Abstract

The influence of a sufficiently nonstationary Raman contribution to nonlinear susceptibility on the dynamics of cnoidal wave propagation in optical fibers is theoretically analyzed. The dependence of the parameter that describes the curvature of the propagation trajectory on the degree of localization of the wave energy is presented.

© 2001 Optical Society of America

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References

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  1. V. Petnikova, V. Shuvalov, and V. Vysloukh, “Multicomponent photorefractive cnoidal waves: stability, localization and soliton asymptotics,” Phys. Rev. E 60, 1–10 (1999).
    [CrossRef]
  2. F. Hioe, “Solitary waves for two and three coupled nonlinear Schrödinger equations,” Phys. Rev. E 58, 6700–6707 (1998).
    [CrossRef]
  3. V. Aleshkevich, V. Vysloukh, and Y. Kartashov, “Self-bending of cnoidal waves in photorefractive medium with drift and diffusion nonlinearity,” Opt. Commun. 173, 277–284 (2000).
    [CrossRef]
  4. V. Kutuzov, V. Petnikova, V. Shuvalov, and V. Vysloukh, “Cross-modulation coupling of incoherent soliton modes in photorefractive crystals,” Phys. Rev. E 57, 6056–6065 (1998).
    [CrossRef]
  5. A. Ankiewicz, W. Krolikowski, and N. Akhmediev, “Partially coherent solitons of variable shape in a slow Kerr-like medium: exact solutions,” Phys. Rev. E 59, 6079–6087 (1999).
    [CrossRef]
  6. F. Mitschke and L. Mollenauer, “Discovery of the soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986).
    [CrossRef] [PubMed]
  7. F. Mitschke and L. Mollenauer, “Experimental observation of interaction forces between solitons in optical fibers,” Opt. Lett. 12, 355–357 (1987).
    [CrossRef] [PubMed]
  8. J. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986).
    [CrossRef] [PubMed]
  9. Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, 510–524 (1987).
    [CrossRef]
  10. R. Stolen, J. Gordon, W. Tomlinson, and H. Haus, “The Raman response function of silica-core fibers,” J. Opt. Soc. Am. B 6, 1159–1166 (1989).
    [CrossRef]
  11. R. Stolen and W. Tomlinson, “Effect of the Raman part of the nonlinear refractive index on propagation of ultrashortoptical pulses in fibers,” J. Opt. Soc. Am. B 9, 565–573 (1992).
    [CrossRef]
  12. E. Marti-Panameno, J. Sánchez-Mondragón, and V. Vysloukh, “Theory of soliton pulse forming in an actively modelocked fiber laser,” IEEE J. Quantum Electron. 30, 822–826 (1994).
    [CrossRef]
  13. V. Afanasjev, V. Vysloukh, and V. Serkin, “Decay and interaction of femtosecond optical solitons induced by the Raman self-scattering effect,” Opt. Lett. 15, 489–491 (1990).
    [CrossRef]
  14. K. Tai, A. Hasegawa, and N. Bekki, “Fission of optical solitons induced by the stimulated Raman effect,” Opt. Lett. 13, 392–394 (1988).
    [CrossRef]
  15. Y. Kivshar, “Dark-soliton dynamics and shock waves induced by the stimulated Raman effect in optical fibers,” Phys. Rev. A 42, 1757–1761 (1990).
    [CrossRef] [PubMed]
  16. Y. Kivshar and V. Afanasjev, “Decay of dark solitons due to the stimulated Raman effect,” Opt. Lett. 16, 285–287 (1991).
    [CrossRef] [PubMed]
  17. I. Uzunov and V. Gerdjikov, “Self-frequency shift of dark solitons in optical fibers,” Phys. Rev. A 47, 1582–1585 (1993).
    [CrossRef] [PubMed]
  18. Y. Kivshar and X. Yang, “Perturbation-induced dynamics of dark solitons,” Phys. Rev. E 49, 1657–1670 (1994).
    [CrossRef]
  19. D. Christodoulides and M. Carvalho, “Compression, self-bending, and collapse of Gaussian beams in photorefractive crystals,” Opt. Lett. 19, 1714–1716 (1994).
    [CrossRef] [PubMed]
  20. Z. Sheng, Y. Cui, N. Cheng, and Y. Wei, “Photorefractive self-trapping and deflection of optical beams,” J. Opt. Soc. Am. B 13, 584–589 (1996).
    [CrossRef]
  21. W. Krolikowski, N. Akhmediev, B. Luther-Davies, and M. Cronin-Golomb, “Self-bending photorefractive solitons,” Phys. Rev. E 54, 5761–5765 (1996).
    [CrossRef]
  22. L. Jinsong and L. Keqing, “Screening-photovoltaic spatial solitons in biased photovoltaic–photorefractive crystals and their self-deflection,” J. Opt. Soc. Am. B 16, 550–555 (1999).
    [CrossRef]

