Abstract

We analyze the propagation of soliton trains with small initial separation in the presence of third-order dispersion. We show that both the amplitudes and the positions can be stabilized, provided that phase modulation is applied and adjacent pulses are initially out of phase.

© 1995 Optical Society of America

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References

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  1. G. P. Agrawal, Nonlinear Fiber Optics (Academic, Orlando, Fla., 1989).
  2. C. Desem and P. L. Chu, "Soliton–soliton interactions," in Optical Solitons—Theory and Experiment, J. R. Taylor, ed. (Cambridge U. Press, Cambridge, 1992), pp. 127–133.
  3. A. S. Gouveiva-Neto, M. E. Faldon, and J. R. Taylor, Opt. Lett. 13, 770 (1988).
    [CrossRef]
  4. A. Hasegawa, T. Yano, and Y. Kodama, "Separation of signal from noise in soliton transmission systems," in Nonlinear Guided-Wave Phenomena, Vol. 15 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 174.
  5. Y. Kodama, M. Romagnoli, S. Wabnitz, and M. Midrio, Opt. Lett. 19, 165 (1994).
    [CrossRef]
  6. I. M. Uzunov, M. Gölles, and F. Lederer, Electron. Lett. 30, 882 (1994).
    [CrossRef]
  7. S. M. J. Kelly, T. Brabec, F. Krausz, and J. N. Elgin, "Suppression of resonances associated with periodically amplified solitons," in Nonlinear Guided-Wave Phenomena, Vol. 15 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 188.
  8. F. W. Wise, I. A. Walmsley, and G. L. Tang, Opt. Lett. 13, 129 (1988).
    [CrossRef]
  9. H. A. Haus, J. D. Moores, and L. E. Nelson, Opt. Lett. 18, 51 (1993).
    [CrossRef] [PubMed]
  10. P. L. Chu and C. Desem, Electron. Lett. 21, 228 (1985).
    [CrossRef]
  11. I. M. Uzunov, V. M. Mitev, and L. M. Kovachev, Opt. Commun. 70, 389 (1989); erratum 108, 392 (1994).
    [CrossRef]
  12. I. M. Uzunov and V. M. Mitev, Opt. Quantum Electron. 21, 507 (1989).
    [CrossRef]
  13. N. J. Smith, W. J. Firth, K. J. Blow, and K. Smith, Opt. Lett. 19, 16 (1994).
    [CrossRef] [PubMed]
  14. S. Wabnitz, Electron. Lett. 29, 1711 (1993).
    [CrossRef]
  15. V. I. Karpman and V. V. Solov'ev, Physica 3D, 487 (1981).
  16. M. Uzunov, M. Gölles, and F. Lederer, "Soliton interaction near the zero-dispersion wavelength," submitted to Phys. Rev. E.
  17. B. A. Malomed, Phys. Rev. E 47, 2874 (1993).
    [CrossRef]
  18. M. Klauder, E. W. Laedke, K. H. Spatschek, and S. K. Turitsyn, Phys. Rev. E 47, R3844 (1993).
    [CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, Orlando, Fla., 1989).

Blow, K. J.

N. J. Smith, W. J. Firth, K. J. Blow, and K. Smith, Opt. Lett. 19, 16 (1994).
[CrossRef] [PubMed]

Brabec, T.

S. M. J. Kelly, T. Brabec, F. Krausz, and J. N. Elgin, "Suppression of resonances associated with periodically amplified solitons," in Nonlinear Guided-Wave Phenomena, Vol. 15 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 188.

Chu, P. L.

P. L. Chu and C. Desem, Electron. Lett. 21, 228 (1985).
[CrossRef]

C. Desem and P. L. Chu, "Soliton–soliton interactions," in Optical Solitons—Theory and Experiment, J. R. Taylor, ed. (Cambridge U. Press, Cambridge, 1992), pp. 127–133.

Desem, C.

C. Desem and P. L. Chu, "Soliton–soliton interactions," in Optical Solitons—Theory and Experiment, J. R. Taylor, ed. (Cambridge U. Press, Cambridge, 1992), pp. 127–133.

