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Nonlinear optical phenomena in silicon waveguides: Modeling and applications

Q. Lin, Oskar J. Painter, and Govind P. Agrawal

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Abstract

Several kinds of nonlinear optical effects have been observed in recent years using silicon waveguides, and their device applications are attracting considerable attention. In this review, we provide a unified theoretical platform that not only can be used for understanding the underlying physics but should also provide guidance toward new and useful applications. We begin with a description of the third-order nonlinearity of silicon and consider the tensorial nature of both the electronic and Raman contributions. The generation of free carriers through two-photon absorption and their impact on various nonlinear phenomena is included fully within the theory presented here. We derive a general propagation equation in the frequency domain and show how it leads to a generalized nonlinear Schrödinger equation when it is converted to the time domain. We use this equation to study propagation of ultrashort optical pulses in the presence of self-phase modulation and show the possibility of soliton formation and supercontinuum generation. The nonlinear phenomena of cross-phase modulation and stimulated Raman scattering are discussed next with emphasis on the impact of free carriers on Raman amplification and lasing. We also consider the four-wave mixing process for both continuous-wave and pulsed pumping and discuss the conditions under which parametric amplification and wavelength conversion can be realized with net gain in the telecommunication band.

©2007 Optical Society of America

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Figures (10)
Fig. 1.
Fig. 1. Rotation of the coordinate system required for SOI waveguides fabricated along the [0 1 ̄ 1] direction.

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Fig. 2.
Fig. 2. Wavelength dependence of β 2 for several waveguide widths simulated with the finite-element method (FEMLAB, COMSOL). Solid and dashed curves correspond to the TE and TM modes, respectively. The black curve shows for comparison the case of bulk silicon, and the inset shows the waveguide geometry.

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Fig. 3.
Fig. 3. (a) SPM-broadened spectra and (b) nonlinear phase shifts showing the impact of FCC. Red curves neglect both FCA and FCC, black curves include FCA but neglect FCC, and green curves include both. (After Ref. [21].)

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Fig. 4.
Fig. 4. Simulated shape (a) and spectrum (b) of input (blue curves) and output (red curves) pulses in the soliton regime. The green curve in (a) shows the output pulse in the absence nonlinear effects. The dashed curve in (b) corresponds to a sech pulse. (After Ref. [17].)

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Fig. 5.
Fig. 5. (a) Measured spectra (blue curves) at the input and output ends for Gaussian pulses. The green and red curves show the Gaussian and ‘sech’ fits to the data. Part (b) shows a numerical fit to the output spectrum. (After Ref. [17].)

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Fig. 6.
Fig. 6. Supercontinuum created inside a 3-mm-long SOI waveguide when a 50-fs pulse excites the third-order soliton (red curve). The blue curve ignores the effects of TPA and FCA are ignored. The dotted curve shows the input pulse spectrum. (After Ref. [16].)

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Fig. 7.
Fig. 7. Signal gain (a) and wavelength-conversion efficiency (b) as a function of signal wavelength for three pump wavelengths in the vicinity of the ZDWL (dashed line) of the TM mode. Input pump intensity is 0.2 GW/cm2 in all cases. (After Ref. [68].)

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Fig. 8.
Fig. 8. Signal gain (a) and conversion efficiency (b) for the TE mode under the same conditions as in Fig. 7. (After Ref. [68].)

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Fig. 9.
Fig. 9. Parametric gain spectra at three pump wavelengths in the mid-infrared region for the waveguide with a cross section of 1.8×0.4 µm2. (After Ref. [137].)

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Fig. 10.
Fig. 10. Normalized photon flux (a) and pair correlation and spectral brightness (b) for the TM mode as a function of pump intensity. The inset shows the waveguide design. (After Ref. [73].)

