FREQUENCY OPTICALLY B.

V.

CONVERSION ACTIVE Bokut'

LIGHT

WAVES

IN

MEDIA and

A.N.

Serdyukov

UDC 535.36

The t h e o r e t i c a l i n v e s t i g a t i o n s c a r r i e d out in [1] for a c l a s s 32 c r y s t a l and for an i s o t r o p i c m e d i u m showed the i m p o r t a n c e of taking into a c c o u n t the r o t a t i o n of the p o l a r i z a t i o n plane for the g e n e r a t i o n of high h a r m o n i c s in o p t i c a l l y a c t i v e c r y s t a l s . H o w e v e r , c o r r e s p o n d i n g r e s u l t s w e r e obtained without the exact solution o f t h e b o u n d a r y value p r o b l e m , and t h e r e f o r e c e r t a i n f e a t u r e s of the n o n l i n e a r p r o c e s s e s in such m e d i a t u r n e d out to h a v e been left out of c o n s i d e r a t i o n . In p a r t i c u l a r this applies to the p o l a r i z a t i o n of c o n v e r t e d w a v e s and, s p e c i f i c a l l y , to the r e l a t i v e intensities of the l e f t - and r i g h t - c i r c u l a r l y - p o l a r i z e d s e c o n d - and t h i r d - h a r m o n i c w a v e s . The s u c c e s s i v e l y i n v a r i a n t m e t h o d [2], when applied to the p r o b l e m of f r e q u e n c y c o n v e r s i o n of light w a v e s by an inactive c r y s t a l [3], allows e a s y d e r i v a t i o n of a g e n e r a l e x p r e s s i o n in explicit f o r m for the inh o m o g e n e o u s w a v e s of the h a r m o n i c s p r o d u c e d by the n o n l i n e a r - p o l a r i z a t i o n w a v e s in an o p t i c a l l y a c t i v e c r y s t a l . In the p r e s e n t p a p e r we s o l v e the p r o b l e m of the g e n e r a t i o n of h a r m o n i c s in g e n e r a l f o r m in the s p e c i f i e d - f i e l d a p p r o x i m a t i o n and c o n s i d e r the f e a t u r e s of this p r o c e s s in c e r t a i n c a s e s . The Solution of the B o u n d a r y Value P r o b l e m . A s s u m e that plane h a r m o n i c e l e c t r o m a g n e t i c w a v e s having the f r e q u e n c i e s w i a r e incident on a n o n l i n e a r o p t i c a l l y a c t i v e c r y s t a l c o n s i s t i n g of an i s o t r o p i c m e dium. As is well known [4], s e t s of two w a v e s having the f r e q u e n c y wi, w h i c h in the g e n e r a l c a s e a r e e l liptica11~, p o l a r i z e d , will be p r o p a g a t e d in the c r y s t a l . The wave v e c t o r s kv~(wi) and the c o m p l e x a m p l i tudes V(i) of the r e f r a c t e d w a v e s can e a s i l y be d e t e r m i n e d [4] via the c o r r e s p o n d i n g c h a r a c t e r i s t i c s of the incident w a v e s (the G r e e k s u b s c r i p t s will indicate the wave m o d e , while the R o m a n s u b s c r i p t s will indicate the f r e q u e n c y ) . The r e f r a c t e d w a v e s will excite n o n l i n e a r p o l a r i z a t i o n in the c r y s t a l : p~l = p (x) + P (0),

where 2

P(~):-2-.,I~

X:

~ p expi[(ka(coi)+kl3(o)j))r--@~+~i)t]

i.i + V(0Vg)*~ , exp i [(k~. (r ~,~=t

- - k~ ((%)) r - - (o~ - - %) t]},

(1)

2

P (0) = ~ -

Oi ~/v(~ p V(Ovexp i [(k s (co~) + k~ (coj) + k,, (col)) r c~,13,~=l i,Ll

--

(r + % + c01)t] + V(i)V(i)V(t)*ex"~ ~ v v i [(k s (o 3 + k~ (%) - - k v (ol)) r

- - (~0~+ % - - o 3 t] + V~)V[~)*V(~')expi [[k~ (o)3 - - k~ (0)j) + k, (o,)) r --(co~--coj+ %)t] + V(~162 r .~ - v -

