Abstract

We present a rate-equation model that accounts for polarization hole burning in erbium-doped fiber amplifiers. This model yields analytical expressions for the polarization sensitivity of the amplifier for arbitrary signal polarization states. We investigate the influence of the birefringence properties of the fiber and calculate the average polarization properties of an amplifier chain.

© 1995 Optical Society of America

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  1. V. P. Lebedev and A. K. Przhevuskii, "Polarized luminescence of rare-earth activated glasses," Sov. Phys. Solid State 19, 1389 (1977).
  2. D. W. Hall and M. J. Weber, "Polarized fluorescence line narrowing measurements of Nd laser glasses: evidence of stimulated emission cross section anisotropy," Appl. Phys. Lett. 42, 157 (1983).
    [CrossRef]
  3. D. W. Hall, R. A. Haas, W. F. Krupke, and M. J. Weber, "Spectral and polarization hole burning in neodymium glass lasers," IEEE J. Quantum Electron. QE-19, 1704 (1983).
    [CrossRef]
  4. J. T. Lin, P. R. Morkel, L. Reekie, and D. N. Payne, "Polarization effects in fiber lasers," in Conference on Lasers and Electro-Optics, Vol. 14 of 1987 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1987), p. 107.
  5. J. T. Lin, L. Reekie, D. N. Payne, and S. B. Poole, "Intensity dependent polarization frequency splitting in an Er3+-doped fiber laser," in Conference on Lasers and Electro-Optics, Vol. 7 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper TuM28.
  6. J. T. Lin, W. A. Gambling, and D. N. Payne, "Modeling of polarization effects in fiber lasers," in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 90.
  7. J. T. Lin and W. A. Gambling, "Polarisation effects in fibre lasers: phenomena, theory and applications," in Fiber Laser Sources and Amplifiers II, M. J. Digonnet and E. Snitzer, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1373, 42 (1990).
    [CrossRef]
  8. S. Bielawski, D. Derozier, and P. Glorieux, "Antiphase dynamics and polarizations effects in the Nd-doped fiber laser," Phys. Rev. A 46, 2811 (1992).
    [CrossRef] [PubMed]
  9. R. Leners, P. L. Francois, and G. Stephan, "Simultaneous effects of gain and loss anisotropies on the thresholds of a bipolarization fiber laser," Opt. Lett. 19, 275 (1994).
    [CrossRef] [PubMed]
  10. R. Leners, P. L. Francois, and G. Stephan, "Polarization cross-saturation in fiber lasers," in International Quantum Electronics Conference, Vol. 9 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 212.
  11. B. Meziane, F. Sanchez, G. Stephan, and P. L. Francois, "Feedback induced polarization switching in a Nd-doped fiber laser," Opt. Lett. 19, 1971 (1994).
    [CrossRef]
  12. M. G. Taylor, "Observation of a new polarisation dependence effect in long haul optically amplified system," in Optical Fiber Communication Conference and International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper PD5.
  13. E. Lichtman, "Limitations imposed by polarization dependent gain and loss on all-optical ultralong communication systems," in Optical Fiber Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 257.
  14. F. Bruyere and O. Audouin, "Penalties in long-haul optical amplifier systems due to polarization dependent loss and gain," IEEE Photon. Technol. Lett. 6, 654 (1994).
    [CrossRef]
  15. E. J. Greer, D. J. Lewis, and W. M. Macaulay, "Polarisation dependent gain in erbium doped fibre amplifiers," Electron. Lett. 30, 46 (1994).
    [CrossRef]
  16. R. W. Keys, S. J. Wilson, M. Healy, S. R. Baker, A. Robinson, and J. E. Righton, "Polarization-dependent gain in erbium-doped fibers," in Optical Fibre Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper FF5.
  17. V. J. Mazurczyk and J. L. Zyskind, "Polarization hole burning in erbium doped fiber amplifiers," in Conference on Lasers and Electro-Optics, Vol. 12 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper CPD26.
  18. V. J. Mazurczyk and J. L. Zyskind, "Polarisation dependent gain in erbium doped fiber amplifiers," IEEE Photon. Technol. Lett. 6, 616 (1994).
    [CrossRef]
  19. N. S. Bergano, "Time dynamics of polarization hole burning in an EDFA," in Optical Fiber Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper FF4.
  20. Y. Fukada, T. Imai, and A. Mamoru, "BER fluctuation suppression in optical in-line amplifier system using polarisation scrambling techniques," Electron. Lett. 30, 432 (1994).
    [CrossRef]
  21. M. G. Taylor and S. J. Penticost, "Improvement in performance of long haul EDFA link using high frequency polarisation modulation," Electron. Lett. 30, 805 (1994).
    [CrossRef]
  22. M. G. Taylor, "Improvement in Q with low frequency polarization modulation on transoceanic EDFA link," IEEE Photon. Technol. Lett. 6, 860 (1994).
    [CrossRef]
  23. N. S. Bergano, V. J. Mazurczyk, and C. R. Davidson, "Polarization scrambling improves SNR performance in a chain of EDFAs," in Optical Fiber Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper ThR2.
  24. V. Letellier, G. Bassier, P. Marmier, R. Morin, R. Uhel, and J. Artur, "Polarisation scrambling in 5 Gbit/s 8100 km EDFA based system," Electron. Lett. 30, 46 (1994).
    [CrossRef]
  25. V. J. Mazurczyk and C. D. Poole, "The effect of birefringence on polarization hole burning in erbium doped fiber amplifiers," in Optical Amplifiers and Their Applications, Vol. 14 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper ThB3.
  26. P. F. Wysocki, "Computer modeling of polarization hole burning in EDFAs," in Optical Fiber Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper FF6.
  27. P. F. Wysocki, "Polarization hole-burning in erbium doped fiber amplifiers with birefringence," in Optical Amplifiers and Their Applications, Vol. 14 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper ThB4.
  28. R. Leners, T. Georges, P. L. Francois, and G. Stephan, "Analytic model of polarization dependent gain in erbium doped fibre amplifier," in Optical Amplifiers and Their Applications, Vol. 14 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper PD1.
  29. R. Leners and G. Stephan, "Rate equations analysis of a multi-mode, bipolarization Nd3+-doped fibre laser," J. Europ. Opt. Soc. (to be published).
  30. T. Georges and E. Delevaque, "Analytic modelling of high-gain erbium doped fiber amplifiers," Opt. Lett. 17, 1113 (1992).
    [CrossRef] [PubMed]
  31. T. Ueda and W. L. Kath, "Stochastic simulation of pulses in nonlinear-optical fibers with random birefringence," J. Opt. Soc. Am. B 11, 818 (1994).
    [CrossRef]
  32. V. J. Matsas, T. P. Newson, D. J. Richardson, and D. N. Payne, "Selfstarting passively mode-locked fibre ring soliton laser exploiting nonlinear polarisation rotation," Electron. Lett. 28, 1391 (1992).
    [CrossRef]
  33. K. Tamura, H. A. Haus, and E. P. Ippen, "Self-starting additive pulse mode-locked erbium fibre ring laser," Electron. Lett. 28, 2226 (1992).
    [CrossRef]
  34. M. E. Fermann, M. J. Andrejco, Y. Silberberg, and M. L. Stock, "Passive mode locking by using nonlinear polarization evolution in a polarization-maintaining erbium-doped fiber," Opt. Lett. 18, 894 (1993).
    [CrossRef] [PubMed]

1994 (19)

R. Leners, P. L. Francois, and G. Stephan, "Simultaneous effects of gain and loss anisotropies on the thresholds of a bipolarization fiber laser," Opt. Lett. 19, 275 (1994).
[CrossRef] [PubMed]

R. Leners, P. L. Francois, and G. Stephan, "Polarization cross-saturation in fiber lasers," in International Quantum Electronics Conference, Vol. 9 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 212.

