accurate determination of the relative strength of n o n r e s o n a n t contributions to ~(o> and of R a m a n and two-photon cross-sections. A generalization of the L y d d a n e - S a c h s - T e l l e r relation for the nonlinear case is also derived.

1. I N T R O D U C T I O N

It is a basic concept of electromagnetic theory that from a microscopic point of view one only has the electromagnetic fields i n v a c u o produced by external sources and by a microscopic distribution of currents from moving charges inside the medium. The latter are related to the microscopic (vacuum) fields by an internal constitutive relationship which may be of arbitrary form and contain linear as well as nonlinear terms. In principle, this microscopic constitutive relationship may be calculated from the quantum mechanical equation of motion obeyed by the particles. The solution of the Schrodinger equation thus yields the microscopic linear and nonlinear polarizabilities of the atomic or molecular constituents. In experiments one measures macroscopic quantities as appropriate averages over the microscopic fields. The relationship between the macroscopic linear polarization and the macroscopic electric field EL and electric displacement vector D has been discussed extensively for linear dielectric media (Lorentz, 1916; De Groot, 1969). These considerations can be extended to the nonlinear'case. It is customary and useful to relate the macroscopic nonlinear polarization to the macroscopic fields EI. that would prevail and propagate in the linear dielectric, although they may have been created by nonlinear source terms inside the nonlinear medium (Bloembergen, 1965). It turns out that the macroscopic nonlinear susceptibility of the third order, )¢o~, contains important contributions from properly retarded fields, created by distant nonlinear polarization sources proportional to the lower order nonlinear susceptibility )¢(2). In addition, there are important local field corrections from the nonlinear polarization of the dielectric in the immediate vicinity of the nonlinear molecule. The physical situation may be clarified by considering first a dilute gas in which the atoms and molecules may be considered as isolated. This means that the fields produced by the other atoms and molecules are negligibly small compared to the vacuum field Ev produced by the external sources. The dipole moment induced in each molecule may be written as a power series in the vacuum field components, p~ = oltjt t )~~v . l + a o~2) kEvaE~.k

271

+ a l jc3),~ k l Z ~ , j E ~,k E ~,l + " " "

(1.1)

272

C. FLYTZANISand N. BLOEMBERGEN

Magnetic dipole and electric quadrupole terms are ignored for the present. T h e y will be discussed briefly in the final section of this paper. The s u m m a t i o n convention o v e r repeated Cartesian indices is implied on the right-hand side of eqn. (1.1). If one multiplies this expression with the particle n u m b e r density N, one obtains immediately the m a c r o s c o p i c nonlinear polarization, correct to terms linear in the density N. This approximation is v e r y inadequate in dense matter. The fields acting on an individual molecule contain important contributions f r o m the fields created b y the surrounding matter. In the linear dielectric medium this leads to a change of the v a c u u m field p r o p a g a t o r to a p r o p a g a t o r appropriate for a medium with a dielectric constant E. This m a y be considered as the effect of the linear polarization f r o m distant material volume elements. The polarization in the immediate vicinity of a selected molecule leads to a local field correction, first discussed b y L o r e n t z and Lorenz. In the nonlinear case, the nonlinear polarization induced in the m e d i u m also leads to radiative retardation effects f r o m distant nonlinear sources, and to local field corrections due to the nonlinear polarization in the immediate vicinity of a given molecule. It is c u s t o m a r y and useful, in the spirit of perturbation treatment of the nonlinearity, to relate the polarization to the field EL which would prevail in the dielectric if it were linear, P = X ")

. E L + X ~2) : E L E L

+ X ts) ! E L E L E L

+ • • •

(1.2)

The problem is to relate these m a c r o s c o p i c susceptibilities to the microscopic polarizabilities in eqn. (1.1). A systematic, but rather formal, d e v e l o p m e n t of this problem, which includes all local field and retardation effects explicitly to third order, was given b y B e d e a u x (1973). The lowest order nonlinearity, proportional to g ~2~,describes the nonlinear optical processes of second harmonic generation, sum and difference f r e q u e n c y mixing and optical rectification. The polarization cubic in the field amplitudes describes a large variety of nonlinear effects such as third harmonic generation, intensity dependent index of refraction and self-focusing, three w a v e mixing and combination f r e q u e n c y generation, stimulated R a m a n scattering, etc. Several reviews of these effects have b e e n published (Bloembergen, 1967; Kaiser, 1973; Kleinman, 1973). The p u r p o s e of the present p a p e r is to discuss in some detail h o w the local field and retardation effects lead to an interference b e t w e e n third-order and second-order nonlinear processes. It will be shown that one c o n s e q u e n c e of this interference is that the third-order polarization describing the generation of radiation at a combination f r e q u e n c y 093 = 209~- 09~ b y two incident light b e a m s o91 and 092 will exhibit strong infrared dispersive characteristics, even though all three frequencies 091,092 and 093 lie in a transparent, nondispersive region of the material. Before proceeding with this main theme of the present paper, it is p r o b a b l y useful to give a simple illustration of such interference effects b y considering the explicit example of third harmonic generation in a cubic crystal of 43m s y m m e t r y . We shall first a s s u m e that a w a v e at f r e q u e n c y 09 is incident normally on a flat surface which is a [001] crystallographic plane. If the electric field is incident along the cubic (100) direction, the induced harmonic polarization is Px (309) = X ~xxEx3(09).

(1.3)

H e r e Ex =- ELx is the m a c r o s c o p i c field p r o d u c e d inside the linear medium. H e n c e f o r t h the subscript L will be dropped. The lowest order nonlinear susceptibility tensor X ~2) has only one nonvanishing element X~,~, ~2~ and the elements derived f r o m it b y p e r m u t a t i o n of the cubic axes. Since E , ( 0 9 ) = E z ( 0 9 ) = 0 , there is no nonlinear polarization induced at the second harmonic f r e q u e n c y in this geometry. The resulting m a c r o s c o p i c polarization at the third harmonic f r e q u e n c y is c o n s e q u e n t l y given b y eqn. (1.3) with ,~

..~

,~

I-e(09)+2-1~e(309)+2

Xxxx~= l-~a .... [

3

J

3

(1.4)

Infrared dispersion of third-order susceptibilities in dielectrics

273

Only the local field corrections due to the polarization components at to and 3to in the surrounding medium need be considered in this case (Armstrong, 1962; Bedeaux, 1973). If the crystal is turned b y 45 ° around its normal, the electric field at to inside the crystal will point along the face diagonal (110). A longitudinal polarization at the s e c o n d

harmonic f r e q u e n c y is created Pz(2to)

= Z X z(2)¢ x,,-

2(o,

to,

¢o)Ex (to)Ey (to) e 2ik''-21"~'

(1.5)

with (2) X zxY=

~T (2) [E(to) + 212 ~(2to) + 2 r~ct x~z[ 3 3

(1.6)

Here kl = toc-lE'/2(to) is the wave number in the linear medium. The second harmonic polarization will induce a longitudinal second harmonic field Ez (2to) =

41rP~ (2to) e (2oJ)

(1.7)

Consider now an atom surrounded by a small sphere cut out of its cubic environment. It experiences an acting field at the second harmonic frequency given by E~(2to)

e ( 2 t o ) + 2 4~'P~(2to) 3 + 3

This Fourier c o m p o n e n t acting at 2to, in combination with the field acting at to, given by ~(to) + 2E~(to) 3 (2)

act

act

induces a dipole moment at 3(o along the ~-axis, proportional to a x,~E~ (2to)Ex (to). In addition, there is, of course, a term proportional to a(3)E3(to). When all local field corrections are properly combined, the macroscopic polarization at the third harmonic f r e q u e n c y is given by P~

(3 to)

=

[3X (3) ~,yEx (to)E,2((o) + x (3) .... E~3(to)] e3,k,.-3,.,

(1.8)

(3) is given by and P, (3(o) = P, (3 to). In eqn. (1.8) X (3) ~,~ i s still given by eqn. (1.4), but X,xy,

(3, ,~ o ) [ e ( t o ) + 2 " ~ 3 ~ ( 3 t o ) + 2 2 (2)2 / X~yy = l,~a~,y[ 3 ] 3t-3 (X~y~) [

47r 47r ¢(2to) + E(2~--) + 2]

(1.9)

where X~x~ is given by eqn. (1.6). This result relates the macroscopic nonlinear susceptibility to the microscopic nonlinear polarizabilities. It is seen that X°)(2to) contains a term proportional to [a (2)(2to)] 2. The result [eqn. (1.9)] is identical to that given in eqn. [3.22] by Bedeaux (1973), if the appropriate value for the longitudinal propagator in the linear medium is substituted, F, ~ -47r/E. Finally, we consider the geometry in which the normal to the crystal is a ( l i d direction. The incident wave is again polarized along (110). N o w the induced second harmonic polarization P~(2to) is transverse to the wave propagation. It produces a transverse electric wave

-41rP~

E~(2to) = + E(2to) - ~(to)

e21~r._2i,ot.

Proceeding in the same way as before one arrives at a contribution to P~(3to) which is formally the same as eqn. (1.8) but in the expression (1.9) for X--yy (3) the last term in the square bracket is replaced by _

)

4~" 4~r E ( 2 t o ) - ¢(to) 4 E(2~-')+2/

because now the propagator corresponding to a transverse wave must be taken. To match the boundary conditions at the surface of the medium, we must introduce, in addition, a free wave at 2w which also propagates normal to surface into the crystal

274

C. FLYTZANIS and N. BLOEMBERGEN

with wave number k2 = 0)c-~'v2(20)). If the reflected wave at 20) is ignored, this free wave has the same amplitude as the forced wave

47rP~

E~(2w) = e(2w--)-~(o~)

eik r~io~t " "

This wave in a second step also leads to a third harmonic polarization

3Xx~yy E~(0))E~ (0)) with (3)-°° _ 2

4~

The third harmonic polarization acquires a spatial modulation with a period 2 z r ( k 2 - 2 k , ) -~ related to the characteristic coherence length for second harmonic generation. The observable intensity at 30) will consequently display a double modulation. The intensity of the M a k e r - T e r h u n e fringes spaced by 2~r(k3- 3k0 -~ is modulated by the second harmonic coherence length. Alternatively, one may say that the two spatial Fourier c o m p o n e n t s of polarization at 30) have two different coherence lengths for third harmonic generation, given by 7r(3k, - k3)-~ and It(k1 + k2 - k3)-' respectively. If we are close to the second harmonic phase matching condition the two-step process proportional to {g~z~}2 will certainly dominate, but even in nonphase-matched condition (3) it may make a considerable contribution to X x~,. This example illustrates how the two-step lower order nonlinear process interferes with the intrinsic cubic nonlinearity. The observable intensity at 30) is proportional to the square of the absolute value of the effective nonlinear susceptibility, [X~3)l2. The results depend sensitively on crystal orientation, as shown by this example. These interference effects have not been studied experimentally for third harmonic generation. A large number of experimental investigations have, however, been reported for similar effects arising in another nonlinear process of the same order. The generation of radiation at the combination f r e q u e n c y 0)3 = 20)i - 0)2 by two incident light beams at 0)1 and 0)2 respectively, has been reported and resonant effects have been seen by Maker (1965), Coffinet (1969) and m a n y others. The interference effects have been studied experimentally in detail by Yablonovitch (1971, 1972), L e v e n s o n (1972, 1974a, b), De Martini (1973) and K r a m e r (1974). In this case of three wave light mixing the two-step process m a y proceed in two different ways. In the first step a field may be created at 20)1, which, in turn, beats with the incident field at 0)2 to give a polarization at the difference f r e q u e n c y 20)~ - 0)2. Alternatively, first a field at the difference frequency 0)~ - 0)2 is created, which then interacts with the field at 0)1 to give again the combination f r e q u e n c y 2 o J , - 0 ) > Either or both of these two-step processes may be resonantly enhanced. For example, 20)~ m a y correspond to a sharp exciton resonance. This process was observed in CuCI (Kramer, 1974). The difference f r e q u e n c y 0)1 - 0)2 may c o r r e s p o n d to an infrared excitation. The most c o m m o n example is that it corresponds to an optical p h o n o n frequency. In that case 20),-0)2 may be considered as an antistokes frequency. This effect has been studied by numerous authors, but perhaps in greatest experimental detail by L e v e n s o n (1974). It occurs in media with inversion s y m m e t r y by coupling to a p h o n o n mode of even parity, a Raman active optical phonon. In media which lack a center of inversion, there is coupling to the vibration of mixed parity and an electric polarization at the difference f r e q u e n c y is also created. In this case one speaks of the coupling with the polariton mode (Coffinet, 1969). This polariton mode, in turn, interacts with a c o m p o n e n t of the incident field at 0)~ to create a polarization at 20), - 0)> Information about infrared vibrational excitations and infrared dispersive effects m a y thus be obtained by experiments which appear to involve laser beams at only to,, 0)z and 0)3 = 2~o, - 0)2 in the transparent, nondispersive region of the material. The interference between the two-step processes near resonance proportional to [a (2)],, and the nonresonant contributions proportional to a (3), is evident in the observed

