BEHAVIOR

OF

WAVES

THE

ON

(AMPLIFYING) ISOTROPIC

HETEROGENEOUS BOUNDARY AND

ELECTROMAGNETIC

BETWEEN

ABSORBING

TRANSPARENT

MEDIA

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

G.

Leshchenko,

UDC 538.421

A n u m b e r of d i f f i c u l t i e s a r i s e in s t u d y i n g the q u e s t i o n of t h e r e f l e c t i o n o r r e f r a c t i o n of p l a n e e l e c t r o m a g n e t i c w a v e s on t h e b o u n d a r y of two m e d i a , of w h i c h one o r b o t h e x h i b i t a b s o r p t i o n , which a r e r e l a t e d with t h e b r e a k d o w n of the u s u a l e n e r g y b a l a n c e f o r t h e f l o w s of e n e r g y on t h e b o u n d a r y and a l s o w i t h t h e i n d e f i n i t e n e s s in s e l e c t i n g a s o l u t i o n f o r t h e r e f r a c t e d w a v e . F o r t h e d e f i n i t i v e d e t e r m i n a t i o n of t h e r e f r a c t e d w a v e t h e M a x w e l l e q u a t i o n s and b o u n d a r y c o n d i t i o n s a p p e a r to be i n s u f f i c i e n t in t h i s c a s e and a d d i t i o n a l p h y s i c a l c o n s i d e r a t i o n s a r e r e q u i r e d . T h e s e d i f f i c u l t i e s do not a r i s e o n l y in the s i m p l e s t c a s e s f o r t h e i n c i d e n c e of a h o m o g e n e o u s p l a n e w a v e f r o m a t r a n s p a r e n t m e d i u m on an a b s o r b i n g m e d i u m [1, 2] and f o r the i n c i d e n c e of a h e t e r o g e n e o u s w a v e of a s p e c i f i c t y p e ( a m p l i t u d e n o r m a l p e r p e n d i c u l a r to t h e b o u n d a r y [3]) f r o m an a b s o r b i n g m e d i u m on a t r a n s p a r e n t m e d i u m - - a n d the p r o b l e m i s c o m p l e t e l y s o l v e d . E v e n f o r a h o m o g e n e o u s w a v e t h e u s e of F r e s n e l ' s w e l l known f o r m u l a s l e a d s to p a r a d o x i c a l r e s u l t s : the i n t e n s i t y of the t r a n s m i t t e d l i g h t , f o r e x a m p l e , a p p e a r s to be g r e a t e r than t h e i n t e n s i t y of the i n c i d e n t l i g h t [4, 5]. The a t t e m p t s o f a n u m b e r of a u t h o r s [6-9] to a n a l y z e the g e n e r a l c a s e h a v e not, to t h i s t i m e , l e d to a s i n g l e o p i n i o n , s p e c i f i c a l l y with r e s p e c t to t h e d e t e r m i n a t i o n of t h e e n e r g y c o e f f i c i e n t s f o r the r e f l e c t i o n and t r a n s m i s s i o n of e l e c t r o m a g n e t i c w a v e s f o r the g i v e n b o u n d a r i e s . M o r e o v e r , t h e s e q u e s t i o n s a c q u i r e a t i m e l i n e s s in c o n n e c t i o n with t h e d e v e l o p m e n t of d i e l e c t r i c w a v e g u i d e s , and a l s o l a s e r d e v i c e s , b a s e d on the r e f l e c t i o n of l i g h t f r o m a m p l i f y i n g m e d i a [10]. -

-

L e t u s c o n s i d e r t h e p r o p a g a t i o n of a p l a n e e l e c t r o m a g n e t i c wave E = E ~ in an a b s o r b i n g ( a m p l i f y i n g ) n o n m a g n e t i c m e d i u m which i s c h a r a c t e r i z e d by a c o m p l e x , d i e l e c t r i c c o n s t a n t g = e + iT, in which ~ > 0 c o r r e s p o n d s to the a b s o r p t i o n and ~ < 0 c o r r e s p o n d s to a m p l i f i c a t i o n . T h i s e l e c t r o m a g n e t i c wave m u s t s a t i s f y the f o l l o w i n g s y s t e m of e q u a t i o n s [3]: eE . . . . . r e •

H = rn v. E,

(1) mE = mH = O, in w h i c h m2 = ~ = ~ +

(2)

ir.

C o n s e q u e n t l y the v e c t o r f o r r e f r a c t i o n of t h e w a v e m m u s t be the c o m p l e x m=m'

(3)

~- ira".

