REFLECTION

OF

WAVES

AT

THE

NONUNIFORM

ELECTROMAGNETIC

INTERFACE

H. B. Boiko, V. and N. S. Petrov

OF

G.

TWO

TRANSPARENT

MEDIA

UDC 535.42

Leshchenko,

It is known that Maxwell equations allow the existence in t r a n s p a r e n t media of two-dimensional nonuniform waves of the kind E = E ~ - t]}, w h e r e m = m ' + ira" are the phase (m') and a m p l i tude (m") n o r m a l s which a r e p e r p e n d i c u l a r to each other. N e v e r t h e l e s s , m o s t problems in phenomenological optics treated so f a r have been, as a rule, r e s t r i c t e d to a consideration of uniform waves. This s i t u a tion changed with the advent of quantum g e n e r a t o r s and d i e l e c t r i c waveguides (light pipes). The effect of wave nonuniformities b e c o m e s significant in p r o b l e m s involving the propagation of electromagnetic waves in l a s e r s and waveguides since different nonuniform waves a r e a s s o c i a t e d with different l o s s e s and a m p l i fications. Even a s m a l l difference in these p a r a m e t e r s can change the generated o r amplified wave mode in l a s e r devices, i.e., give r i s e to the generation of this o r that nonuniform wave. Thus, the behavior of nonuniform waves at the interface of two media has come into importance both from a scientific and a p r a c tical point of view. Here we c o n s i d e r the reflection and r e f r a c t i o n of two-dimensional nonuniform waves at the interface of two t r a n s p a r e n t isotropic media. We r e s t r i c t o u r s e l v e s to the m o s t interesting case when the amplitude n o r m a l of the incident wave lies in the plane of incidence. Let e~ and e2 be the dielectric constants of the f i r s t and second media, respectively. Let us introduce the following unit v e c t o r s : q, the n o r m a l to the interface directed f r o m the f i r s t to the second medium; a = m ' • q / I r a ' • ql, the n o r m a l to the plane of incidence; and b = q x a (Fig. 1). The c h a r a c t e r i s t i c s of the incident, reflected, and transmitted waves a r e denoted by the s u b s c r i p t s 0, 1, and 2, respectively. The r e f r a c t i o n v e c t o r s (see [1]) of these waves can be written as mi=~jb+~bq, ]=0,

w h e r e }j = }j ' + 1}j " " ; ~?j = 77j ' + i ~~.

Here

'

~j' and

"

I, 2,

(I) !

~7~a r e the r e s p e c t i v e projections of the phase mj and

amplitude m ; n o r m a l s . The incident nonuniform wave is assumed to be known, i.e., its amplitude and the direction of its phase n o r m a l as well as the direction and magnitude of its attenuation are given. Let a be the angle between the phase n o r m a l of the incident wave and the n o r m a l to the interface.* As in the case of uniform waves, this angle is called the angle of incidence (0 _ a _< 7~/2). Obviously, f o r any given there are two possible directions of the amplitude n o r m a l of the nonuniform wave. These directions a r e c h a r a c t e r i z e d by angle /3 which the amplitude n o r m a l f o r m s with the interface. At the same time, it follows f r o m the above that/3 = o~ + 7~/2. Thus, for an incident nonuniform wave (see Fig. 1) we have t o = n o , i n ~,, n o = no c o ~ ~,,

(2) ~o - - _+ no cos cz, rio = -T- n~ sin a , /t

.

*

i

Fig. 1. Nonuniform wave at the interface of two t r a n s p a r e n t media.

where the signs of }~ and V~ c o r r e s p o n d to the signs in the e x p r e s s i o n for /3. Here n~ and n 0"a r e , respectively, the r e f r a c t i v e index and attenuation f a c t o r of the incident wave. The latter c h a r a c t e r i z e s the degree of nonuniformity of the wave. In case of nonmagnetic media, the two p a r a m e t e r s are related by [1] *Angles a r e m e a s u r e d counterclockwise from the n o r m a l to the interface q.

