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E H D P J D HJ J EP ()P EE P JE H E - …

5 . 5. 1.. Maxwell . Maxwell-Ampere E H D . P J . D . rotH = J + = J + ( 0E + P ). t t P. = E+ ( E ) + NL ( ). t t . P = 0 E + PNL , J = E ( ). Faraday . H. rotE = 0 ( ). t ( ) ( ) .. rot ( rotE ) = 0 ( rotH ). t E 2 2 PNL. = 0 0 2 E 0 ( ). t t t 2.. 1. rot ( rotE ) = grad ( divE ) 2E. = 2 E.. divE = 0. ( ) . ( ) divE = 0. divD = 0 .. E 2E 2 PNL. 2 E = 0 + 0 2 + 0 ( ). t t t 2.. ( ) . PNL 2 . E . 5. 2. 2 3 . 2 . 4 .. PNL E . 2. Pi ( 1 ) = 0 dijk E (j 2 ) Ek( 3 ) ( ). d 2 (SHG ) 3 . i j k x y z 1 2 3 . j 1 k 2 . i 3 P .. 1 .. 2 paraxial approximation . E A z.

第5章 非線形光学の基礎方程式 5. 1. 非線形波動方程式の導出 Maxwell 方程式から非線形光学の基本波動方程式を導出する。

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Transcription of E H D P J D HJ J EP ()P EE P JE H E - …

1 5 . 5. 1.. Maxwell . Maxwell-Ampere E H D . P J . D . rotH = J + = J + ( 0E + P ). t t P. = E+ ( E ) + NL ( ). t t . P = 0 E + PNL , J = E ( ). Faraday . H. rotE = 0 ( ). t ( ) ( ) .. rot ( rotE ) = 0 ( rotH ). t E 2 2 PNL. = 0 0 2 E 0 ( ). t t t 2.. 1. rot ( rotE ) = grad ( divE ) 2E. = 2 E.. divE = 0. ( ) . ( ) divE = 0. divD = 0 .. E 2E 2 PNL. 2 E = 0 + 0 2 + 0 ( ). t t t 2.. ( ) . PNL 2 . E . 5. 2. 2 3 . 2 . 4 .. PNL E . 2. Pi ( 1 ) = 0 dijk E (j 2 ) Ek( 3 ) ( ). d 2 (SHG ) 3 . i j k x y z 1 2 3 . j 1 k 2 . i 3 P .. 1 .. 2 paraxial approximation . E A z.

2 Z . 2 . 3 . E . E k = 0. 1 2 3 3 . 1. E(i 1 ) ( r , t ) = A1i ( z ) exp i ( 1t k 1 r ) + ( ). 2 . 1. E(j 2 ) ( r , t ) = A 2 j ( z ) exp i ( 2t k 2 r ) + ( ). 2 . 1. E(k 3 ). ( r, t ) = A 3k ( z ) exp i ( 3t k 3 r ) + ( ). 2 . 3. 2 3 . 1 = 3 2 ( ).. ( 1 ). PNL ( r , t ) i 2 { }. = 1 0 d ijk A 3k ( z ) A 2 j ( z ) exp i ( 3 2 ) t ( k 3 k 2 ) r + ( ). PNL 1 ( ) . 3 ( x, z ) . 2 2 . 2 E1( i 1 ) = 2 + 2 E1i( 1 ) ( ). x z . x z z .. 2 ( 1 ) 1 . E1i = 2i A1i ( z ) k1z . z 2. 2 z . 2 . + A1i ( z ) k12z A1i ( z ) exp i ( 1t k1 r ) + ( ). z 2.. x A z . 2 ( 1 ). x 2.

