Abstract

A vector theory of the stimulated Raman scattering process is developed for describing the polarization effects in fiber-based Raman amplifiers. We use this theory to show that polarization-mode dispersion (PMD) induces large fluctuations in an amplified signal. It is found that PMD-induced fluctuations follow a log-normal distribution. We also discuss the random nature of the polarization-dependent gain (PDG) in Raman amplifiers. Using the concept of a PDG vector, we find the probability distribution of PDG in an analytic form and use it to show that both the mean and the standard deviation of PDG depend on the PMD parameter inversely when the effective fiber length is much larger than the PMD diffusion length. We apply our theory to study how PDG can be reduced by scrambling pump polarization randomly and show that the mean value of PDG is directly proportional to the degree of pump polarization.

© 2003 Optical Society of America

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References

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  1. C. V. Raman, “A new radiation,” Indian J. Phys. 2, 387–398 (1928).
  2. E. J. Woodbury and W. K. Ng, “Ruby laser operation in the near IR,” Proc. IRE 50, 2347 (1962).
  3. R. H. Stolen, E. P. Ippen, and A. R. Tynes, “Raman oscillation in glass optical waveguide,” Appl. Phys. Lett. 20, 62–64 (1972).
    [CrossRef]
  4. M. Ikeda, “Stimulated Raman amplification characteristics in long span single-mode silica fibers,” Opt. Commun. 39, 148–152 (1981).
    [CrossRef]
  5. S. Namiki and Y. Emori, “Ultrabroad-band Raman amplifiers pumped and gain-equalized by wavelength-division-multiplexed high-power laser diodes,” IEEE J. Sel. Top. Quantum Electron. 7, 3–16 (2001).
    [CrossRef]
  6. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, San Diego, Calif., 2001).
  7. K. Rottwitt and A. J. Stentz, “Raman amplifiers in lightwave communication systems,” in Optical Fiber Telecommunications IV-A: Components, I. P. Kaminow and T. Li, eds. (Academic, San Diego, Calif., 2002), Chap. 5.
  8. R. H. Stolen, “Polarization effects in fiber Raman and Brillouin lasers,” IEEE J. Quantum Electron. QE-15, 1157–1159 (1979).
    [CrossRef]
  9. D. J. Dougherty, F. X. Kartner, H. A. Haus, and E. P. Ippen, “Measurement of the Raman gain spectrum of optical fibers,” Opt. Lett. 20, 31–33 (1995).
    [CrossRef] [PubMed]
  10. D. Mahgerefteh, H. Yu, D. L. Butler, J. Goldhar, D. Wang, E. Golovchenko, A. N. Phlipetskii, C. R. Menyuk, and L. Joneckis, “Effect of randomly varying birefringence on the Raman gain in optical fibers,” in Conference on Lasers and Electro-Optics, Vol. 11 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), p. 447.
  11. A. Berntson, S. Popov, E. Vanin, G. Jacobsen, and J. Karlsson, “Polarization dependence and gain tilt of Raman amplifiers for WDM systems,” in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), pages MI2–1.
  12. P. Ebrahimi, M. C. Hauer, Q. Yu, R. Khosravani, D. Gurkan, D. W. Kim, D. W. Lee, and A. E. Willner, “Statistics of polarization dependent gain in Raman fiber amplifiers due to PMD,” in Conference on Lasers and Electro-Optics, Vol. 56 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), pp. 143–144.
  13. S. Popov, E. Vanin, and G. Jacobsen, “Influence of polarization mode dispersion value in dispersion-compensating fibers on the polarization dependence of Raman gain,” Opt. Lett. 27, 848–850 (2002).
    [CrossRef]
  14. J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
    [CrossRef] [PubMed]
  15. S. J. Savory and F. P. Payne, “Pulse propagation in fibers with polarization-mode dispersion,” J. Lightwave Technol. 19, 350–357 (2001).
    [CrossRef]
  16. D. Wang and C. R. Menyuk, “Calculation of penalties due to polarization effects in a long-haul WDM system using a Stokes parameter model,” J. Lightwave Technol. 19, 487–494 (2001).
    [CrossRef]
  17. H. Kogelnik, R. M. Jopson, and L. E. Nelson, “Polarization-mode dispersion,” in Optical Fiber Telecommunications IV-B: Systems and Impairments, I. P. Kaminow and T. Li, eds. (Academic, San Diego, Calif., 2002), Chap. 15.
  18. Q. Lin and G. P. Agrawal, “Polarization mode dispersion-induced fluctuations during Raman amplification in optical fibers,” Opt. Lett. 27, 2194–2196 (2002).
    [CrossRef]
  19. Q. Lin and G. P. Agrawal, “Statistics of polarization-dependent gain in fiber-based Raman amplifiers,” Opt. Lett. 27, 227–229 (2003).
    [CrossRef]
  20. R. Hellwarth, J. Cherlow, and T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964–967 (1975).
    [CrossRef]
  21. R. W. Hellwarth, “Third-order optical susceptibilities of liquid and solids,” Prog. Quantum Electron. 5, 1–68 (1977).
    [CrossRef]
  22. P. Ciprut, B. Gisin, N. Gisin, R. Passy, J. P. Von der Weid, F. Prieto, and C. W. Zimmer, “Second-order polarization mode dispersion: impact on analog and digital transmissions,” J. Lightwave Technol. 16, 757–771 (1998).
    [CrossRef]
  23. R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, San Diego, Calif., 1992).
  24. C. W. Gardiner, Handbook of Stochastic Methods, 2nd ed. (Springer-Verlag, New York, 1985).
  25. H. H. Kee, C. R. S. Fludger, and V. Handerek, “Statistical properties of polarization dependent gain in fiber Raman amplifiers,” in Optical Fiber Communication, Vol. 70 of 2002 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2002), pp. 180–181.
  26. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (WCB/McGraw-Hill, New York, 1991).
  27. B. Huttner, C. Geiser, and N. Gisin, “Polarization-induced distortions in optical fiber networks with polarization-mode dispersion and polarization-dependent losses,” IEEE J. Sel. Top. Quantum Electron. 6, 317–329 (2000).
    [CrossRef]
  28. A. Mecozzi and M. Shtaif, “The statistics of polarization-dependent loss in optical communication systems,” IEEE Photon. Technol. Lett. 14, 313–315 (2002).
    [CrossRef]
  29. G. J. Foschini and C. D. Poole, “Statistical theory of polarization dispersion in single mode fibers,” J. Lightwave Technol. 9, 1439–1456 (1991).
    [CrossRef]
  30. Y. Emori, S. Matsushita, and S. Namiki, “Cost-effective depolarized diode pump unit designed for C-band flat-gain Raman amplifiers to control EDFA gain profile,” in Optical Fiber Communication Conference, Vol. 37 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 106–107.
  31. J. Zhang, V. Dominic, M. Missey, S. Sanders, and D. Mehuys, “Dependence of Raman polarization dependent gain on pump degree of polarization at high gain levels,” in Optical Amplifiers and Their Applications, A. Mecozzi, M. Shimizu, and J. Zyskind, eds., Vol. 44 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000), pp. 30–31.
  32. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, New York, 1995), Chap. 6.
  33. 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]

