THE GREEN FUNCTION OF A MACROSCOPIC ELECTROMAGNETIC FIELD IN A DISPERSIVE PLANE-STRATIFIED ANISOTROPIC MEDIUM

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THE GREEN FUNCTION OF A MACROSCOPIC ELECTROMAGNETIC FIELD IN A DISPERSIVE PLANE-STRATIFIED ANISOTROPIC MEDIUM

Ya. Iosilevskii

To cite this version:

Ya. Iosilevskii. THE GREEN FUNCTION OF A MACROSCOPIC ELECTROMAGNETIC FIELD

IN A DISPERSIVE PLANE-STRATIFIED ANISOTROPIC MEDIUM. Journal de Physique Collo-

ques, 1984, 45 (C5), pp.C5-305-C5-309. �10.1051/jphyscol:1984544�. �jpa-00224162�

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JOURNAL DE PHYSIQUE

Colloque C5, supplément a u n04, Tome 45, avril 1984 page ê5-305

T H E GREEN FUNCTION OF A MACROSCOPIC ELECTROMAGNETIC F I E L D IN A D I S P E R S I V E PLANE-STRATIFIED ANISOTROPIC MEDIUM

Ya. A. I o s i l e v s k i i

The Ministry of Science and DeveZoprnent, JerusaZern, IsraeZ

Résumé - La méthode d e s é q u a t i o n s f o n c t i o n n e l l e s d é v e l o p p é e a u p a r a v a n t d a n s l a c o n s t r u c t i o n de l a f o n c t i o n dynamique Green d ' u n champ q u a s i - é l a s t i q u e e t d ' u n champ s c a l a i r e , dans un m i l i e u d i s s i p a t i f , a n i s o t r o p i q u e , s t r a t i f i é p l a n , e s t e t e n d u e B un champ c5lectromagn6tique dans un m i l i e u a n a l o g u e . A b s t r a c t - The method o f f u n c t i o n a l e q u a t i o n s , which was developed p r e - v i o u s l y t o c o n s t r u c t t h e dynamic Green f u n c t i o n o f a q u a s i - e l a s t i c f i e l d and o f a s c a l a r f i e l d , i n a d i s s i p a t i v e p l a n e - s t r a t i f i e d a n i s o t r o p i c mediun, i s e x t e n d e d t o an e l e c t r o m a g n e t i c f i e l d i n an analogous medium.

I n t h i s l e t t e r , we show t h a t , i n t h e a b s e n c e o f an e x t e r n a l s t a t i c magnetic f i e l d , t h e Green f u n c t i o n problem f o r an e l e c t r o m a g n e t i c f i e l d i n a d i s p e r s i v e ( d i s s i p a t i v e ) p l a n e - s t r a t i f i e d a n i s o t r o p i c medium i s reduced t o t h e same c h a i n l i k e s e t o f f u n c t i m a l e q u a t i o n s o f t h e Dyson t y p e , which was e s t a b l i s h e d and s o l v e d p r e v i o u s l y /1-3/ f o r t h e dynamic Green f u n c t i o n o f a q u a s i - e l a s t i c and o f a s c a l a r f i e l d , i n a s i m i l a r medium.

I n t h e w - r e p r e s e n t a t i o n , Maxwell's e q u a t i o n s f o r t h e m a c r o s c o p i c e l e c t r o m a g n e t i c f i e l d w h i c h - i s produced i n any medium b y t h e e x t r a n e o u s s o u r c ë s o f t h e c h a r g e d e n s i t y

- (?;CO) and o f t h e c u r r e n t d e n s i t y

3,;

