Abstract

A fourth-order Runge-Kutta in the interaction picture (RK4IP) method is presented for solving the coupled nonlinear Schrödinger equation (CNLSE) that governs the light propagation in optical fibers with randomly varying birefringence. The computational error of RK4IP is caused by the fourth-order Runge-Kutta algorithm, better than the split-step approximation limited by the step size. As a result, the step size of RK4IP can have the same order of magnitude as the dispersion length and/or the nonlinear length of the fiber, provided the birefringence effect is small. For communication fibers with random birefringence, the step size of RK4IP can be orders of magnitude larger than the correlation length and the beating length of the fibers, depending on the interaction between linear and nonlinear effects. Our approach can be applied to the fibers having the general form of local birefringence and treat the Kerr nonlinearity without approximation. Our RK4IP results agree well with those obtained from Manakov-PMD approximation, provided the polarization state can be mixed enough on the Poincaré sphere.

© 2010 Optical Society of America

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  1. P. K. A. Wai and C. R. Menyuk, "Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence," J. Lightwave Technol. 14, 148-157 (1996).
    [CrossRef]
  2. D. Marcuse, C. R. Manyuk, and P. K. A. Wai, "Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence," J. Lightwave Technol. 15, 1735-1746 (1997).
    [CrossRef]
  3. C. R. Menyuk and B. S. Marks, "Interaction of polarization mode dispersion and nonlinearity in optical fiber transmission systems," J. Lightwave Technol. 24, 2806-2826 (2006).
    [CrossRef]
  4. C. Xie, M. Karlsson, P. A. Andrekson, H. Sunnerud, and J. Li, "Influences of polarization-mode dispersion on soliton transmission systems," J. Sel. Top. Quantum Electron. 8, 575-590 (2002).
    [CrossRef]
  5. E. Alperovich, A. Mecozzi, and M. Shtaif, "PMD penalties in long nonsoliton systems and the effect of inline filtering," IEEE Photon. Technol. Lett. 18, 1179-1181 (2006).
    [CrossRef]
  6. C. R. Menyuk, "Nonlinear pulse propagation in birefringent optical fibers," IEEE J. Quantum Electron. 23, 174-176 (1987).
    [CrossRef]
  7. M. Karlsson, "Modulational instability in lossy optical fibers," J. Opt. Soc. Am. B 12, 2071-2077 (1995).
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    [CrossRef]
  10. M. J. Ablowitz and T. Hirooka, "Nonlinear effects in quasi-linear dispersion-managed pulse transmission," IEEE Photon. Technol. Lett. 13, 1082-1084 (2001).
    [CrossRef]
  11. A. Vannucci, P. Serena, and A. Bononi, "The RP method: a new tool for the iterative solution of the nonlinear Schrodinger equation," J. Lightwave Technol. 20, 1102-1112 (2002).
    [CrossRef]
  12. O. V. Sinkin, R. Holzlohner, J. Zweck, and C. R. Menyuk, "Optimization of the split-step Fourier method in modeling optical-fiber communications systems," J. Lightwave Technol. 21, 61-68 (2003).
    [CrossRef]
  13. E. Ciaramella and E. Forestieri, "Analytical approximation of nonlinear distortions," IEEE Photon. Technol. Lett. 17, 91-93 (2005).
    [CrossRef]
  14. M. Secondini, E. Forestieri, and C. R. Menyuk, "A combined regular-logarithmic perturbation method for signalnoise interaction in amplified optical systems," J. Lightwave Technol. 27, 3358-3369 (2009).
    [CrossRef]
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  22. E. Forestieri, "Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre-and postdetection filtering," J. Lightwave Technol. 18, 1493-1503 (2000).
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  23. J. Wang and J. M. Kahn, "Impact of chromatic and polarization-mode dispersions on DPSK systems using interferometric demodulation and direct detection," J. Lightwave Technol. 22, 362-371 (2004).
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  24. F. Matera, A. Mecozzi, M. Romagnoli, and M. Settembre, "Sideband instability induced by periodic power variation in long-distance fiber links," Opt. Lett. 18, 1499-1501 (1993).
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2009 (1)

2008 (1)