B. Meziane, F. Sanchez, G. Stephan, and P. L. Francois, "Feedback induced polarization switching in a Nd-doped fiber laser," Opt. Lett. 19, 1971 (1994).
[CrossRef]

E. Lichtman, "Limitations imposed by polarization dependent gain and loss on all-optical ultralong communication systems," in Optical Fiber Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 257.

F. Bruyere and O. Audouin, "Penalties in long-haul optical amplifier systems due to polarization dependent loss and gain," IEEE Photon. Technol. Lett. 6, 654 (1994).
[CrossRef]

E. J. Greer, D. J. Lewis, and W. M. Macaulay, "Polarisation dependent gain in erbium doped fibre amplifiers," Electron. Lett. 30, 46 (1994).
[CrossRef]

R. W. Keys, S. J. Wilson, M. Healy, S. R. Baker, A. Robinson, and J. E. Righton, "Polarization-dependent gain in erbium-doped fibers," in Optical Fibre Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper FF5.

V. J. Mazurczyk and J. L. Zyskind, "Polarisation dependent gain in erbium doped fiber amplifiers," IEEE Photon. Technol. Lett. 6, 616 (1994).
[CrossRef]

N. S. Bergano, "Time dynamics of polarization hole burning in an EDFA," in Optical Fiber Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper FF4.

Y. Fukada, T. Imai, and A. Mamoru, "BER fluctuation suppression in optical in-line amplifier system using polarisation scrambling techniques," Electron. Lett. 30, 432 (1994).
[CrossRef]

M. G. Taylor and S. J. Penticost, "Improvement in performance of long haul EDFA link using high frequency polarisation modulation," Electron. Lett. 30, 805 (1994).
[CrossRef]

M. G. Taylor, "Improvement in Q with low frequency polarization modulation on transoceanic EDFA link," IEEE Photon. Technol. Lett. 6, 860 (1994).
[CrossRef]

N. S. Bergano, V. J. Mazurczyk, and C. R. Davidson, "Polarization scrambling improves SNR performance in a chain of EDFAs," in Optical Fiber Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper ThR2.

V. Letellier, G. Bassier, P. Marmier, R. Morin, R. Uhel, and J. Artur, "Polarisation scrambling in 5 Gbit/s 8100 km EDFA based system," Electron. Lett. 30, 46 (1994).
[CrossRef]

V. J. Mazurczyk and C. D. Poole, "The effect of birefringence on polarization hole burning in erbium doped fiber amplifiers," in Optical Amplifiers and Their Applications, Vol. 14 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper ThB3.

P. F. Wysocki, "Computer modeling of polarization hole burning in EDFAs," in Optical Fiber Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper FF6.

P. F. Wysocki, "Polarization hole-burning in erbium doped fiber amplifiers with birefringence," in Optical Amplifiers and Their Applications, Vol. 14 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper ThB4.

R. Leners, T. Georges, P. L. Francois, and G. Stephan, "Analytic model of polarization dependent gain in erbium doped fibre amplifier," in Optical Amplifiers and Their Applications, Vol. 14 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper PD1.

T. Ueda and W. L. Kath, "Stochastic simulation of pulses in nonlinear-optical fibers with random birefringence," J. Opt. Soc. Am. B 11, 818 (1994).
[CrossRef]

1993 (3)

M. E. Fermann, M. J. Andrejco, Y. Silberberg, and M. L. Stock, "Passive mode locking by using nonlinear polarization evolution in a polarization-maintaining erbium-doped fiber," Opt. Lett. 18, 894 (1993).
[CrossRef] [PubMed]

V. J. Mazurczyk and J. L. Zyskind, "Polarization hole burning in erbium doped fiber amplifiers," in Conference on Lasers and Electro-Optics, Vol. 12 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper CPD26.

M. G. Taylor, "Observation of a new polarisation dependence effect in long haul optically amplified system," in Optical Fiber Communication Conference and International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper PD5.

1992 (4)

S. Bielawski, D. Derozier, and P. Glorieux, "Antiphase dynamics and polarizations effects in the Nd-doped fiber laser," Phys. Rev. A 46, 2811 (1992).
[CrossRef] [PubMed]

V. J. Matsas, T. P. Newson, D. J. Richardson, and D. N. Payne, "Selfstarting passively mode-locked fibre ring soliton laser exploiting nonlinear polarisation rotation," Electron. Lett. 28, 1391 (1992).
[CrossRef]

K. Tamura, H. A. Haus, and E. P. Ippen, "Self-starting additive pulse mode-locked erbium fibre ring laser," Electron. Lett. 28, 2226 (1992).
[CrossRef]

T. Georges and E. Delevaque, "Analytic modelling of high-gain erbium doped fiber amplifiers," Opt. Lett. 17, 1113 (1992).
[CrossRef] [PubMed]

1990 (1)

J. T. Lin and W. A. Gambling, "Polarisation effects in fibre lasers: phenomena, theory and applications," in Fiber Laser Sources and Amplifiers II, M. J. Digonnet and E. Snitzer, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1373, 42 (1990).
[CrossRef]

1989 (1)

J. T. Lin, W. A. Gambling, and D. N. Payne, "Modeling of polarization effects in fiber lasers," in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 90.

1988 (1)

J. T. Lin, L. Reekie, D. N. Payne, and S. B. Poole, "Intensity dependent polarization frequency splitting in an Er3+-doped fiber laser," in Conference on Lasers and Electro-Optics, Vol. 7 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper TuM28.

1987 (1)

J. T. Lin, P. R. Morkel, L. Reekie, and D. N. Payne, "Polarization effects in fiber lasers," in Conference on Lasers and Electro-Optics, Vol. 14 of 1987 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1987), p. 107.

1983 (2)

D. W. Hall and M. J. Weber, "Polarized fluorescence line narrowing measurements of Nd laser glasses: evidence of stimulated emission cross section anisotropy," Appl. Phys. Lett. 42, 157 (1983).
[CrossRef]

D. W. Hall, R. A. Haas, W. F. Krupke, and M. J. Weber, "Spectral and polarization hole burning in neodymium glass lasers," IEEE J. Quantum Electron. QE-19, 1704 (1983).
[CrossRef]

1977 (1)

V. P. Lebedev and A. K. Przhevuskii, "Polarized luminescence of rare-earth activated glasses," Sov. Phys. Solid State 19, 1389 (1977).

