Abstract

We present the analysis of parametric and Raman effects in the framework of a single approach. We discuss the transformation of gain spectra for Stokes and anti-Stokes waves under pump wavelength tuning from the region of positive group-velocity dispersion to the region of negative group-velocity dispersion for different types of phase-matching mechanism in a birefringent fiber.

© 1994 Optical Society of America

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  1. A. Hasegawa and W. F. Brinkman, IEEE J. Quantum Electron. QE-16, 694 (1980).
    [CrossRef]
  2. K. Tai, A. Hasegawa, and A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
    [CrossRef] [PubMed]
  3. C. Lin, W. A. Reed, A. D. Pearson, and H. T. Shang, Opt. Lett. 6, 493 (1981).
    [CrossRef] [PubMed]
  4. S. J. Garth and C. Pask, Opt. Lett. 11, 380 (1986).
    [CrossRef] [PubMed]
  5. R. H. Stolen and J. E. Bjorkholm, IEEE J. Quantum Electron. QE-18, 1062 (1982).
    [CrossRef]
  6. S. Wabnitz, Phys. Rev. A 38, 2018 (1988).
    [CrossRef] [PubMed]
  7. S. Trillo and S. Wabnitz, J. Opt. Soc. Am. B 6, 238 (1989).
    [CrossRef]
  8. K. Stenersen and R. K. Jain, Opt. Commun. 51, 121 (1984).
    [CrossRef]
  9. J. E. Rothenberg, Phys. Rev. A 42, 682 (1990).
    [CrossRef] [PubMed]
  10. P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, Opt. Commun. 78, 137 (1990).
    [CrossRef]
  11. J. K. Chee and J. Liu, IEEE J. Quantum Electron. 26, 541 (1990).
    [CrossRef]
  12. E. A. Golovchenko and A. N. Pilipetskii, Sov. Lightwave Commun. 1, 271 (1991).
  13. E. A. Golovchenko, E. M. Dianov, P. V Mamyshev, and A. N. Pilipetskii, JETP Lett. 50, 190 (1989).
  14. E. A. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, IEEE J. Quantum Electron. 26, 1815 (1990).
    [CrossRef]
  15. K. J. Blow and D. Wood, IEEE J. Quantum Electron. 25, 2665 (1989).
    [CrossRef]
  16. Y. Chen, J. Opt. Soc. Am. B 7, 43 (1990).
    [CrossRef]
  17. E. A. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, J. Opt. Soc. Am. B 8, 1626 (1991).
    [CrossRef]
  18. S. Trillo and S. Wabnitz, J. Opt. Soc. Am. B 9, 1061 (1992).
    [CrossRef]
  19. C. R. Menuyk, M. N. Islam, and J. P. Gordon, Opt. Lett. 16, 566 (1991).
    [CrossRef]
  20. R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, J. Opt. Soc. Am. B 6, 1159 (1989).
    [CrossRef]
  21. R. W. Hellwarth, Prog. Quantum Electron. 5, 1 (1977).
    [CrossRef]
  22. R. A. Summut and S. J. Garth, J. Opt. Soc. Am. B 6, 1732 (1989).
    [CrossRef]
  23. S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, Phys. Rev. A 43, 6162 (1991).
    [CrossRef] [PubMed]
  24. P. V. Mamyshev and A. P. Vertikov, in Vol. 13 of the OSA 1992 Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 130.
  25. B. Daino, G. Gregori, and S. Wabnitz, J. Appl. Phys. 58, 4512 (1985).
    [CrossRef]
  26. P. Morgan and J. M. Liu, IEEE J. Quantum Electron. 27, 1011 (1991).
    [CrossRef]

Cavalcanti, S. B.

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, Phys. Rev. A 43, 6162 (1991).
[CrossRef] [PubMed]

Cressoni, J. C.

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, Phys. Rev. A 43, 6162 (1991).
[CrossRef] [PubMed]

da Cruz, H. R.

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, Phys. Rev. A 43, 6162 (1991).
[CrossRef] [PubMed]

Dianov, E. M.

E. A. Golovchenko, E. M. Dianov, P. V Mamyshev, and A. N. Pilipetskii, JETP Lett. 50, 190 (1989).

Golovchenko, E. A.