K. W. Chow, K. K. Y. Wong, and K. Lam, "Modulation instabilities in a system of four coupled, nonlinear Schrodinger equations," Phys. Lett. A 372, 4596-4600 (2008).
[CrossRef]

2007 (1)

2006 (2)

C. R. Menyuk and B. S. Marks, "Interaction of polarization mode dispersion and nonlinearity in optical fiber transmission systems," J. Lightwave Technol. 24, 2806-2826 (2006).
[CrossRef]

E. Alperovich, A. Mecozzi, and M. Shtaif, "PMD penalties in long nonsoliton systems and the effect of inline filtering," IEEE Photon. Technol. Lett. 18, 1179-1181 (2006).
[CrossRef]

2005 (1)

E. Ciaramella and E. Forestieri, "Analytical approximation of nonlinear distortions," IEEE Photon. Technol. Lett. 17, 91-93 (2005).
[CrossRef]

2004 (1)

2003 (1)

2002 (2)

A. Vannucci, P. Serena, and A. Bononi, "The RP method: a new tool for the iterative solution of the nonlinear Schrodinger equation," J. Lightwave Technol. 20, 1102-1112 (2002).
[CrossRef]

C. Xie, M. Karlsson, P. A. Andrekson, H. Sunnerud, and J. Li, "Influences of polarization-mode dispersion on soliton transmission systems," J. Sel. Top. Quantum Electron. 8, 575-590 (2002).
[CrossRef]

2001 (2)

M. J. Ablowitz and T. Hirooka, "Nonlinear effects in quasi-linear dispersion-managed pulse transmission," IEEE Photon. Technol. Lett. 13, 1082-1084 (2001).
[CrossRef]

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "A novel analytical approach to the evaluation of the impact of fiber Parametric Gain on the Bit Error Rate," IEEE Trans. Commun. 49, 2154-2163 (2001).
[CrossRef]

2000 (1)

1999 (1)

A. V. T. Cartaxo, "Small-signal analysis for nonlinear and dispersive optical fibres, and its application to design of dispersion supported transmission systems with optical dispersion compensation," Proc. Inst. Elect. Eng. Optoelectron. 146(5), 213-222 (1999).
[CrossRef]

1997 (2)

D. Marcuse, C. R. Manyuk, and P. K. A. Wai, "Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence," J. Lightwave Technol. 15, 1735-1746 (1997).
[CrossRef]

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber," IEEE J. Select. Areas Commun. 15, 751-765 (1997).
[CrossRef]

1996 (1)

P. K. A. Wai and C. R. Menyuk, "Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence," J. Lightwave Technol. 14, 148-157 (1996).
[CrossRef]

1995 (2)

M. Karlsson, "Modulational instability in lossy optical fibers," J. Opt. Soc. Am. B 12, 2071-2077 (1995).

H. Ghafouri-Shiraz, P. Shum, and M. Nagata, "A novel method for analysis of soliton propagation in optical fibers," IEEE J. Quantum Electron. 31, 190-200 (1995).
[CrossRef]

1993 (1)

1992 (1)

S. G. EvangelidesJr., L. F. Mollenauer, J. P. Gordon, N. S. Bergano, "Polarization multiplexing with solitons," J. Lightwave Technol. 10, 28-35 (1992).
[CrossRef]

1987 (1)

C. R. Menyuk, "Nonlinear pulse propagation in birefringent optical fibers," IEEE J. Quantum Electron. 23, 174-176 (1987).
[CrossRef]

1962 (1)

G. H. Weiss and A. A. Maradudin, "The Baker-Hausdorff formula and a problem in crystal physics," J. Math. Phys. 3, 771-777 (1962).
[CrossRef]

Ablowitz, M. J.

M. J. Ablowitz and T. Hirooka, "Nonlinear effects in quasi-linear dispersion-managed pulse transmission," IEEE Photon. Technol. Lett. 13, 1082-1084 (2001).
[CrossRef]

Alperovich, E.

E. Alperovich, A. Mecozzi, and M. Shtaif, "PMD penalties in long nonsoliton systems and the effect of inline filtering," IEEE Photon. Technol. Lett. 18, 1179-1181 (2006).
[CrossRef]

Andrekson, P. A.

