Abstract

Randomly varying birefringence leads to nonlinear polarization-mode dispersion (PMD) in addition to the well-known linear PMD. Here we calculate the variance of the field fluctuations produced by this nonlinear PMD. Knowing the size of these fluctuations is useful for assessing when nonlinear PMD is important and for its proper incorporation in fast numerical algorithms. We also derive the equilibrium probability distributions for the PMD coefficients, and we track the evolution of the polarization state's probability distribution from its initial delta-function distribution to its steady-state uniform distribution on the Poincaré sphere.

© 1997 Optical Society of America

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References

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  1. D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, New York, 1982).
  2. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).
  3. P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers,” Opt. Lett. 11, 464–466 (1986).
    [Crossref] [PubMed]
  4. C. D. Poole, J. M. Wiesenfeld, A. R. McCormick, and K. T. Nelson, “Broadband dispersion compensation by using the higher-order spatial mode in a two-mode fiber,” Opt. Lett. 17, 985–987 (1992).
    [Crossref] [PubMed]
  5. A. Yariv, D. Fekete, and D. M. Pepper, “Compensation for channel dispersion by nonlinear optical phase conjugation,” Opt. Lett. 4, 52–54 (1979).
    [Crossref] [PubMed]
  6. N. Bergano, “Circulating loop transmission experiments for the study of long-haul transmission systems using erbium-doped fiber amplifiers,” J. Lightwave Technol. 13, 879–888 (1995).
    [Crossref]
  7. A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford U. Press, Oxford, 1995).
  8. J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665–667 (1986).
    [Crossref] [PubMed]
  9. A. Mecozzi, J. Moores, H. Haus, and Y. Lai, “Soliton transmission control,” Opt. Lett. 16, 1841–1843 (1991).
    [Crossref] [PubMed]
  10. Y. Kodama and A. Hasegawa, “Generation of asymptotically stable optical solitons and suppression of the Gordon–Haus effect,” Opt. Lett. 17, 31–33 (1992).
    [Crossref] [PubMed]
  11. C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23, 174–176 (1987).
    [Crossref]
  12. C. R. Menyuk, “Stability of solitons in birefringent optical fibers. I. Equal propagation amplitudes,” Opt. Lett. 12, 614–616 (1987).
    [Crossref] [PubMed]
  13. C. R. Menyuk, “Stability of solitons in birefringent optical fibers. II. Arbitrary amplitudes,” J. Opt. Soc. Am. B 5, 392–402 (1988).
    [Crossref]
  14. 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] [PubMed]
  15. S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
    [Crossref]
  16. C. D. Poole, “Statistical treatment of polarization dispersion in single-mode fiber,” Opt. Lett. 13, 687–689 (1988).
    [Crossref] [PubMed]
  17. C. D. Poole, “Measurement of polarization-mode dispersion in single-mode fibers with random mode coupling,” Opt. Lett. 14, 523–525 (1989).
    [Crossref] [PubMed]
  18. G. J. Foschini and C. D. Poole, “Statistical theory of polarization dispersion in single mode fibers,” J. Lightwave Technol. 9, 1439–1456 (1991).
    [Crossref]
  19. C. R. Menyuk and P. K. A. Wai, “Polarization evolution and dispersion in fibers with spatially varying birefringence,” J. Opt. Soc. Am. B 11, 1288–1296 (1994).
    [Crossref]
  20. P. K. A. Wai and C. R. Menyuk, “Anisotropic diffusion of the state of polarization in optical fibers with randomly varying birefringence,” Opt. Lett. 20, 2493–2495 (1995).
    [Crossref]
  21. 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]
  22. A. J. Barlow and J. J. Ramskov-Ha, “Birefringence and polarization mode-dispersion in spun single-mode fibers,” Appl. Opt. 20, 2962–2968 (1981).
    [Crossref] [PubMed]
  23. D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to the studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. (to be published).
  24. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).
  25. P. K. A. Wai and C. R. Menyuk, “Polarization decorrelation in optical fibers with randomly varying birefringence,” Opt. Lett. 19, 1517–1519 (1994).
    [Crossref] [PubMed]
  26. A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer-Verlag, New York, 1983).
  27. N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1992).
  28. G. C. Papanicolaou, “Introduction to the asymptotic analysis of stochastic equations,” in Modern Modeling of Continuum Phenomena, R. C. DiPrima, ed., Vol. 16 of Lectures in Applied Mathematics (American Mathematical Society, Providence, R.I., 1977), pp. 109–147.
  29. M. H. DeGroot, Probability and Statistics (Addison-Wesley, Menlo Park, Calif., 1975).
  30. L. A. Wainstein and V. D. Zubakov, Extraction of Signals from Noise (Prentice-Hall, Englewood Cliffs, N.J., 1962).
  31. J. Kevorkian and J. D. Cole, Perturbation Methods in Applied Mathematics (Springer-Verlag, New York, 1981).
  32. T. Ueda and W. L. Kath, “Dynamics of pulses in randomly birefringent nonlinear optical fibers,” Physica D 55, 166–181 (1992).
    [Crossref]
  33. 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]
  34. G. Papanicolaou and J. B. Keller, “Stochastic differential equations with applications to random harmonic oscillators and wave propagation in random media,” SIAM J. Appl. Math. 21, 287–305 (1971).
    [Crossref]
  35. M. F. Arend, M. L. Dennis, E. A. Duling, I. N. Golovchenko, A. N. Pilipetskii, and C. R. Menyuk, “Simulations of a nonlinear optical loop mirror demultiplexor using random birefringence fiber,” in Optical Fabrication and Testing, Volume 6 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 226–227.

