Abstract

We show that ultrabroadband parametric generation and wavelength conversion can be realized in silicon waveguides in the wavelength region near 1550 nm by tailoring their zero-dispersion wavelength and launching pump wave close to this wavelength. We quantify the impact of two-photon absorption, free-carrier generation, and linear losses on the process of parametric generation and show that it is difficult to realize a net signal gain and transparent wavelength conversion with a continuous-wave pump. By investigating the transient dynamics of the four-wave mixing process initiated with a pulsed pump, we show that the instantaneous nature of electronic response enables highly efficient parametric amplification and wavelength conversion for pump pulses as wide as 1 ns. We also discuss the dual-pump configuration and show that its use permits multiband operation with uniform efficiency over a broad spectral region extending over 300 nm.

© 2006 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  4. R. L. Espinola, J. I. Dadap, R. M. Osgood, Jr., S. J.  McNab, and Y. A. Vlasov, "Raman amplification in ultrasmall silicon-on-insulator wire waveguides," Opt. Express 12, 3713 (2004).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  7. V. Raghunathan, O. Boyraz, and B. Jalali, "20 dB on-off Raman amplification in silicon waveguides," Proc. CLEO,  1, 349-351 (2005).
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    [CrossRef] [PubMed]
  9. R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, "Anti-stokes Raman conversion in silicon waveguides," Opt. Express 11, 2862-2872 (2003).
    [CrossRef] [PubMed]
  10. V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, "Parametric Raman wavelength conversion in scaled silicon waveguides," J. Lightwave Technol. 23, 2094-2102 (2005).
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  26. D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, "Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides," Appl. Phys. Lett. 86, 071115 (2005).
    [CrossRef]
  27. X. Chen, N. C. Panoiu, and R. M. Osgood, Jr., "Theory of Raman-mediated pulsed amplification in silicon-wire waveguides," IEEE J. Quantum. Electron. 42, 160-170 (2006).
    [CrossRef]
  28. M. E. Marhic, N. Kagi, T. K. Chang, and L. G. Kazovsky, "Broadband fiber optical parametric amplifiers," Opt. Lett. 21, 573-575 (1996).
    [CrossRef] [PubMed]
  29. M. E. Marhic, Y. Park, F. S. Yang, and L. G. Kazovsky, "Broadband fiber-optical parametric amplifiers and wavelength converters with low-ripple Chebyshev gain spectra," Opt. Lett. 21, 1354-1356 (1996).
    [CrossRef] [PubMed]
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    [CrossRef]

2006

2005

Q. Lin, R. Jiang, C. F. Marki, C. J. McKinstrie, R. Jopson, J. Ford, G. P. Agrawal, and S. Radic, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device," IEEE Photon. Technol. Lett. 17, 2376-2378 (2005).
[CrossRef]

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, "Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides," Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

S. F. Preble, Q. Xu, B. S. Schmidt, and M. Lipson, "Ultrafast all-optical modulation on a silicon chip," Opt. Lett. 30, 2891-2893 (2005).
[CrossRef] [PubMed]

V. Raghunathan, O. Boyraz, and B. Jalali, "20 dB on-off Raman amplification in silicon waveguides," Proc. CLEO,  1, 349-351 (2005).

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, "A continuous-wave Raman silicon laser," Nature 433, 725-728 (2005).
[CrossRef] [PubMed]

V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, "Parametric Raman wavelength conversion in scaled silicon waveguides," J. Lightwave Technol. 23, 2094-2102 (2005).
[CrossRef]

R. L. Espinola, J. I. Dadap, R. M. Osgood, Jr., S. J. McNab, and Y. A. Vlasov, "C-band wavelength conversion in silicon photonic wire waveguides," Opt. Express 13, 4341-4349 (2005).
[CrossRef] [PubMed]

H. Fukuda, K. Yamada, T. Shoji, M. Takahashi, T. Tsuchizawa, T. Watanabe, J. Takahashi, and S. Itabashi, "Four-wave mixing in silicon wire waveguides," Opt. Express 13, 4629-4637 (2005).
[CrossRef] [PubMed]

2004

2003

2002

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, "Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength," Appl. Phys. Lett. 80, 416-418 (2002).
[CrossRef]

C. J. McKinstrie and S. Radic, "Parametric amplifiers driven by two pump waves with dissimilar frequencies," Opt. Lett. 27, 1138 (2002).
[CrossRef]