C. Xie, M. Karlsson, P. A. Andrekson, H. Sunnerud, and J. Li, "Influences of polarization-mode dispersion on soliton transmission systems," J. Sel. Top. Quantum Electron. 8, 575-590 (2002).
[CrossRef]

Benedetto, S.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "A novel analytical approach to the evaluation of the impact of fiber Parametric Gain on the Bit Error Rate," IEEE Trans. Commun. 49, 2154-2163 (2001).
[CrossRef]

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber," IEEE J. Select. Areas Commun. 15, 751-765 (1997).
[CrossRef]

Bergano, N. S.

S. G. EvangelidesJr., L. F. Mollenauer, J. P. Gordon, N. S. Bergano, "Polarization multiplexing with solitons," J. Lightwave Technol. 10, 28-35 (1992).
[CrossRef]

Bononi, A.

Bosco, G.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "A novel analytical approach to the evaluation of the impact of fiber Parametric Gain on the Bit Error Rate," IEEE Trans. Commun. 49, 2154-2163 (2001).
[CrossRef]

Carena, A.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "A novel analytical approach to the evaluation of the impact of fiber Parametric Gain on the Bit Error Rate," IEEE Trans. Commun. 49, 2154-2163 (2001).
[CrossRef]

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber," IEEE J. Select. Areas Commun. 15, 751-765 (1997).
[CrossRef]

Cartaxo, A. V. T.

A. V. T. Cartaxo, "Small-signal analysis for nonlinear and dispersive optical fibres, and its application to design of dispersion supported transmission systems with optical dispersion compensation," Proc. Inst. Elect. Eng. Optoelectron. 146(5), 213-222 (1999).
[CrossRef]

Chow, K. W.

K. W. Chow, K. K. Y. Wong, and K. Lam, "Modulation instabilities in a system of four coupled, nonlinear Schrodinger equations," Phys. Lett. A 372, 4596-4600 (2008).
[CrossRef]

Ciaramella, E.

E. Ciaramella and E. Forestieri, "Analytical approximation of nonlinear distortions," IEEE Photon. Technol. Lett. 17, 91-93 (2005).
[CrossRef]

Curri, V.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "A novel analytical approach to the evaluation of the impact of fiber Parametric Gain on the Bit Error Rate," IEEE Trans. Commun. 49, 2154-2163 (2001).
[CrossRef]

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber," IEEE J. Select. Areas Commun. 15, 751-765 (1997).
[CrossRef]

Evangelides, S. G.

S. G. EvangelidesJr., L. F. Mollenauer, J. P. Gordon, N. S. Bergano, "Polarization multiplexing with solitons," J. Lightwave Technol. 10, 28-35 (1992).
[CrossRef]

Forestieri, E.

Gaudino, R.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "A novel analytical approach to the evaluation of the impact of fiber Parametric Gain on the Bit Error Rate," IEEE Trans. Commun. 49, 2154-2163 (2001).
[CrossRef]

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber," IEEE J. Select. Areas Commun. 15, 751-765 (1997).
[CrossRef]

Ghafouri-Shiraz, H.

H. Ghafouri-Shiraz, P. Shum, and M. Nagata, "A novel method for analysis of soliton propagation in optical fibers," IEEE J. Quantum Electron. 31, 190-200 (1995).
[CrossRef]

Gordon, J. P.

S. G. EvangelidesJr., L. F. Mollenauer, J. P. Gordon, N. S. Bergano, "Polarization multiplexing with solitons," J. Lightwave Technol. 10, 28-35 (1992).
[CrossRef]

Hirooka, T.

M. J. Ablowitz and T. Hirooka, "Nonlinear effects in quasi-linear dispersion-managed pulse transmission," IEEE Photon. Technol. Lett. 13, 1082-1084 (2001).
[CrossRef]

Holzlohner, R.

Hult, J.

Kahn, J. M.

Karlsson, M.

C. Xie, M. Karlsson, P. A. Andrekson, H. Sunnerud, and J. Li, "Influences of polarization-mode dispersion on soliton transmission systems," J. Sel. Top. Quantum Electron. 8, 575-590 (2002).
[CrossRef]

M. Karlsson, "Modulational instability in lossy optical fibers," J. Opt. Soc. Am. B 12, 2071-2077 (1995).

Lam, K.

