Abstract

We determine the transient evolution of the probability distribution of the polarization dispersion vector both analytically and numerically, using a physically reasonable model of the fiber birefringence. We show that, for all practical birefringence parameters, the distribution of the differential group delay (DGD), which is the magnitude of the polarization dispersion vector, becomes Maxwellian in just a few kilometers, except in the tail region, where the DGD is large. In this limit, the approach to a Maxwellian distribution takes much longer, of the order of tens of kilometers. In addition, we show that in the transient regime the DGD distribution is very different from Maxwellian. We also find that the probability-distribution function for the polarization-dispersion vector at the output of the fiber depends upon the angle between it and the local birefringence vector on the Poincaré sphere, showing that the DGD remains correlated with the orientation of the local birefringence axes over arbitrarily long distances.

© 2002 Optical Society of America

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  1. G. J. Foschini and C. D. Poole, “Statistical theory of polarization dispersion in single mode fibers,” J. Lightwave Technol. 9, 1439–1456 (1991).
    [Crossref]
  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]
  3. N. Gisin, “Solutions of the dynamic equation for polarization dispersion,” Opt. Commun. 86, 371–373 (1991).
    [Crossref]
  4. F. Curti, B. Daino, G. D. Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1170 (1990).
    [Crossref]
  5. N. Gisin, J.-P. Von der Weid, and J.-P. Pellaux, “Polarization mode dispersion of short and long single-mode fibers,” J. Lightwave Technol. 9, 821–827 (1991).
    [Crossref]
  6. 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]
  7. N. Gisin, R. Passy, P. Blasco, M. O. Van Deventer, R. Distl, H. Gilgen, B. Perny, R. Keys, E. Krause, C. C. Larsen, K. Morl, J. Pelayo, and J. Vobian, “Definition of polarization mode dispersion and first results of the COST 241 round-robin measurements,” Pure Appl. Opt. 4, 511–522 (1995).
    [Crossref]
  8. I. P. Kaminow, “Polarization in optical fibers,” IEEE J. Quantum Electron. QE-17, 15–22 (1981).
    [Crossref]
  9. R. E. Schuh, X. Shan, and A. S. Siddiqui, “Polarization mode dispersion in spun fibers with different linear birefringence and spinning parameters,” J. Lightwave Technol. 16, 1583–1588 (1998).
    [Crossref]
  10. J. Botineau and R. H. Stolen, “Effect of polarization on spectral broadening in optical fibers,” J. Opt. Soc. Am. 72, 1592–1596 (1982).
    [Crossref]
  11. M. N. Islam, “Ultrafast all-optical logic gates based on soliton trapping in fibers,” Opt. Lett. 14, 1257–1259 (1989).
    [Crossref] [PubMed]
  12. A. Galtarossa, L. Palmieri, M. Schiano, and T. Tambosso, “Statistical characterization of fiber random birefringence,” Opt. Lett. 25, 1322–1324 (2000).
    [Crossref]
  13. C. D. Poole, J. H. Winters, and J. A. Nagel, “Dynamical equation for polarization dispersion,” Opt. Lett. 16, 372–374 (1991).
    [Crossref] [PubMed]
  14. L. Arnold, Stochastic Differential Equations, Theory and Applications (Wiley, New York, 1974).
  15. J. W. Zhang, P. K. A. Wai, W. L. Kath, and C. R. Menyuk, “Nonlinear polarization-mode dispersion in optical fibers with randomly varying birefringence,” J. Opt. Soc. Am. B 16, 2967–2979 (1997).
  16. A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1973).
  17. G. Strang, “On the construction and comparison of difference schemes,” SIAM J. Numer. Anal. 5, 506–517 (1968).
    [Crossref]
  18. D. R. Durran, Numerical Methods for Wave Equations in Geophysical Fluid Dynamics (Springer-Verlag, New York, 1999).
  19. H. Ritchie, C. Temperton, A. Simmons, M. Hortal, T. Davies, D. Dent, and M. Hamrud, “Implementation of the semi-Lagrangian method in a high-resolution version of the ECMWF model,” Mon. Weather Rev. 123, 489–514 (1995).
    [Crossref]
  20. A. J. Barlow, D. N. Payne, M. R. Hadley, and R. J. Mansfield, “Production of single-mode fibers with negligible intrinsic birefringence and polarization mode dispersion,” Electron. Lett. 17, 725–726 (1981).
    [Crossref]
  21. 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]
  22. M. J. Li and D. A. Nolan, “Fiber spin-profile designs for producing fibers with low polarization mode dispersion,” Opt. Lett. 23, 1659–1662 (1998).
    [Crossref]
  23. R. E. Schuh, X. Shan, and A. S. Siddiqui, “Polarization mode dispersion in spun fibers with different linear birefringence and spinning parameters,” J. Lightwave Technol. 16, 1583–1588 (1998).
    [Crossref]
  24. T. Ueda and W. L. Kath, “Dynamics of optical pulses in randomly birefringent fibers,” Physica D 55, 166–181 (1992).
    [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]

