Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://shg.phys.msu.ru/educat/nopc/nopc.pdf Äàòà èçìåíåíèÿ: Wed Mar 12 16:11:39 2008 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 19:48:24 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 27

1



I I I
1. 1.1. . . . . . . . . . . . 1.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. . . . . . . . . . . . . . . . . 2.1. . . . . . . . . . . . . . . . . . 2.2. . . . . . . . . . . . . . . . . 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...... ...... ...... ...... ...... ...... ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6 6 7 9 9 9 11 12 3

I I I
14

1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2. 16 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1. 16 2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.1. . . . . . . . 17 3.2. . . . . 18

IV
1. . . . . . . . . . . . . . . . . . . . . 1.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 19 19 21


2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. . . . . . . . . . . . . . . . . . 2.3. . . . . . . . . . . . . . . 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. (111) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. 3.3. . . . . . . . . . . . . . . . . . 3.4. [211]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. [110]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. (111). . . . . 4. . . . . . . . . . . . . . . . . . . . . . . . .

2 23 23 25 26 26 26 27 28 29 30 30 31

V
1. 1.1. 1.2. 1.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. . . . . . . . . . . . . . . . . . . . . . . . 2.2. . . . . . . . . . . . . . . . . . . 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. . . . . . . . . . . . . . . . . . . . . . 4. - . . . . . . . . . . . . . . . 4.1. . . . . . . . . . . . . . . . . 4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 32 32 33 33 33 33 34 35 35 37 37 39 41




I
. - , , , . - , , . . ( ) . ("") . - . . 1. 1.1. . , . . . , . . . . , . . . . . . . () () . ("") . . . "". 1.2. . . . . . . . . . . . . . . . 2. - 2.1. -


I.

4

. . . . 2.2. . . . . , , . . . 2.3. - . . . . . , . - . . . 3. 3.1. . . . , , . . , , . . , , . . 3.2. . . . . 3.3. . . . . . . . 3.4. .


I.

5

. . () . .


I I
1. 1.1. . . 1. ( ) ¨ e, , :
2

-

2m

+ U (r) E (r) = E E (r),

â

1 (r)

â H (r) =

2 H (r). c2

(1)

2. (r, t), - , H(r, t) E(r, t). 3. (r, t) E (r), E . H(r, t) () H (r), - : (r, t) =
E

cE E (r)e

i

E

t

,

H(r, t) =


c H (r)e

i t

.

(2)

4. U (r) (r), ¨ e , , R , : U (r) = U (r + R), (r) = (r + R). (3)

5. k (r) = uk (r)eik·r , Hk (r) = uk (r)eik·r , (4) uk uk . 6. k , . 7. E , ¨ e, ( ) - - k (4) E (k) (k). 8. , , , . (r) , .


II. 1.2.

7

( ) . , . , . , () k |k| = 2 /, - k - , , : = ck , (5) c - . , n, (5) v = c/n = /n. . L . , z , L, E (z + nL) = E (z ), n = 0, ±1..., , E (z = 0) = E (z = L) = 0. E (z )


E (z ) =
m=-

Em e

ikm z

,

(6)

km , 2 m, m = 0, ±1, ±2, .... (7) km = L - Em , :
L/2

dz e
L/2

-ikn z

L/2

E (z ) =
m=-

Em

dz e-

ikn z ikm z

e

.

(8)

L/2

(8) km - kn :
L/2

f (km - kn )

dz ei(k

m

- kn ) z

L/2

= Lsinc ( (m - n)) = Lmn .

(9)

f (km - kn ) m = n. (9) (8), :


Em
m=-

mn

En =

1 L

L/2 L/2

dz e-

ikn z

E (z ).

(10)

, Ex Ey , E(r) =
lmn

El

mn

ei

2 (lx+my +nz )/L

, l, m, n = 0, ±1, ±2, ....

(11)

k k= 2 {l, m, n}, L (12)


II. (11) E(r, t) =
k

8

Ek (t)e

ik·r

,

(13)

k Ek (r, t) = 1 V Ek (t)e
-ik·r

dr.

(14)

V

E Ek t (13,14) exp(-ik t), k = ck . (13) E , , E(r, t) Ek - . - E E-k = Ek . k k - . (12) (k -), 2 /L. , , (2 )3 /V . L (13) : E(r, t) = V (2 )3


E(k, t)e
-

ik·r

dk,

(15)

c k E(k, t) = 1 V


E(r, t)e
-

-ik·r

dr .

(16) dk

, m ... (L/2 )3 (6-9) -,
-

dz eikz = lim Lsinc(k L/2) = 2 (k ).
L

(17)

N , , , V () N ( ) = 2 â V (2 )3
k= /c

dk =
k=0

3V , 3 2 c3

(18)

"2" . (18) . , 0 , k = 0 k = /c. 4 4 3 W ( ) = k 3 = . 3 3c3 (19)

, , v = (2 )3 /V , , , N ( ) = 2 â 4 3 V 3V W ( ) =2· · 3 = 2 3. v 3c3 8 3 c (20)


II.