H e r e m ' i s the v e c t o r of the p h a s e n o r m a l and m" i s t h e v e c t o r f o r the a m p l i t u d e n o r m a l . Since t h e s e v e c t o r s , in t h e u s u a l c a s e , a r e not p a r a l l e l , t h e n t h e p r o p a g a t i n g w a v e s a r e h e t e r o g e n e o u s (see [3]). The v a l u e n' = I m ' l i s the i n d e x of r e f r a c t i o n of the w a v e , w h i c h d e t e r m i n e s i t s p h a s e v e l o c i t y in t h e m e d i u m , and the v a l u e n" = Ira" I d e t e r m i n e s t h e r a p i d i t y of t h e g r e a t e s t c h a n g e in t h e a m p l i t u d e of the w a v e . I n t r o d u c i n g , a s u s u a l , t h e c o m p l e x i n d e x of r e f r a c t i o n of t h e m e d i u m by m e a n s of the r u l e

(4) Translated from Zhurnal Prikladnoi Spektroskopti, Original article submitted October 2, 1972.

Vol. 19, No. 4, pp. 669-674,

October,

1973.

9 1975 Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. No part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any fmTn or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission o f the publisher. A COlO, o f this article is available from the publisher for $15.00.

1315

we g e t f r o m (2) = ~ --=-n ~-(1-- •

(n') ~ -- (n")'

(5)

2n'll" COs

~ =

T -~ 2rt~t,

w h e r e z > 0 c o r r e s p o n d s to a b s o r p t i o n and z < 0 to a m p l i f i c a t i o n , T i s the a n g l e b e t w e e n the a m p l i t u d e and p h a s e n o r m a l s of the w a v e which, in t h e f o l l o w i n g , w i l l b e c a l l e d the a n g l e of h e t e r o g e n e i t y . A s we c a n s e e cos y

2n'n" > 0.

(6)

T h u s , in an a b s o r b i n g m e d i u m (T > 0) t h e a n g l e 3' i s a l w a y s a c u t e (--7r/2 < 3' < w/2), in the a m p l i f y i n g m e d i u m (~ < 0) t h e a n g l e i s o b t u s e 0r/2 < 3' < 37r/2). In t h e c a s e of a t r a n s p a r e n t m e d i u m ~- = 0 and we g e t f r o m (5) n ' n " c o s 3' = 0. F r o m t h i s the known f a c t i m m e d i a t e l y f o l l o w s t h a t only two t y p e s of w a v e s c a n e x i s t in a t r a n s p a r e n t m e d i u m : 1) n" = 0 - - h o m o g e n e o u s w a v e s ; 2) n" ~ 0, c o s 3" = 0, i . e . , 3" = =~7r/2 - - h e t e r o g e n e o u s w a v e s with m " • m ' . Solving s y s t e m (5) with r e s p e c t to n' and n" we find

1l 1' 'i i i--• n' ==---=-_ ~2

....

~!

-7- V / (I--•215

sec'y,

(7)

-!

/~

-- : - :

/

-i- 1<' (: - x V - +

It m u s t be k e p t in m i n d t h a t , b e c a u s e of the n o n l i n e a r i t y of the r e f r a c t i o n v e c t o r m ( m x m * ~ 0) the v e c t o r s E and I-I o f a h e t e r o g e n e o u s w a v e , i n t h e g e n e r a l c a s e , c a n n o t b e l i n e a r s i m u l t a n e o u s l y [3]. T h e r e f o r e l e t u s r e s t r i c t o u r s e l v e s to the s t u d y of only t h e t w o s i m p l e s t c a s e s f o r the p o l a r i z a t i o n of h e t e r o g e n e o u s w a v e s : E - p o l a r i z a t i o n , when the v e c t o r E i s l i n e a r , and H - p o l a r i z a t i o n , when the v e c t o r H i s l i n e a r . L e t u s go d i r e c t l y to the b o u n d a r y p r o b l e m f o r the c a s e of the i n c i d e n c e of a h e t e r o g e n e o u s p l a n e w a v e with an a r b i t r a r y , p e r m i s s i b l e 3 / f r o m an a b s o r b i n g m e d i u m at the b o u n d a r y with a t r a n s p a r e n t m e d i u m . The d i e l e c t r i c c o n s t a n t of t h e f i r s t m e d i u m i s t h e c o m p l e x el = e~ + i~- and f o r t h e s e c o n d - - t r a n s p a r e n t m e d i u m - - i t i s t h e r e a l e 2. L e t u s c o n s i d e r the c a s e when the a m p l i t u d e n o r m a l m " l i e s in the p l a n e of incidence. L e t u s i n t r o d u c e the s i n g l e v e c t o r s : q - - the n o r m a l to the b o u n d a r y ; a = m ' x q / r e ' • q l - - the n o r m a l to the p l a n e of i n c i d e n c e ; b = q x a (Fig. 1). The c h a r a c t e r i s t i c s of the i n c i d e n t , r e f l e c t e d , and r e f r a c t e d wave will be i n d i c a t e d by t h e i n d e x e s 0, 1, and 2 r e s p e c t i v e l y . We will a l s o s e e k f o r the s o l u t i o n f o r the r e f l e c t e d and r e f r a c t e d wave in the f o r m of p l a n e , h e t e r o geneous waves