A. Galtarossa, L. Palmieri, M. Schiano, and T. Tambosso, “Statistical characterization of fiber random birefringence,” Opt. Lett. 25, 1322–1324 (2000).
[Crossref]

1998 (3)

1997 (1)

J. W. Zhang, P. K. A. Wai, W. L. Kath, and C. R. Menyuk, “Nonlinear polarization-mode dispersion in optical fibers with randomly varying birefringence,” J. Opt. Soc. Am. B 16, 2967–2979 (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]

1995 (2)

N. Gisin, R. Passy, P. Blasco, M. O. Van Deventer, R. Distl, H. Gilgen, B. Perny, R. Keys, E. Krause, C. C. Larsen, K. Morl, J. Pelayo, and J. Vobian, “Definition of polarization mode dispersion and first results of the COST 241 round-robin measurements,” Pure Appl. Opt. 4, 511–522 (1995).
[Crossref]

H. Ritchie, C. Temperton, A. Simmons, M. Hortal, T. Davies, D. Dent, and M. Hamrud, “Implementation of the semi-Lagrangian method in a high-resolution version of the ECMWF model,” Mon. Weather Rev. 123, 489–514 (1995).
[Crossref]

1992 (1)

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

1991 (4)

C. D. Poole, J. H. Winters, and J. A. Nagel, “Dynamical equation for polarization dispersion,” Opt. Lett. 16, 372–374 (1991).
[Crossref] [PubMed]

N. Gisin, “Solutions of the dynamic equation for polarization dispersion,” Opt. Commun. 86, 371–373 (1991).
[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. Gisin, J.-P. Von der Weid, and J.-P. Pellaux, “Polarization mode dispersion of short and long single-mode fibers,” J. Lightwave Technol. 9, 821–827 (1991).
[Crossref]

1990 (1)

F. Curti, B. Daino, G. D. Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1170 (1990).
[Crossref]

1989 (1)

1982 (1)

1981 (3)

I. P. Kaminow, “Polarization in optical fibers,” IEEE J. Quantum Electron. QE-17, 15–22 (1981).
[Crossref]

A. J. Barlow, D. N. Payne, M. R. Hadley, and R. J. Mansfield, “Production of single-mode fibers with negligible intrinsic birefringence and polarization mode dispersion,” Electron. Lett. 17, 725–726 (1981).
[Crossref]

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]

1968 (1)

G. Strang, “On the construction and comparison of difference schemes,” SIAM J. Numer. Anal. 5, 506–517 (1968).
[Crossref]

Arnold, L.

L. Arnold, Stochastic Differential Equations, Theory and Applications (Wiley, New York, 1974).

Barlow, A. J.

A. J. Barlow, D. N. Payne, M. R. Hadley, and R. J. Mansfield, “Production of single-mode fibers with negligible intrinsic birefringence and polarization mode dispersion,” Electron. Lett. 17, 725–726 (1981).
[Crossref]

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]

Blasco, P.

N. Gisin, R. Passy, P. Blasco, M. O. Van Deventer, R. Distl, H. Gilgen, B. Perny, R. Keys, E. Krause, C. C. Larsen, K. Morl, J. Pelayo, and J. Vobian, “Definition of polarization mode dispersion and first results of the COST 241 round-robin measurements,” Pure Appl. Opt. 4, 511–522 (1995).
[Crossref]

Botineau, J.

Curti, F.

F. Curti, B. Daino, G. D. Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1170 (1990).
[Crossref]

Daino, B.

