Abstract

We study the effect of third-order dispersion on the interaction between two solitons from different frequency channels in an optical fiber. The interaction may be viewed as an inelastic collision in which energy is lost to continuous radiation owing to nonzero third-order dispersion. We develop a perturbation theory with two small parameters: the third-order dispersion coefficient d3 and the reciprocal of the interchannel frequency difference 1/Ω. In the leading order the amplitude of the emitted radiation is proportional to d3/Ω2, and the source term for this radiation is identical to the one produced by perturbation of the second-order dispersion coefficient. The only other effects up to the third order are shifts in the soliton’s phase and position. Our results show that the statistical description of soliton propagation in a given channel influenced by interaction with a quasi-random sequence of solitons from other channels is similar to the description of soliton propagation in fibers with weak disorder in the second-order dispersion coefficient.

© 2004 Optical Society of America

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    [CrossRef]
  5. J. P. Gordon, “Interaction forces among solitons in optical fibers,” Opt. Lett. 8, 596–598 (1983).
    [CrossRef] [PubMed]
  6. D. Anderson and M. Lisak, “Bandwidth limits due to mutual pulse interaction in optical soliton communication systems,” Opt. Lett. 11, 174–176 (1986).
    [CrossRef] [PubMed]
  7. L. F. Mollenauer, J. P. Gordon, and P. V. Mamyshev, “Solitons in high bit-rate, long-distance transmission,” in Optical Fiber Telecommunications III, I. P. Kaminov and T. L. Koch, eds. (Academic, San Diego, Calif., 1997), Chap. 12, Sect. V-B.
  8. H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
    [CrossRef]
  9. Y. Kodama and K. Nozaki, “Soliton interaction in optical fibers,” Opt. Lett. 12, 1038–1040 (1987).
    [CrossRef] [PubMed]
  10. T. Kano, “Normal form of nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 58, 4322–4328 (1989).
    [CrossRef]
  11. Y. Kodama and S. Wabnitz, “Reduction of soliton interaction forces by bandwidth limited amplification,” Electron. Lett. 27, 1931–1933 (1991).
    [CrossRef]
  12. Y. Kodama, M. Romagnoli, S. Wabnitz, and M. Midrio, “Role of third-order dispersion on soliton instabilities and interactions in optical fibers,” Opt. Lett. 19, 165–167 (1994).
    [CrossRef] [PubMed]
  13. M. Chertkov, Y. Chung, A. Dyachenko, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in weakly disordered optical fibers,” Phys. Rev. E 67, 036615 (2003).
    [CrossRef]
  14. Y. Kodama and A. V. Mikhailov, “Obstacles to asymptotic integrability,” in Algebraic Aspects to Integrable Systems, A. S. Fokas and I. M. Gelfand, eds. (Birkhäuser, Boston, 1997).
  15. Y. Kodama, “On integrable systems with higher-order corrections,” Phys. Lett. 107A, 245–249 (1985).
    [CrossRef]
  16. J. N. Elgin, “Soliton propagation in an optical fiber with 3rd-order dispersion,” Opt. Lett. 17, 1409–1410 (1992).
    [CrossRef]
  17. J. N. Elgin, T. Brabec, and S. M. J. Kelly, “A perturbative theory of soliton propagation in the presence of third order dispersion,” Opt. Commun. 114, 321–328 (1995).
    [CrossRef]
  18. T. P. Horikis and J. N. Elgin, “Soliton radiation in an optical fiber,” J. Opt. Soc. Am. B 18, 913–918 (2001).
    [CrossRef]
  19. D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