Andrejco, M. J.

Artur, J.

V. Letellier, G. Bassier, P. Marmier, R. Morin, R. Uhel, and J. Artur, "Polarisation scrambling in 5 Gbit/s 8100 km EDFA based system," Electron. Lett. 30, 46 (1994).
[CrossRef]

Audouin, O.

F. Bruyere and O. Audouin, "Penalties in long-haul optical amplifier systems due to polarization dependent loss and gain," IEEE Photon. Technol. Lett. 6, 654 (1994).
[CrossRef]

Baker, S. R.

R. W. Keys, S. J. Wilson, M. Healy, S. R. Baker, A. Robinson, and J. E. Righton, "Polarization-dependent gain in erbium-doped fibers," in Optical Fibre Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper FF5.

Bassier, G.

V. Letellier, G. Bassier, P. Marmier, R. Morin, R. Uhel, and J. Artur, "Polarisation scrambling in 5 Gbit/s 8100 km EDFA based system," Electron. Lett. 30, 46 (1994).
[CrossRef]

Bergano, N. S.

N. S. Bergano, "Time dynamics of polarization hole burning in an EDFA," in Optical Fiber Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper FF4.

N. S. Bergano, V. J. Mazurczyk, and C. R. Davidson, "Polarization scrambling improves SNR performance in a chain of EDFAs," in Optical Fiber Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper ThR2.

Bielawski, S.

S. Bielawski, D. Derozier, and P. Glorieux, "Antiphase dynamics and polarizations effects in the Nd-doped fiber laser," Phys. Rev. A 46, 2811 (1992).
[CrossRef] [PubMed]

Bruyere, F.

F. Bruyere and O. Audouin, "Penalties in long-haul optical amplifier systems due to polarization dependent loss and gain," IEEE Photon. Technol. Lett. 6, 654 (1994).
[CrossRef]

Davidson, C. R.

N. S. Bergano, V. J. Mazurczyk, and C. R. Davidson, "Polarization scrambling improves SNR performance in a chain of EDFAs," in Optical Fiber Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper ThR2.

Delevaque, E.

Derozier, D.

S. Bielawski, D. Derozier, and P. Glorieux, "Antiphase dynamics and polarizations effects in the Nd-doped fiber laser," Phys. Rev. A 46, 2811 (1992).
[CrossRef] [PubMed]

Fermann, M. E.

Francois, P. L.

R. Leners, T. Georges, P. L. Francois, and G. Stephan, "Analytic model of polarization dependent gain in erbium doped fibre amplifier," in Optical Amplifiers and Their Applications, Vol. 14 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper PD1.

R. Leners, P. L. Francois, and G. Stephan, "Simultaneous effects of gain and loss anisotropies on the thresholds of a bipolarization fiber laser," Opt. Lett. 19, 275 (1994).
[CrossRef] [PubMed]

R. Leners, P. L. Francois, and G. Stephan, "Polarization cross-saturation in fiber lasers," in International Quantum Electronics Conference, Vol. 9 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 212.

B. Meziane, F. Sanchez, G. Stephan, and P. L. Francois, "Feedback induced polarization switching in a Nd-doped fiber laser," Opt. Lett. 19, 1971 (1994).
[CrossRef]

Fukada, Y.

Y. Fukada, T. Imai, and A. Mamoru, "BER fluctuation suppression in optical in-line amplifier system using polarisation scrambling techniques," Electron. Lett. 30, 432 (1994).
[CrossRef]

Gambling, W. A.

J. T. Lin and W. A. Gambling, "Polarisation effects in fibre lasers: phenomena, theory and applications," in Fiber Laser Sources and Amplifiers II, M. J. Digonnet and E. Snitzer, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1373, 42 (1990).
[CrossRef]

J. T. Lin, W. A. Gambling, and D. N. Payne, "Modeling of polarization effects in fiber lasers," in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 90.

Georges, T.

R. Leners, T. Georges, P. L. Francois, and G. Stephan, "Analytic model of polarization dependent gain in erbium doped fibre amplifier," in Optical Amplifiers and Their Applications, Vol. 14 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper PD1.

T. Georges and E. Delevaque, "Analytic modelling of high-gain erbium doped fiber amplifiers," Opt. Lett. 17, 1113 (1992).
[CrossRef] [PubMed]

Glorieux, P.

S. Bielawski, D. Derozier, and P. Glorieux, "Antiphase dynamics and polarizations effects in the Nd-doped fiber laser," Phys. Rev. A 46, 2811 (1992).
[CrossRef] [PubMed]

Greer, E. J.

E. J. Greer, D. J. Lewis, and W. M. Macaulay, "Polarisation dependent gain in erbium doped fibre amplifiers," Electron. Lett. 30, 46 (1994).
[CrossRef]

Haas, R. A.

D. W. Hall, R. A. Haas, W. F. Krupke, and M. J. Weber, "Spectral and polarization hole burning in neodymium glass lasers," IEEE J. Quantum Electron. QE-19, 1704 (1983).
[CrossRef]

Hall, D. W.

D. W. Hall and M. J. Weber, "Polarized fluorescence line narrowing measurements of Nd laser glasses: evidence of stimulated emission cross section anisotropy," Appl. Phys. Lett. 42, 157 (1983).
[CrossRef]

D. W. Hall, R. A. Haas, W. F. Krupke, and M. J. Weber, "Spectral and polarization hole burning in neodymium glass lasers," IEEE J. Quantum Electron. QE-19, 1704 (1983).
[CrossRef]

Haus, H. A.

K. Tamura, H. A. Haus, and E. P. Ippen, "Self-starting additive pulse mode-locked erbium fibre ring laser," Electron. Lett. 28, 2226 (1992).
[CrossRef]

Healy, M.

R. W. Keys, S. J. Wilson, M. Healy, S. R. Baker, A. Robinson, and J. E. Righton, "Polarization-dependent gain in erbium-doped fibers," in Optical Fibre Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper FF5.

Imai, T.

Y. Fukada, T. Imai, and A. Mamoru, "BER fluctuation suppression in optical in-line amplifier system using polarisation scrambling techniques," Electron. Lett. 30, 432 (1994).
[CrossRef]

Ippen, E. P.

K. Tamura, H. A. Haus, and E. P. Ippen, "Self-starting additive pulse mode-locked erbium fibre ring laser," Electron. Lett. 28, 2226 (1992).
[CrossRef]

Kath, W. L.

Keys, R. W.

R. W. Keys, S. J. Wilson, M. Healy, S. R. Baker, A. Robinson, and J. E. Righton, "Polarization-dependent gain in erbium-doped fibers," in Optical Fibre Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper FF5.

Krupke, W. F.