F. Curti, B. Daino, G. D. Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1170 (1990).
[Crossref]

Davies, T.

H. Ritchie, C. Temperton, A. Simmons, M. Hortal, T. Davies, D. Dent, and M. Hamrud, “Implementation of the semi-Lagrangian method in a high-resolution version of the ECMWF model,” Mon. Weather Rev. 123, 489–514 (1995).
[Crossref]

Dent, D.

H. Ritchie, C. Temperton, A. Simmons, M. Hortal, T. Davies, D. Dent, and M. Hamrud, “Implementation of the semi-Lagrangian method in a high-resolution version of the ECMWF model,” Mon. Weather Rev. 123, 489–514 (1995).
[Crossref]

Distl, R.

N. Gisin, R. Passy, P. Blasco, M. O. Van Deventer, R. Distl, H. Gilgen, B. Perny, R. Keys, E. Krause, C. C. Larsen, K. Morl, J. Pelayo, and J. Vobian, “Definition of polarization mode dispersion and first results of the COST 241 round-robin measurements,” Pure Appl. Opt. 4, 511–522 (1995).
[Crossref]

Durran, D. R.

D. R. Durran, Numerical Methods for Wave Equations in Geophysical Fluid Dynamics (Springer-Verlag, New York, 1999).

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]

Galtarossa, A.

Gilgen, H.

N. Gisin, R. Passy, P. Blasco, M. O. Van Deventer, R. Distl, H. Gilgen, B. Perny, R. Keys, E. Krause, C. C. Larsen, K. Morl, J. Pelayo, and J. Vobian, “Definition of polarization mode dispersion and first results of the COST 241 round-robin measurements,” Pure Appl. Opt. 4, 511–522 (1995).
[Crossref]

Gisin, N.

N. Gisin, R. Passy, P. Blasco, M. O. Van Deventer, R. Distl, H. Gilgen, B. Perny, R. Keys, E. Krause, C. C. Larsen, K. Morl, J. Pelayo, and J. Vobian, “Definition of polarization mode dispersion and first results of the COST 241 round-robin measurements,” Pure Appl. Opt. 4, 511–522 (1995).
[Crossref]

N. Gisin, “Solutions of the dynamic equation for polarization dispersion,” Opt. Commun. 86, 371–373 (1991).
[Crossref]

N. Gisin, J.-P. Von der Weid, and J.-P. Pellaux, “Polarization mode dispersion of short and long single-mode fibers,” J. Lightwave Technol. 9, 821–827 (1991).
[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]

Hadley, M. R.

A. J. Barlow, D. N. Payne, M. R. Hadley, and R. J. Mansfield, “Production of single-mode fibers with negligible intrinsic birefringence and polarization mode dispersion,” Electron. Lett. 17, 725–726 (1981).
[Crossref]

Hamrud, M.

H. Ritchie, C. Temperton, A. Simmons, M. Hortal, T. Davies, D. Dent, and M. Hamrud, “Implementation of the semi-Lagrangian method in a high-resolution version of the ECMWF model,” Mon. Weather Rev. 123, 489–514 (1995).
[Crossref]

Hortal, M.

H. Ritchie, C. Temperton, A. Simmons, M. Hortal, T. Davies, D. Dent, and M. Hamrud, “Implementation of the semi-Lagrangian method in a high-resolution version of the ECMWF model,” Mon. Weather Rev. 123, 489–514 (1995).
[Crossref]

Islam, M. N.

Kaminow, I. P.

I. P. Kaminow, “Polarization in optical fibers,” IEEE J. Quantum Electron. QE-17, 15–22 (1981).
[Crossref]

Kath, W. L.

J. W. Zhang, P. K. A. Wai, W. L. Kath, and C. R. Menyuk, “Nonlinear polarization-mode dispersion in optical fibers with randomly varying birefringence,” J. Opt. Soc. Am. B 16, 2967–2979 (1997).

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

Keys, R.

N. Gisin, R. Passy, P. Blasco, M. O. Van Deventer, R. Distl, H. Gilgen, B. Perny, R. Keys, E. Krause, C. C. Larsen, K. Morl, J. Pelayo, and J. Vobian, “Definition of polarization mode dispersion and first results of the COST 241 round-robin measurements,” Pure Appl. Opt. 4, 511–522 (1995).
[Crossref]

Kogelnik, H.