T h e c o m p l e x v e c t o r s f o r the r e f r a c t i o n of a l l of t h e w a v e s c a n be r e p r e s e n t e d in the f o l l o w i n g way: nij = ~jb -.'- ~ljq,

(9)

w h e r e Cj = ~i + ir 17j = 71i + iil~'. H e r e C.. and ll' a r e c o m p o n e n t s of the v e c t o r s f o r the p h a s e n o r m a l , ~ ' . J J J J J a n d 17--a r e ttie c o m p o n e n t s of t h e v e c t o r s of t h e a m p l i t u d e n o r m a l . The c o m p o n e n t s of t h e v e c t o r f o r the J r e f r a c t i o n of t h e i n c i d e n t w a v e i s t a k e n a s f i x e d and i s d e t e r m i n e d f r o m the r e l a t i o n s h i p s ~0 = n0sina,

+1'0-- n~ cos a,

~o = no sin ~, no = no cos ~, w h e r e a and/7 a r e the a n g l e s of t h e p h a s e and a m p l i t u d e n o r m a l s , r e s p e c t i v e l y , to the n o r m a l to the b o u n d a r y , in w h i c h c a s e / 7 ri + 3'$ and the v a l u e s of n o and a r e found f r o m Eq. (7). In o r d e r to find the r e f l e c t e d and r e f r a c t e d wave we u s e the b o u n d a r y c o n d i t i o n s which, i n t h e c a s e of n o n m a g n e t i c m e d i a , have the f o r m [3] SThe a n g l e s a r e c o u n t e d c o u n t e r c l o c k w i s e : normal.

1316

a and/7 f r o m the n o r m a l s to the b o u n d a r y and 3' f r o m t h e p h a s e

-- H

/ / /

/'/

/

~qT/

//7"

a2 q_

Fig. 1 (E U @ E 1 - E ~ ) x q

= 0,

(1O)

Ho-i H1---H~ = 0 . By s u b s t i t u t i n g s o l u t i o n (8) into (10) we g e t the r e f r a c t i o n r u l e f o r the a m p l i t u d e and p h a s e n o r m a l s f r o m t h e c o n d i t i o n of e q u a l i t y of t h e e x p o n e n t i a l i n d e x e s at any m o m e n t of t i m e So = ~i = ~'-'"

(11)

In o r d e r to find the q - c o m p o n e n t s of t h e r e f r a c t i o n v e c t o r s of t h e c o r r e s p o n d i n g w a v e s , we u s e r e l a t i o n s h i p (2) f o r the f i r s t a n d s e c o n d m e d i u m m 2 = m~ = e 1, m 2 = e2. F r o m t h i s , t a k i n g (11) into a c c o u n t , we g e t f o r the r e f l e c t e d wave ~h =

--

qo,

(12)

and for the refracted wave 1

~-; :: ( •

1 - 7 2 l"'~o. -(~:;)~ + (~;;- • , V [ ~ --(~o)"T(~o)-I .... ~176 ~-' 4(~;)~(~;) ~ ' (13)

~1:: = (=) .

~ -

1

l

,

- - [% - - (~o) 2 + (~of'] -~- V[e,~--

(~0) 2 + (~o)2]~-+4 (~;)2(~;)2,

T h e s i g n s in the e x p r e s s i o n s f o r ~ a n d ~/"2 r e m a i n i n d e f i n i t e s i n c e only a s i n g l e c o n d i t i o n

i s i m p o s e d on t h e s e v a l u e s . The p r o p e r s e l e c t i o n of the s i g n s in (13) c a n o n l y be m a d e by b r i n g i n g in a d ditional physical considerations. F u r t h e r , by s o l v i n g t h e s y s t e m (10) with r e s p e c t to t h e a m p l i t u d e s of t h e r e f l e c t e d a n d r e f r a c t e d w a v e , we find t h e a m p l i t u d e e o e f f i e i e n t s of r e f l e e t i o n r and t r a n s m i s s i o n d: 1) in the e a s e of E - p o l a r i z a t i o n (E~ : E~a) E~~ re -

E~

'1o -- ~h qo + ~1.. '

(15) E~ E00

dE

211o % + %

2) in t h e c a s e of H - p o l a r i z a t i o n (I-Ijo : H~a)

%dH .

