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z ~ ~ rot (Hm e; m z ) = j!" Eme; mz ~ ~ rot Hm = j!" Emz :

z , X Ez = [Am(z)Emz + A;m(z)Emz ] ; Jz:

j!" m , :

h i X ~ ~ ~ ~ E(x y z) = Am(z) Em + A;m(z) E;m ; Jz:

j!" m h i X ~ ~ ~ H(x y z) = Am(z) Hm + A;m(z) H;m :

m 6.

6.1. , , . , , , .

, . , .

, .

6.2. .

~ ~ @H @E ~ ~ rot E = ; rot H = " :

@t @t ~ ~ E H ( ) ~ ~ ~ E = 0 Et = n ~ ~ H = 0 Hn = n n { . j!t ~ ~ E = E(x y z) e j!t ~ ~ H = H(x y z) e ! { .

, ~ ~ ~ ~ rot E = ; j! H rot H = j!"E:

~ ~ ~ H , div E =0, E ~ ~ E + k2E = k2 = !2", { , ~ E. ~ ~ rot rot E ; k2E = 0:

. km k2, .

. ~ ~ Em(x y z) Hm(x y z) .

:

~ ~ rot rot H ; k2H = 0:

, , , . , .

:

~ ~ ~ ~ ~ ~ div (Hm rot Hm) = rot Hm rot Hm ; Hm rot rot Hm = ~ ~ = jrot Hmj2 ; km jHmj2:

Z Z I ~ ~ ~ ~ km jHmj2 dV = jrot Hmj2 dV ; (Hm rot Hm)~ ndS:

V V S R ~ jrot Hmj2 dV V km = R :

~ jHmj2 dV V , . km = !2 ", !2 > 0 ! { . , . :

k !m = ckm = = :

pm m " km km . .

2 , km = kn Z Z ~ ~ ~ ~ Em En dV = 0 Hm Hn dV = V V .

, , ~ ~ Em En:

~ ~ rot rot Em ; kmEm = ~ ~ rot rot En ; knEn = 0:

~ ~ ~ ~ ~ ~ ~ ~ div (En rot Em) ; div (Em rot En) = rot Em rot En ; En rot rot Em ;

~ ~ ~ ~ ~ ~ ~ ~ ; rot En rot Em + Em rot rot En = Em rot rot En ; En rot rot Em:

~ ~ , Em En, 2 ~ ~ ~ ~ ~ ~ div (En rot Em) ; div (Em rot En) = (kn ; km)Em En:

, Z I 2 ~ ~ ~ ~ ~ ~ (kn ; km) Em En dV = (En rot Em ; Em rot En)~ dS:

n V S :

Z 2 ~ ~ (kn ; km) Em En dV = V 2 km = km, Z ~ ~ Em En dV = 0 :

V ~ ~ . Em En , , , , .

~ ~ , Em Hm { , ~ , . Em ~ Hm { , (r) (i (r) ( i ~ ~ ~ ~ ~ ~ Em = Em + Em) Hm = Hm + Hm):

, ~ Em ~ Hm,..

(r) (i) (r) (i) ~ ~ Emt = 0 Emt = 0 Hmn = 0 Hmn = 0 S:

~ ~ Em Hm , (r) (i (i (r) ~ ~ ~ ~ rot Em = ; j!m Hm) rot Em) = ; j!m Hm (i (r) (r) (i ~ ~ ~ ~ rot Hm) = j!m" Em rot Hm = j!m" Em):

(r) (i ~ ~ , Em Hm) , , ~ ~ Em Hm .

(i (r) ~ ~ Em) Hm. , , (r) (i ~ ~ , Em Hm). , . , 90o .

. , , , . { .

km:

Z Z ~ ~ km jHmj2 dV = jrot Hmj2 dV:

V V ~ ~ rot Hm = j!m"Em k2 = !m ", Z Z jHmj2 "jEmjdV = dV 2 V V.. . . , .

6.3. , , . , .

, , , . , , Ex = 0 Ey = 0. , .

, ( ). , , ( =2) (..).

, , , .

j z , , z e, sin z cos z. , .

, , .