K. W. Chow, K. K. Y. Wong, and K. Lam, "Modulation instabilities in a system of four coupled, nonlinear Schrodinger equations," Phys. Lett. A 372, 4596-4600 (2008).
[CrossRef]

Li, J.

C. Xie, M. Karlsson, P. A. Andrekson, H. Sunnerud, and J. Li, "Influences of polarization-mode dispersion on soliton transmission systems," J. Sel. Top. Quantum Electron. 8, 575-590 (2002).
[CrossRef]

Manyuk, C. R.

D. Marcuse, C. R. Manyuk, and P. K. A. Wai, "Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence," J. Lightwave Technol. 15, 1735-1746 (1997).
[CrossRef]

Maradudin, A. A.

G. H. Weiss and A. A. Maradudin, "The Baker-Hausdorff formula and a problem in crystal physics," J. Math. Phys. 3, 771-777 (1962).
[CrossRef]

Marcuse, D.

D. Marcuse, C. R. Manyuk, and P. K. A. Wai, "Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence," J. Lightwave Technol. 15, 1735-1746 (1997).
[CrossRef]

Marks, B. S.

Matera, F.

Mecozzi, A.

E. Alperovich, A. Mecozzi, and M. Shtaif, "PMD penalties in long nonsoliton systems and the effect of inline filtering," IEEE Photon. Technol. Lett. 18, 1179-1181 (2006).
[CrossRef]

F. Matera, A. Mecozzi, M. Romagnoli, and M. Settembre, "Sideband instability induced by periodic power variation in long-distance fiber links," Opt. Lett. 18, 1499-1501 (1993).
[CrossRef] [PubMed]

Menyuk, C. R.

Mollenauer, L. F.

S. G. EvangelidesJr., L. F. Mollenauer, J. P. Gordon, N. S. Bergano, "Polarization multiplexing with solitons," J. Lightwave Technol. 10, 28-35 (1992).
[CrossRef]

Nagata, M.

H. Ghafouri-Shiraz, P. Shum, and M. Nagata, "A novel method for analysis of soliton propagation in optical fibers," IEEE J. Quantum Electron. 31, 190-200 (1995).
[CrossRef]

Poggiolini, P.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "A novel analytical approach to the evaluation of the impact of fiber Parametric Gain on the Bit Error Rate," IEEE Trans. Commun. 49, 2154-2163 (2001).
[CrossRef]

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber," IEEE J. Select. Areas Commun. 15, 751-765 (1997).
[CrossRef]

Romagnoli, M.

Secondini, M.

Serena, P.

Settembre, M.

Shtaif, M.

E. Alperovich, A. Mecozzi, and M. Shtaif, "PMD penalties in long nonsoliton systems and the effect of inline filtering," IEEE Photon. Technol. Lett. 18, 1179-1181 (2006).
[CrossRef]

Shum, P.

H. Ghafouri-Shiraz, P. Shum, and M. Nagata, "A novel method for analysis of soliton propagation in optical fibers," IEEE J. Quantum Electron. 31, 190-200 (1995).
[CrossRef]

Sinkin, O. V.

Sunnerud, H.

C. Xie, M. Karlsson, P. A. Andrekson, H. Sunnerud, and J. Li, "Influences of polarization-mode dispersion on soliton transmission systems," J. Sel. Top. Quantum Electron. 8, 575-590 (2002).
[CrossRef]

Vannucci, A.

Wai, P. K. A.

D. Marcuse, C. R. Manyuk, and P. K. A. Wai, "Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence," J. Lightwave Technol. 15, 1735-1746 (1997).
[CrossRef]

P. K. A. Wai and C. R. Menyuk, "Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence," J. Lightwave Technol. 14, 148-157 (1996).
[CrossRef]

Wang, J.

Weiss, G. H.

G. H. Weiss and A. A. Maradudin, "The Baker-Hausdorff formula and a problem in crystal physics," J. Math. Phys. 3, 771-777 (1962).
[CrossRef]

Wong, K. K. Y.