[( ( k~ (~oi) c - - ok~ (%)i - - k.~(%)l ~ ,

- - o)j - - %) t] l

(2)

a r e the c o m p l e x n o n l i n e a r - p o l a r i z a t i o n v e c t o r s which a r e a s s o c i a t e d r e s p e c t i v e l y with the n o n l i n e a r s u s ceptibility t e n s o r s of the t h i r d r a n k (X) and of the fourth r a n k (0). The p o l a r i z a t i o n w a v e s (1) and (2) in t u r n p r o d u c e e l e c t r o m a g n e t i c r a d i a t i o n in the c r y s t a l , and the field of this r a d i a t i o n s a t i s f i e s the i n h o m o g e n e o u s wave equations [3, 5, 6] T r a n s l a t e d f r o m Z h u r n a l P r i k l a d n o i S p e c t r o s k o p i i , Vol. 12, No. 1, pp. 65-71, J a n u a r y , 1970. O r i g i n a l a r t i c l e s u b m i t t e d J u l y 25, 1969. 9 1972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. }i. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

55

iv Iv.E]] +

e'

=

- - 4n pnl,

(3)

w h e r e e' = ~ + i g • e is the p e r m i t t i v i t y t e n s o r ; g • is an a n t i s y m m e t r i c a l t e n s o r w h i c h is d u a l r e l a t i v e to the g y r a t i o n v e c t o r g. W e n e g l e c t t h e a b s o r p t i o n a n d t h e m a g n e t i c p r o p e r t i e s of the m e d i u m . F o r a n y of t h e p o l a r i z a t i o n w a v e s p(w0) h a v i n g a known f r e q u e n c y w ~ = ~oi + wj (or w ~ = c0 i =~ oJj :~ Wl) a n d the w a v e v e c t o r K ~ = k~(~oi) :~ kfl(~0j) (or K ~ = koflw i) =~ k/~(wj) ~- ky(o~l) t h e g e n e r a l s o l u t i o n of Eq. (3) is [3]

: ~=,,2

exp t ~ - - ( m ~

1-~ y2

m~)rl J

(4)

X exp i (K~r - - co~ - - [e~ e~] E~ [ere2] exp i (K~ - - co~ I §

" -1

-1

H e r e E 0 = 47r~ ~ Po, ~0 b e i n g a t e n s o r w h i c h is t h e r e c i p r o c a l of the t e n s o r ~0 = e + (m~2152 + ig • w h e r e m ~ = k ~ / w ~ i s t h e v e c t o r of t h e p o l a r i z a t i o n w a v e r e f r a c t i o n , w h i l e po is its a m p l i t u d e ; el, 2 = e~ + i y e ~ d a r e t h e p o l a r i z a t i o n v e c t o r s of t h e c o n v e r t e d w a v e h a v i n g the e l l i p t i c i t y y (e ~ c ~ a r e unit p o l a r i z a t i o n v e c t o r s in t h e a b s e n c e of a c t i v i t y ) ; K s = mo~w ~ is t h e w a v e v e e t o r of t h e c o n v e r t e d w a v e . T h e a m p l i t u d e s Ao~ d e t e r m i n e d f r o m t h e b o u n d a r y c o n d i t i o n s t u r n out to e q u a l A1"2 =

Eo [et, ezl.qe~. 2 +_ (1 + y2) E~ [qez.,] (1 + y~-)q [ele2]

'

(5)

w h e r e q is t h e unit v e c t o r of t h e n o r m a l to t h e i n t e r f a c e s u r f a c e , w h i c h is d i r e c t e d into t h e n o n l i n e a r m e dium, E~ = [m', aid 1 + a2dz] ' m' [dtd2] a1'2

--

Eo [e1%] ( mOe[2 _ qel.2"m,.z[ele2] / q[e,ezl.Eoe[2 1 + y~ q [et%] / + ~la.z 1+ u '

(6)

dL 2 = [eL2rn,] _~ m2,1[e,%] [qeLz]" q [et%] !

