Abstract

A general theory is used to describe the correlation properties of polarization mode dispersion (PMD) in a birefringent, linear, dispersive medium such as optical fibers. The theory includes the effects of frequency dependence of birefringence on all orders, and it is capable of providing statistical information about second- and higher-order correlations among the polarization and PMD vectors. We apply the general theory to study pulse broadening induced by different-order PMD and PMD-induced pulse distortion through the third- and fourth-order temporal moments (related to skewness and flatness, respectively). Our analytic results are in good agreement with numerical simulations.

© 2003 Optical Society of America

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References

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  1. C. D. Poole, “Polarization effects in lightwave systems,” in Optical Fiber Telecommunications III A, I. P. Kaminow and T. L. Koch, eds. (Academic, San Diego, Calif., 1997), Chap. 6.
  2. H. Kogelnik, R. Jopson, and L. Nelson, “Polarization-mode dispersion,” in Optical Fiber Telecommunications IV B, I. P. Kaminow and T. L. Koch, eds. (Academic, San Diego, Calif., 2002), Chap. 15.
  3. 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]
  4. C. D. Poole and R. E. Wagner, “Phenomenological approach to polarisation dispersion in long single-mode fibers,” Electron. Lett. 22, 1029-1030 (1986).
    [CrossRef]
  5. G. J. Foschini and C. D. Poole, “Statistical theory of polarization dispersion in single mode fibers,” J. Lightwave Technol. 9, 1439-1456 (1991).
    [CrossRef]
  6. G. J. Foschini, R. M. Jopson, L. E. Nelson, and H. Kogelnik, “The statistics of PMD-induced chromatic fiber dispersion,” J. Lightwave Technol. 17, 1560-1565 (1999).
    [CrossRef]
  7. G. J. Foschini, L. E. Nelson, R. M. Jopson, and H. Kogelnik, “Probability densities of second-order polarization mode dispersion including polarization dependent chromatic fiber dispersion,” IEEE Photon. Technol. Lett. 12, 293-295 (2000).
    [CrossRef]
  8. G. J. Foschini, L. E. Nelson, R. M. Jopson, and H. Kogelnik, “Statistics of second-order PMD depolarization,” J. Lightwave Technol. 19, 1882-1886 (2001).
    [CrossRef]
  9. M. Karlsson and J. Brentel, “Autocorrelation function of the polarization-mode dispersion vector,” Opt. Lett. 24, 939-941 (1999).
    [CrossRef]
  10. M. Shtaif, A. Mecozzi, and J. A. Nagel, “Mean-square magnitude of all orders of polarization mode dispersion and the relation with the bandwidth of the principal states,” IEEE Photon. Technol. Lett. 12, 53-55 (2000).
    [CrossRef]
  11. J. P. Gordon and H. Kogelnik, “PMD fundamentals: polar-ization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541-4550 (2000).
    [CrossRef]
  12. G. P. Agrawal, Fiber Optic Communication Systems, 3rd ed. (Wiley, New York, 2002), Chap. 2.
  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. F. Curti, B. Daino, G. de Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162-1166 (1990).
    [CrossRef]
  15. N. Gisin and J. P. Pellaux, “Polarization mode dispersion: time versus frequency domains,” Opt. Commun. 89, 316-323 (1992).
    [CrossRef]
  16. H. Risken, The Fokker-Planck Equation, 2nd ed. (Springer-Verlag, New York, 1989).
  17. C. W. Gardiner, Handbook of Stochastic Methods, 2nd ed. (Springer-Verlag, New York, 1985).
  18. 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]
  19. M. Karlsson, J. Brentel, and P. A. Andrekson, “Long-term measurement of PMD and polarization drift in installed fibers,” J. Lightwave Technol. 18, 941-951 (2000).
    [CrossRef]
  20. M. Karlsson, “Polarization mode dispersion-induced pulse broadening in optical fibers,” Opt. Lett. 23, 688-690 (1998).
    [CrossRef]
  21. H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Analytical theory for PMD-compensation,” IEEE Photon. Technol. Lett. 12, 50-52 (2000).
    [CrossRef]
  22. H. Bu¨low, “System outage probability due to first- and second-order PMD,” IEEE Photon. Technol. Lett. 10, 696–698 (1998).
    [CrossRef]
  23. M. C. Wang and G. E. Uhlenbeck, “On the theory of the Brownian motion II,” in Selected Papers on Noise and Stochastic Processes, N. Wax, ed. (Dover, New York, 1954), pp. 113–132.