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Equations (137)

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P ˜ i ( 3 ) ( r , ω i ) = 3 ε 0 4 ( 2 π ) 2 ∬ χ ijkl ( 3 ) ( − ω i ; ω j , − ω k , ω l ) E ˜ j ( r , ω j ) E ˜ k * ( r , ω k ) E ˜ l ( r , ω l ) d ω j d ω k ,
χ ijkl R ( − ω i ; ω j , − ω k , ω l ) = g ′ H ˜ R ( ω l − ω k ) ∑ v = x , y , z ℜ ij v ℜ kl v + g ′ H ˜ R ( ω j − ω k ) ∑ v = x , y , z ℜ il v ℜ jk v ,
H ˜ R ( Ω ) = Ω R 2 Ω R 2 − Ω 2 − 2 i Γ R Ω .
ℜ ij x = δ iy δ jz + δ iz δ jy , ℜ ij y = δ ix δ jz + δ iz δ jx , ℜ ij z = δ ix δ jy + δ iy δ jx ,
χ ijkl R ( − ω i ; ω j , − ω k , ω l ) = g ′ H ˜ R ( ω l − ω k ) ( δ ik δ jl + δ il δ jk − 2 δ ijkl )
+ g ′ H ˜ R ( ω j − ω k ) ( δ ik δ jl + δ ij δ kl − 2 δ ijkl ) ,
χ ijkl e = χ 1122 e δ ij δ kl + χ 1212 e δ ik δ jl + χ 1221 e δ il δ jk + χ d e δ ijkl ,
χ ijkl e = χ 1111 e [ ρ 3 ( δ ij δ kl + δ ik δ jl + δ il δ jk ) + ( 1 − ρ ) δ ijkl ] ,
ω c n 2 ( ω ) + i 2 β T ( ω ) = 3 ω 4 ε 0 c 2 n 0 2 ( ω ) χ 1111 e ( − ω ; ω , − ω , ω ) ,
P i f ( r , t ) = N e ( r , t ) 〈 p i e ( r , t ) 〉 + N h ( r , t ) 〈 p i h ( r , t ) 〉 ,
ϒ v ( ω ) = q 2 τ v ε 0 m v * ( − 1 ω ( ω τ v + i ) ) .
P ˜ i f ( r , ω ) = ε 0 ∫ χ ˜ f ( ω , ω ′ , N ˜ e , N ˜ h ) E ˜ i ( r , ω ′ ) d ω ′ ,
χ ˜ f ( ω , ω ′ , N ˜ e , N ˜ h ) ≡ ϒ e ( ω ′ ) N ˜ e ( r , ω − ω ′ ) + ϒ h ( ω ′ ) N ˜ h ( r , ω − ω ′ ) .
P i f ( r , t ) = ε 0 ∑ u χ f ( ω u , N e , N h ) E i ( r , ω u , t ) ,
χ f ( ω u , N e , N h ) = ϒ e ( ω u ) N e ( r , t ) + ϒ h ( ω u ) N h ( r , t ) .
χ f = 2 n 0 [ n f + i c α f ⁄ ( 2 ω ) ] ,
n f ( ω , N e , N h ) = − q 2 2 ε 0 n 0 ω 2 ( N e m e * + N h m h * ) ,
α f ( ω , N e , N h ) = q 3 ε 0 c n 0 ω 2 ( N e μ e m e * 2 + N h μ h m h * 2 ) ,
n f ( ω r , N e , N h ) = − ( 8.8 × 10 − 4 N e + 8.5 N h 0.8 ) × 10 − 18 ,
α f ( ω r , N e , N h ) = ( 8.5 N e + 6.0 N h ) × 10 − 18 ,
