MATERIAL P.

A.

EQUATIONS

FOR

Apanasevich

and

LINEAR N.

V.

GYROTROPIC Martinovich

MEDIA UDC 535.51

In the phenomenological a p p r o a c h the m a t e r i a l equations of e l e c t r o d y n a m i c s a r e not defined uniquely [1]. As a r e s u l t of this, these equations differ c o n s i d e r a b l y in f o r m in the t h e o r y of n a t u r a l optical g y r o t r o p y , in p a r t i c u l a r (optical activity) [1-14], and different authors even hold v e r y different points of view as r e g a r d s fundamental questions in the e l e c t r o d y n a m i c s of g y r o t r o p i c m e d i a [2, 4-8, 11-14]. Various f o r m s of the m a t e r i a l equations can e a s i l y be c o m p a r e d inside homogeneous m e d i a by using M a x w e l l ' s equations and t h e i r i n v a r i a n c e r e l a t i v e to c e r t a i n field t r a n s f o r m a t i o n s . It can then be decided where they differ in fact, and which of the d i f f e r e n c e s a r e p u r e l y f o r m a l . If, in addition, we m a k e use of the concepts of field e n e r g y c o n s e r v a t i o n (in t r a n s p a r e n t media) and s y m m e t r y of the kinetic coefficients, then c e r t a i n g e n e r a l r e q u i r e m e n t s can be f o r m u l a t e d which m u s t be s a t i s f i e d by the m a t e r i a l equations [11, 12]. However, all this turns out to be insufficient in o r d e r f o r the choice of a p a r t i c u l a r s t r u c t u r e f o r these equations to be fully justified and unique in the c a s e of g y r o t r o p i c m e d i a . P a r t i c u l a r l y g r a v e difficulties in solving this p r o b l e m a r e encom~tered in the c a s e of inhomogeneous g y r o t r o p i e m e d i a and t h e i r b o u n d a r i e s . The m i c r o t h e o r y of the m a t e r i a l equations of g y r o t r o p i c m e d i a can, without doubt, m a k e a l a r g e contribution to solving these p r o b l e m s . Such a theory is a l s o invoked to explain the connection between individual t e r m s in the m a t e r i a l equations and the p h y s i c a l c h a r a c t e r i s t i c s of the m e d i u m m i c r o s t r u c t u r e , which is v e r y i m p o r t a n t f o r a deep understanding of the content of t h e s e equations. Up to the p r e s e n t a c o m p a r a t i v e l y l a r g e n u m b e r of p a p e r s (see, f o r e x a m p l e , [15-17]) has been devoted to the m i c r o t h e o r y of n a t u r a l optical g y r o t r o p y . H o w e v e r , full use has still not been m a d e of the p o s s i b i l i t i e s of m i c r o t h e o r y in e s t a b l i s h i n g the g e n e r a l p r o p e r t i e s of the m a t e r i a l equations in g y r o t r o p i c media. In addition, p r a c t i c a l l y all p a p e r s on the m i c r o t h e o r y of g y r o t r o p y m a k e c o n s i d e r a b l e use of the r e p r e s e n t a t i o n of the e l e c t r o m a g n e t i c field in the f o r m of a plane wave, as a r e s u l t of which it r e q u i r e s a s p e c i a l t r e a t m e n t in o r d e r to g e n e r a l i z e the r e l a t i o n s obtained in these p a p e r s to the c a s e of inhomogeneous m e d i a . The p r e s e n t p a p e r is devoted to eliminating t h e s e p r o b l e m s f r o m the existing m i c r o t h e o r y of g y r o t r o p y . It is well known that l i n e a r optical activity is connected with the fact that the m e d i u m has a nonlocal s p a tial r e s p o n s e to the action of e l e c t r o m a g n e t i c radiation. In m e d i a consisting of e l e c t r i c a l l y n e u t r a l weakly mobile m o l e c u l e s , the nonlocal nature of the r e s p o n s e is c h a r a c t e r i z e d by e l e c t r i c quadrupole and magnetic dipole m o m e n t s of the m o l e c u l e s . Strictly speaking, the d e r i v a t i o n of the m a t e r i a l equations f o r g y r o t r o p i c m e d i a c o m e s down to the contributions m a d e by these m o m e n t s to the r e s p o n s e of the m e d i u m . As a b a s i s f o r calculating this r e s p o n s e we take the M a x w e l l - - L o r e n z equations for m i c r o f i e l d s , t o g e t h e r with c h a r g e and c u r r e n t densities defined by the r e l a t i o n s i