2000 (1)

V. Aleshkevich, V. Vysloukh, and Y. Kartashov, “Self-bending of cnoidal waves in photorefractive medium with drift and diffusion nonlinearity,” Opt. Commun. 173, 277–284 (2000).
[CrossRef]

1999 (3)

V. Petnikova, V. Shuvalov, and V. Vysloukh, “Multicomponent photorefractive cnoidal waves: stability, localization and soliton asymptotics,” Phys. Rev. E 60, 1–10 (1999).
[CrossRef]

A. Ankiewicz, W. Krolikowski, and N. Akhmediev, “Partially coherent solitons of variable shape in a slow Kerr-like medium: exact solutions,” Phys. Rev. E 59, 6079–6087 (1999).
[CrossRef]

L. Jinsong and L. Keqing, “Screening-photovoltaic spatial solitons in biased photovoltaic–photorefractive crystals and their self-deflection,” J. Opt. Soc. Am. B 16, 550–555 (1999).
[CrossRef]

1998 (2)

F. Hioe, “Solitary waves for two and three coupled nonlinear Schrödinger equations,” Phys. Rev. E 58, 6700–6707 (1998).
[CrossRef]

V. Kutuzov, V. Petnikova, V. Shuvalov, and V. Vysloukh, “Cross-modulation coupling of incoherent soliton modes in photorefractive crystals,” Phys. Rev. E 57, 6056–6065 (1998).
[CrossRef]

1996 (2)

Z. Sheng, Y. Cui, N. Cheng, and Y. Wei, “Photorefractive self-trapping and deflection of optical beams,” J. Opt. Soc. Am. B 13, 584–589 (1996).
[CrossRef]

W. Krolikowski, N. Akhmediev, B. Luther-Davies, and M. Cronin-Golomb, “Self-bending photorefractive solitons,” Phys. Rev. E 54, 5761–5765 (1996).
[CrossRef]

1994 (3)

Y. Kivshar and X. Yang, “Perturbation-induced dynamics of dark solitons,” Phys. Rev. E 49, 1657–1670 (1994).
[CrossRef]

D. Christodoulides and M. Carvalho, “Compression, self-bending, and collapse of Gaussian beams in photorefractive crystals,” Opt. Lett. 19, 1714–1716 (1994).
[CrossRef] [PubMed]

E. Marti-Panameno, J. Sánchez-Mondragón, and V. Vysloukh, “Theory of soliton pulse forming in an actively modelocked fiber laser,” IEEE J. Quantum Electron. 30, 822–826 (1994).
[CrossRef]

1993 (1)

I. Uzunov and V. Gerdjikov, “Self-frequency shift of dark solitons in optical fibers,” Phys. Rev. A 47, 1582–1585 (1993).
[CrossRef] [PubMed]

1992 (1)

1991 (1)

1990 (2)

Y. Kivshar, “Dark-soliton dynamics and shock waves induced by the stimulated Raman effect in optical fibers,” Phys. Rev. A 42, 1757–1761 (1990).
[CrossRef] [PubMed]

