Abstract

We present a generic approach to determine the phase mismatch for any optical nonlinear process. When applying this approach, which is based on the evaluation of local phase changes, to Raman- and Kerr-based four-wave mixing in silicon waveguides, we obtain an expression for the phase mismatch which is more accurate as compared to the conventional definition; and which contains additional contributions due to the dispersion of the four-wave-mixing processes. Furthermore, starting from the general propagation equations for the involved pump, Stokes and anti-Stokes waves, we investigate the impact of this four-wave-mixing dispersion in silicon waveguides and examine how it is influenced by changing the frequency difference between the pump and Stokes input waves. We show by means of numerical simulations that, by detuning this frequency difference slightly away from Raman resonance, the four-wave-mixing conversion efficiency can be more than doubled, but can also lead to a decrease in efficiency of more than 10 dB. We also discuss how the pump-Stokes frequency difference that is optimal for wavelength conversion varies with the length of the silicon waveguides and with their dispersion characteristics. Finally, starting from the newly introduced phase mismatch formula we simplify the set of propagation equations such that they are less computationally intensive to solve while still giving accurate estimates of the optimal pump-Stokes frequency difference and the corresponding wavelength conversion efficiency.

© 2011 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Anti-Stokes Raman conversion in silicon waveguides,” Opt. Express 11, 2862–2872 (2003).
    [CrossRef] [PubMed]
  2. D. Dimitropoulos, V. Raghunathan, R. Claps, and B. Jalali, “Phase-matching and nonlinear optical processes in silicon waveguides,” Opt. Express 12, 149–160 (2004).
    [CrossRef] [PubMed]
  3. 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]
  4. 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]
  5. R. Espinola, J. Dadap, J. Osgood, S. McNab, and Y. Vlasov, “C-band wavelength conversion in silicon photonic wire waveguides,” Opt. Express 13, 4341–4349 (2005).
    [CrossRef] [PubMed]
  6. Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultrabroadband parametric generation and wavelength conversion in silicon waveguides,” Opt. Express 14, 4786–4799 (2006).
    [CrossRef] [PubMed]
  7. Y. Kuo, H. Rong, V. Sih, S. Xu, M. Paniccia, and O. Cohen, “Demonstration of wavelength conversion at 40 Gb/s data rate in silicon waveguides,” Opt. Express 14, 11721–11726 (2006).
    [CrossRef] [PubMed]
  8. M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
    [CrossRef] [PubMed]
  9. 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).
    [CrossRef] [PubMed]
  10. W. Mathlouthi, H. Rong, and M. Paniccia, “Characterization of efficient wavelength conversion by four-wave mixing in sub-micron silicon waveguides,” Opt. Express 16, 16735–16745 (2008).
    [CrossRef] [PubMed]
  11. P. Koonath, D. R. Solli, and B. Jalali, “High efficiency CARS conversion in silicon,” in Conference on Lasers and Electro-Optics and on Quantum Electronics and Laser Science, (2008), pp. 1–2.
  12. N. Vermeulen, C. Debaes, and H. Thienpont, “The behavior of CARS in anti-Stokes Raman converters operating at exact Raman resonance,” IEEE J. Quantum Electron. 44, 1248–1255 (2008).
    [CrossRef]
  13. A. C. Turner-Foster, M. A. Foster, R. Salem, A. L. Gaeta, and M. Lipson, “Frequency conversion over two-thirds of an octave in silicon nanowaveguides,” Opt. Express 18, 1904–1908 (2010).
    [CrossRef] [PubMed]
  14. S. Gao, E. Tien, Q. Song, Y. Huang, and O. Boyraz, “Ultra-broadband one-to-two wavelength conversion using low-phase-mismatching four-wave mixing in silicon waveguides,” Opt. Express 18, 11898–11903 (2010).
    [CrossRef] [PubMed]
  15. N. Vermeulen, J. E. Sipe, Y. Lefevre, C. Debaes, and H. Thienpont, “Wavelength conversion based on Raman-and non-resonant four-wave mixing in silicon nanowire rings without dispersion engineering,” IEEE J. Sel. Top. Quantum Electron. 17, 1078–1091 (2011).
    [CrossRef]
  16. G. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, 2001).
  17. E. Golovchenko, P. Mamyshev, A. Pilipetskii, and E. Dianov, “Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 26, 1815–1820 (1990).
    [CrossRef]
  18. B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 412–421 (2006).
    [CrossRef]
  19. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007).
    [CrossRef] [PubMed]
  20. Y. Chen, “Combined processes of stimulated Raman scattering and four-wave mixing in optical fibers,” J. Opt. Soc. Am. B 7, 43–52 (1990).
    [CrossRef]
  21. G. Cappellini and S. Trillo, “Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B 8, 824–838 (1991).
    [CrossRef]
  22. Y. Chen and A.W. Snyder, “Four-photon parametric mixing in optical fibers: effect of pump depletion,” Opt. Lett. 14, 87–89 (1989).
    [CrossRef] [PubMed]
  23. T. Sylvestre, H. Maillotte, E. Lantz, and P. Tchofo Dinda, “Raman-assisted parametric frequency conversion in a normally dispersive single-mode fiber,” Opt. Lett. 24, 1561–1563 (1999).
    [CrossRef]
  24. 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]
  25. ITU-T Rec. G.694.1, “Spectral grids for WDM applications: DWDM frequency grid” (2002).

