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NONLINEAR SURFACE ELECTROMAGNETIC WAVES AT METAL-SEMICONDUCTOR
INTERFACES
Y. Chen, G. Carter
To cite this version:
Y. Chen, G. Carter. NONLINEAR SURFACE ELECTROMAGNETIC WAVES AT METAL-
SEMICONDUCTOR INTERFACES. Journal de Physique Colloques, 1984, 45 (C5), pp.C5-261-C5-
267. �10.1051/jphyscol:1984539�. �jpa-00224157�
JOURNAL D E PHYSIQUL
Colloque C5, supplCment au n04, Tome 45, a v r i l 1984 page C5-261
NONLINEAR SURFACE ELECTROMAGNETIC WAVES AT METAL-SEMICONDUCTOR INTERFACES
Y . J . Chen and G.M. C a r t e r
GTE Laboratories, Inc., 40 Sy Zvan Road, WaZtham, Massachusetts 02254, U.S.A.
Resume
L ' e x i s t e n c e d'une grande s u s c e p t i b i l i t s n o n - l i n e a i r e du t r o i s i e m e ordre, X ( 3 ) , dans l e s semiconducteurs (en p a r t i c u l i e r , dans l e s compo- ses 111-V) p r o d u i t de nouveaux types d'ondes PlectromagnPtiques de surface l o c a l i s e e s aux i n t e r f a c e s metal-semiconducteur. La r e l a t i o n de d i s p e r s i o n de ces ondes e s t f o n c t i o n de l e u r i n t e n s i t e . Nous avons observe recemment pour l a premiere f o i s , ces ondes electromagnetiques de surface n o n - l i n e a i r e s aux i n t e r f a c e s Ag-S1 e t Ag-GaAs. Dans ce papier, nous donnons un b r e f aperqu s u r l e s considerations e x p e r i - mentales e t s u r l e t r a v a i l t h e o r i q u e correspondant.
A b s t r a c t
T h e existence o f a l a r g e t h i r d o r d e r nonlinear susceptibility, x ( ~ ) , in semiconductors (e.g., I l l - V compounds) g i v e s r i s e t o new t y p e s o f surface electromagnetic waves (SEWs) a t t h e metal-semiconductor interfaces whose dispersion relations a r e i n t e n s i t y dependent.
Recently, we o b s e r v e d f o r t h e f i r s t time, nonlinear SEWs a t A g - S i a n d Ag-GaAs interfaces. I n t h i s paper we g i v e a b r i e f summary of t h e experimental considerations a n d related theoretical w o r k .
INTRODUCTION
T h e degenerate t h i r d o r d e r nonlinear s u s c e p t i b i l i t y x ( ~ ) (w;w,-w,w), defined as x ( ~ ) w , is a v e r y i n t e r e s t i n g and technologically important optical p r o p e r t y . / l / T h e dielectric function, c(w), f o r isotropic materials w i t h a nonzero x ( ~ ) changes l i n e a r l y w i t h t h e l i g h t i n t e n s i t y , 1,:
c(w,l) = c O ( w ) + 4i7 x(3) ( @ ) I
T h i s nonlinear ( i n t e n s i t y dependent) behavior of t h e dielectric f u n c t i o n c contributes t o many well-known optical effects as optical b i s t a b i l i t y , self- focusing, s e l f - t r a p p i n g a n d self-bending o f l i g h t , f o u r wave mixing, phase conjugation, etc. /2-4
/
These effects a r e essential t o all optical signal processing. /4-5/ T h e dispersion relations o f t h e optical normal modes, e. g . surface electromagnetic waves and waveguide modes, f o r systems containing nonlinear materials a r e also i n t e n s i t y dependent as t h e dispersion relations of t h e normal modes a r e generally f u n c t i o n s o f t h e dielectric functions. I n t h i s w o r k we shall l i m i t o u r discussion t o surface electromagnetic waves.T h e r e has been considerable i n t e r e s t i n nonlinear surface electromagnetic waves./6/ I t has been shown t h a t a s-polarized nonlinear surface electromagnetic wave exists /7-8/ and t h a t a p-polarized surface electromagnetic wave is allowed a t a uniaxial nonlinear interface even t h o u g h t h e zero f i e l d dielectric functions, c,, of b o t h media a r e positive./9/ It
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984539
C5-262 J O U R N A L DE PHYSIQUE
