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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2007 (30)

L. Yin, Q. Lin, and G. P. Agrawal, "Soliton fission and supercontinuum generation in silicon waveguides," Opt. Lett. 32, 391-393 (2007).
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J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, "Optical solitons in a silicon waveguide," Opt. Express 15, 7682-7688 (2007).
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P. Koonath, D. R. Solli, and B. Jalali, "Continuum generation and carving on a silicon chip," Appl. Phys. Lett. 91, 061111 (2007).
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R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, "All-optical regeneration on a silicon chip," Opt. Express 15, 7802-7809 (2007).
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R. Dekker, N. Usechak, M. Först, and A. Driessen, "Ultrafast nonlinear all-optical processes in silicon-oninsulator waveguides," J. Phys. D: Appl. Phys. 40, R249-R271 (2007).
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L. Yin and G. P. Agrawal, "Impact of two-photon absorption on self-phase modulation in silicon waveguides," Opt. Lett. 32, 2031-2033 (2007).
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N. Suzuki, "FDTD analysis of two-photon absorption and free-carrier absorption in Si high-index-contrast waveguides," J. Lightwave Technol. 25, 2495-2501 (2007).
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I-W. Hsieh, X. Chen, X. Liu, J. I. Dadap, N. C. Panoiu, C-Y. Chou, F. Xia,W. M. Green, Y. A. Vlasov, and R. M. Osgood, Jr., "Supercontinuum generation in silicon photonic wires," Opt. Express 15, 15242-15248 (2007).
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H. Rong, S. Xu, Y. Kuo, V. Sih, O. Cohen, O. Raday, and M. Paniccia, "Low-threshold continuous-wave Raman silicon laser," Nature Photon. 1, 232-237 (2007).
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X. Yang and C.W. Wong, "Coupled-mode theory for stimulated Raman scattering in high-Q/Vm silicon photonic band gap defect cavity lasers," Opt. Express 15, 4763-4780 (2007).
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V. Sih, S. Xu, Y. Kuo, H. Rong, M. Paniccia, O. Cohen, and O. Raday, "Raman amplification of 40 Gb/s data in low-loss silicon waveguides," Opt. Express 15, 357-362 (2007).
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V. Raghunathan, H. Renner, R. R. Rice, and B. Jalali, "Self-imaging silicon Raman amplifier," Opt. Express 15, 3396-3408 (2007).
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F. De Leonardis and V. M. N. Passaro, "Modelling of Raman amplification in silicon-on-insulator optical microcavities," New J. Phys. 9, 25 (2007).
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F. De Leonardis and V. M. N. Passaro, "Modeling and performance of a guided-wave optical angular-velocity sensor based on Raman effect in SOI," IEEE J. Lightwave Technol. 25, 2352-2366 (2007).
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V. Raghunathan, D. Borlaug, R. R. Rice, and B. Jalali, "Demonstration of a mid-infrared silicon Raman ampli-fier," Opt. Express 15, 14355-14362 (2007).
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I-W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, R. M. Osgood, Jr., S. J. McNab, and Y. A. Vlasov, "Crossphase modulation-induced spectral and temporal effects on co-propagating femtosecond pulses in silicon photonic wires," Opt. Express 15, 1135-1146 (2007).
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N. Vermeulen, C. Debaes, and H. Thienpont, "Mitigating heat dissipation in near- and mid-infrared silicon-based Raman lasers using CARS," IEEE J. Sel. Top. Quantum Electron. 13, 770-787 (2007).
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S. Ayotte, H. Rong, S. Xu, O. Cohen, and M. Paniccia, "Multichannel dispersion compensation using a silicon waveguide-based optical phase conjugator," Opt. Lett. 32, 2393-2395 (2007).
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M. A. Foster, A. C. Turner, R. Salem, M. Lipson, and A. L. Gaeta, "Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides," Opt. Express 15, 12949-12958 (2007).
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A. D. Bristow, N. Rotenberg, and H. M. van Driel, "Two-photon absorption and Kerr coefficients of silicon for 850-2200 nm," Appl. Phys. Lett. 90, 191104 (2007).
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Q. Lin, J. Zhang, G. Piredda, R.W. Boyd, P. M. Fauchet, and G. P. Agrawal, "Dispersion of silicon nonlinearities in the near-infrared region," Appl. Phys. Lett. 90, 021111 (2007).
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J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, "Anisotropic nonlinear response of silicon in the near-infrared region," Appl. Phys. Lett. 90, 071113 (2007).
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E. Tien, N. S. Yuksek, F. Qian, and O. Boyraz, "Pulse compression and modelocking by using TPA in silicon waveguides," Opt. Express 15, 6500-6506 (2007).
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T. Kagawa and S. Ooami, "Polarization dependence of two-photon absorption in Si avalanche photodiodes," Jpn. J. Appl. Phys. 46, 664-668 (2007).
[CrossRef]