P. L. Chu and C. Desem, Electron. Lett. 21, 228 (1985).
[CrossRef]

Elgin, J. N.

S. M. J. Kelly, T. Brabec, F. Krausz, and J. N. Elgin, "Suppression of resonances associated with periodically amplified solitons," in Nonlinear Guided-Wave Phenomena, Vol. 15 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 188.

Faldon, M. E.

A. S. Gouveiva-Neto, M. E. Faldon, and J. R. Taylor, Opt. Lett. 13, 770 (1988).
[CrossRef]

Firth, W. J.

N. J. Smith, W. J. Firth, K. J. Blow, and K. Smith, Opt. Lett. 19, 16 (1994).
[CrossRef] [PubMed]

Gölles, M.

I. M. Uzunov, M. Gölles, and F. Lederer, Electron. Lett. 30, 882 (1994).
[CrossRef]

M. Uzunov, M. Gölles, and F. Lederer, "Soliton interaction near the zero-dispersion wavelength," submitted to Phys. Rev. E.

Gouveiva-Neto, A. S.

A. S. Gouveiva-Neto, M. E. Faldon, and J. R. Taylor, Opt. Lett. 13, 770 (1988).
[CrossRef]

Hasegawa, A.

A. Hasegawa, T. Yano, and Y. Kodama, "Separation of signal from noise in soliton transmission systems," in Nonlinear Guided-Wave Phenomena, Vol. 15 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 174.

Haus, H. A.

H. A. Haus, J. D. Moores, and L. E. Nelson, Opt. Lett. 18, 51 (1993).
[CrossRef] [PubMed]

Karpman, V. I.

V. I. Karpman and V. V. Solov'ev, Physica 3D, 487 (1981).

Kelly, S. M. J.

S. M. J. Kelly, T. Brabec, F. Krausz, and J. N. Elgin, "Suppression of resonances associated with periodically amplified solitons," in Nonlinear Guided-Wave Phenomena, Vol. 15 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 188.

Klauder, M.

M. Klauder, E. W. Laedke, K. H. Spatschek, and S. K. Turitsyn, Phys. Rev. E 47, R3844 (1993).
[CrossRef]

Kodama, Y.

A. Hasegawa, T. Yano, and Y. Kodama, "Separation of signal from noise in soliton transmission systems," in Nonlinear Guided-Wave Phenomena, Vol. 15 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 174.

Y. Kodama, M. Romagnoli, S. Wabnitz, and M. Midrio, Opt. Lett. 19, 165 (1994).
[CrossRef]

Kovachev, L. M.

I. M. Uzunov, V. M. Mitev, and L. M. Kovachev, Opt. Commun. 70, 389 (1989); erratum 108, 392 (1994).
[CrossRef]

Krausz, F.

S. M. J. Kelly, T. Brabec, F. Krausz, and J. N. Elgin, "Suppression of resonances associated with periodically amplified solitons," in Nonlinear Guided-Wave Phenomena, Vol. 15 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 188.

Laedke, E. W.

M. Klauder, E. W. Laedke, K. H. Spatschek, and S. K. Turitsyn, Phys. Rev. E 47, R3844 (1993).
[CrossRef]

Lederer, F.

M. Uzunov, M. Gölles, and F. Lederer, "Soliton interaction near the zero-dispersion wavelength," submitted to Phys. Rev. E.

I. M. Uzunov, M. Gölles, and F. Lederer, Electron. Lett. 30, 882 (1994).
[CrossRef]

Malomed, B. A.

B. A. Malomed, Phys. Rev. E 47, 2874 (1993).
[CrossRef]

Midrio, M.

Y. Kodama, M. Romagnoli, S. Wabnitz, and M. Midrio, Opt. Lett. 19, 165 (1994).
[CrossRef]

Mitev, V. M.

I. M. Uzunov and V. M. Mitev, Opt. Quantum Electron. 21, 507 (1989).
[CrossRef]

I. M. Uzunov, V. M. Mitev, and L. M. Kovachev, Opt. Commun. 70, 389 (1989); erratum 108, 392 (1994).
[CrossRef]

Moores, J. D.

