Dispersion management and the direct scattering transform

Y. Xudous

doi:10.1103/PhysRevE.69.036609

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Dispersion management and the direct scattering transform

Y. Xudous^*

Centre d’Optique, Photonique et Laser (COPL), Département de Physique, Université Laval, Québec (Québec), Canada G1K 7P4

^*Electronic address: yxudous@phy.ulaval.ca

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Phys. Rev. E 69, 036609 – Published 30 March, 2004

DOI: https://doi.org/10.1103/PhysRevE.69.036609

Abstract

It is shown that the direct scattering transform may be used to derive a perturbative theory of quasiperiodic field propagation in nonlinear fiber lines managed through zero-average, piecewise-constant dispersion maps. In this scheme, the field is propagated exactly and it is the quasiperiodicity property which is approximated. The resulting quasiperiodicity conditions are shown to agree with the dispersion managed nonlinear Schrödinger equation up to order 4.

References (19)

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M. Suzuki, I. Morita, N. Edgawa, S. Yammamoto, H. Taga, and S. Akiba, Electron. Lett. 31, 2027 (1995).
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M.J. Ablowitz and G. Biondini, Opt. Lett. 23, 1668 (1998).
J.H.B. Nijhof, S.K. Turitsyn, and N.J. Doran, Nonlinear Guided Waves and their Applications (OSA, Washington, D.C., 1999), p. 313.
A.V. Mikhailov and V.Y. Novokshenov, JETP Lett. 73, 250 (2001).
S. Novikov, S.V. Manakov, L.P. Pitaevskii, and V.E. Zakharov, Theory of Solitons-The Inverse Scattering Transform (Consultants Bureau, New York, 1984).
M.J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM, Philadelphia, 1981).
I. Gabitov, E.G. Shapiro, and S.K. Turitsyn, Phys. Rev. E 55, 3624 (1997).
C. Paré and P.A. Bélanger, Opt. Commun. 168, 103 (1999).
M.J. Ablowitz, T. Hirooka, and T. Inoue, J. Opt. Soc. Am. B 19, 2876 (2002).
J.H.B. Nijhof, W. Forysiak, and N.J. Doran, IEEE J. Quantum Electron. 6, 330 (2000).
Note that $I_{s}^{(2 n - 1)} (k)$ and $I_{c}^{(2 n - 1)} (k)$ can be shown to be real by complex conjugating and changing variables according to $τ_{j} \leftrightarrow τ_{2 n - 1 - j},$ $p_{j} \leftrightarrow p_{2 n - 1 - j} .$
Note that the Riemann-Lebesgue lemma cannot be used to simplify $I_{c}^{(3)} (k)$ since in that case, integrating the exponentials produces the limit of a nonintegrable kernel.

[c1] G.M. Carter, J.M. Jacob, C.R. Menyuk, E.A. Golovchenko, and A.N. Pilipetskii, Opt. Lett. 22, 513 (1997).

[c2] M. Suzuki, I. Morita, N. Edgawa, S. Yammamoto, H. Taga, and S. Akiba, Electron. Lett. 31, 2027 (1995).

[c3] N.J. Smith, W. Forysiak, and N.J. Doran, Electron. Lett. 32, 2085 (1996).

[c4] C. Kurtzke, IEEE Photonics Technol. Lett. 5, 1250 (1993).

[c5] R.W. Tkach, A.R. Chraplyvy, F. Forghieri, A.H. Gnauck, and R.M. Derosier, J. Lightwave Technol. 13, 841 (1995).

[c6] I. Gabitov and S.K. Turitsyn, Opt. Lett. 21, 327 (1996).

[c7] C. Paré and P.A. Bélanger, Opt. Lett. 225, 331 (1997).

[c8] C. Paré, V. Roy, F. Lesage, P. Mathieu, and P.A. Bélanger, Phys. Rev. E 60, 4836 (1999).

[c9] M.J. Ablowitz and G. Biondini, Opt. Lett. 23, 1668 (1998).

[c10] J.H.B. Nijhof, S.K. Turitsyn, and N.J. Doran, Nonlinear Guided Waves and their Applications (OSA, Washington, D.C., 1999), p. 313.

[c11] A.V. Mikhailov and V.Y. Novokshenov, JETP Lett. 73, 250 (2001).

[c12] S. Novikov, S.V. Manakov, L.P. Pitaevskii, and V.E. Zakharov, Theory of Solitons-The Inverse Scattering Transform (Consultants Bureau, New York, 1984).

[c13] M.J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM, Philadelphia, 1981).

[c14] I. Gabitov, E.G. Shapiro, and S.K. Turitsyn, Phys. Rev. E 55, 3624 (1997).

[c15] C. Paré and P.A. Bélanger, Opt. Commun. 168, 103 (1999).

[c16] M.J. Ablowitz, T. Hirooka, and T. Inoue, J. Opt. Soc. Am. B 19, 2876 (2002).

[c17] J.H.B. Nijhof, W. Forysiak, and N.J. Doran, IEEE J. Quantum Electron. 6, 330 (2000).

[c18] Note that $I_{s}^{(2 n - 1)} (k)$ and $I_{c}^{(2 n - 1)} (k)$ can be shown to be real by complex conjugating and changing variables according to $τ_{j} \leftrightarrow τ_{2 n - 1 - j},$ $p_{j} \leftrightarrow p_{2 n - 1 - j} .$

[c19] Note that the Riemann-Lebesgue lemma cannot be used to simplify $I_{c}^{(3)} (k)$ since in that case, integrating the exponentials produces the limit of a nonintegrable kernel.

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Dispersion management and the direct scattering transform

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