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)

N. Bergano, “Circulating loop transmission experiments for the study of long-haul transmission systems using erbium-doped fiber amplifiers,” J. Lightwave Technol. 13, 879–888 (1995).
[Crossref]

P. K. A. Wai and C. R. Menyuk, “Anisotropic diffusion of the state of polarization in optical fibers with randomly varying birefringence,” Opt. Lett. 20, 2493–2495 (1995).
[Crossref]

1994 (3)

1992 (4)

Y. Kodama and A. Hasegawa, “Generation of asymptotically stable optical solitons and suppression of the Gordon–Haus effect,” Opt. Lett. 17, 31–33 (1992).
[Crossref] [PubMed]

C. D. Poole, J. M. Wiesenfeld, A. R. McCormick, and K. T. Nelson, “Broadband dispersion compensation by using the higher-order spatial mode in a two-mode fiber,” Opt. Lett. 17, 985–987 (1992).
[Crossref] [PubMed]

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[Crossref]

T. Ueda and W. L. Kath, “Dynamics of pulses in randomly birefringent nonlinear optical fibers,” Physica D 55, 166–181 (1992).
[Crossref]

1991 (3)

1989 (1)

1988 (2)

1987 (2)

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23, 174–176 (1987).
[Crossref]

C. R. Menyuk, “Stability of solitons in birefringent optical fibers. I. Equal propagation amplitudes,” Opt. Lett. 12, 614–616 (1987).
[Crossref] [PubMed]

1986 (2)

1981 (1)

1979 (1)

1971 (1)

G. Papanicolaou and J. B. Keller, “Stochastic differential equations with applications to random harmonic oscillators and wave propagation in random media,” SIAM J. Appl. Math. 21, 287–305 (1971).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).

Arend, M. F.

M. F. Arend, M. L. Dennis, E. A. Duling, I. N. Golovchenko, A. N. Pilipetskii, and C. R. Menyuk, “Simulations of a nonlinear optical loop mirror demultiplexor using random birefringence fiber,” in Optical Fabrication and Testing, Volume 6 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 226–227.

Barlow, A. J.

Bergano, N.

N. Bergano, “Circulating loop transmission experiments for the study of long-haul transmission systems using erbium-doped fiber amplifiers,” J. Lightwave Technol. 13, 879–888 (1995).
[Crossref]

Bergano, N. S.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Chen, H. H.

Cole, J. D.

J. Kevorkian and J. D. Cole, Perturbation Methods in Applied Mathematics (Springer-Verlag, New York, 1981).

DeGroot, M. H.

M. H. DeGroot, Probability and Statistics (Addison-Wesley, Menlo Park, Calif., 1975).

Dennis, M. L.

M. F. Arend, M. L. Dennis, E. A. Duling, I. N. Golovchenko, A. N. Pilipetskii, and C. R. Menyuk, “Simulations of a nonlinear optical loop mirror demultiplexor using random birefringence fiber,” in Optical Fabrication and Testing, Volume 6 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 226–227.

Duling, E. A.