2003

Q. Lin and G. P. Agrawal, “Statistics of polarization-dependent gain in fiber-based Raman amplifiers,” Opt. Lett. 27, 227–229 (2003).
[CrossRef]

2002

2001

2000

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

B. Huttner, C. Geiser, and N. Gisin, “Polarization-induced distortions in optical fiber networks with polarization-mode dispersion and polarization-dependent losses,” IEEE J. Sel. Top. Quantum Electron. 6, 317–329 (2000).
[CrossRef]

1998

1996

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

1991

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

1981

M. Ikeda, “Stimulated Raman amplification characteristics in long span single-mode silica fibers,” Opt. Commun. 39, 148–152 (1981).
[CrossRef]

1979

R. H. Stolen, “Polarization effects in fiber Raman and Brillouin lasers,” IEEE J. Quantum Electron. QE-15, 1157–1159 (1979).
[CrossRef]

1977

R. W. Hellwarth, “Third-order optical susceptibilities of liquid and solids,” Prog. Quantum Electron. 5, 1–68 (1977).
[CrossRef]

1975

R. Hellwarth, J. Cherlow, and T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964–967 (1975).
[CrossRef]

1972

R. H. Stolen, E. P. Ippen, and A. R. Tynes, “Raman oscillation in glass optical waveguide,” Appl. Phys. Lett. 20, 62–64 (1972).
[CrossRef]

1962

E. J. Woodbury and W. K. Ng, “Ruby laser operation in the near IR,” Proc. IRE 50, 2347 (1962).

1928

C. V. Raman, “A new radiation,” Indian J. Phys. 2, 387–398 (1928).

Agrawal, G. P.