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]

Krause, E.

N. Gisin, R. Passy, P. Blasco, M. O. Van Deventer, R. Distl, H. Gilgen, B. Perny, R. Keys, E. Krause, C. C. Larsen, K. Morl, J. Pelayo, and J. Vobian, “Definition of polarization mode dispersion and first results of the COST 241 round-robin measurements,” Pure Appl. Opt. 4, 511–522 (1995).
[Crossref]

Larsen, C. C.

N. Gisin, R. Passy, P. Blasco, M. O. Van Deventer, R. Distl, H. Gilgen, B. Perny, R. Keys, E. Krause, C. C. Larsen, K. Morl, J. Pelayo, and J. Vobian, “Definition of polarization mode dispersion and first results of the COST 241 round-robin measurements,” Pure Appl. Opt. 4, 511–522 (1995).
[Crossref]

Li, M. J.

Mansfield, R. J.

A. J. Barlow, D. N. Payne, M. R. Hadley, and R. J. Mansfield, “Production of single-mode fibers with negligible intrinsic birefringence and polarization mode dispersion,” Electron. Lett. 17, 725–726 (1981).
[Crossref]

Marchis, G. D.

F. Curti, B. Daino, G. D. Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1170 (1990).
[Crossref]

Matera, F.

F. Curti, B. Daino, G. D. Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1170 (1990).
[Crossref]

Menyuk, C. R.

J. W. Zhang, P. K. A. Wai, W. L. Kath, and C. R. Menyuk, “Nonlinear polarization-mode dispersion in optical fibers with randomly varying birefringence,” J. Opt. Soc. Am. B 16, 2967–2979 (1997).

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]

Morl, K.

N. Gisin, R. Passy, P. Blasco, M. O. Van Deventer, R. Distl, H. Gilgen, B. Perny, R. Keys, E. Krause, C. C. Larsen, K. Morl, J. Pelayo, and J. Vobian, “Definition of polarization mode dispersion and first results of the COST 241 round-robin measurements,” Pure Appl. Opt. 4, 511–522 (1995).
[Crossref]

Nagel, J. A.

Nayfeh, A. H.

A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1973).

Nolan, D. A.

Palmieri, L.

Passy, R.

N. Gisin, R. Passy, P. Blasco, M. O. Van Deventer, R. Distl, H. Gilgen, B. Perny, R. Keys, E. Krause, C. C. Larsen, K. Morl, J. Pelayo, and J. Vobian, “Definition of polarization mode dispersion and first results of the COST 241 round-robin measurements,” Pure Appl. Opt. 4, 511–522 (1995).
[Crossref]

Payne, D. N.

A. J. Barlow, D. N. Payne, M. R. Hadley, and R. J. Mansfield, “Production of single-mode fibers with negligible intrinsic birefringence and polarization mode dispersion,” Electron. Lett. 17, 725–726 (1981).
[Crossref]

Pelayo, J.

N. Gisin, R. Passy, P. Blasco, M. O. Van Deventer, R. Distl, H. Gilgen, B. Perny, R. Keys, E. Krause, C. C. Larsen, K. Morl, J. Pelayo, and J. Vobian, “Definition of polarization mode dispersion and first results of the COST 241 round-robin measurements,” Pure Appl. Opt. 4, 511–522 (1995).
[Crossref]

Pellaux, J.-P.

N. Gisin, J.-P. Von der Weid, and J.-P. Pellaux, “Polarization mode dispersion of short and long single-mode fibers,” J. Lightwave Technol. 9, 821–827 (1991).
[Crossref]

Perny, B.

N. Gisin, R. Passy, P. Blasco, M. O. Van Deventer, R. Distl, H. Gilgen, B. Perny, R. Keys, E. Krause, C. C. Larsen, K. Morl, J. Pelayo, and J. Vobian, “Definition of polarization mode dispersion and first results of the COST 241 round-robin measurements,” Pure Appl. Opt. 4, 511–522 (1995).
[Crossref]

Poole, C. D.

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

C. D. Poole, J. H. Winters, and J. A. Nagel, “Dynamical equation for polarization dispersion,” Opt. Lett. 16, 372–374 (1991).
[Crossref] [PubMed]

Ramskov-Ha, J. J.

Ritchie, H.