9

D( ) , D( ) = N ( )/ , 2V D( ) = 2 3 (21) c + d . , 2 . 1.3. d, , U= 4 2 2 2 |d| D( ). 3V (22)

P ( ), P = U / , 4|d|2 3 D( ). (23) P= 3 c3 , : 2¨ I = 3 |d|2 , (24) 3c , , : 4 4 2 U= |d| , (25) 3 c3 (22). , ( ), , . , () . . 2. 2.1. . (5) = ck , c - . (k) . . , E(z , t) 4 2 2 E 2E = . (z ) z 2 t2 (26)


II.

10

(z ) - a, (z + a) = (z ), 2 m -1 (z ) = z, (27) m exp i a m=- - -m = . , m ( ) , Ek (z , t) = uk (z ) exp (i(k z - k t)) , (28) uk , a, uk (z + a) = uk (z ). (29)

k Ek (z , t), k (26). Ek (z , t)


Ek (z , t) =
m=-

Em exp i k +

2 m a

z - ik t .

(30)

(27), -1 (z ) 0 + 1 exp i (26) 1 k + 2(m - 1) a
2

2 z a

+ -1 exp -i

2 z, a

(31)

Em-1 +

-1

k+

2(m + 1) a

2

Em+1
2

2 k 2m - 0 k + 2 c a

Em .

(32)

m = 0, (32) : E0 m = -1: E-1 c2 2 (k - 0 c2 (k -
2 2 a) 1/2

c2 2 (k - 0 c2 k 2 )

1 k -

2 a

2

E-1 + -1 k +

2 a

2

E1

,

(33)

)

1 k -

4 a

2

E-2 + -1 k 2 E0

.

(34)

k 0 ck |k | |k - 2 /a|, E0 E-1 - (30). , , (33) (34) E0 E-1 :
2 (k - 0 c2 k 2 )E0 - 1 c2 k -

2 a
2

2

E-1 = 0, E-1 = 0. (35)

-1 c2 k 2 E0 +

2 k - 0 c2 k -

2 a


II.

11

(35) ,
2 k - 0 c2 k --1 c2 k 2 2 -1 c2 k - 2a 2 k - 0 c2 k - 2a 2 2

.

(36)

|k | |k - 2 /a| : ± (k ) = c ac 0 ± 1 ± a 1 2 - 0 2 1 2 k- a
2

.

(37)

|k | = /a, c c 0 - 1 < < 0 + 1 a a (38)

(35) , .. , . , ± /2, , , , , , . 2.2. . vg vg = . k (39)

k /a v
g±

(k ) = ±

2ac 1

2 - 0

2 1 2

k-

, a

(40)

± . k /a vg 0. , , . , ( ). , vg+ vg- . , (37) |k | /a - (31), (z ) . , , , .


II. 3.

12

(5) ( ). , , , 2m /a, m - , . a [- /a, /a], . a1 a2 ( ), () a1 = (a1 , 0) a2 = (0, a2 ). b1 = 2 , 0 , b2 = a1 0, 2 a2 , (41)

, b1 b2 . k = (kx , ky ) kx [- /a1 , /a1 ] ky [- /a2 , /a2 ]. 2 /a1 2 /a2 . - = (0, 0), X = ( /a1 , 0) M = ( /a1 , /a2 ). , M , X - . a, : a 3a a1 = (a, 0) , a2 = , , (42) 22 b1 = 2 2 , - a 3a , b2 = 4 0, 3a . (43)

4 /3a. = (0, 0), K = (4 /3a, 0) M = ( /a, - / 3a). , K T , M - . , . : a1 = (a, 0, 0) , a2 = (0, a, 0) , a3 = (0, 0, a) , (44) a - . b1 = 2 , 0, 0 , b2 = a 0, 2 , 0, a , b3 = 0, 0, 2 a . (45)

2 /a. , = (0, 0, 0), X = (0, 0, /a) M = ( /a, /a, 0).


II.

13

R = ( /a, /a, /a), M . , X , X R - S , R M - T , R - , M - . : a1 = a/2 (0, 1, 1) , a2 = a/2 (1, 0, 1) , a3 = a/2 (1, 1, 0) , a - . b1 = 2 2 2 (-1, 1, 1) , b2 = (1, -1, 1) , b3 = (1, 1, -1) . a a a (47) (46)

, . (±1, 0, 0), (0, ±1, 0) (0, 0, ±1) , (±1, ±1, ±1) - . , , , X , , .