M. F. Arend, M. L. Dennis, E. A. Duling, I. N. Golovchenko, A. N. Pilipetskii, and C. R. Menyuk, “Simulations of a nonlinear optical loop mirror demultiplexor using random birefringence fiber,” in Optical Fabrication and Testing, Volume 6 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 226–227.

Evangelides, S. G.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[Crossref]

Fekete, D.

Foschini, G. J.

G. J. Foschini and C. D. Poole, “Statistical theory of polarization dispersion in single mode fibers,” J. Lightwave Technol. 9, 1439–1456 (1991).
[Crossref]

Golovchenko, I. N.

M. F. Arend, M. L. Dennis, E. A. Duling, I. N. Golovchenko, A. N. Pilipetskii, and C. R. Menyuk, “Simulations of a nonlinear optical loop mirror demultiplexor using random birefringence fiber,” in Optical Fabrication and Testing, Volume 6 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 226–227.

Gordon, J. P.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[Crossref]

J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665–667 (1986).
[Crossref] [PubMed]

Hasegawa, A.

Haus, H.

Haus, H. A.

Kath, W. L.

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]

T. Ueda and W. L. Kath, “Dynamics of pulses in randomly birefringent nonlinear optical fibers,” Physica D 55, 166–181 (1992).
[Crossref]

Keller, J. B.

G. Papanicolaou and J. B. Keller, “Stochastic differential equations with applications to random harmonic oscillators and wave propagation in random media,” SIAM J. Appl. Math. 21, 287–305 (1971).
[Crossref]

Kevorkian, J.

J. Kevorkian and J. D. Cole, Perturbation Methods in Applied Mathematics (Springer-Verlag, New York, 1981).

Kodama, Y.

Lai, Y.

Lee, Y. C.

Lichtenberg, A. J.

A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer-Verlag, New York, 1983).

Lieberman, M. A.

A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer-Verlag, New York, 1983).

Marcuse, D.

D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, New York, 1982).

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to the studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. (to be published).

McCormick, A. R.

Mecozzi, A.

Menyuk, C. R.

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 and C. R. Menyuk, “Anisotropic diffusion of the state of polarization in optical fibers with randomly varying birefringence,” Opt. Lett. 20, 2493–2495 (1995).
[Crossref]

C. R. Menyuk and P. K. A. Wai, “Polarization evolution and dispersion in fibers with spatially varying birefringence,” J. Opt. Soc. Am. B 11, 1288–1296 (1994).
[Crossref]

P. K. A. Wai and C. R. Menyuk, “Polarization decorrelation in optical fibers with randomly varying birefringence,” Opt. Lett. 19, 1517–1519 (1994).
[Crossref] [PubMed]

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] [PubMed]

C. R. Menyuk, “Stability of solitons in birefringent optical fibers. II. Arbitrary amplitudes,” J. Opt. Soc. Am. B 5, 392–402 (1988).
[Crossref]

C. R. Menyuk, “Stability of solitons in birefringent optical fibers. I. Equal propagation amplitudes,” Opt. Lett. 12, 614–616 (1987).
[Crossref] [PubMed]

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23, 174–176 (1987).
[Crossref]

P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers,” Opt. Lett. 11, 464–466 (1986).
[Crossref] [PubMed]

M. F. Arend, M. L. Dennis, E. A. Duling, I. N. Golovchenko, A. N. Pilipetskii, and C. R. Menyuk, “Simulations of a nonlinear optical loop mirror demultiplexor using random birefringence fiber,” in Optical Fabrication and Testing, Volume 6 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 226–227.

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to the studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. (to be published).

Mollenauer, L. F.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[Crossref]

Moores, J.

Nelson, K. T.

Papanicolaou, G.

G. Papanicolaou and J. B. Keller, “Stochastic differential equations with applications to random harmonic oscillators and wave propagation in random media,” SIAM J. Appl. Math. 21, 287–305 (1971).
[Crossref]

Papanicolaou, G. C.

G. C. Papanicolaou, “Introduction to the asymptotic analysis of stochastic equations,” in Modern Modeling of Continuum Phenomena, R. C. DiPrima, ed., Vol. 16 of Lectures in Applied Mathematics (American Mathematical Society, Providence, R.I., 1977), pp. 109–147.

Pepper, D. M.

Pilipetskii, A. N.