.,

i

where ej, m l , ~j, r j , arid rj a r e the c h a r g e , m a s s , spin magnetic m o m e n t , coordinate, and velocity of the j - t h e l e m e n t a r y c h a r g e . The s u m m a t i o n in Eq. (1) is c a r r i e d out o v e r all the c h a r g e s going to m a k e up t h e m e d i u m u n d e r c o n s i d e r a t i o n . Expanding Eq. (1) in a T a y l o r s e r i e s with r e s p e c t to c h a r g e d i s p l a c e m e n t s rj~ = r3--r-~ r e l a t i v e to the c e n t e r of m a s s ~ of the m o l e c u l e s , and a v e r a g i n g the M a x w e l l - - L o r e n z equations o v e r the motion of all the c h a r g e s , we obtain the following equations for the m e a n fields without difficulty: T r a n s l a t e d f r o m Zhurnal Prikladnoi Spektroskopii, Vol. 25, No. 3, p p . 4 9 3 - 4 9 9 , S e p t e m b e r , 1976. Original a r t i c l e s u b m i t t e d S e p t e m b e r 5, 1975. I This material is protected by copyright registered in the name of Plenum Publishing Corporation, 227 West 17th Street, New York, N.Y. 10011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the publisher for $ 7.50.

1151

[V, B] --(I/c) O1~Ot= (4z~/c)of/at + 4~t [~7, M], vE = _ 4~vP '

(2a) (2b)

'-~-

-0,4

[V, /~1 + (t/c) OB/Ot -- 0, VB = 0.

(2c)

With a d e g r e e of a c c u r a c y which takes into account the t h r e e lowest m o l e c u l a r m o m e n t s , the e l e c t r i c dipole m o m e n t

-~=~ e,r~,

(3)

= (1/2) ~ e j ~ . ~

(4)

the e l e c t r i c quadrupole m o m e n t

and the magnetic dipole moment m ----(1/4c) ~] eja {[~,

rM ~

-

[rj~, z. ;M} +~,

(5)

the quantities ~ and M appearing in Eqs. (2), which a r e the e l e c t r i c and magnetic polarizations of the medium, a r e defined as

= N < Sp (pp) > - - v~N< Sp (p//)>), ~I = N < Sp (pro)>.

(6a) (6b)

Here N is the density of m o l e c u l e s , p is the density m a t r i x c h a r a c t e r i z i n g the motion of the quantized d e g r e e s of f r e e d o m of the molecule, S p ( p . . . ) is the s y m b o l denoting an a v e r a g e with r e s p e c t to these degrees of f r e e dom, and < . . . > is the s y m b o l which denotes an a v e r a g e with r e s p e c t to the orientation of the m o l e c u l e s , the local i n t e r m o l e c u l a r fields, and o t h e r p a r a m e t e r s , for which the m o l e c u l a r distribution is described c l a s s i cally. The s u m m a t i o n in Eqs. (3)-(5) is c a r r i e d out o v e r all the c h a r g e s in the molecule, the dot between the v e c t o r s in Eq. (4) indicates that they a r e multiplied d i r e c t l y , while in Eqs. (1) and (3)-(5) the quantities r j a and rjc ~ a r e r e g a r d e d as coordinate and velocity o p e r a t o r s of the c h a r g e s in the molecule. Equations (6) are written for m o l e c u l e s of the one type. In the case of media which a r e solutions and m i x t u r e s , a summation o v e r all the types of molecules m u s t be included. The polarizations P and M r e p r e s e n t the r e s p o n s e of the medium to the light-wave field. The light-wave field c a u s e s the c h a r g e s of the medium to change t h e i r motion, which, in turn, means that the density m a t r i x p and the distributions of o t h e r p a r a m e t e r s , with r e s p e c t to which the averaging in Eq. (6) is c a r r i e d out, become dependent on this field. In the linear approximation this dependence can be contained only in p and is d e t e r m i n e d by solving the equation

Opij/Ot = [V, p~ u - 7iJPi~,

(7)

where pO = po6ii is the density m a t r i x in the absence of the light field, and Yij is the s p e c t r a l width of the lj • J m o l e c u l a r t r a n s i t i o n i -~ j ;

V=

-

(pE'-~mB'~ ~q:v.E')/ih.

