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5 5.. Maxwell Maxwell-Ampere E H D P J D roth = J+ = J+ E+ P ( ε P = σe+ εe + (5. ( NL P= ε χe+ P NL, J = σe (5. Faraday rot = µ H E (5. (5. (5. ( E ( roth rot rot = µ NL µσ E µε µ P E (5.4 =

( = grad ( div rot rote E E = E div E = ( ( div E = div D = µσ E εµ E µ PNL E = + + (5.5 (5.5 P NL E 5.. 4 PNL E

P d E E ω ( ω ( ( i = ε ω ijk j k (5.6 d (SHG i j k x y z j ω k ω i ω P paraxial approximation E A z z E Ek = ω ω ω ( ω Ei ( r, t = i( z expi( ωt A k r +cc.. (5.7 ( ω Ej ( r, t = j( z expi( ωt A k r + cc.. (5.8 ( ω Ek ( r, t = k ( z expi( ωt A k r + cc.. (5.9

ω = ω ω (5. 5.6 { cc} ( ω NL (, t εdijk k ( z j ( z expi ( ω ω t ( P r = A A i k k r +.. (5. PNL (5.5 ( x, z ( ω ( ω E i E i (5. x z = + x z z z ( ω E i = i A i( z k z z + A i( z k z i( z exp i( ωt + cc.. z A k r (5. x A z x ( ω E i = { A i( z kxexp i( ωt k r + c. c.} (5.4 ( ω z ( z E i = i A i k z 4

+ A i( z( k x + k z i( z exp i( ωt + cc.. z A k r (5.5 4 (Slow varying envelop approximation SVEA z A( k A( z (5.6 z z A z A λ A A z λ A (5.5 A z ( ω E = i ( z ( z( i + + z i z i A k A k x k z ( ω } exp i t k r + cc.. (5.7 (5.5 5

( ( r exp +.. iµσω ω µε A i z i ωt k cc +µ P NL (5.8 k c ωµε = µε= k = k k ( x + z n P NL = ( ω ω ε d A ( z A ( z ijk k j ( ω ω t ( k k r cc } exp i +.. = ijk { Ak A j z ( ω ε d ( z ( { ( } exp i ω t + cc.. k k r (5.9 ( ω ( z A ik e = iωσµ A ( z e z i ik r ik r i + µωε ( ( ( i k k r dijkak z A j z e (5. 6

c µ i ωσµ = i σ k = i n σ εεµ r k µ = iσ k ε A ( ω µ µ = σ i ω ε d e A A A (5. i i ijk k j z ε ε ( i k k k r E ( ω ( ω E PNL ( ω E( ω A ( ω i µ µ i( k+ k k r = σ A j + i ω εd jikaia ke (5. z ε ε E ( ω E ( ω P ( E ( ω NL ω A ( ω µ µ = σ i ω ε d e A A A (5. k k kij i j z ε ε ( i k + k k r 7

6 second harmonic generation, SHG 6. ( ω = ω = ω ω = ω + ω = ω ω (5. (5. ω ω ω ω ω z colinear (5. (5. (5. (5. (5. A ( ω µ = iω ε d A A i kr (6. ( ω ( ω exp( k ijk i j z ε 8

ω k = k k A L µ i kr k ( L = iω εdkij i j e d ε A A A z = iω µ ε d ε AA kij i j i kl e i k (6. L (6. εµ A A A ( L = ω ( d i kl i kl ( e e + ( k k kij i j n ε µ = ω n εµ = ω n ( d A A cos kl ( k kij i j ( d A A sin ( k kij i j kl εµ = ω n εµ = ω ( d A A 4sin ( k kij i j ( d sin kij A i Aj L n kl (6. S 9

ε S = A k µ ( L sin = ω ( ε ( µ ( ( dkij A i Aj L ω n sin kl µ ( ( = ω ε dkij i j L A A ω ε n (6.4 ε ε S A A p (6.5 p ( p = p = n ω p µ µ (6.4 S sin µ = ω ε ( ω ( ω { } ( ε d kij S is jl kl n n (6.6 ( sin ( 6. ω ω ω ω ω sin k

( d d d ( SHG (4 (5.4 SHG η ( ( { } ( sin L µ k S SS i j η = = ω ε dkij L S ω ω ε n n S ( ( { } ( sin L µ k = ω ε dkij cosαi sinα jl S ε ω ω n n (6.7 η η 5 (5.

6. ( (5.4 k phase matching S sin kl µ = ω ( ω ( ω ε { } ( ε d kij S is jl n n (6.6 sin ( ω ( ω k = k k ω ω = n c c ( ω ( ω ( ω { n n } ω = = c (6.8 n ( ω ( ω = n (6.9

SHG (6.6 sin ( k (6. L SHG 6. SHG sin π π k L c = L c π = = k ω c π ( ω ( ω { n n } λ = 4 n n ( ω ( ω { } (6. λ L c SHG k k L c SHG SHG SHG k

6. SHG SHG k cm L c 5 57 ( ω ( ω δ n= n n =.7 4 57.7 5.7.7 6 KDP LiNbO λ µ m n n n n ( e.6.498.4599 4.4 4.554.5.5.475 4.5 4. δ n.85.6.9.776 e SHG ( L c L SH 6.4 6.4 4

( δ k = 9 x y z ( x + y + z = (6. n ne n n n n n > n < e e KDP LN k z x k z k n n k n n k z θ e e e n e ( θ = cosθ sinθ + n n e (6. 5

n ( ω ( ω = n ( n n > e 6.5 6.5 k z k z k (6.8 n ( ω e ( θ = cosθ sinθ ( ω + ( ω n n e ( ω = n (6.4 θ θ sin ω ω ( n ( n ω ω ( ne ( ne θ = (6.5 ( ω, ωω, ( e,, (, ee, e 6

x z ( ω ( ω ( ω k + k = k ( ω ( ω ( ω n + n = n e ( ω ne = n + n e ( ω ( ω ( (6.6 7

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