2001 (1)

2000 (5)

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

M. Karlsson, J. Brentel, and P. A. Andrekson, “Long-term measurement of PMD and polarization drift in installed fibers,” J. Lightwave Technol. 18, 941-951 (2000).
[CrossRef]

H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Analytical theory for PMD-compensation,” IEEE Photon. Technol. Lett. 12, 50-52 (2000).
[CrossRef]

M. Shtaif, A. Mecozzi, and J. A. Nagel, “Mean-square magnitude of all orders of polarization mode dispersion and the relation with the bandwidth of the principal states,” IEEE Photon. Technol. Lett. 12, 53-55 (2000).
[CrossRef]

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

1999 (2)

1998 (3)

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]

1992 (1)

N. Gisin and J. P. Pellaux, “Polarization mode dispersion: time versus frequency domains,” Opt. Commun. 89, 316-323 (1992).
[CrossRef]

1991 (2)

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]

1990 (1)

F. Curti, B. Daino, G. de Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162-1166 (1990).
[CrossRef]

1986 (1)

C. D. Poole and R. E. Wagner, “Phenomenological approach to polarisation dispersion in long single-mode fibers,” Electron. Lett. 22, 1029-1030 (1986).
[CrossRef]

Andrekson, P. A.

M. Karlsson, J. Brentel, and P. A. Andrekson, “Long-term measurement of PMD and polarization drift in installed fibers,” J. Lightwave Technol. 18, 941-951 (2000).
[CrossRef]

H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Analytical theory for PMD-compensation,” IEEE Photon. Technol. Lett. 12, 50-52 (2000).
[CrossRef]

Brentel, J.

Bu¨low, H.

H. Bu¨low, “System outage probability due to first- and second-order PMD,” IEEE Photon. Technol. Lett. 10, 696–698 (1998).
[CrossRef]

Ciprut, P.

Curti, F.

F. Curti, B. Daino, G. de Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162-1166 (1990).
[CrossRef]

Daino, B.

F. Curti, B. Daino, G. de Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162-1166 (1990).
[CrossRef]

de Marchis, G.

F. Curti, B. Daino, G. de Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162-1166 (1990).
[CrossRef]

Foschini, G. J.

G. J. Foschini, L. E. Nelson, R. M. Jopson, and H. Kogelnik, “Statistics of second-order PMD depolarization,” J. Lightwave Technol. 19, 1882-1886 (2001).
[CrossRef]

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

G. J. Foschini, R. M. Jopson, L. E. Nelson, and H. Kogelnik, “The statistics of PMD-induced chromatic fiber dispersion,” J. Lightwave Technol. 17, 1560-1565 (1999).
[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]

Gisin, B.

Gisin, N.

Gordon, J. P.

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

Jopson, R. M.

G. J. Foschini, L. E. Nelson, R. M. Jopson, and H. Kogelnik, “Statistics of second-order PMD depolarization,” J. Lightwave Technol. 19, 1882-1886 (2001).
[CrossRef]

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

G. J. Foschini, R. M. Jopson, L. E. Nelson, and H. Kogelnik, “The statistics of PMD-induced chromatic fiber dispersion,” J. Lightwave Technol. 17, 1560-1565 (1999).
[CrossRef]

Karlsson, M.

Kogelnik, H.

G. J. Foschini, L. E. Nelson, R. M. Jopson, and H. Kogelnik, “Statistics of second-order PMD depolarization,” J. Lightwave Technol. 19, 1882-1886 (2001).
[CrossRef]

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

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

G. J. Foschini, R. M. Jopson, L. E. Nelson, and H. Kogelnik, “The statistics of PMD-induced chromatic fiber dispersion,” J. Lightwave Technol. 17, 1560-1565 (1999).
[CrossRef]

Matera, F.