    [CrossRef] [PubMed]
  20. A. Peleg, M. Chertkov, and I. Gabitov, “Inter-channel interaction of optical solitons,” Phys. Rev. E 68, 026605 (2003).
    [CrossRef]
  21. The dimensionless z in Eq. (1) is z=x(αP0/2), where x is the actual position, P0 is the peak soliton power, and α is the Kerr nonlinearity coefficient. The dimensionless retarded time is t=τ/τ0, where τ is the retarded time and τ0 is the soliton width. The spectral width ν0 is given by ν0= 1/(π2τ0), and the channel spacing is given by Δν= Ων0. Ψ=E/P0, where E is the actual electric field. The dimensionless second- and third-order dispersion coefficients are given by d=−1=β2/(αP0τ02) and d3= β3/(3αP0τ03), where β2 and β3 are the second- and third-order chromatic dispersion coefficients, respectively.
  22. Effects of fiber losses can be neglected (see, for instance, Ref. 23) if values of Δν, β2, β3, and τ0 satisfy the following two conditions: τ0≪β2/γ and Δν≫(γτ02)/β3.
  23. F. G. Omenetto, Y. Chung, D. Yarotski, T. Schaefer, I. Gabitov, and A. J. Taylor, “Phase analysis of nonlinear femtosecond pulse propagation and self-frequency shift in optical fibers,” Opt. Commun. 208, 191–196 (2002).
    [CrossRef]
  24. L. F. Mollenauer and P. V. Mamyshev, “Massive wavelength-division multiplexing with solitons,” IEEE J. Quantum Electron. 34, 2089–2102 (1998).
    [CrossRef]
  25. Y. Kodama, A. V. Mikhailov, and S. Wabnitz, “Input pulse optimization in wavelength-division-multiplexed soliton transmission,” Opt. Commun. 143, 53–56 (1997).
    [CrossRef]
  26. Note that it was found in Ref. 17 that even if the pulse launched into the fiber is not exactly of the form given by Eq. (3) it evolves into this form after a transient.
  27. L. F. Mollenauer, P. V. Mamyshev, and M. J. Neubelt, “Demonstration of soliton WDM transmission at 6 and 7× 10 Gbit/s, error free over transoceanic distances,” Electron. Lett. 32, 471–474 (1996).
    [CrossRef]
  28. L. F. Mollenauer and P. V. Mamyshev, “Wavelength-division-multiplexing channel energy self-equalization in a soliton transmission line by guiding filters,” Opt. Lett. 20, 1658–1660 (1996).
  29. P. K. A. Wai, C. R. Menyuk, H. H. Chen, and Y. C. Lee, “Soliton at the zero-group-dispersion wavelength of a single-modal fiber,” Opt. Lett. 12, 628–630 (1987).
    [CrossRef] [PubMed]
  30. A. S. Gouveia-Neto, M. E. Faldon, and J. R. Taylor, “Solitons in the region of the minimum group-velocity dispersion of single-mode optical fibers,” Opt. Lett. 13, 770–772 (1988).
    [CrossRef] [PubMed]
  31. P. K. A. Wai, H. H. Chen, and Y. C. Lee, “Radiation by soli-tons at the zero group-dispersion wavelength of single-mode fibers,” Phys. Rev. A 41, 426–439 (1990).
    [CrossRef] [PubMed]
  32. M. Nakazawa, E. Yoshida, K. Suzuki, T. Kitoh, and M. Kawachi, “80 Gbit/s soliton data transmission over 500 km with unequal amplitude solitons for timing clock extraction,” Electron. Lett. 30, 1777–1778 (1994).
    [CrossRef]
  33. M. Nakazawa, K. Suzuki, E. Yoshida, E. Yamada, T. Kitoh, and M. Kawachi, “160 Gbit/s soliton data transmission over 200 km,” Electron. Lett. 31, 565–566 (1995).
    [CrossRef]
  34. M. Nakazawa, K. Suzuki, E. Yoshida, E. Yamada, T. Kitoh, and M. Kawachi, “160 Gbit/s (80 Gbit/s×2 channels) WDM soliton transmission over 10000 km using in-line synchronous modulation,” Electron. Lett. 35, 1358–1359 (1999).
    [CrossRef]

2003 (2)

