Abstract

This paper is concerned with the evolution of nonlinear pulses driven by random polarization mode dispersion (PMD). The evolution of the slowly varying envelopes is governed by the stochastic Manakov equation, which has been derived as the limit of the Manakov PMD equation. The aim in this work is to investigate the effect of the PMD on Manakov’s solitons and soliton wave-train propagation. I also study the statistical property of the differential group delay (DGD), and, using Monte Carlo simulations, I compute its probability density function. For linear pulses with zero group-velocity dispersion, I propose an algorithm, based on importance sampling, to estimate the outage probability, i.e., the probability that the value of the DGD exceeds an acceptance level.

© 2013 Optical Society of America

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References

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  1. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).
  2. C. R. Menyuk and A. Galtarossa, eds., Polarization Mode Dispersion (Springer, 2005) [Originally published in Opt. Fiber Commun. Rep. 1, 312–344 (2004)].
  3. J. Garnier, J. Fatome, and G. L. Meur, “Statistical analysis of pulse propagation driven by polarization-mode dispersion,” J. Opt. Soc. Am. B 19, 1968–1977 (2002).
    [CrossRef]
  4. D. Marcuse, P. K. A. Wai, and C. R. Menyuk, “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]
  5. 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]
  6. S. Wabnitz and K. Turitsyn, “Mitigation of nonlinear and PMD impairments by bit-synchronous polarization scrambling,” J. Lightwave Technol. 30, 2494–2501 (2012).
    [CrossRef]
  7. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973).
    [CrossRef]
  8. C. R. Menyuk, “Stability of solitons in birefringent optical fibers. I: equal propagation amplitudes,” Opt. Lett. 12, 614–616 (1987).
    [CrossRef]
  9. C. R. Menyuk, “Stability of solitons in birefringent optical fibers. II. Arbitrary amplitudes,” J. Opt. Soc. Am. B 5, 392–402 (1988).
    [CrossRef]
  10. L. F. Mollenauer, K. Smith, J. P. Gordon, and C. R. Menyuk, “Resistance of solitons to the effects of polarization dispersion in optical fibers,” Opt. Lett. 14, 1219–1221 (1989).
    [CrossRef]
  11. P. K. A. Wai, C. R. Menyuk, and H. H. Chen, “Stability of solitons in randomly varying birefringent fibers,” Opt. Lett. 16, 1231–1233 (1991).
    [CrossRef]
  12. Y. Chung, V. V. Lebedev, and S. S. Vergeles, “Interaction of solitons through radiation in optical fibers with randomly varying birefringence,” Opt. Lett. 29, 1245–1247 (2004).
    [CrossRef]
  13. V. Chernyak, M. Chertkov, I. Gabitov, I. Kolokolov, and V. Lebedev, “PMD-induced fluctuations of bit-error rate in optical fiber systems,” J. Lightwave Technol. 22, 1155–1168 (2004).
    [CrossRef]
  14. T. Ueda and W. L. Kath, “Stochastic simulation of pulses in nonlinear-optical fibers with random birefringence,” J. Opt. Soc. Am. B 11, 818–825 (1994).
    [CrossRef]
  15. A. de Bouard and M. Gazeau, “A diffusion approximation theorem for a nonlinear PDE with application to random birefringent optical fibers,” Ann. Appl. Probab. 22, 2460–2504 (2012).
    [CrossRef]
  16. M. Gazeau, “Analyse de modèles mathématiques pour la propagation de la lumière dans les fibers optiques en présence de biréfringence aléatoire,” Thèse de Doctorat (Ecole Polytechnique, 2012).
  17. 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]
  18. M. Gazeau, “Strong order of convergence of a semi-discrete scheme for the stochastic Manakov equation,” preprint (2013), available at http://hal.archives-ouvertes.fr/hal-00850617.
  19. A. Hasegawa, “Effect of polarization mode dispersion in optical soliton transmission in fibers,” Physica D 188, 241–246 (2004).
    [CrossRef]
  20. J. Yang, “Multisoliton perturbation theory for the Manakov equations and its applications to nonlinear optics,” Phys. Rev. E 59, 2393–2405 (1999).
    [CrossRef]
  21. J. Yang, “Suppression of Manakov soliton interference in optical fibers,” Phys. Rev. E 65, 036606 (2002).
    [CrossRef]
  22. R. Radhakrishnan, M. Lakshmanan, and J. Hietarinta, “Inelastic collision and switching of coupled bright solitons in optical fibers,” Phys. Rev. E 56, 2213–2216 (1997).
    [CrossRef]
  23. J. Yang, Nonlinear Waves in Integrable and Non-integrable Systems (Society for Industrial and Applied Mathematics, 2010).
  24. A. Kebaier, “Statistical Romberg extrapolation: a new variance reduction method and applications to option pricing,” Ann. Appl. Probab. 15, 2681–2705 (2005).
    [CrossRef]
  25. M. B. Giles, “Multilevel Monte Carlo path simulation,” Oper. Res. 56, 607–617 (2008).
    [CrossRef]
  26. K. A. Cliffe, M. B. Giles, R. Scheichl, and A. L. Teckentrup, “Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients,” Comput. Vis. Sci. 14, 3–15 (2011).
    [CrossRef]
  27. J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97, 4541–4550 (2000).
    [CrossRef]
  28. G. J. Foschini, L. E. Nelson, R. M. Jopson, and H. Kogelnik, “Probability densities of second-order polarization mode dispersion including polarization dependent chromatic fiber dispersion,” IEEE Photon. Technol. Lett. 12, 293–295 (2000).
    [CrossRef]
  29. J. Garnier and R. Marty, “Effective pulse dynamics in optical fibers with polarization mode dispersion,” Wave Motion 43, 544–560 (2006).
    [CrossRef]
  30. N. J. Newton, “Variance reduction for simulated diffusions,” SIAM J. Appl. Math. 54, 1780–1805 (1994).
    [CrossRef]
  31. E. Fournie, J. Lebuchoux, and N. Touzi, “Small noise expansion and importance sampling,” Asymptot. Anal. 14, 361–376 (1997).
  32. J.-P. Fouque and T. A. Tullie, “Variance reduction for Monte Carlo simulation in a stochastic volatility environment,” Quant. Finance 2, 24–30 (2002).
    [CrossRef]