T r a n s l a t e d f r o m Zhurnal Prikladnoi Spektroskopi[, VoL 11, No. 1, pp. 103-109, J a n u a r y , 1975. Original article submitted June 4, 1973. 9 Plenum Publishing Corporation, 22 7 West 17th Street, New York, A~ Y. 10011. No part o f 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 o f the publisher. A copy o f this article is available from the pubfisher for $15.00.

80

/20--/'/0

~

(3)

El,

which follows from the conditions of the existence of nonzero solutions of Maxwell equations. the boundary conditions for two-dimensional waves [1] we have

Considering (4)

m0 • 2 1 5 C o n s i d e r i n g (1) w e g e t

(5) ~0 = ~ = ~.

T o find the n o r m a l c o m p o n e n t s o f the r e f r a c t i o n v e c t o r s w e u s e the e q u a t i o n s d e r i v e d in [1]: 2

2

2

(6) H e n c e , f o r the r e f l e c t e d w a v e w e h a v e

(7)

nl = - - ~]0T h e c o r r e s p o n d i n g c o m p o n e n t s of the r e f r a c t e d w a v e a r e then ~=(•

+

(f + A) , n:--~(_+. )

(8)

(--f + A) ,

where f = n 2 - - sin2a + k~ cos 2cc, A = V r" + k2 (I + k"-)sin2 2~, n=

'

V~'

and, a s f o l l o w s f r o m (5), (6)

n~ n~ = no n~.

(10)

T h e s i g n s of .7~ and *75 a r e f o r the t i m e b e i n g u n d e t e r m i n e d s i n c e the only c o n d i t i o n they m u s t o b e y i s the c o n d i t i o n (10). W e c a n only s a y t h a t w h e n fi = a - n / 2 (the i n c i d e n t - w a v e a m p l i t u d e n o r m a l d i r e c t e d t o w a r d s the interface)~7~ and 7~ h a v e the s a m e s i g n (~7~7~ > 0), and w h e n f i = a + 7r/2 (the a m p i i t u d e n o r m a l of the i n c i d e n t w a v e d i r e c t e d f r o m the i n t e r f a c e ) , the s i g n s a r e d i f f e r e n t (~2.72 , , < 0). T h e p r o b l e m i s to c h o o s e the s i g n s s o t h a t they a c c u r a t e l y d e s c r i b e e x p e r i m e n t a l f a c t s . In c a s e of u n i f o r m w a v e s (n~ = 0), s e l e c t i o n of the s i g n s of .7~ and 75 is s i m p l e , s i n c e in this c a s e ! tt tt *7272 = 0. E q u a t i o n (8) i n d i c a t e s t h a t then f o r a n g l e s of i n c i d e n c e ~ < s 0 (sinflo = n), 72 = 0, i . e . , the r e f r a c t e d w a v e i s u n i f o r m and i s d i r e c t e d f r o m the i n t e r f a c e (7~ > 0). F o r a n g l e s of i n c i d e n c e a > a0, w h i c h c a n t a k e p l a c e if el > e2, - 7 2 = 0 (*72 # 0 ) , i . e . , the r e f r a c t e d w a v e is n o n u n i f o r m and i s p r o p a g a t e d a l o n g the i n t e r f a c e . F o r t r a n s p a r e n t m e d i a , a p h y s i c a l l y a c c e p t a b l e c o n d i t i o n i s t h a t w h i l e the w a v e p r o p a g a t e s i n s i d e the s e c o n d m e d i u m i t s a m p l i t u d e i s a t t e n u a t e d . Since the p a r a m e t e r .7~ e n t e r s the e x p r e s s i o n f o r N w a v e a m p l i t u d e [Eg ~ e x p { - ( w / c ) ~ 2 z }, z = q r > 0], '72 > 0 s a t i s f i e s this c o n d i t i o n w h e n z > 0. In c a s e of a c r i t i c a l i n c i d e n c e a n g l e , ~7~ = .7~ = 0. F o r n o n u n i f o r m w a v e s the p r o d u c t 72*72 ' ~ (10) i s a l w a y s d i f f e r e n t f r o m z e r o w i t h the e x c e p t i o n of two l i m i t i n g c a s e s of n o r m a l i n c i d e n c e (7~ = 0) and g l a n c i n g i n c i d e n c e (7~ = 0). In a l l o t h e r c a s e s the p r o d u c t is e i t h e r g r e a t e r o r s m a l l e r than z e r o d e p e n d i n g on the d i r e c t i o n of the a m p l i t u d e n o r m a l of the i n c i d e n t w a v e in r e l a t i o n to the i n t e r f a c e . F r o m t h i s it f o l l o w s t h a t in c a s e of n o n u n i f o r m w a v e s t h e r e i s no a n g l e of i n c i d e n c e f o r w h i c h the p h a s e n o r m a l of the r e f r a c t e d w a v e i s d i r e c t e d a l o n g the i n t e r f a c e . T h u s , d e p e n d ! i n g of the s i g n of the n o r m a l c o m p o n e n t of *72 of the p h a s e n o r m a l v e c t o r , the r e f r a c t e d w a v e is a l w a y s d i r e c t e d e i t h e r f r o m o r t o w a r d s the i n t e r f a c e . ?