I I I
1. . . , .. . , . , , . . , - .. . 1.1. : â E(r, t) = - 1 B(r, t) , c t 1 D(r, t) â H(r, t) = , c t · D(r, t) = 0, · B(r, t) = 0, (48) (49) (50) (51)

E, H, D B. (51) j . B(r, t) = µH(r, t), D(r, t) = (r)H(r, t). (52) µ , (r) (r + ai ) = (r), (53)

ai , i = 1, 2, 3. (51) (52) 1 (r) â â( 1 (r) â E(r, t)) = - â H(r, t) 1 2 E(r, t) , c2 t2 1 2 H(r, t) =- 2 . c t2 (54) (55)


III.

15

(55) E(r, t) H(r, t). E(r, t) = E(r)eit , H(r, t) = H(r)eit . (56) (), E(r) H(r) - (55), . E H E E(r) H H(r) 1 (r) â( â E(r)) , (57) (58)

â

1 (r)

â H(r, t) .

(56), (55) 2 E(r), c2 2 H H(r) = 2 H(r). c E E(r) = (59) (60)

(r), (60) Ekn (r) = ukn (r)ei
k·r

, Hkn (r) = vkn (r)e

ik·r

,

(61)

k n. , ukn (r) vkn (r), xi , ai : ukn (r + ai ) = ukn (r), vkn (r + ai ) = vkn (r) (62) bi , i = 1, 2, 3, , : ai · bj = 2 ij , (63) ij - . ,
3

G=
j =1

lj bj ,

(64)

c lj . (r) , 1 = (r) (G)e
G iG·r

.

(65)

(r) , - (G) (-G) = (G) : (G) = 1 V 1 e (r)
-iG·r

dr,

(66)

V

V - () . , ukn (r) vkn (r), ,


III.

16

Ekn (r) Hkn (r). (61) : Ekn (r) =
G

Ekn (G)e

i(k+G)·r

, .

(67) (68)

Hkn (r) =
G

Hkn (G)e

i(k+G)·r

(55), (56) (68), , k + G. (planewave expansion method). (65) (68), (60) : -
G

(G - G )(k + G ) â (k + G ) â Ekn (G ) = (G - G )(k + G ) â (k + G ) â Hkn (G ) =
G

2 kn Ekn (G), c2 2 kn Hkn (G). c2

(69) (70)

-

(70) 2 (G - G ), , kn (k), .. . . , (70) G . N - (G - G ). N 103 . 3N , (70), k. , , , , , . - G . 1.2. 2. 2.1. , , . . , - . ra , a - . b . - r . 1 1 = + (r) b 1 1 - a b S (r), (71)


III. S (r) = 1, |r| ra . 0, |r| > ra

17

(72)

(71) 1/ (66) - (G), : (G) = (G) =
V0

1 1 (G) + b V0

1 1 - a b
iG·r

(G),

(73)

S (r)e-

dr.

(74)

(G) (r, , ) = 0 G. G = 0
r
a



(G) = 2
0

dr
0

dr2 sin e

-iGr cos

=

(75) (76)

= G = 0

4 (sin Gra - Gra cos Gra ) , G3 (G) =

3 4 ra . (77) 3 , :

f=

3 4 r a . 3V0

(78)

- : (0) = f f -1 + , a b 1 1 (G) = 3f - a b

sin Gra cos Gra - (Gra )3 (Gra )2

(79) .

, a b , ( ), . , .. , . , . , . , , . "" 2.2. . 2.3. 3. 3.1. , ra , a.


III.

18

a , b . G - r , . , (71) (72): 1 1 = + (r) b S2 (r) = 1, |r| ra . 0, |r| > ra (81) 1 1 - a b S2 (r), (80)

1/(r) G - (G) = 2 (G) =
0

1 1 (G) + b 0

1 1 - a b
-iG·r

2 (G),

(82)

S2 (r)e

dr.

(83)

0 (82) (83) - , , , 0 = a2 . (r, ) , G, 2 (G) :
ra 2

2 (G) =

dr
0 0

dre-iGr

cos

,

(84)

G = |G|. Jl (x) l


Jl (x)e
l=-

il sin

=e

ix sin

,

(85)

,
ra 2

2 (G) =

dr
0 0 ra

dr
l=-

Jl (Gr)eil(

-

2

)=

(86) (87)

= 2
0

rJ0 (Gr)dr =

2 ra J1 (Gra ), G

: (xJ1 (x)) = xJ0 (x). 2 a , , , f = r0 =
2 ra a2

, - 1 1 - a b f f -1 (0) = + . a b (G) = 2f J1 (Gra ) , G=0 Gra

(88)

3.2.


IV
1. 1.1. - E+ exp(i(k r - t)) k 0 0 0 , 0 , N , dj , , j nj = , j = 1...N . z , xz . . , , , , , z ( ): E (z , t) = j E+ exp[(+ik j E- exp[(-ik j
z ,j z ,j (z - zij ) + (ikx x - i t)]+ (z - zij ) + (ikx x - i t)].