Q. Lin and G. P. Agrawal, “Statistics of polarization-dependent gain in fiber-based Raman amplifiers,” Opt. Lett. 27, 227–229 (2003).
[CrossRef]

Q. Lin and G. P. Agrawal, “Polarization mode dispersion-induced fluctuations during Raman amplification in optical fibers,” Opt. Lett. 27, 2194–2196 (2002).
[CrossRef]

Cherlow, J.

R. Hellwarth, J. Cherlow, and T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964–967 (1975).
[CrossRef]

Ciprut, P.

Dougherty, D. J.

Emori, Y.

S. Namiki and Y. Emori, “Ultrabroad-band Raman amplifiers pumped and gain-equalized by wavelength-division-multiplexed high-power laser diodes,” IEEE J. Sel. Top. Quantum Electron. 7, 3–16 (2001).
[CrossRef]

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]

Geiser, C.

B. Huttner, C. Geiser, and N. Gisin, “Polarization-induced distortions in optical fiber networks with polarization-mode dispersion and polarization-dependent losses,” IEEE J. Sel. Top. Quantum Electron. 6, 317–329 (2000).
[CrossRef]

Gisin, B.

Gisin, N.

B. Huttner, C. Geiser, and N. Gisin, “Polarization-induced distortions in optical fiber networks with polarization-mode dispersion and polarization-dependent losses,” IEEE J. Sel. Top. Quantum Electron. 6, 317–329 (2000).
[CrossRef]

P. Ciprut, B. Gisin, N. Gisin, R. Passy, J. P. Von der Weid, F. Prieto, and C. W. Zimmer, “Second-order polarization mode dispersion: impact on analog and digital transmissions,” J. Lightwave Technol. 16, 757–771 (1998).
[CrossRef]

Gordon, J. P.

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

Haus, H. A.

Hellwarth, R.

R. Hellwarth, J. Cherlow, and T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964–967 (1975).
[CrossRef]

Hellwarth, R. W.

R. W. Hellwarth, “Third-order optical susceptibilities of liquid and solids,” Prog. Quantum Electron. 5, 1–68 (1977).
[CrossRef]

Huttner, B.

B. Huttner, C. Geiser, and N. Gisin, “Polarization-induced distortions in optical fiber networks with polarization-mode dispersion and polarization-dependent losses,” IEEE J. Sel. Top. Quantum Electron. 6, 317–329 (2000).
[CrossRef]

Ikeda, M.

M. Ikeda, “Stimulated Raman amplification characteristics in long span single-mode silica fibers,” Opt. Commun. 39, 148–152 (1981).
[CrossRef]

Ippen, E. P.

D. J. Dougherty, F. X. Kartner, H. A. Haus, and E. P. Ippen, “Measurement of the Raman gain spectrum of optical fibers,” Opt. Lett. 20, 31–33 (1995).
[CrossRef] [PubMed]

R. H. Stolen, E. P. Ippen, and A. R. Tynes, “Raman oscillation in glass optical waveguide,” Appl. Phys. Lett. 20, 62–64 (1972).
[CrossRef]

Jacobsen, G.

Kartner, F. X.

Kogelnik, H.

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

Lin, Q.

Q. Lin and G. P. Agrawal, “Statistics of polarization-dependent gain in fiber-based Raman amplifiers,” Opt. Lett. 27, 227–229 (2003).
[CrossRef]

Q. Lin and G. P. Agrawal, “Polarization mode dispersion-induced fluctuations during Raman amplification in optical fibers,” Opt. Lett. 27, 2194–2196 (2002).
[CrossRef]

Mecozzi, A.

A. Mecozzi and M. Shtaif, “The statistics of polarization-dependent loss in optical communication systems,” IEEE Photon. Technol. Lett. 14, 313–315 (2002).
[CrossRef]

Menyuk, C. R.