( ) L Hz = (k2 ; ) e;j z:

-:

z ( z=z =0) Hz = (k2 ; ) (e;j z ; ej z ) = ; j(k2 ; ) sin z:

, @ @ Hx = ; j e;j z Hy = ; j e;j z :

@x @y 1 @ @ j z Hx = ; j (e;j z + e ) = ; j cos z 2 @x @x 1 @ @ j z Hy = ; j (e;j z + e ) = ; j cos z:

2 @y @y @ @ Ex = ; j! e;j z Ey = j! e;j z :

@y @x 1 @ @ j z Ex = ; j! (e;j z ; e ) = ; ! sin z 2 @y @y 1 @ @ j z Ey = j! (e;j z ; e ) = ; ! sin z:

2 @x @x , sin L = 0 L = l..

= l L = l l = 1 2 : : : :

, , l 2 0 2 2 = k2 ; gmn2 L2 = l2 = :

L , l 2 kmnl = gmn2 + :

L , l = 0 , , =0 ( L = 0).

:

Ez = (k2 ; ) cos z @ Ex = ; sin z @x @ Ey = ; sin z @y @ Hx = j!" cos z @y @ Hy = ; j!" cos z:

@x z = L sin L = 0,.. L = l, l 2 kmnl = gmn + :

L l = 0 (.. = 0) . Ex Ey. , . 2 kmn0 = gmn.. .

6.4. , 2 n m gnm = + a b 2 n m l knml = + + :

a b L , , , z.

(H) (E) gnm = gnm , ( ). , , .

, , . .

:

t0 2 l (H) mn kmnl 2 = + a L tmn 2 l (E) kmnl 2 = + :

a L , , m = 0 ( ). ,.. m =0.

E010 H01n.

, (E) 2: . t01 2:4. k010, a 2 2:4 , = 2:6 a = 1:3 d, d 0:75 = a ( 0.764 ), d { .

(..) , , .

E010 :

Ez = J0(kr) r 1 dEz k " H' = = J0(kr) = j J1 (kr):

j! dr j! H01 2 t0 2 3:(H k011) 2 = + = + :

a L a L , . .

, , . . , .

:

L = n = 2n 2 n = 1 2 3 : : :

L L = (2n + 1) n = 0 1 2 : : :

L L = 2n n = 1 2 3 : : :

L . , . , . , { , . . , . .

6.5. , . , , : , , , .

6.5.1. . , , { . , .

@H @t . @E ", @t ~ ~ rot H = J:

, , . , . .

6.5.2. : , \ { " .

r S , , d S S (..). 1 d, { 2a S. dS "0E2 "0E2 a2"0 U2 CU WE = V = a2d = = 2 2 d 2 a2" C =, U = dE.

d Z HWH = dV VH I H =, dV =2 r dS. , 2 r Z Z Z H2 I2 I2 dS LI0 0 W = 2 r dS = 2 r dS = = 2 2 4 r2 4 r SH SH SH R dS L =.

4 r SH C L:

v v Z Z u u p a2"0 dS 1 dS u 0 u = 2 c LC = 2 c = 2 a :

t t d 2 r 2d r SH SH Z dS S = :

r r SH L C, s S = 2 a :

2dr S 2dr.. d S ( dr S).

{ \ { ", (..).

. h a ( ), . l I H = :

h d I a = LI = HS = a2 = I:

0 0 h h aL = :

h () lh C = "0 :

d :

r r a2 lh l = 2 c "0 = 2 a :

h d d 6.5.3. , , . \" . . , .

. { , . 1 2 (.

.).

, , i..

Z1 i1 ZWE1 + WE2 = WH1 + WH2:

ii (WH1 ; WE1 ) + (WH2 ; WE2) = 0:

:

1 WH1 ; WE1 = Z1 jI1j2j! 1 WH2 ; WE2 = Z2 jI2j2:

2j! , Z1 jI1j2 + Z2 jI2j2 = Z1 Z2 { . , jI1j = jI2j . ( ):

Z1 + Z2 = 0:

Y1 + Y2 = 0:

Z1 Z2 , . . , .