It w o u l d s e e m t h a t the s a m e p r o c e d u r e c a n b e e m p l o y e d f o r finding the s i g n s of 72 and .7~ in c a s e of n o n u n i f o r m w a v e s , i . e . , b y r e q u i r i n g t h a t in the s e c o n d m e d i u m the w a v e a l w a y s d e c a y s w h i l e p r o p a g a t i n g in it. (It s h o u l d be noted t h a t in the d i r e c t i o n of i t s p r o p a g a t i o n in a t r a n s p a r e n t m e d i u m a n o n u n i f o r m w a v e i s n e i t h e r a t t e n u a t e d n o r a m p l i f i e d . ) T h e n , in c a s e fi = c~ + ~ / 2 , w h e n the a m p l i t u d e n o r m a l of the i n c i d e n t w a v e i s d i r e c t e d f r o m the i n t e r f a c e , w e s h o u l d a s s u m e .7~ > 0 and .7~ -: 0 in the e n t i r e i n t e r v a l of v a r i a t i o n s of the a n g l e of i n c i d e n c e a . H o w e v e r , in t h i s c a s e the s o l u t i o n s f o r the f i e l d of r e f r a c t e d n o n u n i f o r m w a v e s in a t r a n s p a r e n t m e d i u m do not t u r n into the w e l l - k n o w n s o l u t i o n s f o r u n i f o r m w a v e s in the s a m e m e d i u m . T h e c o n d i t i o n of c o n t i n u o u s t r a n s i t i o n of the s o l u t i o n s f o r n o n u n i f o r m w a v e s to the known s o l u t i o n s f o r

81

uniform waves when ~ ~ 0 is physically natural. However, from the condition of such continuity it follows that the solutions for the field in the second medium should significantly depend on the sign of the expression for f [see (8)]. The angle of incidence for which this expression turns into zero is called the critica[ angle .F

n~t§ k~

acrit = arc sin [// i - ~ 2 k~'

(II)

It can be easily seen that f o r uniform waves this angle is the same as the limiting angle of total reflection C~ 0 .