has also been shown t h a t u n d e r l a r g e f i e l d conditions, t h e d i s p e r s i o n o f nonlinear s u r f a c e electromagnetic waves e x h i b i t s m u l t i p l e states a n d t h e r e i s a g r e a t deal o f i n t e r e s t i n o p t i c a l b i s t a b i l i t y phenomena associated w i t h these m u l t i p l e states./7,10-14/ Experimentally, nonlinear s u r f a c e electromagnetic waves were o n l y r e c e n t l y o b s e r v e d b y Chen and C a r t e r f o r t w o Ag-GaAs a n d A g - S i s i n g l e i n t e r f a c e systems a t wavelengths near l u m /15/ T h e lack o f experimental r e s u l t s can b e a t t r i b u t e d , i n p a r t , t o t h e l a r g e f i e l d ( a n d t h u s t h e l a r g e i n p u t laser i n t e n s i t y ) r e q u i r e d t o i n d u c e a measurable nonlinear d i s p e r s i o n f o r a normal nonlinear material ( e . 9 . CS,). T h e l a r g 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 i n semiconductors makes t h e measurement o f t h e nonlinear d i s p e r s i o n o f SEWS a t t h e metal-semiconductor ( o r u n d e r special conditions, e v e n d i e l e c t r i c - s e m i c o n d u c t o r ) i n t e r f a c e s possible. It also opens u p a new avenue f o r s t u d y i n g t h e n o n l i n e a r o p t i c a l phenomena a t semiconductor-metal interfaces. I n t h i s p a p e r we w i l l g i v e a b r i e f summary of t h e experimental a n d related t h e o r e t i c a l w o r k on t h e s u b j e c t .
NONLINEAR DISPERSION
OF
SURFACE ELECTROMAGNETIC WAVEST h e nonlinear polarization f i e l d Pi c r e a t e d a t t h e f r e q u e n c y w b y a field ( o r f i e l d s ) of t h e same f r e q u e n c y , f o r an i s o t r o p i c medium, i s g i v e n b y /16/
Pi(w,E)=C(u)[21E.(w) 12Ei(w) + E , ~ ( w ) E ~
*
(w)]J (1)
w h e r e t h e Einstein summation notation is used f o r repeated indices. T h e second t e r m i n Eq(1) allows m i x i n g o f e l e c t r i c f i e l d at d i f f e r e n t polarizations.
A g r a n o v i c h e t . al. /17/ h a v e a r g u e d t h a t it can lead t o t h e decay o f n o n l i n e a r SEW'S t h r o u g h t h e i r c o u p l i n g t o bulk waves. I f one i g n o r e s t h e second t e r m i n Eq (11, t h e f i e l d d e p e n d e n t t e r m o f t h e nonlinear d i e l e c t r i c f u n c t i o n can b e expressed i n t h e f o l l o w i n g simple f o r m :
z N L i ( w , E ) = 4 r x ( ~ ) ( w ) E ~ ( w )
I
( 2 )F o r s i m p l i c i t y we w i l l c o n s i d e r a metal-semiconductor s i n g l e - i n t e r f a c e system o f w h i c h t h e metal (medium a) has a l i n e a r d i e l e c t r i c f u n c t i o n , ca(w) = coalw) < 0 a n d t h e semiconductor (medium b ) has a nonlinear d i e l e c t r i c f u n c t i o n , r (w) b = cbo(w) + c N L b ( u , ~ ) > 0 . T h e f i e l d i n d u c e d change of d i e l e c t r i c c o n s t a n t i s normally v e r y small compared w i t h t h e zero f i e l d t e r m
( i . e sbo >> r N L b ( w , ~ ) ) . T h u s t h e change of t h e d i s p e r s i o n o f a SEW, i n terms o f t h e change of t h e p r o p a g a t i o n constant Ak// d u e t o t h e p e r t u r b a t i o n o f t h e d i e l e c t r i c constant, Ar = rNLb, o f t h e system can b e obtained via t h e v a r i a t i o n theorem /18/:
w h e r e P E ( c / ~ T I ) If ( E x H ) da p o w e r c a r r i e d b y t h e SEW which is p r o p a g a t i n g along x; z is normal t o t h e i n t e r f a c e a n d t h e i n t e g r a t i o n i n Eq. (3) is o v e r t h e medium b w h i c h i s assumed t o o c c u p y t h e z>0 space ( w h e r e bz=cNL). I f we make t h e f o l l o w i n g assumptions: (1) l1z: >> r o b , so t h a t most o f t h e e n e r g y o f t h e SEW i s i n t h e nonlinear medium b a n d