⁼^Tex(?;w) can b e w r i t t e n , u s i n g Pex - Pex

Gaussian u n i t s ,

i j k j k -1 k k

e V B = c [ - i w ~ ~ + 4 1 r ( ~ ~ ~ + ~ ~ ~ ) ] / a / , V B =O / b / , r 3 ,

i j k j k k k ^{i l )}

e

v

E = i c - ' w ~ ~ / V E = 4 n ( p . +P 1 / d / ,

+ +, l3 el

,

where E = E(r;w) i s t h e macroscopic e l e c t r i c f i e l d s t r e n g t h , B = B(r;w) i s t h e macro- s c o p i c maggetic f i e l d s t r e n g t h ( t r a d i t i o n a l l y c a l l e d t h e m a g n e t i c i n d u c t i o n ) ;

pin = p . (r;w) i s t h e m a c r o s c o p i c d e n s i t y o f i n t r i n s i c c h a r g e s i n t h e medium;

+ l n +

fin = 3 . (r;w) i s t h e macroscopic d e n s i t y v e c t o r o f i n t r i n s i c c u r r e n t s i n t h e medium;

i n i j k .

e 1 s t h e c o m p l e t e l y a n t i s y m m e t r i c u n i t p s e u d o t e n s o r ; L a t i n s u p e r s c r i p t s d e n o t e corn p o n e n t s o f a v e c t o r o r o f a t e n s o r i n an 2 r b i t r a r y C a r t e s i a n c o o r d i n a t e system, which i s a t r e s t r e l a t i v e t o t h e medium; r = ( r f , r Z , r 3 ) = ( ~ , ~ , z ) i s t h e p o s i t i o n v e c t o r o f a v a r i a b l e s p a c e p o i n t w i t h r e s p e c t t o t h a t c o o r d i n a t e s y s t e m ; w i s a v a r i a b l e f r e q u e n c y ; c i s t h e s p e e d o f l i g h t i n vacuum. Each L a t i n s u p e r s c r i p t assumes t h e v a l u e s 1 , 2 , 3 ( o r x , y , z ) . C o r r e s p o n d i n g l y , a r e p e a t e d L a t i n s u p e r s ~ r i p t i m p l i e s a sum- mation o v e r tha: s u p e r s c r i p t from 1 t o 3. The t - r e p r e s e n t a t i o n $ ( r , t ) and t h e w-re- p r e s e n t a t i o n $ ( r ; w ) , o f a q u a n t i t y $ a r e r e l a t e d by

where t i s t h e t i m e a c c o r d i n g t o a c l o c k which i s a t r e s t r e l a t i v e t o t h e medium. We suppose t h a t t h e e x t r a t i e o u s c h a r g e s and c u r r e n t s do n o t b e l o n g t o t h e g i v e n medi%m, i . e . t h e $ a r e b r o u g h t i n from o u t s i d e . A c c o r d i n g l y , we s y p o s e t h a t , f i r s t l y , p e x ( r ; w ) and j,,(r;w) t a r e given (known, p r e s c r i b e d ) f u n c t i o n s o f r and w, and t h a t , s e c o n d l y ,

-S.

e a c h p a i r p and

Tex,

^and^p. ^{and Y.} s a t i s f i e s an i n d e p e n d e n t c o n t i n u i t y

e x l n l n

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984544

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C5-306 J O U R N A L DE PHYSIQUE

e q u a t i o n , i . e .

which a r e t h e c o n s e r v a t i o n laws f o r t h e extraneous and f o r t h e i n t r i n s i c charges, r e - s p e c t i v e l y . Equation (3a) i s p o s t u l a t e d . Equation (3b) can be derived by applying t h e o p e r a t o r - i w - l v i t o both s i d e s of eq. ( l a ) and by using eqs. (Id) and (3a). The spec- t r a l composition of t h e extraneous c u r r e n t s i s supposed t o be such t h a t

Equation (4b) follows from e q . ( 4 a ) , i f n o n - l i n e a r e f f e c t s f o r t h e e l e c t r o m a g n e t i c f i e l d i n q u e s t i o n a r e n e g l i g i b l e .