D. Wang and C. R. Menyuk, “Calculation of penalties due to polarization effects in a long-haul WDM system using a Stokes parameter model,” J. Lightwave Technol. 19, 487–494 (2001).
[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]

Namiki, S.

S. Namiki and Y. Emori, “Ultrabroad-band Raman amplifiers pumped and gain-equalized by wavelength-division-multiplexed high-power laser diodes,” IEEE J. Sel. Top. Quantum Electron. 7, 3–16 (2001).
[CrossRef]

Ng, W. K.

E. J. Woodbury and W. K. Ng, “Ruby laser operation in the near IR,” Proc. IRE 50, 2347 (1962).

Passy, R.

Payne, F. P.

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]

Popov, S.

Prieto, F.

Raman, C. V.

C. V. Raman, “A new radiation,” Indian J. Phys. 2, 387–398 (1928).

Savory, S. J.

Shtaif, M.

A. Mecozzi and M. Shtaif, “The statistics of polarization-dependent loss in optical communication systems,” IEEE Photon. Technol. Lett. 14, 313–315 (2002).
[CrossRef]

Stolen, R. H.

R. H. Stolen, “Polarization effects in fiber Raman and Brillouin lasers,” IEEE J. Quantum Electron. QE-15, 1157–1159 (1979).
[CrossRef]

R. H. Stolen, E. P. Ippen, and A. R. Tynes, “Raman oscillation in glass optical waveguide,” Appl. Phys. Lett. 20, 62–64 (1972).
[CrossRef]

Tynes, A. R.

R. H. Stolen, E. P. Ippen, and A. R. Tynes, “Raman oscillation in glass optical waveguide,” Appl. Phys. Lett. 20, 62–64 (1972).
[CrossRef]

Vanin, E.

Von der Weid, J. P.

Wai, P. K. A.

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, D.

Woodbury, E. J.

E. J. Woodbury and W. K. Ng, “Ruby laser operation in the near IR,” Proc. IRE 50, 2347 (1962).

Yang, T.

R. Hellwarth, J. Cherlow, and T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964–967 (1975).
[CrossRef]

Zimmer, C. W.

Appl. Phys. Lett.

R. H. Stolen, E. P. Ippen, and A. R. Tynes, “Raman oscillation in glass optical waveguide,” Appl. Phys. Lett. 20, 62–64 (1972).
[CrossRef]

IEEE J. Quantum Electron.

R. H. Stolen, “Polarization effects in fiber Raman and Brillouin lasers,” IEEE J. Quantum Electron. QE-15, 1157–1159 (1979).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

B. Huttner, C. Geiser, and N. Gisin, “Polarization-induced distortions in optical fiber networks with polarization-mode dispersion and polarization-dependent losses,” IEEE J. Sel. Top. Quantum Electron. 6, 317–329 (2000).
[CrossRef]

S. Namiki and Y. Emori, “Ultrabroad-band Raman amplifiers pumped and gain-equalized by wavelength-division-multiplexed high-power laser diodes,” IEEE J. Sel. Top. Quantum Electron. 7, 3–16 (2001).
[CrossRef]

IEEE Photon. Technol. Lett.

A. Mecozzi and M. Shtaif, “The statistics of polarization-dependent loss in optical communication systems,” IEEE Photon. Technol. Lett. 14, 313–315 (2002).
[CrossRef]

Indian J. Phys.

C. V. Raman, “A new radiation,” Indian J. Phys. 2, 387–398 (1928).

J. Lightwave Technol.

Opt. Commun.

M. Ikeda, “Stimulated Raman amplification characteristics in long span single-mode silica fibers,” Opt. Commun. 39, 148–152 (1981).
[CrossRef]

Opt. Lett.

Phys. Rev. B

R. Hellwarth, J. Cherlow, and T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964–967 (1975).
[CrossRef]

Proc. IRE

E. J. Woodbury and W. K. Ng, “Ruby laser operation in the near IR,” Proc. IRE 50, 2347 (1962).

Proc. Natl. Acad. Sci. USA

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

Prog. Quantum Electron.