2011 (1)

N. Vermeulen, J. E. Sipe, Y. Lefevre, C. Debaes, and H. Thienpont, “Wavelength conversion based on Raman-and non-resonant four-wave mixing in silicon nanowire rings without dispersion engineering,” IEEE J. Sel. Top. Quantum Electron. 17, 1078–1091 (2011).
[CrossRef]

2010 (2)

2008 (2)

W. Mathlouthi, H. Rong, and M. Paniccia, “Characterization of efficient wavelength conversion by four-wave mixing in sub-micron silicon waveguides,” Opt. Express 16, 16735–16745 (2008).
[CrossRef] [PubMed]

N. Vermeulen, C. Debaes, and H. Thienpont, “The behavior of CARS in anti-Stokes Raman converters operating at exact Raman resonance,” IEEE J. Quantum Electron. 44, 1248–1255 (2008).
[CrossRef]

2007 (2)

2006 (4)

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[CrossRef] [PubMed]

B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 412–421 (2006).
[CrossRef]

Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultrabroadband parametric generation and wavelength conversion in silicon waveguides,” Opt. Express 14, 4786–4799 (2006).
[CrossRef] [PubMed]

Y. Kuo, H. Rong, V. Sih, S. Xu, M. Paniccia, and O. Cohen, “Demonstration of wavelength conversion at 40 Gb/s data rate in silicon waveguides,” Opt. Express 14, 11721–11726 (2006).
[CrossRef] [PubMed]

2005 (3)

2004 (1)

2003 (2)

1999 (1)

1991 (1)

1990 (2)

Y. Chen, “Combined processes of stimulated Raman scattering and four-wave mixing in optical fibers,” J. Opt. Soc. Am. B 7, 43–52 (1990).
[CrossRef]

E. Golovchenko, P. Mamyshev, A. Pilipetskii, and E. Dianov, “Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 26, 1815–1820 (1990).
[CrossRef]

1989 (1)

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, 2001).

Agrawal, G. P.

Boyraz, O.

Cappellini, G.

Chen, Y.

Claps, R.

Cohen, O.

Dadap, J.

Debaes, C.

N. Vermeulen, J. E. Sipe, Y. Lefevre, C. Debaes, and H. Thienpont, “Wavelength conversion based on Raman-and non-resonant four-wave mixing in silicon nanowire rings without dispersion engineering,” IEEE J. Sel. Top. Quantum Electron. 17, 1078–1091 (2011).
[CrossRef]

N. Vermeulen, C. Debaes, and H. Thienpont, “The behavior of CARS in anti-Stokes Raman converters operating at exact Raman resonance,” IEEE J. Quantum Electron. 44, 1248–1255 (2008).
[CrossRef]

Dianov, E.

E. Golovchenko, P. Mamyshev, A. Pilipetskii, and E. Dianov, “Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 26, 1815–1820 (1990).
[CrossRef]

Dimitropoulos, D.

Espinola, R.

Fauchet, P. M.