2012 (2)

S. Wabnitz and K. Turitsyn, “Mitigation of nonlinear and PMD impairments by bit-synchronous polarization scrambling,” J. Lightwave Technol. 30, 2494–2501 (2012).
[CrossRef]

A. de Bouard and M. Gazeau, “A diffusion approximation theorem for a nonlinear PDE with application to random birefringent optical fibers,” Ann. Appl. Probab. 22, 2460–2504 (2012).
[CrossRef]

2011 (1)

K. A. Cliffe, M. B. Giles, R. Scheichl, and A. L. Teckentrup, “Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients,” Comput. Vis. Sci. 14, 3–15 (2011).
[CrossRef]

2008 (1)

M. B. Giles, “Multilevel Monte Carlo path simulation,” Oper. Res. 56, 607–617 (2008).
[CrossRef]

2006 (2)

J. Garnier and R. Marty, “Effective pulse dynamics in optical fibers with polarization mode dispersion,” Wave Motion 43, 544–560 (2006).
[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]

2005 (1)

A. Kebaier, “Statistical Romberg extrapolation: a new variance reduction method and applications to option pricing,” Ann. Appl. Probab. 15, 2681–2705 (2005).
[CrossRef]

2004 (3)

2002 (3)

J. Yang, “Suppression of Manakov soliton interference in optical fibers,” Phys. Rev. E 65, 036606 (2002).
[CrossRef]

J. Garnier, J. Fatome, and G. L. Meur, “Statistical analysis of pulse propagation driven by polarization-mode dispersion,” J. Opt. Soc. Am. B 19, 1968–1977 (2002).
[CrossRef]