P r a c t i c a l l y , the condition of continuous transition of solutions m e a n s that f o r angles of incidence a < ~ c r i t (f > 0) it is always n e c e s s a r y to take V~ > 0 (outgoing wave). At the same time, ~2 will have the s a m e sign as ~ . On the other hand, f o r angles of incidence ~ > a c r i t (f < 0) one has to take solutions with # ! 772 > 0. The sign of ~2 is then the s a m e as that of ~?~. Since ~ > 0 c o r r e s p o n d s to a r e f r a c t e d wave receding from the interface, and V~ < 0 to a wave approaching the interface, the n o r m a l component ~ of the p h a s e n o r m a l v e c t o r of the r e f r a c t e d wave has a discontinuity at an angle of incidence o~ = a c r i t . Thus, in p a s s ing through a c r i t , the outgoing r e f r a c t e d wave b e c o m e s an incoming wave. The behavior of a r e f r a c t e d wave in such conditions is somewhat unusual: Unlike in the conventional c a s e , its phase n o r m a l is directed towards the interface. In this the wave is s i m i l a r to the incident wave. However, it differs f r o m the incident wave in that, firstly, no independent source is n e c e s s a r y for its f o r m a t i o n and, secondly, it does not generate a reflected wave which would spread in the same medium. This seemingly paradoxical behavior of the refracted wave will, obviously, find an explanation when the real problem of reflection of a spatially limited beam is solved. It should be noted that the a b o v e - d i s c u s s e d case of a r e f r a c t e d wave approaching the i n t e r f a c e is not unique. A s i m i l a r situation is encountered in the problem of reflection of e l e c t r o m a g n e t i c waves f r o m a boundary with an amplifying medium [2-4], and in p r o b l e m s on the propagation of light in absorbing w a v e guides. This c i r c u m s t a n c e has stimulated the publication of this work. Thus, the behavior of a nonuniform wave at the interface of two t r a n s p a r e n t media depends on the d i r e c t i o n of the amplitude n o r m a l of the incident wave with r e s p e c t to this interface. If this n o r m a l is directed f r o m the interface (fi = c~ + u / 2 ) , the incident nonuniform wave has some s i m i l a r i t y with a unif o r m wave in that f o r ~ < a c r i t the r e f r a c t e d wave r e c e d e s f r o m the interface (~ > 0) being attenuated in the d i r e c t i o n towards the interface; at a = a c r i t the components 77~ and ~ of the r e f r a c t e d wave abruptly change their signs so that for incidence angles a _> C~crit the r e f r a c t e d wave approaches the interface being attenuated inward the second medium. In this case, when the amplitude n o r m a l of the incident wave is directed towards the interface (fi = a - v / 2 ) , the r e f r a c t e d wave is always, i.e., for any angle of incidence a , an outgoing wave which is attenuated in a direction f r o m the interface. In this c a s e , the c r i t i c a l angle as such does not exist. Consequently, w h e r e a s in the f i r s t c a s e a phenomenon s i m i l a r to total r e flection should be o b s e r v e d , no such effect exists in the second case. Let us now c o n s i d e r e n e r g y relations a s s o c i a t e d with nonuniform waves at the interface of two t r a n s p a r e n t media. Boundary conditions indicate that t i m e - a v e r a g e d e n e r g y fluxes at the interface are related by (see, e.g., [1]) Poq q-, Plq q- Poiq - - P2q ---- 0,

(12)

w h e r e Pj = ( 1 / 2 ) R e {Ej x H~}, and P01 = ( 1 / 2 ) R e {E 0 x H~ + E1 x H~} is the s o - c a l l e d interference flux. Dividing by P0q and introducing the e n e r g y coefficients of reflection R = - P l q / P o q and t r a n s m i s s i o n D = (1/P0q)(P2q - P01q), which a c c o r d i n g to [5] includes the i n t e r f e r e n c e flux, we get 1 - - R - - D = O.

(13)

F o r the m o s t simple polarization modes of the uniform wave ~ and H modes [1]), the e x p r e s s i o n for e n e r g y reflection coefficients have the f o r m [5]

R~=1~lo--~[~i (14) RH =

82

e,qo __ ei~l~

2.

C~crit~eg

z

1,0

"

~q .

~

35

~ ~ 60 ~o~,deg0 gO qO 60 ~ , deg

0

I

I

~

t

0,2

o,q

0,6

~a

Fig. 2

k

0

Fig. 3

03

0//

O~

O~

k

Fig. 4

F i g . 2. E n e r g y r e f l e c t i o n c o e f f i c i e n t s of E - and H - p o l a r i z e d w a v e s a s a f u n c t i o n of a n g l e of i n c i d e n c e f o r n = 0.67 and k = 0.03 (fi = oz + v / 2 ) (1), 0 (2), and 0.03 (fi = a - ~ / 2 ) (3). F i g . 3. C r i t i c a l a n g l e a s a f u n c t i o n of w a v e a t t e n u a t i o n f a c t o r k f o r d i f f e r e n t v a l u e s of the r e l a t i v e r e f r a c t i o n i n d e x : n = 0.80 (1), 1/~f2 (2), 0.67 (3), and 0.60 (4). F i g . 4. R e f l e c t i o n c o e f f i c i e n t of H - (1) and E - p o l a r i z e d a s a f u n c t i o n of the a t t e n u a t i o n f a c t o r k f o r n = 0.67.