H. Ritchie, C. Temperton, A. Simmons, M. Hortal, T. Davies, D. Dent, and M. Hamrud, “Implementation of the semi-Lagrangian method in a high-resolution version of the ECMWF model,” Mon. Weather Rev. 123, 489–514 (1995).
[Crossref]

Schiano, M.

Schuh, R. E.

Shan, X.

Siddiqui, A. S.

Simmons, A.

H. Ritchie, C. Temperton, A. Simmons, M. Hortal, T. Davies, D. Dent, and M. Hamrud, “Implementation of the semi-Lagrangian method in a high-resolution version of the ECMWF model,” Mon. Weather Rev. 123, 489–514 (1995).
[Crossref]

Stolen, R. H.

Strang, G.

G. Strang, “On the construction and comparison of difference schemes,” SIAM J. Numer. Anal. 5, 506–517 (1968).
[Crossref]

Tambosso, T.

Temperton, C.

H. Ritchie, C. Temperton, A. Simmons, M. Hortal, T. Davies, D. Dent, and M. Hamrud, “Implementation of the semi-Lagrangian method in a high-resolution version of the ECMWF model,” Mon. Weather Rev. 123, 489–514 (1995).
[Crossref]

Ueda, T.

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

Van Deventer, M. O.

N. Gisin, R. Passy, P. Blasco, M. O. Van Deventer, R. Distl, H. Gilgen, B. Perny, R. Keys, E. Krause, C. C. Larsen, K. Morl, J. Pelayo, and J. Vobian, “Definition of polarization mode dispersion and first results of the COST 241 round-robin measurements,” Pure Appl. Opt. 4, 511–522 (1995).
[Crossref]

Vobian, J.

N. Gisin, R. Passy, P. Blasco, M. O. Van Deventer, R. Distl, H. Gilgen, B. Perny, R. Keys, E. Krause, C. C. Larsen, K. Morl, J. Pelayo, and J. Vobian, “Definition of polarization mode dispersion and first results of the COST 241 round-robin measurements,” Pure Appl. Opt. 4, 511–522 (1995).
[Crossref]

Von der Weid, J.-P.

N. Gisin, J.-P. Von der Weid, and J.-P. Pellaux, “Polarization mode dispersion of short and long single-mode fibers,” J. Lightwave Technol. 9, 821–827 (1991).
[Crossref]

Wai, P. K. A.

J. W. Zhang, P. K. A. Wai, W. L. Kath, and C. R. Menyuk, “Nonlinear polarization-mode dispersion in optical fibers with randomly varying birefringence,” J. Opt. Soc. Am. B 16, 2967–2979 (1997).

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]

Winters, J. H.

Zhang, J. W.

J. W. Zhang, P. K. A. Wai, W. L. Kath, and C. R. Menyuk, “Nonlinear polarization-mode dispersion in optical fibers with randomly varying birefringence,” J. Opt. Soc. Am. B 16, 2967–2979 (1997).

Appl. Opt. (1)

Electron. Lett. (1)

A. J. Barlow, D. N. Payne, M. R. Hadley, and R. J. Mansfield, “Production of single-mode fibers with negligible intrinsic birefringence and polarization mode dispersion,” Electron. Lett. 17, 725–726 (1981).
[Crossref]

IEEE J. Quantum Electron. (1)

I. P. Kaminow, “Polarization in optical fibers,” IEEE J. Quantum Electron. QE-17, 15–22 (1981).
[Crossref]

J. Lightwave Technol. (6)

R. E. Schuh, X. Shan, and A. S. Siddiqui, “Polarization mode dispersion in spun fibers with different linear birefringence and spinning parameters,” J. Lightwave Technol. 16, 1583–1588 (1998).
[Crossref]

F. Curti, B. Daino, G. D. Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1170 (1990).
[Crossref]

N. Gisin, J.-P. Von der Weid, and J.-P. Pellaux, “Polarization mode dispersion of short and long single-mode fibers,” J. Lightwave Technol. 9, 821–827 (1991).
[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]

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

R. E. Schuh, X. Shan, and A. S. Siddiqui, “Polarization mode dispersion in spun fibers with different linear birefringence and spinning parameters,” J. Lightwave Technol. 16, 1583–1588 (1998).
[Crossref]

J. Opt. Soc. Am. (1)

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

J. W. Zhang, P. K. A. Wai, W. L. Kath, and C. R. Menyuk, “Nonlinear polarization-mode dispersion in optical fibers with randomly varying birefringence,” J. Opt. Soc. Am. B 16, 2967–2979 (1997).