Infrared dispersion of third-order susceptibilities in dielectrics

275

intensity at to3 = 2to1 - to2, which is proportional to [X°)]2. In the remainder of this paper a detailed analysis of the infrared dispersive behavior of X °) will be presented. This type of analysis is essential in the interpretation of nonlinear spectroscopic data obtained in the cited experimental papers. In our treatment the Schrtdinger equation will be replaced b y classical equations of motion. Indeed for the restricted f r e q u e n c y region which extends below the electronic transitions only the movement of the ions is relevant and the adiabatic approximation ensures us that the dielectric properties there, can be described in terms of a set of quantities which are analytical functions of the nuclear coordinates for small deviations from their values at the equilibrium configuration. Further, for such small displacements of the ions, their m o v e m e n t can be treated classically and the change of the polarization of the medium can be obtained as a function of the fields. A similar approach can be used to include exciton effects. We shall first consider, in detail, the case where the small displacements of the ions can be described in terms of independent normal modes, and the expressions for the intrinsic susceptibilities of the second and third order, X~2)(tol, to2) and X°)(oJi, to2, to3), respectively, will be obtained when all the frequencies involved are below the electronic transitions. The necessary modifications which follow from electrostatics are indicated in Section 2.2 for the case of second-order susceptibility, where we also present a generalization of the Lyddame--Sachs-Teller relations to the nonlinear regime. In Section 2.3 effects arising from inclusion of retardation are discussed and the expression for the effective third-order susceptibility is given in some special cases. The same approach is used to discuss the behavior of X~3~(to~,to~,-to2) near exciton states; the necessary modifications are indicated in Section 3 without going into the details. In particular the behavior of X°)(tol, to,, -to2) when to~ - to2 and 2to~ are near a phonon and an exciton frequency, respectively, is discussed. This allows one to determine accurately the relative magnitudes of Raman and two-photon absorption cross-sections and illustrates some of the possible pitfalls in the usual two-photon absorption spectroscopy. 2. D I S P E R S I O N ON N O N L I N E A R S U S C E P T I B I L I T I E S NEAR PHONON RESONANCES

We first consider the case where all frequencies are below the onset of electronic transitions in a crystal. In this f r e q u e n c y range there is no contribution from real electronic transitions and the electronic dispersion can be neglected. The total induced polarization arises only from the redistribution of the electronic density by the applied fields and lattice displacements. Since only long wavelength phonons come into play, the nuclear displacements are uniform throughout a volume of dimensions smaller than the wavelengths. H e n c e the total induced polarization can be written as a sum of identical effective dipoles each within a unit cell. Such a dipole will be written as a power series of the applied electric fields; the coefficients of this expansion are the macroscopic polarizabilities. The susceptibilities can be expressed in terms of these coefficients. In the following we shall first consider in detail the case where the small displacements of the ions can be described in terms of independent modes. Then the necessary modifications which follow from electrostatics and retardation will be discussed. 2.1. Independent According to absence of any described by the

M o d e Scheme ; Dispersion M o d e s the adiabatic approximation (Born and Huang, 1954, p. 166), in the macroscopic electromagnetic fields the weak nuclear motion is hamiltonian, normalized to a unit cell, Ho(X) = T~ + ok(X)

(2.1)

where TN designates the kinetic energy of the nuclei and ~b(X) their potential energy in the field of the electrons, and X characterizes the nuclear configuration.

276

C. FLYTZANISand N. BLOEMBERGEN

In the p r e s e n c e of electromagnetic fields the energy of the system is modified and becomes H ( ~ , X) = Ho(X) - # ( X ) . g - ½a (X): ~

- '13 (X) ~

- ~3' (X) ~

(2.2)

where ~ (X) is the lattice dipole m o m e n t and a , fl and y are the linear, second- and third-order polarizabilities for an arbitrary position of the ions. All quantities are normalized to a unit cell. We neglect for the m o m e n t any effects arising f r o m spatial dispersion. For inclusion of these effects in the linear case see Maradudin et al. (1971), p. 269 and in the nonlinear case see Section 5. To the energy density per unit cell given by (2.2) c o r r e s p o n d s an induced dipole m o m e n t given b y a H ( ~ , X) 0~

P=

(2.3)

U n d e r the action of the electromagnetic fields the ions are displaced f r o m their equilibrium positions designated globally b y Xo. T h e s e displacements, u = X - Xo, assumed small, can be e x p r e s s e d in t e r m s of the normal m o d e amplitudes q ( k / ~ ) where k is the w a v e vector and o" designates the mode. Only the m o d e s with k = 0 will be driven b y the electromagnetic fields (dipole approximation) and we designate b y q~ the amplitude and b y to~ the principal f r e q u e n c y of the m o d e o- for k = 0. Clearly effects arising f r o m acoustic m o d e s are left out of consideration in the present discussion. An a t t e m p t to include such effects was m a d e recently b y L a x and Nelson (1971). H e r e , and in the subsequent sections, we a s s u m e that the frequencies are well a b o v e any elastic resonances. Then in terms of these normal m o d e amplitudes the hamiltonian (2.2) describing the system of ions in an internal m a c r o s c o p i c field ~ ( t ) can be written 2

1

+~..

1

~

1

=

2

1

~(" ~2)

p. ~,~.m~q~

-

1

~,(r'

<3) • o-,o-',~"

lot,~gigj

1

1

,)

1

1

1

here pc is the canonically conjugated m o m e n t u m of the o~-mode. S u m m a t i o n o v e r the p h o n o n indices is explicitly indicated while s u m m a t i o n o v e r repeated cartesian indices i, j . . . is assumed. According to (2.3) this energy density is equivalent to postulating the following d e p e n d e n c e of the polarization of the field and ion coordinates: Oh P~ = - Og, = ~ E

m

+ ~

m ~

o.

1

-,~' /*~..~'q~.' 1 tr, tr'

~,

. ,, Ix~,,.,.,,.,,q,.q,.,q,.,,(3~

a o,,,~' gJq,~q,.'

c,

a ~E, /3 ~, and y e,~ are the purely electronic unit cell polarizabilities of first, second and third order, respectively, for the equilibrium configuration of the ions: they are related to the distortion of the electronic distribution, the lattice being held fixed in its (1) equilibrium configuration. The coefficients a ~-p,,,.,a ~=~,s.~',/3,,~,. are defined as follows: i~,~= Oq~,'

(2.6)

I n f r a r e d dispersion of third-order susceptibilities in dielectrics

277

O 20lij

(2)

Ol fj,o.o.' -~-

(.

_

Oq~Oq,,,'

(2.7)

O/31j~

(2.8)

where the derivatives are evaluated at q~ = O; the coefficients (2.6-8) account for the electron polarization-nuclear displacement interaction. Finally Ix~(', Ix (2) ~.~, and Ix (3) ~.o~ are the first-, second- and third-order lattice dipole coefficients; /.t ¢1) ~.~is proportional to the effective charge of the m o d e (r while Ix (2~ ~.~, and Ix (3) ~.~,~,, describe the modulation of the effective charge b y one and two phonons respectively. In writing (2.5) and all the previous expressions the essential assumption was m a d e that the electronic polarizabilities are f r e q u e n c y independent. This is sufficiently well fulfilled for frequencies below the electronic transitions. Further, it was assumed that the frequencies are a b o v e the elastic resonances of the system. An extension of the present theory to include the latter resonances has been given by Maradudin and Burstein (1968) and extended b y L a x and Nelson (1971). E x p r e s s i o n (2.5) relates the dipole m o m e n t per unit cell with the electric field and with the normal coordinates of the ion displacements. We are only interested in the connection b e t w e e n the induced dipole m o m e n t and the electric fields. Such a connection can be obtained if it is taken into account that those q~ which differ f r o m zero appear themselves as a result of the action of the electric field. Thus we must obtain the connection b e t w e e n the normal coordinates and the electric fields. The equation of motion for q~ is obtained f r o m the relation OH

Oq,. which according to (2.4) is of the f o r m /~ +to~q~ +F~4~ + ~1 ~

6~,,.,,q,.,q,.,,+ l~,,.w,.,.,,~,,,.,,,q,.,q,.,,q,.,,, ~(')

o-' ,tr"

• tr',

=

(I)co

,(r"

Ix i'trl~i 4- E er'

(2) ~ i,.,1o-' ~ 1 ~~, (3) Ix~.,.o-' + -~ Ixl,...'.."~iq,.'q,."

~ o",o'"

1 (,) 1 <. (2) ~i%q~,'+~/3~jk.~,%%, + ~a,j.~,~j +~1 ~ a,j.~,

(2.9)

where a damping constant F~ has been introduced phenomonologically for each m o d e separately; this will be termed the independent m o d e approximation. The F~'s arise f r o m the interaction of the m o d e cr with all the other modes. We solve iteratively the equations (2.9) up to and including terms cubic in the fields or

q~ = q•l)

q,(2) + q~(3) 4-

+

(2.10)

...

where the superscript indicates the order of the field. The field is assumed to be a superposition of m o n o c h r o m a t i c plane transverse w a v e s of frequencies to,, to2, to3. . . . or

~ ( t ) = ~'~ E(to~) e -i'~''.

(2.11)

s

Introducing (2.10) into (2.9) one obtains p __ pO) 4- p(2) -4- p(3) 4- . . .

for the induced dipole in a unit cell, where p(" is the part of the dipole m o m e n t linear in the fields, P 1(2)

E o-

(i)(2)

I

Ixt.==,q~, q=, o.,o.'

~ o-

+/3~k%~k

(2.12)

278

C. FLYTZANIS and N. BLOEMBERGEN

is part of the dipole m o m e n t quadratic in the fields and p , < , ) = ~ ~.~i,o'qcr .) ,3,__1 + 1 ~,~ (3) -'~-2 E /z~.oo, (2) q~(2) q~, (I) (1)(1) -~ /z~,~,~,,q~(1) q~,q~,, o~

o',(7'

o',

"

+ G ~;.~qd2' + G ~l~.~,~,q~'l'o~ ~+ ~ ~l;d.~,~qo ~1~ E

"F ~t ijkl~1~k~l

(2.13)

is the part of the dipole m o m e n t cubic in the fields. Introducing the Fourier c o m p o n e n t s one obtains

[30k(co~, co2)Ei(co,)Ei(co2)

pi(2)(CO, + co2) =

(2.14)

and

YOk,(CO,,CO2,co3)Ej(col)E~(co2)E,(co2)

p/3)(col "4- 602-1- £03) =

(2.15)

which constitute the definitions of the second- and third-order polarizabilities per unit cell. C o m p a r i n g (2.14) with (2.12) we arrive at the following e x p r e s s i o n for/3,~(co~, cod (Genkin et al., 1967): fl~k..F [31"i iJk(col,

flijk (COl) CO2) =

£02) ~"

(2.16)

H' N ljk(col, 0)2) "~- ~ 'jk(co 1, CO2)

where H 1 jI3 Ijk(CO 1, CO2) = ~ ~

( |

(1)

(1)

~J~i#rOlJk'~r

(1)

(1)

(I) 1~ (I)

-r ]~J.~r ik,cr

I~k,~rl-,li~.~

"I

[D(co~, co, + co2) D(co~, co,) ~- D(co~, co~)J' !

r ~" ]JJ (2) i)(~r°"]'~ I,(I) (:r]=1~(I) k'°'' (I) (2) (I) fl H',j,(co,,coz) = _i ~ + 2 ~.~. tD(co~, co~)D(co~., co2) D(co~, co, + co:)D(co~, co:)

+ D(co~, co, + co2)D(co~,, co,

'

/x ~,~/x(1) ~.~,/x(1) ~.~,, /3 ,~k(CO,,Nco~) = --~ #.#'~.~"~b~.~ D (co~, cot + co2)D (co~., col)D (co~.., co2) 1 1

(I)

where D (co~, co,) = col - co,2_ ico,F,.

(2.17)

Similarly for the third-order polarizability one obtains I

E

r

(,~ (i)

(i)~ (i)

7,Jk,(co,, co2, co3) = y,jkl + ~ ~ LD(co~, co, + co2 + co3) -~ D(co~, co,) (1) ~ ( 1 )

( 1 ) ~ (I)

"~

-f D ( c o . , co~) - D ( c o ~ , co,)J

3 ~ tD(co,,, co2 + co3) "+ D(co,., co, +co3) )-,--,co,.,co,~co2)I +1.-,~.~

' o r (') /.t(') i.,dz"j.,,, kl,,,~'

a jk.~,.' (2~ /X ") ~,,./~(1) k,,.'