(2) n o n u n i f o r m w a v e s f o r n o r m a l i n c i d e n c e

D e v e l o p i n g t h e s e e x p r e s s i o n s and t a k i n g into c o n s i d e r a t i o n (2), (8), w e h a v e f o r a l l a n g l e s of i n c i d e n c e in c a s e fi = a - ~ / 2 , and f o r a n g l e s of i n c i d e n c e c~ < a c r i t in c a s e fi = a + ~ / 2 , RE =

R~ =

k 2 + cos~a + A - - B k s + cos~ e. + A + B n4 (kS + c~ a) + A - - n2B n ~ (kS + cos~a) + A + n~B '

(15)

where B ----V - 2 {]/(1 + k ~) (f + A) cos cr + k t / If p = a + 7r/2 and ~ -> a c r i t ,

f + A sin a}.

the r e f l e c t i o n c o e f f i c i e n t s b e c o m e

~'~- ~1 ,

R'n---

~.i

(16)

T h u s , w h e n the r e f r a c t e d w a v e i s o u t g o i n g (V~ > 0), the r e f l e c t i o n c o e f f i c i e n t s a r e shown by (15) to be l e s s than unity. On the o t h e r h a n d , w h e n the r e f r a c t e d w a v e a p p r o a c h e s the i n t e r f a c e the r e f l e c t i o n c o e f f i c i e n t s a r e g r e a t e r than unity a s i n d i c a t e d by (16). F i g u r e 2 s h o w s R E and RH a s f u n c t i o n s of a f o r v a r i o u s a t t e n u a t i o n f a c t o r s k. It is s e e n that the g r e a t e r the w a v e n o n u n i f o r m i t y the m o r e p r o n o u n c e d the d i f f e r e n c e in the b e h a v i o r o f r e f l e c t i o n c o e f f i c i e n t s of n o n u n i f o r m and u n i f o r m w a v e s . S e v e r a l r e m a r k s c o n c e r n i n g the r e f l e c t i o n c o e f f i c i e n t s R > 1 a r e h e r e in o r d e r . The p o i n t i s t h a t the e x p r e s s i o n s f o r R E and R H d e r i v e d f r o m d i f f e r e n t i a l M a x w e l l e q u a t i o n s a l s o a r e d i f f e r e n t i a l i . e . , they r e f e r to p o i n t s on the i n t e r f a c e w h e r e the r a t i o of the a m p l i t u d e of a w a v e r e c e d i n g f r o m the c o n s i d e r e d p o i n t in the d i r e c t i o n of the r e f l e c t e d b e a m to the a m p l i t u d e of a w a v e i n c i d e n t a t t h i s p o i n t i s d e t e r m i n e d . T h e v a l u e R > 1 i n d i c a t e s t h a t the o u t g o i n g w a v e a m p l i t u d e i s g r e a t e r than the i n c i d e n t w a v e a m p l i t u d e . T h i s h o w e v e r d o e s not e n t a i l a n y c o n t r a d i c t i o n w i t h the law of c o n s e r v a t i o n . T h e p o i n t i s t h a t in p r a c t i c e w e a l w a y s d e a l w i t h l i m i t e d b e a m d i s t o r t e d a t the e d g e s . T h e o b t a i n e d r e s u l t s thus r e f e r to the c e n t r a l ( u n d i s t o r t e d ) p o r t i o n of s u c h a b e a m . P r a c t i c a l b e a m s a l w a y s h a v e r e g i o n s w h e r e the l o c a l r e f l e c t i o n c o e f f i c i e n t s a r e l e s s t h a n unity s o t h a t the a v e r a g e r e f l e c t i o n c o e f f i c i e n t of the b e a m c a n n o t e x c e e d unity. A c c o r d i n g to [5], the t r a n s m i s s i o n into a c c o u n t the i n t e r f e r e n c e flux. T h u s , a p p r o a c h i n g the i n t e r f a c e , R > 1 a n d D < e n e r g y flux f r o m the s e c o n d m e d i u m into s e c o n d a t any p o i n t of the i n t e r f a c e .