(89)

kx = |k | sin 0 0 , , kz,j = |k | cos j j - . j E+ E- (89) j j . , exp(ikx x - i t) . i- j - (i < j ) zij

E (zij + 0) = E+ + E- j j j E (zij - 0) = E+ + E- , i i i

(90)

E E i j , : E+ 1/tij rij /tij E+ i j = , (91) E- rij /tij 1/tij E- i j tij rij , i- , ij - . E+ E- j j E+ j Ej = , (92) E- j ij - Mij : Mij = 1/tij rij /tij rij /tij 1/tij . (93)


IV. Ei = Mij · Ej . Mij ij - . zj zj + j - j ( ): j ( ) = Ej (zj + ) = j ( ) · Ej (zj ) 2 â 2 : TT
(N +1)0 exp(ikz ,j ) 0 0 exp(-ikz ,j )

20

(94)

,

(95)

(96)

= M(

N +1)N

N ...M10 ,

(97)

m = m (dm ). , , ( ) N + 1 , 0 - 1 , : T 0 = T11 T21 T12 T22 1 R . (98)

R R = -T 21 /T 22 , (99)

T 21 , T 22 - T. T T11 T22 - T12 T21 T= . (100) T22 , E = (1, R), 0 : E (z ) = Tj (z )E = 0 j exp(ikz ,j z ) 0 · (101) 0 exp(-ikz ,j z ) Mj
(j -1)

(j

-1)

...M10

1 R

.

. , dj j , (1.1) (1.1) . c q , , d1 d2 n1 n2 , [3]: cos (q d) = cos (k1 d1 ) cos (k2 d2 ) - 1 2 n1 n + n2 n
2 1

sin (k1 d1 ) sin (k2 d2 ) .

(102)


IV.

21

"" k

- kx c , kx - , , d = d1 + d2 - . t1 r1 . N , 1 , :
1(2)

=



1(2)

t

N

= cos (N q d) - H 1 2t1

sin (N q d) ,r sin (q d)

N

=

r1 sin (N q d) tN , t1 sin (q d)

(103)

H=

2 2 t2 - r1 - 1 cos (q d) + i t2 - r1 + 1 sin (q d) . 1 1

(104)

1 , (103) () , tI (rI ) tI I (rI I ). , : TN = N = 1 + (rI - rI I )rN + rI rI
I 2 t2 - rN . N

tN · tI · t N

II

2

,

(105)

(106)

TN ( ) , , ( n1 n2 ). , . , q (102). 1.2. , N a b, , z . = a + b, L = N . , : d2 2 + 2 (z ) = 0, dz 2 c (107)

(z ) - . r t . (107) (z = 0) (z = L) : 1 + r = (0), t = (L), d (0) d (L) i (1 - r ) = , i t = . c dz c dz (108)


IV.

22

, , r t I r I t I , (z ) = E (z )/E , r = E /E , t = (E /E )ei(/c)L . r t , . , , , , - . , , L, , , , . : L 1 c2 d (z ) 2 dz , (109) (z )| (z )|2 + 2 2Lc 0 dz , 1/c. (108), (109) = 1 Lc
L 0

(z )| (z )|2 dz -

1 Im(r ). L

(110)

, kef f ( ). , kr ( ), dkr ( ) = d (111)

kr ( ) = 0. , ki ( ), -, : 2 ki ( ) = - P
0

kr () d, 2 - 2

(112)

, - kr (- ) = -kr ( ). , k
ef f

( ) = kr ( ) + iki ( ).
ef f

(113)

k nef f , kef f ( ) = n c
ef f

-

=

(nr ( ) + ini ( )) . c

(114)

nr ( ) , dnr ( )/d < 0 . ni .


IV. 2.

23

2.1. ki , 1 , 2, , , . z , , xz , . ki = (kx , k1y , 0) = (k1 sin , k1 cos , 0) k1 = 1 /c. x, (n) (n) , , kr kt (n = 0, ±1, ±2...), (0) (0) , xy . kr kt (±1) (±1) , kr , kt .. () ±1.... n (n) (n ( kr,x) = kt,x = kxn) = kx + Gn , (115) Gn = 2 n/a, n = ±1, ±2... (116)

a. (115) , , , xz x. n k1 k3 = 3 /c. (n) kr,y - k 2 - (k (n) )2 , k |k (n) | x x 1 1 (n) kr,y = . (117) -i (k (n) )2 - k 2 , k < |k (n) |
x 1 1 x

, k 2 - (k (n) )2 , k |k (n) | x x 3 (n) 3 kt,y = . (118) (n) 2 -i (k ) - k 2 , k < |k (n) | x x 1 3 , , E z E z . . , E z . 1 :


E1z (r) = E0 e

iki ·r

+
n=-

Rn e

ikr ·r

(n)

,

(119)

E0 - , Rn - n, r - , .