2001

R. Loudon, "The Raman effect in crystals," Adv. in Phys. 50, 813-864 (2001).
[CrossRef]

1996

1989

1980

H. H. Li, "Refractive-index of silicon and germanium and its wavelength and temperature derivatives," J. Phys. Chem. Ref. Data,  9, 561-658 (1980).
[CrossRef]

1969

J. J. Wynne, "Optical third-order mixing in GaAs, Ge, Si, and InAs," Phys. Rev. 178, 1295 (1969).
[CrossRef]

1965

P. D. Maker and R. W. Terhune, "Study of optical effects due to an induced polarization third order in the electric field strength," Phys. Rev. 137, A801-A818 (1965).
[CrossRef]

Y. R. Shen and N. Bloembergen, "Theory of stimulated Brillouin and Raman scattering," Phys. Rev. 137, A1787-A1805 (1965).
[CrossRef]

Agrawal, G. P.

L. Yin, Q. Lin, and G. P. Agrawal, "Dispersion tailoring and soliton propagation in silicon waveguides," Opt. Lett. 31, 1295-1297 (2006).
[CrossRef] [PubMed]

Q. Lin, R. Jiang, C. F. Marki, C. J. McKinstrie, R. Jopson, J. Ford, G. P. Agrawal, and S. Radic, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device," IEEE Photon. Technol. Lett. 17, 2376-2378 (2005).
[CrossRef]

C. HeadleyIII and G. P. Agrawal, "Unified description of ultrafast stimulated Raman scattering in optical fibers," J. Opt. Soc. Am. B 13, 2170-2177 (1996).
[CrossRef]

Almeida, V. R.

Asghari, M.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, "Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength," Appl. Phys. Lett. 80, 416-418 (2002).
[CrossRef]

Barrios, C. A.

Bloembergen, N.

Y. R. Shen and N. Bloembergen, "Theory of stimulated Brillouin and Raman scattering," Phys. Rev. 137, A1787-A1805 (1965).
[CrossRef]

Boyraz, O.

V. Raghunathan, O. Boyraz, and B. Jalali, "20 dB on-off Raman amplification in silicon waveguides," Proc. CLEO,  1, 349-351 (2005).

Chang, T. K.

Chen, X.

X. Chen, N. C. Panoiu, and R. M. Osgood, Jr., "Theory of Raman-mediated pulsed amplification in silicon-wire waveguides," IEEE J. Quantum. Electron. 42, 160-170 (2006).
[CrossRef]

Claps, R.

Cohen, O.

Dadap, J. I.

Day, I. E.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, "Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength," Appl. Phys. Lett. 80, 416-418 (2002).
[CrossRef]

Dimitropoulos, D.

Drake, J.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, "Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength," Appl. Phys. Lett. 80, 416-418 (2002).
[CrossRef]

Espinola, R. L.

Fang, A.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, "A continuous-wave Raman silicon laser," Nature 433, 725-728 (2005).
[CrossRef] [PubMed]

Ford, J.

Q. Lin, R. Jiang, C. F. Marki, C. J. McKinstrie, R. Jopson, J. Ford, G. P. Agrawal, and S. Radic, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device," IEEE Photon. Technol. Lett. 17, 2376-2378 (2005).
[CrossRef]

Foster, M. A.

Fukuda, H.

Gaeta, A. L.

Hak, D.

Han, Y.

Harpin, A.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, "Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength," Appl. Phys. Lett. 80, 416-418 (2002).
[CrossRef]

Headley, C.

Itabashi, S.

Jalali, B.

Jhaveri, R.

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, "Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides," Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

Jiang, R.

Q. Lin, R. Jiang, C. F. Marki, C. J. McKinstrie, R. Jopson, J. Ford, G. P. Agrawal, and S. Radic, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device," IEEE Photon. Technol. Lett. 17, 2376-2378 (2005).
[CrossRef]

Jones, R.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, "A continuous-wave Raman silicon laser," Nature 433, 725-728 (2005).
[CrossRef] [PubMed]

Jopson, R.

Q. Lin, R. Jiang, C. F. Marki, C. J. McKinstrie, R. Jopson, J. Ford, G. P. Agrawal, and S. Radic, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device," IEEE Photon. Technol. Lett. 17, 2376-2378 (2005).
[CrossRef]

Kagi, N.

Kazovsky, L. G.

Kuo, Y.

Li, H. H.