D. W. Hall, R. A. Haas, W. F. Krupke, and M. J. Weber, "Spectral and polarization hole burning in neodymium glass lasers," IEEE J. Quantum Electron. QE-19, 1704 (1983).
[CrossRef]

Lebedev, V. P.

V. P. Lebedev and A. K. Przhevuskii, "Polarized luminescence of rare-earth activated glasses," Sov. Phys. Solid State 19, 1389 (1977).

Leners, R.

R. Leners, P. L. Francois, and G. Stephan, "Simultaneous effects of gain and loss anisotropies on the thresholds of a bipolarization fiber laser," Opt. Lett. 19, 275 (1994).
[CrossRef] [PubMed]

R. Leners, P. L. Francois, and G. Stephan, "Polarization cross-saturation in fiber lasers," in International Quantum Electronics Conference, Vol. 9 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 212.

R. Leners, T. Georges, P. L. Francois, and G. Stephan, "Analytic model of polarization dependent gain in erbium doped fibre amplifier," in Optical Amplifiers and Their Applications, Vol. 14 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper PD1.

R. Leners and G. Stephan, "Rate equations analysis of a multi-mode, bipolarization Nd3+-doped fibre laser," J. Europ. Opt. Soc. (to be published).

Letellier, V.

V. Letellier, G. Bassier, P. Marmier, R. Morin, R. Uhel, and J. Artur, "Polarisation scrambling in 5 Gbit/s 8100 km EDFA based system," Electron. Lett. 30, 46 (1994).
[CrossRef]

Lewis, D. J.

E. J. Greer, D. J. Lewis, and W. M. Macaulay, "Polarisation dependent gain in erbium doped fibre amplifiers," Electron. Lett. 30, 46 (1994).
[CrossRef]

Lichtman, E.

E. Lichtman, "Limitations imposed by polarization dependent gain and loss on all-optical ultralong communication systems," in Optical Fiber Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 257.

Lin, J. T.

J. T. Lin and W. A. Gambling, "Polarisation effects in fibre lasers: phenomena, theory and applications," in Fiber Laser Sources and Amplifiers II, M. J. Digonnet and E. Snitzer, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1373, 42 (1990).
[CrossRef]

J. T. Lin, W. A. Gambling, and D. N. Payne, "Modeling of polarization effects in fiber lasers," in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 90.

J. T. Lin, L. Reekie, D. N. Payne, and S. B. Poole, "Intensity dependent polarization frequency splitting in an Er3+-doped fiber laser," in Conference on Lasers and Electro-Optics, Vol. 7 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper TuM28.

J. T. Lin, P. R. Morkel, L. Reekie, and D. N. Payne, "Polarization effects in fiber lasers," in Conference on Lasers and Electro-Optics, Vol. 14 of 1987 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1987), p. 107.

Macaulay, W. M.

E. J. Greer, D. J. Lewis, and W. M. Macaulay, "Polarisation dependent gain in erbium doped fibre amplifiers," Electron. Lett. 30, 46 (1994).
[CrossRef]

Mamoru, A.

Y. Fukada, T. Imai, and A. Mamoru, "BER fluctuation suppression in optical in-line amplifier system using polarisation scrambling techniques," Electron. Lett. 30, 432 (1994).
[CrossRef]

Marmier, P.

V. Letellier, G. Bassier, P. Marmier, R. Morin, R. Uhel, and J. Artur, "Polarisation scrambling in 5 Gbit/s 8100 km EDFA based system," Electron. Lett. 30, 46 (1994).
[CrossRef]

Matsas, V. J.

V. J. Matsas, T. P. Newson, D. J. Richardson, and D. N. Payne, "Selfstarting passively mode-locked fibre ring soliton laser exploiting nonlinear polarisation rotation," Electron. Lett. 28, 1391 (1992).
[CrossRef]

Mazurczyk, V. J.

V. J. Mazurczyk and C. D. Poole, "The effect of birefringence on polarization hole burning in erbium doped fiber amplifiers," in Optical Amplifiers and Their Applications, Vol. 14 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper ThB3.

N. S. Bergano, V. J. Mazurczyk, and C. R. Davidson, "Polarization scrambling improves SNR performance in a chain of EDFAs," in Optical Fiber Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper ThR2.

V. J. Mazurczyk and J. L. Zyskind, "Polarisation dependent gain in erbium doped fiber amplifiers," IEEE Photon. Technol. Lett. 6, 616 (1994).
[CrossRef]

V. J. Mazurczyk and J. L. Zyskind, "Polarization hole burning in erbium doped fiber amplifiers," in Conference on Lasers and Electro-Optics, Vol. 12 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper CPD26.

Meziane, B.

B. Meziane, F. Sanchez, G. Stephan, and P. L. Francois, "Feedback induced polarization switching in a Nd-doped fiber laser," Opt. Lett. 19, 1971 (1994).
[CrossRef]

Morin, R.

V. Letellier, G. Bassier, P. Marmier, R. Morin, R. Uhel, and J. Artur, "Polarisation scrambling in 5 Gbit/s 8100 km EDFA based system," Electron. Lett. 30, 46 (1994).
[CrossRef]

Morkel, P. R.

J. T. Lin, P. R. Morkel, L. Reekie, and D. N. Payne, "Polarization effects in fiber lasers," in Conference on Lasers and Electro-Optics, Vol. 14 of 1987 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1987), p. 107.

Newson, T. P.

V. J. Matsas, T. P. Newson, D. J. Richardson, and D. N. Payne, "Selfstarting passively mode-locked fibre ring soliton laser exploiting nonlinear polarisation rotation," Electron. Lett. 28, 1391 (1992).
[CrossRef]

Payne, D. N.

V. J. Matsas, T. P. Newson, D. J. Richardson, and D. N. Payne, "Selfstarting passively mode-locked fibre ring soliton laser exploiting nonlinear polarisation rotation," Electron. Lett. 28, 1391 (1992).
[CrossRef]

J. T. Lin, W. A. Gambling, and D. N. Payne, "Modeling of polarization effects in fiber lasers," in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 90.

J. T. Lin, L. Reekie, D. N. Payne, and S. B. Poole, "Intensity dependent polarization frequency splitting in an Er3+-doped fiber laser," in Conference on Lasers and Electro-Optics, Vol. 7 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper TuM28.

J. T. Lin, P. R. Morkel, L. Reekie, and D. N. Payne, "Polarization effects in fiber lasers," in Conference on Lasers and Electro-Optics, Vol. 14 of 1987 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1987), p. 107.

Penticost, S. J.

M. G. Taylor and S. J. Penticost, "Improvement in performance of long haul EDFA link using high frequency polarisation modulation," Electron. Lett. 30, 805 (1994).
[CrossRef]

Poole, C. D.

V. J. Mazurczyk and C. D. Poole, "The effect of birefringence on polarization hole burning in erbium doped fiber amplifiers," in Optical Amplifiers and Their Applications, Vol. 14 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper ThB3.

Poole, S. B.