E. A. Golovchenko, E. M. Dianov, P. V Mamyshev, and A. N. Pilipetskii, JETP Lett. 50, 190 (1989).

Gouveia-Neto, A. S.

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, Phys. Rev. A 43, 6162 (1991).
[CrossRef] [PubMed]

Mamyshev, P. V

E. A. Golovchenko, E. M. Dianov, P. V Mamyshev, and A. N. Pilipetskii, JETP Lett. 50, 190 (1989).

Pilipetskii, A. N.

E. A. Golovchenko, E. M. Dianov, P. V Mamyshev, and A. N. Pilipetskii, JETP Lett. 50, 190 (1989).

Bjorkholm, J. E.

R. H. Stolen and J. E. Bjorkholm, IEEE J. Quantum Electron. QE-18, 1062 (1982).
[CrossRef]

Blow, K. J.

K. J. Blow and D. Wood, IEEE J. Quantum Electron. 25, 2665 (1989).
[CrossRef]

Brinkman, W. F.

A. Hasegawa and W. F. Brinkman, IEEE J. Quantum Electron. QE-16, 694 (1980).
[CrossRef]

Chee, J. K.

J. K. Chee and J. Liu, IEEE J. Quantum Electron. 26, 541 (1990).
[CrossRef]

Chen, Y.

Y. Chen, J. Opt. Soc. Am. B 7, 43 (1990).
[CrossRef]

Daino, B.

B. Daino, G. Gregori, and S. Wabnitz, J. Appl. Phys. 58, 4512 (1985).
[CrossRef]

Dianov, E. M.

E. A. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, IEEE J. Quantum Electron. 26, 1815 (1990).
[CrossRef]

E. A. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, J. Opt. Soc. Am. B 8, 1626 (1991).
[CrossRef]

Drummond, P. D.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, Opt. Commun. 78, 137 (1990).
[CrossRef]

Dudley, J. M.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, Opt. Commun. 78, 137 (1990).
[CrossRef]

Garth, S. J.

S. J. Garth and C. Pask, Opt. Lett. 11, 380 (1986).
[CrossRef] [PubMed]

R. A. Summut and S. J. Garth, J. Opt. Soc. Am. B 6, 1732 (1989).
[CrossRef]

Golovchenko, E. A.

E. A. Golovchenko and A. N. Pilipetskii, Sov. Lightwave Commun. 1, 271 (1991).

E. A. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, J. Opt. Soc. Am. B 8, 1626 (1991).
[CrossRef]

E. A. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, IEEE J. Quantum Electron. 26, 1815 (1990).
[CrossRef]

Gordon, J. P.

R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, J. Opt. Soc. Am. B 6, 1159 (1989).
[CrossRef]

C. R. Menuyk, M. N. Islam, and J. P. Gordon, Opt. Lett. 16, 566 (1991).
[CrossRef]

Gregori, G.

B. Daino, G. Gregori, and S. Wabnitz, J. Appl. Phys. 58, 4512 (1985).
[CrossRef]

Harvey, J. D.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, Opt. Commun. 78, 137 (1990).
[CrossRef]

Hasegawa, A.

A. Hasegawa and W. F. Brinkman, IEEE J. Quantum Electron. QE-16, 694 (1980).
[CrossRef]

K. Tai, A. Hasegawa, and A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
[CrossRef] [PubMed]

Haus, H. A.

R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, J. Opt. Soc. Am. B 6, 1159 (1989).
[CrossRef]

Hellwarth, R. W.

R. W. Hellwarth, Prog. Quantum Electron. 5, 1 (1977).
[CrossRef]

Islam, M. N.

C. R. Menuyk, M. N. Islam, and J. P. Gordon, Opt. Lett. 16, 566 (1991).
[CrossRef]

Jain, R. K.

K. Stenersen and R. K. Jain, Opt. Commun. 51, 121 (1984).
[CrossRef]

Kennedy, T. A. B.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, Opt. Commun. 78, 137 (1990).
[CrossRef]

Leonhardt, R.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, Opt. Commun. 78, 137 (1990).
[CrossRef]

Lin, C.

C. Lin, W. A. Reed, A. D. Pearson, and H. T. Shang, Opt. Lett. 6, 493 (1981).
[CrossRef] [PubMed]

Liu, J.

J. K. Chee and J. Liu, IEEE J. Quantum Electron. 26, 541 (1990).
[CrossRef]

Liu, J. M.