I E z > > E ~ ~ { ; b a n d (2) t h a t t h e f i e l d p r o f i l e o f t h e SEW i n t h e nonlinear medium does n o t change a p p r e c i a b l y f r o m t h a t o f t h e z e r o f i e l d case, t h e n f o r t h e lowest p e r t u r b a t i o n o r d e r , Eb = Eb(0) exp(-abz) w h e r e o 2
b = Ik// (0) - cob ( W / C ) ~ ] ~ / ~ a n d k//(O) = ( w / c ) [roazob/ (zo a + c 0b)]1/2. F r o m E q . (3) we
can r e a d i l y o b t a i n t h e following simple r e s u l t f o r Ak// /15,19-20/
We note t h a t even f o r t h e simple nonlinear d i e l e c t r i c f u n c t i o n shown i n Eq. ( 2 ) , t h e nonlinear d i s p e r s i o n o f t h e SEW cannot b e solved a n a l y t i c a l l y . However, it can b e r e a d i l y solved u s i n g numerical m e t h o d s . / l l / We f u r t h e r n o t e t h a t f o r a u n i a x i a l nonlinear case, w h e r e E~
Y 4 7 ~ ' ~ ) [ I E X l 2 +
1
E y 2I ,
t h e n o n l i n e a r dispersion can b e d e r i v e d a n a l y t i c a l l y . /6,9/ However, t h e u n i a x i a l system i s somewhat a r t i f i c i a l a n d i t i s h a r d t o f i n d a nonlinear material w i t h t h e uniaxial nonlinear p r o p e r t y .F i g u r e 1 shows t h e nonlinear d i s p e r s i o n o f a SEW calculated b y numerical approaches. T h e calculation was c a r r i e d o u t b y t r e a t i n g t h e nonlinear medium as m u l t i l a y e r s o f i d e n t i c a l nonlinear media each w i t h a c o n s t a n t d i e l e c t r i c f u n c t i o n d e p e n d i n g o n t h e e l e c t r i c f i e l d s t r e n g t h o f t h a t l a y e r . / l l / F o r l a r g e changes o f t h e nonlinear d i e l e c t r i c constant ( ~ I T x ( ~ ) .
~~1~
> 6 x IO-~), t w o SEW branches were obtained. T h e presence o f t w o possible SEW states f o r a g i v e n f i e l d s t r e n g t h has raised t h e p o s s i b i l i t y o f optical b i s t a b i l i t y f o r t h e system./lO/ We w a n t t o p o i n t o u t t h a t e x a c t l y w h i c h of t h e t w o states can b e reached i n an experimental ( e i t h e r b y v a r y i n g t h e i n p u t l i g h t i n t e n s i t y o r i n c i d e n t c o u p l i n g angle) is dependent on t h e i n i t i a l s t a t e o f t h e system.U n d e r "pulsed" conditions, ( i . e . , t h e nonlinear d i e l e c t r i c is allowed t o relax back t o t h e zero f i e l d state, as t h e i n p u t c o n d i t i o n ( i n c i d e n t angle o r t h e i n c i d e n t laser i n t e n s i t y i s changed), t h e h i g h g a i n state ( t h e u p p e r c u r v e ) cannot b e reached. O n l y u n d e r "cw" conditions, ( i . e . , t h e nonlinear d i e l e c t r i c constant does n o t relax as t h e i n p u t condition i s changed), t h e h i g h q a i n s t a t e can b e reached. Since most nonlinear materials h a v e a r e l a t i v e l y
-
small t h i r d o r d e r nonlinear s u s c e p t i b i l i t y ( 5 1 x 1 0 - ~ ~ ( e s u ) ) , it i s n e a r l y impossible t o achieve t h e cw c o n d i t i o n b y u s i n g a cw laser source since t h e l a r g e laser power w i l l damage t h e sample. F o r nonlinear materials w h i c h h a v e
, -,
a l a r g e xL" one may attempt t o o b s e r v e t h e o p t i c a l b i s t a b i l i t y u s i n g a cw laser source. However, a ( q u a s i - ) cw condition may b e achieved f o r "slow"