F. Curti, B. Daino, G. de Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162-1166 (1990).
[CrossRef]

Mecozzi, A.

M. Shtaif, A. Mecozzi, and J. A. Nagel, “Mean-square magnitude of all orders of polarization mode dispersion and the relation with the bandwidth of the principal states,” IEEE Photon. Technol. Lett. 12, 53-55 (2000).
[CrossRef]

Menyuk, C. R.

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148-157 (1996).
[CrossRef]

Nagel, J. A.

M. Shtaif, A. Mecozzi, and J. A. Nagel, “Mean-square magnitude of all orders of polarization mode dispersion and the relation with the bandwidth of the principal states,” IEEE Photon. Technol. Lett. 12, 53-55 (2000).
[CrossRef]

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

Nelson, L. E.

G. J. Foschini, L. E. Nelson, R. M. Jopson, and H. Kogelnik, “Statistics of second-order PMD depolarization,” J. Lightwave Technol. 19, 1882-1886 (2001).
[CrossRef]

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

G. J. Foschini, R. M. Jopson, L. E. Nelson, and H. Kogelnik, “The statistics of PMD-induced chromatic fiber dispersion,” J. Lightwave Technol. 17, 1560-1565 (1999).
[CrossRef]

Passy, R.

Pellaux, J. P.

N. Gisin and J. P. Pellaux, “Polarization mode dispersion: time versus frequency domains,” Opt. Commun. 89, 316-323 (1992).
[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]

C. D. Poole and R. E. Wagner, “Phenomenological approach to polarisation dispersion in long single-mode fibers,” Electron. Lett. 22, 1029-1030 (1986).
[CrossRef]

Prieto, F.

Shtaif, M.

M. Shtaif, A. Mecozzi, and J. A. Nagel, “Mean-square magnitude of all orders of polarization mode dispersion and the relation with the bandwidth of the principal states,” IEEE Photon. Technol. Lett. 12, 53-55 (2000).
[CrossRef]

Sunnerud, H.

H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Analytical theory for PMD-compensation,” IEEE Photon. Technol. Lett. 12, 50-52 (2000).
[CrossRef]

Von der Weid, J. P.

Wagner, R. E.

C. D. Poole and R. E. Wagner, “Phenomenological approach to polarisation dispersion in long single-mode fibers,” Electron. Lett. 22, 1029-1030 (1986).
[CrossRef]

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]

Winters, J. H.

Zimmer, C. W.

Electron. Lett. (1)

C. D. Poole and R. E. Wagner, “Phenomenological approach to polarisation dispersion in long single-mode fibers,” Electron. Lett. 22, 1029-1030 (1986).
[CrossRef]

IEEE Photon. Technol. Lett. (4)

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

M. Shtaif, A. Mecozzi, and J. A. Nagel, “Mean-square magnitude of all orders of polarization mode dispersion and the relation with the bandwidth of the principal states,” IEEE Photon. Technol. Lett. 12, 53-55 (2000).
[CrossRef]

H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Analytical theory for PMD-compensation,” IEEE Photon. Technol. Lett. 12, 50-52 (2000).
[CrossRef]

H. Bu¨low, “System outage probability due to first- and second-order PMD,” IEEE Photon. Technol. Lett. 10, 696–698 (1998).
[CrossRef]

J. Lightwave Technol. (7)

G. J. Foschini, L. E. Nelson, R. M. Jopson, and H. Kogelnik, “Statistics of second-order PMD depolarization,” J. Lightwave Technol. 19, 1882-1886 (2001).
[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]

G. J. Foschini, R. M. Jopson, L. E. Nelson, and H. Kogelnik, “The statistics of PMD-induced chromatic fiber dispersion,” J. Lightwave Technol. 17, 1560-1565 (1999).
[CrossRef]

F. Curti, B. Daino, G. de Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162-1166 (1990).
[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]

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]