D ( c o ~ , co~ + co~ + co~)D(co~., co,) + D (co ~ , co, + co~ + co~)D(co~., co2) (1)

-t

(1)(~ (2)

O/ (2)

/z i.~/z ~.~ jk.~'

q

(I)

(1)

0,~,'/z k.,,/x I.,,'

D (co~, col + co2 + co3)D(co~,, co3) D(co~,co2)D(co~,,co3) i k , ~ ' / - L (j,o-~ 0 t (2) i) (I,~' l)

(1) (1) t0g x (2) iL,r,r'lA, l.,r]~ k.~'

•

-4 D(co~, co,)D(co~,, co3) -~ D(co~, co,)D(co~,, co2)~ .~. 1 ~'~

(I)

(2)

a(l)

D(co,,, co, + co2 + co3)D(co,,,, co~ + co~) (1)

(2)

CI(I)

/-(, i,o-/-~k,,r,7' jl,~'

+ D(co~, co~ + co~+ co3)D(co~, co~ + co3)

Infrared dispersion Of third-order susceptibilities in dielectrics (I)

(I)

(2)

(2)

~¢o~~l,,,,-,iu. i.,.=,

/z

#z ~,~,/.~

(I)

279

(l)

i,i,.ott,l,,.,

+ D ( m : , o.I, + m ~ + o.l~)D(ai:,, m, + o.l~) + D ( a l : , oJt)D(o.l:,, o~ + m~) (2)

(1)

(1)

(2)

/~ ~ - ' l u , ~ , . a i~,~ '

(t) ~ (1)

lu. ~,o='/~ z.<.' ik,='

+ D(o.l:, al:)D(o.l:,, o~, + m~) + D ( a l : , o.l~)D(oJ:,, oJ, + ~o~) '"

'"

">

?',L

+

aO,,.~k,=,/X~.,,=,

. txi, ,/z,

+/~"(w:, a l i + m~)D(w,,,, m2) (l)

( l ) , (2)

(t)

Ol ik,o,l.~ I,o.' ~ J,o~o.'

+/)(m:,

,

D ( o : , oJ, + m,)D(m:,, o~,) (2)

--(I)

Ol ij,=~.l~ k,am" ~ l,o'

+ o.l~)D (m<,,, o.l~)+ D ( ~ : , mi + ~o~)D(aJ:,, m~)

o~, (I)

(2)

(1)

(1)

+ D'(oi,,, m, + mi)D(m,,., o~,)

(I)

-~

(o.l:, m, + mi)D(o~,,., o.l~)

, (1),

__1 X "

(2)

(i)~(1)

At(3)

~,=e's,=' kl.,."w<.<.,<.3<.,~,~. D(al:, all + oJ2 + oJ~)D(~o<,., o~,)D(oJ:., ojz + m3) (1), (1) ~ ( l ) I/.(3) / £ i , = ~ k , o ' u it,,,'"'4" o.o.'o."

+

D(o~o, o~ + o~2+ o~3)D(o)=,, w2)O(w=,,,O) 1 -~ (I) ~ ( 1 )

(I)

0.33)

i(3)

/z i.<.~ ~,=, ~,<..q~,,~,<.,,

¢o3)D (o.Jo-,, o~3)D (m,,--, ~o, + oJ2)

+ D'(oJo., m, + ~2 +

{3/(1) (1) (1) ~ ( 3 ) ii,,r/~k,o'/-i,i,o-"~po~,'~"

+ D(al,,--

m2 + w )D(oJ,,., oJ~)D(oo<,.., oJ~) (I)

(l)

(1) ~t(3)

(1) (1) ~t(3) "1 ot (1) ,~,,,-/z i,~, p- ,~,<.-(p o-,.,<.,, + D(m:, o~, + oo~)D(~o:,, o~,)D(oo:,,, m~) D(m:, o~, + oo~)D(m:,, m,)D(~o:., o~)

+

ix a,.<.~ ~,,.,lu. ~.,.,,(p o=,<>-,,

+ 3 <,.<,,,:. tO(o,<.,

<,,,) (1)

+

J

(3)

(1)

(1)

/.* ~,,,/.i, i.<,<,,,~,~ k.<,,/.i, i,,~,,

D(m,,, aJt + al2 + co~)D(m,,,, aJ2)D(co,,,,, co~) (t)

O)

(3)

(t)

+ D(o~,,, m, + m2 + o.l~)D(o.l<,,, o , ) D ( o J , : , o.l~) (I)

fl)

(D

(3)

/~ ~.<,/-i, j.<,,/J, k.<:/~ ~,,,=.<,,,

+ D(m<,, oJ, + m: + o.13)D(o.l,~,, oJi)D (al,,,,, m2)

tD(m~, oo, + ~o2)D(m~,, ~o,)D(~o~., cod /.*(2) i,,,~,/~ (1) j,=,/~(2) k,,,~,,b~(I) ,,=o +D(~o~, w, + m2)D(¢o~,, o~,)D(m~,,, o~3) (2)

(2)

(1)

(1)

/.~l,o,.'/.~ l,,,=-/x k,,.'/Z I,,," + /~(W,,, m2 + ro3)D(m,,,, ¢o2)D(o,,., m3) (1)

(2)

(1)

(2)

~ ~.<,V, j.<,=,/.L k.<,-V, ~.<,,,,-

+D'(m,,, m, + oJ2+ 0 3 ) D ( o , . , , ¢0, + ~3)D(oJ~,-, ~2) (1)

(2)

(2)

(I)

+ D(~o~, oJ~ + co~ + ~o~)D(~o.,, ~o, + co~)D(o~,,, oJ~) (1)

(1)

(2)

(2)

"t

D ( m ~ , m, + m~ + oJ~)D(¢o~,, ¢o~ + m~)D(m~., ¢o,)j

280

C. FLYTZANIS

and N. BLOEMBERGEN

1 _,,) (1) (,) ~o ~,~,,~,,,/x (1) ~.~/xj,~,p~ k.~,,/x(1) ~,~,,, 3 ~.~,...~- D(to~, 0`), + 0`)2+ to3)D(to~., to,)D(to~,,, to2)D(to~,,,, to3) 1 ~ (3) (,) (1) (1) (2) N-. 1" 3 ~.=,,~,,,=,,,~D(to.. to, + to:)D(to~,, toj + 0`)2+ to3)D(to=,,, to,)D (to~,,,, 0`)2) /)(3)

(,)

(2)

(,)

(1)

+ D(to~, 0`)2+ to3)D(to~,, 0`), + to2+ to3)D(to~,,, to2)D(to~,,,, to3) (b(3)

(,)

(2)

(,

+ D(to=, 0) 1 "4- to3)D(to=., 0`), + to2 + to3)D(to~.,,, to,)D(to~,,., to3)

C D (2) ( (,)3 (1) )

(1)

1

+ D(to~, 0`), + to2+ to3)D(to~,, to,)D(to~,,, to2)D(to~,,,, 0`)3) --(3)

+

.

(1)

(,)

(1)

(,)

,,,,.,.,,~,,.,,,.,,,,,,~D(to,,,, 0`)1-]- 0,)2 "~ to3)D(to,,,,, to,)D(to,,., to2)D (to~,~, 0`)3) 1

×

t(3)

1

D(to~,tol+to2)

A D(to,~,to2+to3)

1 } +D(to~,to,+to3) "

(2.18)

This c o n c l u d e s the d e r i v a t i o n of the e x p r e s s i o n s f o r /3~jk(to,, 0`)2) and y~jkt(to,, 0`)2, 0`)3) at f r e q u e n c i e s b e l o w the e l e c t r o n i c r e s o n a n c e s of the s y s t e m . W e p r o c e e d n o w to give the e x p r e s s i o n s f o r the susceptibilities. T h e p o l a r i z a t i o n per unit v o l u m e is given b y the e x p r e s s i o n p = lp

v

= Np

(2.19)

w h e r e v = N -1 is the v o l u m e of a unit cell and N the n u m b e r of unit cells p e r unit v o l u m e . U s i n g (2.19) one obtains p = L ( p ( 1 ) q_ p(2) .+ p(3)) ---- p ( l ) _~_e(2) .q._p(3) ,/2

where p,l) : l p , ) , /)

pa, = lp(2,

and

p,3, = lp(3~

13

V

or using the F o u r i e r c o m p o n e n t s f o r P") and p a ) and p(3)

P['(to) =

a,s(to)Es(to) = X~s('(0`))E j( 0`)),

pi(2)(0`) 1 "~- 092) : 1 ~ijk (O),, t o 2 ) E / ( 0 ` ) l ) E k (092) = X I~)( 0`)1 , to2)E} (0`)1)Ek (to2),

(2.20) (2.21)

1

p o)(to, + 0`).,+ to~) = v 3,,jk,(0`),, to~, to~)Ej (0`)0E,, (to:)E, (to~) (3), = )('ijkd.to,, 0`)2,

to3)Ej(tol)Ek(toJ)El(to3)

(2.22)

w h i c h c o n s t i t u t e also the definitions of the first-, s e c o n d - and third-order susceptibilities respectively. We have Xl~(to~, 0`)2) = 1/3,jk (to1, to2),

(2.23)

Xl~l(to,, 0`)2, 0`)3) = u1 3',jk,(to~, 0`)2, 0`)3).

(2.24)

This c o n c l u d e s the d e r i v a t i o n of the e x p r e s s i o n s f o r X a) and X ~3) in the i n f r a r e d region of the s p e c t r u m b e l o w the electronic transitions. W e r e m i n d the r e a d e r that the

Infrared dispersion of third-order susceptibilitiesin dielectrics

281

linear susceptibility in the same region, defined by (2.20), is given by x

~l'(to)

(2.25)

= 1 a , (,o) I)

where (1)

(1)

a,(to ) = a ~ + ~ to,,lZ~'~txto

(2.26)

In the above derivation of the expression for the induced polarization of a given order in the fields, the wavevector dependence of this quantity was not explicitly indicated. We only point out here that the wavevector of this polarization is the algebraic sum of the wavevectors of the fields involved. 2.2. Independent Mode Scheme; Natural Modes The dispersion frequencies to~, previously introduced, are not in general the k = 0 optical mode natural frequencies of the crystal. H o w e v e r , in order to relate the nonlinear polarizabilities to the fundamental properties of the material we must introduce the natural frequencies of the medium. In order to incorporate them the equation of motion (2.9) and the constitutive relation (2.5) must be supplemented by the relation between the amplitude of the macroscopic field and the polarization V • (~ + 4~-P) = 0.

(2.27)

Omitting for the moment retardation effects one obtains from electrostatics the condition V x ~ = 0.

(2.28)

We express g and P in terms of their Fourier components and now we indicate explicitly the spatial dependence of their phases by writing

* ( r , t) = ~, E(to,) e i(klr--wlt) ,

(2.29)

P(r, t) = ~ P(to,)e i<~''-''.

(2.30)

!

1

From (2.27), (2.28), (2.29) and (2.30) one obtains

E (oJ,) = -4~rk, (/~,. P(to,)

(2.31)

where k~ is a unit vector in the direction of k,. Hence, in addition to the primary incident fields, there will be in the crystal internal fields determined by the polarization state of the crystal; for each Fourier component of the polarization induced in the crystal there will instantaneously be a Fourier component in the field given by (2.31). These additional "material" components of the field will act on the charges and their effect will be to renormalize the frequencies and the coefficients previously introduced. In fact the equation of motion (2.9) and the constitutive relation (2.5) are still valid. Since P = Np and p is given by (2.5), when the additional fields (2.31) are introduced in (2.5) and (2.9), b y proper rearrangement and redefinition of the terms the new equations for p and q, can be recast in their initial form, (2.5) and (2.9) respectively. This is a power series of the normal coordinates q`, and the external fields, with coefficients which are functions of the old coefficients w~, ~3> . . m a ~ ), fl~ and the directions I~t.The expressions for these coefficients will in general be very complicated. Below we outline the derivation of the expressions for a and fl taking into account (2.27) and (2.31). From (2.5), (2.9) and (2.31) we see that the linear polarizability is determined by the two equations ~o'o"cr".

p,m(to0 = .~_~/x"'l..q~m-'l-

•

•

•

Jt£`,

,

.

.

•

,

•

•

•

,

•

•

•

.l--,~-~°t~lE'(toO-4"n'Nk"~"P"t'(to')l

(2.32,

282

C . FLYTZANIS and N. BLOEMBERGEN

and q¢"'+ to, q J " = ~ tzl,'~{E,(oJ,)-47rN/~, ~/~,p[1)(oJ~)}

(2.33)

where in the equation of motion of q¢ for simplicity we have neglected the damping term F¢q~ '). After rearrangement of the terms the two eqns. (2.32) and (2.33) can be written as pi(1)((.O1) __

E ~ S,. i m , ,tx.l~,..~.q~ i ~', (1, (1,

. a~(/¢l)E(oJl)

+

(2.34)

and ~ (1) ..~ ~

2 ^ °')~.~'(kl)q

(1)'~'= Z

^ (1) S..(kdl~,..~E,(w,)

Z

(2.35)

respectively, where (Maradudin et al., 1971, p. 251) E ^ o~~(i) = ~ ot,mS,.,(k) m

Eo(~)

,~,,,, A,i

K,~j~

(2.36)

with 4 " / T N .^

S o ( f O = 6o - "-'--"7-k~ ~

e~(k)

^

E

(2.37)

k,,a,,,j

and ~.(/~) = 1 + 47rN ~]/~a~/~. O

One can find a unitary transformation V(t~) which diagonalizes the matrix {~o]~,} by solving the associated eigenvalue problem det ]w#(t~)G~,- o~]~,(t~) I = 0;

(2.38)

we will call the w=(k)'s the natural frequencies of the system. Introducing then the new normal coordinates

(2.39)

q=(k) = ~ V~,(k)q~, cr

eqns. (2.34) and (2.35) can be written as pi(l)(o),)

= p [ 1 ) ( / ¢ 1 , (O,) = Z

(') ^ (1) (kl) ^ /x,,~(k,)q~

_1_ Z j

a ,E( k ^0 E j (wl),

/j~,,(/~,) + w d ( f c O q , , , ( f c , ) = ~ / ~ I~(/~I)E, (~o,)

(2.40) (2.41)

respectively, where ("i"

mo-'

V

"i"

(,, o

"i"

The linear polarizability is defined by p(')(oJ0 -~ p m(~,, ~ol) = ~ a,(/~,, oJ1)Ej(w,).