-

(8)

The e n e r g y o p e r a t o r for the i n t e r a c t i o n of the field with the molecule is written with an a c c u r a c y which takes into account the m o m e n t s _ ~ and_~. Fields acting on the molecules in the medium a p p e a r in Eq. (8), and this is denoted by a p r i m e on E and B. In the g e n e r a l c a s e they a r e different f r o m the mean Maxwell fields which a p p e a r on the left-hand sides of Eqs. (2). The solution of Eq. (7), which has the form t

9t",1= .f [Vt', p~176 0

1152

f o r the r e s p o n s e of the m e d i u m to the action of the m o n o c h r o m a t i c field [E, B ~ exp(--iwt)], is substituted into Eq. (6), to give the final equations Ptj,

~ - - O3 " =- %aloE?, @ vaaf~vaE~' - - VaVaaf~E~ "@• (O 9

6)

9

M~ = ~,~B~ +

~, (!)

"

(9a)

(9b)

where a , /3, and g a r e indices d e n u m e r a t i n g the c o o r d i n a t e s , x, y , z. defined by the r e l a t i o n s

X~ =

"

--o)

The t e n s o r s a p p e a r i n g in Eq. (9) a r e

F~ (p~, p~), rl~ = Fo (ms, m~),

(10a)

--(j)

v~o6 = F~o(p~, qo~), V~o 6-~ = F~ (q=o, P~), • = F,. (p=, m~), ~ = F,o (m,. p~),

F.(a. b)= ((N/h) Z (p~i__p0) [ a,*bi, + biia~, i> j O)i]--O) -[- iYii ~ou + ~o + i V u ] > "

(10b) (10e)

(11)

We note s o m e s y m m e t r y p r o p e r t i e s of the t e n s o r s (10) which r e s u l t f r o m t h e i r definition and the p r o p e r t i e s of the functions F~ (a,b). Since the quadrupole m o m e n t t e n s o r ~ is s y m m e t r i c , the t e n s o r s (10b) a r e also s y m m e t r i c with r e s p e c t to the c o r r e s p o n d i n g indices. It follows f r o m Eq. (11) that the function Fw (a,b) has the p r o p e r t y F * ( a , b ) = F_w(a,b). In the windows where the m e d i u m is t r a n s p a r e n t (for Iwij--wr >> 7ij) the r e l a t i o n Fw (b,a) = F_w(a,b) is also valid, and, consequently, the equation F * (a,b) = F w ( b , a ) is also valid. Taking this into account, we have the following r e l a t i o n s for the t e n s o r s (10): ~(~,~= X~, • Vcta~ ~

= ~, Vacua - -

~1~ = ~1~,

(12a)

~4b

(12b)

--

V=o~= v ~ , ~r

=

--

* Vf~aa

(12c)

in the windows where the m e d i u m is t r a n s p a r e n t . Thus, the r e a l p a r t s of the t e n s o r s ~.afl. and ~ a ~ , a r e mutually adjoint ( ~ t ~ = ~ ' ~ , ) , while t h e i r i m a g i n a r y p a r t a r e anti-adjoint (~'~/3 = - ~ a ) " Szmflarly, ua(rfl = vflaa and ~ u~ ~o = - - ~ 'p ~ 9 A~nd, finall Y ' the r e a l P a r t of the t e n s o r u,~,,~a ~ ~ is a n t i s y m m e t ~ i c , while the i m a g i n a r y p a r t is s y m m e t r i c with r e s p e c t to the e x t r e m e indices. G e n e r a l l y speaking, the t e r m s in Eq. (9) e o n t a i n i n g t h e t e n s o r s V, 7,, • and ~ make contributions which a r e of the s a m e o r d e r of magnitude as the p o l a r i z a t i o n s P and M. It follows f r o m t h e i r definition that in nonmagnetic m e d i a (in m e d i a with V = 0) t h e s e contributions a r e a p p r o x i m a t e l y r e l a t e d to the basic t e r m in the p o l a r i z a t i o n P (containing X) as a/X, where a a r e the d i m e n s i o n s of p a r t i c l e s of the m e d i u m , and X is the w a v e length of the light. Equations (9) define the dependence of the e l e c t r i c P and magnetic M p o l a r i z a t i o n s of the m e d i u m on the local e l e c t r o m a g n e t i c field strength, It is well known that in condensed m e d i a this field can differ a p p r e c i a b l y f r o m the m e a n Maxwell field a p p e a r i n g on the left-hand side of E q s . (2J. In o r d e r to obtain m a t e r i a l equations f r o m E q s . (9) w_.~eh close the s y s t e m (2), the effective fields E ' and B' in Eqs. (9) m u s t be r e p l a c e d by the a v e r a g e fields E and B. Finding the connection between the m e a n and effective fields in the g e n e r a l e a s e p o s e s a v e r y difficult p r o b l e m [18]. It is usually solved a p p r o x i m a t e l y on the b a s i s of the L o r e n z r e l a t i o n E ' _ z E + 4 ~ / 3 . On substituting this r e l a t i o n in the f i r s t t e r m of Eq. (9a) and solving the r e s u l t i n g equation for P, we find [19] ^