M. Chertkov, Y. Chung, A. Dyachenko, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in weakly disordered optical fibers,” Phys. Rev. E 67, 036615 (2003).
[CrossRef]

A. Peleg, M. Chertkov, and I. Gabitov, “Inter-channel interaction of optical solitons,” Phys. Rev. E 68, 026605 (2003).
[CrossRef]

2002 (1)

F. G. Omenetto, Y. Chung, D. Yarotski, T. Schaefer, I. Gabitov, and A. J. Taylor, “Phase analysis of nonlinear femtosecond pulse propagation and self-frequency shift in optical fibers,” Opt. Commun. 208, 191–196 (2002).
[CrossRef]

2001 (1)

1999 (1)

M. Nakazawa, K. Suzuki, E. Yoshida, E. Yamada, T. Kitoh, and M. Kawachi, “160 Gbit/s (80 Gbit/s×2 channels) WDM soliton transmission over 10000 km using in-line synchronous modulation,” Electron. Lett. 35, 1358–1359 (1999).
[CrossRef]

1998 (1)

L. F. Mollenauer and P. V. Mamyshev, “Massive wavelength-division multiplexing with solitons,” IEEE J. Quantum Electron. 34, 2089–2102 (1998).
[CrossRef]

1997 (1)

Y. Kodama, A. V. Mikhailov, and S. Wabnitz, “Input pulse optimization in wavelength-division-multiplexed soliton transmission,” Opt. Commun. 143, 53–56 (1997).
[CrossRef]

1996 (3)

L. F. Mollenauer, P. V. Mamyshev, and M. J. Neubelt, “Demonstration of soliton WDM transmission at 6 and 7× 10 Gbit/s, error free over transoceanic distances,” Electron. Lett. 32, 471–474 (1996).
[CrossRef]

L. F. Mollenauer and P. V. Mamyshev, “Wavelength-division-multiplexing channel energy self-equalization in a soliton transmission line by guiding filters,” Opt. Lett. 20, 1658–1660 (1996).

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
[CrossRef]

1995 (2)

J. N. Elgin, T. Brabec, and S. M. J. Kelly, “A perturbative theory of soliton propagation in the presence of third order dispersion,” Opt. Commun. 114, 321–328 (1995).
[CrossRef]

M. Nakazawa, K. Suzuki, E. Yoshida, E. Yamada, T. Kitoh, and M. Kawachi, “160 Gbit/s soliton data transmission over 200 km,” Electron. Lett. 31, 565–566 (1995).
[CrossRef]

1994 (2)

M. Nakazawa, E. Yoshida, K. Suzuki, T. Kitoh, and M. Kawachi, “80 Gbit/s soliton data transmission over 500 km with unequal amplitude solitons for timing clock extraction,” Electron. Lett. 30, 1777–1778 (1994).
[CrossRef]

Y. Kodama, M. Romagnoli, S. Wabnitz, and M. Midrio, “Role of third-order dispersion on soliton instabilities and interactions in optical fibers,” Opt. Lett. 19, 165–167 (1994).
[CrossRef] [PubMed]

1992 (1)

1991 (1)

Y. Kodama and S. Wabnitz, “Reduction of soliton interaction forces by bandwidth limited amplification,” Electron. Lett. 27, 1931–1933 (1991).
[CrossRef]

1990 (2)

D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
[CrossRef] [PubMed]

P. K. A. Wai, H. H. Chen, and Y. C. Lee, “Radiation by soli-tons at the zero group-dispersion wavelength of single-mode fibers,” Phys. Rev. A 41, 426–439 (1990).
[CrossRef] [PubMed]

1989 (1)

T. Kano, “Normal form of nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 58, 4322–4328 (1989).
[CrossRef]

1988 (1)

1987 (2)

1986 (1)

1985 (1)

Y. Kodama, “On integrable systems with higher-order corrections,” Phys. Lett. 107A, 245–249 (1985).
[CrossRef]

1983 (1)

1981 (1)

V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the 2-soliton systems,” Phys. D 3, 487–502 (1981).
[CrossRef]

Anderson, D.