H. H. Li, "Refractive-index of silicon and germanium and its wavelength and temperature derivatives," J. Phys. Chem. Ref. Data,  9, 561-658 (1980).
[CrossRef]

Liang, T. K.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, "Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength," Appl. Phys. Lett. 80, 416-418 (2002).
[CrossRef]

Lin, Q.

L. Yin, Q. Lin, and G. P. Agrawal, "Dispersion tailoring and soliton propagation in silicon waveguides," Opt. Lett. 31, 1295-1297 (2006).
[CrossRef] [PubMed]

Q. Lin, R. Jiang, C. F. Marki, C. J. McKinstrie, R. Jopson, J. Ford, G. P. Agrawal, and S. Radic, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device," IEEE Photon. Technol. Lett. 17, 2376-2378 (2005).
[CrossRef]

Lipson, M.

Liu, A.

Loudon, R.

R. Loudon, "The Raman effect in crystals," Adv. in Phys. 50, 813-864 (2001).
[CrossRef]

Maker, P. D.

P. D. Maker and R. W. Terhune, "Study of optical effects due to an induced polarization third order in the electric field strength," Phys. Rev. 137, A801-A818 (1965).
[CrossRef]

Marhic, M. E.

Marki, C. F.

Q. Lin, R. Jiang, C. F. Marki, C. J. McKinstrie, R. Jopson, J. Ford, G. P. Agrawal, and S. Radic, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device," IEEE Photon. Technol. Lett. 17, 2376-2378 (2005).
[CrossRef]

McKinstrie, C. J.

Q. Lin, R. Jiang, C. F. Marki, C. J. McKinstrie, R. Jopson, J. Ford, G. P. Agrawal, and S. Radic, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device," IEEE Photon. Technol. Lett. 17, 2376-2378 (2005).
[CrossRef]

C. J. McKinstrie and S. Radic, "Parametric amplifiers driven by two pump waves with dissimilar frequencies," Opt. Lett. 27, 1138 (2002).
[CrossRef]

McNab, S. J.

McNab, S. J.

Moss, D. J.

Osgood, R. M.

Ouzounov, D. G.

Panepucci, R. R.

Paniccia, M.

Panoiu, N. C.

X. Chen, N. C. Panoiu, and R. M. Osgood, Jr., "Theory of Raman-mediated pulsed amplification in silicon-wire waveguides," IEEE J. Quantum. Electron. 42, 160-170 (2006).
[CrossRef]

Park, Y.

Preble, S. F.

Radic, S.

Q. Lin, R. Jiang, C. F. Marki, C. J. McKinstrie, R. Jopson, J. Ford, G. P. Agrawal, and S. Radic, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device," IEEE Photon. Technol. Lett. 17, 2376-2378 (2005).
[CrossRef]

C. J. McKinstrie and S. Radic, "Parametric amplifiers driven by two pump waves with dissimilar frequencies," Opt. Lett. 27, 1138 (2002).
[CrossRef]

Raghunathan, V.

Roberts, S. W.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, "Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength," Appl. Phys. Lett. 80, 416-418 (2002).
[CrossRef]

Rong, H.

Schmidt, B. S.

Shen, Y. R.

Y. R. Shen and N. Bloembergen, "Theory of stimulated Brillouin and Raman scattering," Phys. Rev. 137, A1787-A1805 (1965).
[CrossRef]

Shoji, T.

Sipe, J. E.

Takahashi, J.

Takahashi, M.

Terhune, R. W.

P. D. Maker and R. W. Terhune, "Study of optical effects due to an induced polarization third order in the electric field strength," Phys. Rev. 137, A801-A818 (1965).
[CrossRef]

Tsang, H. K.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, "Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength," Appl. Phys. Lett. 80, 416-418 (2002).
[CrossRef]

Tsuchizawa, T.

van Driel, H. M.

Vlasov, Y. A.

Watanabe, T.

Wong, C. S.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, "Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength," Appl. Phys. Lett. 80, 416-418 (2002).
[CrossRef]

Woo, J. C. S.

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, "Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides," Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

Wynne, J. J.

J. J. Wynne, "Optical third-order mixing in GaAs, Ge, Si, and InAs," Phys. Rev. 178, 1295 (1969).
[CrossRef]

Xu, Q.

Yamada, K.

Yang, F. S.

Yin, L.

Adv. in Phys.