6.5.4. , 11111111111 Z0. C . (..).

l Z = jZ0 tg kl ! k =, c ! Z = jZ0 tg l:

c 1 ! + jZ0 tg l = 0:

j!C c ! ctg l = Z0!C:

c ! l = x, c ctg x = Ax:

Z0Cc C A = =, C1 { .

l C1 l (.

.).

x1 x2 : : :, , = l:

n xn , , C x , x1 x2 x .

.

Z Z 01 ll l1 l !l1 !ljZ01 tg + jZ02 tg = c c Z02 !l2 !ltg = ; tg :

Z01 c c !l= x c Z02 ltg x = ; tg x:

Z01 l (..).

x, .

, . . xx x, (..).

() 1 + + j!C = 0:

!l1 !ljZ01 tg jZ02 tg c c Z C Z 01 , 0000 1111 E01, 0000 1111 h ,.. 0000 1111 0000 1111 (..).

0000 1111 1 Ez = J0(k1 r).

2a 2b 1 @Ez k1 0 kH' = = J0(k1 r) = ; J1 (k1 r):

j! @r j! 1 j! r = a kI = 2 a H' = ; 2 a J1(k1 a):

j! 1 Ez h hj! J0(k1a) Z1 = = ; :

I 2 a k1 J1 (k1a) Ez = AJ0(k2r) + BN0(k2r):

A=B AJ0(k2b) + BN0(k2b) = B J0(k2b) = ; :

A N0(k2b) Ez, J0(k2r) N0(k2b) ; J0(k2b) N0(k2r) Ez = A :

N0(k2b) 0 1 @Ez k2 J0(k2r) N0(k2b) ; J0(k2b) N0(k2r) H' = = A :

j! @r j! 2 N0(k2b) 2 hEz hj! J0(k2a) N0(k2b) ; J0(k2b) N0(k2a) Z2 = = :

0 2 a H' r=a 2 a k2 J0(k2a) N0(k2b) ; J0(k2b) N0(k2a) Z1 + Z2 = Z1 Z2, r r J0(k2r) N0(k2b) ; J0(k2b) N0(k2r) J0(k1 a) 2 ; = 0:

0 "2 J0(k2r) N0(k2b) ; J0(k2b) N0(k2r) "1 J1 (k1 a) , . , , , .

6.5.5. . R jrot Hj2 dV R k2 = jHj2 dV .

~ H, , ~ H, ~ ~ rot rot H ; k2H = ~ ~ rot H = 0. , n ~ , H .

. Z Z ~ k2 jHj2 dV = jrot Hj2 dV:

V V ~ ~ H, H . , ( ) . Z Z Z ~ ~ ~ ~ k2 jHj2 dV + 2k2 H HdV = 2 rot H rot HdV:

V V V ~ ~ ~ ~ ~ ~ div ( H rot H) = rot H rot H ; H rot rot H ~ ~ ~ ~ ~ ~ rot H rot H = H rot rot H + div ( H rot H):

, Z Z Z ~ ~ ~ ~ ~ ~ k2 jHj2 dV = 2 (rot rot H ; k2H) HdV + div ( H rot H) dV:

V V V , Z Z I ~ ~ ~ ~ ~ ~ k2 jHj2 dV = 2 (rot rot H ; k2H) HdV ; (~ rot H) HdS:

n V V S , k2 =0 ~ H :

~ ~ 1)rot rot H ; k2H = 0 V ~ 2)~ rot H = 0 S :

n , .

.

~ , k2 H, .

, , .

{ . . , .

, , , V V1 V2 (.

.). , V k1 :

V1 V2 R ~ jrot H(2)j2 dV Vk2 = R ~ jH(2)j2 dV V~ H(2) { . ~ ~ k2, H(2) H(1) { . :

R R R ~ ~ ~ jrot H(1)j2 dV jrot H(1)j2 dV ; jrot H(1)j2 dV V1 V Vk2 = = R R R ~ ~ ~ jH(1)j2 dV jH(1)j2 dV ; jH(1)j2 dV V1 V V8 R R ~ ~ jH(1)j2 dV jrot H(1)j2 dV > > < = V2 V= k1 1 + R ; R = > ~ ~ > jH(1)j2 dV jrot H(1)j2 dV :

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