Brabec, T.

J. N. Elgin, T. Brabec, and S. M. J. Kelly, “A perturbative theory of soliton propagation in the presence of third order dispersion,” Opt. Commun. 114, 321–328 (1995).
[CrossRef]

Chen, H. H.

P. K. A. Wai, H. H. Chen, and Y. C. Lee, “Radiation by soli-tons at the zero group-dispersion wavelength of single-mode fibers,” Phys. Rev. A 41, 426–439 (1990).
[CrossRef] [PubMed]

P. K. A. Wai, C. R. Menyuk, H. H. Chen, and Y. C. Lee, “Soliton at the zero-group-dispersion wavelength of a single-modal fiber,” Opt. Lett. 12, 628–630 (1987).
[CrossRef] [PubMed]

Chertkov, M.

M. Chertkov, Y. Chung, A. Dyachenko, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in weakly disordered optical fibers,” Phys. Rev. E 67, 036615 (2003).
[CrossRef]

A. Peleg, M. Chertkov, and I. Gabitov, “Inter-channel interaction of optical solitons,” Phys. Rev. E 68, 026605 (2003).
[CrossRef]

Chung, Y.

M. Chertkov, Y. Chung, A. Dyachenko, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in weakly disordered optical fibers,” Phys. Rev. E 67, 036615 (2003).
[CrossRef]

F. G. Omenetto, Y. Chung, D. Yarotski, T. Schaefer, I. Gabitov, and A. J. Taylor, “Phase analysis of nonlinear femtosecond pulse propagation and self-frequency shift in optical fibers,” Opt. Commun. 208, 191–196 (2002).
[CrossRef]

Dyachenko, A.

M. Chertkov, Y. Chung, A. Dyachenko, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in weakly disordered optical fibers,” Phys. Rev. E 67, 036615 (2003).
[CrossRef]

Elgin, J. N.

Faldon, M. E.

Gabitov, I.

M. Chertkov, Y. Chung, A. Dyachenko, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in weakly disordered optical fibers,” Phys. Rev. E 67, 036615 (2003).
[CrossRef]

A. Peleg, M. Chertkov, and I. Gabitov, “Inter-channel interaction of optical solitons,” Phys. Rev. E 68, 026605 (2003).
[CrossRef]

F. G. Omenetto, Y. Chung, D. Yarotski, T. Schaefer, I. Gabitov, and A. J. Taylor, “Phase analysis of nonlinear femtosecond pulse propagation and self-frequency shift in optical fibers,” Opt. Commun. 208, 191–196 (2002).
[CrossRef]

Gordon, J. P.

Gouveia-Neto, A. S.

Haus, H. A.

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
[CrossRef]

Horikis, T. P.

Kano, T.

T. Kano, “Normal form of nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 58, 4322–4328 (1989).
[CrossRef]

Karpman, V. I.

V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the 2-soliton systems,” Phys. D 3, 487–502 (1981).
[CrossRef]

Kaup, D. J.

D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
[CrossRef] [PubMed]

Kawachi, M.

M. Nakazawa, K. Suzuki, E. Yoshida, E. Yamada, T. Kitoh, and M. Kawachi, “160 Gbit/s (80 Gbit/s×2 channels) WDM soliton transmission over 10000 km using in-line synchronous modulation,” Electron. Lett. 35, 1358–1359 (1999).
[CrossRef]

M. Nakazawa, K. Suzuki, E. Yoshida, E. Yamada, T. Kitoh, and M. Kawachi, “160 Gbit/s soliton data transmission over 200 km,” Electron. Lett. 31, 565–566 (1995).
[CrossRef]

M. Nakazawa, E. Yoshida, K. Suzuki, T. Kitoh, and M. Kawachi, “80 Gbit/s soliton data transmission over 500 km with unequal amplitude solitons for timing clock extraction,” Electron. Lett. 30, 1777–1778 (1994).
[CrossRef]

Kelly, S. M. J.