The m a t e r i a l e q u a t i o n f o r j! i s supposed t o have t h e form i n

where ⁺ i j ⁺ . + ' +

j p ( r ; w ) = a ( r ; w ) ~ ' ( r ; w ) a ~ ~ ( ; ; w ) = ~ ~ j ( ; ; w ) B ' ( r ; w ) / b / ; ( 6 )

"

t t

J i s t h e d e n s i t y v e c t o r o f t h e i n t r i c s i c charge c u r r e n t ; j i s t h e d e n s i t y v e c t o r of t h e i n t r i n s i c c h a r g e l e s s c u r r e n t ; M i s , f h e magnetic momenT of t h e medium p e r u n i t volume;

dJ

i s t h e c o n d u c t i v i t y t e n s o r ;

xlJ

i s t h e magnetic s u s c e p t i b i l i t y t e n s o r ; a l 1 t h e above q u a n t i t i e s . a r e taken i n t h e w - r e p r e s e n s a t i o n , i n accordance with eqs,(2).

The f a c t t h a t alJ and a r e supposed t o depend on r , s i g n i f i e s t h a t , i n g e n e r a l , t h e 3edium under z o n s i d e r a t i o n can be s p a t i a l l y inhomogeneous. Also, we suppose t h a t 0'1 (r;,) and ( r ; w ) , when regarded a s f u n c t i o n ~ o f t h e complex v a r i a b l e w , have+no p o l e s i n t h e upper semirp13ne. Sybs$ituting o l J (r;w) and

xlJ

(r;w) i n t u r n s f o r q(r;w) i n eq. ( 2 a ) , we o b t a i n a l J ( r , t ) = x l J ( r , t ) = O f o r t<O. As a consequence, i n t h e t - r e p r e - s e n t a t i o n eqs

.

( 6 ) become

Thus, e q s . (6a) and (6b) s u b j e c t t o 3he ab$ve assumptions a r e t h e most g e n e r a l l i n e a r cause-and-effect r e l a t i o n s between E and J e ' and between

3

and & r e s p e c t i v e l y , which allow f o r t h e frequency d i s p e r s i o n of t h e medium, b u t which l e a v e o u t of account a p o s s i b l e space d i s p e r s i o n o f t h e medium. A macroscopic c h a r g e l e s s c u r r e n t i s a s s o c i - a t e d w i t h t h e corresponding microscopic s p i n c u r r e n t , and i s , hence, e s s e n t i a l o n l y i f t h e medium has a magnetic s t r u c t u r e , i . e . i s e i t h e r a ferromagnet o r an a n t i f e r r o - magnet. I f t h e mediu? has no magnetic s t r u c t u r e , we can s e t xlJ=O, thus n e g l e c t i n g

a s compared t o

1

f e

1 .

S u b s t i t u t i o n o f e q s . ( 5 ) i n t o eq. ( l a ) , and o f e q . (3b) i n t o eq. (Id) y i e l d s

where, by d e f i n i t i o n ,

+ +

D i s the v e c t o r of e l e c t r i c induction; H i s t h e v e c t o r of magnetic i n d u c t i o n ( t r a d i - t i o n a l l y c a l l e d t h e magnetic f i e l d s t r e n g t h ) . I n view of e q s . ( 6 ) , e q s . ( 8 ) become

i j ⁺ i j ^-t

E (r;w) i s the d i e l e c t r i c p e r 3 e a b i l i t y t e n s o r ; (r;w) i s t h e inveyçe+of t h e mag- n e f i s p e r m e a b i l i t y t e n s o r u l J ( r ; w ) . Denoting, a l s o , t h e i n v e r s e o f ~ l J ( r ; w ) by

<'J (r;w)

,

we have, by d e f i n i t i o n ,

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A ~ q o s d i n g t o t h e d e f i n i t i o n of o i j (r;w) and of X i j (?;CO), t h e q u a n t i t i e s c i j (;;w) * and

$1 (r;w) have no p o l e s i n t h e upper semi-glane o f t h e complex v a r i a b l e w. Note t h a t u s u a l l y t h e t e n s o r plJ (,;a), and n o t rllJ (r;w)

,

i s supposed t o be r e g u l a r i n t h e up- p e r semi-plane of w .