P. Morgan and J. M. Liu, IEEE J. Quantum Electron. 27, 1011 (1991).
[CrossRef]

Mamyshev, P. V.

E. A. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, J. Opt. Soc. Am. B 8, 1626 (1991).
[CrossRef]

P. V. Mamyshev and A. P. Vertikov, in Vol. 13 of the OSA 1992 Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 130.

E. A. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, IEEE J. Quantum Electron. 26, 1815 (1990).
[CrossRef]

Menuyk, C. R.

C. R. Menuyk, M. N. Islam, and J. P. Gordon, Opt. Lett. 16, 566 (1991).
[CrossRef]

Morgan, P.

P. Morgan and J. M. Liu, IEEE J. Quantum Electron. 27, 1011 (1991).
[CrossRef]

Pask, C.

S. J. Garth and C. Pask, Opt. Lett. 11, 380 (1986).
[CrossRef] [PubMed]

Pearson, A. D.

C. Lin, W. A. Reed, A. D. Pearson, and H. T. Shang, Opt. Lett. 6, 493 (1981).
[CrossRef] [PubMed]

Pilipetskii, A. N.

E. A. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, J. Opt. Soc. Am. B 8, 1626 (1991).
[CrossRef]

E. A. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, IEEE J. Quantum Electron. 26, 1815 (1990).
[CrossRef]

E. A. Golovchenko and A. N. Pilipetskii, Sov. Lightwave Commun. 1, 271 (1991).

Reed, W. A.

C. Lin, W. A. Reed, A. D. Pearson, and H. T. Shang, Opt. Lett. 6, 493 (1981).
[CrossRef] [PubMed]

Rothenberg, J. E.

J. E. Rothenberg, Phys. Rev. A 42, 682 (1990).
[CrossRef] [PubMed]

Shang, H. T.

C. Lin, W. A. Reed, A. D. Pearson, and H. T. Shang, Opt. Lett. 6, 493 (1981).
[CrossRef] [PubMed]

Stenersen, K.

K. Stenersen and R. K. Jain, Opt. Commun. 51, 121 (1984).
[CrossRef]

Stolen, R. H.

R. H. Stolen and J. E. Bjorkholm, IEEE J. Quantum Electron. QE-18, 1062 (1982).
[CrossRef]

R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, J. Opt. Soc. Am. B 6, 1159 (1989).
[CrossRef]

Summut, R. A.

R. A. Summut and S. J. Garth, J. Opt. Soc. Am. B 6, 1732 (1989).
[CrossRef]

Tai, K.

K. Tai, A. Hasegawa, and A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
[CrossRef] [PubMed]

Tomita, A.

K. Tai, A. Hasegawa, and A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
[CrossRef] [PubMed]

Tomlinson, W. J.

R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, J. Opt. Soc. Am. B 6, 1159 (1989).
[CrossRef]

Trillo, S.

S. Trillo and S. Wabnitz, J. Opt. Soc. Am. B 9, 1061 (1992).
[CrossRef]

S. Trillo and S. Wabnitz, J. Opt. Soc. Am. B 6, 238 (1989).
[CrossRef]

Vertikov, A. P.

P. V. Mamyshev and A. P. Vertikov, in Vol. 13 of the OSA 1992 Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 130.

Wabnitz, S.

S. Trillo and S. Wabnitz, J. Opt. Soc. Am. B 6, 238 (1989).
[CrossRef]

S. Wabnitz, Phys. Rev. A 38, 2018 (1988).
[CrossRef] [PubMed]

S. Trillo and S. Wabnitz, J. Opt. Soc. Am. B 9, 1061 (1992).
[CrossRef]

B. Daino, G. Gregori, and S. Wabnitz, J. Appl. Phys. 58, 4512 (1985).
[CrossRef]

Wood, D.