Foster, M. A.

Fukuda, H.

Gaeta, A. L.

Gao, S.

Golovchenko, E.

E. Golovchenko, P. Mamyshev, A. Pilipetskii, and E. Dianov, “Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 26, 1815–1820 (1990).
[CrossRef]

Han, Y.

Huang, Y.

Itabashi, S.

Jalali, B.

Koonath, P.

P. Koonath, D. R. Solli, and B. Jalali, “High efficiency CARS conversion in silicon,” in Conference on Lasers and Electro-Optics and on Quantum Electronics and Laser Science, (2008), pp. 1–2.

Kuo, Y.

Lantz, E.

Lefevre, Y.

N. Vermeulen, J. E. Sipe, Y. Lefevre, C. Debaes, and H. Thienpont, “Wavelength conversion based on Raman-and non-resonant four-wave mixing in silicon nanowire rings without dispersion engineering,” IEEE J. Sel. Top. Quantum Electron. 17, 1078–1091 (2011).
[CrossRef]

Lin, Q.

Lipson, M.

Maillotte, H.

Mamyshev, P.

E. Golovchenko, P. Mamyshev, A. Pilipetskii, and E. Dianov, “Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 26, 1815–1820 (1990).
[CrossRef]

Mathlouthi, W.

McNab, S.

Osgood, J.

Painter, O. J.

Paniccia, M.

Pilipetskii, A.

E. Golovchenko, P. Mamyshev, A. Pilipetskii, and E. Dianov, “Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 26, 1815–1820 (1990).
[CrossRef]

Raghunathan, V.

Rong, H.

Salem, R.

Schmidt, B. S.

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[CrossRef] [PubMed]

Sharping, J. E.

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[CrossRef] [PubMed]

Shoji, T.

Sih, V.

Sipe, J. E.

N. Vermeulen, J. E. Sipe, Y. Lefevre, C. Debaes, and H. Thienpont, “Wavelength conversion based on Raman-and non-resonant four-wave mixing in silicon nanowire rings without dispersion engineering,” IEEE J. Sel. Top. Quantum Electron. 17, 1078–1091 (2011).
[CrossRef]

Snyder, A.W.

Solli, D. R.

P. Koonath, D. R. Solli, and B. Jalali, “High efficiency CARS conversion in silicon,” in Conference on Lasers and Electro-Optics and on Quantum Electronics and Laser Science, (2008), pp. 1–2.

Song, Q.

Sylvestre, T.

Takahashi, J.

Takahashi, M.

Tchofo Dinda, P.

Thienpont, H.

N. Vermeulen, J. E. Sipe, Y. Lefevre, C. Debaes, and H. Thienpont, “Wavelength conversion based on Raman-and non-resonant four-wave mixing in silicon nanowire rings without dispersion engineering,” IEEE J. Sel. Top. Quantum Electron. 17, 1078–1091 (2011).
[CrossRef]

N. Vermeulen, C. Debaes, and H. Thienpont, “The behavior of CARS in anti-Stokes Raman converters operating at exact Raman resonance,” IEEE J. Quantum Electron. 44, 1248–1255 (2008).
[CrossRef]

Tien, E.

Trillo, S.

Tsuchizawa, T.

Turner, A. C.

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).
[CrossRef] [PubMed]

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[CrossRef] [PubMed]

Turner-Foster, A. C.

Vermeulen, N.

N. Vermeulen, J. E. Sipe, Y. Lefevre, C. Debaes, and H. Thienpont, “Wavelength conversion based on Raman-and non-resonant four-wave mixing in silicon nanowire rings without dispersion engineering,” IEEE J. Sel. Top. Quantum Electron. 17, 1078–1091 (2011).
[CrossRef]

N. Vermeulen, C. Debaes, and H. Thienpont, “The behavior of CARS in anti-Stokes Raman converters operating at exact Raman resonance,” IEEE J. Quantum Electron. 44, 1248–1255 (2008).
[CrossRef]

Vlasov, Y.

Watanabe, T.

Xu, S.

Yamada, K.

Zhang, J.