Multiplying eq. ( l c ) by ,iml, and t h e n making use of eq. (9b), we have Hm=-icw-l mn nkQ k Q

ri e V E . (12)

I f we i n s e r t e q s . (9a) and (12) i n t o eq. ( 7 a ) , we o b t a i n

i k * k + i +

K (r;-iV,w)E (r;w)=-4~riwj ( r ; w ) , ex (13)

i k

*

2 i k +

( r ; - i ~ , w ) = w E (r;w) +vjzijk' (?;w)~', (14)

Z i j k Q * 2 i j m m * nkQ

(r;w)=c e

n

[r;w)e

.

⁽¹⁵⁾

A s i m i l a r e q u a t i o n can be d e r i v e d f o r

3 .

To t h i s end, we i n s e r t eq. (Sa) i n t o eq. ( 7 a ) , and we m u l t i p l y t h e e q u a t i o n s o obtained by t h e t e n s o r Cml, e q . ( l l a ) . This g i v e s

)

.

e x (16)

i j ^'

Applying t h e o p e r a t o r e

9'

t o both s i d e s of eq. (16), and then making use o f eqs. ( 1 4 (9b), and ( l l b ) , we o b t a i n

i k * k * i *

K m (r;-iV,w)H (r;w)=@ ( r ; w ) , (17)

2 i k * j i j k Q * Q

~?(;;-iV,w)=w (r;w)+V Lm (r;w)V

,

(18)

Z;jkR(= ,c2eijmSmn(;r;w) .nkQ, (19)

. .

Q~ (?;w)=-4rrce49j~""'(?;u) jex(?;w). . .

- -

⁽²⁰⁾

II

We d e f i n e t h e e l e c t r o m a g n e t i c Green f u n c t i o n E .

. .

. O(?,?o;u) of t h e E-type and t h e e l e c -

l L 0

*

tromagnetic Green f u n c t i o n H (r,ro;w) o f t h e H-type a s t h e s o l u t i o n s o f two independent e q u a t i o n s :

i n t h e whole i n f i n i t e s p a c e , where

*

r and i a r e parameters ( i = 1 , 2 , 3 ) . Since e q s . ( l 3 ) and (17) a r e l i n e a r , t h e i r s o l u t i o n s can be w r i t t e n

The v e c t o r s D~ and B~ can t h e n be found by e q s . (9) and ( l l b ) . S u b s t i t u t i o n of eq. (23) inTo eq. (12), and of eq. (24) i n t o eq. (16) enables us t o e x p r e s s H' i n terms of

EiiO, and E~ i n terms o f H ilo

.

Hence, o n l y one o f t h e two Green f u n c t i o n s i s a c t t a l l y r e q u i r e d .

We r e w r i t e e q s . (21) and (22) a s one e q u a t i o n o f t h e form k i. O + + +

*

ⁱⁱ

Kik (:;-iv,u)~ ( r , r o ;u) =6 ( r - r ) 6 O ,

where ^O

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JOURNAL DE PHYSIQUE

Z i j k l ( r ; u ) = c e -+ 2 i j m mn B ( r ; u ) e ⁺ nkQ

.

, i i ii

oZE O i f .ik=,ik i.k i k

,

B ^=ri / a / ; ii i i

oZH O i f .ik- i k 8 i k , . i k

-u

³ ^{/ b / .}

I n t h e a b s e n c e o f a s t a t i c . e t e r n a l m a p n e t i c f i e l d , t h e t e n s o r s ûik and Bik a r e sym- r n e t r i c . S i n c e t h e t e n s o r e l i E i s a n t i s y m m e t r i c w i t h r e s p e c t t o t h e p e r m u t a t i o n o f any two i n d i c e s , we have

Zi j k ~ - ~ k L i - j - -- Z j i k L = - Z i jkk (29)