K. W. Chow, K. K. Y. Wong, and K. Lam, "Modulation instabilities in a system of four coupled, nonlinear Schrodinger equations," Phys. Lett. A 372, 4596-4600 (2008).
[CrossRef]

Xie, C.

C. Xie, M. Karlsson, P. A. Andrekson, H. Sunnerud, and J. Li, "Influences of polarization-mode dispersion on soliton transmission systems," J. Sel. Top. Quantum Electron. 8, 575-590 (2002).
[CrossRef]

Zweck, J.

IEEE J. Quantum Electron. (2)

C. R. Menyuk, "Nonlinear pulse propagation in birefringent optical fibers," IEEE J. Quantum Electron. 23, 174-176 (1987).
[CrossRef]

H. Ghafouri-Shiraz, P. Shum, and M. Nagata, "A novel method for analysis of soliton propagation in optical fibers," IEEE J. Quantum Electron. 31, 190-200 (1995).
[CrossRef]

IEEE J. Select. Areas Commun. (1)

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber," IEEE J. Select. Areas Commun. 15, 751-765 (1997).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

E. Ciaramella and E. Forestieri, "Analytical approximation of nonlinear distortions," IEEE Photon. Technol. Lett. 17, 91-93 (2005).
[CrossRef]

M. J. Ablowitz and T. Hirooka, "Nonlinear effects in quasi-linear dispersion-managed pulse transmission," IEEE Photon. Technol. Lett. 13, 1082-1084 (2001).
[CrossRef]

E. Alperovich, A. Mecozzi, and M. Shtaif, "PMD penalties in long nonsoliton systems and the effect of inline filtering," IEEE Photon. Technol. Lett. 18, 1179-1181 (2006).
[CrossRef]

IEEE Trans. Commun. (1)

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "A novel analytical approach to the evaluation of the impact of fiber Parametric Gain on the Bit Error Rate," IEEE Trans. Commun. 49, 2154-2163 (2001).
[CrossRef]

J. Lightwave Technol. (10)

S. G. EvangelidesJr., L. F. Mollenauer, J. P. Gordon, N. S. Bergano, "Polarization multiplexing with solitons," J. Lightwave Technol. 10, 28-35 (1992).
[CrossRef]

E. Forestieri, "Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre-and postdetection filtering," J. Lightwave Technol. 18, 1493-1503 (2000).
[CrossRef]

J. Wang and J. M. Kahn, "Impact of chromatic and polarization-mode dispersions on DPSK systems using interferometric demodulation and direct detection," J. Lightwave Technol. 22, 362-371 (2004).
[CrossRef]

A. Vannucci, P. Serena, and A. Bononi, "The RP method: a new tool for the iterative solution of the nonlinear Schrodinger equation," J. Lightwave Technol. 20, 1102-1112 (2002).
[CrossRef]

O. V. Sinkin, R. Holzlohner, J. Zweck, and C. R. Menyuk, "Optimization of the split-step Fourier method in modeling optical-fiber communications systems," J. Lightwave Technol. 21, 61-68 (2003).
[CrossRef]

M. Secondini, E. Forestieri, and C. R. Menyuk, "A combined regular-logarithmic perturbation method for signalnoise interaction in amplified optical systems," J. Lightwave Technol. 27, 3358-3369 (2009).
[CrossRef]

J. Hult, "A fourth-order Runge-Kutta in the interaction picture method for simulating supercontinuum generation in optical fibers," J. Lightwave Technol. 25, 3770-3775 (2007).
[CrossRef]

P. K. A. Wai and C. R. Menyuk, "Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence," J. Lightwave Technol. 14, 148-157 (1996).
[CrossRef]

D. Marcuse, C. R. Manyuk, and P. K. A. Wai, "Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence," J. Lightwave Technol. 15, 1735-1746 (1997).
[CrossRef]

C. R. Menyuk and B. S. Marks, "Interaction of polarization mode dispersion and nonlinearity in optical fiber transmission systems," J. Lightwave Technol. 24, 2806-2826 (2006).
[CrossRef]

J. Math. Phys. (1)

G. H. Weiss and A. A. Maradudin, "The Baker-Hausdorff formula and a problem in crystal physics," J. Math. Phys. 3, 771-777 (1962).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Sel. Top. Quantum Electron. (1)