T h e v e c t o r E 0 is t h e a m p l i t u d e of t h e r e f l e c t e d w a v e a t t h e f r e q u e n c y oJ ~ h a v i n g t h e r e f r a c t i o n v e c t o r m ' . The p a r a m e t e r ~ d e t e r m i n e d f r o m t h e r e l a t i o n s h i p m ~ = m ~ + v a q s a t i s f i e s the e q u a t i o n of the n o r m a l Is + m~X"-}- ig• = 0. T h e v e c t o r s e ~ a r e d e t e r m i n e d f r o m t h e r e l a t i o n %.e~ = (1 § 7~) •215162 w h e r e t h e s u b s c r i p t "t" d e n o t e s the t r a c e . In t h e c a s e of d i r e c t i o n s c l o s e to t h e d i r e c t i o n of p h a s e m a t c h i n g w e h a v e b y a n a l o g y w i t h [3] the f o l l o w i n g r e l a t i o n s h i p f o r t h e i n c r e a s i n g p o r t i o n of t h e a m p l i t u d e of the c o n v e r t e d w a v e : Eoe~

1--expi

coO r

(m~

w h e r e o = - - ( m ~ q• -}-q•176215 5 = ( • 2 1 5 u n i t y f o r x -~ 0.

4~ico~ 'e~Po ~ -- ce~(r% ( +

( o - ~c) - t - o •

rj~'

~

i--

)

~l~qr' qr, r

-

(0•162 t h e function ~2(x) = (1 - exp ( - x ) ) / x t e n d s to

T h u s , f r o m (4) w e d i r e c t l y d e r i v e t h e s o l u t i o n w h i c h is a l s o v a l i d when t h e p h a s e m a t c h i n g c o n d i t i o n if f u l f i l l e d . H e r e , a s in the c a s e of an o p t i c a l l y i n a c t i v e c r y s t a l [3], the a m p l i t u d e of one of the s t i m u l a t e d w a v e s w h o s e w a v e v e c t o r K s c o i n c i d e s w i t h t h e w a v e v e c t o r K ~ of t h e c o r r e s p o n d i n g n o n l i n e a r - p o l a r i z a tion w a v e (K ~ = K s ) t u r n s out to b e p r o p o r t i o n a l to q r . Of c o u r s e , t h e l a t t e r s t a t e m e n t is v a l i d w i t h a f a i r d e g r e e of a c c u r a c y d u r i n g the i n i t i a l s t a g e of t h e p r o c e s s o f h i g h e r = h a r m o n i c g e n e r a t i o n , so t h a t the d e r i v e d s o l u t i o n (4) d e s c r i b e s c o r r e s p o n d i n g n o n l i n e a r p r o c e s s e s in c r y s t a l s h a v i n g a f a i r l y s m a l l t h i c k n e s s . G e n e r a t i o n of t h e S e c o n d H a r m o n i c a l o n g t h e O p t i c A x i s . L e t us c o n s i d e r t h e c a s e when the f u n d a m e n t a l r a d i a t i o n a t t h e f r e q u e n c y ~ is p r o p a g a t e d a l o n g t h e o p t i c a x i s c of a u n i a x i a l o p t i c a l l y a c t i v e c r y s t a l . T h e c o r r e s p o n d i n g e q u a t i o n of n o r m a l s [7] (the b a r a b o v e a q u a n t i t y d e n o t e s a t e n s o r w h i c h is the r e c i p r o c a l of t h e g i v e n one) n4nen + n 2 {[ng] 2 - - n (~ - - ~ n} - - geg + [ e l = 0 h a s the f o l l o w i n g s o l u t i o n f o r nil c: /7"2 ~ 1,2

56

~o +

cg.