IEEE J. Quantum Electron. (2)

E. Golovchenko, P. Mamyshev, A. Pilipetskii, and E. Dianov, “Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 26, 1815–1820 (1990).
[CrossRef]

N. Vermeulen, C. Debaes, and H. Thienpont, “The behavior of CARS in anti-Stokes Raman converters operating at exact Raman resonance,” IEEE J. Quantum Electron. 44, 1248–1255 (2008).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

N. Vermeulen, J. E. Sipe, Y. Lefevre, C. Debaes, and H. Thienpont, “Wavelength conversion based on Raman-and non-resonant four-wave mixing in silicon nanowire rings without dispersion engineering,” IEEE J. Sel. Top. Quantum Electron. 17, 1078–1091 (2011).
[CrossRef]

B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 412–421 (2006).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (2)

Nature (1)

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441, 960–963 (2006).
[CrossRef] [PubMed]

Opt. Express (12)

Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultrabroadband parametric generation and wavelength conversion in silicon waveguides,” Opt. Express 14, 4786–4799 (2006).
[CrossRef] [PubMed]

Y. Kuo, H. Rong, V. Sih, S. Xu, M. Paniccia, and O. Cohen, “Demonstration of wavelength conversion at 40 Gb/s data rate in silicon waveguides,” Opt. Express 14, 11721–11726 (2006).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007).
[CrossRef] [PubMed]

W. Mathlouthi, H. Rong, and M. Paniccia, “Characterization of efficient wavelength conversion by four-wave mixing in sub-micron silicon waveguides,” Opt. Express 16, 16735–16745 (2008).
[CrossRef] [PubMed]

A. C. Turner-Foster, M. A. Foster, R. Salem, A. L. Gaeta, and M. Lipson, “Frequency conversion over two-thirds of an octave in silicon nanowaveguides,” Opt. Express 18, 1904–1908 (2010).
[CrossRef] [PubMed]

S. Gao, E. Tien, Q. Song, Y. Huang, and O. Boyraz, “Ultra-broadband one-to-two wavelength conversion using low-phase-mismatching four-wave mixing in silicon waveguides,” Opt. Express 18, 11898–11903 (2010).
[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. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Anti-Stokes Raman conversion in silicon waveguides,” Opt. Express 11, 2862–2872 (2003).
[CrossRef] [PubMed]

D. Dimitropoulos, V. Raghunathan, R. Claps, and B. Jalali, “Phase-matching and nonlinear optical processes in silicon waveguides,” Opt. Express 12, 149–160 (2004).
[CrossRef] [PubMed]

R. Espinola, J. Dadap, J. Osgood, S. McNab, and Y. 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]

Opt. Lett. (2)

Other (3)

ITU-T Rec. G.694.1, “Spectral grids for WDM applications: DWDM frequency grid” (2002).

G. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, 2001).

P. Koonath, D. R. Solli, and B. Jalali, “High efficiency CARS conversion in silicon,” in Conference on Lasers and Electro-Optics and on Quantum Electronics and Laser Science, (2008), pp. 1–2.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Spectrum of a typical wavelength conversion set-up. A strong pump and a weak Stokes spectral component, of which the angular frequencies ωp and ωs are detuned by ΔΩ = ωp ωs , are used as input (full lines). Due to FWM interactions an anti-Stokes component will be generated at ωa = ωp + ΔΩ (broken line).

Fig. 2
Fig. 2

The variation for both the Stokes and the anti-Stokes waves of (a) the FWM dispersion factors –Im [G FWM,j e iΔϕ ] and (b) the FWM intensity gain factor Re [G FWM,j e iΔϕ ] versus the phase difference Δϕ, as computed for TE-polarized light at perfect Raman resonance and λp =1550 nm using the silicon nonlinear parameters given in Table 1. As can be seen in (a), the FWM dispersion due to each wave drives the Δϕ towards the value –phase [GFWM,j ] that corresponds to that wave.

Fig. 3
Fig. 3

(a) The initial FWM anti-Stokes intensity gain |GFWM,a | and (b) the FWM phase difference ΔϕFWM between the complex FWM gains GFWM,j which characterizes the FWM dispersion, both in function of the frequency resonance off-set ΔΩ – Ω R and the Stokes wavelength resonance off-set λs λ s0, where λ s0 is the Stokes wavelength at perfect Raman resonance, as computed for TE-polarized light at a fixed λp =1550 nm using the silicon nonlinear parameters given in Table 1.