(2.43)

In view of the similarity of eqns. (2.40) and (2.41) with the linear parts of eqns. (2.5) and (2.9), respectively, the expression for or(k, w ) has the same form as the expression for a ( w ) given by (2.26). Similarly, from (2.5), (2.9) and (2.31) one obtains that the second-order polarizability

Infrared dispersion of third-order susceptibilitiesin dielectrics

283

is determined by ,~.

(,)

(2, + l E E S,m(ko)//, (2)

ra

(')-(')

~ tr.tr '

+ ~.~ ~ S,.( o)Ol.,,.~E,(to )-41rN[~, ~. f~.pm(to,)q(,,

'"'{

1

+ ~ Z,.,,S,(ko)/3:.{E,,,(to')-4"n'Nf~ ~,r/~,p re(to,)} X {E.(to")-4"n'N/~ ~ x,p, c.,, (,,.t.to . . . )~] and /j(,)+todqf2)+ y~ v" ,ho, o, = - 4 7 r N ~ /x (,,c x-, kompm ^ (')t~tot + to2) ~'-"q (,, ~' q ~" i,~,KoiL ~r',tr"

i

+ ~ , g ~., [ E , ( t o ' ) - 47rN/~

m

/~'p,.(')(to')}q~?

1

+ 2 ~ al~l.~{E , ( t o ' ) - 47rN~I ~ / ~ ' P , f ) ( t o ' ) }

x { El(to ") - 47rN~'I "ff]i~"p.")(to")} where to' = to,, to,, to" = to,, to,, /~o is the unit vector in the direction of k, + k2 and the bars above the products of fields and normal coordinates mean that all the components at f r e q u e n c y to, + to2 will be kept and only these. Taking into account (2.40), (2.41) and (2.39) the previous two equations b e c o m e

pY~(to, + to~) -= pY~(~o, to1 + to~) = ~ a,.~t om~ t oj +

1~,

(2)

^

^

, /x,.,,~'(ko; k',/~")q,,m(/~')qg)(/~")

+

(2.44)

i")Ej(to')E~(~")

+ ~/3.~(io, i', jk

and

~f'(~o) + to/(~o)qf'(~o) + ~ ..~-'".~,~,,,.~o,":~'~")qd'(~')q ~,,'(~") cr',cr"

= ~ p,~,2~,(/~,;/~o, k")E,(to')q~?(k") hr'

+~'~ ct Cl),~,,.~tK,/~";/~o)E,(to')E~(to") (2.45)

where /3,,(ko, k',

- ~

( k^') & . S,,(~)/3 ,~.Sj~ ~

(2.46)

( k"" ),

Iron

" " ~ i,~ ' ; k^" ) = ~ S,, ( ~ ){a,~.~, ., E .... ot,.~ - 4 ¢ r N E /3,m./~.E/~s Imp'

n

x Z tx,.,,,S,,(k (1)

t

^it

^p ^tp )}S,,, (k)V,,,,.(k ),

s

(2.47)

284

C.

(~) (/~ "' /~") =

and N.

FLYTZANIS

BLOEMBERGEN

( ,{~,,,.,.,.,-4"~'N E a , . . . . +(4,rN) ~

r

s

x {k"X k " X p. 2;, , s,~ (k")}} t

~

)

^, ~, k~,. 2~ p, ~J.~S.,(1~')} 13e.m~{k,. mn

,/,'~' el" k^' , i")= o~,~-.-,

/x,.,,o,,t

v~ (£-') v.,.,(~")

(2.48)

u

{ (;b~..,. -

41rN E / x '~' ,..... - G ~~ kr X

*-r '1-"

nt

r

(41rN)2 ~

X ] d , s(,) . r S r s ( k ) +~

s

ol t" . . . . ,,

rtln

x{k,, Y,k~Y,p,~.,~r,k r

kt]~l~...,S,.(k )}

)Yt~. t

s

u

')tn - (4zrN) 3 ~ , / 3 ~,~{/~. E/~, X/x (,..~,, t"k'" )t Irnn

r

s

k 0 E / x ~..,,O~wt )j'j,

x U.(fOU~,.,(fC)U..,,.,,(fC').

(2.49)

The s e c o n d - o r d e r polarizability /3 is defined by

13isk([CotO,+ tO2; k,tO,, k~tO2)Ej(tO,)Ek(tO2).

p[:~(tO~ + tOz) --=p[2~(l~o, tO, + tO2) = ~

(2.50)

p<

In view of the similarity of eqns. (2.40) and (2.41) with the second-order parts of eqns. (2.5) and (2.9) respectively the expression f o r / 3 (/~otO,+ o~2; /~to,, /~2to2) has the s a m e f o r m as the expression f o r / 3 (to,, too given b y (2.16). F r o m the previous formulae it is clear that in general the dispersion frequencies to~ and the natural frequencies tO, (/~) of the k = 0 optical m o d e s do not coincide. The same is also true for the other coefficients defined b y (2.4), (2.4) and the corresponding ones defined above. In addition b y comparing the two sets of expressions, we see that the values of the limiting optical m o d e natural frequencies and coefficients at kl = 0 depend on the directions /~ along which the points ki = 0 are reached. H o w e v e r , for crystals whose s y m m e t r y and structure are such t h a t / ~ ( " is isotropic, and the m o d e s are pure longitudinal and pure transverse the second term on the right-hand side of the eqn. (2.36) vanishes for t r a n s v e r s e modes, and the frequencies of these m o d e s coincide with the corresponding dispersion frequencies. The solutions of eqns. (2.38) are difficult to obtain in a c o m p l e t e l y general form. H o w e v e r , for diatomic cubic crystals of the zinc-blende structure, eqns. (2.38) simplify considerably, and a complete solution is possible. For this class of crystals one has a doubly degenerate t r a n s v e r s e m o d e of f r e q u e n c y tOT and a longitudinal m o d e of f r e q u e n c y tOE. Further, for the k = 0 optical m o d e s one can take as normal coordinate for these m o d e s the relative displacement of the a t o m s per unit cell. Then instead of /XT(" and /XLm it is convenient to introduce the effective charges e~ and e*. F r o m (2.36) and (2.42) one obtains the L y d d a n e - S a c h s - T e l l e r relations 2

[ml ~ 1 - ,

e~ = e?/e~.

(2.52)

2

~o

F r o m (2.47), (2.48) and (2.49) one obtains, putting K~ = (to2 _ tOT2)/tOT2, the relations OL~(1)= (XTm(1 - K,=~----'~.),

(2.53)

Infrared dispersion of third-order susceptibilitiesin dielectrics

(

/~,,~, =/x~- 1 - (K~ + K~,)~--~2+

6,,,,,,,,,= c h ~

1)

285 (2.54)

( 1--(K,, + K,,, + K,~,)-'~3+(K,,K,,, c2 + K,,,,+ K,,,,K,,)

× -C~ -~ K=K~' K="-~3

(2.55)

where tr, tr', tr" = L, T and C1 =

e ~ o t T 'l)

2MtoT:[3 '~'

Cz =

[ e-~ ~2/z~ \ M ~ v 2) 2[3 r,

[ e*t '~3~b~-r C3 = \Mto.r2 } 2[3 ~"

(2.56)

The relations (2.53), (2.54) and (2.55) are actually a generalization to the nonlinear regime of the L y d d a n e - S a c h s - T e l l e r relations (2.51) and (2.52); they allow one to obtain the coefficients a ~ ' , tz ~), and tb ~,~., in terms of those for the transverse modes and they play an important role in the theory of two-phonon spectra of solids (Flytzanis, 1972a, b). The above approach can also be extended to derive the expressions of the third- and higher-order terms of the induced polarization when eqn. (2.27) is included. In the following, however, we shall also be interested in the retardation effects in the third-order polarization; therefore it is preferable to obtain these expressions within the more general context provided by the complete system of Maxwell's equations. This will be taken up in the next section. We stop here to point out that throughout this discussion we have disregarded mode damping. As can be easily seen, if the damping term F~q~ were kept in (2.33) and the subsequent equations for q~, mode superposition through damping may occur and the algebra becomes exceedingly complicated. This problem is taken up below in the simple case of a system with two modes. 2.3. Retardation Effects; Polaritons Up to this point we have described the electronic interaction between ions by means of the Coulomb force, which has been assumed to act instantaneously between the ions. H o w e v e r , the Coulomb interaction does not act instantaneously but propagates with the speed of light in the crystal. Just as the long wavelength longitudinal part of the polarization in a polar crystal can give rise to a longitudinal electrostatic field, it is found that when the retardation of the Coulomb interaction is taken into account the long wavelength transverse part of the polarization can give rise to a transverse electromagnetic field. Further, just as the longitudinal field altered the frequencies of the longitudinal modes, the transverse field also will act back on the transverse modes and will alter their frequencies from what they would be in its absence. But at the same time it may interact with other fields to induce nonlinear terms in the polarization of a given order in the fields and of a given frequency. Accordingly the total polarization of a given order and a given frequency besides the direct contribution will contain indirect contributions arising from many-step processes involving, at the intermediate stage, generation of lower-order nonlinearities. Even longitudinal fields and polarizations may be involved at the intermediate and final steps. Hence the total nonlinear susceptibility describing the total polarization of a given order and a given frequency will contain, besides the genuine nonlinear susceptibility of the same order and f r e q u e n c y combination, sums of appropriate products of lower-order susceptibilities. Since these lowerorder susceptibilities may be associated with either longitudinal or transverse polarizations these additional terms in the total susceptibility will be strongly dependent on the w a v e vector direction. In order to incorporate retardation effects in the total susceptibility we must drop the equation (2.28), which is only valid in the electrostatic approximation, and use the complete set of Maxwell's equations, which for a nonmagnetic medium in the absence

286

C. FLYTZANISand N. BLOEMBERGEN

o f free c h a r g e s and c u r r e n t s are written as V .D =0,

10B

VxE

. . . . c Ot'

/

V.H=0,

V

x

1 0D c 0t

H-

(2.57)

with D = E + 4~rP.

(2.58)

T h e e q u a t i o n o f m o t i o n (2.9) and the constitutive relation (2.5), on the other hand, r e m a i n u n c h a n g e d . Eliminating B f r o m M a x w e l l ' s equations one obtains 1 ~92D Vx Vx E . . . . c dt

(2.59)

or separating the linear and nonlinear parts in P P -- PL ÷ PNL

where PL = p~n and PNL = pC2) + pt3) + . . . o n e obtains 1

2

2

V × V × E + _ _ . ~ a ~ D L -47ra PNL c at - c - ~

(2.60)

w h e r e DL = E + 4~rPL. I n t r o d u c i n g the F o u r i e r c o m p o n e n t s of the fields and polarizations

E(r, t ) = ~ E ( r , to,) e -i~,, P ( r , t) = ~ P ( r , tol) e -io't I

l

and the relation DL(r, to) = ~(to). E ( r , to)

(2.61)

o n e obtains in the dipole a p p r o x i m a t i o n V x V x E ( r , to) + ~(to)to2 C2 E ( r , to) -----4,trto - - 7 -2 PNL(r, to)-

(2.62)

E q u a t i o n s (2.5), (2.9) and (2.62) fully solve the stated p r o b l e m . H o w e v e r , with the inclusion of eqn. (2.62) the p r o b l e m of p h a s e m a t c h i n g m e n t i o n e d in the I n t r o d u c t i o n c a n no longer be dissociated f r o m the s u b s e q u e n t analysis of the nonlinear susceptibilities. In fact the solution o f eqn. (2.62) c a n be written E ( r , to) = E ' ( r , to) + E"(r, to)

(2.63)

w h e r e E ' ( r , to) is the solution o f the h o m o g e n e o u s or free e q u a t i o n V xV x E'(r, to)+

to~E(to) ¢2

E ' ( r , to) = 0

(2.64)

and E"(r, to) is a particular solution of the i n h o m o g e n e o u s or f o r c e d eqn. (2.62). I n t r o d u c i n g plane w a v e s E ' ( r , to) =

E(k', to) e ik''"

E"(r, to) = E ( k " , to) e i~'" P ( r , to) = P ( k " , to) e iv'"

Infrared dispersion of third-order susceptibilitiesin dielectrics

287

one has

k ' × ( k ' × E ( k ' , t o ) ) + t°2~(t°) C2 Et~k, ,to)=O

(2.65)

tk" , to) k"×(k"×E(k",to))-4 t°2E(t°) c2 E ( k " , t o ) = - 41rt°2P -~ SL~

(2.66)

and

from (2.64) and (2.62) respectively. The amplitude of E(k", to) can be immediately obtained by solving the system of the three linear equations (2.66); this solution can be easily obtained for the cubic and the hexagonal cases. The amplitude of the field E(k', to) has to be determined from the boundary conditions. The phase-matching problem arises because, in general, the wavevector k" of the induced polarization source PNL which generates the field E " is different from the wavevector k' with which a monochromatic field of f r e q u e n c y is allowed to propagate freely in the medium according to (2.6). The Maker fringes arise from the spatial interference of the fields E ' and E". We want now to consider the effects that follow as a consequence of the presence of the "material" fields (2.66) in the crystal along with the incident fields. The equation of motion (2.9) and the constitutive relation (2.5) are still valid. However, the fields (2.63) generated by the nonlinear terms of the polarization must also be included in (2.11) along with the initially incident fields; these "material" fields act back on the charges of the medium and modify their motion. The approach followed in the previous Section 2.2, when the electrostatic condition was imposed, can again be used; one only needs to replace everywhere the tensor S(/¢), given by (2.37), by the tensor

T,j = C ? / - -

41rn 2 .l n ~ - 1 ~ ~. 8.,.