F r o m this we find the wave n u m b e r s of r i g h t - and l e f t - c i r c u l a r l y p o l a r i z e d w a v e s (7)

k,,~ = k o • 9,

w h e r e k 0 = n0w/c is the w a v e n u m b e r in the a b s e n c e of a c t i v i t y , and p = c0cg/2cn 0 is the s p e c i f i c rotation. F o r p r o p a g a t i o n of c i r c u l a r l y p o l a r i z e d r a d i a t i o n (y = ~1) along the optic axis of uniaxial c r y s t a l s t r a n s v e r s e q u a d r a t i c p o l a r i z a t i o n d e v e l o p s only in c r y s t a l s having t r i g o n a l syngony. The s e c o n d - h a r m o n i c w a v e g e n e r a t e d by it l i k e w i s e t u r n s out to be c i r c u l a r l y p o l a r i z e d , but in the o p p o s i t e d i r e c t i o n [8]. M o r e o v e r , it is not difficult to v e r i f y the fact that in t h e s e c r y s t a l s no o p p o s i t e l y p o l a r i z e d e l e c t r o m a g n e t i c w a v e s i n t e r a c t , and c o n s e q u e n t l y the c o r r e s p o n d i n g n o n l i n e a r - p o l a r i z a t i o n wave does not develop. As a r e s u l t , f o r uniaxial c r y s t a l s it t u r n s out in p r i n c i p l e to be p o s s i b l e to have only two p h a s e - m a t c h i n g c o n ditions f o r s e c o n d - h a r m o n i c g e n e r a t i o n (SHG) along the optic axis: AksH G = + [29 ((o) + p (2(o)],

(8)

w h e r e AkSH G = K0(2r - 2k0(w); p(c0) and p(2w) a r e the s p e c i f i c r o t a t i o n s at the f r e q u e n c i e s w and 2r r e s p e c t i v e l y . The conditions c o r r e s p o n d to the c a s e when the r e f r a c t i o n v e c t o r s of the n o n l i n e a r - p o l a r i z a t i o n w a v e excited by one of the two c i r c u l a r l y p o l a r i z e d f u n d a m e n t a l - r a d i a t i o n w a v e s and the o p p o s i t e l y p o l a r ized s e c o n d - h a r m o n i c w a v e g e n e r a t e d by it coincide. Equations (8)were d e r i v e d in [1] for a c r y s t a l of c l a s s 32. L e t us l i k e w i s e c o n s i d e r an o p t i c a l l y a c t i v e c r y s t a l of c l a s s 23 which has cubic syngony. In the c o o r d i n a t e s y s t e m w h i c h is stipulated, as usual [9], by t h r e e m u t u a l l y o r t h o g o n a l unit v e c t o r s c i, e 2, c 3 dir e c t e d along the twofold a x e s , the n o n l i n e a r s u s c e p t i b i l i t y t e n s o r of the t h i r d r a n k has n o n v a n i s h i n g c o m p o n e n t s XI~3 = X23i = 9 9 9 = • [10]. In the s y s t e m w h i c h is r o t a t e d r e l a t i v e to it and h a s the c o o r d i n a t e vectors 1 'V-32--1 el -- V~3- (. cl

c: :

|..'-3+ l c 2 -]- C~) , 2

1 ( - ] / - 7 + 1 Ct-~- |/-3---1 - C~ +C.~') , ]/3- , 2 2 ~

1 the n o n v a n i s h i n g c o m p o n e n t s of the t e n s o r X a r e the s a m e as t h e y a r e f o r a c r y s t a l of c l a s s 3, but with the additional c o n s t r a i n t that X ~ w a r e s y a n m e t r i c a l r e l a t i v e to any p e r m u t a t i o n of the s u b s c r i p t s and a l s o that ! T = ? Xlll =X222 --X333/2~ T h u s , the g e n e r a t i o n of the s e c o n d h a r m o n i c in a cubic c r y s t a l of c l a s s 23 along the c~ d i r e c t i o n d i f f e r s in no w a y in p r i n c i p l e f r o m the c o r r e s p o n d i n g p r o c e s s in c r y s t a l s having t r i g o n a l s y n g o n y when the f u n d a m e n t a l r a d i a t i o n p r o p a g a t e s along the optic axis (the t h r e e f o l d axis). C o n s e q u e n t l y , the conditions f o r p h a s e m a t c h i n g will have the f o r m (8) in this c a s e a l s o . O n e o f t h e s e conditions w a s c o n s i d e r e d in [11] for g e n e r a t i o n of the s e c o n d h a r m o n i c of a r i g h t - c i r c u l a r l y - p o l a r i z e d f u n d a m e n t a l - r a d i a t i o n w a v e . The c o r r e s p o n d i n g e x p e r i m e n t a l i n v e s t i g a t i o n s on NaC10 3 and N a B r O 3 c r y s t a l s belonging to c l a s s 23 have been d e s c r i b e d in [8]. F o r n o r m a l incidence of an e l l i p t i c a l l y p o l a r i z e d plane wave h a v i n g the f r e q u e n c y w and the ellipticity '~0"