n f = σ n ( ω ) N , α f = σ a ( ω ) N ,
∇ 2 E ˜ i ( r , ω ) + ω 2 c 2 n 0 2 ( ω ) E ˜ i ( r , ω ) = − μ 0 ω 2 [ P ˜ i f ( r , ω ) + P ˜ i ( 3 ) ( r , ω ) ] .
E ˜ i ( r , ω ) ≈ F ˜ i ( x , y , ω ) A ˜ i ( z , ω ) ,
∂ 2 A ˜ i ∂ z 2 + β i 2 ( ω ) A ˜ i = − μ 0 ω 2 ∬ F ˜ i * [ P ˜ i f + P ˜ i ( 3 ) ] dx dy ∬ ∣ F ˜ i ∣ 2 dx dy ,
β i 2 ( ω ) = ω 2 c 2 ∬ n 0 2 ( ω ) ∣ F ˜ i ∣ 2 dx dy ∬ ∣ F ˜ i ∣ 2 dx dy + ∬ F ˜ i * ∇ T 2 F ˜ i dx dy ∬ ∣ F ˜ i ∣ 2 dx dy ,
∂ 2 ∂ z 2 + β i 2 = ( ∂ ∂ z + i β i ) ( ∂ ∂ z − i β i ) ≈ 2 i β i ( ∂ ∂ z − i β i ) .
∂ A ˜ i ∂ z = i β i ( ω ) A ˜ i + i μ 0 ω 2 2 β i ( ω ) ∬ F ˜ i * [ P ˜ i f + P ˜ i ( 3 ) ] dx dy ∬ [ F ˜ i ] 2 dx dy .
∂ A ˜ i ∂ z = i β i ( ω ) A ˜ i + i ∫ β ˜ i f ( ω , ω ′ , N ˜ e , N ˜ h ) A ˜ i ( z , ω ′ ) d ω ′
+ i 4 π 2 ∬ γ ijkl ( − ω ; ω j , − ω k , ω l ) A j ( z , ω j ) A k * ( z , ω k ) A l ( z , ω l ) d ω j d ω k ,
γ ijkl ( − ω i ; ω j , − ω k , ω l ) = 3 ω i η ijkl 4 ε 0 c 2 a ¯ ( n i n j n k n l ) 1 ⁄ 2 χ ijkl ( 3 ) ( − ω i ; ω j , − ω k , ω l ) ,
a ¯ ≡ ( a i a j a k a l ) 1 ⁄ 4 , a v = [ ∬ ∣ F v ˜ ∣ 2 dx dy ] 2 ∬ ∣ F ˜ v ∣ 4 dx dy ,
η ijkl ≡ ∬ F ˜ i * F ˜ j F ˜ k * F ˜ l dx dy [ Π v = i , j , k , l ∫ ∣ F ˜ v ∣ 4 dx dy ] 1 ⁄ 4 .
β ˜ i f ( ω , ω ′ , N ˜ e , N ˜ h ) = ω 2 c n i ( ω ) ∬ χ ˜ f ( ω , ω ′ , N ˜ e , N ˜ h ) ∣ F ˜ i ∣ 2 dx dy ∬ ∣ F ˜ i ∣ 2 dx dy ,
β ˜ i f = ω 2 c n i ( ω ) χ ˜ f ( ω , ω ′ , N ¯ ˜ e , N ¯ ˜ h ) , N ¯ ˜ v = ∬ N ˜ v ∣ F ˜ i ∣ 2 dx dy ∬ ∣ F ˜ i ∣ 2 dx dy .
β i f ( ω u , N ¯ e , N ¯ h ) = n 0 ( ω u ) n i ( ω u ) [ ω u c n f ( ω u , N ¯ e , N ¯ h ) + i 2 α f ( ω u , N ¯ e , N ¯ h ) ] ,
∂ N v ∂ t = G − N v τ v ′ + D v ∇ 2 N v − s v μ v ∇ · ( N v E dc ) ,
∬ ∣ F ˜ i ∣ 2 [ D v ∇ 2 N v − s v μ v ∇ · ( N v E dc ) ] dx dy ∬ ∣ F ˜ i ∣ 2 dx dy = − N ¯ v τ v * ,
∂ N ¯ v ∂ t = G ¯ − N ¯ v τ 0 , G ¯ = ∬ G ∣ F ˜ i ∣ 2 dx dy ∬ ∣ F ˜ i ∣ 2 dx dy ,