3 1. 2. {. E1i = A1i ( z ) k12x exp i ( 1t k1 r ) + } ( ).. 1 . 2 E1( i 1 ) = 2i A1i ( z ) k1z . 2 z . 4. 2 . ( ). + A1i ( z ) k12x + k12z . z 2. A1i ( z ) exp i ( 1t k1 r ) + ( ).. 4 (Slow varying envelop approximation SVEA).. 2 . A( z) k A(z) ( ). z 2. z . A A.. z .. A z . A . A . ( ) A z 2 . 1 . (. 2 E1( i 1 ) = 2i A1i ( z ) k1z + A1i ( z ) k12x + k12z . 2 z . ). exp i ( 1t k 1 r ) + } ( ). ( ) . 5. 1. i 0 1 12 0 A1i ( z ) exp i ( 1t k1 r ) + 2. 2 PNL. + 0 ( ). t 2.. 2. k c . 0 = 1 0 = k 12 = k 12x + k 12z 2. 1 ( ). n1 .. 2 PNL. t 2. 1. 2. 2. {. = ( 3 2 ) 0 dijk A 3k ( z ) A 2 j ( z ).

4 Exp i ( 3 2 ) t ( k 3 k 2 ) r + }. = . 1. 2. 2. {. ( 1 ) 0 dijk A3k ( z ) A 2 j ( z ). exp i 1t ( k 3 k 2 ) r + } ( ).. A1( i 1 ) ( z ) ik r 1. e 1 = i 1 0 A1i ( z ) e 1. ik r ik 1 . z 2. 1 . + 0 12 0 dijk A 3k ( z ) A 2 j ( z ) e ( 3 2 ). i k k r ( ).. 2.. 6. c 0. i 1 0 = i k1 = i k1. n1 1r 0 0. 0. = i k 1 1.. A1( i 1 ) 1 0 1 . A1i i 1 0 0 dijk A3k A 2 j e ( 3 2 1 ). i k k k r = ( ). z 2 1 2 1. E1 ( 1 ) E3 ( 3 ) PNL ( 2 ) E2 ( 2 ) .. A (2 i 2 ) 1 0 1 0. = A 2 j + i 2 0 d jik A1i A 3k e i(k1 +k 2 k 3 ) r ( ). z 2 2 2 2. E1 ( 1 ) E2 ( 2 ) PNL ( 3 ) E3 ( 3 ).

5 A(3k3 ).. 1 0 1 . A3k i 3 0 0 d kij A1i A 2 j e ( 1 2 3 ). i k + k k r = ( ). z 2 3 2 3. 3 3 2 . 2 .. 1 2 .. 7. 6 2 second harmonic generation, SHG . 6. 1 2 . 2 2. ( 1 = 2 = ) 3 = 1 + 2 = 2 3 . 2 2 . 3 . 1 . ( ) ( ) 1 2 .. 2 2 .. 2 .. 3 2 z . colinear .. 3 ( ) ( ) ( ) ( ) . ( ) . 2 . A(3k3 ).. 1 . = i 3 0 0 dijk A1( i 1 ) A1( j 1 ) exp ( i kr ) ( ). z 2 3. 8. k = k 3 2k1 . A1 . 0 L. A3k ( L ) = i 0 d kij A1i A1 j ei k r dz 3 0. 0 ei kL 1. = i 0 d kij A1i A1 j ( ). 3 i k L ( ) . 0 0 (. 2 ei kL + e i kL ). A 3k ( L ) = 2 ( d kij ) A1i 2 2 2 2. A1 j ( k ). 2 2.

6 N 3. 0 0 2 2 cos kL. (d ). 2 2 2. = 2 A1i A1 j ( k ). 2 kij 2. n 3. 1 . 2 2 1 2sin 2 kL . 0 0 2 . (d ) . 2 2 2. = 2 A1i A1 j ( k ). 2 kij 2. n 3. 1 . 4sin 2 kL . 0 0. (d ) 2 . 2 2 2. = 2 A1i A1 j ( k ). 2 kij 2. n 3. 1 . sin 2 kL . 0 0. (d ) 2 . 2 2 2. = 2 2 kij A1i A1 j L2 2. ( ). n 1 . kL . 3. 2 .. 2 S 3 . 9. 1 3. A 3k ( L ). 2. S3 =. 2 0. 1 . sin 2 kL . (d ) 2 . 3 1. 1 1. = 2 ( 0 ) ( 0 ). 2 2 2. 2 2. ( 2 ) kij A1i A1 j L2 2. 2 n3 1 . kL . 2 . 1 . 1. sin 2 kL . 1 1. = 2 0 ( 2 ) ( 0 d kij ) A1i 2 . 2 2 2 2. A1 j L2 ( ). 2 0 n3 1 . 2. kL . 2 .. 1 p 2 1 0 ( ) 2.