Fig. 4
Fig. 4

Normalized intensity evolutions of (a) anti-Stokes Ia /Is (0), (b) Stokes Is /Is (0) and (c) the pump Ip /Ip (0), and evolutions of the corresponding (d) phase difference Δϕ and (e) the phase mismatch, and this for several frequency differences ΔΩ as simulated for CW TE-polarized light, with pump and Stokes input intensities of Ip (0) = 0.2 GW/cm2 and Is (0) = 0.2 MW/cm2, along a SOI nano-waveguide with β 2,p = −1.14 × 10−4 ps2/cm and β 4,p = −8.93 × 10−8 ps4/cm, at the fixed pump wavelength λp =1550 nm. The nonlinear parameters of Table 1 were used together with a linear loss α =0.2 dB/cm [6]. In (e) the phase mismatch is both computed as κ′, including the FWM dispersion (full lines), and according to the conventional definition as κ, excluding the FWM dispersion (dotted lines).

Fig. 5
Fig. 5

Evolution of ΔΩ Opt – Ω R , the frequency resonance off-set that gives the optimal CW wavelength conversion after a certain distance, in function of that distance, for several SOI nano-waveguides of which the dispersion characteristics are given in Table 2. The pump wavelength is fixed at λp =1550 nm, with the same input and material parameters as before. ΔΩ Opt is determined by comparing at each position z the anti-Stokes intensities as numerically simulated for a set of ΔΩ with a resolution of 0.5 GHz.

Fig. 6
Fig. 6

The approximation error relative to the solutions found with the weak-pump description which is depicted in Fig. 4 of the anti-Stokes intensity evolutions, under the same conditions as those in Fig. 4, as made by numerically solving on the one hand the set of three complex Eqs. (34)(36) using 2000 equally spaced spatial points (dash-dotted lines), and on the other hand the set of the four real Eqs. (25), (37)(39) using 200 equally spaced spatial points (full lines).

Tables (2)

Tables Icon

Table 1 List of Values of Nonlinear Silicon Parameters near λ = 1550 nm for TE-Polarized Light

Tables Icon

Table 2 Dispersion Characteristics of the Numerically Simulated Waveguides

Equations (39)

Equations on this page are rendered with MathJax. Learn more.