J. T. Lin, L. Reekie, D. N. Payne, and S. B. Poole, "Intensity dependent polarization frequency splitting in an Er3+-doped fiber laser," in Conference on Lasers and Electro-Optics, Vol. 7 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper TuM28.

Przhevuskii, A. K.

V. P. Lebedev and A. K. Przhevuskii, "Polarized luminescence of rare-earth activated glasses," Sov. Phys. Solid State 19, 1389 (1977).

Reekie, L.

J. T. Lin, L. Reekie, D. N. Payne, and S. B. Poole, "Intensity dependent polarization frequency splitting in an Er3+-doped fiber laser," in Conference on Lasers and Electro-Optics, Vol. 7 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper TuM28.

J. T. Lin, P. R. Morkel, L. Reekie, and D. N. Payne, "Polarization effects in fiber lasers," in Conference on Lasers and Electro-Optics, Vol. 14 of 1987 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1987), p. 107.

Richardson, D. J.

V. J. Matsas, T. P. Newson, D. J. Richardson, and D. N. Payne, "Selfstarting passively mode-locked fibre ring soliton laser exploiting nonlinear polarisation rotation," Electron. Lett. 28, 1391 (1992).
[CrossRef]

Righton, J. E.

R. W. Keys, S. J. Wilson, M. Healy, S. R. Baker, A. Robinson, and J. E. Righton, "Polarization-dependent gain in erbium-doped fibers," in Optical Fibre Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper FF5.

Robinson, A.

R. W. Keys, S. J. Wilson, M. Healy, S. R. Baker, A. Robinson, and J. E. Righton, "Polarization-dependent gain in erbium-doped fibers," in Optical Fibre Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper FF5.

Sanchez, F.

B. Meziane, F. Sanchez, G. Stephan, and P. L. Francois, "Feedback induced polarization switching in a Nd-doped fiber laser," Opt. Lett. 19, 1971 (1994).
[CrossRef]

Silberberg, Y.

Stephan, G.

R. Leners, T. Georges, P. L. Francois, and G. Stephan, "Analytic model of polarization dependent gain in erbium doped fibre amplifier," in Optical Amplifiers and Their Applications, Vol. 14 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper PD1.

B. Meziane, F. Sanchez, G. Stephan, and P. L. Francois, "Feedback induced polarization switching in a Nd-doped fiber laser," Opt. Lett. 19, 1971 (1994).
[CrossRef]

R. Leners, P. L. Francois, and G. Stephan, "Polarization cross-saturation in fiber lasers," in International Quantum Electronics Conference, Vol. 9 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 212.

R. Leners, P. L. Francois, and G. Stephan, "Simultaneous effects of gain and loss anisotropies on the thresholds of a bipolarization fiber laser," Opt. Lett. 19, 275 (1994).
[CrossRef] [PubMed]

R. Leners and G. Stephan, "Rate equations analysis of a multi-mode, bipolarization Nd3+-doped fibre laser," J. Europ. Opt. Soc. (to be published).

Stock, M. L.

Tamura, K.

K. Tamura, H. A. Haus, and E. P. Ippen, "Self-starting additive pulse mode-locked erbium fibre ring laser," Electron. Lett. 28, 2226 (1992).
[CrossRef]

Taylor, M. G.

M. G. Taylor, "Improvement in Q with low frequency polarization modulation on transoceanic EDFA link," IEEE Photon. Technol. Lett. 6, 860 (1994).
[CrossRef]

M. G. Taylor and S. J. Penticost, "Improvement in performance of long haul EDFA link using high frequency polarisation modulation," Electron. Lett. 30, 805 (1994).
[CrossRef]

M. G. Taylor, "Observation of a new polarisation dependence effect in long haul optically amplified system," in Optical Fiber Communication Conference and International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper PD5.

Ueda, T.

Uhel, R.

V. Letellier, G. Bassier, P. Marmier, R. Morin, R. Uhel, and J. Artur, "Polarisation scrambling in 5 Gbit/s 8100 km EDFA based system," Electron. Lett. 30, 46 (1994).
[CrossRef]

Weber, M. J.

D. W. Hall, R. A. Haas, W. F. Krupke, and M. J. Weber, "Spectral and polarization hole burning in neodymium glass lasers," IEEE J. Quantum Electron. QE-19, 1704 (1983).
[CrossRef]

D. W. Hall and M. J. Weber, "Polarized fluorescence line narrowing measurements of Nd laser glasses: evidence of stimulated emission cross section anisotropy," Appl. Phys. Lett. 42, 157 (1983).
[CrossRef]

Wilson, S. J.

R. W. Keys, S. J. Wilson, M. Healy, S. R. Baker, A. Robinson, and J. E. Righton, "Polarization-dependent gain in erbium-doped fibers," in Optical Fibre Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper FF5.

Wysocki, P. F.

P. F. Wysocki, "Computer modeling of polarization hole burning in EDFAs," in Optical Fiber Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper FF6.

P. F. Wysocki, "Polarization hole-burning in erbium doped fiber amplifiers with birefringence," in Optical Amplifiers and Their Applications, Vol. 14 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper ThB4.

Zyskind, J. L.

V. J. Mazurczyk and J. L. Zyskind, "Polarisation dependent gain in erbium doped fiber amplifiers," IEEE Photon. Technol. Lett. 6, 616 (1994).
[CrossRef]

V. J. Mazurczyk and J. L. Zyskind, "Polarization hole burning in erbium doped fiber amplifiers," in Conference on Lasers and Electro-Optics, Vol. 12 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper CPD26.

Appl. Phys. Lett. (1)

D. W. Hall and M. J. Weber, "Polarized fluorescence line narrowing measurements of Nd laser glasses: evidence of stimulated emission cross section anisotropy," Appl. Phys. Lett. 42, 157 (1983).
[CrossRef]

Electron. Lett. (6)

E. J. Greer, D. J. Lewis, and W. M. Macaulay, "Polarisation dependent gain in erbium doped fibre amplifiers," Electron. Lett. 30, 46 (1994).
[CrossRef]

Y. Fukada, T. Imai, and A. Mamoru, "BER fluctuation suppression in optical in-line amplifier system using polarisation scrambling techniques," Electron. Lett. 30, 432 (1994).
[CrossRef]

M. G. Taylor and S. J. Penticost, "Improvement in performance of long haul EDFA link using high frequency polarisation modulation," Electron. Lett. 30, 805 (1994).
[CrossRef]

V. Letellier, G. Bassier, P. Marmier, R. Morin, R. Uhel, and J. Artur, "Polarisation scrambling in 5 Gbit/s 8100 km EDFA based system," Electron. Lett. 30, 46 (1994).
[CrossRef]

V. J. Matsas, T. P. Newson, D. J. Richardson, and D. N. Payne, "Selfstarting passively mode-locked fibre ring soliton laser exploiting nonlinear polarisation rotation," Electron. Lett. 28, 1391 (1992).
[CrossRef]