M. Karlsson, J. Brentel, and P. A. Andrekson, “Long-term measurement of PMD and polarization drift in installed fibers,” J. Lightwave Technol. 18, 941-951 (2000).
[CrossRef]

Opt. Commun. (1)

N. Gisin and J. P. Pellaux, “Polarization mode dispersion: time versus frequency domains,” Opt. Commun. 89, 316-323 (1992).
[CrossRef]

Opt. Lett. (3)

Proc. Natl. Acad. Sci. USA (1)

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

Other (6)

G. P. Agrawal, Fiber Optic Communication Systems, 3rd ed. (Wiley, New York, 2002), Chap. 2.

C. D. Poole, “Polarization effects in lightwave systems,” in Optical Fiber Telecommunications III A, I. P. Kaminow and T. L. Koch, eds. (Academic, San Diego, Calif., 1997), Chap. 6.

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

H. Risken, The Fokker-Planck Equation, 2nd ed. (Springer-Verlag, New York, 1989).

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

M. C. Wang and G. E. Uhlenbeck, “On the theory of the Brownian motion II,” in Selected Papers on Noise and Stochastic Processes, N. Wax, ed. (Dover, New York, 1954), pp. 113–132.

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Figures (4)

Fig. 1
Fig. 1

Probability distribution of skewness at a distance of 1000 km. The skewness was normalized to σ03. The rms width of the input Gaussian is σ0=14.14 ps. The PMD parameter of the fiber is 0.1414 ps/km. We realized 5000 random rounds. The solid curve represents the analytic results and the histogram represents numerical simulations.

Fig. 2
Fig. 2

Probability distribution of skewness at a distance of 5000 km. The skewness was normalized to σ03. The rms width of the input Gaussian is σ0=3.54 ps. The PMD parameter is the same as in Fig. 1. We realized 5000 random rounds. The solid curve represents the analytic results and the histogram represents numerical simulations.

Fig. 3
Fig. 3

Probability distribution of pulse flatness at a distance of 1000 km. The flatness was normalized to σ04. The rms width of the input Gaussian pulse is σ0=14.14 ps. The PMD parameter is the same as in Fig. 1. The solid curve represents the analytic results and the histogram represents numerical simulations.

Fig. 4
Fig. 4

Ensemble-averaged pulse flatness changes with propagation distance. The flatness is normalized to σ04. The PMD parameter is the same as in Fig. 1. The solid curve represents the analytic theoretical results and the dashed curve represents the numerical simulations.

Equations (68)