H. A. Haus, J. D. Moores, and L. E. Nelson, Opt. Lett. 18, 51 (1993).
[CrossRef] [PubMed]

Nelson, L. E.

H. A. Haus, J. D. Moores, and L. E. Nelson, Opt. Lett. 18, 51 (1993).
[CrossRef] [PubMed]

Romagnoli, M.

Y. Kodama, M. Romagnoli, S. Wabnitz, and M. Midrio, Opt. Lett. 19, 165 (1994).
[CrossRef]

Smith, K.

N. J. Smith, W. J. Firth, K. J. Blow, and K. Smith, Opt. Lett. 19, 16 (1994).
[CrossRef] [PubMed]

Smith, N. J.

N. J. Smith, W. J. Firth, K. J. Blow, and K. Smith, Opt. Lett. 19, 16 (1994).
[CrossRef] [PubMed]

Solov'ev, V. V.

V. I. Karpman and V. V. Solov'ev, Physica 3D, 487 (1981).

Spatschek, K. H.

M. Klauder, E. W. Laedke, K. H. Spatschek, and S. K. Turitsyn, Phys. Rev. E 47, R3844 (1993).
[CrossRef]

Tang, G. L.

F. W. Wise, I. A. Walmsley, and G. L. Tang, Opt. Lett. 13, 129 (1988).
[CrossRef]

Taylor, J. R.

A. S. Gouveiva-Neto, M. E. Faldon, and J. R. Taylor, Opt. Lett. 13, 770 (1988).
[CrossRef]

Turitsyn, S. K.

M. Klauder, E. W. Laedke, K. H. Spatschek, and S. K. Turitsyn, Phys. Rev. E 47, R3844 (1993).
[CrossRef]

Uzunov, I. M.

I. M. Uzunov, V. M. Mitev, and L. M. Kovachev, Opt. Commun. 70, 389 (1989); erratum 108, 392 (1994).
[CrossRef]

I. M. Uzunov and V. M. Mitev, Opt. Quantum Electron. 21, 507 (1989).
[CrossRef]

I. M. Uzunov, M. Gölles, and F. Lederer, Electron. Lett. 30, 882 (1994).
[CrossRef]

Uzunov, M.

M. Uzunov, M. Gölles, and F. Lederer, "Soliton interaction near the zero-dispersion wavelength," submitted to Phys. Rev. E.

Wabnitz, S.

Y. Kodama, M. Romagnoli, S. Wabnitz, and M. Midrio, Opt. Lett. 19, 165 (1994).
[CrossRef]

S. Wabnitz, Electron. Lett. 29, 1711 (1993).
[CrossRef]

Walmsley, I. A.

F. W. Wise, I. A. Walmsley, and G. L. Tang, Opt. Lett. 13, 129 (1988).
[CrossRef]

Wise, F. W.

F. W. Wise, I. A. Walmsley, and G. L. Tang, Opt. Lett. 13, 129 (1988).
[CrossRef]

Yano, T.

A. Hasegawa, T. Yano, and Y. Kodama, "Separation of signal from noise in soliton transmission systems," in Nonlinear Guided-Wave Phenomena, Vol. 15 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 174.

Other

G. P. Agrawal, Nonlinear Fiber Optics (Academic, Orlando, Fla., 1989).

C. Desem and P. L. Chu, "Soliton–soliton interactions," in Optical Solitons—Theory and Experiment, J. R. Taylor, ed. (Cambridge U. Press, Cambridge, 1992), pp. 127–133.

A. S. Gouveiva-Neto, M. E. Faldon, and J. R. Taylor, Opt. Lett. 13, 770 (1988).
[CrossRef]

A. Hasegawa, T. Yano, and Y. Kodama, "Separation of signal from noise in soliton transmission systems," in Nonlinear Guided-Wave Phenomena, Vol. 15 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 174.