K. Tamura, H. A. Haus, and E. P. Ippen, "Self-starting additive pulse mode-locked erbium fibre ring laser," Electron. Lett. 28, 2226 (1992).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. W. Hall, R. A. Haas, W. F. Krupke, and M. J. Weber, "Spectral and polarization hole burning in neodymium glass lasers," IEEE J. Quantum Electron. QE-19, 1704 (1983).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

M. G. Taylor, "Improvement in Q with low frequency polarization modulation on transoceanic EDFA link," IEEE Photon. Technol. Lett. 6, 860 (1994).
[CrossRef]

V. J. Mazurczyk and J. L. Zyskind, "Polarisation dependent gain in erbium doped fiber amplifiers," IEEE Photon. Technol. Lett. 6, 616 (1994).
[CrossRef]

F. Bruyere and O. Audouin, "Penalties in long-haul optical amplifier systems due to polarization dependent loss and gain," IEEE Photon. Technol. Lett. 6, 654 (1994).
[CrossRef]

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

Opt. Lett. (4)

Phys. Rev. A (1)

S. Bielawski, D. Derozier, and P. Glorieux, "Antiphase dynamics and polarizations effects in the Nd-doped fiber laser," Phys. Rev. A 46, 2811 (1992).
[CrossRef] [PubMed]

Sov. Phys. Solid State (1)

V. P. Lebedev and A. K. Przhevuskii, "Polarized luminescence of rare-earth activated glasses," Sov. Phys. Solid State 19, 1389 (1977).

Other (16)

R. Leners, P. L. Francois, and G. Stephan, "Polarization cross-saturation in fiber lasers," in International Quantum Electronics Conference, Vol. 9 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 212.

J. T. Lin, P. R. Morkel, L. Reekie, and D. N. Payne, "Polarization effects in fiber lasers," in Conference on Lasers and Electro-Optics, Vol. 14 of 1987 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1987), p. 107.

J. T. Lin, L. Reekie, D. N. Payne, and S. B. Poole, "Intensity dependent polarization frequency splitting in an Er3+-doped fiber laser," in Conference on Lasers and Electro-Optics, Vol. 7 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper TuM28.

J. T. Lin, W. A. Gambling, and D. N. Payne, "Modeling of polarization effects in fiber lasers," in Conference on Lasers and Electro-Optics, Vol. 11 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 90.

J. T. Lin and W. A. Gambling, "Polarisation effects in fibre lasers: phenomena, theory and applications," in Fiber Laser Sources and Amplifiers II, M. J. Digonnet and E. Snitzer, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1373, 42 (1990).
[CrossRef]

M. G. Taylor, "Observation of a new polarisation dependence effect in long haul optically amplified system," in Optical Fiber Communication Conference and International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper PD5.

E. Lichtman, "Limitations imposed by polarization dependent gain and loss on all-optical ultralong communication systems," in Optical Fiber Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 257.

R. W. Keys, S. J. Wilson, M. Healy, S. R. Baker, A. Robinson, and J. E. Righton, "Polarization-dependent gain in erbium-doped fibers," in Optical Fibre Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper FF5.

V. J. Mazurczyk and J. L. Zyskind, "Polarization hole burning in erbium doped fiber amplifiers," in Conference on Lasers and Electro-Optics, Vol. 12 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper CPD26.

N. S. Bergano, V. J. Mazurczyk, and C. R. Davidson, "Polarization scrambling improves SNR performance in a chain of EDFAs," in Optical Fiber Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper ThR2.

N. S. Bergano, "Time dynamics of polarization hole burning in an EDFA," in Optical Fiber Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper FF4.

V. J. Mazurczyk and C. D. Poole, "The effect of birefringence on polarization hole burning in erbium doped fiber amplifiers," in Optical Amplifiers and Their Applications, Vol. 14 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper ThB3.

P. F. Wysocki, "Computer modeling of polarization hole burning in EDFAs," in Optical Fiber Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper FF6.

P. F. Wysocki, "Polarization hole-burning in erbium doped fiber amplifiers with birefringence," in Optical Amplifiers and Their Applications, Vol. 14 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper ThB4.

R. Leners, T. Georges, P. L. Francois, and G. Stephan, "Analytic model of polarization dependent gain in erbium doped fibre amplifier," in Optical Amplifiers and Their Applications, Vol. 14 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper PD1.

R. Leners and G. Stephan, "Rate equations analysis of a multi-mode, bipolarization Nd3+-doped fibre laser," J. Europ. Opt. Soc. (to be published).

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Figures (17)

Fig. 1
Fig. 1

Three-level system of an EDFA.

Fig. 2
Fig. 2

Poincaré sphere and Stokes parameters.

Fig. 3
Fig. 3

Absorption and amplification of the pump (solid curve) and the signal (dashed curve) fields along the fiber.

Fig. 4
Fig. 4

Longitudinal variation of the average population inversion D0.

Fig. 5
Fig. 5

Longitudinal variation of D1 (long-dashed curve) and D2 (short-dashed curve) for ϕs = 0 and ϕp = 0.

Fig. 6
Fig. 6

Longitudinal variation of D1 (long-dashed curve) and D2 (short-dashed curve) for ϕs = π/4 and ϕp = 0.

Fig. 7
Fig. 7

Longitudinal variation of D1 (long-dashed curve) and D2 (short-dashed curve) for ϕs = 0 and ϕp = π/4.

Fig. 8
Fig. 8

Longitudinal variation of D1 (long-dashed curve) and D2 (short-dashed curve) for ϕs = π/4 and ϕp = π/4.

Fig. 9
Fig. 9

Gain difference ΔG as a function of ϕs for five values of ϕp.

Fig. 10
Fig. 10

Gain difference ΔG as a function of P s in for ϕp = 0° and five values of ϕs.

Fig. 11
Fig. 11

Gain difference ΔG as a function of P s in for ϕp = 45° and three values of ϕs.

Fig. 12
Fig. 12

Gain G (dashed curve) and average gain difference Δ G ¯ (solid curve) as a function of P s in.

Fig. 13
Fig. 13

Relative average gain difference Δ G ¯ / G as a function of G.

Fig. 14
Fig. 14

Relative standard deviation of ΔG as a function of P s in for six values of ϕp.

Fig. 15
Fig. 15

Average gain difference Δ G ¯ as a function of P s in for P p in = 10 mW.

Fig. 16
Fig. 16

Average gain difference Δ G ¯ as a function of P s in for P p in = 20 mW.

Fig. 17
Fig. 17

Average gain difference Δ G ¯ as a function of P s in for P p in = 30 mW.