J. N. Elgin, T. Brabec, and S. M. J. Kelly, “A perturbative theory of soliton propagation in the presence of third order dispersion,” Opt. Commun. 114, 321–328 (1995).
[CrossRef]

Kitoh, T.

M. Nakazawa, K. Suzuki, E. Yoshida, E. Yamada, T. Kitoh, and M. Kawachi, “160 Gbit/s (80 Gbit/s×2 channels) WDM soliton transmission over 10000 km using in-line synchronous modulation,” Electron. Lett. 35, 1358–1359 (1999).
[CrossRef]

M. Nakazawa, K. Suzuki, E. Yoshida, E. Yamada, T. Kitoh, and M. Kawachi, “160 Gbit/s soliton data transmission over 200 km,” Electron. Lett. 31, 565–566 (1995).
[CrossRef]

M. Nakazawa, E. Yoshida, K. Suzuki, T. Kitoh, and M. Kawachi, “80 Gbit/s soliton data transmission over 500 km with unequal amplitude solitons for timing clock extraction,” Electron. Lett. 30, 1777–1778 (1994).
[CrossRef]

Kodama, Y.

Y. Kodama, A. V. Mikhailov, and S. Wabnitz, “Input pulse optimization in wavelength-division-multiplexed soliton transmission,” Opt. Commun. 143, 53–56 (1997).
[CrossRef]

Y. Kodama, M. Romagnoli, S. Wabnitz, and M. Midrio, “Role of third-order dispersion on soliton instabilities and interactions in optical fibers,” Opt. Lett. 19, 165–167 (1994).
[CrossRef] [PubMed]

Y. Kodama and S. Wabnitz, “Reduction of soliton interaction forces by bandwidth limited amplification,” Electron. Lett. 27, 1931–1933 (1991).
[CrossRef]

Y. Kodama and K. Nozaki, “Soliton interaction in optical fibers,” Opt. Lett. 12, 1038–1040 (1987).
[CrossRef] [PubMed]

Y. Kodama, “On integrable systems with higher-order corrections,” Phys. Lett. 107A, 245–249 (1985).
[CrossRef]

Kolokolov, I.

M. Chertkov, Y. Chung, A. Dyachenko, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in weakly disordered optical fibers,” Phys. Rev. E 67, 036615 (2003).
[CrossRef]

Lebedev, V.

M. Chertkov, Y. Chung, A. Dyachenko, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in weakly disordered optical fibers,” Phys. Rev. E 67, 036615 (2003).
[CrossRef]

Lee, Y. C.

P. K. A. Wai, H. H. Chen, and Y. C. Lee, “Radiation by soli-tons at the zero group-dispersion wavelength of single-mode fibers,” Phys. Rev. A 41, 426–439 (1990).
[CrossRef] [PubMed]

P. K. A. Wai, C. R. Menyuk, H. H. Chen, and Y. C. Lee, “Soliton at the zero-group-dispersion wavelength of a single-modal fiber,” Opt. Lett. 12, 628–630 (1987).
[CrossRef] [PubMed]

Lisak, M.

Mamyshev, P. V.

L. F. Mollenauer and P. V. Mamyshev, “Massive wavelength-division multiplexing with solitons,” IEEE J. Quantum Electron. 34, 2089–2102 (1998).
[CrossRef]

L. F. Mollenauer, P. V. Mamyshev, and M. J. Neubelt, “Demonstration of soliton WDM transmission at 6 and 7× 10 Gbit/s, error free over transoceanic distances,” Electron. Lett. 32, 471–474 (1996).
[CrossRef]

L. F. Mollenauer and P. V. Mamyshev, “Wavelength-division-multiplexing channel energy self-equalization in a soliton transmission line by guiding filters,” Opt. Lett. 20, 1658–1660 (1996).

Menyuk, C. R.

Midrio, M.