^

where ~ d is the r i g h t - h a n d side of Eq. (9a) without the f i r s t t e r m , and = (I + 8n~/3)(I - - 4n~/3) -1

(14)

is the p e r m i t t i v i t y t e n s o r of the m e d i u m . Equation (13) is substituted in the L o r e n z r e l a t i o n , and a f t e r ne~ e c t i n g t h e s m a l l q u a n t i t y containing ~ d , we obtain the equation ~ ' = (g + 2)E/3 which can be u s e d to r e p l a c e E ' c o m p l e t e l y by E in E q s . (13) and (9b). As r e g a r d s the magnetic field, we can s e t B' = B in nonmagnetic media.

1153

T h u s , when the effective field is repl_~aced by_,the m e a n f i e l d t h e e x p r e s s i o n s f o r the p o l a r i z a t i o n s P and M again r e d u c e to the f o r m of Eq. (9) with E ' and B' r e p l a c e d by E and B, and with c o r r e s p o n d i n g l y o v e r d e t e r mined connection t e n s o r s . In this c a s e the t e n s o r )~ is r e p l a c e d by ( r = (~ + 2)~/3 = (1--4v~/3)-1~, the t e n s o r ~ is r e p l a c e d by (~ + 2) ~ (~ + 2)/9, ~ is r e p l a c e d by (e + 2)~t/3, ~ by ~(~ + 2 ) / 3 , and, finally, the t e n s o r ~ r e m a i n s unchanged.

.-.r

Denoting the o v e r d e t e r m i n e d t e n s o r s by the s a m e l e t t e r s as the o r i g i n a l t e n _ s o r s , a_.nd introducing the e l e c t r i c d i s p l a c e m e n t v e c t o r D = E + 4nP and magnetic field s t r e n g t h v e c t o r H = B --4~M, we finally obtain the r e q u i r e d m a t e r i a l equations: D,~ ~ e~.~E~+ 4 ~ , . v , ~ E ~ --4nV~ot~E ~ + 4z~• H~ = ,a~-~B~ - - 4 a ~ E ~ .

(15a) (15b)

H e r e Pail = (6aft --47n?afl )" Introduction of the v e c t o r s D and H r e d u c e s Eqs. (2a) and (2b) to the f o r m