J.-P. Fouque and T. A. Tullie, “Variance reduction for Monte Carlo simulation in a stochastic volatility environment,” Quant. Finance 2, 24–30 (2002).
[CrossRef]

2000 (2)

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97, 4541–4550 (2000).
[CrossRef]

G. J. Foschini, L. E. Nelson, R. M. Jopson, and H. Kogelnik, “Probability densities of second-order polarization mode dispersion including polarization dependent chromatic fiber dispersion,” IEEE Photon. Technol. Lett. 12, 293–295 (2000).
[CrossRef]

1999 (1)

J. Yang, “Multisoliton perturbation theory for the Manakov equations and its applications to nonlinear optics,” Phys. Rev. E 59, 2393–2405 (1999).
[CrossRef]

1997 (3)

R. Radhakrishnan, M. Lakshmanan, and J. Hietarinta, “Inelastic collision and switching of coupled bright solitons in optical fibers,” Phys. Rev. E 56, 2213–2216 (1997).
[CrossRef]

D. Marcuse, P. K. A. Wai, and C. R. Menyuk, “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]

E. Fournie, J. Lebuchoux, and N. Touzi, “Small noise expansion and importance sampling,” Asymptot. Anal. 14, 361–376 (1997).

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]

1994 (2)

1991 (1)

1989 (1)

1988 (1)

1987 (1)

1973 (1)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

Chen, H. H.

Chernyak, V.

Chertkov, M.

Chung, Y.

Cliffe, K. A.

K. A. Cliffe, M. B. Giles, R. Scheichl, and A. L. Teckentrup, “Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients,” Comput. Vis. Sci. 14, 3–15 (2011).
[CrossRef]

de Bouard, A.

A. de Bouard and M. Gazeau, “A diffusion approximation theorem for a nonlinear PDE with application to random birefringent optical fibers,” Ann. Appl. Probab. 22, 2460–2504 (2012).
[CrossRef]

Fatome, J.

Foschini, G. J.

G. J. Foschini, L. E. Nelson, R. M. Jopson, and H. Kogelnik, “Probability densities of second-order polarization mode dispersion including polarization dependent chromatic fiber dispersion,” IEEE Photon. Technol. Lett. 12, 293–295 (2000).
[CrossRef]

Fouque, J.-P.

J.-P. Fouque and T. A. Tullie, “Variance reduction for Monte Carlo simulation in a stochastic volatility environment,” Quant. Finance 2, 24–30 (2002).
[CrossRef]

Fournie, E.

E. Fournie, J. Lebuchoux, and N. Touzi, “Small noise expansion and importance sampling,” Asymptot. Anal. 14, 361–376 (1997).

Gabitov, I.

Garnier, J.

J. Garnier and R. Marty, “Effective pulse dynamics in optical fibers with polarization mode dispersion,” Wave Motion 43, 544–560 (2006).
[CrossRef]

J. Garnier, J. Fatome, and G. L. Meur, “Statistical analysis of pulse propagation driven by polarization-mode dispersion,” J. Opt. Soc. Am. B 19, 1968–1977 (2002).
[CrossRef]

Gazeau, M.

A. de Bouard and M. Gazeau, “A diffusion approximation theorem for a nonlinear PDE with application to random birefringent optical fibers,” Ann. Appl. Probab. 22, 2460–2504 (2012).
[CrossRef]

M. Gazeau, “Strong order of convergence of a semi-discrete scheme for the stochastic Manakov equation,” preprint (2013), available at http://hal.archives-ouvertes.fr/hal-00850617.

M. Gazeau, “Analyse de modèles mathématiques pour la propagation de la lumière dans les fibers optiques en présence de biréfringence aléatoire,” Thèse de Doctorat (Ecole Polytechnique, 2012).

Giles, M. B.