Mikhailov, A. V.

Y. Kodama, A. V. Mikhailov, and S. Wabnitz, “Input pulse optimization in wavelength-division-multiplexed soliton transmission,” Opt. Commun. 143, 53–56 (1997).
[CrossRef]

Mollenauer, L. F.

L. F. Mollenauer and P. V. Mamyshev, “Massive wavelength-division multiplexing with solitons,” IEEE J. Quantum Electron. 34, 2089–2102 (1998).
[CrossRef]

L. F. Mollenauer, P. V. Mamyshev, and M. J. Neubelt, “Demonstration of soliton WDM transmission at 6 and 7× 10 Gbit/s, error free over transoceanic distances,” Electron. Lett. 32, 471–474 (1996).
[CrossRef]

L. F. Mollenauer and P. V. Mamyshev, “Wavelength-division-multiplexing channel energy self-equalization in a soliton transmission line by guiding filters,” Opt. Lett. 20, 1658–1660 (1996).

Nakazawa, M.

M. Nakazawa, K. Suzuki, E. Yoshida, E. Yamada, T. Kitoh, and M. Kawachi, “160 Gbit/s (80 Gbit/s×2 channels) WDM soliton transmission over 10000 km using in-line synchronous modulation,” Electron. Lett. 35, 1358–1359 (1999).
[CrossRef]

M. Nakazawa, K. Suzuki, E. Yoshida, E. Yamada, T. Kitoh, and M. Kawachi, “160 Gbit/s soliton data transmission over 200 km,” Electron. Lett. 31, 565–566 (1995).
[CrossRef]

M. Nakazawa, E. Yoshida, K. Suzuki, T. Kitoh, and M. Kawachi, “80 Gbit/s soliton data transmission over 500 km with unequal amplitude solitons for timing clock extraction,” Electron. Lett. 30, 1777–1778 (1994).
[CrossRef]

Neubelt, M. J.

L. F. Mollenauer, P. V. Mamyshev, and M. J. Neubelt, “Demonstration of soliton WDM transmission at 6 and 7× 10 Gbit/s, error free over transoceanic distances,” Electron. Lett. 32, 471–474 (1996).
[CrossRef]

Nozaki, K.

Omenetto, F. G.

F. G. Omenetto, Y. Chung, D. Yarotski, T. Schaefer, I. Gabitov, and A. J. Taylor, “Phase analysis of nonlinear femtosecond pulse propagation and self-frequency shift in optical fibers,” Opt. Commun. 208, 191–196 (2002).
[CrossRef]

Peleg, A.

A. Peleg, M. Chertkov, and I. Gabitov, “Inter-channel interaction of optical solitons,” Phys. Rev. E 68, 026605 (2003).
[CrossRef]

Romagnoli, M.

Schaefer, T.

F. G. Omenetto, Y. Chung, D. Yarotski, T. Schaefer, I. Gabitov, and A. J. Taylor, “Phase analysis of nonlinear femtosecond pulse propagation and self-frequency shift in optical fibers,” Opt. Commun. 208, 191–196 (2002).
[CrossRef]

Solov’ev, V. V.

V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the 2-soliton systems,” Phys. D 3, 487–502 (1981).
[CrossRef]

Suzuki, K.

M. Nakazawa, K. Suzuki, E. Yoshida, E. Yamada, T. Kitoh, and M. Kawachi, “160 Gbit/s (80 Gbit/s×2 channels) WDM soliton transmission over 10000 km using in-line synchronous modulation,” Electron. Lett. 35, 1358–1359 (1999).
[CrossRef]

M. Nakazawa, K. Suzuki, E. Yoshida, E. Yamada, T. Kitoh, and M. Kawachi, “160 Gbit/s soliton data transmission over 200 km,” Electron. Lett. 31, 565–566 (1995).
[CrossRef]

M. Nakazawa, E. Yoshida, K. Suzuki, T. Kitoh, and M. Kawachi, “80 Gbit/s soliton data transmission over 500 km with unequal amplitude solitons for timing clock extraction,” Electron. Lett. 30, 1777–1778 (1994).
[CrossRef]

Taylor, A. J.