IV. 3 :


24

E3z (r) =
n=-

Tn e

ikt

(n)

·(r-L)

,

(120)

Tn - n, L = (0, L), L - . Rn Tn , ( 2), : E E2z (r) 1 (r) 2 2 +2 x2 y + 2 c2 E2z (r) = 0, (121)

- . (121) fE (x, y ) : (n) 1 fE (x, y ) {y Tn + (L - y ) (n0 E0 + Rn )} eikx x , (122) L n=- n0 - . fE (x, y ) , E1z E3z : fE (x, 0) = E1z (x, 0), fE (x, L) = E3z (x, L). , E (x, y ) = E2z (x, y ) - fE (x, y ), E (x, 0) = E (x, L) = 0, (121) E E (x, y ) = -E fE (x, y ). E (x, y ) 1/(x, y ) :


(123)

(124)

(125)

(126)

E (x, y ) =
n=- m=1

Anm e


ikx x

(n)

sin

m y, L
y

(127)

1 = (x, y )

nm ei(G

n

x+

m L

).

(128)

n=- m=-

(122) fE (x, y ) y 2 = L
m=1

(-1)m-1 m 2 sin y, 1 = m L

m=1

1 - (-1)m m sin y m L

(129)

C (127) (128), (126) n m : 2 2 m 2 ( Anm + kxn ) + n-n ,m+m - n-n ,|m-m | An m = c2 L 2 2 (-1)m-1 Tn + Rn + n0 E0 + =- 2 c m 2 2 ( + kxn ) n-n ,|m-m | - n
n =- m =1 n =- m =1

(130) (-1)
-n ,m+m m -1

Tn + Rn + n 0 E0 . m


IV.

25

Anm , Rn Tn (x) ,



m=1

(n mAnm = iLkr,y) + 1 Rn - Tn + n0 E0 (iLk1,y + 1)

(131)

y = 0



m=1

m(-1)m Anm = Rn + iLkt,y - 1 Tn + n0 E0

(n)

(132)

y = L. p- . 2.2. () - nm (128) (x, y ). nm . ... . , , , nm . , N 0 y L : 1 1 = + (x, y ) b 1 1 - a b
2 N -1

S (r - uj (l, l )).
j =1 l=- j =0

(133)

S (r) (81) S (r) = 1, |r| ra , 0, |r| > ra (134)

uj (l, l ) l l : u1 (l, l ) = (al, al + ra + d), u2 (l, l ) = (al, -al - ra - d). (135) (136)

, L () d : a (137) ra + d = , L = N a. 2 - nm (128) nm = 1 2aL
a L

dx
0 -L

dy

1 e-i(G (x, y )

n

x+

m L

y

).

(138)

y , -L y 0, nm


IV. 0 y L. nm f + 1-f, a b 1 1 J1 (Gnm ra ) 2f - , a b Gnm ra nm = 2f 1 - 1 (-1)j J1 (Gnm ra ) , a b Gnm ra 0 : n = 0, m = 0 n = 0, m = 0 m = 2j N

26

(139)

2.3. 3. 3.1. (111) (a-SiO2 ), . a-SiO2 , , . , , (111), - [111]. , , .. , , L. (111) , G111 , . d - [111], |G111 | = 2 /d. , (111) |G
111

|2 = -2k · G111 ,

(140)

k - . (140) (111), , . , P B G , P
BG

= 2dnef f cos ,

(141)

nef f - , - . R a-SiO2 d = d[111] = R 8/3 1.63R. . : ef f ( ) = a S ( ) + b (1 - S ( )). (142)


IV.

27

a b , . S ( ) , z = , a-SiO2 . (105), [111]. , (103,104,105) b r1 t1 , . (111) . , (141), = P B G . (111) . , , = P B G , (141). - (111) . , , (111) , , , , ( ) . k - k , G111 , , , (111) . , , . 3.2. . , k , k : k = k + G, (143)

G - . , |k | = |k|. , : I (k, k ) = C S (k - k)|k
-k

|2 ,

(144)
-k

C , k - (r) , k
-k

-

=

1 V0

dr(r)e-iG · r.
V0

(145)

S (k - k) :
2

1 S (k - k) = N

e
j

-i(k -k)·rj

=

1 N

e
j,j

-i(k -k)·(rj -rj )

,

(146)


IV.

28

rj - - a-SiO2 . ai (i = 1, 2, 3) (146) S (k - k) rj = i ai li , li - . , (144) , (143) , . , , G, (143), () Gi G = i Gi mi , mi - . (111) . , , . , S (k - k) = S (k) : S (k) = S (k)S (k, p), (147)

p - , . S , . 3.3. S a1 a2 . S (146) : S (k) =
i=1,2

S ,i (k) =
i=1,2

1 sin2 (Ni k · ai /2) , Ni sin2 (k · ai /2)

(148)

Ni - ai . Ni S ,i (k) : 1 sin2 (Ni k · ai /2) lim = 2 (k · ai - 2 mi ), (149) Ni Ni sin2 (k · ai /2) m
i

m1 m2 - . - (149) - . K, , k · ai = 2 mi . (150)

, ex , ey ez , ez k, xz . k () : (151) k = ef f ((ex cos + ey sin ) cos + ez sin ) . c


IV.