n o n - l i n e a r i t y cases b y u s i n g a h i g h r e p e t i t i o n r a t e p u l s e d laser w h e r e t h e relaxation time of t h e n o n l i n e a r i t y is much l o n g e r t h a n t h e p u l s e d i n t e r v a l . T h e calculated r e s u l t s o f scanning t h e i n c i d e n t angle at f i x e d i n p u t i n t e n s i t i e s have been r e p o r t e d elsewhere /11/ a n d w i l l not b e discussed h e r e . F i g u r e 2 shows t h e change o f r e f l e c t e d i n t e n s i t y b y v a r y i n g t h e i n p u t l i g h t i n t e n s i t y a t a f i x e d i n c i d e n t c o u p l i n g angle ( 0 = 60.2') U n d e r t h e "pulsed" condition, t h e r e f l e c t e d i n t e n s i t y (normalized t o t h e i n c i d e n t i n t e n s i t y ) follows t h e s o l i d c u r v e f o r b o t h i n c r e a s i n g i n t e n s i t y a n d decreasing i n t e n s i t y cases. T h i s means t h a t it is o n l y possible t o couple i n t o t h e low g a i n s t a t e . U n d e r t h e
"cw" condition, t h e normalized r e f l e c t e d i n t e n s i t y c u r v e follows t h e s o l i d c u r v e as t h e i n t e n s i t y is increased f r o m 0 w h i l e t h e r e f l e c t e d i n t e n s i t y follows t h e dashed c u r v e as t h e i n t e n s i t y is r e d u c e d f r o m 1 x 1 0 - ~ a n d a h y s t e r e s i s loop opens. When an i n c i d e n t angle closer t o t h e zero f i e l d resonant angle (259.4') was used, t h e resonant c o u p l i n q o c c u r e d at a smaller value o f 47
-
x (3) E~~ ( i . e . , smaller i n p u t i n t e n s i t y ) and, t h e h y s t e r e s i s window was r e d u c e d a c c o r d i n g l y . B o t h calculations were done f o r a p r i s m - A g - n o n l i n e a r d i e l e c t r i c c o n f i g u r a t i o n u n d e r t h e f o l l o w i n g c o n d i t i o n s : X = 3 . 3 9 urn;
= 16.28;
A 9 = -570 + i62 ; E~ = 11.67 a n d s i l v e r t h i c k n e s s o f 300 A . E X P E K I M E N T L CONSIDERATIONS AND DISCUSSION
T h e experimental s e t u p used f o r measuring t h e nonlinear dispersion o f SEW has been d e s c r i b e d i n detail before, /15/ a n d w i l l o n l y b e b r i e f l y mentioned
J O U R N A L DE PHYSIQUE
F i g u r e 1. Field-dependent dispersion relation of SEWS at a metal- nonlinear m e d i u m interface.
h e r e . Basically, c o u p l i n g t o t h e SEW was b y attenuated t o t a l r e f l e c t i o n ( A T R ) techniques ( e i t h e r v i a a p r i s m o r a g r a t i n g ) . T h e nonlinear d i s p e r s i o n was determined b y measuring t h e change o f c o u p l i n g a n g l e w i t h
,m,
i n p u t laser i n t e n s i t y . From t h e nonlinear d i s p e r s i o n t h e values o f x l J ) o f Si a n d GaAs a t Xelpm were determined. B y u s i n g a collimated laser beam, a n d a h i g h a n g u l a r r e s o l u t i o n spectrogonirneter we c o u l d measure x ( ~ ) t o b e t t e r
F i g u r e
2.
Reflected intensity versus E . ; ' f o r a prism-metal-non- linear medium ATR configuration.t h a n 2 x 1 0 - l 1 e s u Furthermore, f r o m t h e d i r e c t i o n o f t h e change o f Akl/
w i t h laser i n t e n s i t y , we also established t h e s i g n o f x ( ~ ) r e l a t i v e t o t o f o r GaAs a n d Si. T h i s additional i n f o r m a t i o n can b e q u i t e valuable i n s t u d y i n g t h e kinematics of t h e n o n l i n e a r i t y a t resonance conditions w h e r e more t h a n o n e mechanism may b e i n v o l v e d .
F i g u r e 3 shows a t y p i c a l a n g u l a r scan o f t h e normalized r e f l e c t e d i n t e n s i t y n e a r t h e resonant c o u p l i n g angle f o r t h e A g - S i system a t an i n c i d e n t w a v e l e n g t h o f l . l l u m f o r b o t h h i g h i n p u t laser e n e r g y (=70uj/pulse) a n d low i n p u t laser e n e r g y ( = l O ~ j / p u l s e ) . T h e resonant c o u p l i n g angle was o b s e r v e d t o increase w i t h i n p u t laser i n t e n s i t y . Since a - 1 g r a t i n g c o u p l i n g c o n d i t i o n was used, t h e experimental r e s u l t s showed t h a t Ak,, decreases as t h e laser
. .