.

H~ Moo

co% -- ~lq? ~;10 + e l ~

H~ . . H~

2e2,q0 e~rlo 4,- el+12

(16)

T h e s e f o r m u l a s c o i n c i d e in f o r m with t h e known F r e s n e l f o r m u l a s , only h e r e t h e v a l u e s of 7/0, ~2, and e 1 a r e c o m p l e x in t h e g e n e r a l c a s e . The l a t t e r l e a d s to an a d d i t i o n a l c h a n g e in the p h a s e of the h e t e r o g e n e o u s w a v e , a s c o m p a r e d with the c a s e of two t r a n s p a r e n t m e d i a , f o r r e f l e c t i o n a n d t r a n s m i s s i o n t h r o u g h t h e b o u n d a r y b e t w e e n an a b s o r b i n g and a t r a n s p a r e n t m e d i u m .

1317

In o r d e r to find t h e e n e r g y c o e f f i c i e n t s f o r r e f l e c t i o n and t r a n s m i s s i o n of a h e t e r o g e n e o u s wave, it i s n e c e s s a r y t o s t u d y the b a l a n c e of the n o r m a l c o m p o n e n t s of the e n e r g y f l o w s on the b o u n d a r y . F o r the a v e r a g e , with r e s p e c t to t i m e , d e n s i t y v e c t o r of the e n e r g y flow we u s e t h e u s u a l e x p r e s s i o n

P == 1----Re { e x U*}.

(17)

2 p r o c e e d i n g 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 i t c a n be shown t h a t in t h e g e n e r a l e a s e the f o l l o w i n g r e l a t i o n s h i p s o c c u r at the b o u n d a r y : Poq -t- Pxq q- Polq - - Poq = 0,

(18)

w h e r e P0i = (1/2)Re{E0 x tt~ + El x tI0*} i s t h e s o - c a l l e d i n t e r f e r e n c e flow which i s the r e s u l t of the s i m u l t a n e o u s p r e s e n e e of an i n c i d e n t and r e f l e c t e d wave in the f i r s t m e d i u m . D i v i d i n g (18) by P0q and i n t r o d u c ing the c o e f f i c i e n t s R = - - P i q / P 0 q , D = P2q/P0q, and I = - P 0 , q / P 0 q we c o m e to t h e r e l a t i o n s h i p R -i- D -i- I = I,

(19)

w h e r e t h e v a l u e s of R, D, a n d I a r e d e t e r m i n e d by the f o l l o w i n g e x p r e s s i o n s . for E-polarization

rio + %

~1"2 ] de I" ~10

, De

& -

,10

t '

4~1;1n0/2i2

(20)

% I ~lo + lh % -

'~ );

for H-polarization [2

RH = I rn 7 =-DH =

rl;I e~ Is

8,.~1o- - q ~ , 8~1o -}- 81~1.o ! '

[d,~ 12 =:

to. (sin; + ~G)

482 1~1 J~"'1; f~lo]2

(e1~1; + ~n;)i e~t0 -- qn~ I ~ ' ~(81G - *n;)

(21)

(r. - r5 ).

The p r e s e n c e of an i n t e r f e r e n c e flow in (18) c h a n g e s the u s u a l e n e r g y b a l a n c e at the b o u n d a r y . In c o n n e c t i o n with t h i s t h e q u e s t i o n a r i s e s a s to the p r o p e r a s s i g n m e n t of t h i s i n t e r f e r e n c e t e r m and the c o r r e s p o n d i n g r e d e t e r m i n a t i o n of t h e e n e r g y c o e f f i c i e n t s f o r r e f l e c t i o n and t r a n s m i s s i o n . In s o l v i n g t h i s p r o b l e m we will p r o c e e d f r o m t h e s e c o n d p r i n c i p l e of t h e r m o d y n a m i c s , a c c o r d i n g to which the e n e r g y c o e f f i c i e n t s of t r a n s m i s s i o n , f o r t h e i n c i d e n c e of the c o r r e s p o n d i n g w a v e s on the b o u n d a r y f r o m one s i d e o r t h e o t h e r , s h o u l d be t h e s a m e , i . e . , en ~ Den U12 21-

(22)