29

ef f (142). , , , a-SiO2 , 0.74. = = 0. ex , ey , ez , (150) (cos sin ) a1x + (cos - 1) a1z = m1 / a ef f , (152) (cos sin ) a2x + (cos - 1) a2z = m2 / a ef f = 2 c/ - a = 2R - R. 6m . , , , (111) [111] /3. , , , , . , , a-SiO2 , [211], [110]. 3.4. [211] [211], .. , a1 a2 ex , ey , ez : a1 = aex , a2 = - (a/2) ex + a 3/2 ez . (153) (152) : cos sin = m1 / a ef f . (154) 3/2 (cos - 1) = (m1 /2 + m2 )/ a ef f (154) = /2 , (0,-1), .. m1 = 0 m2 = -1. 0 < < a 3ef f , (), , : () = a/2 3
ef f

(1 - cos ) .

(155)

, [211] < 211 = a 3ef f , max z y . (155): , . = 211 max . > 211 (154) (155) max


IV.

30

. , . , < 211 (154) c max m1 m2 , , , (1, 0), (0, 2), (1, 1) .. , a. a 250 nm, , , - . 3.5. [110] [110], k - K . a1 a2 : a1 = a 3/2 ex + (a/2) ez , a2 = - a 3/2 ex + (a/2) ez . (156)

(152) : 3/2 (cos sin ) + 1/2 (cos - 1) = m1 / a ef f . (157) - 3/2 (cos sin ) + 1/2 (cos - 1) = m2 / a ef f (157) , (0,1) (-1,0). < 110 = 3a ef f /2, max z x z y . || ||. , , || , || - . = 110 , x, max = 0 = . > 110 (157) max , . , z y . 3.6. (111). (111). (111). , . , 6 , [111] /3. [111] - 2 /3,


IV.

31

[211]. (154), , z y . , (111). k = k + G111 , k G111 , (111). 4.


V
1. 1.1. Pext (r, t), (r). ( ) : 1 H(r, t) â E(r, t) = - , c t 1 â H(r, t) = ((r)E(r, t) + P c t · ((r)E(r, t) + Pext (r, t)) = 0, · H(r, t) = 0,

ext

(r, t)) ,

(158)

µ , (r) (r + ai ) = (r), (159) ai , i = 1, 2, 3. Q(r, t) (r)E(r, t) (160) W , Q W Q(r, t) 1 (r) â â 1 (r) Q(r, t) . (161)

W . . (158) H(r, t), Q(r, t) , Pext (r, t): 1 1 2 2 + W Q(r, t) = - Pext (r, t). (162) c2 t2 c2 (r) t2 (162) , G(r, r , t), : 1 2 +W c2 t2 G(r, r , t) = 0, t < 0, . (162) :


G(r, r , t - t ) = -I (r - r ) (t - t ),

(163)

(164)

Q(r, t) =
V

dr
-

dt G(r, r , t - t )

1 c2

2 P (r ) t 2

ext

(r , t ),

(165)


V.

33

V - . , (165). W , (T ) . Qkn (r) W
(T ) (T ) Qkn

(T ) (r) = kn Qkn (r). 2 c
(T )

(T )2

(166)

kn - W Qkn (r). (165) : Q(r, t) = - Pext (r, t) (r) + 1 V Qkn (r)
kn (T ) t

dr
V -

dt

Qk

(T ) n

(r ) · P

ext

(r , t )

(r )

kn sin kn (t - t ).

(T )

(T )

(167)

, (r, t) : E(r, t) + Pext (r, t) 1 = (r) V Ekn (r)
kn (T ) t

dr
V -

dt Ekn (r ) · Pext (r , t )kn sin kn (t - t ).

(T )

(T )

(T )

(168)

1.2. 1.3. : 2. 2.1. d, - r0 . Pext Pd (r, t) Pd (r, t) = d (r - r0 )e-it+t , (169) et . (169) (168), Ed , : Ed (r, t) + Pd (r, t) e-i = (r) 2V â
t kn

kn Ekn (r) Ekn (r0 ) · d â 1 + kn + i
(T )

(T )

(T )

(T )

-

1 - kn + i
(T )

(170) .