C. Xie, M. Karlsson, P. A. Andrekson, H. Sunnerud, and J. Li, "Influences of polarization-mode dispersion on soliton transmission systems," J. Sel. Top. Quantum Electron. 8, 575-590 (2002).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. A (1)

K. W. Chow, K. K. Y. Wong, and K. Lam, "Modulation instabilities in a system of four coupled, nonlinear Schrodinger equations," Phys. Lett. A 372, 4596-4600 (2008).
[CrossRef]

Proc. Inst. Elect. Eng. Optoelectron. (1)

A. V. T. Cartaxo, "Small-signal analysis for nonlinear and dispersive optical fibres, and its application to design of dispersion supported transmission systems with optical dispersion compensation," Proc. Inst. Elect. Eng. Optoelectron. 146(5), 213-222 (1999).
[CrossRef]

Other (5)

J. Butcher, Numerical Methods for Ordinary Differential Equations (Wiley, 2003).
[CrossRef]

J. N. Damask, Polarization Optics in Telecommunications (Springer 2005).

A. Galtarossa and C. R. Menyuk, Polarization Mode Dispersion (Springer 2005).
[CrossRef]

J. M. Wiesenfeld, L. D. Garrett, M. Shtaif, M. H. Eiselt, and R. W. Tkach, "Effects of DGE bandwidth on nonlinear ULH systems," in Proc. Optical Fiber Communications Conf., Anaheim, CA, 2005, paper OWA2.

E. Ibragimov, C. Menyuk, andW. Kath, "PMD-induced reduction of nonlinear penalties in terrestrial optical fiber transmission," in Proc. OFC, 2000, pp. 195-197, Paper WL3.

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Figures (6)

Fig. 1.
Fig. 1.

(a) and (b): Electrically filtered pulse as a function of time (Tb = 200ps), with CD=17 ps/nm·km and fiber birefringence being neglected. The nonlinear coefficient γ = 1.26/(W·km) is obtained using effective mode area Aeff = 80 μm2. The input mark power Eb/Tb is 20 mW for L=100 km (a) and 2 mW for L=500 km (b). There is no visible difference between RK4IP solution [Eq. (17)] and SSFM solution for Eq. (19). (c): Optical power as a function of time (before the optical filter). Other parameters are same as those in (b). (d): Electrically filtered pulse as a function of time (no optical filter). Related parameters shown in (d) were given by Ref. [2] [p. 1740, Fig. 4(c)].

Fig. 2.
Fig. 2.

The filtered photoelectric current (W) vs time (Tb = 100 ps) in a NRZ-OOK system consisting of 5 100-km spans of transmission fiber [CD100=17ps/(nm·km), Aeff =80 μm2]. Each span is followed by a 13-km dispersion compensation fiber [CD13=-120ps/(nm·km), Aeff=30 μm2]. Before the 1st 100-km fiber, there is a precompensation of -120×6 ps/nm, whereas the 5th 100-km fiber is followed by the compensation of -120×5 ps/nm. The input mark power is 10 mW. Birefringence parameters are DPMD=2.0 ps/(km)1/2, Lcorr = 10 m, and Λbeat = 50 m. There is very little difference between the RK4IP solution [Eq. (17)] and the M-PMD approx [Eq. (8)].

Fig. 3.
Fig. 3.

The filtered photoelectric current (W) vs time (Tb = 100 ps) in a RZ-OOK system using 50 100-km eLEAF fibers [average CD100=4.5ps/(nm·km), Aeff =72 μm2], precom-pensation of -(3 × 39) ps/nm before the first 100-km fiber, -(11 × 39) ps/nm compensation per span, and -(6 × 39) ps/nm compensation after the 50th 100-km fiber. The input mark power is 2 mW and the PMD coefficient is DPMD=0.40 ps/(km)1/2.

Fig. 4.
Fig. 4.

Same as Fig. 3, except that CD100=6.0ps/(nm·km) and that each span is followed by a -(15 × 39) ps/nm dispersion compensation and a -(10 × 39) ps/nm compensation after the 50th 100-km fiber.

Fig. 5.
Fig. 5.