M. F. Arend, M. L. Dennis, E. A. Duling, I. N. Golovchenko, A. N. Pilipetskii, and C. R. Menyuk, “Simulations of a nonlinear optical loop mirror demultiplexor using random birefringence fiber,” in Optical Fabrication and Testing, Volume 6 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 226–227.

Poole, C. D.

Ramskov-Ha, J. J.

Ueda, T.

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]

T. Ueda and W. L. Kath, “Dynamics of pulses in randomly birefringent nonlinear optical fibers,” Physica D 55, 166–181 (1992).
[Crossref]

Van Kampen, N. G.

N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1992).

Wai, P. K. A.

Wainstein, L. A.

L. A. Wainstein and V. D. Zubakov, Extraction of Signals from Noise (Prentice-Hall, Englewood Cliffs, N.J., 1962).

Wiesenfeld, J. M.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Yariv, A.

Zubakov, V. D.

L. A. Wainstein and V. D. Zubakov, Extraction of Signals from Noise (Prentice-Hall, Englewood Cliffs, N.J., 1962).

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23, 174–176 (1987).
[Crossref]

J. Lightwave Technol. (4)

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[Crossref]

G. J. Foschini and C. D. Poole, “Statistical theory of polarization dispersion in single mode fibers,” J. Lightwave Technol. 9, 1439–1456 (1991).
[Crossref]

N. Bergano, “Circulating loop transmission experiments for the study of long-haul transmission systems using erbium-doped fiber amplifiers,” J. Lightwave Technol. 13, 879–888 (1995).
[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]

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

Opt. Lett. (12)

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] [PubMed]

C. R. Menyuk, “Stability of solitons in birefringent optical fibers. I. Equal propagation amplitudes,” Opt. Lett. 12, 614–616 (1987).
[Crossref] [PubMed]

P. K. A. Wai and C. R. Menyuk, “Anisotropic diffusion of the state of polarization in optical fibers with randomly varying birefringence,” Opt. Lett. 20, 2493–2495 (1995).
[Crossref]

C. D. Poole, “Statistical treatment of polarization dispersion in single-mode fiber,” Opt. Lett. 13, 687–689 (1988).
[Crossref] [PubMed]

C. D. Poole, “Measurement of polarization-mode dispersion in single-mode fibers with random mode coupling,” Opt. Lett. 14, 523–525 (1989).
[Crossref] [PubMed]

J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665–667 (1986).
[Crossref] [PubMed]

A. Mecozzi, J. Moores, H. Haus, and Y. Lai, “Soliton transmission control,” Opt. Lett. 16, 1841–1843 (1991).
[Crossref] [PubMed]

Y. Kodama and A. Hasegawa, “Generation of asymptotically stable optical solitons and suppression of the Gordon–Haus effect,” Opt. Lett. 17, 31–33 (1992).
[Crossref] [PubMed]

P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers,” Opt. Lett. 11, 464–466 (1986).
[Crossref] [PubMed]

C. D. Poole, J. M. Wiesenfeld, A. R. McCormick, and K. T. Nelson, “Broadband dispersion compensation by using the higher-order spatial mode in a two-mode fiber,” Opt. Lett. 17, 985–987 (1992).
[Crossref] [PubMed]

A. Yariv, D. Fekete, and D. M. Pepper, “Compensation for channel dispersion by nonlinear optical phase conjugation,” Opt. Lett. 4, 52–54 (1979).
[Crossref] [PubMed]

P. K. A. Wai and C. R. Menyuk, “Polarization decorrelation in optical fibers with randomly varying birefringence,” Opt. Lett. 19, 1517–1519 (1994).
[Crossref] [PubMed]

Physica D (1)

T. Ueda and W. L. Kath, “Dynamics of pulses in randomly birefringent nonlinear optical fibers,” Physica D 55, 166–181 (1992).
[Crossref]

SIAM J. Appl. Math. (1)

G. Papanicolaou and J. B. Keller, “Stochastic differential equations with applications to random harmonic oscillators and wave propagation in random media,” SIAM J. Appl. Math. 21, 287–305 (1971).
[Crossref]

Other (12)

M. F. Arend, M. L. Dennis, E. A. Duling, I. N. Golovchenko, A. N. Pilipetskii, and C. R. Menyuk, “Simulations of a nonlinear optical loop mirror demultiplexor using random birefringence fiber,” in Optical Fabrication and Testing, Volume 6 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 226–227.

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to the studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. (to be published).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer-Verlag, New York, 1983).