V. Afanasjev, V. Vysloukh, and V. Serkin, “Decay and interaction of femtosecond optical solitons induced by the Raman self-scattering effect,” Opt. Lett. 15, 489–491 (1990).
[CrossRef]

1989 (1)

1988 (1)

1987 (2)

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, 510–524 (1987).
[CrossRef]

F. Mitschke and L. Mollenauer, “Experimental observation of interaction forces between solitons in optical fibers,” Opt. Lett. 12, 355–357 (1987).
[CrossRef] [PubMed]

1986 (2)

Afanasjev, V.

Akhmediev, N.

A. Ankiewicz, W. Krolikowski, and N. Akhmediev, “Partially coherent solitons of variable shape in a slow Kerr-like medium: exact solutions,” Phys. Rev. E 59, 6079–6087 (1999).
[CrossRef]

W. Krolikowski, N. Akhmediev, B. Luther-Davies, and M. Cronin-Golomb, “Self-bending photorefractive solitons,” Phys. Rev. E 54, 5761–5765 (1996).
[CrossRef]

Aleshkevich, V.

V. Aleshkevich, V. Vysloukh, and Y. Kartashov, “Self-bending of cnoidal waves in photorefractive medium with drift and diffusion nonlinearity,” Opt. Commun. 173, 277–284 (2000).
[CrossRef]

Ankiewicz, A.

A. Ankiewicz, W. Krolikowski, and N. Akhmediev, “Partially coherent solitons of variable shape in a slow Kerr-like medium: exact solutions,” Phys. Rev. E 59, 6079–6087 (1999).
[CrossRef]

Bekki, N.

Carvalho, M.

Cheng, N.

Christodoulides, D.

Cronin-Golomb, M.

W. Krolikowski, N. Akhmediev, B. Luther-Davies, and M. Cronin-Golomb, “Self-bending photorefractive solitons,” Phys. Rev. E 54, 5761–5765 (1996).
[CrossRef]

Cui, Y.

Gerdjikov, V.

I. Uzunov and V. Gerdjikov, “Self-frequency shift of dark solitons in optical fibers,” Phys. Rev. A 47, 1582–1585 (1993).
[CrossRef] [PubMed]

Gordon, J.

Hasegawa, A.

K. Tai, A. Hasegawa, and N. Bekki, “Fission of optical solitons induced by the stimulated Raman effect,” Opt. Lett. 13, 392–394 (1988).
[CrossRef]

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, 510–524 (1987).
[CrossRef]

Haus, H.

Hioe, F.

F. Hioe, “Solitary waves for two and three coupled nonlinear Schrödinger equations,” Phys. Rev. E 58, 6700–6707 (1998).
[CrossRef]

Jinsong, L.

Kartashov, Y.

V. Aleshkevich, V. Vysloukh, and Y. Kartashov, “Self-bending of cnoidal waves in photorefractive medium with drift and diffusion nonlinearity,” Opt. Commun. 173, 277–284 (2000).
[CrossRef]

Keqing, L.

Kivshar, Y.

Y. Kivshar and X. Yang, “Perturbation-induced dynamics of dark solitons,” Phys. Rev. E 49, 1657–1670 (1994).
[CrossRef]

Y. Kivshar and V. Afanasjev, “Decay of dark solitons due to the stimulated Raman effect,” Opt. Lett. 16, 285–287 (1991).
[CrossRef] [PubMed]

Y. Kivshar, “Dark-soliton dynamics and shock waves induced by the stimulated Raman effect in optical fibers,” Phys. Rev. A 42, 1757–1761 (1990).
[CrossRef] [PubMed]

Kodama, Y.

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, 510–524 (1987).
[CrossRef]

Krolikowski, W.