Y. Kodama, M. Romagnoli, S. Wabnitz, and M. Midrio, Opt. Lett. 19, 165 (1994).
[CrossRef]

I. M. Uzunov, M. Gölles, and F. Lederer, Electron. Lett. 30, 882 (1994).
[CrossRef]

S. M. J. Kelly, T. Brabec, F. Krausz, and J. N. Elgin, "Suppression of resonances associated with periodically amplified solitons," in Nonlinear Guided-Wave Phenomena, Vol. 15 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 188.

F. W. Wise, I. A. Walmsley, and G. L. Tang, Opt. Lett. 13, 129 (1988).
[CrossRef]

H. A. Haus, J. D. Moores, and L. E. Nelson, Opt. Lett. 18, 51 (1993).
[CrossRef] [PubMed]

P. L. Chu and C. Desem, Electron. Lett. 21, 228 (1985).
[CrossRef]

I. M. Uzunov, V. M. Mitev, and L. M. Kovachev, Opt. Commun. 70, 389 (1989); erratum 108, 392 (1994).
[CrossRef]

I. M. Uzunov and V. M. Mitev, Opt. Quantum Electron. 21, 507 (1989).
[CrossRef]

N. J. Smith, W. J. Firth, K. J. Blow, and K. Smith, Opt. Lett. 19, 16 (1994).
[CrossRef] [PubMed]

S. Wabnitz, Electron. Lett. 29, 1711 (1993).
[CrossRef]

V. I. Karpman and V. V. Solov'ev, Physica 3D, 487 (1981).

M. Uzunov, M. Gölles, and F. Lederer, "Soliton interaction near the zero-dispersion wavelength," submitted to Phys. Rev. E.

B. A. Malomed, Phys. Rev. E 47, 2874 (1993).
[CrossRef]

M. Klauder, E. W. Laedke, K. H. Spatschek, and S. K. Turitsyn, Phys. Rev. E 47, R3844 (1993).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Decay of the three-soliton bound state: a, r(0) = 6, γ = 0.05; b, r(0) = 8, γ= 0.05.

Fig. 2
Fig. 2

Minimal amplitude of phase modulation αmin versus TOD (γ) for r(0) = 6. In-phase pulses: solid curve, beam propagation method; dashed curve, perturbation approach. Out-of-phase pulses: dotted curve, beam propagation method; dashed-dotted curve, perturbation approach.

Fig. 3
Fig. 3

Multisoliton interaction in the presence of third-order dispersion; r(0) = 6, γ = 0.05. (a) Decay of the bound state of in-phase [ψ(0) = 0] solitons; α = 0. (b) Repulsion of out-of-phase [ψ(0) = π] pulses, stabilization of the amplitudes; α = 0. (c) Stabilization of the positions of in-phase pulses with phase modulation applied; α = 0.075, Ω = π/3. (d) Suppression of interaction and stabilization of both the amplitudes and the positions for out-of-phase pulses and phase modulation applied; α = 0.075, Ω = π/3.

Equations (6)

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i u z + 1 2 2 u t 2 + | u | 2 u = i γ 3 u t 3 + α cos ( Ω t ) u ,
u ( z , t ) = n = 1 2 2 ν n sech [ 2 ν n ( t ξ n ) ] exp [ i 2 μ n ( t ξ n ) + i δ n ] ,
d μ n d z = ( 1 ) n 16 ν 3 exp ( 2 ν r ) cos Φ + α Ω Θ n 2 sin ( Ω ξ n ) sinh Θ n ,
d ν n d z = ( 1 ) n 16 ν 3 exp ( 2 ν r ) sin Φ ,
d ξ n d z = 2 μ n + 4 ν exp ( 2 ν r ) sin Φ + 4 γ ( ν n 2 + 3 μ n 2 ) ,
d δ n d z = 2 ( ν n 2 + μ n 2 ) + 8 ν exp ( 2 ν r ) × ( μ sin Φ + 3 ν cos Φ ) + 16 γ μ n ( ν n 2 μ n 2 ) α Θ n 2 cos ( Ω ξ n ) cosh Θ n sinh 2 Θ n ,

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