c o e f f i c i e n t D of a n o n u n i f o r m w a v e should be d e t e r m i n e d t a k i n g in a c c o r d a n c e w i t h (13), D = 1 - R. In c a s e of r e f r a c t e d w a v e s 0. T h i s c a n b e e a s i l y e x p l a i n e d c o n s i d e r i n g t h a t in t h i s c a s e the the f i r s t e x c e e d s the e n e r g y flux f r o m the f i r s t m e d i u m into the

C e r t a i n o t h e r p e c u l a r i t i e s of the b e h a v i o r of n o n u n i f o r m w a v e s a t the i n t e r f a c e of two t r a n s p a r e n t m e d i a a r e b r i e f l y m e n t i o n e d b e l o w . In p a r t i c u l a r , the c r i t i c a l a n g l e ( a c r i t ) f o r w h i c h the r e f l e c t i o n

83

coefficient of one case of nonuniform wave incidence (~ = ~ + ~ / 2 ) has a discontinuity depends, according to (11), on the d e g r e e of wave nonuniformity k. F o r uniform waves this angle coincides with the c r i t i c a l angle of total reflection, approaching a c r i t = 45~ with i n c r e a s i n g nonuniformity. Only when the relative r e f r a c t i o n index of the two media is n = 1 / ~ 2 is the c r i t i c a l angle always equal to 45 ~ r e g a r d l e s s of the value of k. F i g u r e 3 shows C~crit as a function of k for different n. Of p a r t i c u l a r i n t e r e s t is the behavior of the reflection coefficients of E - and H-polarized nonuniform waves in case of n o r m a l incidence. The reflection coefficients are given then by

RO ( V n2-t- k2 --1./'-]"-'~ 1~, -

(17)

which coincide only for k = 0, i.e., in case of uniform waves. At the same time R~. = R~ = [(n - 1)/(n + 1)] 2. The behavior of these e x p r e s s i o n s when k (or n~) i n c r e a s e s is different: R ~ tends to the limiting value of [(n2 - 1 ) / ( n 2 + 1)] 2 when k ~ oo, w h e r e a s R~ ~ 0. Figure 4 shows the c o r r e s p o n d i n g curves. This difference in the b e h a v i o r of the reflection coefficients is due to physical differences in the s t r u c t u r e of differently polarized nonuniform waves: In an E - p o l a r i z e d wave the E v e c t o r is linear while the H v e c t o r d e s c r i b e s an ellipse in a plane n o r m a l to E; in case of H-polarization the situation is r e v e r s e d . Thus, the behavior of nonuniform e l e c t r o m a g n e t i c waves at the interface of two t r a n s p a r e n t i s o tropic media shows c e r t a i n distinctive features not observed in uniform waves. In p a r t i c u l a r , nonuniform waves have no analog of the B r e w s t e r angle, which can be associated with the experimentally observed residual reflection [6]. M o r e o v e r , an effect s i m i l a r to total reflection does not always o c c u r in nonuniform waves but, possibly, in the case a >- C~crit, only in a c e r t a i n definite direction of the amplitude n o r m a l with r e s p e c t to the interface. The c r i t i c a l angle is not constant but depends on the degree of nonuniformity of the wave. The latter also affects the reflection coefficients in case of n o r m a l incidence, and E - and H polarized waves behave differently. All these pecularities are the m o r e pronounced the higher the degree of wave nonuniformity. LITERATURE i. 2. 3. 4. 5. 6.

84

CITED

F. I. F e d o r o v , Optics of Anisotropic Media [in Russian], Izd. AN BelSSR, Minsk (1958). Ch. J. Koester, IEEE J. Quant. E l e c t r . , 2, LXIII (1966). / B. Ya. Kogan, V. M. Volkov, and S. A. Lebedev, P i s ' m a Zh. Eksp. Teor. Fiz., 16, No. 3, 114 (1972). B. B. Boiko, N. S. Petrov, and I. Z. Dzhilavdari, Zh. PrikL Spektrosk., 1_~8, No. 4, 725 (1973). B. B. Boiko, V. G. Leshchenko, and N. S. P e t r o v , Zh. Prikl. Spektrosk., 19, No. 4, 669 (1973). L. L Mandel'shtam, L e c t u r e s in Optics, T h e o r y of Relativity, and Quantum Mechanics [in Russian], Nauka, Moscow (1972).