which is the f o r m they a r e usually given in the s y s t e m of Maxwell's equations. In the r e g i o n s of t r a n s p a r e n c y , the t e n s o r s e~and ~ a p p e a r i n g in Eq. (15) a r e r e a l and s y m m e t r i c . All the r e m a i n i n g t e n s o r s have the s y m m e t r y p r o p e r t i e s given by E q s . (12b). The t r a n s i t i o n f r o m the effective fields to the a v e r a g e fields c a r r i e d out above only leads to a change in the n u m e r i c a l values of the e l e m e n t s of t h e s e t e n s o r s and does not affect t h e i r s y m m e t r y at a11. It can e a s i l y be seen that this is so if we note that the t e n s o r (e + 2)/3 is s y m m e t r i c . As shown above, the m a t e r i a l equations in the f o r m (15) r e s u l t n a t u r a l l y f r o m expanding the c u r r e n t and c h a r g e d e n s i t i e s (1), as well as the i n t e r a c t i o n e n e r g y (8), in m o l e c u l a r m u l t i p o l e s . This is the m o s t g e n e r a l and, in o u r opinion, the m o s t p h y s i c a l f o r m of the l i n e a r m a t e r i a l equations f o r g y r o t r o p i c media (media ~4th a s p a t i a l l y nonlocal r e s p o n s e to the f i r s t o r d e r in a/X). The contributions to the g y r o t r o p i c m e d i u m connected with e l e c t r i c q u a d r u p o l e ( t e r m s with ~ and g) and magnetic dipole (terms with ~t and ~) m o m e n t s a r e taken into account s e p a r a t e l y in t h e s e equations, and in the g e n e r a l c a s e a r e c h a r a c t e r i z e d by 36 complex (72 r e a l ) p a r a m e t e r s . In s p e c i f i c m e d i a , b e c a u s e they a r e i s o t r o p i c or have a definite c r y s t a l l i n e s y m m e t r y , the n u m b e r of t h e s e p a r a m e t e r s can be c o n s i d e r a b l y l e s s . Some f o r m s of the m a t e r i a l equations, f a m i l i a r f r o m p a p e r s on the phen_.omen_?log~al theory o f g y r o t r o p y , can be obt_Mnedcomparatively e a s i l y f r o m Eq._Q5). F o r e.xa._.mple, r e p l a c i n g D by D' = D--(4~rc/iw)[V, ~E] and H by H' = H + 4~6E and using the equation B = (c/io:) [V, E] which c o m e s f r o m Eq. (2c) f o r fields with a t i m e dependence of the f o r m exp(--iwt), Eq. (15) r e d u c e s to the f o r m D~ = e~.~E~+ 7~,.voE~ ~ V~{~a~E~,

(17a) (17b)

d i s c u s s e d in [12].

The t e n s o r s a p p e a r i n g in Eq. (17) a r e connected with the t e n s o r s of Eq. (15) by the r e l a t i o n

7l~a~ --- ~?-~o~~ 4.~v~ + 4n (c/io)) •

(18)

w h e r e epcrfl is the L e v i - - C i v i t a t e n s o r , w h i c h is a n t i s y m m e t r i c with r e s p e c t to all the indices. Equation (12b) has been taken into account in writing Eq. (18). It was shown in [12] that the f i r s t p a r t of Eq. (18) f o r r e a l tens o r s r e s u l t s f r o m the s y m m e t r y p r i n c i p l e f o r kinetic coefficients. The field v e c t o r substitutio__n ca_fried out above m e a n s e s s e n t i a l l y that the v e c t o r s P_~and M in Eq. (2a) a r e o v e r d e t e r m i n e d . The quantity IV, ~E], a f t e r it has been reduced to the f_orm (1/io:)IV, ~E], is included in the t e r m 0 P / a t (and, consequently, in 3D/at also), and not in the t e r m IV,M], as was done in the derivation of Eq. (15). It can e a s i l y be seen that this o v e r d e t e r m i n a t i o n does not change the f o r m of Maxwell's equations. In homogeneous media, the~ t e n s o r ~/o~afl can be r e m o v e d f r o m u n d e r the differentiation sign, and r e m e m b e r i n g that the t e n s o r Y~crfl'--Y~r is a n t i s y m m e t r i c with r e s p e c t to the end indices, we can reduce Eq. (17a) to the f o r m D"a = e~E~ + i : ~ v o E ~ + [~V, E]~, (19) w h e r e ~r~xcrfland 75(r a r e r e a l t e n s o r s equal to (20)