ω a ω p = ω p ω s Δ Ω .
E ˜ ( r , ω ) = F ˜ ( x , y , ω ) A ˜ ( z , ω ) 1 x .
A ( z , t ) = 1 2 j = p , s , a ( A j ( z , t ) e i ω j t + A j * ( z , t ) e i ω j t ) .
A p z = i β 0 , p A p 1 2 ( α p + α f , p ) A p i ω p c n f , p A p i ( γ K + γ R , p ) | A p | 2 A p ,
A s z = i β 0 , s A s 1 2 ( α s + α f , s ) A s i ω s c n f , s A s i ( 2 γ K + γ R , s + γ R , s H R ( Δ Ω ) ) | A p | 2 A s i ( γ K + γ R , s H R ( Δ Ω ) ) A p 2 A a * ,
A a z = i β 0 , a A a 1 2 ( α a + α f , a ) A a i ω a c n f , a A a i ( 2 γ K + γ R , a + γ R , a H R ( Δ Ω ) ) | A p | 2 A a i ( γ K + γ R , a H R ( Δ Ω ) ) A p 2 A s * .
γ K = ξ K ( n 2 ω p c i β T 2 ) .
α f , j = σ a , j N ,
n f , j = σ n e , j N + σ n h , j N 0.8 ,
N = β T τ c 2 h ¯ ω p | A p | 4 ,
H R ( Δ Ω ) = Ω R 2 Ω R 2 Δ Ω 2 + 2 i Γ R Δ Ω ,
γ R , j = ξ R g R , r e f Γ R Ω R ω j ω r e f .
G F W M , j i ( γ K + γ R , j H R ( ω j ω p ) ) .
I s z | F W M = 2 Re [ G F W M , s A p 2 A a * A s * ] ,
I a z | F W M = 2 Re [ G F W M , a A p 2 A a * A s * ] .
κ = Δ β 0 + 1 c ( n f , s ω s + n f , a ω a 2 n f , p ω p ) + I p Re [ 2 γ K + ( γ R , s + γ R , a ) H R ( Δ Ω ) ] .
Δ β 0 = β 2 , p Δ Ω 2 + β 4 , p 12 Δ Ω 4 + ,
κ β 2 , p Δ Ω 2 + β 4 , p 12 Δ Ω 4 + 1 c ( n f , s ω s + n f , a ω a 2 n f , p ω p ) + I p Re [ 2 γ K + ( γ R , s + γ R , a ) H R ( Δ Ω ) ] .
A j ( z ) = | A j ( z ) | e i ϕ j ( z ) ,
A j z = i ϕ j z A j + | A j | z A j | A j | .
β l o c , j = ϕ j z ,
β l o c , j = Im [ 1 A j A j z ] .
κ = β l o c , s + β l o c , a 2 β l o c , p
= β 2 , p Δ Ω 2 + β 4 , p 12 Δ Ω 4 + 1 c ( n f , s ω s + n f , a ω a 2 n f , p ω p ) + I p Re [ 2 γ K + ( γ R , s + γ R , a ) H R ( Δ Ω ) ] Im [ G F W M , s A p 2 A a * A s ] Im [ G F W M , a A p 2 A s * A a ] .
Δ ϕ = 2 ϕ p ϕ s ϕ a .
Δ ϕ z = κ
= β 2 , p Δ Ω 2 + β 4 , p 12 Δ Ω 4 + 1 c ( n f , s ω s + n f , a ω a 2 n f , p ω p ) + I p Re [ 2 γ K + ( γ R , s + γ R , a ) H R ( Δ Ω ) ] Im [ G F W M , s e i Δ ϕ ] I p | A a | | A s | Im [ G F W M , a e i Δ ϕ ] I p | A s | | A a | ,
I s z | F W M = 2 Re [ G F W M , s e i Δ ϕ ] I p | A a | | A s | ,
I a z | F W M = 2 Re [ G F W M , a e i Δ ϕ ] I p | A s | | A a | .
A a z ( z 0 ) = G F W M , a A p 2 A s * .
p h a s e [ A a ( z 0 ) ] = p h a s e [ G F W M , a A p 2 A s * ] .
Δ ϕ F W M = p h a s e [ G F W M , a ] ( p h a s e [ G F W M , s ] ) = p h a s e [ G F W M , s G F W M , a * ] .
I a z ( z 0 ) = 2 Real [ G F W M , a A p 2 A s * A a * ] = 2 | G F W M , a | I p | A s | | A a | .
A ¯ p z = 1 2 ( α p + α f , p ) A ¯ p i ω p c n f , p A ¯ p i ( γ K + γ R , p ) | A ¯ p | 2 A ¯ p ,
A ¯ s z = 1 2 ( α s + α f , s ) A ¯ s i ω s c n f , s A ¯ s i ( 2 γ K + γ R , s + γ R , s H R ( Δ Ω ) ) | A ¯ p | 2 A ¯ s i ( γ K + γ R , s H R ( Δ Ω ) ) e i Δ β 0 z A ¯ p 2 A ¯ a * ,
A ¯ a z = 1 2 ( α a + α f , a ) A ¯ a i ω a c n f , a A ¯ a i ( 2 γ K + γ R , a + γ R , a H R ( Δ Ω ) ) | A ¯ p | 2 A ¯ a i ( γ K + γ R , a H R ( Δ Ω ) ) e i Δ β 0 z A ¯ p 2 A ¯ s * ,
I p z = ( α p + α f , p ) I p ξ K β T I p 2 ,
| A s | z = 1 2 ( α + α f , s ) | A s | + ( ξ K β T + γ R , s Im [ H R ( Δ Ω ) ] ) I p | A s | + Re [ G F W M , s e i Δ ϕ ] I p | A a | ,
| A a | z = 1 2 ( α + α f , a ) | A a | + ( ξ K β T + γ R , a Im [ H R ( Δ Ω ) ] ) I p | A a | + Re [ G F W M , a e i Δ ϕ ] I p | A s | .

Metrics