∂ A i ∂ z = ∑ m = 0 ∞ i m + 1 β im m ! ∂ m A i ∂ t m + i β i f ( ω 0 , N ¯ e , N ¯ h ) A i + i ( 1 + i ξ ∂ ∂ t ) P i NL ,
P i NL ( z , t ) = A j ( z , t ) ∫ − ∞ ∞ R ijkl ( 3 ) ( t − τ ) A k * ( z , τ ) A l ( z , τ ) d τ ,
R ijkl ( 3 ) ( τ ) = γ e ( ω 0 ) δ ( τ ) [ ρ 3 ( δ ij δ kl + δ ik δ jl + δ il δ jk ) + ( 1 − ρ ) δ ijkl ]
+ γ R h R ( τ ) ( δ ik δ jl + δ il δ jk − 2 δ ijkl ) ,
γ e ( ω 0 ) ≡ γ 1111 e ( − ω 0 ; ω 0 , − ω 0 , ω 0 ) ≡ γ 0 ( ω 0 ) + i 2 β T ′ ( ω 0 ) ,
h R ( t ) = Ω R 2 τ 1 e − t ⁄ τ 2 sin ( t ⁄ τ 1 ) ,
R ′ ijkl ( 3 ) ( τ ) = γ e δ ( τ ) [ ρ 3 ( δ ij δ kl + δ ik δ jl + δ il δ jk ) + ( 1 − ρ ) ∑ s M si M sj M sk M sl ]
+ h R ( τ ) ( δ ik δ jl + δ il δ jk − 2 ∑ s M si M sj M sk M sl ) .
M = ( 1 0 0 0 1 ⁄ 2 − 1 ⁄ 2 0 1 ⁄ 2 1 ⁄ 2 ) .
R ′ xxxx ( 3 ) ( τ ) = γ e δ ( τ ) , R ′ yyyy ( 3 ) ( τ ) = γ e δ ( τ ) ( 1 + ρ ) ⁄ 2 + γ R h R ( τ ) ,
R ′ yxxy ( 3 ) ( τ ) = R ′ xyyx ( 3 ) ( τ ) , R ′ xyyx ( 3 ) ( τ ) = γ e ρ δ ( τ ) ⁄ 3 + γ R h R ( τ ) .
ξ ≡ 1 ω 0 + 1 χ e ( ω 0 ) d χ e d ω ∣ ω 0 − 1 a ¯ ( ω 0 ) d a ¯ d ω ∣ ω 0 ,
∂ A ∂ z = i β f ( ω 0 , N ¯ ) A + i γ e ∣ A ∣ 2 A ,
G ¯ = − 1 2 h ¯ ω 0 a ¯ ∂ P ∂ z = β T ∣ A ∣ 4 2 h ¯ ω 0 a 2 ¯ .
N ¯ ( z , τ ) = β T 2 h ¯ ω 0 a ¯ 2 ∫ − ∞ τ e − ( τ − τ ′ ) ⁄ τ 0 ∣ A ( z , τ ′ ) ∣ 4 d τ ′ .
N ¯ m = β T 2 h ¯ ω 0 a ¯ 2 ∫ − ∞ ∞ ∣ A ( z , τ ′ ) ∣ 4 d τ ′ .
N ¯ m = π β T P 0 2 T 0 2 2 h ¯ ω 0 a ¯ 2 .
r a ≡ α fm α Tm = n 0 σ a ℰ p 2 2 h ¯ ω 0 n a ¯ ,
∂ Φ K ∂ z = γ 0 ∣ A ∣ 2 , ∂ Φ f ∂ z = n 0 ω 0 σ n cn N ¯ ,
∂ ( δ ω K ) ∂ z = − γ 0 ∂ ∣ A ∣ 2 ∂ τ , ∂ ( δ ω f ) ∂ z = − n 0 ω 0 σ n cn ∂ N ¯ ∂ τ .
∂ N ¯ ∂ τ = β T 2 h ¯ ω 0 a ¯ 2 [ ∣ A ∣ 4 − 1 τ 0 ∫ − ∞ τ e − ( τ − τ ′ ) ⁄ τ 0 ∣ A ( z , τ ′ ) ∣ 4 d τ ′ ] .
∂ ( δ ω f ) ∂ z ≈ − n 0 σ n β T ∣ A ∣ 4 2 cn h ¯ a ¯ 2 .
∂ ( δ ω fm ) ∂ z ≈ − n 0 σ n β T P 0 2 2 cn h ¯ a ¯ 2 .