Tables (1)

Tables Icon

Table 1 Amplifier Parameters

Equations (105)

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

E Q ( r , t ) = ½ ψ q ( r ) j = x , y E q j ( z , t ) exp [ i ( ω q t - k q j z ) ] e ^ j + c . c . ½ ψ q ( r ) j = x , y E q j ( z , t ) U q j ( z ) exp ( i ω q t ) + c . c . ½ ψ q ( r ) E q ( z , t ) exp ( i ω q t ) + c . c .             ( Q = P , S ;             q = p , s ) ,
U q j ( z ) = exp ( - i k q j z ) e ^ j , E q ( z , t ) = j = x , y E q j ( z , t ) exp ( - i k q j z ) e ^ j .
( z + 1 V q t ) E q j = N Γ q 2 E q · Σ q e D - Σ q a ( 1 - D ) · U q j * ,
d D d t = - D τ - q = p , s 1 2 ɛ 0 c n q N Γ q ω q × E q * · [ Σ q e D - Σ q a ( 1 - D ) ] · E q .
Σ μ ( α ) = R ( α ) Σ μ R - 1 ( α )             ( μ = q a , q e ) ,
Σ μ = [ σ μ ( 1 ) 0 0 σ μ ( 2 ) ]
R ( α ) = [ cos α - sin α sin α cos α ] .
F ( α ) = 0 2 π d α 2 π F ( α ) .
Γ q = 0 d r r ψ q ( r ) 2 f ( r ) 0 d r r ψ q ( r ) 2 ,             Γ q = 0 d r r ψ q ( r ) 2 f ( r ) 0 d r r f ( r ) ,
Σ μ ( α ) = σ μ 1 + 1 - β μ 1 + β μ S ( α ) ,
S ( α ) = [ cos 2 α sin 2 α sin 2 α - cos 2 α ] .
( z + 1 V q t ) I q = N Γ q [ ( σ q a + σ q e ) D - σ q a ] I q ,
d D d t = - D τ - q N Γ q ω q [ ( σ q a + σ q e ) D - σ q a ] I q ,
I q = ½ ɛ 0 c n q E q 2 .
E q ( z , t ) = E q ( z , t ) [ a q ( z , t ) U q x ( z ) + b q ( z , t ) U q y ( z ) ] .
a q 2 + b q 2 = 1.
s 1 q = a q 2 - b q 2 ,
s 2 q = a q b q * + a q * b q ,
s 3 q = 1 i ( a q b q * - a q * b q ) .
a q = cos ϕ q ,
b q = sin ϕ q ,
( z + 1 V q t ) I q = N Γ q σ q ( D 0 + c 1 q D 1 + c 2 q D 2 - d q ) I q g q I q ,
D 0 = D ,
D 1 = D cos 2 α ,
D 2 = D sin 2 α ;
c 1 q = 1 - β q 1 + β q ( a q 2 - b q 2 ) = 1 - β q 1 + β q s 1 q ,
c 2 q = 1 - β q 1 + β q [ a q b q * exp ( - φ q ) + a q * b q exp ( i φ q ) ] = 1 - β q 1 + β q ( s 2 q cos φ q + s 3 q sin φ q ) .
φ q = 2 π z L q , beat ,
d q = σ q a σ q a + σ q e .
D ( z , t , α ) = D 0 ( z , t ) + 2 D 1 ( z , t ) cos 2 α + 2 D 2 ( z , t ) sin 2 α .
τ d d t ( D 0 D 1 D 2 ) = - ( D 0 D 1 D 2 ) - q = p , s [ 1 c 1 q c 2 q c 1 q 2 1 0 c 2 q 2 0 1 ] ( D 0 - d q D 1 D 2 ) P q P q sat .
P q = A q I q ,
P q sat = A ω q Γ q σ q τ ,
d J q d z = g q J q .
D 0 = c 0 d 0 - ½ ( c 1 d 1 + c 2 d 2 ) c 0 2 - ½ ( c 1 2 + c 2 2 ) ,
D 1 = d 1 - c 1 D 0 2 c 0 ,
D 2 = d 2 - c 2 D 0 2 c 0 ,
c 0 = 1 + q J q , c 1 = q c 1 q J q , c 2 = q c 2 q J q , d 0 = q d q J q , d 1 = q c 1 q d q J q , d 2 = q c 2 q d q J q .
D 0 = d 0 c 0 + 0 ( 2 ) ,
D 1 = 0 ( ) ,
D 2 = 0 ( ) .
g q = g q ( 0 ) + g q ( 1 ) ,
E ˜ s ( z ) = E ˜ s ( z ) [ a ˜ s ( z ) U q x ( z ) + b ˜ s ( z ) U q y ( z ) ] .
G ˜ s = 0 L d z g ˜ s = N Γ s σ s 0 L d z ( D 0 - d s ) + N Γ s σ s 1 - β s 1 + β s × 0 L d z [ s ˜ 1 s D 1 + ( s ˜ 2 s cos φ s + s ˜ 3 s sin φ s ) D 2 ] .
G = N Γ s σ s 0 L d z ( D 0 - d s ) + N Γ s σ s 1 - β s 1 + β s × 0 L d z [ s 1 s D 1 + ( s 2 s cos φ s + s 3 s sin φ s ) D 2 ] ,
G = N Γ s σ s 0 L d z ( D 0 - d s ) - N Γ s σ s 1 - β s 1 + β s × 0 L d z [ s 1 s D 1 + ( s 2 s cos φ s + s 3 s sin φ s ) D 2 ] .
Δ G = G - G = - 2 N Γ s σ s 1 - β s 1 + β s × 0 L d z [ s 1 s D 1 + ( s 2 s cos φ s + s 3 s sin φ s ) D 2 ] .
D 1 = 1 2 q c 1 q ( d q - D 0 ) J q 1 + r J r ,
D 2 = 1 2 q c 2 q ( d q - D 0 ) J q 1 + r J r ,
g q = N Γ q σ p ( D 0 - d q ) + 0 ( 2 ) .
D 1 = - 1 2 N q c 1 q Γ q σ q g q J q 1 + r J r + 0 ( 3 ) = - 1 2 N q 1 - β q 1 + β q s 1 q Γ q σ q J q 1 + r J r + 0 ( 3 ) ,
D 2 = - 1 2 N q 1 - β q 1 + β q s 2 q cos φ q + s 3 q sin φ q Γ q σ q J q 1 + r J r + 0 ( 3 ) .
Δ G = q γ q 0 L [ s 1 s s 1 q ( s 2 s cos φ s + s 3 s sin φ s ) × ( s 2 q cos φ q + s 3 q sin φ q ) ] J q d z 1 + r J r + 0 ( 4 ) ,
γ q = ( 1 - β s 1 + β s ) ( 1 - β q 1 + β q ) Γ s σ s Γ q σ p .
Δ G = q γ q ( s 1 s s 1 q + s 2 s s 2 q ) 0 L J q d z 1 + r J r ,
Δ G q γ q [ s 1 s s 1 q + 0 L d z L ( s 2 s cos φ s + s 3 s sin φ s ) × ( s 2 q cos φ q + s 3 q sin φ q ) ] 0 L J q d z 1 + r J r .