We o b s e r v e t h a t e q . ( 2 5 ) s u b j e c t r ~ e q . ( 2 6 ) tu r n s i n t o t h e e q u a t i o n f o r t h e a c o u s t i c

Ill,

'

^-*

v i b r a t i o n f i e l d Green f u n c t i o n D ( r , r ; a ) o f a s p z t i a l l y inhomogeneous e l a s t i c o - v i s c o u s a n i s o t r h p i c medium i i i t h t h e mas: d e n s i t y p ( r ) and w i t h t h e complex e l a s t i c i t y t e n s o r cijkR(?;w) ( s e e r e f s . / l , 2 / ) i f i n e q . ( 2 6 ) we f o r m a l l y s e t

ii i i

,i jkk-,ijkQ +

G O = D O , a i k = p ( ? ) ô i k , - ( r ; u ) . (30)

In t h i s c a s e , i n s t e a d o f e q s . ( 3 2 ) , we have , i j k Q = , k l i j = , j i k Q = , i j l k

(31) Suppose now t h a t t h e medium u n d e r c o n s i d e r a t i o n c o n s i s t s o f n + l homogeneous a n i s o - t r o p i c p l a n e - p a r a l l e l l a y e r s o f a r b i t r a r y t h i c k n e s s e s i n c o q t a c t . .W a l i g n t h e a x i s x w i t h a normal t o t h e i n t e r f a c e s , and d e n o t e by alk (w) , B F ( a ) , Z ; J ~ ' . ( ~ ) t h e c o r r e - s p o n d i n g m a t e r i a l c h a r a c t e r i s t i c s o f t h e i n d i v i d u a y l a y c r s ; p = O , l . .

.

, n . In t h i s c a s e , we mav w r i t e

L L 3 -k

t h e v a r i a b l e a b e i n g o m i t t e d . The q w n t i t y G O ( x , x o ; k l , ) i s c a l l e d t h e (x,k ) - r e p r e - .--O

s e n t a t i o n , o r i n words, t h e mixe c o o r d i n a t e - p r g g a t i o v c c t o r - r e p r e s e n t a t i o n , o f t h e t o t a l Grccn f u n c t i o n . I f ~ i e = o ~ k ( u ) and Z1~"=Z:jk';u) i n t h e whoie i n f i n i t e

P P

l10

' '

s p a c e , t n e c o r r c s p o n d i n g s o l i i t i o n o f e q . ( 2 5 ) i s d c n o t a d by G,, ( r - r ) and i s c a l l e d t!!e - d a r d Green f u n c t i o n o f t h e 11th k i n d . The ( x , k l l ) -r e p r e s e n t a ? i o n

G l l ~ -f

(x-xo;kll) o f t h i s f u n c t i o n i s t h e s o l u t i o n o f t h c e q u a t i o n

u

2 i k i j k l k i o i i

[m U, (u)+zIi (a) a:2:]Gu ( x ) = S ( x ) 6 O , a i = 6 j i o + i k j ax 11 , - a < x < a . (33) From h e r e on t h e v a r i a b l e

2

o f e v e r y f u n c t i o n i n t h e ( x , g ) - r e p r e s e n t a t i o n w i l l he