r c ≡ ∣ ∂ ( δ ω fm ) ⁄ ∂ z ∣ ∣ ∂ ( δ ω Km ) ⁄ ∂ z ∣ ≈ n 0 ∣ σ n ∣ ℰ p 4 π 3 ⁄ 2 F n cn h ¯ a ¯ .
∂ A ∂ z = − α l 2 A + i γ e ∣ A ∣ 2 A ,
P ( z , τ ) = P ( 0 , τ ) exp ( − α l z ) 1 + β T P ( 0 , τ ) α l a [ 1 − exp ( − α l z ) ] .
Φ K ( L , τ ) = γ 0 a ¯ β T ln [ 1 + β T P ( 0 , τ ) a ¯ L eff ] ,
∂ A ∂ z + α l 2 A + i β 2 2 ∂ 2 A ∂ τ 2 = i γ e ∣ A ∣ 2 A .
∂ A p ∂ z − i ∑ m = 0 ∞ i m β mp m ! ∂ m A p ∂ t m = − α lp 2 A p + i β p f A p + i { γ pp ( 0 ) ∣ A p ∣ 2 + [ γ ps e + γ ps ( 0 ) ] ∣ A s ∣ 2 } A p
+ i γ ps R A s ∫ − ∞ t h R ( t − t ′ ) e i Ω ps ( t − t ′ ) A s * ( z , t ′ ) A p ( z , t ′ ) d t ′ ,
∂ A s ∂ z − i ∑ m = 0 ∞ i m β ms m ! ∂ m A s ∂ t m = − α ls 2 A s + i β s f A s + i { γ ss ( 0 ) ∣ A s ∣ 2 + [ γ sp e + γ sp ( 0 ) ] ∣ A p ∣ 2 } A s
+ i γ sp R A p ∫ − ∞ t h R ( t − t ′ ) e i Ω sp ( t − t ′ ) A p * ( z , t ′ ) A s ( z , t ′ ) d t ′ ,
g R ( ω u ) = 3 ω u g ′ Ω R η uv 2 ε 0 c 2 n u n v Γ R ,
γ uv e = γ 1111 e ( − ω u ; ω v , − ω v , ω u ) ( 1 + ρ ) ⁄ 2 .
∂ A p ∂ z + β 1 p ∂ A p ∂ t = i β p f A p + i ( γ pp e ∣ A p ∣ 2 + 2 γ ps e ∣ A s ∣ 2 ) A p ,
∂ A s ∂ z + β 1 s ∂ A s ∂ t = i β s f A s + i ( γ ss e ∣ A s ∣ 2 + 2 γ sp e ∣ A p ∣ 2 ) A s ,
∂ P p ∂ z + β 1 p ∂ P p ∂ t = − β Tpp ′ P p 2 − 2 β Tps ′ P s P p ,
∂ P s ∂ z + β 1 s ∂ P s ∂ t = − β Tss ′ P s 2 − 2 β Tsp ′ P s P p ,
β Tps ′ ω p = β Tsp ′ ω s ,
G ¯ = β Tpp ∣ A p ∣ 4 2 h ¯ ω p a ¯ pp 2 + β Tss ∣ A s ∣ 4 2 h ¯ ω s a ss 2 ¯ + 2 β Tps ∣ A p ∣ 2 ∣ A s ∣ 2 h ¯ ω p a ¯ ps 2 ,
∂ Φ K ∂ z = 2 Re ( γ sp e ) ∣ A p ∣ 2 , ∂ Φ f ∂ z = n 0 s ω s cn s σ ns N ¯ .
∂ Φ fm ∂ z = π n 0 s ω s σ ns β Tpp P 0 2 T 0 2 2 cn s h ¯ ω p a ¯ pp 2 .
r x ≡ ∣ ∂ Φ fm ⁄ ∂ z ∣ ∣ ∂ Φ Km ⁄ ∂ z ∣ = n 0 s ω s ∣ σ ns ∣ β Tpp ℰ p 4 2 cn s Re ( γ sp e ) h ¯ ω p a ¯ pp 2 ≈ π r c 2 2 ,
∂ A p ∂ z = i β p A p − α lp 2 A p + i β p f A p + i { γ pp ( 0 ) ∣ A p ∣ 2 + [ γ ps ( 0 ) + γ ps ( Ω ps ) ] ∣ A s ∣ 2 } A p ,