Y. Liu and H. K. Tsang, "Time dependent density of free carriers generated by two photon absorption in silicon waveguides," Appl. Phys. Lett. 90, 211105 (2007).
[CrossRef]

M. Först, J. Niehusmann, T. Plötzing, J. Bolten, T. Wahlbrink, C. Moormann, and H. Kurz, "High-speed alloptical switching in ion-implanted silicon-on-insulator microring resonators," Opt. Lett. 32, 2046-2048 (2007).
[CrossRef] [PubMed]

T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, and M. Notomi, "Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities," Appl. Phys. Lett. 90, 031115 (2007).
[CrossRef]

T. Torounidis and P. Andrekson, "Broadband single-pumped fiber-optic parametric amplifiers," IEEE Photon. Technol. Lett. 19, 650-652 (2007).
[CrossRef]

J. M. Chavez Boggio, J. D. Marconi, S. R. Bickham, and H. L. Fragnito, "Spectrally flat and broadband double-pumped fiber optical parametric amplifiers," Opt. Express 15, 5288-5309 (2007).
[CrossRef] [PubMed]

Q. Lin, F. Yaman, and G. P. Agrawal, "Photon-pair generation in optical fibers through four-wave mixing: Role of Raman scattering and pump polarization," Phys. Rev. A 75, 023803 (2007).
[CrossRef]

2006 (39)

Q. Lin, F. Yaman, and G. P. Agrawal, "Photon-pair generation by four-wave mixing in optical fibers," Opt. Lett. 31, 1286-1288 (2006).
[CrossRef] [PubMed]

K. F. Lee, J. Chen, C. Liang, X. Li, P. L. Voss, and P. Kumar, "Generation of high-purity telecom-band entangled photon pairs in dispersion-shifted fiber," Opt. Lett. 31, 1905-1907 (2006).
[CrossRef] [PubMed]

B. Jalali, V. Raghunathan, R. Shori, S. Fathpour, D. Dimitropoulos, and O. Stafsudd, "Prospects for silicon Mid-IR Raman Lasers," IEEE J. Sel. Top. Quantum Electron. 12, 1618-1627 (2006).
[CrossRef]

M. Dinu, D. C. Kilper, H. R. Stuart, "Optical performance monitoring using data stream intensity autocorrelation," IEEE J. Lightwave Technol. 24, 1194-1202 (2006).
[CrossRef]

S. Fathpour, K. K. Tsia, and B. Jalali, "Energy harvesting in silicon Raman amplifiers," Appl. Phys. Lett. 89, 061109 (2006).
[CrossRef]

K. K. Tsia, S. Fathpour, and B. Jalali, "Energy harvesting in silicon wavelength converters," Opt. Express 14, 12327-12333 (2006).
[CrossRef] [PubMed]

Y. Liu and H. K. Tsang, "Nonlinear absorption and Raman gain in helium-ion-implanted silicon waveguides," Opt. Lett. 31, 1714-1716 (2006).
[CrossRef] [PubMed]

P. St. J. Russell, "Photonic crystal fibers," IEEE J. Lightwave Technol. 24, 4729-4749 (2006).
[CrossRef]

A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, "Tailored anomalous group-velocity dispersion in silicon channel waveguides," Opt. Express 14, 4357-4362 (2006).
[CrossRef] [PubMed]

T. K. Liang, L. R. Nunes, M. Tsuchiya, K. S. Abedin, T. Miyazaki, D. Van Thourhout, W. Bogaerts, P. Dumon, R. Baets, and H. K. Tsang, "High speed logic gate using two-photon absorption in silicon waveguides," Opt. Commun. 265, 171-174 (2006).
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J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
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V. Raghunathan, R. Shori, O. M. Stafsudd, B. Jalali, "Nonlinear absorption in silicon and the prospects of midinfrared silicon Raman lasers," Physica Status Solidi A 203, R38-R40 (2006).
[CrossRef]

T. J. Johnson,M. Borselli, and O. Painter, "Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator," Opt. Express 14, 817-831 (2006).
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Q. Lin, F. Yaman, and G. P. Agrawal, "Photon-pair generation in optical fibers through four-wave mixing: Role of Raman scattering and pump polarization," Phys. Rev. A 75, 023803 (2007).
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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.

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.

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].)

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].)

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].)

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].)

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].)

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].)

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].)

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].)

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 ij v kl v + g H ˜ R ( ω j ω k ) v = x , y , z il v jk v ,
H ˜ R ( Ω ) = Ω R 2 Ω R 2 Ω 2 2 i Γ R Ω .
ij x = δ iy δ jz + δ iz δ jy , ij y = δ ix δ jz + δ iz δ jx , 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 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 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 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