N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1992).

G. C. Papanicolaou, “Introduction to the asymptotic analysis of stochastic equations,” in Modern Modeling of Continuum Phenomena, R. C. DiPrima, ed., Vol. 16 of Lectures in Applied Mathematics (American Mathematical Society, Providence, R.I., 1977), pp. 109–147.

M. H. DeGroot, Probability and Statistics (Addison-Wesley, Menlo Park, Calif., 1975).

L. A. Wainstein and V. D. Zubakov, Extraction of Signals from Noise (Prentice-Hall, Englewood Cliffs, N.J., 1962).

J. Kevorkian and J. D. Cole, Perturbation Methods in Applied Mathematics (Springer-Verlag, New York, 1981).

D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, New York, 1982).

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford U. Press, Oxford, 1995).

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Figures (5)

Fig. 1
Fig. 1

Comparison between the numerical and the theoretical estimates for the variance of 0l(z12-)dz for several values of hfiber/LB when the ratio is small. Each curve represents the results of 5000 solutions of Eq. (5). The theoretical curve is Eq. (51) with σE2=845hE2 and hE=LB2/(12π2hfiber).

Fig. 2
Fig. 2

Numerically determined probability distribution of (z12-)dz for l/hfiber=2.5, 5.0, 10.0, and 20.0, showing the approach to a Gaussian distribution for large l. Here LB/hfiber=10 (so hE0.84hfiber), and each curve is the result of 50,000 runs.

Fig. 3
Fig. 3

Comparison between numerical and theoretical estimates for the variance of 0l(z12-)dz for several values of hfiber/LB when the ratio is large. Each curve represents the results of 5000 solutions of Eq. (5), and the theoretical curve is Eq. (61) with R(0)=4/45 and hE=112hfiber.

Fig. 4
Fig. 4

Comparison between numerical and uniform theoretical estimates for the variance of 0l(z12-)dz for several values of hfiber/LB. Each curve represents the results of 5000 solutions of Eq. (5). In this case two theoretical curves are shown, namely, Eqs. (51) and (61), with R(0)=4/45 and hE replaced by hU. The uniform mixing length hU is given by Eq. (65).

Fig. 5
Fig. 5

Numerical estimates for the variance of x1dz for various values of hfiber/LB. Each curve represents the results of 5000 solutions of Eq. (5). Here σE2=hE2.

Equations (128)