K. J. Blow and D. Wood, IEEE J. Quantum Electron. 25, 2665 (1989).
[CrossRef]

Other

A. Hasegawa and W. F. Brinkman, IEEE J. Quantum Electron. QE-16, 694 (1980).
[CrossRef]

K. Tai, A. Hasegawa, and A. Tomita, Phys. Rev. Lett. 56, 135 (1986).
[CrossRef] [PubMed]

C. Lin, W. A. Reed, A. D. Pearson, and H. T. Shang, Opt. Lett. 6, 493 (1981).
[CrossRef] [PubMed]

S. J. Garth and C. Pask, Opt. Lett. 11, 380 (1986).
[CrossRef] [PubMed]

R. H. Stolen and J. E. Bjorkholm, IEEE J. Quantum Electron. QE-18, 1062 (1982).
[CrossRef]

S. Wabnitz, Phys. Rev. A 38, 2018 (1988).
[CrossRef] [PubMed]

S. Trillo and S. Wabnitz, J. Opt. Soc. Am. B 6, 238 (1989).
[CrossRef]

K. Stenersen and R. K. Jain, Opt. Commun. 51, 121 (1984).
[CrossRef]

J. E. Rothenberg, Phys. Rev. A 42, 682 (1990).
[CrossRef] [PubMed]

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, Opt. Commun. 78, 137 (1990).
[CrossRef]

J. K. Chee and J. Liu, IEEE J. Quantum Electron. 26, 541 (1990).
[CrossRef]

E. A. Golovchenko and A. N. Pilipetskii, Sov. Lightwave Commun. 1, 271 (1991).

E. A. Golovchenko, E. M. Dianov, P. V Mamyshev, and A. N. Pilipetskii, JETP Lett. 50, 190 (1989).

E. A. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, IEEE J. Quantum Electron. 26, 1815 (1990).
[CrossRef]

K. J. Blow and D. Wood, IEEE J. Quantum Electron. 25, 2665 (1989).
[CrossRef]

Y. Chen, J. Opt. Soc. Am. B 7, 43 (1990).
[CrossRef]

E. A. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, J. Opt. Soc. Am. B 8, 1626 (1991).
[CrossRef]

S. Trillo and S. Wabnitz, J. Opt. Soc. Am. B 9, 1061 (1992).
[CrossRef]

C. R. Menuyk, M. N. Islam, and J. P. Gordon, Opt. Lett. 16, 566 (1991).
[CrossRef]

R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, J. Opt. Soc. Am. B 6, 1159 (1989).
[CrossRef]

R. W. Hellwarth, Prog. Quantum Electron. 5, 1 (1977).
[CrossRef]

R. A. Summut and S. J. Garth, J. Opt. Soc. Am. B 6, 1732 (1989).
[CrossRef]

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, Phys. Rev. A 43, 6162 (1991).
[CrossRef] [PubMed]

P. V. Mamyshev and A. P. Vertikov, in Vol. 13 of the OSA 1992 Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 130.

B. Daino, G. Gregori, and S. Wabnitz, J. Appl. Phys. 58, 4512 (1985).
[CrossRef]

P. Morgan and J. M. Liu, IEEE J. Quantum Electron. 27, 1011 (1991).
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

(a) Dependence of the parametric generation frequency shift on pump wavelength. The dashed curve is defined by the condition Δk = 0, and the solid curves are given for the pump intensity Ip = 5 × 1010 W/cm2. (b), (c) Gain spectra of modulational instability in the region of (b) positive GVD (∂2k/∂ω2 > 0) and (c) negative GVD (∂2k/∂ω2 < 0) for Ip = 5 × 1010 W/cm2. The gain value is normalized to RIp.

Fig. 2
Fig. 2

Raman gain in the polarization parallel to that of the pump at the frequency shift Ω = 440 cm−1 versus the parameter ξ = (4 × 1010 W/cm2)/Ip for different input pump wavelengths λ. The gain value is normalized to the value of Re[K(Ω = 440 cm−1)] in the absence of Stokes and anti-Stokes wave parametric interaction (Δk → ∞). The solid curves are given for the value γ = Ω/ωp, and the dashed curves are given for γ = 0.

Fig. 3
Fig. 3

Typical amplification spectrum in the same polarization as the pump wave for pump wavelength Ip = 1.29 μm and Ip = 8 × 109 W/cm2. The gain value is normalized to the value of Raman gain Re[K(Ω)] at the frequency shift Ω = 440 cm−1 in the absence of parametric interaction (Δk → ∞).

Fig. 4
Fig. 4

Dependence of the Stokes and anti-Stokes pair frequency shifts on the value of pump wavelength for three cases: isotropic (nonbirefringent) fiber (solid curve), pump wave coupled into the slow-fiber axis (Δn = 3 × 10−5; dashed curves), and pump wave coupled into the fast-fiber axis (Δn = 3 × 10−5; dotted curves).