The optical power (W) vs time (before the optical filter, Tb = 100 ps), with Lcorr=10 m (a) and Lcorr=100 m (b). Other related parameters are same as those in Fig. 2. Here, the agreement between the two approaches is not very sensitive to the value of Lcorr.

Fig. 6.
Fig. 6.

The optical power (W) vs time (before the optical filter, Tb = 100 ps), with Lcorr=10 m (a) and Lcorr=100 m (b). Other related parameters are the same as those in Fig. 4. The agreement between the two approaches is affected by the value of Lcorr because of the strong linear-nonlinear interaction.

Tables (1)

Tables Icon

Table 1. Step size Δz using RK4IP and SSFM of Ref. [2] obtained with given LN and fiber length L. The dispersion length LD ~ 240 km and the computational accuracy < 1% of the pulse peak.

Equations (46)

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e A e B = e A + B + 1 2 [ A , B ] + 1 12 [ A B , [ A , B ] ] + .
j u z + Σ [ Δ β u + j Δ β ω u t ] β ω ω 2 u tt + γ [ 5 6 u u u + 1 6 σ 3 u u σ 3 u + 1 3 v ] = 0 ,
j u z + Σ [ Δ β u + j Δ β ω u t ] β ω ω 2 u tt + γ [ u u u 1 3 σ 31 u u σ 13 u ] = 0 ,
σ 1 = ( 0 1 1 0 ) , σ 2 = ( 0 j j 0 ) , σ 3 = ( 1 0 0 1 ) .
Σ β 0 ( z ) . σ = ( β 3 β 1 j β 2 β 1 + j β 2 β 3 ) = σ 3 cos θ + σ 1 sin θ cos ϕ + σ 2 sin θ sin ϕ
T = ( t 11 t 12 t 12 * t 11 * ) , T 1 = T = ( t 11 * t 12 t 12 * t 11 ) , t 11 2 + t 12 2 = 1 ,
j u z + Σ [ Δ β u + j Δ β ω u t ] β ω ω 2 u tt + γ 8 9 u u u = γ 3 [ 1 3 u u u σ 31 u u σ 13 u ] = 0 ,
j u z + Σ [ Δ β u + j Δ β ω u t ] β ω ω 2 u tt + γ 8 9 u u u = 0 .
j u z β ωω 2 2 u t 2 + γ u 2 u = 0 .
j u z + ( D ̂ + N ̂ ) u = 0 , D ̂ = β ωω 2 2 t 2 , N ̂ = γ u 2 .
j u I z + N ̂ I u I = 0 .
u I z = j N ̂ I u I f ( z , u I )
N ̂ I = exp [ j ( z z 0 ) D ̂ ] N ̂ exp [ j ( z z 0 ) D ̂ ] I ̃ D N ̂ I ̂ D .
u ( z n + h , t ) = exp [ j h 2 D ̂ ] [ u n I + k 1 h / 6 + k 2 h / 3 + k 3 h / 3 ] + k 4 h / 6
u n I = exp [ j h 2 D ̂ ] u ( z n , t )
k 1 = f ( z n , u n I ) = j exp [ j h 2 D ̂ ] N ̂ ( u ( z n , t ) ) u ( z n , t )
k 2 = f ( z n + h 2 , u n I + h 2 k 1 ) = j N ̂ ( u n I + h 2 k 1 ) [ u n I + h 2 k 1 ]
k 3 = f ( z n + h 2 , u n I + h 2 k 2 ) = j N ̂ ( u n I + h 2 k 2 ) [ u n I + h 2 k 2 ]