F o r t h e i n c i d e n c e of a wave with a r e f r a c t i o n v e c t o r m 0 f r o m the f i r s t m e d i u m on to the b o u n d a r y , a r e f l e e t e d and a r e f r a c t e d wave o r i g i n a t e with r e f r a c t i o n v e c t o r s of m 1 and m2, r e s p e c t i v e l y . Now if a wave with a r e f r a c t i o n v e c t o r m 3 = - - m 2 f r o m t h e s e c o n d m e d i u m i m p i n g e s on the b o u n d a r y , t h e n a r e f l e c t e d wave a r i s e s with a r e f r a c t i o n v e c t o r m 4 and a r e f r a c t e d w a v e a r i s e s with a r e f r a c t i o n v e c t o r m 5 = - - m 0. In t h i s e a s e Eq. (19) s h o u l d be f u l f i l l e d in both e a s e s . C o n s e q u e n t l y R~2 + DI~ q- Ii2 = R21 + D,, + I~i.

(23)

Since ~73 = --~2 a n d ~5 = --~/0, then f o r E - p o l a r i z a t i o n ~lo + ~h

~I3 + *la l

The s a m e c a n b e p r o v e d f o r H - p o l a r i z a t i o n in an a n a l o g o u s way. Thus, s i n c e It2 ~ I2i ( s p e c i f i c a l l y , the c a s e m a y o c c u r when I21 = 0 and 112 ;e 0) then, in the g e n e r a l c a s e Di2 # D21 a l s o and Eq. (22) c a n only be f u l f i l l e d if it i s a s s u m e d t h a t t h e e n e r g y c o e f f i c i e n t of t r a n s m i s s i o n i n c l u d e s an i n t e r f e r e n c e t e r m in it, i . e . , 1318

D~ := D -i I.

(24)

This also follows f r o m the indirect c a l c u l a t i o n s of the values D and t and of Eq. (22). The d e t e r m i n a t i o n of the e n e r g y coefficient of t r a n s m i s s i o n , a c c o r d i n g to (24) avoids the contradiction which is due to not taking the i n t e r f e r e n c e t e r m into account. Thus, a c c o r d i n g to calculation [4] f o r the n o r m a l incidence of light on the s i l v e r - a i r boundary (the light i m p i n g e s f r o m the metal) the e n e r g y c o e f f i cient of t r a n s m i s s i o n , d e t e r m i n e d a s D = P2q/P0q, is equal to 20, Taking into account the i n t e r f e r e n c e t e r m leads, a c c o r d i n g to (24), to a value of D en = 0.05. It is p e r t i n e n t to note that the conclusion made about the d e t e r m i n a t i o n of the e n e r g y coefficient of t r a n s m i s s i o n for a h e t e r o g e n e o u s wave a g r e e s with the data in [6] in wh[ch, in o r d e r to avoid the difficult i e s that w e r e mentioned, a special, f o r m a l p r o c e d u r e was suggested f o r calculating the e n e r g y c o e f f i c i ents for the reflection and t r a n s m i s s i o n , the validity of which, however, is not t h e o r e t i c a l l y founded and is only v e r i f i e d by r e f e r e n c e s to e x p e r i m e n t s . The e x p r e s s i o n s which w e r e found for the e n e r g y coefficients f o r the reflection and t r a n s m i s s i o n of h e t e r o g e n e o u s waves have a g e n e r a l c h a r a c t e r and a r e valid for any t s o t r o p i e a b s o r b i n g (amplifying)media.

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

8. 9. I0.

CITED

A . J . Mahan, JOSA, 4__66, No. 11, 913 (1956). A . P . P r i s h i v a l k o , Reflection of Light f r o m Absorbing Media [in Russian], Minsk, Izd. Akad. Nauk B e l o r u s s . SSR (1963). F . I . F e d o r o v , Optics of Anisotroptc Media [in Russian], Minsk, Izd. Akad. Nauk B e l o r u s s . SSR (195S). A. Va~i~ek, Chekhostovatskii Fizich. Zhurnal, A16, 555 (1966). I. Santavy, Atti Fondaz, "G. R o n c h i " e contrib. I s t naz ottica, 2._55, No. 2, 201 (1970). 13. Dold, Optik, 2.~2, H. 9, 615 (1965). H. Toman, Wissenschaftliche Z e i t s c h r i f t tier technischen Hochschnle Otto yon Gueriche, Magdeburg, 12, H. 2/3, 291 (1968). L. Dunaiskii, Chekhoslovatskii Fizich. Zhurnal, ]312, No. 9, 665 (1962). A. Va~iShek, Optik, 19, 327 (1962). ]3. Ya. Kogan, V. M. Volkov, and S. A. Lebedev, Pis'ma Zh. I~ks. Teor. Fiz., 16, No. 3, 144 (1972).

1319