R. W. Hellwarth, “Third-order optical susceptibilities of liquid and solids,” Prog. Quantum Electron. 5, 1–68 (1977).
[CrossRef]

Other

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

K. Rottwitt and A. J. Stentz, “Raman amplifiers in lightwave communication systems,” in Optical Fiber Telecommunications IV-A: Components, I. P. Kaminow and T. Li, eds. (Academic, San Diego, Calif., 2002), Chap. 5.

H. Kogelnik, R. M. Jopson, and L. E. Nelson, “Polarization-mode dispersion,” in Optical Fiber Telecommunications IV-B: Systems and Impairments, I. P. Kaminow and T. Li, eds. (Academic, San Diego, Calif., 2002), Chap. 15.

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, San Diego, Calif., 1992).

C. W. Gardiner, Handbook of Stochastic Methods, 2nd ed. (Springer-Verlag, New York, 1985).

H. H. Kee, C. R. S. Fludger, and V. Handerek, “Statistical properties of polarization dependent gain in fiber Raman amplifiers,” in Optical Fiber Communication, Vol. 70 of 2002 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2002), pp. 180–181.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (WCB/McGraw-Hill, New York, 1991).

Y. Emori, S. Matsushita, and S. Namiki, “Cost-effective depolarized diode pump unit designed for C-band flat-gain Raman amplifiers to control EDFA gain profile,” in Optical Fiber Communication Conference, Vol. 37 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 106–107.

J. Zhang, V. Dominic, M. Missey, S. Sanders, and D. Mehuys, “Dependence of Raman polarization dependent gain on pump degree of polarization at high gain levels,” in Optical Amplifiers and Their Applications, A. Mecozzi, M. Shimizu, and J. Zyskind, eds., Vol. 44 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000), pp. 30–31.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, New York, 1995), Chap. 6.

D. Mahgerefteh, H. Yu, D. L. Butler, J. Goldhar, D. Wang, E. Golovchenko, A. N. Phlipetskii, C. R. Menyuk, and L. Joneckis, “Effect of randomly varying birefringence on the Raman gain in optical fibers,” in Conference on Lasers and Electro-Optics, Vol. 11 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), p. 447.

A. Berntson, S. Popov, E. Vanin, G. Jacobsen, and J. Karlsson, “Polarization dependence and gain tilt of Raman amplifiers for WDM systems,” in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), pages MI2–1.

P. Ebrahimi, M. C. Hauer, Q. Yu, R. Khosravani, D. Gurkan, D. W. Kim, D. W. Lee, and A. E. Willner, “Statistics of polarization dependent gain in Raman fiber amplifiers due to PMD,” in Conference on Lasers and Electro-Optics, Vol. 56 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), pp. 143–144.

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Figures (7)

Fig. 1
Fig. 1

(a) Average gain and (b) standard deviation of signal fluctuations at the output of a Raman amplifier as a function of the PMD parameter for forward and backward pumping. The solid and dashed curves correspond to copolarized and orthogonally polarized signals, respectively.

Fig. 2
Fig. 2

(a) Average gain and (b) level of signal fluctuations as a function of amplifier length for a fiber with Dp=0.05 ps/km. The solid and dashed curves correspond to copolarized and orthogonally polarized signals, respectively. The two curves nearly coincide in the case of backward pumping.

Fig. 3
Fig. 3

(a) Average gain and (b) level of signal fluctuations plotted as a function of pump-signal detuning. The solid and dashed curves correspond to copolarized and orthogonally polarized cases, respectively. The thin solid curve is for a backward-pumped Raman amplifier and does not change much with signal polarization.

Fig. 4
Fig. 4

Probability density of an amplified signal for three values of Dp (Dp in units of ps/km) in the cases of (a) copolarized and (b) orthogonally polarized signals. The amplified signal is normalized to the input signal power.

Fig. 5
Fig. 5

Probability distribution of PDG as a function of Dp under conditions of Fig. 1. The PDG value is normalized to the average gain Gav.

Fig. 6
Fig. 6

(a) Mean PDG and (b) variance σΔ (both normalized to the average gain Gav) as a function of PMD parameter under forward- and backward-pumping conditions.

Fig. 7
Fig. 7

(a) Mean PDG and (b) variance σΔ (both normalized to the average gain Gav) as a function of amplifier length under forward and backward pumping. The solid and dashed curves correspond to Dp=0.05 and 0.15 ps/km, respectively.