∂ A s ∂ z = i β s A s − α ls 2 A s + i β s f A s + i { γ ss ( 0 ) ∣ A s ∣ 2 + [ γ sp ( 0 ) + γ sp ( Ω sp ) ] ∣ A p ∣ 2 } A s .
∂ P p ∂ z = − ( α lp + α fp ) P p − β Tpp ′ P p 2 − 2 β Tps ′ P s P p − 2 γ ps R Im [ H ˜ R ( Ω ps ) ] P s P p ,
∂ P s ∂ z = − ( α ls + α fs ) P s − β Tss ′ P s 2 − 2 β Tsp ′ P p P s − 2 γ sp R Im [ H ˜ R ( Ω sp ) ] P p P s ,
α fs = n 0 s σ as β Tpp τ 0 P p 2 2 h ¯ ω p n s a ¯ pp 2 .
r a = n 0 s σ as β Tpp P p τ 0 a ¯ sp 4 h ¯ ω 0 n s β Tsp a ¯ pp 2 .
( g R − 2 β Tsp ) P p a ¯ sp − n 0 s σ as β Tpp τ 0 P p 2 2 h ¯ ω p n s a ¯ pp 2 − α ls > 0 .
τ 0 < τ th ≡ h ¯ ω p n s a ¯ pp 2 ( g R − 2 β Tsp ) 2 2 α ls σ as n 0 s β Tpp a ¯ sp 2 .
( g R − 2 β Tsp ) a ¯ sp ∫ 0 L P p dz − n 0 s σ as β Tpp τ 0 2 h ¯ ω p n s a ¯ pp 2 ∫ 0 L P p 2 dz − α ls L > 0 .
P th = ω p ω s n gp n gs V m 2 c 2 ( g R − 2 β Tsp ) Q ep Q ts Q tp 2 τ th τ 0 [ 1 − ( 1 − τ 0 τ th ) 1 ⁄ 2 ] ,
P th = ω p ω s n gp n gs V m 4 c 2 ( g R − 2 β Tsp ) Q ep Q ts Q tp 2 ,
P m = ω p ω s n gp n gs V m 2 c 2 ( g R − 2 β Tsp ) Q ep Q ts Q tp 2 τ th τ 0 [ 1 + ( 1 − τ 0 τ th ) 1 ⁄ 2 ] .
β R = g R P p Γ R Ω R a ¯ sp Ω R 2 − Ω 2 ( Ω R 2 − Ω 2 ) 2 + 4 Γ R 2 Ω 2 .
τ g = g R 2 Γ R a ¯ sp ∫ 0 L P p ( z ) dz .
∂ A ∂ z = ∑ m = 0 ∞ i m + 1 β m m ! ∂ m A ∂ t m + i γ e ∣ A ∣ 2 A + i γ f A ∫ − ∞ t e − ( t − t ′ ) ⁄ τ 0 ∣ A ( z , t ′ ) ∣ 4 dt ′ ,
γ f = β T 2 h ¯ ω 0 a ¯ 2 n 0 ( ω 0 ) n ( ω 0 ) [ ω 0 c σ n ( ω 0 ) + i 2 σ a ( ω 0 ) ] .
∂ A p ∂ z − i ∑ m = 0 ∞ i m β mp m ! ∂ m A p ∂ t m = i γ e ∣ A p ∣ 2 A p + i γ f A p ∫ − ∞ t e − ( t − t ′ ) ⁄ τ 0 ∣ A p ( z , t ′ ) ∣ 4 dt ′ ,
∂ A s ∂ z − i ∑ m = 0 ∞ i m β ms m ! ∂ m A s ∂ t m = 2 i γ e ∣ A p ∣ 2 A s + i γ e A p 2 A i * + i γ f A s ∫ − ∞ t e − ( t − t ′ ) ⁄ τ 0 ∣ A p ∣ 4 dt ′ ,
+ 2 i γ f A p ∫ − ∞ t e − ( t − t ′ ) ⁄ τ 0 e i Ω sp ( t − t ′ ) ∣ A p ∣ 2 ( A p * A s + A p A i * ) dt ′ ,