.,.

kma ",C,j

4~rn2 + n---~_1 N ~ t~ r.,C,, k,

(2.67)

rnn

where

C,J=n21-~_l[n2&~-~j]

and

E,~(to)=l+41rNaij(to)

and alj(to) is given by (2.26). When this is done the analog of eqns. (2.36) and (2.38) give the frequencies of the modes when retardation is taken into account (see, for instance, Maradudin et al., 1971). These apparent frequencies are different from the ones defined in Section 2.1; in particular they are strongly wavevector direction dependent. Clearly the nonlinear terms in the polarization may again be related to the initially incident fields with the help of nonlinear susceptibilities. However, in general the expressions for these quantities will not be identical with the ones derived in Section 2.2; because of the intermediately generated fields (2.66) these susceptibilities will be strongly wavevector direction dependent. In the following they will be referred to as effective susceptibilities to be distinguished from the intrinsic ones defined in Section 2.2. The effect of the retardation on the linear properties of the crystal is well known and leads to the polariton concept (Huang, 1950). In the case of the second-order polarization the problem of retardation is closely related to that of phase matching (Bloembergen, 1965) and the relation between the intrinsic and effective second-order susceptibilities is easily derived. The dispersion of X(2~(to,, to2) when to2 is near a phonon resonance while to, + to2 and to1 are well above the phonon resonances was obtained by Faust and H e n r y (1966) in GaP, a crystal with 43m-symmetry, and found to be of the form I.LT(I)o~T(1) X(:~(to~, to2) = l f l ~ + V(tOT2 - to22+ ito2F)

(2.68)

which can also be derived from (2.17). This expression was found to have a minimum ,a,o.r.

4 3

i-

288

C. FLYTZANISand N. BLOEMBERGEN

absolute value for a value of ogz smaller than tot and this allows accurate measurement of the ratio ~ T ( 1 ) O / T ( 1 ) / O g T 2 f l E . The situation is different in the case of the third-order polarization at frequency ogs + co, + to, induced by three fields of frequencies cos, to,, to, and wave vectors ks, k,, k, respectively. If the crystal possesses inversion symmetry the relation between intrinsic and effective susceptibilities is easily derived and the third-order polarization is given by (2.18). Recently the dispersion of the third-order susceptibility X(3~(o91,o91, o92) has been observed for crystals with inversion s y m m e t r y like diamong (Levenson et al., 1972) and calcite (Levenson et al., 1973) when o91 - to: is near a Raman active mode while to~, o92 and 2o9~- to: are below the exciton resonances and above the p h o n o n resonances. One obtains from (2.18) (3) t

X iikltogl, O9,,

- - 0) 2)

(3)

= X,~k,E + ~

(I) (1) .a(1) (1) a ij,o.ol kt,o- ~ Ol ik,o-O[ lj,~r

l

~

( t o 2 _ (tO, -- to2) 2 + i(to, - to2)F,)"

(2.69)

By determining the values of to, - o92where this quantity has a minimum value one is able to determine the sign and magnitude of XE°) if the Raman cross-section is known. Distinctly additional effects caused by the retardation, and not associated with the overall phase matching problem, arise in the case of three-wave mixing in noncentrosymmetric crystals. They are associated with two-photon resonant terms in the third-order susceptibility such as R a m a n effect from polaritons (Coffinet and de Martini, 1969; Wynne, 1972), two-photon absorption (see Section 3) and three-wave mixing far from resonances (Yablonovitch et al., 1972). There have been many theoretical studies of the R a m a n effect from polariton-phonons, most of them based on the initial work of Shen (1965). Here we shall follow a different approach which explicitly shows the direct and indirect contribution to the third-order polarization in a crystal without inversion symmetry. In such a crystal, besides the direct contribution given by (2.22) and (2.18), there will be additional contributions to the total third-order polarization at the f r e q u e n c y ogs + to, + to. which arises in the following way. Because the crystal lacks inversion symmetry, fields at frequencies ogs + co,, co, + to, and co, + ogs may be generated by two-wave mixing. As can be seen from (2.63) these fields will contain a free and a forced part; the expression for the latter is given by (2.66) where PNL now is the second-order polarization at f r e q u e n c y o9, + to,, o9, + to, and to, + o9~ respectively. These fields m a y beat with the fields at frequencies o9,, o9~ and to, respectively to generate polarizations at to, + o9, + to,. In the second two-wave mixing process both the free and forced part of (2.63) are involved. However, the contribution from the free part is in general negligible. Under these conditions the total third-order polarizationt at to, + o9, + to. can be written: p(3)(to~ + to, + to,) = X<3)(to,, to,, t o , ) E ( k , to,)E(k,, to,)E(k,, to,)

+ X(z)(to, + to,, to,,)E(k, + kt, to., + tot)E(k,,, to.) + x(to,. + to., to,)E(k. + ks, to. + to.)E(k,, to,) __ ~<3)(to,, to,, to,,)E(k,, to,)E(k,, to,)E(k,,, to,).

(2.70)

This last expression defines the effective third-order susceptibility. Explicitly it is derived from (2.18) and (2.66) after PNL has been expressed in terms of (2.21) and (2.17). Clearly because of the indirect process ~(3~ is strongly w a v e v e c t o r direction dependent. In a crystal with inversion s y m m e t r y clearly ~ 3 ) = X~3). The expression for ~o) in a crystal of arbitrary s y m m e t r y is rather involved. Considerable simplifications occur in the case of a cubic crystal. There (2.66) can be easily solved

E ( k " , to) = --

4"rro92

{

c~(k,2_ k,,2) PNL(k", t0)--

tSee the discussion in the Appendix.

k"(k". ~

k,2

:k" to))

}

(2.71)

Infrared dispersion of third-order susceptibilities in dielectrics

289

or, separating out longitudinal and transverse parts,

to)

47r/~"(k". PNL(k",

E (k", to) =

e(to )

47r(PNL(k", to) -- k"(k" . PNL(k", to)) -~

(_ck__~,,)2_ ~ (to)

(2.72)

where e(to) = (ck'/to) 2. In order to illustrate the main points we consider the case of t h r e e - w a v e mixing where the f r e q u e n c y combination to3 = 2to~ - to2 is created in a cubic diatomic crystal; the frequencies to~, to2 and to3 are optical frequencies in the t r a n s p a r e n c y region of the crystal b e t w e e n the electronic transitions and the phonon frequencies. We assume that there are two w a v e s at f r e q u e n c y to~ with wave vectors k, and k~ respectively. This situation permits the observation of three-wave mixing under p h a s e - m a t c h e d conditions since the coherence length l~ = 1r/(k' - k") = " l T/ (k3 -I- k s - k~ - k~) m a y be v e r y long. Further the f r e q u e n c y combination to~-to2 can be swept across the phonon f r e q u e n c y spectrum of the crystal b y changing one of the two incident frequencies, to~ or

to2.

In a crystal of the NaCl-structure, the f r e q u e n c y 2to1 - to2 can only be generated b y the electronic third-order nonlinearity

X~)= N y

E.

In the diamond structure, however, the R a m a n terms in (2.18) m a y also contribute. An interference b e t w e e n the R a m a n term, which is resonant when to1-to2 = toT, the p h o n o n frequency, and the nonresonant term XE~3) has b e e n observed in diamond ( L e v e n s o n et al., 1972). The nonlinear susceptibility in this case is

X~x~x(tol, to,, -to2) = XCx~x~E (3) ~

x

(3)

Xxyxyttot, tol, --to2J -- )(xyxyE

(2.73)

d" 1 3 M(toT

x r Ol (I) IN xy.zOl (I) xy,z 2 -- (to1 -- to2) 2 --

i(to~ -

to2)F)'

(2.74)

only the x y x y - c o m p o n e n t shows the resonant behavior. For crystals in the zinc-blende structure, e.g. GaAs, additional contributions to the three-wave mixing 2to1-to2 must be considered. Because of the lack of inversion s y m m e t r y t w o - w a v e mixing is allowed in these crystals. Accordingly an electromagnetic w a v e at the second harmonic 2to1, can be generated and it m a y beat again with to2 to produce a polarization at 2to~- to2. Additionally, second-order mixing can generate a polariton w a v e at to~ - 0)2 which m a y mix again with to~ to produce 2to1 - to2; this last process has been used to o b s e r v e the polariton under m o m e n t u m - m a t c h i n g conditions (Coffinet and de Martini, 1969). The fields at 2tol and to~ - to2 are determined by (2.63) with PNL(r, 2toO = X(2)(to,, toOE(r, to,)E(r, to,) and PNL(r, OJ,--to2) = X~2)(tol,- to2)E(r, t o O E ( r , - to~) respectively; they are of the f o r m (2.63). When these fields beat with the incident fields at -to2 and to2 respectively to create 2to, - to2, both parts of (2.63) will be involved. H o w e v e r , the free part will in general be phasem i s m a t c h e d with the incident w a v e and the corresponding contribution will in general be negligible if the direct process is phase-matched. On the other hand, when the driven part of (2.63) beats against the incident wave, then a polarization with w a v e vector k, + k l - k2 will be produced, i.e. the same as in the direct three-wave mixing, and accordingly p h a s e - m a t c h e d ; the second step of the indirect process is always able to take full advantage of the phase-matching of the direct process. The total third-order susceptibility under these conditions is X--(3) x x x x/[ t o l , t o l , __0.)2) ~ ,]( (3) xxxxE C3)" ( t o ,, to,, ~?~,(to,, to,, -to~) = xx,~y~to,,°",to,, -to~) + Xx,~,,

(2.75) --to2)

(2.76)

where XxyxAto,,(3)'to,, , -to2) is given b y (2.74) and Xxyxytto,,°r,to,, -to2) using (2.71) is given b y

290

C. FLYTZANISand N. BLOEMBERGEN

X o.~.Ato,, r. to,, - o J D = -

-

~)'2to (~" k~"l c .4"rr(2to)~ ( k , : _ k:)X.,=t , , - to" :)X:.Ato,, to,) x { 1 --k-;-~

8~r(~Xto) ~

c~(--~:(~)~)

,~),

X.z~to,

-

to~,

(~)

to,)X~xy(to,,

_to:)f~ 1

~j(Ak~)2"[

where k' and k" are the wavevectors at frequencies 2to, and to, - to2 = Ato respectively, Ak = k , - k2 and k = 2k,, where we assume for simplicity k, = k~. We shall now cast expression (2.76) in a more useful form by using expression (2.72) instead of (2.71) for the intermediately generated field at f r e q u e n c y Ato = t o , - to2. Further, since to,, to2, 2 t o , - to2 and 2to, are in the transparency region between the exciton states and the p h o n o n resonances of the crystal, we are allowed to put rtzyx(to', )¢~)~(2to,. -to2) -~ /./2).

(3)') ~

x/(2) ]( xyz E

and (2) ..,

-to~)

-

(2)

X~(to,,

-to~)

N e ~:a :'~,~

(2)

= Xx~E +

2M(toT ~-- Ato ~ + iAto F)"

Then using (2.74) and (2.72) expression (2.76) becomes (3, .