k 4 = exp [ j h 2 D ̂ ] f ( z n + h , u n I + h k 3 ) = j N ̂ ( exp [ j h 2 D ̂ ] [ u n I + h k 3 ] ) exp [ j h 2 D ̂ ] [ u n I + h k 3 ]
j u z + ( D ̂ 2 + N ̂ 2 ) u = 0 ,
D ̂ 2 = β 0 ( z ) · σ ( Δ β + j Δ β ω t ) β ωω 2 2 t 2 , N ̂ 2 = γ [ u u + 1 3 σ 13 u σ 13 u ] .
u I ( z ) z = j N ̂ 2 I ( z ) u I ( z ) f ( z , u I ) , N ̂ 2 I ( z ) = I ̂ ( z , z 0 ) N ̂ 2 ( z ) I ̂ ( z , z 0 ) .
u n + 1 = I ̂ ( z n + 1 , z 0 ) u n + 1 I = d ̂ ( h 2 ) M ̂ 2 [ u n I + k 1 h 6 + k 2 h 3 + k 3 h 3 ] + k 4 h 6
u n I = I ̂ ( z n , z 0 ) u n = d ̂ ( h 2 ) M ̂ L u n = d ̂ ( h 2 ) M ̂ 1 u n
k 1 = f ( z n , u n I )
= j γ d ̂ ( h 2 ) M ̂ 1 [ u n u n + σ 13 3 u n σ 13 u n ] u n
k 2 = f ( z n + h 2 , u n I + h 2 k 1 )
= j γ [ u n + h k 1 2 u n + h k 1 2 + σ 13 3 u n I + h k 1 2 σ 13 u n I + h k 1 2 ] u n I + h k 1 2
k 3 = f ( z n + h 2 , u n I + h 2 k 2 )
= j γ [ u n + h k 2 2 u n + h k 2 2 + σ 13 3 u n I + h k 2 2 σ 13 u n I + h k 2 2 ] u n I + h k 2 2
k 4 = d ̂ ( h 2 ) M ̂ 2 f ( z n + h , u n I + h k 3 )
= j γ [ v n v n + σ 13 3 v n σ 13 v n ] v n , v n = d ̂ ( h 2 ) M ̂ 2 u n I + h k 3 ,
M ̂ 1 m ̂ N h / 2 m ̂ N h / 2 1 m ̂ 2 · m ̂ 1 , M ̂ 2 m ̂ N h · m ̂ N h 1 m ̂ N h / 2 + 2 · m ̂ N h / 2 + 1 ,
M ̂ L m ̂ 1 ( ) · m ̂ 2 ( ) m ̂ N h / 2 ( ) = m ̂ 1 · m ̂ 2 m ̂ N h / 2 ,
u ( z n + 1 , t ) F 1 [ e j β ωω 2 ω 2 h U ˜ ] , u ˜ = u ( z n , t ) e j γ u ( z n ) 2 h ,
j z u ( z , t ) + β 0 ( z ) · σ ( Δ β + j Δ β ω t ) u ( z , t ) β ωω 2 2 t 2 u ( z , t ) = 0
j z U ( z , ω ) + [ β 0 ( z ) · σ ( Δ β Δ β ω ω ) + β ωω 2 ω 2 ] U ( z , ω ) = 0 ,
U ( z , ω ) = M ( z , ω ) e j β ωω 2 ω 2 ( z z 0 ) U ( z 0 , ω ) I ( z , z 0 ) U ( z 0 , ω ) ,
j M ( z , ω ) z + β 0 ( z ) · σ ( Δ β Δ β ω ω ) M ( z , ω ) = 0 .
M ( z , ω ) = m N h · m N h 1 m 2 · m 1
m i = e j δ i η ( β 0 ( z i ) · σ ) ( i = 1 , , N h , η Δ β Δ β ω ω )
m 1 = cos ( δ 1 η ) I + j sin ( δ 1 η ) β 0 ( z 1 ) · σ μ 0 ( 1 ) I + j μ ( 1 ) · σ ,
m 1 m 2 = [ μ 0 ( 1 ) μ 0 ( 2 ) μ ( 1 ) · μ ( 2 ) ] I + j σ · [ μ 0 ( 1 ) μ ( 2 ) + μ 0 ( 2 ) μ ( 1 ) ( μ ( 1 ) × μ ( 2 ) ) ] μ 0 ( 12 ) I + j σ · μ ( 12 ) ,
I ̂ ( z , z 0 ) = M ̂ ( z , t ) d ̂ ( z z 0 )
d ̂ ( z z 0 ) z = z 0 + h = e jh β ωω 2 2 t 2
M ̂ ( z , t ) = m ̂ N h · m ̂ N h 1 m ̂ 2 · m ̂ 1 , m ̂ i = e j δ i ( Δ β + j Δ β ω t ) ( β 0 ( z i ) · σ ) .

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