A. Ankiewicz, W. Krolikowski, and N. Akhmediev, “Partially coherent solitons of variable shape in a slow Kerr-like medium: exact solutions,” Phys. Rev. E 59, 6079–6087 (1999).
[CrossRef]

W. Krolikowski, N. Akhmediev, B. Luther-Davies, and M. Cronin-Golomb, “Self-bending photorefractive solitons,” Phys. Rev. E 54, 5761–5765 (1996).
[CrossRef]

Kutuzov, V.

V. Kutuzov, V. Petnikova, V. Shuvalov, and V. Vysloukh, “Cross-modulation coupling of incoherent soliton modes in photorefractive crystals,” Phys. Rev. E 57, 6056–6065 (1998).
[CrossRef]

Luther-Davies, B.

W. Krolikowski, N. Akhmediev, B. Luther-Davies, and M. Cronin-Golomb, “Self-bending photorefractive solitons,” Phys. Rev. E 54, 5761–5765 (1996).
[CrossRef]

Marti-Panameno, E.

E. Marti-Panameno, J. Sánchez-Mondragón, and V. Vysloukh, “Theory of soliton pulse forming in an actively modelocked fiber laser,” IEEE J. Quantum Electron. 30, 822–826 (1994).
[CrossRef]

Mitschke, F.

Mollenauer, L.

Petnikova, V.

V. Petnikova, V. Shuvalov, and V. Vysloukh, “Multicomponent photorefractive cnoidal waves: stability, localization and soliton asymptotics,” Phys. Rev. E 60, 1–10 (1999).
[CrossRef]

V. Kutuzov, V. Petnikova, V. Shuvalov, and V. Vysloukh, “Cross-modulation coupling of incoherent soliton modes in photorefractive crystals,” Phys. Rev. E 57, 6056–6065 (1998).
[CrossRef]

Sánchez-Mondragón, J.

E. Marti-Panameno, J. Sánchez-Mondragón, and V. Vysloukh, “Theory of soliton pulse forming in an actively modelocked fiber laser,” IEEE J. Quantum Electron. 30, 822–826 (1994).
[CrossRef]

Serkin, V.

Sheng, Z.

Shuvalov, V.

V. Petnikova, V. Shuvalov, and V. Vysloukh, “Multicomponent photorefractive cnoidal waves: stability, localization and soliton asymptotics,” Phys. Rev. E 60, 1–10 (1999).
[CrossRef]

V. Kutuzov, V. Petnikova, V. Shuvalov, and V. Vysloukh, “Cross-modulation coupling of incoherent soliton modes in photorefractive crystals,” Phys. Rev. E 57, 6056–6065 (1998).
[CrossRef]

Stolen, R.

Tai, K.

Tomlinson, W.

Uzunov, I.

I. Uzunov and V. Gerdjikov, “Self-frequency shift of dark solitons in optical fibers,” Phys. Rev. A 47, 1582–1585 (1993).
[CrossRef] [PubMed]

Vysloukh, V.

V. Aleshkevich, V. Vysloukh, and Y. Kartashov, “Self-bending of cnoidal waves in photorefractive medium with drift and diffusion nonlinearity,” Opt. Commun. 173, 277–284 (2000).
[CrossRef]

V. Petnikova, V. Shuvalov, and V. Vysloukh, “Multicomponent photorefractive cnoidal waves: stability, localization and soliton asymptotics,” Phys. Rev. E 60, 1–10 (1999).
[CrossRef]

V. Kutuzov, V. Petnikova, V. Shuvalov, and V. Vysloukh, “Cross-modulation coupling of incoherent soliton modes in photorefractive crystals,” Phys. Rev. E 57, 6056–6065 (1998).
[CrossRef]

E. Marti-Panameno, J. Sánchez-Mondragón, and V. Vysloukh, “Theory of soliton pulse forming in an actively modelocked fiber laser,” IEEE J. Quantum Electron. 30, 822–826 (1994).
[CrossRef]

V. Afanasjev, V. Vysloukh, and V. Serkin, “Decay and interaction of femtosecond optical solitons induced by the Raman self-scattering effect,” Opt. Lett. 15, 489–491 (1990).
[CrossRef]

Wei, Y.