Equations (121)

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

P(r, t)=P(1)(r, t)+P(2)(r, t)+P(3)(r, t)+,
P(3)(r, t)
=ε02 σ[E(r, t)E(r, t)]E(r, t)+E(r, t)0ε0a(τ)[E(r, t-τ)E(r, t-τ)]dτ+E(r, t)0ε0b(τ)E(r, t-τ)E(r, t-τ)dτ,
E=Re[Epexp(-iωpt)+Esexp(-iωst)],
P(3)=Re[Ppexp(-iωpt)+Psexp(-iωst)],
Pj(ωj)=ε08 [σ+2b˜(0)](Ej  Ej)Ej*+ε04 [σ+2a˜(0)+b˜(0)](Ej*  Ej)Ej+ε04 [σ+2a˜(0)+b˜(ωj-ωm)](Em*  Em)Ej+ε04 [σ+b˜(0)+b˜(ωj-ωm)](Em  Ej)Em*+ε04 [σ+2a˜(ωj-ωm)+b˜(0)]×(Em*  Ej)Em,
a˜(ω)=0a(τ)exp(iωτ)dτ,
b˜(ω)=0b(τ)exp(iωτ)dτ
2Ej+ωj2c2 εjEj=-ωj2ε0c2Pj,
Ep(r)=Fp(x, y)|Apexp(ikpz),
Es(r)=Fs(x, y)|Asexp(iksz),
(ωj2/c2)εj=(kj+iαj/2)2σ0-kjωjβσ,
σ1=100-1,σ2=0110,σ3=0-ii0.
ξ d|Ajdz=-αj2 |Aj-i2 ωjβ  σ|Aj+iγjj32Aj|Aj+κbκa |Aj*Aj*||Aj+2iγjm3 [(1+δb)Am|Am+(1+δa)|AmAm|+(κb/κa+δb)|Am*Am*|]|Aj+ζ2 [g2(Am|Am+|Am*Am*|)+g1|AmAm|]|Aj,
κa=σ+2a˜(0)+b˜(0),κb=σ+2b˜(0),
δa=2{Re[a˜(ΩR)]-a˜(0)}/κa,
δb={Re[b˜(ΩR)]-b˜(0)}/κa,
γjm=3ωj2ωmκa/(8c4kjkmAeff)
g1=ωs2ωpIm[a˜(ΩR)]/(c4kpksAeff),
g2=ωs2ωpIm[b˜(ΩR)]/(2c4kpksAeff).
P=Ap|σ|ApP1e^1+P2e^2+P3e^3,
S=As|σ|AsS1e^1+S2e^2+S3e^3,
|A*A*|=|AA|-A|σ3|Aσ3,
|AA|={A|A+A|σ|Aσ}/2,
ξ dPdz=-αpP-ωp2ωs g1[(1+3μ)S0P+(1+μ)P0S-2μP0S3]+(ωpβ+Wp)×P,
dSdz=-αsS+g12 [(1+3μ)P0S+(1+μ)S0P-2μS0P3]+(ωsβ+Ws)×S,
Wp=23 [γppP3+2γps(1+δb)S3-γps(2+δa+δb)S],
Ws=23 [γssS3+2γsp(1+δb)P3-γsp(2+δa+δb)P].
dRdz=ξωpβ×R.
ξ dPdz=-αpP-ωp2ωs g1[(1+3μ)S0P+(1+μ/3)P0S]-εpsS×P,
dSdz=-αsS+g12 [(1+3μ)P0S+(1+μ/3)S0P]-(ΩRB+εspP)×S,
V=exp-εsp0zP0(z)dzpˆ×V,
dSdz=-αsS+g12 [(1+3μ)P0S+(1+μ/3)S0P]-ΩRb×S,
b(z)=0,b(z1)b(z2)=13 Dp2Iδ(z2-z1),
S=sexp0zg12 (1+3μ)P0(z)-αsdz,
ds0dz=gR2 P0(z)pˆs,
dsdz=gR2 P0(z)s0pˆ-ΩRb×s,
Gav=S0(L)S0(0),σs2=S02(L)S0(L)2-1.
ds0dz=gR2 P0(z)s0cos θ,