i n t e n s i t y increases, a n d t h u s , x ( ~ ) a n d e-
"
h a v e opposite signs. T h e magnitude o f x ( ~ ) was d e r i v e d f r o m t h e amount o f change dk,/ w i t h t h e g i v e n f i e l d s t r e n g t h o f t h e SEW a t t h e A g - S i i n t e r f a c e a n d was estimated t o b e 1 x 1 0 - ~ esu.T h e nonlinear measurement t e c h n i q u e can also b e used f o r time resolved pump a n d p r o b e studies, as shown i n F i g u r e 4. Note t h a t t h e p u m p a n d p r o b e beams a r e c o - l i n e a r so t h a t b o t h beams a r e i n c i d e n t on t h e same sample s p o t a t t h e same i n c i d e n t angle. If t h e time delay between t h e pump a n d p r o b e beam, T , i s s h o r t e r t h a n t h e r e l a x a t i o n time o f t h e n o n l i n e a r i t y , z , t h e p r o b e beam (a low i n t e n s i t y beam) i s u n d e r a quasi-cw c o n d i t i o n (i.e., t h e system i s s t i l l u n d e r t h e i n f l u e n c e o f t h e p u m p beam) a n d t h e d i s p e r s i o n r e l a t i o n measured b y t h e p r o b e beam w i l l b e similar t o t h a t o f t h e p u m p beam ( w h i c h has a much l a r g e r laser i n t e n s i t y ) . If T >> s , t h e p r o b e beam i s u n d e r a p u l s e d c o n d i t i o n ( i . e . , t h e system has r e t u r n e d t o t h e zero f i e l d state) a n d t h e d i s p e r s i o n relation measured b y t h e p r o b e beam w i l l b e d i f f e r e n t f r o m t h a t o f t h e p u m p beam. T h e r e f o r e , b y u s i n g t h e p u m p and p r o b e t e c h n i q u e one can s t u d y n o t o n l y t h e nonlinear d i s p e r s i o n o f SEWs, b u t also t h e dynamics o f t h e n o n l i n e a r i t y . T o i n c l u d e t h e dynamics o f t h e SEW, one m u s t b e i n t h e time scale w h e r e (a) t h e SEW has n o t decayed a n d ( b ) has n o t p r o p a g a t e d away. T h e pump a n d p r o b e s e t u p can also b e modified so t h a t t h e pump a n d p r o b e beams a r e n o t co-linear ( i . e . , t h e t w o beams have d i f f e r e n t i n c i d e n t angles). I n t h i s case one can use t h e pump beam t o c o n d i t i o n t h e nonlinear system a n d then, b e f o r e t h e system relaxes, p r o b e t h e system w i t h a n o t h e r i n t e n s i t y beam a t a s l i g h t l y d i f f e r e n t angle. T h i s way, one can i n i t i a t e t h e cw c o n d i t i o n w i t h p u l s e d laser beams a n d couple t o t h e h i g h gain s t a t e a t c o u p l i n g angles w h e r e t h e t w o n o n l i n e a r SEW b r a n c h e s a r e s t a r t i n g t o s p l i t .
F i n a l l y , we w a n t t o p o i n t o u t t h a t t h e n o n l i n e a r d i s p e r s i o n r e l a t i o n f o r an uniaxial system i s d i f f e r e n t f r o m t h a t o f a n i s o t r o p i c system. T h u s one can use t h e measured nonlinear d i s p e r s i o n t o s t u d y t h e n a t u r e o f o p t i c a l n o n l i n e a r i t v a t t h e metal-semiconductor i n t e r f a c e s . One case o f articular i n t e r e s t is t h e G ~ A S / G ~ ~ - ~ A I ~ A ~ m u l t i p l e q u a n t u m well (MQW) s t r u c t u r e . 2 1 A t wavelengths n e a r t h e two-dimensional exciton, MQW s t r u c t u r e s have been - shown t o h a v e a l a r g e x ( ~ ) i n t h e QW plane. I n t h e d i r e c t i o n normal t o t h e p l a n e ( z ) , t h e optical nonlinearity has not been well studied. However, f r o m t h e l a r g e d i f f e r e n c e s i n t h e l i n e a r e l e c t r o n i c a n d e x c i t o n i c p r o p e r t i e s , we - . .
believe t h a t xZ (3) can b e q u i t e d i f f e r e n t f r o m x,'~). T h e nonlinear d i s p e r s i o n r e l a t i o n o f SEW w i l l address t h i s q u e s t i o n . F u r t h e r m o r e , t h e x (3) o f t h e MQW s t r u c t u r e s i s a f a c t o r o f 10 l a r g e r t h a n t h a t o f t h e silicon, t h u s 5 it is also a p r o m i s i n g system f o r s t u d y i n g t h e optical b i s t a b i l i t y phenomena of nonlinear SEWs.