Yang, X.

Y. Kivshar and X. Yang, “Perturbation-induced dynamics of dark solitons,” Phys. Rev. E 49, 1657–1670 (1994).
[CrossRef]

IEEE J. Quantum Electron. (2)

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, 510–524 (1987).
[CrossRef]

E. Marti-Panameno, J. Sánchez-Mondragón, and V. Vysloukh, “Theory of soliton pulse forming in an actively modelocked fiber laser,” IEEE J. Quantum Electron. 30, 822–826 (1994).
[CrossRef]

J. Opt. Soc. Am. B (4)

Opt. Commun. (1)

V. Aleshkevich, V. Vysloukh, and Y. Kartashov, “Self-bending of cnoidal waves in photorefractive medium with drift and diffusion nonlinearity,” Opt. Commun. 173, 277–284 (2000).
[CrossRef]

Opt. Lett. (7)

Phys. Rev. A (2)

Y. Kivshar, “Dark-soliton dynamics and shock waves induced by the stimulated Raman effect in optical fibers,” Phys. Rev. A 42, 1757–1761 (1990).
[CrossRef] [PubMed]

I. Uzunov and V. Gerdjikov, “Self-frequency shift of dark solitons in optical fibers,” Phys. Rev. A 47, 1582–1585 (1993).
[CrossRef] [PubMed]

Phys. Rev. E (6)

Y. Kivshar and X. Yang, “Perturbation-induced dynamics of dark solitons,” Phys. Rev. E 49, 1657–1670 (1994).
[CrossRef]

W. Krolikowski, N. Akhmediev, B. Luther-Davies, and M. Cronin-Golomb, “Self-bending photorefractive solitons,” Phys. Rev. E 54, 5761–5765 (1996).
[CrossRef]

V. Kutuzov, V. Petnikova, V. Shuvalov, and V. Vysloukh, “Cross-modulation coupling of incoherent soliton modes in photorefractive crystals,” Phys. Rev. E 57, 6056–6065 (1998).
[CrossRef]

A. Ankiewicz, W. Krolikowski, and N. Akhmediev, “Partially coherent solitons of variable shape in a slow Kerr-like medium: exact solutions,” Phys. Rev. E 59, 6079–6087 (1999).
[CrossRef]

V. Petnikova, V. Shuvalov, and V. Vysloukh, “Multicomponent photorefractive cnoidal waves: stability, localization and soliton asymptotics,” Phys. Rev. E 60, 1–10 (1999).
[CrossRef]

F. Hioe, “Solitary waves for two and three coupled nonlinear Schrödinger equations,” Phys. Rev. E 58, 6700–6707 (1998).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Input intensity distributions of the cnoidal (a) cn, (b) sn, and (c) dn waves for various values of localization parameter k.

Fig. 2
Fig. 2

Propagation dynamics of (a) a cn wave with k=0.85; (b) a dn wave with k=0.5, and (c) a sn wave with k=0.95 in the medium with delayed nonlinear response. β=0.2, δ=0.25, and μ=0.1.

Fig. 3
Fig. 3

Parabolic coefficient a versus localization parameter k for a cn wave: 1, results obtained with the aid of the finite-number harmonic approximation; 2, results of numerical integration by the split-step Fourier method. β=0.2, δ=0.25, and μ=0.1.

Fig. 4
Fig. 4

Parabolic coefficient a versus localization parameter k for a sn wave: 1, results obtained with the aid of the finite-number harmonic approximation; 2, results of numerical integration by the split-step Fourier method. β=0.2, δ=0.25, and μ=0.1.

Fig. 5
Fig. 5

Parabolic coefficient a versus localization parameter k for a dn wave: 1, results obtained with the aid of the finite-number harmonic approximation; 2, results of numerical integration by the split-step Fourier method. β=0.2, δ=0.25, and μ=0.1.