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

i Ψ¯z±122Ψ¯t2+89|Ψ¯|2Ψ¯=-ibσ¯ Ψ¯t-13(Nˆ-Nˆ),
σ¯=x1x2-ix3x2+ix3-x1,
Nˆ1=z12(2|V¯|2-|U¯|2)U¯-z1(z2-iz3)×(2|U¯|2-|V¯|2)V¯-z1(z2+iz3)U¯2V¯*-(z2-iz3)2V¯2U¯*,
Nˆ2=z12(2|U¯|2-|V¯|2)V¯+z1(z2+iz3)×(2|V¯|2-|U¯|2)U¯+z1(z2-iz3)V¯2U¯*-(z2+iz3)2U¯2V¯*,
Nˆ1=(2|V¯|2-|U¯|2)U¯,
Nˆ2=(2|U¯|2-|V¯|2)V¯.
ddzxjyjzj=0gθ0-gθ02b0-2b0xjyjzj
dAdz=ibΣA,
σ1=0110,σ2=0-ii0,σ3=100-1
Ψ=UV=cos θ/2sin θ/2-sin θ/2cos θ/2A1A2,
i dΨdz+Σ˜Ψ=0,Σ˜=b-igθ/2igθ/2-b.
ddzS˜1S˜2S˜3=0gθ0-gθ02b0-2b0S˜1S˜2S˜3.
i dTdz+Σ˜T=0,T(0)=I
Ψ(z, t)=T(z)Ψ¯.
T(z)u1-u2*u2u1*
x1=|u1|2-|u2|2,x2+ix3=-2u1u2,
y1=u1u2*+u1*u2,y2+iy3=u12-u22,
z1=i(u1u2*-u1*u2),z2+iz3=i(u12+u22).
Q=x1x2x3y1y2y3z1z2z3,
dQdz=WQ,
W=0gθ0-gθ02b0-2b0.
dθdz=gθ(z),
gθ(z)=0,gθ(z)gθ(z)=σθ2δ(z-z).
pz=12σθ2 2pθ2,
p(θ, z)=12π+1πn=1 exp-12σθ2n2zcos n(θ-θ0),
cos(θ-θ0)=exp(-½σθ2z),
ddzxj=-½σθ2xj,
ddzyj=-½σθ2yj-2b(zj),
ddzzj=2byj,
ddzxjxk=-σθ2(xjxk-yjyk),
ddzyjyk=σθ2(xjxk-yjyk)-4b½(yjzk+ykzj),
ddzzjzk=4b½(yjzk+ykzj),
ddz½(yjzk+ykzj)
=2b(yjyk-zjzk)-½σθ2½(yjzk+ykzj),
xjS˜1,yjS˜2,zjS˜3,
xjxkS˜12,yjykS˜22,zjzkS˜32,
½(yjzk+ykzj)S˜2S˜3.
z0z0+lf(ζ)dζ=z0z0+lf(ζ)dζ,
Varz0z0+lf(ζ)dζ=z0z0+l[f(ζ)-f(ζ)]dζ×z0z0+l[f(ζ)-f(ζ)]dζ=z0z0+lz0z0+lf(ζ)-f(ζ)]×[f(ζ)-f(ζ)]dζdζ,
z0z0+lf(ζ)dζz0z0+lf¯dζ=f¯l.
Varz0z0+lf(ζ)dζ
=z0z0+lz0z0+l[f(ζ)-f¯][f(ζ)-f¯]dζdζ
=z0z0+lz0z0+lR(ζ-ζ)dζdζ,
z0z0+lz0-ζz0+l-ζR(ξ)dξdζ=20l(l-ξ)R(ξ)dξ,
Varz0z0+lf(ζ)dζ2l0R(ξ)dξ-20ξR(ξ)dξ,
2R(0)hE2lhE+exp-lhE-1,
Varz0z0+lf(ζ)dζ2R(0)hEl-2R(0)hE2,
Varz0z0+lf(ζ)dζ2[f(ζ)-f¯]2hEl=2R(0)hEl,
hE1R(0)0R(ξ)dξ.
[f(z0)-f¯][f(z0+ξ)-f¯]|f(z0)
=[f(z0)-f¯][f(z0+ξ)-f¯]|f(z0).
R(ξ)=[f(z0)-f¯][f(z0+ξ)-f¯]|f(z0).
½(yjzk+ykzj)2bσθ2[δjk-3zjzk].
ddzzjzk24b2σθ2[δjk-zjzk].
z1, z2, z3=(cos φ,sin φ cos ψ,sin φ sin ψ)
z12-()=cos2 φ-(1/3),
2z1z2=2 cos φ sin φ cos ψ=sin 2φ cos ψ,
2z1z3=2 cos φ sin φ sin ψ=sin 2φ sin ψ,
z22-z32=sin2 φ(cos2 ψ-sin2 ψ)=sin2 φ cos 2ψ,
2z2z3=2 sin2 φ cos ψ sin ψ=sin2 φ sin 2ψ.
z12-=14π02π0π(cos2 φ-)sin φdφdψ=0,
z1z2=z2z3=z2z3=14π02π0π(cos φ sin φ cos ψ)sin φdφdψ=0,
zjzk-δjk|zj(z0)zk(z0)
=[zj(z0)zk(z0)-]exp-24b2σθ2(z-z0),
R(ξ)=[zj(z0)zk(z0)-δjk]2exp-24b2σθ2ξ.