R. Loudon, "The Raman effect in crystals," Adv. in Phys. 50, 813-864 (2001).
[CrossRef]

Appl. Phys. Lett.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, "Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength," Appl. Phys. Lett. 80, 416-418 (2002).
[CrossRef]

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, "Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides," Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

IEEE J. Quantum. Electron.

X. Chen, N. C. Panoiu, and R. M. Osgood, Jr., "Theory of Raman-mediated pulsed amplification in silicon-wire waveguides," IEEE J. Quantum. Electron. 42, 160-170 (2006).
[CrossRef]

IEEE Photon. Technol. Lett.

Q. Lin, R. Jiang, C. F. Marki, C. J. McKinstrie, R. Jopson, J. Ford, G. P. Agrawal, and S. Radic, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device," IEEE Photon. Technol. Lett. 17, 2376-2378 (2005).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. B

J. Phys. Chem. Ref. Data

H. H. Li, "Refractive-index of silicon and germanium and its wavelength and temperature derivatives," J. Phys. Chem. Ref. Data,  9, 561-658 (1980).
[CrossRef]

Nature

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, "A continuous-wave Raman silicon laser," Nature 433, 725-728 (2005).
[CrossRef] [PubMed]

Opt. Express

R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, "Anti-stokes Raman conversion in silicon waveguides," Opt. Express 11, 2862-2872 (2003).
[CrossRef] [PubMed]

R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, "Observation of stimulated Raman amplification in silicon waveguides," Opt. Express 11, 1731-1739 (2003).
[CrossRef] [PubMed]

R. L. Espinola, J. I. Dadap, R. M. Osgood, Jr., S. J.  McNab, and Y. A. Vlasov, "Raman amplification in ultrasmall silicon-on-insulator wire waveguides," Opt. Express 12, 3713 (2004).
[CrossRef] [PubMed]

A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, "Net optical gain in a low loss silicon-on-insulator waveguide by stimulated Raman scattering," Opt. Express 12, 4261 (2004).
[CrossRef] [PubMed]

Q. Xu, V. R. Almeida, and M. Lipson, "Time-resolved study of Raman gain in highly confined silicon-oninsulator waveguides," Opt. Express 12, 4437 (2004).
[CrossRef] [PubMed]

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Figures (9)

Fig. 1.
Fig. 1.

(a) Structure of the SOI waveguide with a 50% etching depth; two insets show spatial profiles for the TE and TM modes. (b) Second- (blue) and third-order (red) dispersion parameters for the TE and TM modes.

Fig. 2.
Fig. 2.

(a) Structure of the SOI waveguide with 100% etching depth; two insets show spatial profiles for the TE and TM modes. (b) Second- (blue) and third-order (red) dispersion parameters for the TE and TM modes.

Fig. 3.
Fig. 3.

Signal gain (a) and conversion efficiency (b) as a function of signal wavelength at three pump wavelengths in the vicinity of the ZDWL of the TM mode. Input pump intensity is 0.2 GW/cm2 in all cases. The dashed vertical line shows the location of ZDWL.

Fig. 4.
Fig. 4.

(a) Gs (dashed curves) and Gi (solid curves) as a function of input pump intensity for several values of carrier lifetime. (b) Upper limit of carrier lifetime τ0 and linear propagation loss αs for efficient FWM. In the TE case, pump-signal detuning is set to be 15.54 THz, where the contribution of Raman nonlinearity is maximum.

Fig. 5.
Fig. 5.

(a) Signal gain (a) and conversion efficiency (b) for the TE mode under the same conditions as in Fig. 3.

Fig. 6.
Fig. 6.

Signal gain (a) and conversion efficiency (b) for the TM mode at three pump wavelengths for pump pulses with a peak intensity of 0.6 GW=cm2 when FCA is neglected.

Fig. 7.
Fig. 7.

Input and output temporal profiles of (a) pump and (b) signal for a carrier lifetime of τ0=1 ns; red curve shows the idler pulse. The 16.7-ps pump pulses at 1571.3 nm have a peak intensity of 0.6 GW/cm2. Both signal and idler profiles are normalized by the input signal intensity.

Fig. 8.
Fig. 8.

Temporal profiles of (a) pump, (b) idler, and (c) signal pulses for three values of carrier lifetime. All Other parameters are the same as in Fig. 7. (d) Signal temporal profiles plotted on a log scale.

Fig. 9.
Fig. 9.