-to2) =

,3)

XxyxY'to"tol'

87r(A/~) 2, (2) .~ "~

+

+

2

8rr(1 - k~) 3[(-~)

-- e(2tol)J

. (:) .2

N[o~iY)] ~ (A/~)2 3M(toL2 -- Ato 2 + iAto F)

16w(1 - A/~2)

NOIT '')2

(2)

2

~T (2) s~ (1) 1 ~ ~ x y z E e ~C~T

[X~y~E]+ M(toT2_ Ato2 + iAtoF)

[(cAk~ 2

(" e~ is the transverse effective charge and tOE and aL (" where ~= = 1 + 4~'aE, aT "~ = Ctxy.z, are the longitudinal p h o n o n f r e q u e n c y and R a m a n tensor given by (2.51) and (2.53) respectively. Expression (2.77) clearly shows that in general X°)(to,, to,, -to:) have resonances at both the longitudinal and transverse p h o n o n frequencies. Pure longitudinal p h o n o n resonance behavior is obtained for k, and k2 both parallel along the z-axis, or A/~ -- 1, and could also be accounted for by the approach of Section 2.2; the longitudinal resonance behavior is always present because it is insensitive to phase matching conditions which are mainly associated with retardation effects. Pure transverse p h o n o n r e s o n a n c e behavior is obtained for k, and k2 both in the xy-plane or A/~, = 0; it is more complex than the longitudinal because it is drastically affected by retardation effects. In particular it is not easily observable unless one is near the dispersion curve (c Ak/Ato) 2 = e(Ato). This is the situation in the experiments of Cottinet and de Martini (1969) and of W y n n e (1972); however, by proper choice of the g e o m e t r y of the propagation vectors the longitudinal p h o n o n resonance behavior can also be seen. As can be seen from (2.75), because of crystal symmetry, the x x x x - c o m p o n e n t is unaffected b y the indirect (two-step) process. Further, the contribution (2.77) from the indirect process is strongly w a v e v e c t o r direction dependent, while that of the direct process, eqn. (2.74), is not. When 2to, and t o , - to2 are far from resonances these two features of the indirect process m a y be used to extract the value of XE(3) both in sign and magnitude with the same a c c u r a c y as {XE(2)}2 (Yablonovitch et al., 1972). • (3) (2) 2 Thus for GaAs one obtains Xxyx~E/(X,~,O = 70 --+7. One also obtains the true value of the ratio X~y,rEIX*~*E o) • o) in the crystal. This quantity, when expressed in terms of bond

Infrared dispersion of third-order susceptibilities in dielectrics

291

polarizabilities (Flytzanis, 1970), directly reflects the anisotropy of the charge distribution along the bond axis and transversally to this axis in this crystal, an important structural feature in covalent crystals. F o r G a A s the m e a s u r e d value of this ratio is 0.52, the same as in Ge, which is consistent with the bonds being roughly ellipsoidal in f o r m with an axis ratio of about 2/3. We point out that such information cannot be obtained b y other means. It would be particularly interesting to p e r f o r m the same experiment in CuCl and determine the sign and magnitude of the ratio Xt~)xyxy~tX'xx~°~r. Because of the large contribution of d-states in the valence band of this crystal the ratio m a y be negative, as one can see using group theoretical arguments. W h e n to~ - to2 is near a p h o n o n resonance and 2to, is below the electronic transitions the n o n r e s o n a n t terms in (2.77) m a y be neglected and one recovers the formulae of the stimulate R a m a n effect derived by Shen (1965) and others. T h e y describe also the experiments of Coffinet and de Martini (1969) and W y n n e (1972). One can similarly treat higher-order nonlinearities. It is easily seen that in a nonlinear process of the n th order, besides the direct process, any many-step (indirect) process involving a succession of nonlinearities of orders nj such that n -- 1 = ~ (n, - 1)

(2.78)

i

will contribute to the total polarization of order n. The set of all ordered partitions of n - 1 determines all possible successions of nonlinearities that contribute to the n th order polarization. Accordingly the total n th order susceptibility )~") can be written in the f o r m X'"'=X'"'+

~ {hi--l}

I-Ix'")f(n,)

(2.79)

i

where the s u m m a t i o n goes o v e r all possible ordered partitions of n - 1 (relation (2.78)); the f(n) are functions containing linear coefficients and are determined through the solution of (2.66). The s y m m e t r y restrictions on the n u m b e r of independent c o m p o n e n t s of )~") are the same as for X ~"). H o w e v e r , because of corresponding s y m m e t r y restrictions also on the Xt'~'s in (2.79) in general the indirect processes m a y not contribute to some c o m p o n e n t s of ~"~; for these c o m p o n e n t s )~") = X ~"~.This was the case with the xxxx-component in the case of the zinc-blende structure. It is e a s y to convince oneself that in general the orders of magnitude of the genuine susceptibility X ~") and of any non-zero term of the sum in (2.79) are the same. In the previous discussion, and in the case of the electrostatic approximation, we have neglected m o d e damping effects. When these are included the problem b e c o m e s v e r y complicated and m a n y modifications of the previous a p p r o a c h are necessary. One of these results f r o m the problem of m o d e superposition through damping; we discuss this problem briefly in Section 4. 3. DISPERSION OF NONLINEAR SUSCEPTIBILITIES NEAR EXCITON RESONANCES W h e n some of the frequencies involved a p p r o a c h the region of electronic transitions the previous discussion must be modified to include electronic dispersion as well. Such a dispersion will show up in quantities like a E,/3 E . . . ; but also a quantity like a (1) ij,~ will show strong electronic dispersion and is responsible for the resonant R a m a n effect o b s e r v e d in crystals. In order to account for this dispersion in crystals one must in general m a k e use of the complete formalism of band theory. H o w e v e r , for frequencies near the exciton states below the edge of the forbidden gap EB, the previous formalism, appropriately modified and extended, can be used. Indeed in this f r e q u e n c y region the linear polarizability (2.26) is purely electronic and represented b y c~,j(co) = a ~ ( o o )

= ajj +

2 toe--to

2

• +l¢O'y~

(3.1)

292

C. FLYTZANISand N. BLOEMBERGEN

where a E' is a constant containing contributions f r o m the continuum of the band states only; to,, % and IX~., are the frequencies, damping and coupling constants respectively of an a s s e m b l y of harmonic oscillators representing the exciton states. Clearly for to ,~ toe but still a b o v e the p h o n o n r e s o n a n c e s a~(0) =-- ~,iE = a ,E' +

E

IXi,eiXj,e

r

e

(3.2)

toe

where a E is the total purely electronic linear polarizability defined b y (2.26). E x p r e s s i o n (3.2) is similar in f o r m to (2.26) and in analogy with the p h o n o n case we a s s u m e that the behavior of the nonlinear susceptibilities for frequencies near the exciton states can be described b y a system of anharmonic oscillator equations of the type (2.9) and a constitutive relation of the type (2.5), where q~ stands for the oscillator coordinate representing the exciton state o'. Quantities like aE, fl E, y E . . . will be replaced b y new constant terms a w,/3 w, 3 w . . . , respectively, containing contributions only f r o m the continuum of the band states the exciton contributions being taken care of b y the equations of motion. With tr, t r ' . . , standing for exciton states, coefficients like a~.!~,~.~,~'°(',~k.~• • •, a !?~,,,~,represent cross-terms b e t w e e n different higher bands and the exciton states of the forbidden gap, IX~"~is proportional to an exciton effective charge and IX~,, ~b~,,.,... are coupling coefficients for excitons of a single band. Clearly the exciton and band p a r a m e t e r s depend on the lattice coordinates X and, for small deviations f r o m the equilibrium configuration No, it will be assumed that they can be e x p a n d e d in p o w e r series of the p h o n o n coordinates. H e n c e anharmonic coupling b e t w e e n exciton and p h o n o n oscillators occur. Because both phonon and exciton oscillators satisfy the same type of eqns. (2.9) and are also coupled through e x c i t o n - p h o n o n cross t e r m s we shall use the same indices indifferently to label excitons or phonons. Then the complete b e h a v i o r of the intrinsic susceptibilities X ~) and X (~ for frequencies below the forbidden gap E is given by expressions (2.17) and (2.18) respectively, where o-, ~ r ' . . . n o w stand for excitons or phonons. The discussion of sections 2.2 and 2.3 concerning modifications arising f r o m electrostatics and retardation effects can be t a k e n over to the exciton case without any change since it is based on m a c r o s c o p i c considerations. With these preliminaries we proceed to discuss explicitly some special cases connected with X°~(to,, to~,-to2) and X ~ S ) ( t o ~ , - t o , , to2) which are analogous to the ones discussed in connection with phonons. Since the incident and created frequencies, to,, to2 and 2to,-to2, are assumed to fall, as before, in the t r a n s p a r e n c y region a b o v e all p h o n o n r e s o n a n c e s and below all exciton states, the pertinent intermediate f r e q u e n c y for excitons is 2to, or to, + to2 instead of t o , - to2; therefore in the following we shall consider dispersion effects arising f r o m this combination f r e q u e n c y being near an exciton state. The dispersion of Xm(to,, to2) when to, + to2 is near an exciton state " e " has been o b s e r v e d b y H a u e i s e n and Mahr (1971) in CuC1, a crystal with 4 3 m - s y m m e t r y , and was found to be well accounted for b y an expression (2)r to ' 1 E 1 X ,ik, .,to2)=V/31ik(to,,to2)=

[d,i.ebik,e

b,k +

V[to,:-(to,+to:):+i3,e(to,+w:)]

(3.3)

where b~jk and b~j., are constants. This expression can be obtained f r o m (2.16) by separating the resonant f r o m the non-resonant terms; one obtains: b~k.e =

a~!, +

liX~.ee'iXk.,' '

. IX~."'IX~."i + (.Oe'

.'~

---r---r-

.

e',e"

¢J) e'O.)e"

.1

and a similar expression for b,k containing all non-resonant terms in (2.16). In the case of CuCI only the x y z - c o m p o n e n t is different f r o m zero. E x p r e s s i o n (3.14) is similar to the one o b s e r v e d b y F a u s t - H e n r y near a p h o n o n frequency. In analogy to the lattice case, then, b,j., plays the role of a R a m a n - t y p e tensor, or more correctly of a t w o - p h o t o n transition m o m e n t for excitons. Its magnitude is determined by the relative signs of three terms proportional to a !p..e, tX(2),.ee'and ~b~e,e,.respectively; the first arises f r o m the

Infrared dispersion of third-order susceptibilitiesin dielectrics

293

coupling of the exciton state " e " to the different higher bands and the other two from the anharmonic coupling of the excitons within the band to which they are assigned. The dispersion behavior of X°)(0),, 0)~, -0)2) in centrosymmetric crystals for 20), = 0), but 0)ph.... ~ 0)~ - 0)2, 0),, 0)2, 20)j - 0)2 ~ 0), is given by C'~%

X~I(O,, 0),, --0)2) =

Cilk, + V(0) 2 _ (2o)1)2+ 2i0),y,)

(3.5)

with

C"jkz = b~j.,b~.,

(3.6)

where b~j., is given by (3.4) and clearly shows the analogy with the Raman-type dispersion behavior observed in crystals. A dispersion of the type (3.5) could be observed in crystals like CuO2. For crystals without inversion symmetry the retardation effects must be included. Again for cubic crystals like CuCI the problem can be easily solved. The dispersion of XxyM0),,(3),0),, - 0)z) when 20)1 ~ (De in CuCI has been studied by K r a m e r et al. (1973). Using (3.3) and (3.5) together with Maxwell's equations one obtains for CuCI ) i X O,,yxy~0)t, to,,

C ",x~

^2 87rk~ {bx,~ 2

b2,L

e" \ V ] + ~ 2 3 V ( 0 ) ~ L - ( 2 0 ) , ) : + 2 i 0 ) , y )

8 ~ ( 1 - k,2....~)

b:r

bx~,ix,b,T

l (_~2) 2

+3{(ck'~2-e(20)O)\\20),l [ ' - - "

+ V(O)'2-(20)~)2+2io)"Y)

~(ck~2_

+ 4v(0),z - ( 2 0 ) 0 ' + 2 i 0 ) , v ) ( \ 2 0 ) , /

e"

}t

(3.7)

where we assume that there are two fields at 0)~ propagating in two different directions with wavevectors k, and k~ respectively and Ikd = Ik~l; k = k, + k~, ~ is a unit vector along k, • " = 1 + 4~-a E,, 0), is the true frequency for the transverse exciton 0),i, is the f r e q u e n c y for the longitudinal exciton and is related to 0), by a L y d d a n e - S a c h s - T e l l e r relation of the type (2.51), Ix, is proportional to an effective charge for the transverse exciton, b = b~,~, b,r and b,L are the two-photon transition moments (3.4) for the transverse and longitudinal excitons and they are related by a relation of the type (2.53). Expression (3.7) clearly shows that X~x,(0)~, 0)~, -0)2) will have resonances at both the longitudinal and transverse exciton frequencies. The one due to the longitudinal exciton is always present unless/~, = 0 (k, and k~ are both in the xy-plane). On the other hand, the resonant behavior at the transverse exciton is more complicated and drastically modified by the retardation effects; in particular it will be mainly observable for w a y • v e c t o r s k~ and k~ such that (ck/20)~) 2= e(20)~), the dispersion curve for the exciton-polariton. This behavior for Xc3)(0)~,0)~, -0)z) should be compared with the one observed by Haueisen and Mahr (1971) in X~2)(0)~,0)2), to which it is closely related. Also closely related to this dispersion is the two-photon absorption effect at the same exciton state observed in CuC1 by Frfhlich et al. (1971). This p h e n o m e n o n is described by the imaginary part of Xcs)(0)~, 0)~, -0)2) with 20), = 0),. Putting 0), = 0)2 in (3.7) one obtains Im X~3}(0),, 0),, - 0 ) 0 = Im /~2

b 2

(to2L- (20),)2 + 2i0)y)

+ 3[(k\20),/ck ,~2_ • (2~o~)] t.\v / + 0(0),2 - (20),)2 + 2i0)~y)

+ 419(0)e 2 -- (20)1) 2 +

.r,,c,,v ]}]

2i0)y) L\20),1 - • "

(3.8)

where we have assumed for simplicity that the damping constant is the same for the longitudinal and transverse exciton. From (3.8) one can obtain the two-photon absorption coefficient from this exciton. One finds that for ~ = 0 the two-photon absorption coefficient obtained from (3.8) is identical to the one derived by Boggett

294

C. FLYTZANISand N. BLOEMBERGEN

and L o u d o n (1972). H o w e v e r , expression (3.8) is more general since it also contains the longitudinal c o m p o n e n t . The advantage of the present a p p r o a c h is that it clearly exhibits the different approximations involved in (3.8) and shows that consistent use of the intrinsic susceptibilities and retardation effects completely accounts for the o b s e r v e d behavior and m a k e s redundant the use of the artificial polariton-concept. One can similarly treat the other f r e q u e n c y regions and in particular the cases where some frequencies are near the exciton states and others near p h o n o n resonances. Such a situation arises, for instance, in the resonant R a m a n effect. H e r e we shall only give the pertinent expressions for the case where 2to,-~ to, and A(0 = t o , - to2-to, while (0~ < tot, (02, 2(0, - (02 < (0,. F o r centrosym'metric crystals one obtains f r o m (2.18) X o),m((0t,tot, -(02) =

bil.ebkt.e Ol o) O,o'O! (1) k,.o" Ci,k, + 3 v ((0,2 _ (2(01)2 + 2i(0t y) + 3 v ((0 2 _ A(02 + iAto~ F).