, S, : S(r, t) = [(E(r, t) + E (r, t)) â (H(r, t) + H (r, t))] . (171)

, Ed , , Hd , e-it , : S(r, t) = [(Ed (r, t) + Ed (r, t)) â (Hd (r, t) + Hd (r, t))]. (172)


V. : S(r, t) = [Ed (r, t) â Hd (r, t)] + [Ed (r, t) â Hd (r, t)] . : · S(r, t) = (Hd · [ â Ed ]) - (Ed · [ â Hd ]) + + (Hd · [ â Ed ]) - (Ed · [ â Hd ]) . (158), : · S(r, t) = i (Ed · Pd - Ed · Pd ) . (170) , 1 = - 0 ± i - 0 i ( - 0 ),

34

(173)

(174)

(175)

(176)

- . Pd Ed , (169),(170) (176), : · S(r, t) = 2 (r - r0 ) V d · Ekn (r0 ) - k
kn (T ) 2 (T ) n

.

(177)

2.2. S1 , . U S1 U=
S1

Sn (r, t) dS,

(178)

dS - S1 Sn - S1 . , S1 V1 , S1 : Sn (r, t) dS = · S(r, t) dr.
V1

(179)

S1

(178) : U= V
2 kn

d · Ekn (r0 ) - kn

(T )

2

(T )

.

(180)

D( ) , k n . (180), U : U 2 (T ) d · Ekn (r0 ) V
2 0

D( ) - d =

2 2 (T ) d · Ekn (r0 ) D( ). V

(181)

(181) (22). . , , .


V. 3. 3.1.

35

, . , . (r), , , . (bf r) . (r) (r). , (T ) Eext (r) Ekn , .. . , (r, t) (T ) Ekn : (1) (T ) Pst (r, t) = (r)Ekn (r)e(-i+)t . (182) () . (182) (168) , , (1) Pst (r, t): E(1) (r, t) = e
(-i + )t

2V

k
kn

(T ) n

Ek

(T ) n

(r)
V

dr (r )Ek 1
(T ) n

(T ) n

(r ) · Ekn (r )â 1 (183) + i .
(T ) n

(T )

â

+ k

+ i

-

- k

(183) V . (183) . a. V = a3 Nx Ny Nz , Nj - , . , x y , z - l = anz , nz - . , (r) , (r). (61) (62) , (183) : dr (r )Ek
(T ) n

V

(r ) · Ekn (r ) = dr (r )uk
(T ) n

(T )

Nx -1 Ny -1 nz -1 j1 =0 j2 =0 j3 =0 V0

=

(r ) · ukn (r )e

(T )

i(k-k )·(r +aj)

(184) ,

j = (j1 , j2 , j3 ) V0 = a3 - . (T ) (T ) Ek n Ekn , akx j1 , aky j2 , akx j1 aky j2 2 .


V. (184) :
Nx -1 Ny -1 nz -1

36

e
j1 =0 j2 =0 j3 =0

ia(k-k )·j

= Nx Ny k

x kx

k

y ky

1 - eianz k 1 - eiakz

z

= Nx Ny

kx k

x



ky k

y

e

ia(nz -1)kz /2

sin (anz kz /2) , · sin (akz /2)

(185)

kz = kz - kz . (185) , j1 j2 , kx = kx ky = ky . (T ) F1 (k, n), ( ) Ekn : F1 (k, n) = 1 V0 (r) Ekn (r) dr.
(T ) 2

(186)

V0

. -, (183) (T ) (T ) , k n = kn , .. n = n. , nz (185) kz 0, sin (anz kz /2) lim = nz . (187) kz 0 sin (akz /2) , (183) kz kz , kx = kx ky = ky . , (183) : E(1) (r, t) kn Nx Ny V0 F1 (k, n) (T ) Ekn (r)e(-i+)t â 2V 1 1 â â - (T ) (T ) + k n + i - k n + i kz sin (anz kz /2) âeia(nz -1)kz /2 · . sin (akz /2)
(T )

(188)

kz , Ekn z (T ) kn . (189) vg (k, n) = kz kz = 0 (187), (188) E
(1)

(T )

Nx Ny nz V0 F1 (k, n) (T ) (r, t) - kn Ekn (r)e(-i+)t â 2V (T ) dk n aNz (T ) â - i - k n . 2 vg (k , n) - (T )
kn

(T )

(190)

, , , : E(1) (r, t) kn l Ekn e
(T ) (-i + )t

,

(191)

kn ikn F1 (k, n) = . 2vg (k, n)
(T )

(192)


V.
(1)

37

(191) . Pst (r, t) (T ) Ekn . , (2) Pst (r, t), E(1) (r, t): Pst (r, t) = (r)E(1) (r, t),
(2) (2)

(193)

.. E(2) (r, t), Pst (r, t), E(1) (r, t) , (191): E(2) (r, t) 1 2 2 (T ) l Ekn e 2 kn
(-i + )t

,

(194)

1/2 E(1) (r, t) l. , E(j ) (r, t) 1/j !.


E(r, t) =
j =0

E(j ) (r, t) =

1 j j (T ) l Ekn (r)e j ! kn =
(T ) Ekn

-i

(T ) kn

+ t

= (195) .