∂ A i ∂ z − i ∑ m = 0 ∞ i m β mi m ! ∂ m A i ∂ t m = 2 i γ e ∣ A p ∣ 2 A i + i γ e A p 2 A s * + i γ f A i ∫ − ∞ t e − ( t − t ′ ) ⁄ τ 0 ∣ A p ∣ 4 dt ′
+ 2 i γ f A p ∫ − ∞ t e − ( t − t ′ ) ⁄ τ 0 e i Ω ip ( t − t ′ ) ∣ A p ∣ 2 ( A p * A i + A p A s * ) dt ′ ,
∂ A p ∂ z = i β p A p + i γ e ∣ A p ∣ 2 A p + i γ f τ 0 ∣ A p ∣ 4 A p ,
∂ A s ∂ z = i β s A s + 2 i γ e ∣ A p ∣ 2 A s + i γ e A p 2 A i *
+ i γ f τ 0 ∣ A p ∣ 2 [ ∣ A p ∣ 2 A s + 2 1 + i Ω ps τ 0 ( ∣ A p ∣ 2 A s + A p 2 A i * ) ] ,
∂ A i ∂ z = i β i A i + 2 i γ e ∣ A p ∣ 2 A i + i γ e A p 2 A s *
+ i γ f τ 0 ∣ A p ∣ 2 [ ∣ A p ∣ 2 A i + 2 1 + i Ω pi τ 0 ( ∣ A p ∣ 2 A i + A p 2 A s * ) ] ,
r FWM = 2 γ f τ 0 ∣ A p ∣ 2 γ e ( 1 + i Ω ps τ 0 ) .
r FWM ≈ n 0 σ n τ 0 ∣ A p ∣ 2 hcn a ¯ F n ( 1 + i Ω ps τ 0 ) .
Δ κ f = 4 Re ( γ f ) τ 0 ∣ A p ∣ 4 1 + ( Ω ps τ 0 ) 2 .
r κ = 2 Re ( γ f ) τ 0 ∣ A p ∣ 2 Re ( γ e ) [ 1 + ( Ω ps τ 0 ) 2 ] .
∂ A p ∂ z + β 1 p ∂ A p ∂ t = i β p A p + i γ e ∣ A p ∣ 2 A p + i γ f T 0 ∣ A p ∣ 4 A p ,
∂ A s ∂ z + β 1 s ∂ A s ∂ t = i β s A s + 2 i γ e ∣ A p ∣ 2 A s + i γ e A p 2 A i * + i γ f T 0 ∣ A p ∣ 2 ( 3 ∣ A p ∣ 2 A s + 2 A p 2 A i * ) ,
∂ A i ∂ z + β 1 i ∂ A i ∂ t = i β i A i + 2 i γ e ∣ A p ∣ 2 A i + i γ e A p 2 A s * + i γ f T 0 ∣ A p ∣ 2 ( 3 ∣ A p ∣ 2 A i + 2 A p 2 A s * ) ,
r FWM = 2 γ f T 0 ∣ A p ∣ 2 ⁄ γ e , r κ = 2 Re ( γ f ) T 0 ∣ A p ∣ 2 ⁄ Re ( γ e ) .
∂ A p ∂ z = [ i ( β p + β p f ) − α lp 2 ] A p + i ( γ p e P p + 2 γ ps e P s + 2 γ pi e P i ) A p + 2 i γ pspi e A s A i A p * ,
∂ A s ∂ z = [ i ( β s + β s f ) − α ls 2 ] A s + i ( γ s e P s + 2 γ sp e P p + 2 γ si e P i ) A s + i γ spip e A p 2 A i * ,
∂ A i ∂ z = [ i ( β i + β i f ) − α li 2 ] A i + i ( γ i e P i + 2 γ ip e P p + 2 γ is e P s ) A i + i γ ipsp e A p 2 A p * ,
γ pspi e = 3 ω p η pspi χ 1111 e ( − ω p ; ω s , − ω p , ω i ) 4 ε 0 c 2 a ¯ pspi n p n s n i ,