U (~o) --

et~+ iYoe~ U exp i (kz - - o)t) vl +v~

(9)

on an o p t i c a l l y a c t i v e c r y s t a l along the z d i r e c t i o n w h i c h coincides with the t h r e e f o l d a x i s , the r a d i a t i o n r e f r a c t e d in the c r y s t a l can be r e p r e s e n t e d in the f o r m of the s u m of two c i r c u l a r l y p o l a r i z e d w a v e s : V(~,~) (~) : _Un(1 + y0)(e~ ie~) e x p i ( k l , 2 z - - ~ (n + nl,~) ~, 1 + Vo ~ w h e r e n is the r e f r a c t i v e index of the w a v e s in the i s o t r o p i c m e d i u m . v e c t o r in a c r y s t a l of c l a s s 3 is equal to

(10)

Theeomplex nonlinear-polarization

P (20)) = p(1) (2(o) + pc".)(2o)),

57

p(l,~)(2o ) = 2Uen" (Zm +- ix~.~z)(1 ~ y0)~(e~ _-Z-ie~) expi (K~

- - 2or),

(n + n~.,) ~"(1 + ~,~) in a c c o r d a n c e with the explicit f o r m of the t e n s o r X [10]; h e r e K~, 2 = 2k2,1. Then, taking a c c o u n t of the fact that nI[q and ne~ = 0, we obtain f r o m (4)-(6) the r e f l e c t e d and t r a n s m i t t e d s e c o n d h a r m o n i c w a v e , r e spectively: E' (2o) = (E '~ -I-E'(~)) exp i (-- K'z - - 2~ot), E,(~,_,)= - - 8nU"n~ (Xm • i7.~) (1 -T- %)"-(e~ _+ ie~ (rtz,~ --k n)" (N~,e -4- N~,2) (N,,~4--N) (1 + y~) ' e (2o) = E (~) (2~0) + e (~) (2~),

E("2)(2~ =

8~U2n~(~,n• ~(e~176 (n2. , q- n) z (N, . q- N~ .2)(1 q- y~)

[ ~ _

/2i~ h a [-~-~l,.2z)

(11)

1 ]expi( K~.~z NL2+ N

2or).

(12)

I{ere N is the refractive index oi~the second harmonic in the isotropic medium; Ni,2are the refractive indices of the two circularly polarized second-harmonic waves in the crystal; N~ = = n2,I and ~1,2 = [NI,2 - N~ From the expressions derived it follows that for incidence of circularly polarized radiation ("/0 = ~1) the transmitted second-harmonic wave likewise turns out to be circularly polarized, only in the opposite direction; the reflected second-harmonic wave has the same polarization as the fundamental radiation. If the incident radiation is elliptically polarized, then the reflected second-harmonic wave is likewise elliptically polarized. However, the axes of the ellipse of the reflected second harmonic turn out to be rotated relative to the axes of the ellipse of the incident fundamental radiation. This rotation is caused by the presence of the component X222of the nonlinear susceptibility tensor of the third rank in (12).

K~

Assuming in (11) and (12) that X222---- 0 , We obtain the corresponding results for a crystal of class 32. In this case the reflected second harmonic is a plane wave