Fig. 5
Fig. 5

(a) Regions of parametric generation for the pump wave in the slow-fiber polarization axis and for Δn = 3 × 10−5. The spectral band of MI is confined between the curves Δk = 0 (dashed curves) and the curves corresponding to the boundary of Re(K) ≥ 0 for Ip = 5 × 1010 W/cm2 (solid curves). (b) Gain spectra of MI for the pump wavelength λ = 1.274 μm and for Ip = 25 × 108 W/cm2 (curves 1), Ip = 5 × 109 W/cm2 (curve 2), and Ip = 5 × 1010 W/cm2 (curve 3). The gain value is normalized to RIp.

Fig. 6
Fig. 6

(a) Spectral band of parametric generation depending on pump wavelength for the pump wave in the fast-fiber polarization axis and for Δn = 6 × 10−6. The spectral band of MI amplification is confined between the curves Δk = 0 (dotted curves) and the curves corresponding to the boundary of Re(K ≥ 0 for Ip = 5 × 1010 W/cm2 (solid curves). (b) Gain spectra of MI for the pump wavelength λp = 1.31 μm and for Ip = 1010 W/cm2 (curve 1) and Ip = 5 × 1010 W/cm2 (curve 2). The gain value is normalized to RIp.

Fig. 7
Fig. 7

(a) Same as Fig. 6(a) but for Δn = 1.2 × 10−5. (b) MI gain spectrum for Ip = 5 × 109 W/cm2 (curve 1) and Ip = 5 × 1010 W/cm2 (curve 2) (Δn = 1.2 × 10−5). The gain value is normalized to RIp.

Fig. 8
Fig. 8

Dependence of Raman gain on pump intensity in the polarizations parallel (g Ω, = 440 cm−1) and orthogonal (g, Ω = 110 cm−1) to the pump for Δn ≈ 10−5. The gain values are normalized to the value of Re[K(Ω = 440 cm−1)] calculated from Eq. (9) in the absence of parametric interaction (Δk → ∞).

Fig. 9
Fig. 9

Dependence of the gain value at Ω = 110 cm−1 in the polarization orthogonal to the pump wave on the parameter η = δ/RIp. The gain value is normalized to the value of peak Raman gain in the polarization parallel to the pump (similar to Fig. 8).

Fig. 10
Fig. 10

Amplification spectra in the polarization orthogonal to the pump wave for Δn ≈ 2 × 10−4 and Ip = 4 × 1010 W/cm2. The gain value is normalized to the value of peak Raman gain in the polarization parallel to the pump (similar to Figs. 8 and 9).

Fig. 11
Fig. 11

Dependencies of Stokes and anti-Stokes frequency shifts on pump wavelength defined by the conditions Δk(1) = 0 (dashed curves) and Δk(2) = 0 (solid curves) for Δn = 10−4 and for a pump wave equally split between the principal fiber polarization axes.

Fig. 12
Fig. 12

MI gain spectral dependence for the two positive roots of Eq. (17), where we have assumed that k4 = 0 in the region of negative GVD. q = RIp/|k2s2, Ωs/2πc = 150 cm−1, and the gain value is normalized to RIp.

Fig. 13
Fig. 13

(a) MI gain spectrum for Δn = 10−4, λp = 1.28 μm, Ip = 25 × 108 W/cm2 (solid curves), and Ip = 1010 W/cm2 (dashed curves). (b) Same as (a) but for λp = 1.3 μm, Ip = 7 × 109 W/cm2 (solid curves), and Ip = 25 × 109 W/cm2 (dashed curves). The gain value is normalized to RIp, and both roots of Eq. (13) are shown together.

Fig. 14
Fig. 14

Output spectra shapes for different input pump intensities: (a) q = RIp/|k2s2 = 5 × 10−4, (b) q = 0.15, (c) q = 0.6, (d) q = 5, and (e) q = 15. All the dependencies are given for the value of Ωs/2πc = 150 cm−1 and the region of positive GVD. The gain is normalized to RIp, and the spectrum is shown in arbitrary units.