χ 1111 e ( − ω s ; ω p , − ω i , ω p ) = χ 1111 e ( − ω i ; ω p , − ω s , ω p ) = [ χ 1111 e ( − ω p ; ω s , − ω p , ω i ) ] * .
γ spip e ω s = γ ipsp e ω i = γ pspi e * ω p .
1 ω s ∂ P s ∂ z = 1 ω i ∂ P i ∂ z = − 1 2 ω p ∂ P p ∂ z .
G ¯ = ∑ v = p , s , i β Tvv ∣ A p ∣ 4 2 h ¯ ω v a ¯ vv 2 + ∑ u , v = p , s , i u ≠ v β Tuv ∣ A u ∣ 2 ∣ A v ∣ 2 h ¯ ω u a ¯ uv 2 ,
∂ A p ∂ z = [ i ( β p + β p f ) − α lp 2 ] A p + i γ p e P p A p ,
∂ A s ∂ z = [ i ( β s + β s f ) − α ls 2 ] A s + 2 i γ sp e P p A s + i γ spip e A p 2 A i * ,
∂ A i ∂ z = [ i ( β i + β i f ) − α li 2 ] A i + 2 i γ ip e P p A i + i γ ipsp e A p 2 A s * .
κ = Δ β + Δ β f + 2 P p Re ( γ sp e + γ ip e * − γ p e ) ,
Δ β f = N ¯ σ np ω p 2 c 2 ( n 0 s β s + n 0 i β i − 2 n 0 p β p ) = 2 N ¯ σ np ω p 2 c 2 ∑ m = 1 ∞ ζ 2 m ( 2 m ) ! Ω sp 2 m ,
β 2 = N ¯ σ np ω p 2 ζ 2 ⁄ c 2 , β 4 = N ¯ σ np ω p 2 ζ 4 ⁄ ( c 2 ) .
2 [ Re ( γ spip e ) − β Tsp ′ ] P p − n 0 s σ as β Tpp τ 0 P p 2 2 h ¯ ω p n s a ¯ pp 2 − α ls > 0 ,
τ 0 < 2 h ¯ ω p n s a ¯ pp 2 α 1 s σ as n 0 s β Tpp [ Re ( γ spip e ) − β Tsp ′ ] 2 ≈ 2 h ¯ ω p n s β Tpp α 1 s σ as n 0 s ( 2 π F n − 1 ) 2 ,
∂ A p ∂ z = [ i ( β p + β p f ) − α lp 2 ] A p + i γ p ( 0 ) P p A p ,
∂ A s ∂ z = [ i ( β s + β s f ) − α 1 s 2 ] A s + i [ γ sp ( 0 ) + γ sp ( Ω sp ) ] P p A s + i γ spip ( Ω sp ) A p 2 A i * ,
∂ A i ∂ z = [ i ( β i + β i f ) − α 1 i 2 ] A i + i [ γ ip ( 0 ) + γ ip ( Ω ip ) ] P p A i + i γ ipsp ( Ω ip ) A p 2 A s * ,
∂ A ̂ s ∂ z = [ i ( β s + β s f ) − 1 2 α 1 s ] A ̂ s + i [ γ sp ( 0 ) + γ sp ( Ω sp ) ] P p A ̂ s + i γ spip ( Ω sp ) A p 2 A ̂ i †
+ m ^ l ( z , ω s ) + m ^ f ( z , ω s ) + m ^ T ( z , ω s ) A p + i m ^ R ( z , Ω sp ) A p .
[ m ̂ j ( z 1 , ω u ) , m ̂ j † ( z 2 , ω v ) ] = 2 π D j δ ( ω u − ω v ) δ ( z 1 − z 2 ) ,
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