JOURNAL DE PHYSIQUE
LOW -
-
INCIDENT ANGLE
Figure 3. Anpular scan of the reflected intensities around the resonant coupling angles for the Si-Ag case. The arrows shown mark the excitation angle (in degrees) for the two cases.
SPECTROGENIOMETER
DETECTOR PUMP PROBE 1
0 LENS REFERENCE
PULSED LASER SYSTEM
PROBE BEAM
Figure 4. Schematics for time resolved (pump and probe) nonlinear
coupling measurement set-up.
A C K N O W L E D G E M E E
We l i k e t o acknowledge Professor J.M. B a l l a n t y n e a n d G.J. Sonek a t t h e National Research a n d Resource F a c i l i t y f o r Submicron S t r u c t u r e a t Cornell U n i v e r s i t y a n d D r . N. Economour a t M I T Lincoln L a b o r a t o r y f o r s u p p l y i n g us w i t h t h e submicron g r a t i n g samples u s e d i n t h e experiments.
References
1. D . Frohlich, i n Advances i n Solid State Physics: Festkoer Probleme (Pergarnon, New Y o r k , 1981), V o l . XXI, p . 363.
2. Y . R . S h e n , R e v . M o d . Phys. @ , I (1976) 3. D. M. Pepper, O p t . Eng.
2,
156 (1982).4. See, f o r example, "Optical Phase Conjugation," e d i t e d b y R. A . F i s h e r (Academic Press, New Y o r k , 1983).
5. P.
W.
Smith a n d W. J . Tomlinson, IEEE Spectrum18,
26 (1981).6. See, f o r example, a r e c e n t r e v i e w p a p e r b y A . A . M a r a d u d i n ( p r e p r i n t ) , t o b e p u b l i s h e d .
7. W. J. Tomlinson, Optical L e t t e r s
5,
323 (1980).8 . A . A . Maradudin, Z. Physics,
841,
341 (1981).9. Y . R . Alanakyan, Soviet Phys. - Tech. Phys. 12, 587 (1967); V . M.
A g r a n o v i c h , Psi'ma Z h . E k s p . T e o r . Fiz.
2,
5 3 n (1980).10. G . M. Wysin, H. J. Simon, a n d R. T. Deck, Optical Letters
6,
3 0(1981).
11. Y . J . Chen a n d G . M. C a r t e r , Solid State Commun.
41,
277 (1983).12. V . K. Fedyania, D . Minhalache, Z. Phys.
847,
167 (1982) 13. D . Sarid, A p p l . P h y s . L e t t .2 ,
889 (1981).14. G. I . Stegeman, A p p l . Phys. L e t t . 4J, 214 (1982).
15. Y . J. Chen a n d G. M. C a r t e r , A p p l . P h y s . L e t t .
41,
307 (1982).16. P. D . Maker a n d R. W. Terhune, Phys. Rev.
137,
A801 (1965) 17. V. M. Agranovich, V . S. Babichenko, a n d V. Ya Chernyak, SovietPhysics - - JETP L e t t . 3_1, 512 (1980).
18. H. Kogelnik, i n I n t e g r a t e d Optics, e d i t e d b y T . Tarnir ( S p r i n g e r , New Y o r k , 1979), Chap. 2.
19. V.M. A g r a n o v i c h a n d V . Ya. C h e r n y a k , Solid State Commun.
9,
1309 (1982).20. T h e same r e s u l t was o b t a i n e d i n d e p e n d e n t l y b y Chen a n d C a r t e r ( R e f . 15) a n d A g r a n o v i c h a n d C h e r n y a k ( R e f . 19). T h e expression g i v e n i n Ref.15 (Eq. 2) used an e f f e c t i v e e l e c t r i c f i e l d E d e f i n e d as Eo/J2 w h i c h accounts f o r t h e f a c t o r o f 2 d i f f e r e n c e .
21. D . A . B . Miller, D. S. Chemlla, D. J . Eilenberger, P. W. Smith, A . C . Gossard, a n d W. T . Tsang, A p p l . Phys. L e t t .