1154

(T' = Re~, 7" = ImT). The t e n s o r ;raa ~ is s y m m e t r i c with r e s p e c t to the end indices. In the g e n e r a l c a s e of n0nlocal r e s p o n s e 27 p a r a m e t e r s a r e t a k e n into account in Eq. (19) (18 components of the t e n s o r naofi and 9 of the t e n s o r 75a). It was shown in [1, 4-12] that the g y r o t r o p y of m e d i a , in the s e n s e that they h a v e the capacity to r o t a t e the p o l a r i z a t i o n of light fluxes, is d e s c r i b e d by the l a s t t e r m in Eq. (19). The middle t e r m in Eq. (19) is a c o r r e c t i o n to the t e n s o r e a ~ , and, consequently, it is also a c o r r e c t i o n to the refra_ctive index caused b y , h e nonloeal nature of p o l a r i z a t i o n of the m e d i u m . F o r a plane wave with wave v e c t o r k this c o r r e c t i o n is ~a~r~ka, i . e . , s y m m e t r i c like ~. However, as distinct f r o m ~ it changes sign when the direction of k is r e v e r s e d . This m e a n s that the r e f r a c t i v e indices f o r opposite d i r e c t i o n s in g y r o t r o p i c m e d i a can differ by an amount of o r d e r 217raa~kal. It should also be noted that in the t r a n s i t i o n f r o m the m a t e r i a l equations (15) to Eq. (17) the boundary conditions f o r the magnetic field m u s t also be changed. Using the f i r s t of E q s . (16) it can be shown [20]that in the g e n e r a l c a s e the following equation is valid: [H~--Hm n] = [HI - - H,,, n] + 4~ [~ E~ - - ~,,Ex~, hi,

(21)

where n is the n o r m a l to the interface- of m e d i a I and II. F o r H and D f r o m E q s . (16) and (15a) which a r e n6t o v e r d e t e r m i n e d , it follows, at the boundary of homogeneous m e d i a , that

= (4~/c)(v~E~I__v~a~E~H)n~, (v~a~v~E~1 ~v~v~E~II)no, ~ if= ~.

(22a) (22b)

The boundary condition f o r D' coincides with Eq. (22b). The boundary conditions f o r the o t h e r fields r e m a i n unchanged. T h e y c o i n c i d e with the boundary conditions f o r n o n g y r o t r o p i c m e d i a , i . e . , the tangential c o m p o nents of the field E and the n o r m a l c o m p o n e n t s of the field B a r e continuous. LITERATURE

1. 2. 3. 4. 5. 6. 7~

8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

CITED

L . D . Landau and E. M. L i f s h i t s , The E l e c t r o d y n a m i c s of Continuous Media, Addison-Wesley (1960). V . M . Agranovich and V. L. Ginzburg, C r y s t a l Optics with Spatial D i s p e r s i o n and the T h e o r y of Excitons [in Russian], Nauka, Moscow (1965). H.B.G. C a s i m i r , Philips R e s . R e p . , 21, 417 (1966). B . V . Bokut' and A. N. Serdyukov, Zh. l~ksp. T e o r . F i z . , 61, 1808 (1971). F . I . F e d o r o v , Usp. Fiz. Nauk, 108, 762 (1972). B . V . Bokut', N. S. Kazak and A. N. Serdyukov, Nonlinear Optical Activity [in Russian], P r e p r i n t , Inst. Fiz. Akad. Nauk BelorusSSR, Minsk (1972). V. M. Agranovic:h and V. L. Ginzburg, Zh. l~ksp. T e o r . F i z . , 6__33, 836 (1972). V . L . Ginzburg, Usp. Fiz. Nauk, 108, 749 (1972). V . M . Agranovich and V. I. Yudson, Opt. C o m m u n . , 5, 422 (1972). V . M . A g r a n o v i c h and V. I. Yudson, Opt. C o m m u n . , 9 , 58 (1973). B . V . Bokut', A. N. Serdyukov, and F. I. F e d o r o v , Zh. P r i k l . S p e k t r o s k . , 19, 377 (1972). B . V . Bokut' and A. N. Serdyukov, Zh. P r i k l . S p e k t r o s k . , 20, 677 (1974). U. Schlagheck, Z. P h y s . , 258, 223 (1973). A . P . Khapalyuk, Vestn. B e l o r u s . Gos. U n i v . , 3, 86 (1975). V . A . K i z e l ' , The Reflection of Light [in Russian], Nauka, Moscow (1973). R . M . H o r n r e i c h and S. Shtrikman, P h y s . R e v . , 171, 1065 (1968). H. Nakano and H. K i m u r a , J o u r n . P h y s . Soc. J a p a n . , 2__77, 519 (1969). V . M . A g r a n o v i c h , Usp. Fiz. Nauk, 122, 143 (1974). N. B l o e m b e r g e n , Nonlinear Optics, W. A. B e n j a m i n (1965). F . I . F e d o r o v , The Optics of Anisotropic Media [in Russian], Izd. Akad. Nauk BelorusSSR (1958).

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