(r)e

kn l- ikn + t

(T )

kn : Re [kn ] = - Im [] kn F1 (k, n) . 2vg (k, n)
(T )

(196)

, . , , , , , . , Im [] < 0, . Im [] > 0, . 4. - 4.1. , (2) (r) ^ , (r). a V = a3 Nx Ny Nz . , z l = anz , nz - . , (T ) (T ) Ek1 n1 (r) Ek2 n2 (r) k1 n1 k2 n2 . PN L (r, t), Ek1 n1 (r) Ek2 n2 (r), (T ) (T ) 1 (-ik n -ik n + )t (T ) (T ) 11 22 ^ . (197) PN L (r, t) = A2 (2) (r) : Ek1 n1 (r)Ek2 n2 (r)e 2
(T ) (T ) (T ) (T )


V.

38

(197) A- , , 1/2 . , : F2 (kn, k1 n1 , k2 n2 ) = 1 V0
V0

Ekn (r) · ^

(T )

(2)

(r) : Ek1 n1 (r)Ek2 n2 (r)dr.

(T )

(T )

(198)

(183) (184) , EN L , PN L , : EN L (r, t) = A2 V0 e
(-ik
(T ) 1 n1

-i

(T ) k2 n2

+ ) t kn

4V eia(k
1

kn Ekn (r)â (199) .

(T )

(T )

Nx -1 Ny -1 nz -1 +k2 -k)·j j1 =0 j2 =0 j3 =0

F2 (kn, k1 n1 , k2 n2 )â - 1
(T ) k1 n1

â

1
(T ) k1 n
1

+

(T ) k2 n2

+

(T ) kn

+ i

+

(T ) k2 n

2

- kn + i

(T )

j :
Nx -1 Ny -1 nz -1

e
j1 =0 j2 =0 j3 =0

ia(k1 +k2 -k)·j

= Nx Ny

kx ,k1x +k

2x



ky ,k1y +k

2y

e kz = k
1z

ia(nz -1)kz /2

sin (anz kz /2) · , sin (akz /2) 1, k = k + 2 j /a 0.

(200)

+k

2z

- kz - z
kk

=

(201)

kz , sin (anz kz /2) lim = nz (202) 2 j sin (akz /2) kz - a . (200) k = k1 + k2 + G, (203)

G - . (203) . (199): 1
(T ) k1 n
1

2 i

(T ) (T ) + k2 n2 + kn (T ) (T ) k1 n1 + k2 n2

+ i -

-

1
(T ) k1 n
1

+

(T ) kn

(T ) k2 n2

- kn + i

(T )



(204)

.

(199) , : kx = k
1x

+ k2x +
y

ky = k1y + k2 kz k
1z

+k

2z

2 p , a 2 q + , a 2 j + , a

(205)


V.

39

p q - . k n (199) , : (T ) iaA2 kn F2 (kn, k1 n1 , k2 n2 ) EN L (r, t) â 4vg (k, n) (206) (T ) sin (anz kz /2) (T ) âEkn e-ikn t eia(nz -1)kz /2 · , sin (akz /2) vg - z . : (T )2 a2 A4 kn |F2 (kn, k1 n1 , k2 n2 )|2 |EN L (r, t)|2 â 2 16vg (k, n) (207) 2 (T ) 2 sin (anz kz /2) . â Ekn sin2 (akz /2) (207) , : (1) , (2) (3) F2 . (1) (2) , . (3) , (2) , , ^ () . , , (207), , , , . . 4.2. , . , , . = /0 , = z /0 = ct/0 , z 0 , - 0 = 2 /0 . : i 2 E ( , ) = E- E + i ( - 1)E + i8 2 (2) E2 E , 2 4 i 2 2 E2 - E2 + i (2 - 1)2E2 + i8 2 2(2) E . 2 E2 ( , ) = 4 2 2


(208)

(208) L , L (114). 2 , , , : d A = i def f A2 A eikef f z , dz nr ( )c d A2 = i def f A2 e-ikef f z , dz nr (2 )c

(209)


V.

40

kef f = kr (2 ) - 2kr ( ) , def f = 1 L
L

d
0

(2)

(z )| (z )|2 |2 (z )|dz .

(210)




41



[1] J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals, Prinston University Press, 1995. [2] K. Sakoda, Optical Properties of Photonic Crystals, Springer, 2001. [3] A. Yariv, P.Yeh, Optical Waves in Crystals, Wiley, New York, 1984. [4] .. -, .. -, .. , 170, 697 (2000). [5] Photonic Band Structures, Topical Issue of J. Mod. Opt. 41, 171-404 (1994). [6] Development and Applications of Materials Exhibiting Photonic Band Gaps, Topical Issue of J. Opt. Soc. Am. B 10, 279-413 (1993). [7] Nonlinear Optics of Photonic Crystals, Topical Issue of J. Opt. Soc. Am. B 19, 2079-2356 (2003).