K. A. Cliffe, M. B. Giles, R. Scheichl, and A. L. Teckentrup, “Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients,” Comput. Vis. Sci. 14, 3–15 (2011).
[CrossRef]

M. B. Giles, “Multilevel Monte Carlo path simulation,” Oper. Res. 56, 607–617 (2008).
[CrossRef]

Gordon, J. P.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97, 4541–4550 (2000).
[CrossRef]

L. F. Mollenauer, K. Smith, J. P. Gordon, and C. R. Menyuk, “Resistance of solitons to the effects of polarization dispersion in optical fibers,” Opt. Lett. 14, 1219–1221 (1989).
[CrossRef]

Hasegawa, A.

A. Hasegawa, “Effect of polarization mode dispersion in optical soliton transmission in fibers,” Physica D 188, 241–246 (2004).
[CrossRef]

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973).
[CrossRef]

Hietarinta, J.

R. Radhakrishnan, M. Lakshmanan, and J. Hietarinta, “Inelastic collision and switching of coupled bright solitons in optical fibers,” Phys. Rev. E 56, 2213–2216 (1997).
[CrossRef]

Jopson, R. M.

G. J. Foschini, L. E. Nelson, R. M. Jopson, and H. Kogelnik, “Probability densities of second-order polarization mode dispersion including polarization dependent chromatic fiber dispersion,” IEEE Photon. Technol. Lett. 12, 293–295 (2000).
[CrossRef]

Kath, W. L.

Kebaier, A.

A. Kebaier, “Statistical Romberg extrapolation: a new variance reduction method and applications to option pricing,” Ann. Appl. Probab. 15, 2681–2705 (2005).
[CrossRef]

Kogelnik, H.

G. J. Foschini, L. E. Nelson, R. M. Jopson, and H. Kogelnik, “Probability densities of second-order polarization mode dispersion including polarization dependent chromatic fiber dispersion,” IEEE Photon. Technol. Lett. 12, 293–295 (2000).
[CrossRef]

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97, 4541–4550 (2000).
[CrossRef]

Kolokolov, I.

Lakshmanan, M.

R. Radhakrishnan, M. Lakshmanan, and J. Hietarinta, “Inelastic collision and switching of coupled bright solitons in optical fibers,” Phys. Rev. E 56, 2213–2216 (1997).
[CrossRef]

Lebedev, V.

Lebedev, V. V.

Lebuchoux, J.

E. Fournie, J. Lebuchoux, and N. Touzi, “Small noise expansion and importance sampling,” Asymptot. Anal. 14, 361–376 (1997).

Marcuse, D.

D. Marcuse, P. K. A. Wai, and C. R. Menyuk, “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.

Marty, R.

J. Garnier and R. Marty, “Effective pulse dynamics in optical fibers with polarization mode dispersion,” Wave Motion 43, 544–560 (2006).
[CrossRef]

Menyuk, C. R.

Meur, G. L.

Mollenauer, L. F.

Nelson, L. E.

G. J. Foschini, L. E. Nelson, R. M. Jopson, and H. Kogelnik, “Probability densities of second-order polarization mode dispersion including polarization dependent chromatic fiber dispersion,” IEEE Photon. Technol. Lett. 12, 293–295 (2000).
[CrossRef]

Newton, N. J.

N. J. Newton, “Variance reduction for simulated diffusions,” SIAM J. Appl. Math. 54, 1780–1805 (1994).
[CrossRef]

Radhakrishnan, R.

R. Radhakrishnan, M. Lakshmanan, and J. Hietarinta, “Inelastic collision and switching of coupled bright solitons in optical fibers,” Phys. Rev. E 56, 2213–2216 (1997).
[CrossRef]

Scheichl, R.

K. A. Cliffe, M. B. Giles, R. Scheichl, and A. L. Teckentrup, “Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients,” Comput. Vis. Sci. 14, 3–15 (2011).
[CrossRef]

Smith, K.

Tappert, F.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973).
[CrossRef]

Teckentrup, A. L.