o m i t t e d . II !I

I n r c f s . / l , 2 / , we showed t h a t t h e a c o u s t i c v i b r a t i o n f i e l d Green f u n c t i o n i n t h e (x,!,, ) - r e p r e s e n t a t i o n , o f t h e p l a n e - s t r a t i f i e d medium a s m e n t i o n e d , s a t i s f i e s a c h a i n l i k e s e t o f n + l 3 x 3 - m a t r i x f u n c t i o n a l e q u û t i o n s o f t h e Byson t y p e , which i n - c l u d e b a t h t h e e q u a t i o n o f motion and t h c c o r r e s p o n d i n f i boundary c o n d i t i o n s a t e a c h i n t e r f a c e . I n d e r i v i n g t h a t s e t o f e q u a t i o n s , i t was e s s e n t i a l t h a t t h e e l a s t i c i t y t e n s o r was i n v a r i a n t u n d e r t h e p e r m u t a t i o n i j 4 , k Q ( s e e e q s . ( 3 0 ) and ( 3 1 ) ) . S i n c e t h e m a t e r i a l t e n s o r z l l k R a s d e f i n e d hy e q . ( 2 7 ) p o s s e s s e s t h e same symmetry p r o p e r t y ( s e e e q s . ( 2 9 ) ) . t h e s e t o f f u n c t i o n a l e q u a t i o n s o f r e f s . / l , 2 / remain.~, v a l i d a l s o i n t h e c a s e o f an e l e c t r o m a g n e t i c f i e l d . Onc s h o u l d o n l y s i l b s t i t i i t e C 110 ( x , x O ) f o r

D l l ~ "0 110

( x , x o ) , and (; _Ii (x) f o r Il ( x ) (u=O, 1 , .

. .

^{, n )}

.

Li i i o

Thus, i\re have t h e f o l l o w i n r i s e t o f e q u a t i o n s f o r G [ x . x 1 :

(6)

ii (u-1) iio ( ~ + l ) T i o ii s,, ( 4 G O (x, xo) + B ~ ( ~ - d ~ - ~ ,xo) - B

1-i (x-d x ) = s (x )G O (x-xo) ,

1 - i Y o 1 - i O 1 - i

x=d i s t h e ç e p a r a t i o n p l a n e between t h e u t h and t h e ( p + l ) t h l a y e r . The z e r o t h (p=b) and t h e l a s t (p=n) l a y e r a r e supposed t o b e s e m i - i n f i n i t e spaceç x<do and

~ > d , - ~ , r e s p e c t i v e l y (d-l=-m, cin=-). Equations (38) a r e t h e usual boundary condi- t i o n s of c o n t i n u i t y f o r t h e t a n g e n t i a l components of

8

and

2

( o r correspondingly

8

and Ï$+ i n t h e c a s e , when

3

i s t h e Green f u n c t i o n of t h e E-type ( o r correspondingly, when H i s t h e Green f u n c t i o n o f - t h e H-type). A t t h e same time, i t follows from

i l

eqs. (271, ( 3 9 ) , and (40) t h a t P1lO(r,~o)=O and T (x)=O. I n view o f t h e l a s t e q p a t i o n , e q . (37) does n o t involve ^Fi

G1lo

(dv+O,x ) . These p e c u l i a r i t i e s of e q s . (34)-(37) r e f l e c t t h e f a c t t h a t t h e e l e c - t r o m a g n e t i c O f i e l d i s t r a n s v e r s e . I n t h i ç connection, we may n o t i c e t h a t eqs. (34)-

(37) a r e , i n a s e n s e , complementary t o t h e corresponding e q u a t i o n s f o r t h e a c o u ç t i c v i b r a t i o n f i e l d Green f u n c t i o n i n a s t r a t i f i e d i d e a l f l u i d . The l a t t e r f i e l d i ç g u r e l y l o n g i t u d i n a l . Equations (34) (36) can b e çolved l a r g e l y by t h e same methods, which were suggested i n refç.11-3/. The Green f u n c t i o n t h e o r y o f e l e c t r o m a g n e t i c f i e l d s i n s t r a t i f i e d media w i l l be d i s c u s s e d i n d e t a i l i n f u r t h e r p u b l i c a t i o n s . This t h e o r y w i l l , a l s o , i n c l u d e t h e c a s e , when a s t r a t i f i e d medium i ç i n a s t a t i c external magnetic f i e l d .

References

1 . IOSILEVSKII YA. A . , Ann. Phys. (N.Y.),

107

(1976) 360.

2 . IOSILEVSKII YA. A . , Phys. Rev. (1979) 1473.

3 . IOSILEVSKII Y A . A . , Phys. Rev. (1979) 856.