(3.9)

F o r n o n c e n t r o s y m m e t r i c crystals b e c a u s e of the retardation effects the problem is quite involved. For a CuCI, one obtains ^2

(3) [ X xyxy~to t, (0t,

C'yxy

2

87rkz (bx,z~ b:L E; \ V } +/~z23V((0~L-(2(00:+2itotv)

8~r(1 - / ~ ) )

+ (to2 _ ( 2 ( 0 f + 2ioJty)

+ 3[ k2(0t} ( ck '~24v ((0, - (2(0t) 2 + 2i(0,3') [ \2(0t ] - e" ,

N[,~,.('q"

+ (A/~,) 3M((0L: -- A(0 2 + iA(0 F)

~- (A/~')' . . . . N[a("]~ ~ln t(0L -- A(0: + iA(0F)

xr (21 ~ (1) 16"rr (1 - A/c'~z) (2) 2 l~¢XxyzEeTOtw -~ [ ( ~~ 3 .--E(A(0)] ... (X~''E) + M ( ( 0 T 2 - A ( 0 2 + i A ( 0 F )

NaT (`)'

[(cAk'~2

,~]}+

(--..-~l 16"rr : A~'~2--~ ) L\.~t.u

,-

(2)

x2

/

d

xr (2) l~X xyzEe ~~OtT(1)

× ~tXx~E) + M ( ( 0 T , _ A ( 0 2 + i A ( 0 F ) NotT`':

. [{c Ak'~ 2 _

+4M((0r,_ato,+ia(0r)[\A(0}

,=]}

(3.10)

where k = k t + k ~ , k = k t - k 2 a n d k t = k ~ - k 2 . In both cases we see that X(3)((0t, tot, -(02) has the contributions f r o m two resonant terms, one associated with exciton and the other with p h o n o n resonances. T h e s e t e r m s m a y b e c o m e positive or negative independently of each other, depending on the values of the frequencies. H e n c e interference b e t w e e n the two m a y occur and f r o m the zero o f (3.9) or (3.10) one m a y determine the magnitude of the t w o - p h o t o n absorption cross-section f r o m excitons relative to the R a m a n cross-section f r o m phonons. H o w e v e r , (3.9) and (3.10) exhibits some of the pitfalls in t w o - p h o n o n absorption s p e c t r o s c o p y . As was pointed out previously, t w o - p h o t o n absorption cross-sections are determined b y the imaginary part of X(3)((0t, -tot, (02) when tot + 0)2 I-t/e.H o w e v e r , the same quantity also has an imaginary part when tot - (02 = (0p, a p h o n o n frequency, even without (01+ (o2 = (0~, and this m a y lead to incorrect exciton state assignments b y t w o - p h o t o n absoption spectroscopy. In other cases it m a y lead to incorrect values for the cross-sections or affect the line form. Clearly this type of effects can equally well be o b s e r v e d in molecules in solution where a lot of t w o - p h o t o n spectroscopical studies h a v e already been p e r f o r m e d . We stress again that the a b o v e m e n t i o n e d effect has no relation to the resonant R a m a n effect in crystals (see, for instance, Burstein et al., 1969). =

Infrared dispersion of third-order susceptibilities in dielectrics

295

4. COUPLED M~DE SCHEME In the previously derived expressions of the dipole second- and third-order polarizations in a dielectric it was assumed that to a good approximation the linear susceptibility in the rest strahlen region is quite well reproduced b y a formula involving the sum of contributions f r o m independent classical oscillators. There are notable exceptions, however. The high dielectric constant materials BaTiO3, SiTiO3 and KaTiO3 provide a striking example. Barker and Hopfield (1964) were able to reproduce the linear susceptibility in these materials b y postulating m o d e superposition. We investigate here the implications of m o d e superposition in X (5). L e t us define a t w o - m o d e dielectric response function as a response function having two pairs of poles. The m o s t general such dielectric response function which can be generated f r o m a set of equations of motion of two variables, consistent with all motions of the variables being dissipative, can be written

q,. + y, dt,~ + (oJ,~2 + o~],,~)q,. - (o~(~,,,, + ~_~ $~)~,,..,q~,,q~,,, = I.*~")g + E tz~)~'q ~' ~ + ~a,,")gg o-',o."

o.'

while the dipole m o m e n t per unit cell is

p = ~cr t.~o(')q. + ~ o~. , o . ' ~)~,q,,q~,, + ~o. a ) " q . g + a ~ g + 1 3 ~ where we have assumed for simplicity a unidimensional medium, that (r, or', (r" = 1, 2, and the restriction is imposed that y, and y5 are positive and all the coefficients are real with oJ~2= o~2,. The usual dielectric model for two independent oscillators is identical to the a b o v e model with oJ,= = 0. The a b o v e equations of motion can be written in the following equivalent way:

Q, + (F, + Fo)Q, - FoQj + I),SQ, + ~, d~dQjQk = M,(')~ + ~, M~)Qfig + ½A[')~g~ i,k J

(4.1)

and the dipole m o m e n t per unit cell is

p = ~ M [ ' Q , + ½ ~ M~)Q~Qs + ~ A,")Q, fg +aEf~ + / 3 E * * i,j j

(4.2)

with i, j = 1,2 and the equivalence given b y (cos0 T = k-sin 0

sin00) cos

(r, +r,2 M2")] = T(/xz(" )

r,2 r, j =

0

0 oJ, T-' 1122 = T \ +2°J~2/ O)12 Attt)-

a,")

A f t ) ) = T(ot 2°) )

and

I)(3) ilk

Tlo.Tjo,Tko.,,(~, o..

Such a unitary transformation which diagonalizes the linear force constant matrix can always be found and 0 is given b y 2 2 c o t 20 +O)2 --001 2 cot 0 - 1 = 0. (4.3) (O12 To solve the coupled eqns. (4.1) we use the iterative solution method and write

Q,

=

Q,(, + Q c2, + . . .

296

C. FLYTZANIS and

N. BLOEMBERGEN

I n t r o d u c i n g the F o u r i e r c o m p o n e n t s of the field ~ ( t ) = ~ E(to,) e -~''' s

and inserting in (2.78) we obtain m(')'to Q[')(to~) = re, t ,)"E"t t o , ) ", Qi(2)(to~ + t o , ) = O,(2)(tos + to,)E(to,)E(to,) where •

~

•

i(1)

M i ( i ) ..} ltos l i l l V l j

D(f~j, co,)

O,(')(to:) =

D (fl,, to~ ) -t

(4.4)

to~2F'~ D (l)s, tos)

and ~" qb~k)Qi"(co~)Ok("(to,) + ~ Ml~)Qi"(to.) + ~IA,(') 0,re(to, + to,) = '~ J'" 2 2 (to, + to,) F,j D(I),, toe + to,) + D(flj, toe + to,) i(to, + to,)F0

-

{,~k -'-(3)/'"), ,,'~"' .t t o., ) .+ .N' ~.~ ~v,~,a)'~(')''td, tto,) w j , k ~ , toJs)t~k z_~

+ ½Aj(') }

D (flu to~ + tot)

+

D ( f l , , co, + to,) +

(to, + to,) 2 F,~2

D (fls, to~ + to,)

(4.5)

with to. = to,, to, and D ( f L , to)

=

~l

2 -

0)2 _

ito (F, + F~j).

(4.6)

T h e dipole m o m e n t per unit cell (4.2) can similarly be written

p = p(1)+pa)+... where

p (') = a EE + ~ / M [ ' Q , "),

(4.7)

p(2) = /3EEE + ~ A'(')Q'(')E +

. M'(t)Q'(2) + 2 v

~"o

~

'-¢s

•

(4.8)

I n t r o d u c i n g the F o u r i e r c o m p o n e n t s one has p(')(to~) = ~ (tos)E(to~), p (2)(toe + to, ) = / 3 (to~, to,)E (to,)E (to,)• T h e s e e x p r e s s i o n s define also the linear and s e c o n d - o r d e r polarizabilities of a unit cell• T h e c o r r e s p o n d i n g susceptibilities are

X(')(to~) = l a (to~) and

x(~)~to

to,) = l / 3 ( t o , ,

respectively. The expression for X ")= (t(1964). T h e e x p r e s s i o n f o r Xm(to. to,) is

to,)

1)/4~r w a s given b y B a r k e r and H o p f i e l d

I f E 1.--. Xa)(to,, to,)=-~t# +~ 2~ A,('){Q,("(w,)+ Q,(')(to,)} +

,to..

(4.9)

Infrared dispersion of third-order susceptibilities in dielectrics

297

Similarly one can proceed to obtain the expression for X c3~(tos,to,, to,) in the coupled mode scheme. These expressions, however, are too complicated and will not be reproduced here. We conclude this section by noticing that in the coupled mode scheme the dispersion of X ~ near the modes has a different behavior from that expected in the independent mode scheme. In the latter case the dispersion is essentially a superposition of harmonic oscillator dispersion curves. It has been found that the former fits better the infrared dispersion of the linear dielectric constant in the ferroelectric like BaTiO3 (Barker and Hopfield, 1964) than the latter. The extension of the coupled mode scheme to the nonlinear case, given above, will allow one to account for similar effects in the dispersion of the nonlinear susceptibilities. As can be seen from (2.9) this is reflected in the Raman scattering experiments by such modes (see, for instance, Scott, 1970, 1971) or in the dispersion of X~3~(tol,to~,-to2) when tol-to2 approach the frequencies of these coupled modes. The above formalism, however, does not include the case where the two modes become a bound system (for instance, biexcitons, b i p h o n o n s . . . ) . Such bound elementary excitations may show additional dispersion effects. The inclusion of such effects in the susceptibility formalism is at present hampered b y considerable mathematical difficulties. 5. S P A T I A L D I S P E R S I O N

In the phenomenological approach used in the previous section to derive the f r e q u e n c y dispersion of X ~2~and X ~3~near isolated resonances it was assumed that the relation between the electric polarization and the electric field was local; the polarization at a point in the medium was completely determined by the value of the electric field at the same point. This was reflected in the use of the dipole approximation. In general, however, the relation between polarization and electric field is nonlocal and the value of the first at a point depends on the values that the field takes in a neighborhood of the point. Such a nonlocal relation is necessary to account for the spatial dispersion observed in linear optics and corresponding effects may also be observed in nonlinear optics. This is the case of the observed second harmonic generation in calcite by Terhune et al. (1962). In this section we briefly describe a model to include spatial dispersion in the case of isolated resonances. The model is extensively discussed in the book of Agranovich et al. (1966) for the linear case. Here we extend it to the nonlinear case. For this reason we generalize the case treated in Section 2.1 by introducing nonlocal relations between fields and displacements

02u~(r) + f K~(r, at 2 J

r ' ) u ~ ( r ' ) d r ' + f ~ K~,,,,,(r, r', r")u~,(r')u~,,(r") dr' dr" J tr'~r"

+ ~'f A~(r, r', r")~(r')~(r") dr' dr",

f

,(,)= +

Zf

o',~'

(5.1)

d,"

f As(r, r ' , r")~(r')u~(r") d r ' dr" J

+ J B(r, r ' , r")~(r')~(r") dr' dr"

(5.2)

where for simplicity we have suppressed in (5.1) the dissipation term

F Ou~(r). Ot Since the different quantities M, K, A and B cannot be reduced to ~-functions with respect to the arguments r - r', r - r " . . . the force at a point r is determined by the magnitude of the electric field not only at r but also in its environment. Equations

298

C. FLYTZANISand N. BLOEMBERGEN

( E l . I ) t h e r e f o r e a c c o u n t f o r spatial dispersion. F o r an u n b o u n d e d m e d i u m and w e a k spatial dispersion we m a y put K ( r , r ' ) = K ( r - r'), K ( r , r ' , r") = K ( r - r ' , r - r") and similarly f o r the o t h e r quantities. T h e r e f o r e b y e x p a n d i n g u and g in a series near r = r ' or r = r" and restricting o u r s e l v e s to the l o w e s t o r d e r terms we h a v e 02u~ ~ - l ~ f u ~ + -

Ot

Ou~ + K=,kl d 02 U~ + . . .