K. A. Cliffe, M. B. Giles, R. Scheichl, and A. L. Teckentrup, “Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients,” Comput. Vis. Sci. 14, 3–15 (2011).
[CrossRef]

Touzi, N.

E. Fournie, J. Lebuchoux, and N. Touzi, “Small noise expansion and importance sampling,” Asymptot. Anal. 14, 361–376 (1997).

Tullie, T. A.

J.-P. Fouque and T. A. Tullie, “Variance reduction for Monte Carlo simulation in a stochastic volatility environment,” Quant. Finance 2, 24–30 (2002).
[CrossRef]

Turitsyn, K.

Ueda, T.

Vergeles, S. S.

Wabnitz, S.

Wai, P. K. A.

D. Marcuse, P. K. A. Wai, and C. R. Menyuk, “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]

P. K. A. Wai, C. R. Menyuk, and H. H. Chen, “Stability of solitons in randomly varying birefringent fibers,” Opt. Lett. 16, 1231–1233 (1991).
[CrossRef]

Yang, J.

J. Yang, “Suppression of Manakov soliton interference in optical fibers,” Phys. Rev. E 65, 036606 (2002).
[CrossRef]

J. Yang, “Multisoliton perturbation theory for the Manakov equations and its applications to nonlinear optics,” Phys. Rev. E 59, 2393–2405 (1999).
[CrossRef]

J. Yang, Nonlinear Waves in Integrable and Non-integrable Systems (Society for Industrial and Applied Mathematics, 2010).

Ann. Appl. Probab. (2)

A. de Bouard and M. Gazeau, “A diffusion approximation theorem for a nonlinear PDE with application to random birefringent optical fibers,” Ann. Appl. Probab. 22, 2460–2504 (2012).
[CrossRef]

A. Kebaier, “Statistical Romberg extrapolation: a new variance reduction method and applications to option pricing,” Ann. Appl. Probab. 15, 2681–2705 (2005).
[CrossRef]

Appl. Phys. Lett. (1)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973).
[CrossRef]

Asymptot. Anal. (1)

E. Fournie, J. Lebuchoux, and N. Touzi, “Small noise expansion and importance sampling,” Asymptot. Anal. 14, 361–376 (1997).

Comput. Vis. Sci. (1)

K. A. Cliffe, M. B. Giles, R. Scheichl, and A. L. Teckentrup, “Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients,” Comput. Vis. Sci. 14, 3–15 (2011).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

G. J. Foschini, L. E. Nelson, R. M. Jopson, and H. Kogelnik, “Probability densities of second-order polarization mode dispersion including polarization dependent chromatic fiber dispersion,” IEEE Photon. Technol. Lett. 12, 293–295 (2000).
[CrossRef]

J. Lightwave Technol. (5)

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]

V. Chernyak, M. Chertkov, I. Gabitov, I. Kolokolov, and V. Lebedev, “PMD-induced fluctuations of bit-error rate in optical fiber systems,” J. Lightwave Technol. 22, 1155–1168 (2004).
[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]

S. Wabnitz and K. Turitsyn, “Mitigation of nonlinear and PMD impairments by bit-synchronous polarization scrambling,” J. Lightwave Technol. 30, 2494–2501 (2012).
[CrossRef]

D. Marcuse, P. K. A. Wai, and C. R. Menyuk, “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]

J. Opt. Soc. Am. B (3)

Oper. Res. (1)

M. B. Giles, “Multilevel Monte Carlo path simulation,” Oper. Res. 56, 607–617 (2008).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. E (3)

J. Yang, “Multisoliton perturbation theory for the Manakov equations and its applications to nonlinear optics,” Phys. Rev. E 59, 2393–2405 (1999).
[CrossRef]

J. Yang, “Suppression of Manakov soliton interference in optical fibers,” Phys. Rev. E 65, 036606 (2002).
[CrossRef]

R. Radhakrishnan, M. Lakshmanan, and J. Hietarinta, “Inelastic collision and switching of coupled bright solitons in optical fibers,” Phys. Rev. E 56, 2213–2216 (1997).
[CrossRef]