ds0cos θdz=gR2 P0(z)s0-ηs0cos θ,
Gav=a[g1(1+3μ)PinLeff/2-αsL],
Gav=[cosh(κL/2)+sinh(κL/2)(gRPincos θ0+η)/κ]×exp{[g1(1+3μ)Pin-η-2αs]L/2},
ds02dz=gRP0(z)s02cos θ,
ds02cos θdz=-ηs02cos θ+gR2 P0(z)[s02+s02cos2 θ],
ds02cos2 θdz=-3ηs02cos2 θ+ηs02+gRP0(z)s02cos θ.
GdB=a lnS0(L)S0(0)=ag12 (1+3μ)PinLeff-αsL+a2 gR0LP0(z)[pˆ(z)sˆ(z)]dz,
dsˆdz=gR2 P0(z)[pˆ-(pˆsˆ)sˆ]-ΩRb×sˆ.
p[S0(L)]=[ln(σs2+1)]-1/2S0(L)2πexp-12 ln(σs2+1)×ln2S0(L)σs2+1S0(L),
dΔdz=gR2 Δ cothΔ2a[P-(PΔˆ)Δˆ]+agR(PΔˆ)Δˆ-ΩRb×Δ.
Δ cothΔ2a2a+Δ26a.
dΔdz=agRP-ΩRb×Δ.
Δ(L)=agRR(L)0LR-1(z)P(z)dz,
dΔ2dz=2agRP0(z)pˆΔ,
dΔdz=-ηΔ+agRP0(z)pˆ,
dCdz=-3ηC+η[Δ2I-ΔΔ].
Δ=agRPinpˆη-αp [1-αpLeff-exp(-ηL)],
Δ2=2(agRPin)2η2-αp2 [(1-αpLeff)exp(-ηL)-1+(αp+η)Leff(1-αpLeff/2)].
p(Δ)=(2π)-3/2σσ2exp-(Δ1-Δ0)22σ2-Δ22+Δ322σ2,
σ2=η0L[Δ2-Δ2]exp[-3η(L-z)]dz,
σ2=η0LΔ2exp[-3η(L-z)]dz.
p(Δ)=Δ2σσexp-Δ2(r-1)-rΔ022σ2×erfΔ(r-1)+rΔ02σ+erfΔ(r-1)-rΔ02σ,
Δ4agRPinπDp|ΩR| [Leff(1-αpLeff/2)]1/2,
σΔ[(3π/8-1)]1/2Δ.
[pˆ(z1)-pˆ][pˆ(z2)-pˆ]
Γ(z2-z1)=Γ0Γ(z2-z1),
V=exp-εsp0zP0(z)dzpˆ×V,
Δ(L)b=agR0LP0(z)pˆ(z)exp[-η(L-z)]dz,
Δ2(L)b=2agR0LP0(z)pˆ(z)Δ(z)bdz,=2(agR)20Ldz10z1dz2P0(z1)P0(z2)pˆ(z1)pˆ(z2)exp[-η(z1-z2)],
Δ(L)bp=agRPinpˆη-αp [1-αpLeff-exp(-ηL)],
Δ2(L)bp=2(agRPin)2Tr(Γ0)(γc+η)2-αp2× {(1-αpLeff)exp[-(γc+η)L]-1+Leff(γc+η+αp)(1-αpLeff/2)}+2(agRPin)2dp2η2-αp2 {(1-αpLeff)exp(-ηL)-1+Leff(η+αp)(1-αpLeff/2)},
dCdz=-3ηC+η[Δ2I-ΔΔ]+Hp,
Hp=agRP0(z)[pˆ(z)Δ(z)bp+Δ(z)b pˆ(z)p-pˆΔ-Δpˆ].
=2(agRPin)2Γ0γc+η-αp {exp(-2αpz)-exp[-(γc+η+αp)z]}.
Hp2(agRPin)2Γ0lc(1-αpLeff)2,
Δ2(L)bp2(agRPindp)2η2-αp2 [(1-αpLeff)exp(-ηL)-1+Leff(η+αp)(1-αpLeff/2)].
ξ dPdz=-αpP-ωpg12ωs [(1+3μ)S0P+(1+μ)P0S-2μP0R-1S3]+Wp×P,
dSdz=-αsS+g12 [(1+3μ)P0S+(1+μ)S0P-2μS0R-1P3]+(ΩB+Ws)×S,