F. G. Omenetto, Y. Chung, D. Yarotski, T. Schaefer, I. Gabitov, and A. J. Taylor, “Phase analysis of nonlinear femtosecond pulse propagation and self-frequency shift in optical fibers,” Opt. Commun. 208, 191–196 (2002).
[CrossRef]

Taylor, J. R.

Wabnitz, S.

Y. Kodama, A. V. Mikhailov, and S. Wabnitz, “Input pulse optimization in wavelength-division-multiplexed soliton transmission,” Opt. Commun. 143, 53–56 (1997).
[CrossRef]

Y. Kodama, M. Romagnoli, S. Wabnitz, and M. Midrio, “Role of third-order dispersion on soliton instabilities and interactions in optical fibers,” Opt. Lett. 19, 165–167 (1994).
[CrossRef] [PubMed]

Y. Kodama and S. Wabnitz, “Reduction of soliton interaction forces by bandwidth limited amplification,” Electron. Lett. 27, 1931–1933 (1991).
[CrossRef]

Wai, P. K. A.

P. K. A. Wai, H. H. Chen, and Y. C. Lee, “Radiation by soli-tons at the zero group-dispersion wavelength of single-mode fibers,” Phys. Rev. A 41, 426–439 (1990).
[CrossRef] [PubMed]

P. K. A. Wai, C. R. Menyuk, H. H. Chen, and Y. C. Lee, “Soliton at the zero-group-dispersion wavelength of a single-modal fiber,” Opt. Lett. 12, 628–630 (1987).
[CrossRef] [PubMed]

Wong, W. S.

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
[CrossRef]

Yamada, E.

M. Nakazawa, K. Suzuki, E. Yoshida, E. Yamada, T. Kitoh, and M. Kawachi, “160 Gbit/s (80 Gbit/s×2 channels) WDM soliton transmission over 10000 km using in-line synchronous modulation,” Electron. Lett. 35, 1358–1359 (1999).
[CrossRef]

M. Nakazawa, K. Suzuki, E. Yoshida, E. Yamada, T. Kitoh, and M. Kawachi, “160 Gbit/s soliton data transmission over 200 km,” Electron. Lett. 31, 565–566 (1995).
[CrossRef]

Yarotski, D.

F. G. Omenetto, Y. Chung, D. Yarotski, T. Schaefer, I. Gabitov, and A. J. Taylor, “Phase analysis of nonlinear femtosecond pulse propagation and self-frequency shift in optical fibers,” Opt. Commun. 208, 191–196 (2002).
[CrossRef]

Yoshida, E.

M. Nakazawa, K. Suzuki, E. Yoshida, E. Yamada, T. Kitoh, and M. Kawachi, “160 Gbit/s (80 Gbit/s×2 channels) WDM soliton transmission over 10000 km using in-line synchronous modulation,” Electron. Lett. 35, 1358–1359 (1999).
[CrossRef]

M. Nakazawa, K. Suzuki, E. Yoshida, E. Yamada, T. Kitoh, and M. Kawachi, “160 Gbit/s soliton data transmission over 200 km,” Electron. Lett. 31, 565–566 (1995).
[CrossRef]

M. Nakazawa, E. Yoshida, K. Suzuki, T. Kitoh, and M. Kawachi, “80 Gbit/s soliton data transmission over 500 km with unequal amplitude solitons for timing clock extraction,” Electron. Lett. 30, 1777–1778 (1994).
[CrossRef]

Electron. Lett. (5)

Y. Kodama and S. Wabnitz, “Reduction of soliton interaction forces by bandwidth limited amplification,” Electron. Lett. 27, 1931–1933 (1991).
[CrossRef]

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Other (8)

Note that it was found in Ref. 17 that even if the pulse launched into the fiber is not exactly of the form given by Eq. (3) it evolves into this form after a transient.