Physica D (1)

A. Hasegawa, “Effect of polarization mode dispersion in optical soliton transmission in fibers,” Physica D 188, 241–246 (2004).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A. (1)

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97, 4541–4550 (2000).
[CrossRef]

Quant. Finance (1)

J.-P. Fouque and T. A. Tullie, “Variance reduction for Monte Carlo simulation in a stochastic volatility environment,” Quant. Finance 2, 24–30 (2002).
[CrossRef]

SIAM J. Appl. Math. (1)

N. J. Newton, “Variance reduction for simulated diffusions,” SIAM J. Appl. Math. 54, 1780–1805 (1994).
[CrossRef]

Wave Motion (1)

J. Garnier and R. Marty, “Effective pulse dynamics in optical fibers with polarization mode dispersion,” Wave Motion 43, 544–560 (2006).
[CrossRef]

Other (5)

J. Yang, Nonlinear Waves in Integrable and Non-integrable Systems (Society for Industrial and Applied Mathematics, 2010).

M. Gazeau, “Strong order of convergence of a semi-discrete scheme for the stochastic Manakov equation,” preprint (2013), available at http://hal.archives-ouvertes.fr/hal-00850617.

M. Gazeau, “Analyse de modèles mathématiques pour la propagation de la lumière dans les fibers optiques en présence de biréfringence aléatoire,” Thèse de Doctorat (Ecole Polytechnique, 2012).

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

C. R. Menyuk and A. Galtarossa, eds., Polarization Mode Dispersion (Springer, 2005) [Originally published in Opt. Fiber Commun. Rep. 1, 312–344 (2004)].

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Figures (8)

Fig. 1.
Fig. 1.

Evolution along the fiber of both nonlinear and linear pulses, which are, respectively, solutions of Eqs. (2.2) and (3.10). The input profile is a Manakov soliton, and the effective parameter is γ=2.6082e02.

Fig. 2.
Fig. 2.

Evolution along the fiber of the DoP, the energy, the time displacement, and the pulse width. The input profile is a Manakov soliton.

Fig. 3.
Fig. 3.

Evolution over the Poincaré sphere of the Stokes vector for nonlinear pulses (red curve) and for linear pulses (blue curve).

Fig. 4.
Fig. 4.

Evolution along the fiber of both nonlinear and linear pulses, which are, respectively, solutions of Eqs. (2.2) and (3.10). The input profile is a Manakov soliton and the effective PMD parameter is γ=0.1044.

Fig. 5.
Fig. 5.

Evolution along the fiber of a 2-soliton. The polarization vectors are chosen orthogonal so that the changes of polarization are induced only by the random birefringence. The level of noise is γ=0.0261.

Fig. 6.
Fig. 6.

Statistics of the mean center, the virial and the root mean square computed by the SR method. The green curves correspond to the exact formulas in the linear case in absence of GVD [Eq. (4.1)].

Fig. 7.
Fig. 7.

Average curves of the DGD and the square DGD for nonlinear pulses. The last row displays the empirical p.d.f. of the DGD and the square DGD. The green curves correspond to the exact formulas (4.1). These quantities are computed using the Relaxation scheme and the SR method on a fiber of 40 km length and at the frequency ω=0.157.

Fig. 8.
Fig. 8.

Segments of the DGD’s p.d.f. obtained by IS (red dots) on a logarithmic scale. The solid line represents the Maxwellian distribution.