R(0)=[zj(z0)zk(z0)-δjk]2=14π02π0π(zjzk-δjk)2 sin φdφdψ.
(cos2 φ-)2=4/45,
(cos φ sin φ sin ψ)2=1/15,
(sin2 φ cos 2ψ)2=4/15,
Varz0z0+l(z12-)dζ845hE(l-hE),
Varz0z0+lzjzkdζ215hE(l-hE)
Varz0z0+lz22-z32dζ815hE(l-hE).
VarσE2lhE+exp-lhE-1,
p(u)=p(φ)sin φ dφdu=14u+1/3, -1/3<u<2/3.
α=xjxk-½yjyk+zjzk,
β=yjyk-zjzk,
γ=yjzk+ykzj,
ddzαβγ=-32σθ2¾σθ20σθ2-½σθ2-4b04b-½σθ2αβγ
zjzk=δjk-α-½β.
βγ=cos 4βξ-sin 4bξsin 4bξcos 4bξβγ,
dαdξ=-32σθ2α+¾σθ2(β cos 4bξ-γ sin 4bξ),
dβdξ=σθ2α cos 4bξ-½σθ2β,
dγdξ=-σθ2 sin 4bξ-½σθ2γ.
αα(z0)exp(-32σθ2ξ),
ββ(z0)exp(-½σθ2ξ),
γγ(z0)exp(-½σθ2ξ).
zjzk|(xl, yl, zl)z0-δjk
=-[xjxk-½(yjyk+zjzk)]z0 exp(-32σθ2ξ)-½[yjyk-zjzk]z0 exp(-½σθ2ξ)cos 4bξ+½[yjzk+ykzj]z0 exp(-½σθ2)sin 4bξ.
zjzk|(xl, yl, zl)z0-δjk
½(δjk-xjxk)z0 exp(-32σθ2ξ).
Varz0z0+l(zjzk-)dζ
=2R(0)hE2lhE+4 exp-l4hE-4,
Varz0z0+l(z12-)dζ845hE(l-4hE),
Varz0z0+lzjzkdζ215hE(l-4hE)
Varz0z0+l(z22-z32)dζ815hE(l-4hE).
hU=LB212π2hfiber+112hfiber,
Ψ¯z=-bσ¯ Ψ¯t,
Ψ¯(z0+l, t)Ψ¯(z0, t)-bz0z0+lσ¯dz Ψ¯t(z0, t).
ΔτU=bz0z0+lσ¯dzU.
Δτ2=b2j=13z0z0+lxjdz2.
τD2=4b2j=13z0z0+lxjdz2.
z0z0+lx1dz2=hE2lhE+exp-lhE-1hE(l-hE),
τD2=8b2hE2lhE+exp-lhE-1,
Pz=12σθ2S˜2 S˜1-S˜1 S˜22-2bS˜3 S˜1-S˜1 S˜3P.
S˜1=R sin ϕ cos ψ,S˜2=R sin ϕ sin ψ,
S˜3=R cos ϕ.
Pz=12σθ2 2Pψ2-2bcos ϕ cos ψsin ϕPψ+sin ψ Pϕ,
P=P0(ϕ, ψ, ζ, ζ˜)+P1(ϕ, ψ, ζ, ζ˜)+2P2(ϕ, ψ, ζ, ζ˜)+,
P0ζ=2P0ψ2,
P0=-pn(ϕ, ζ˜)exp(inψ-n2ζ).
P1ζ-2P1ψ2=-cos ϕ cos ψsin ϕP0ψ+sin ψ P0ϕf1(ϕ, ψ, ζ)=-fn(1) exp(inψ).
limζf0(k)(ϕ, ζ)=limζ 12π02πfk(ϕ, ψ, ζ)dψ=0.
-2P1ψ2=-sin ψ p0ϕ,
P1-sin ψ p0ϕ.
P2ζ-2P2ψ2=P0ζ˜-cos ϕ cos ψsin ϕP1ψ+sin ψ P1ϕf2(ϕ, ψ, ζ)=-fn(2) exp(inψ).
p0ζ˜=121sin ϕϕsin ϕ p0ϕ.
Pζ=P0ζ+2 P0ζ˜+
Pz=12σθ2 2Pψ2+4b2σθ21sin ϕϕsin ϕ Pϕ.
ddzS˜1S˜2S˜3=0gθg1-gθ0g2-g1-g20S˜1S˜2S˜3,
gj(z)=0,gj(z)gk(z)=8b2σθ2δ(z-z)δjk.
Pz=12σθ2+4b2σθ21sin2 ϕ-1 2Pψ2+4b2σθ21sin ϕϕsin ϕ Pϕ.
S˜1S˜2S˜3=1000cos 2bzsin 2bz0-sin 2bzcos 2bzR1R2R3.
ddζR1R2R3=h(ζ)0cos ζsin ζ-cos ζ00-sin ζ00R1R2R3,
dRidζ=jBij(ζ)Rj,
dPdζ=2i,j,k,lBijBklRj RiRl PRk.
dPdz=σθ24R1 R2-R2 R12+R1 R3-R3 R12P.
ddzR1R2R3=0g1g2-g100-g200R1R2R3,
gj(z)=0,gj(z)gk(z)=(σθ2/2)δ(z-z)δjk.

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