Equations (20)

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E ¯ = i = 1 2 e ˆ i A i ( z , t ) exp ( i ω 0 t i k i z ) + c . c . , i = 1 , 2 ,
i A i z + i 1 υ i A i t + 1 2 2 k ω 2 2 A i t 2 i 6 3 k ω 3 3 A i t 3 1 24 4 k ω 4 4 A i δ t 4 = α R | A i | 2 A i + ( 1 α ) R A i 0 | A i ( t θ ) | 2 [ a ( θ ) + b ( θ ) ] d θ + 2 3 α R | A 3 i | 2 A i + ( 1 α ) R A i 0 | A 3 i ( t θ ) | 2 a ( θ ) d ( θ ) + 1 2 ( 1 α ) R A 3 i 0 A i ( t θ ) A 3 i * ( t θ ) b ( θ ) d θ + 1 3 α R exp ( i δ i z ) A i * A 3 i 2 + 1 2 ( 1 α ) R A 3 i × exp ( i δ i z ) 0 A i * ( t θ ) A 3 i ( t θ ) b ( θ ) d θ , δ i = 2 k i 2 k 3 i i = 1 , 2 .
χ 1111 R ( Ω ) = 0 [ a ( θ ) + b ( θ ) ] exp ( i Ω θ ) d θ , χ 1122 R ( Ω ) = 0 a ( θ ) exp ( i Ω θ ) d θ , χ 1221 R ( Ω ) = χ 1212 R ( Ω ) = 1 2 0 b ( θ ) exp ( i Ω θ ) d θ .
A i ( z , t ) = A i p ( z ) + u ¯ i ( z ) exp ( i Ω t ) + υ ¯ i ( z ) exp ( i Ω t ) ,
Δ k = 2 k p k s k a = 0 ,
Δ k k 2 Ω 2 1 12 k 4 Ω 4 .
( K i 2 γ R I p ) 2 = ( Δ k 2 ) 2 + Δ k R I p γ 2 R 2 I p 2 ,
Δ β = Δ k 2 n 2 I p ω p / c ,
Δ k = 2 k p 0 k s 0 k a 0 , k p = k p 0 + n 2 I p ω p / c , k s = k s 0 + 2 n 2 I p ω s / c , k a = k a 0 + 2 n 2 I p ω a / c .
{ K i γ R I p [ 1 + χ ( Ω ) ] 2 = Δ k 2 / 4 + Δ k R I p χ ( Ω ) [ γ R I p χ ( Ω ) ] 2 ,
( K 2 3 i γ R I p ) 2 = ( Δ k / 2 ) 2 1 3 Δ k R I p 1 9 γ 2 R 2 I p 2 ,
Δ k = 2 δ k 2 Ω 2 1 12 k 4 Ω 4 ,
Δ β = Δ k + 2 3 n 2 I p ω p / c = 0.
Ω 1 , 2 2 = 6 k 2 k 4 ± [ ( 6 k 2 k 4 ) 2 12 k 4 ( 2 3 R I p + 2 δ ) ] 1 / 2 ,
Ω 2 = ( 1 / k 2 ) ( 1 3 R I p + 2 δ ) .
( K i γ R I p { 2 3 α + ( 1 α ) [ χ 1122 R ( 0 ) + χ 1212 R ( Ω ) ] } ) 2 = ( Δ k / 2 + d R I p ) 2 ( Δ k / 2 + d R I p ) × [ 1 3 α ( 1 α ) χ 1212 R ( Ω ) ] R I p + R 2 I p 2 × { 4 3 α ( 1 α ) χ 1212 R ( Ω ) γ 2 [ 1 3 α ( 1 α ) χ 1212 R ( Ω ) ] 2 } ,
Δ k ( 1 ) = 2 k p x k s x k a x = 2 k p y k s y k a y , Δ k ( 2 ) = k p x + k p y k s , a x k s , σ y , Δ k ( 3 ) = 2 k p x k s y k a y , Δ k ( 4 ) = 2 k p y k s x k a x .
K 2 = B p 2 ± ( 4 B p 2 + C 2 ) ,
p 2 = ( Δ n Ω / 2 c ) 2 , B = ( k 2 Ω 2 + 1 12 k 4 Ω 4 ) 2 / 4 + ( k 2 Ω 2 + 1 12 k 4 Ω 4 ) R I p / 2 , C = 2 3 ( k 2 Ω 2 + 1 12 k 4 Ω 4 ) R I p / 2 .
Ω s = Δ / | k 2 | ,

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