Spectra of parametric gain (blue curve) and conversion efficiency (red curve) for the TM mode pumped with two waves of a same input intensity of 0.3 GW/cm2. The gain exceeds 10 dB over a 350-nm bandwsidth when two pump wavelengths are 255.3 nm apart.

Equations (23)

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E ( z , t ) = Re [ A p ( z , t ) e i ω p t + A s ( z , t ) e i ω s t + A i ( z , t ) e i ω i t ] ,
A p z + β 1 p A p t + i β 2 p 2 2 A p t 2 = 1 2 [ α p + α fp ( z , t ) ] A p + i β 0 p A p + i ( γ e + γ R ) I p A p ,
A s z + β 1 s A s t + i β 2 s 2 2 A s t 2 = 1 2 [ α s + α fs ( z , t ) ] A s + i β 0 s A s + i ( 2 γ e + γ R ) I p A s + i γ e A p 2 A i *
+ i γ R A p t h R ( t τ ) e i Ω sp ( t τ ) [ A p * ( z , τ ) A s ( z , τ ) + A p ( z , τ ) A i * ( z , τ ) ] d τ ,
γ e = ξ e ( γ 0 + i β T 2 ) , γ R = ξ R g R Γ R Ω R , H ˜ R ( Ω ) = Ω R 2 Ω R 2 Ω 2 2 i Γ R Ω ,
N eh t = ξ e β T A p ( z , t ) 4 2 ω p N eh τ 0 ,
A s z + β 1 s A s t + i β 2 s 2 2 A s t 2 = 1 2 [ α s + α fs ( z , t ) ] A s + i β 0 s A s
+ i [ 2 γ e + γ R + γ R H ˜ R ( Ω sp ) ] A p 2 A s + i [ γ e + γ R H ˜ R ( Ω sp ) ] A p 2 A i * .
κ = Δ β 0 + 2 A p 2 Re [ γ e + γ R H ˜ R ( Ω sp ) ] ,
Δ β 0 = β 2 p Ω sp 2 + β 4 p 12 Ω sp 4 + ,
G j = 10 log 10 [ A j ( L ) 2 A s ( 0 ) 2 ] ( j = s , i ) ,
η f 2 I p Re [ γ e + γ R H ˜ R ( Ω sp ) ] 2 ξ e β T I p σ s I p 2 α s ,
A l z + β 1 l A l t + i β 2 l 2 2 A l t 2 = 1 2 [ α l + α f l ( z , l ) ] A l + i β 0 l A l + i ( γ e + γ R ) A l 2 A l
+ i ( 2 γ e + γ R ) A h 2 A l + i γ R A h t h R ( t τ ) e i Ω l h ( t τ ) A h * ( z , τ ) A l ( z , τ ) d τ ,
A s z + β 1 s A s t + i β 2 s 2 2 A s t 2 = 1 2 [ α s + α fs ( z , t ) ] A s + i β 0 s A s + i ( 2 γ e + γ R ) I p A s + 2 i γ e A l A h A i *
+ i γ R j , k = l , h A j t h R ( t τ ) e i Ω s j ( t τ ) [ A j * ( z , τ ) A s ( z , τ ) + A k ( z , τ ) A i * ( z , τ ) ] d τ ,
A l z + β 1 l A l t + i β 2 l 2 2 A l t 2 = 1 2 [ α l + α f l ( z , t ) A l + i β 0 l A l
+ i ( γ e + γ R ) A l 2 + A l + i [ 2 γ e + γ R + γ R H ˜ R ( Ω l h ) ] A h 2 A l ,
A s z + β 1 s A s t + i β 2 s 2 2 A s t 2 = 1 2 [ α s + α fs ( z , t ) ] A s + i β 0 s A s
+ i [ 2 γ e + γ R + γ R H ˜ R ( Ω sl ) ] A l 2 A s + i [ 2 γ e + γ R + γ R H ˜ R ( Ω sh ) ] A h 2 A s
+ i [ 2 γ e + γ R H ˜ R ( Ω sl ) + γ R H ˜ R ( Ω sh ) ] A l A h A i * .
κ = Δ β 0 + I p Re [ γ e + γ R H ˜ R ( Ω sl ) + γ R H ˜ R ( Ω sh ) γ R H ˜ R ( Ω hl ) ] ,
Δ β 0 = [ β 2 c Ω sc 2 + β 4 c 12 Ω sc 4 ] [ β 2 c Ω d 2 + β 4 c 12 Ω d 4 ] + .

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