Fig. 6
Fig. 6

Propagation dynamics of (a) cn, (b) dn, and (c) sn waves in the medium with delayed nonlinear response. β=0.2, δ=0.25, and μ=0.5; localization parameter, k=0.8.

Fig. 7
Fig. 7

Parabolic coefficient as a function of parameter μ for localization parameter k=0.95. Calculations in the framework of 1, the perturbation model; 2, the relaxation model; curve 3, the oscillator model. β=0.2 and δ=0.25.

Equations (35)

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

i qξ=±12 2qη2-(1-β)|q|2q-βwq,
μ2 2wη2+2δμ wη+w=|q|2.
qcn(η, ξ)=k cn(η, k)exp[iξ(k2-1/2)],
qdn(η, ξ)=dn(η, k)exp[iξ(1-k2/2)]
qsn(η, ξ)=k sn(η, k)exp[iξ(k2+1)/2]
qcn(η, ξ)|k0=k cos(η)exp(-iξ/2),
qcn(η, ξ)|k1=sech(η)exp(iξ/2),
qsn(η, ξ)|k0=k sin(η)exp(iξ/2),
qsn(η, ξ)|k1=tanh(η)exp(iξ),
qdn(η, ξ)|k0=exp(iξ),
qdn(η, ξ)|k1=sech(η)exp(iξ/2).
k cn(η, k)=2πK(k) m=0 gm+1/2(1+g2m+1)-1 cos (2m+1)π2K(k)η,
qcn(η, ξ)=A(ξ)exp(iωη)+B(ξ)exp(-iωη),
i dAdξ=12ω2A-(1-β)A(AA*-2BB*)-βA{AA*+[1+L*(ω)]BB*},
i dBdξ=12ω2B-(1-β)B(BB*-2AA*)-βB{BB*+[1+L(ω)]AA*},
L(ω)=(1-4iμδω-4µ2ω2)-1,
QA=Q{1+exp[-2β Im L(ω)Q2ξ]}-1/2,
QB=Q{1+exp[2β Im L(ω)Q2ξ]}-1/2,
ψ=(1-β)+β Re L(ω)β Im L(ω) ln{cosh[-β Im L(ω)Q2ξ]}.
a=β2ω Im L(ω)[(1-β)+β Re L(ω)]Q4,
k sn(η, k)=2πK(k) m=0 gm+1/2(1-g2m+1)-1×sin (2m+1)π2K(k)η.
dn(η, k)=π2K(k)+2πK(k) m=1×gm(1+g2m)-1 cos πmK(k)η.
q(η, ξ)=A(ξ)exp(2iωη)+B(ξ)+C(ξ)exp(-2iωη),
i dAdξ=2ω2A-(1-β)(2UA-A2A*+B2C*)-β[UA+L*(ω)(ABB*+B2C*)],
i dBdξ=-(1-β)(2UB-B2B*+2ACB*)-β[UB+L*(ω)(ACB*+BCC*)+L(ω)×(ACB*+BAA*)],
i dCdξ=2ω2C-(1-β)(2UC-C2C*+B2A*)-β[UC+L(ω)(CBB*+B2A*)],
QAQA(0)[1+2β Im L(ω)Q2ξ],QBQ,
QCQC(0)[1-2β Im L(ω)Q2ξ],
ψC-ψBψB-ψAβ[2(1-β)+β Re L(ω)]Im L(ω)Q4ξ2,
a=(β/ω)[2(1-β)+β Re L(ω)]Im L(ω)Q4,
u0 d2dξ2ηcenter=-β- |q(η)|2η dη-G(η-ζ)|q(ζ)|2dζ,
ηcenter(ξ)=1u0 -qηq*dη,
G(η)=μ-1(1-δ2)-1/2 exp(-δη/μ)sin[(1-δ2)1/2η/μ]
2δμ wη+w=|q|2.
i qξ=-12 2qη2-q|q|2+2βδμq η|q|2.

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