E' (2o) =

--

1 ~- yg

+ t~ %) -

(1 - - %),2 (1 -t- %).2 (n2 +. n) ~-(N~ 5 n2) (N~ --k N) + (nx -k n) ~ (N2 -b nO (g~ + N)

exp i (-- K'z -- 2or)

having the e l l i p t i c i t y

y, = ( 1 - - y0) ~ (n~-k n) 2 (N2+ n~) (N~ + N ) - ( I + %)2 (n2 ~_ n)'-'(N~+ n~) (N, + N) (1 --%)~ (n~--kn) (N2 + nO (N~.+ N) § + %)~ (nz+n) "2(N~-t-n.,) (N~-~ N) ' it being obvious that the axes of its e l l i p s e c o i n c i d e with the a x e s of the ellipse of the incident w a v e . The r e s u l t s obtained a r e a l s o valid for c r y s t a l s of c l a s s 43m in which t h e r e is no optical a c t i v i t y , and l i k e w i s e f o r c r y s t a l s of c l a s s 3m in which no a c t i v i t y is m a n i f e s t e d for p r o p a g a t i o n of r a d i a t i o n along the t h r e e f o l d axis [7]. Since u n d e r t h e s e conditions n 1 = n 2 = no, N1 = N 2 = No, K 1 = K 2 = K0, and~ h = ~72 = ~?, it follows that the s e c o n d - h a r m o n i c w a v e (12) can be r e p r e s e n t e d in the f o r m E (2o) =

(no + n)~ (No + no)

"(2 ( , - - / - ,

No 2-{_N- exp i (Koz - - 2(or),

(13)

w h e r e 3' = - 2 " / o / 1 + ~ is the eliiptieity of this wave. F r o m (13) it follows that the e n e r g y d e n s i t y of the s e c o n d h a r m o n i c inside the c r y s t a l in the e a s e c o n s i d e r e d is W(2o) ~ 1 -b ~'~ --

1 -{- 6 ~ + y~ (l q-.y~)2

Hence it is not difficult to show that for the s a m e p o w e r of the incident r a d i a t i o n the g e n e r a t i o n of the s e c ond h a r m o n i c is m o s t efficient when the incident r a d i a t i o n is c i r c u l a r l y p o l a r i z e d ("/0 = ~1) and l e a s t efficient when the incident r a d i a t i o n is l i n e a r l y p o l a r i z e d ("/0 = 0). G e n e r a t i o n of the T h i r d H a r m o n i c . L e t us c o n s i d e r the g e n e r a t i o n of the t h i r d h a r m o n i c in an o p t i c a l ly a c t i v e c r y s t a l of c l a s s 4 for p r o p a g a t i o n of the f u n d a m e n t a l r a d i a t i o n along the optic axis of the c r y s t a l . Using the explicit f o r m of the t e n s o r O [9], as well as Eqs. (9) and (10), the n o n l i n e a r - p o l a r i z a t i o n v e c t o r is w r i t t e n thus-

58

4

P (3o)) = X p(a)e x p i ( K ~

--

30)0'

3U~n ~ (0mr + 0x~z2) (1 - - ~1~)(1 • %) (e~ + i%~

p(l,2)

(14)

(n~.~ + n) ~ (n~,~ + n) (I + v~)~n U3n a (Omt - - 30nzz ~ 4iOan) (1 N yo)3 (e~ 4- ie o) (nzA + n) a (1 -[- y~)3/2 w h e r e K~ = 2kl, 2 + k2,1, K~ = 3~,_i. The r e f l e c t e d and t r a n s m i t t e d t h i r d - h a r m o n i c w a v e s under t h e s e conditions will r e s p e c t i v e l y be (coo - 3w) 4

E' (30)) = X E'(~) exp i (-- K'z -- 30)0, ~=,

E'(~)=

- - 4aP(a) (N~ + N ~ (N~ + N)

and 2

E (3o) = X E(a) exp i (Kc, z -- 30)t), 0;=t

E O,2)= Nl.~ + N~

\,

c

/

N,,2 + N

NLo. -[-N~

c

~

c

./

N1.z+ N

where N~ = K~ ~?1,2 = N1,2 - Nl~ ?73,4 = NI,2 - N~ 9 F o r c r y s t a l s belonging to c l a s s e s 422, 223, and 432 it is n e c e s s a r y to p l a c e 0~lil = 0 in (14). As d e r i v e d f r o m (15) and (7), in o p t i c a l l y a c t i v e c r y s t a l s h a v ing t e t r a g o n a l s y n g o n y arid in cubic c r y s t a l s it is i n p r i n c i p l e p o s s i b l e to have four p h a s e - m a t c h i n g conditions for t h i r d - h a r m o n i c g e n e r a t i o n (THG) along the f o u r f o l d o r twofold a x e s :

AkTHG = =- [39 (0)) + 9 (30))],

akzHG'= +- [O (0)) --

(16)

~ (30))],

w h e r e AkTH G = K0(3w ) - 3k0(w ). T h e f i r s t two conditions have as yet not been c o n s i d e r e d . p h a s e - m a t c h i n g conditions w e r e d e r i v e d in [1].