Mon. Weather Rev. (1)

H. Ritchie, C. Temperton, A. Simmons, M. Hortal, T. Davies, D. Dent, and M. Hamrud, “Implementation of the semi-Lagrangian method in a high-resolution version of the ECMWF model,” Mon. Weather Rev. 123, 489–514 (1995).
[Crossref]

Opt. Commun. (1)

N. Gisin, “Solutions of the dynamic equation for polarization dispersion,” Opt. Commun. 86, 371–373 (1991).
[Crossref]

Opt. Lett. (4)

Physica D (1)

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

Pure Appl. Opt. (1)

N. Gisin, R. Passy, P. Blasco, M. O. Van Deventer, R. Distl, H. Gilgen, B. Perny, R. Keys, E. Krause, C. C. Larsen, K. Morl, J. Pelayo, and J. Vobian, “Definition of polarization mode dispersion and first results of the COST 241 round-robin measurements,” Pure Appl. Opt. 4, 511–522 (1995).
[Crossref]

SIAM J. Numer. Anal. (1)

G. Strang, “On the construction and comparison of difference schemes,” SIAM J. Numer. Anal. 5, 506–517 (1968).
[Crossref]

Other (3)

D. R. Durran, Numerical Methods for Wave Equations in Geophysical Fluid Dynamics (Springer-Verlag, New York, 1999).

A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1973).

L. Arnold, Stochastic Differential Equations, Theory and Applications (Wiley, New York, 1974).

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Figures (4)

Fig. 1
Fig. 1

(a)–(g) Probability-distribution function p(τ, Z) of the differential group delay at various distances when β1. The dotted curves in (f) and (g) are Maxwellian distributions. The distance is normalized to the fiber correlation length hfiber, and the DGD is normalized to 2bhfiber. (h) Contour plot showing the angular dependence of the probability distribution P for the dispersion vector Ω¯ at Z=100. The contour levels have been magnified 40 000 times.

Fig. 2
Fig. 2

Comparison of the true DGD probability distribution (solid curves) with a Maxwellian approximation (dashed curves) at various distances with a log scale to emphasize the behavior in the tails of the distributions. Differences at large DGD values are clearly seen, showing that hundreds of correlation lengths can be required for the distribution’s tail to become Maxwellian.

Fig. 3
Fig. 3

Simulation result for the full Fokker–Planck equation (6) with β=10 and Z=30. (a) DGD probability distribution p(τ) (squares). Also plotted in this figure is the same quantity from simulation of the reduced Fokker–Planck equation (16) (solid curves) for comparison [see Fig. 1(g)]. (b) Contour plot of P(τ, Θ, Φ) at τ=τmax=6.5, where p(τ) reaches maximum [see (a)]. The contour levels have been magnified 10 000 times.

Fig. 4
Fig. 4

DGD probability distribution p(τ, Z) at various distances for β=1. In (c) and (d), the Maxwellian distribution (31) (dashed curves) is also shown for comparison.

Equations (42)