Y. Kodama and A. V. Mikhailov, “Obstacles to asymptotic integrability,” in Algebraic Aspects to Integrable Systems, A. S. Fokas and I. M. Gelfand, eds. (Birkhäuser, Boston, 1997).

The dimensionless z in Eq. (1) is z=x(αP0/2), where x is the actual position, P0 is the peak soliton power, and α is the Kerr nonlinearity coefficient. The dimensionless retarded time is t=τ/τ0, where τ is the retarded time and τ0 is the soliton width. The spectral width ν0 is given by ν0= 1/(π2τ0), and the channel spacing is given by Δν= Ων0. Ψ=E/P0, where E is the actual electric field. The dimensionless second- and third-order dispersion coefficients are given by d=−1=β2/(αP0τ02) and d3= β3/(3αP0τ03), where β2 and β3 are the second- and third-order chromatic dispersion coefficients, respectively.

Effects of fiber losses can be neglected (see, for instance, Ref. 23) if values of Δν, β2, β3, and τ0 satisfy the following two conditions: τ0≪β2/γ and Δν≫(γτ02)/β3.

L. F. Mollenauer, J. P. Gordon, and P. V. Mamyshev, “Solitons in high bit-rate, long-distance transmission,” in Optical Fiber Telecommunications III, I. P. Kaminov and T. L. Koch, eds. (Academic, San Diego, Calif., 1997), Chap. 12, Sect. V-B.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1995).

A. C. Newell and J. V. Moloney, Nonlinear Optics of Advanced Topics in the Interdisciplinary Mathematical Sciences Series (Addison-Wesley, Redwood City, Calif., 1992).

G. P. Agrawal, Fiber-Optic Communication Systems (Wiley, New York, 1997).

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Figures (1)

Fig. 1
Fig. 1

Schematic description of the collision between two solitons from different frequency channels.

Equations (12)

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izΨ+t2Ψ+2|Ψ|2Ψ=id3t3Ψ,
η exp[iα+iΩ(t-y)+i(η2-Ω2)z]cosh[η(t-y-2Ωz)],
ΨΩ(t, z)=[Ψ˜Ω0(hΩ)+Ψ˜Ω1(hΩ)+]exp(iχΩ),
Ψtwo=Ψ0+ΨΩ+Φ0+ΦΩ+Φ-Ω+Φ2Ω+O(1/Ω3),
zΦ˜0-i[(t2-η02)Φ˜0+4|Ψ0|2Φ˜0+2Ψ˜02Φ˜0*]=4i[|ΨΩ|2Ψ˜0+|ΨΩ|2Φ˜0+Ψ˜0(ΨΩΦΩ*+ΨΩ*ΦΩ)+Ψ˜0*ΨΩΦ-Ω+12ΨΩ2Φ2Ω*+Ψ˜0|Φ0|2+12Ψ˜0*Φ˜02+Ψ˜0|ΦΩ|2+Φ˜0(ΨΩΦΩ*+ΨΩ*ΦΩ)]+d3t3Φ˜0,
zΦ˜01(0)=4i|ΨΩ0|2Ψ˜00=4iη0ηΩ2cosh(h0)cosh2(hΩ).
Φ˜01(0)(t, z)=-2iη0ηΩ(1+3d3Ω)1/2(1+3d3Ω/2)Ω tanh(hΩ)cosh(h0).
zΦ˜03(1)R=d3t3Φ˜01(0).
Φ˜03(1)R(t, z0+z˜/|β|)=-iBh02Ψ˜00(h0),
B=6η03ηΩ(1+3d3Ω)1/2d3(1+3d3Ω/2)2Ω|Ω|.
E0192η05ηΩ2(1+3d3Ω)d325(1+3d3Ω/2)4Ω4.
Δxic=πTτ032(1+3d3Ω/2)β3|Ω|2s,

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