Wp=23 [γppR-1P3+2γps(1+δb)R-1S3-γps(2+δa+δb)S],
Ws=23 [γssR-1S3+2γsp(1+δb)R-1P3-γsp(2+δa+δb)P].
R=cos θ-sin θ cos φ0sin θ sin φ0sin θ cos φcos θ cos φ0cos φ-sin φ0sin φ-cos θ sin φ0cos φ-cos φ0sin φsin θ sin φcos θ cos φ0sin φ+sin φ0cos φcos φ0cos φ-cos θ sin φ0sin φ,
R-1S3=R-1e^3(e^3  S)=R-1e^3(e^3R)S.
R-1S3=13S,R-1P3=13P.
dW=0,dWdW=13 Dp2Idz,
dWdW=Dp2dz.
ds=s(z+dz)-s(z)=-ΩRdW×s(z+dz/2)
s(z+dz/2)=s(z)+dz2ds(z)dz+=s(z)-ΩR2dW×s(z)+ .
ds=-ΩRdW×s(z)+ΩR22dW×[dW×s(z)]+ .
dW×(dW×s)=dW(dW  s)-s(dWdW)
ds=-ΩRdW×s(z)-13 Dp2ΩR2s(z)dz
ds0=gR2Psdz,
ds=gR2Ps0dz-ηsdz+ΩRs×dW,
ds02=gRP(s0s)dz,
d(s0s)=gR2 [P(ss)+Ps02]dz-η(s0s)dz+ΩR(s0s)×dW,
d(ss)=gR2 (Ps0s+s0sP)-3η(ss)dz+ηs02Idz+ΩR(ss)×dW-ΩRdW×(ss).
d|Adz=M(z)|A[m0(z)+m(z)σ]|A.
dT(z)dz=[m0(z)+m(z)σ]T(z).
A|Ain=A|[T(z)T(z)]-1|Aout.
d[T(z)T(z)]dz=dt0(z)dz+dt(z)dzσ.
t(z)=u(z)exp0z[m0(z)+m0*(z)]dz.
du0dz=2mr  u,
dudz=2mru0-2mi×u,
[T(z)T(z)]-1=[u0(z)-u(z)σ]×exp-0zdz[m0(z)+m0*(z)].
S0(0)=S0(L)[u0(L)-u(L)s^out]×exp-0L[m0(z)+m0*(z)]dz,
G=a lnS0(L)S0(0)=a0L[m0(z)+m0*(z)]dz-a ln[u0(L)-u(L)s^out].
ΔuˆΔ=auˆ lnu0+uu0-u,
dΔdz=du0dzΔu0+dudzu(Δ)=2Δ cothΔ2a[mr-(mrΔˆ)Δˆ]+4a(mrΔˆ)Δˆ-2mi×Δ,
dΔ=agRPdz-ηΔdz+ΩRΔ×dW.
dΔ2=2agRP  Δdz,
d(ΔΔ)=agR(PΔ+ΔP)dz-3η(ΔΔ)dz+ηΔ2Idz+ΩR(ΔΔ)×dW-ΩRdW×(ΔΔ).
dΔ=agRPdz-ηΔdz,
dΔ2=2agRPΔdz,
dΔΔ=agR(PΔ+ΔP)dz-3ηΔΔdz+ηΔ2Idz.
dC=dΔΔ-(dΔ)Δ-Δ(dΔ),
dC=dΔΔ-(dΔ)Δ-Δ(dΔ).
dΔ=agRPdz-ηΔdz,
dΔ2=2agRPΔbpdz,
dΔΔ=agR(PΔbp+ΔbPp)dz-3ηΔΔdz+ηΔ2Idz.
dS=-εspP0(z)δpˆ(z+dz/2)×S(z+dz/2)dz
dS=-εspP0(z)δpˆ(z+dz/2)×S(z)+dz2dSdz+dz.
dS=-εspP0(z)δpˆ(z+dz/2)×S(z)dz+12 [εspP0(z)dz]2{δpˆ(z)δpˆ(z+dz/2)S(z)-[δpˆ(z)δpˆ(z+dz/2)]S(z)}+
dS12 [εspP0(z)dz]2[Γ0-Tr(Γ0)I]S(z)+=O[(dz)2]

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