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Ω(z, ω)z=W(z, ω)ω+W(z, ω)×Ω(z, ω),
gθ(z)=0,gθ(z)gθ(z)=σ2δ(z-z).
H1(z)=cos θsin θ0-sin θcos θ0001,
z Ω˜1Ω˜2Ω˜3=Ω˜2-Ω˜10g(z)+2b-2bΩ˜32bΩ˜2.
Pz+2b PΩ˜1-2bΩ˜3 PΩ˜2-Ω˜2 PΩ˜3
-12σ2Ω˜2 Ω˜1-Ω˜1 Ω˜22P=0.
Z=12σ2z=zhfiber,β=4bσ2=4πhfiberLB,
Ω˜k=Ω˜k4b/σ2=Ω˜k2bhfiber(k=1, 2, 3),
PZ+PΩ˜1-βΩ˜3 PΩ˜2-Ω¯2 PΩ˜3
-Ω˜2 Ω˜1-Ω˜1 Ω˜22P=0.
---PdΩ¯1dΩ¯2dΩ¯3=1,
|Ω|2=|Ω¯|2=2[exp(-Z)+Z-1],
H2(z)=1000cos 2bzsin 2bz0-sin 2bzcos 2bz,
z Ωˆ1Ωˆ2Ωˆ3=g(z)Ωˆ2 cos 2bz-Ωˆ3 sin 2bz-Ωˆ1 cos 2bzΩˆ1 sin 2bz+2b00.
Pz=12σ2(Ωˆ2 cos 2bz-Ωˆ3 sin 2bz)2PΩˆ1Ωˆ1+Ωˆ12(cos2 2bzPΩˆ2Ωˆ2+sin2 2bzPΩˆ3Ωˆ3-2(Ωˆ1Ωˆ2P)Ωˆ1Ωˆ2 cos2 2bz-2(Ωˆ1Ωˆ3P)Ωˆ1Ωˆ3 sin2 2bz+sin 4bz[(Ωˆ1Ωˆ3P)Ωˆ1Ωˆ2+(Ωˆ1Ωˆ2P)Ω¯1Ωˆ3-Ωˆ12PΩˆ2ωˆ3]+(Ωˆ1P)Ωˆ1+cos2 2bz(Ωˆ2P)Ωˆ2+sin2 2bz(Ωˆ3P)Ωˆ3-12 sin 4bz(Ωˆ3PΩˆ2+Ωˆ2PΩˆ3)-2bPΩˆ1.
Pz=σ24 Ωˆ1 Ωˆ2-Ωˆ2 Ωˆ12+Ωˆ1 Ωˆ3-Ωˆ3 Ωˆ12P-2b PΩˆ1.
PZ=12 Ωˇ1 Ωˇ2-Ωˇ2 Ωˇ12+Ωˇ1 Ωˇ3-Ωˇ3 Ωˇ12P-PΩˇ1.
Ωˇ1=τ cos ϕ,Ωˇ2=τ sin ϕ cos ψ,Ωˇ3=τ sin ϕ sin ψ,
PZ=12 sin ϕ ϕ sin ϕ Pϕ+12 1sin2 ϕ-1 2Pψ2-cos ϕ Pτ-sin ϕτ Pϕ.
PZ=12 sin ϕ ϕ sin ϕ Pϕ-cos ϕ Pτ-sin ϕτ Pϕ.
PZ=12 sin ϕ ϕ sin ϕ Pϕ-cos ϕ Pτˆ-sin ϕτˆ Pϕ.
P=P0(τˆ, ϕ, Z, Z2)+P1(τˆ, ϕ, Z, Z2)+2P2+,
P0Z=12 sin ϕ ϕ sin ϕ P0ϕ.
P0=n=0P0n(τˆ, Z2)exp[-n(n+1)Z/2]Ln(cos ϕ),
P1Z-12 sin ϕ ϕ sin ϕ P1ϕ=-cos ϕ P0τˆ.
0π sin ϕ cos ϕ P0τˆdϕ=0.
P1=-cos ϕ P0τˆ.
P2Z-12 sin ϕ ϕ sin ϕ P2ϕ
=-P0Z2+cos ϕ P1τˆ-sin ϕτˆ P1ϕ.
0πP0Z2+cos ϕ P1τˆ-sin ϕτˆ P1ϕsin ϕdϕ=0.
P0Z2=13 2P0τˆ2+2τˆ P0τˆ.
P0Z=13 2P0τ2+2τ P0τ.
P0(τ, Z)=34πZ3/2 exp-3τ24Z,Z1.
P(τ, ϕ, Z)P0(τ, Z)-cos ϕ P0(τ, Z)τ,Z1,
p(τ, Z)=2πτ20πP sin ϕdϕ.
p(τ, Z)33τ22πZ3/2 exp-3τ24Z=pM(τ, Z),Z1.
P(τ, ϕ, Z)P0(τ-cos ϕ, Z),Z1,
P|Z=0=D3π3/2 exp(-D2|Ω¯|2),
Ω¯1=τ sin Θ cos Φ,Ω¯2=τ sin Θ sin Φ,Ω¯3=τ cos Θ.
PZ=2PΦ2.
Φ×Θ×τ=[0, 2π)×[0, π]×[0, τ1],
p(τ, Z)=02π0πPτ2 sin ΘdΘdΦ,

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