The l a t t e r two

F o r c r y s t a l s having t r i g o n a l and h e x a g o n a l syngony, 02m = 0 and 01n 1 = 30n~ 2. T h e r e f o r e in (14) one should p l a c e p(3, t) = 0 so that the f i r s t p h a s e - m a t c h i n g conditions (16) do n o t h o l d f o r t h e s e c r y s t a l s . In the a b s e n c e of optical a c t i v i t y ( c r y s t a l s belonging to c l a s s e s 3, 3 m , 3 m , 6, 6m2, 6ram, and 43m) we obtain E(3(o) = f r o m (14) and (15).

64nUana0mx( 1 - Y~)(e~ + iY0e~ (n o ~- n)3(No + N ~ (1 T y~)a/2

(3i0)Z~li

[~_ ~

\,

c

1

]

]

N O+ N

expi (Koz-- 30)0.

The e n e r g y d e n s i t y of the third h a r m o n i c ~

I i - vo~ ~2

~, l+v0 j in this c a s e t u r n s out to be h i g h e s t for l i n e a r p o l a r i z a t i o n of the incident r a d i a t i o n (Y0 = 0). F o r c r y s t a l s having no optical a c t i v i t y all of the p h a s e - m a t c h i n g conditions d e g e n e r a t e into one condition Ak = 0. F o r the i n t e n s i t y of the second h a r m o n i c g e n e r a t e d in a c r y s t a l of c l a s s 32 we h a v e the following r e sult on the b a s i s of (12): (1--%)~ IsMc ~ ~ 2 + n) 4 (N1 + N?)2

[

1 N t + ~ - + 4 (N~ + N)

sin 2 0) Tllz e ] B~

sin ~ ~

~-

(I + Yo)~

[

I

' (n~+n)'(N 2+N~ ~ N2+N

~-4(NO +N)

c

~1~

q2z

1.

The c a l c u l a t i o n of the i n t e n s i t y of the s e c o n d h a r m o n i c f o r Y0 = 0 which w a s c a r r i e d out in [1] t u r n s out to h a v e b e e n a p p r o x i m a t e , s i n c e t h e r e : 1) the d i f f e r e n c e between the r e f r a c t i v e indices of l e f t - and r i g h t c i r c u l a r l y - p o l a r i z e d w a v e s w a s not taken into a c c o u n t , and 2) it w a s a s s u m e d that the h a r m o n i c field w a s

59

absent on the crystal boundary (z = 0). The same also applies to the third-harmonic intensity calculated in [1]. The solution of the boundary value problems refines the corresponding results. LITERATURE

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

60

CITED

H. Rabin and P. Bey, Phys. Rev., 156, 1010 (1967). F . I . Fedorov, Optics of Anisotropic Media [in Russian], Minsk (1968). B . V . Bokut' and A. G. Khatkevich, Dokl. Akad. Nauk BSSR, 8, 713 (1964). B . V . Bokut' and F. I. Fedorov, Opt. i Spektr., 15, 798 (1963~. I . A . Armstrong, N. Bloembergen, I. Ducuing, and P. S. Pershan, Phys. Rev., 127, 1918 (1962). D . A . Kleinman, Phys. Rev., 128, 1761 (1962). B . V . Bokut' and F. I. Fedorov, Opt. i Spektr., 6~ 537 (1969). H. Simon and N. Bloembergen, Phys. Rev., 171, 1105 (1968). J. Nye, Physical Properties of Crystals [Russiantranslation], Izd. Mir (1967). S . A . Akhmanov and V. I. Zharikov, Pis'ma Zh. Eksperim. i Teor. Fiz., 6, 644 (1967).