K~"k ~X,

Ox~Ox~

..,3, u ¢,U=, + ~ ' , ~ , , ~-2x {-~, u~,,} + • • • + N~'o,,,o,,,

=t~

-+

--+..-+tx~,~u~,+tx~,,~ ,,.k Oxk

{~u~}+... (5.3)

+ ½ ~ 2 " ~ + ½~ ~.'~o-~{~ } + • •

w h e r e r = {x,, x2, xs}. T h e t e n s o r s appearing in this e x p a n s i o n are a s s u m e d to be i n d e p e n d e n t of to in the f r e q u e n c y range c o n s i d e r e d . T h e n u m b e r o f i n d e p e n d e n t c o m p o n e n t s is d e t e r m i n e d b y the t y p e o f the r e s o n a n c e and the crystal s y m m e t r y . T h e solution of e q u a t i o n (5.3) m a y be s o u g h t in the f o r m o f plane w a v e s u~(r, t) = u~(00, k) e 'tk'-~'),

(5.4)

• (r, t) = ~ E(00,, ki) e i~'" ~'~.

(5.5)

W e use the iterative a p p r o a c h of the p r e v i o u s sections and put u~(r, t) = u¢')(r, t ) + u22'(r, t ) + . . .

(5.6)

Substituting (5.6) in (5.3) we find f o r the terms linear in the field A('~(00, k ) ~ , , ~ k-~ ~t00, k)

,,2"(00, k)

(5.7)

w h e r e ( A g r a n o v i c h and Ginzburg, 1966) A(')" ~t00, k) = ix~(" + ikox~.]- k,k,,tx~.,,,, A~(00, k ) = f i r -

002 + ik~K~.~ - k&,,K~.t,,.

(5.8) (5.9)

Similarly f o r the term quadratic in the fields one obtains u (2),

t00~,

k~, to,,

k,) = A ~(="t00,)'k~ ;tot, k,)E(00,, k~)E(00,, k,) A~(00o, ko)

(5.10)

w h e r e 000 = 00, + 00, and ko the c o r r e s p o n d i n g w a v e v e c t o r : Aa~(00~k~ ; 00tk,) = - ~] (~2~),=,, + i(k~ + k, ), ~o=,~,,,, k ~ (I)(3)

-(k,

+ k,),(k~ +

,.~

" 1~ ~("tr"t00s, " X A ")" ~rt00s, k s.~) ks)

=,~,,sa~(~.,k~ ) A~,,(00. k~)

+ ~ {tz ~ , + i(k, + k, ) i t L,2,~ . , -

(k~ + k,),(k~ + k , ) ~ o . .,~, .,...

: A °) " k x o) x / ~ :'t00,_-,) A'"~'(00,tk,)]+½(a <,,+i(k ~ + k ' a < " t A~(00~, k~) + A~,(00,, k,) J 'J' ~'~ - (k~ + k,),(k~ + k,)..o~ ~,,ms"~ ~

(5.11)

T h e i n d u c e d dipole is given b y eqn. (5.2); the part linear in the fields is po,(00, k) = ~ A"~(00, k )u,,"'(00, k) = a(00, k )E(00, k) and the part quadratic in the fields is

(5.12)

Infrared dispersion of third-order susceptibilities in dielectrics

299

p'2)(too, ko) = ~ AC')#(too, ko)u~(2I(tO,, k, ;tO,, k,) o"

n

tO k

xu o),.

k ~

o),.

tr.tr"

+~A(1)/ ~, ~tO,~,, ~. co, k,)u,, (i)(tO,, k,)E(tO,, k,) o"

+ B(co, k, ; tO,k,)E(to, k,)E(to, kt).

(5.13)

For a more rigorous approach of the problem we refer to the article by Agranovich et al. (1966). It is easy to convince oneself that the above expressions reduce to the ones

derived in Section 2.1 when spatial dispersion is neglected (all k~ = 0). In order to observe spatial dispersion effects narrow exciton lines must be used. CONCLUSION

In the present work we have discussed the dispersion behavior of the nonlinear susceptibilities in the infrared below the band-to-band electronic transitions. Retardation effects were explicitly taken into account and the relation between the intrinsic and effectively observed nonlinear susceptibilities was obtained for some cases. Only the dispersion arising from excitons and phonons was considered. H o w e v e r , the approach is quite general and can be extended to the case of other elementary excitations (magnons, p l a s m o n s . . . ) . In conclusion the simplifications introduced in the course of the discussion are recalled. These are the neglect of acoustic phonon contributions and of the free field in (2.70) when discussing the two-photon resonant terms in X °). The justification of the latter is not well established and cases where this is not allowed could be found. The former is justified as long as all frequencies are above the acoustic phonon resonances. However, acoustic phonon contributions may become important when lower frequencies are involved and in fact acoustic phonon second harmonic generation has been observed (Boyd et al., 1971). Acknowledgements--Throughout the course of this work the authors benefited from interesting discussions with Drs. M. D. Levenson, F. G. Parsons and E. Yablonovitch. REFERENCES AGRANOVICH, V. M. and GINZBURG, V. L. (1966) Spatial Dispersion in Crystal Optics and the Theory of Excitons, Interscience Publishing Co., New York, N.Y. AGRANOVlCH,V. M., OVANDER,L. N. and TOSHICN, B. S. (1966) Zh. Eksp. Teor. Fiz. 50, 1332 0966, JETP-Sov. Phys. 23, 885). ARMSTRONG, J. A., BLOEMBERGEN, N. DUCUING, J. and PERSHAN, P. S. (1962) Phys. Rev. 127, 1918. BARKER, S. A. and HOPFIELD, J. J. (1964) Phys. Rev. 135, 1732. BARKER, S. A. and LOODON, R. (1972) Rev. Mod. Phys. 44, 78. BEDEAUX, D. and BLOEMBERGEN, N. (1973) Physica (Amsterdam) 69, 57. BLOEMBERGEN, N. (1965) Nonlinear Optics, W. A. Benjamin Inc., New York, N.Y. BLOEMBERGEN, N. (1967) Am. J. Physics, 35, 987. BOGGET, D. and LOUDON, R. (1972) Phys. Rev. Lett. 28, I051. BORN, M. and HUANG, K., Dynamical Theory of Crystal Lattices, Clarendon Press, Oxford. BOYD, G. D., NASH, F. R. and NELSON, D. F. (1970) Phys. Rev. Lett. 24, 1928. BURSTEIN, E., MILLS, D. L., PINCZUK, A. and USHIODA, S. (1969) Phys. Rev. 22, 348. COFHNET, J. P., and DE MARTINI, F. (1969) Phys. Rev. Lett. 22, 60. DE GROOT, S. R. (1969) The Maxwell Equations, North-Holland Publishing Co., Amsterdam. DE MARTINI, F., SIMONI, F. and SANTAMATO, E. (1973) Optics Comm. 9, 176. FAUST, W. L. and HENRY, C. H. (1966) Phys. Rev. Lett. 17, 1265. FLYTZANIS, C. (1970) Phys. Lett. 31A, 273. FLYTZANIS, C. (1972) Phys. Rev. B6, 1264. FLYTZANIS, C. (1972) Phys. Rev. Lett. 29, 772. FR6HLICH, D., MOHLER, E. and WIESNER, P. (1971) Phys. Rev. Lett. 26, 554. GENKIN, G. M., FAIN, V. M. and YASCHIN, F. G. (1967) Zhur. Eksp. Teor. Fiz. 52, 897 (1967, JETP-Sov. Phys. 25, 592). HAUEISEN, D. C. and MAHR, H. (1971a) Phys. Rev. Lett. 26, 838. HAUEISEN, D. C. and MAHR, H. (1971b) Phys. Lett. 36A, 433. KAISER, W. and MAIER, M. (1973) in Laser Handbook, edited by F. T. ARECCHI and E. O. SCI-IULZ-DUBOIS, North-Holland Publishing Co., Amsterdam, p. 1229. KRAMER, S. D., PARSONS, F. G. and BLOEMBERGEN N. (1974) Phys. Rev. B9, 1853. LAX, M. and NELSON, D. F. (1971) Phys. Rev. B4, 3694.

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C. FLYTZANIS and N. BLOEMBERGEN

LEVENSON, M. D., FLYTZANIS, C. and BLOEMBERGEN,N. (1972) Phys. Rev. B6, 3962. LEVENSON, M. D. and BLOEMBERGEN,N. (1974) J. Chem. Phys. 60, 1323. LEVENSON, M. D. and BLOEMBERGEN,N. (1974) Phys. Rev. B10, 4447. LORENTZ, H. A. (1916) The Theory of Electrons, Teubner, Leipzig. MAKER, P. D. and TERHUNE, P. W. (1965) Phys. Rev. 137, A803. MARADUDIN, A. A. and BURSTEIN, E. (1968) Proc. Int. Conf. of Semiconductors, Moscow, p. 1008. MARADUDIN,A. A., MONTROLL,E. W., WEISS,G. H. and IPANTOVA,I. P. (1971) Theory of Lattice Dynamics in the Harmonic Approximation, Academic Press, New York, N.Y. SCOTT, J. C. (1970) Phys. Rev. Lett. 240, 1107. ScoTT, J. C. (1971) In Light Scattering in Solids, edited by M. BALKANSKI,Flammarion Sciences, Paris, p. 387. SHEN, Y. R. (1965) Phys. Rev. 138, A1741. TERHUNE, R. W., MAKER, P. D. and SAVAGE,C. M. (1962) Phys. Rev. Lett. 8, 404. WANG, C. C. and BAARDSEN,E. L. (1969) Appl. Phys. Left. 15, 396. WYNNE, J. J. (1972) Phys. Rev. Lett. 29, 650. YABLONOVITCH,E., BLOEMBERGEN,N. and WYNNE, J. J. (1971) Phys. Rev. B3, 2060. YABLONOVITCH,E., FLYTZANIS,C. and BLOEMBERGEN,N. (1972) Phys. Rev. Lett. 29, 865. APPENDIX I n t h e m a i n t e x t , w e h a v e p r e s e n t e d a g e n e r a l a p p r o a c h w h i c h a l l o w s o n e to i n c o r p o r a t e r e t a r d a t i o n e f f e c t s in t h e s u s c e p t i b i l i t y f o r m a l i s m , a n d in o r d e r t o s i m p l i f y t h e p r e s e n t a t i o n w e h a v e m a i n l y d i s c u s s e d r e t a r d a t i o n e f f e c t s in c r y s t a l s w i t h c u b i c s y m m e t r y . F o r c r y s t a l s w i t h a s y m m e t r y o t h e r t h a n c u b i c , t h e a l g e b r a is m o r e i n v o l v e d ; in p a r t i c u l a r t h e s e p a r a t i o n i n t o t r a n s v e r s e a n d l o n g i t u d i n a l m o d e s is n o t p o s s i b l e except for certain directions, and the number of independent susceptibility components is l a r g e r t h a n in t h e c u b i c c a s e , w h e r e X <" b e h a v e s as a s c a l a r , X <2) h a s o n l y o n e (2) (3) i n d e p e n d e n t c o m p o n e n t g~y~, a n d X (3) h a s t w o , X (3) ~ a n d Xx,,,~. In general, once the independent susceptibility components have been obtained and t h e s y s t e m o f t h e t h r e e l i n e a r a l g e b r a i c e q u a t i o n s (2.66) h a s b e e n s o l v e d , it is q u i t e s t r a i g h t f o r w a r d to s e t u p t h e t o t a l t h i r d o r d e r p o l a r i z a t i o n (2.70) a n d s t u d y r e t a r d a t i o n e f f e c t s in c r y s t a l s w i t h s y m m e t r y o t h e r t h a n c u b i c . H e r e w e o n l y g i v e t h e m a i n e l e m e n t s n e e d e d f o r t h e s t u d y o f r e t a r d a t i o n e f f e c t s in X °> in c r y s t a l s p o s s e s s i n g h e x a g o n a l s y m m e t r y (e.g. w u r z i t e s t r u c t u r e ) . T h e d i e l e c t r i c c o n s t a n t , in t h e d i a g o n a l f o r m , h a s t w o i n d e p e n d e n t c o m p o n e n t s e l = ex~ a n d ell = ezz o r e = (e±, e l , e0, t h e s e c o n d - o r d e r s u s c e p t i b i l i t y g t2) h a s f o u r , X(z2z~, (2) X,~,z <2) a n d X*= (2) X~=, a n d t h e t h i r d - o r d e r s u s c e p t i b i l i t y X <3~h a s t e n , X c3) <3> Xxyyx, <3) Xxyxy, <3) ..... Xx,,~, (3)

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w h e r e E " = E ( k " , to), P " = PNL(k", to), Ae = e H - e± a n d el,, e± a n d Ae a r e c a l c u l a t e d at a f r e q u e n c y to. U s i n g t h e a b o v e e x p r e s s i o n f o r t h e i n t e r m e d i a t e field a n d t h e i n d e p e n d e n t s u s c e p t i b i l i t y c o m p o n e n t s e n u m e r a t e d p r e v i o u s l y o n e c a n set u p t h e e x p r e s s i o n f o r t h e t h i r d - o r d e r p o l a r i z a t i o n (2.70) a n d s t u d y r e t a r d a t i o n e f f e c t s f o r d i f f e r e n t g e o m e t r i e s . T h e b e h a v i o r a l o n g t h e z - a x i s is t h e e a s i e s t to s t u d y ; in p a r t i c u l a r a l o n g this axis o n e c a n s e p a r a t e t h e m o d e s a n d t h e fields i n t o t r a n s v e r s e a n d l o n g i t u d i n a l p a r t s a n d o b t a i n a b e h a v i o r s i m i l a r to t h e o n e o b t a i n e d in t h e c u b i c c a s e .