7 Sp = Ap = np A p ( ). 2 0 2 0. ( ) . 1 . 3. sin 2 kL .. 2 ( 0 kij ). S3 = 2 2 0 . 2. 1. d 2. S1i S1 j L2 2 ( ). 0 n3( 2 ) n1( ) { }. 2. 1 . kL . 2 .. (1) sin ( ) . 2 . 2 2 2 . 2 . sin .. k 0 2 .. 2 . 10.. (2) 2 d . d 1 1 . d .. (3) 2 SHG 2 .. (4) ( ) SHG . 1 . 3. sin 2 kL .. 2 ( 0 kij ). =. S3. = 2 2 0 . 2. 1. d 2 S1i S1 j 2. L 2 . { }. 0 n3( 2 ) n1( ). 2. S1 S1 1 . kL . 2 . 1 . 3. sin 2 kL .. 2 ( 0 kij ). = 2 2 0 . 2. 1. d 2. S1 cos i sin j L2 2 ( ). { }. 0 n3( 2 ) n1( ). 2. 1 . kL . 2 .. 30 50 .. 100 .. ( ) .. 11. 6. 2 . (1) . ( ) 2 . k 0 . phase matching.

8 1 . 3. sin 2 kL .. 2 ( 0 kij ). S3 = 2 2 0 . 2. 1. d 2. S1i S1 j L2 2 . { }. 0 n3( 2 ) n1( ). 2. 1 . kL . 2 . ( ) . 1 . sin 2 kL . 2 . 2. 1 . kL . 2 . 1 2 .. k = k 3( 2 ) 2k1( ). 2 ( ). = 2 n c c . 2 ( 2 ). = . c . n n{ ( ). =0 } ( ). 2 2 .. n( 2 ) = n( ) ( ). 12.. SHG .. ( ) . 1 . sin 2 kL . 2 ( ). ( k ). 2. L SHG . SHG 1 .. sin . 2. 1 . kLc =. 2 2.. Lc = =. 2 ( 2 ). k c {. n n( ) }. 0. = ( ). 4 n { ( 2 ). n ( ) }. 0 Lc .. SHG 1 k . 1. Lc SHG . k SHG 1 . SHG k . 13. SHG 2 . SHG k . 1cm . Lc 1570 100 1 . 1500 . n = n( 2 ) n( ) = 10 4. 157 . 10 3 1 1 . 10 2 10 6 100 1.

9 KDP LiNbO3. ( m) n0 ne n0 ne .532 n SHG .. (2) Lc . L SH .. 14. (3) . k = 0 .. 90 2 . 2 2 .. 1 . x y z .. 2 (. x + y2 ) + 2 z2 = 1. 1 2 1. ( ). n0 ne n0 ne . n0 < ne n0 > ne . KDP LN . k z x . k z . k n0 ne .. k n0 ne . 1 k z .. 1. ne ( ) = ( ). 2 2. cos sin . + . n0 ne . 15.. n( 2 ) = n( ).. ( n0 > ne ) .. k z . 2 k z . k . ( ) . (2 ) 1 ( ). ne ( ) = 2 2. = n0 ( ). cos sin . ( 2 ) + (2 ) . n n . 0 e . 0 0 . (n ) (n ). 2 2. 2 . sin 0 =. 0 0. 2. ( ). (n ) (n ). 2 2. 2 2 . e e . 2 . 2 ( 2 , , ) ( e, 0, 0 ) . ( 0, e, e ) .. 2 e 0 . 16. 1 x z .. k1( ) + k1( ) = k 3( 2 ).

10 ( ) ( ) (2 ). n0 + n0 = 2ne (2 ). ne =. 2. (. 1 ( ) ( ). n0 + ne ) ( ).. 17.

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