0 L d z L cos φ s sin φ q θ 0 L d z L sin φ s cos φ q ,
0 L d z L cos φ s cos φ q ½ δ s q 0 L d z L sin φ s sin φ q ,
Δ G γ s [ s 1 s 2 + ½ ( s 2 s 2 + s 3 s 2 ) ] 0 L J s d z 1 + r J r + γ p s 1 p s 1 s 0 L J p d z 1 + r J r = q γ q s 1 s s 1 q + δ s q 1 + δ s q 0 L J q d z 1 + r J r ,
Δ G = q γ q cos 2 ϕ s cos 2 ϕ q + δ s q 1 + δ s q 0 L J q d z 1 + r J r .
Δ G q γ q cos 2 ϕ s cos 2 ϕ q + δ s q 1 + δ s q J q out - J q in 1 + r J r out .
s 1 q = sin χ q cos ψ q ,
s 2 q = sin χ q sin ψ q ,
s 3 q = cos χ q .
Δ G ¯ = 1 4 π 0 2 π d ψ s 0 π d χ s sin χ s Δ G .
Δ G ¯ = 2 3 ( 1 - β s 1 + β s ) 2 J s out - J s in 1 + r J r out .
( Δ G 2 ¯ - Δ G ¯ 2 ) 1 / 2 Δ G = 1 2 5 { 1 + 15 [ ( Γ s σ s Γ p σ p ) ( 1 - β p 1 + β p ) × ( 1 + β s 1 - β s ) cos 2 ϕ p ( J p out - J p in J s out - J s in ) ] 2 } 1 / 2 .
lim J s in ( Δ G 2 ¯ - Δ G ¯ 2 ) 1 / 2 Δ G = 1 2 5 [ 1 + 15 ( ( 1 - β p 1 + β p ) × ( 1 + β s 1 - β s ) cos 2 ϕ p 1 - N Γ p σ p L d s J p in { 1 - exp [ - N Γ p σ p L ( 1 - d s ) ] } ) ] 1 / 2 .
f q z = 1 L q , scat [ 1 sin χ q χ q ( sin χ q f q χ q ) + ( 1 sin 2 χ q - 1 ) 2 f q ψ q 2 ] .
s j q ¯ ( z = 0 ) = s j q in             ( j = 1 , 2 , 3 ) .
lim z s j q ( z ) = 0 ,
lim z s j q 2 ¯ ( z ) = 1 / 3.
lim z s j q s j s ¯ = 0.
L q , scat 2 L q , beat 2 π 2 L corr .
Δ G q γ q 0 L d z s 1 s s 1 q ¯ + δ s q 1 + δ s q J q 1 + r J r = 2 3 ( 1 - β s 1 + β s ) 2 0 L J s d z 1 + r J r + q γ q 0 L d z s 1 s s 1 q ¯ - δ s q 1 + δ s q J q 1 + r J r .
Δ G = ( 1 - β s 1 + β s ) 2 1 + cos 2 2 ϕ s 2 0 L J s d z 1 + r J r .
Δ G pp = 1 2 ( 1 - β s 1 + β s ) 2 0 L J s d z 1 + r J r .
Δ G = ( 1 - β s 1 + β s ) 2 0 L d z 1 + s 1 s 2 ( ϕ s ) ¯ 2 J s d z 1 + r J r .
Δ G pp = 1 2 ( 1 - β s 1 + β s ) 2 0 L d z [ s 1 s 2 ( 0 ° ) ¯ - s 1 s 2 ( 45 ° ) ¯ ] J s 1 + r J r .
1 2 ( 1 - β s 1 + β s ) 2 0 L d z [ s 1 s 2 ( 0 ° ) ¯ - s 1 s 2 ( 45 ° ) ¯ ] J s 1 + r J r < 1 2 ( 1 - β s 1 + β s ) 2 0 L J s d z 1 + r J r .
0 L J q d z 1 + r J r = 1 1 + ½ r ( J r in - J r out ) 0 L J q d z 1 + j ,
j = t J t - ½ ( J t in + J t out ) 1 + ½ r ( J r in + J r out ) .
0 L J q d z 1 + r J r = 1 1 + ½ r ( J r in + J r out ) × n = 0 ( - 1 ) n 0 L J q j n d z = J q out - J q in 1 + ½ r ( J r in + J r out ) + 1 1 + ½ r ( J r in + J r out ) × n = 1 ( - 1 ) n 0 L J q j n d z .
0 L J q d z 1 + r J r J q out - J q in 1 + ½ r ( J r in + J r out ) .
Δ G q γ q s 1 s s 1 q + δ s q 1 + δ s q J q out - J q in 1 + ½ r ( J r in + J r out ) .
0 L d z ( 1 + r J r ) s 1 s D 1 = - 1 2 N q 1 - β q 1 + β q s 1 s s 1 q Γ q σ q × ( J q out - J q in ) .
0 L d z ( 1 + r J r ) D 1 = ( 1 - r J r out ) 0 L d z D 1 - 0 L d z r J r 0 L d z D 1 .
0 L d z s 1 s D 1 s 1 s 2 N q 1 - β q 1 + β q s 1 q Γ q σ q J q out - J q in 1 + r J r out .
0 L d z ( 1 + r J r ) ( s 2 s cos φ s + s 3 s sin φ s ) D 2 = - 1 2 N q 1 - β q 1 + β q δ s q 2 s 2 s s 2 q + s 3 s s 3 q Γ q σ q ( J q out - J q in ) ,
0 L d z ( 1 + r J r ) ( s 2 s cos φ s + s 3 s sin φ s ) D 2 ( 1 + r J r out ) 0 L d z ( s 2 s cos φ s + s 3 s sin φ s ) D 2 ,
0 L d z ( s 2 s cos φ s + s 3 s sin φ s ) D 2 - 1 2 N q 1 - β q 1 + β q δ s q 2 s 2 s s 2 q + s 3 s s 3 q Γ q σ q J q out - J q in 1 + r J r out .
Δ G q γ q s 1 s s 1 q + δ s q 1 + δ s q J s out - J s in 1 + r J r out .
Δ G = Δ G p + Δ G s ,
Δ G p = Γ s σ s Γ p σ p ( 1 - β p 1 + β p ) ( 1 - β s 1 + β s ) × cos 2 ϕ p cos 2 ϕ s J p out - J p in 1 + r J r out ,
Δ G s = ( 1 - β s 1 + β s ) 2 1 + cos 2 2 ϕ s 2 J s out - J s in 1 + r J r out .
Δ G ( 2 ) = - Δ G p + Δ G s .
Δ G ( 3 ) = Δ G s .
Δ G ( 1 ) 135 mdB ,
Δ G ( 2 ) 0 mdB ,
Δ G ( 3 ) 70 mdB ,
Δ G ( 3 ) = ½ [ Δ G ( 1 ) + Δ G ( 2 ) ] .
J s out - J s in 1 + r J r out J s out - J s in J s out = 1 - exp ( - G ) 1 ,
Δ G ( 3 ) = ( 1 - β s 1 + β s ) 2 1 + cos 2 2 ϕ s 2 .
β s = 0.8             for             ϕ s = 0 ° ,
β s = 0.7             for             ϕ s = 45 ° .

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