Tables (1)

Tables Icon

Table 1. Comparison of Relative Errors between the Two Methods

Equations (39)

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

iΦz+Σ(θz,z)Φ+ibσ3Φt+d022Φt2+56|Φ|2Φ+16(Φ*σ3Φ)σ3Φ+13(Φ¯1Φ22e4ibzΦ¯2Φ12e4ibz)=0,
Σ(θz,z)=(0i2dθ(z)dze2ibzi2dθ(z)dze2ibz0),
σ1=(0110),σ2=(0ii0),σ3=(1001).
izX(z)+d022X(z)t2+F(X(z))+iγk=13σkX(z)tξ˙k(z)=0,
χkn=ξk((n+1)Δz)ξk(nΔz)Δz,k=1,2,3,
i(Xjn+1Xjn)+rΔXjn+1/2+iγr2k=13σkXjn+1/2χkn+89Φjn+1/2Xjn+1/2Δz=0,
{ΔXjn+1/2=Xj1n+1/22Xjn+1/2+Xj+1n+1/2Xjn+1/2=Xj+1n+1/2Xj1n+1/2.
{i(Y^kn+1X^kn)=mk2(Y^kn+1+X^kn)Xjn+1=exp(i89|Yjn+1|2Δz)Yjn+1,
mk=(Δzhk2+γΔzhkl=13σlχln)
X(z,t)=98(cosΘ/2eiϕ1sinΘ/2eiϕ2)ηsechη(tτ(z))eiφ(z),
X(0,t)=j=12cjηjsechηj(tτj0)eikj(tτj0)+iαj0,
cj=98(cosΘj/2exp(iϕj1)sinΘj/2exp(iϕj2)).
Xn2=Δtj=0M+1|X1,jn|2+|X2,jn|2.
errN=maxn1,N|Xn2X02X02|.
errlN=maxn1,N|Xln2Xl(0)2Xl(0)2|.
H(X)=12R|Xt|2dt29R|X|4dt.
H(X(z))=H(X0)+γ89k=130z|X|2X,σkXtξ˙(z)dz.
H=Δtj=0M|Xj+1nXjnΔt|289Njn,
s^(z,ω)=X^*σ⃗X^(z,ω)s^0(z,ω),z0,
Dp=(S12+S22+S32)1/2S0
S(z)=RX*σ⃗X(z,t)dt.
{Tc=t|X(z,t)|2dtTw=t2|X(z,t)|2dtσrms2=TwTc2τ=4(R^(z,ω)R^(0,ω)),
R^(z,ω)|F(tX(z,t))X^(z,ω)|2=|X^(z,ω)X^(z,ω)|2,
izX+iγk=13σkXtξ˙(z)=0,
Q=f(Xjn)Il,jm,nf(Xlm)+Il,jm,nf(Xlm).
1Nfk=1Nff(Xj,kn)Il,jm,nf(Xl,km)+1Nck=1NcIl,jm,nf(X^l,km),
{τ(z)=32πγzτ(z)=12γzTw2(z)=Tw(0)+3γz.
fX(x,3;0)=x2π(4γz)3/2exp(x8γz).
{dTc2(z)dz=γDp2dV(Tw(z))dz=12γTc(z).
izXϵ+iγϵk=13σkXϵtξ˜˙(z)=0,
errτ=maxn0,N|μ(τn)τ(zn)τ(zn)|,errτ2=maxn0,N|μ(τn)τ(zn)τ(zn)|.
errTc=maxn0,N|μ(Tcn)Tc(zn)Tc(zn)|,errTw=maxn0,N|μ(Twn)Tw(zn)Tw(zn)|.
zτ(z,ω)=2γωξ˙×τ(z,ω)+2γξ˙,
zτ(z,ω)=2γσ(τ)ξ˙4γω2τ,
σ(τ)=2γ(1ωτ3ωτ2ωτ31ωτ1ωτ2ωτ11).
Mz=exp(0zh(s),ξ˙(s)ds+120z|h(s)|2ds).
h(z)=2γμτ(z)(1pq)2cτ(z)exp((c+μ)8γ(Zz))π(Zz)μqcosh(μc4γ(Zz)),
P(τ(Z)c)1Nk=1N1τ(h)(Z)cMZ,
zτ(h)(z,ω)=2γσ(τ(h))(ξ˜˙h(z))4γω2τ(h),

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