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

A˜(z, ω)=12π-+A(z, t)exp(iωt)dt,
A˜(z, ω)zzA˜xA˜y=iβ0-b1/2-b2/2+ib3/2-b2/2-ib3/2β0+b1/2A˜xA˜y.
A˜(z, ω)ziβ0(z, ω)-12b(z, ω)βA˜(z, ω),
σ1=100-1,σ2=0110,σ3=0-ii0.
S(z, ω)ω=iτ(z, ω)-12Ω(z, ω)βS(z, ω),
S(z, ω)z=b(z, ω)×S(z, ω),
Ω(z, ω)z=b(z, ω)×Ω(z, ω)+bω(z, ω),
b(z, ω)=f(ω)b(z),
b(z)=0,b(z1)b(z2)=η2Iδ(z2-z1),
S(z, ω)z=f(ω)b(z)×S(z, ω),
Ω(z, ω)z=f(ω)b(z)×Ω(z, ω)+fω(ω)b(z).
G=η22 {-2f12Ω11-2f22Ω22+[(f1ω2+f12Ω12)I-f12Ω1Ω1]:11+[(f2ω2+f22Ω22)I-f22Ω2Ω2]:22+2[f1ωf2ωI+f1f2[(Ω1Ω2)I-Ω2Ω1]]:21+2(f2f1ωΩ2-f1f2ωΩ1)(2×1)},
dΩ1Ω2dz=G(Ω1Ω2)=-η2(f1-f2)2Ω1Ω2+3η2f1ωf2ω,
dΩ1Ω2dz=-η2(f12+f22)Ω1Ω2-η2f1f2Ω2Ω1+η2[f1ω f2ω+f1 f2Ω1Ω2]I.
Ω1Ω2=3f1ω f2ω(f1-f2)2 {1-exp[-η2z(f1-f2)2]},
Ω1Ω2=f1ω f2ωI(f1-f2)2 {1-exp[-η2z(f1-f2)2]}.
S1S2=S01S02exp[-η2z(f1-f2)2],
S1S2S01S02exp[-Ω2(ω2-ω1)2/3],
Ω1Ω23(ω2-ω1)2 {1-exp [-Ω2(ω2-ω1)2/3]},
(S1Ω1)(S2Ω2)=S01S02f1ωf2ω(f1-f2)2×{1-exp[-η2z(f1-f2)2]}.
tˆ=-+tA(z, t)A(z, t)dt=-i-+A˜(z, ω)A˜ω(z, ω)dω,
t2^=-+t2A(z, t)A(z, t)dt=-+A˜ω(z, ω)A˜ω(z, ω)dω,
-+A(z, t)A(z, t)dt=-+A˜(z, ω)A˜(z, ω)dω=1.
σ2=σ02+σdisp2+σPMD2,
σPMD2=14 [Ω02¯-(Ω0S0¯)2]-[(τ+θω)(Ω0S0)¯-(τ¯+θω¯)(Ω0S0)¯].
σPMD2=14 [Ω02¯-(Ω0S0¯)2].
Ω0(z, ω)=n=0+1n!Ω0(n)ωn,
Ω0(n)=nΩ0/ωn|ω=ω0
σPMD2=14N=0+ωN¯m=0NΩ0(m)Ω0(N-m)m!(N-m)!-m=0Nωm¯ωN-m¯m!(N-m)!S0Ω0(m)Ω0(N-m)S0,
Ω0(m)Ω0(N-m)=(-1)N-mΩ0(N)Ω0(0),
m=0NΩ0(m)Ω0(N-m)m!(N-m)!
=Ω0(N)Ω0(0)m=0N(-1)N-mm!(N-m)!=0
σPMD2=14(Ω0(0))2-N=0+(S0Ω0(2N)Ω0(0)S0)×m=0Nω2m¯ω(2N-2m)¯(2m)!(2N-2m)!.
μ3-+(t-tˆ)3A(z, t)A(z, t)dt=t^3-3tˆt^2+2(tˆ)3,
μ4-+(t-tˆ)4A(z, t)A(z, t)dt=t^4-4tˆt^3+6t^2(tˆ)2-3(tˆ)4,
t^3=-+t3A(z, t)A(z, t)dt=-i-+A˜ω(z, ω)A˜ωω(z, ω)dω,
t^4=-+t4A(z, t)A(z, t)dt=-+A˜ωω(z, ω)A˜ωω(z, ω)dω.
μ3=18 {2(Ω×Ωω)S¯-Ω2(ΩS)¯+3Ω2¯ (ΩS¯)-2(ΩS¯)3}-12-+[(aω)2-2aaωω]×(ΩS)dω-(ΩS)¯-+(aω)2dω,
μ4=116 {Ω4¯+4Ωω2¯+8(ΩS¯)(Ω×Ωω)S¯-4Ω2(ΩS)¯(ΩS¯)+6Ω2¯(ΩS¯)2-3(ΩS¯)4}+12-+Ω2[(aω)2-2aaωω]dω+3(ΩS¯)2-+(aω)2dω-2(ΩS¯)-+(ΩS)×[(aω)2-2aaωω]dω+μ40,
(Ω×Ωω)S=limω2ω1ω2 (Ω1×Ω2)S1=0.
μ318 {3Ω2(ΩS)-Ω2(ΩS)-2(ΩS)3}=14 Ω0x(Ω0y2+Ω0z2),
P(μ3)=2π3Ω23/20+dyy×exp-32Ω2y+4μ3y2.
μ4μ40+116 {Ω4-4Ω2(ΩS)(ΩS)+6Ω2(ΩS)2-3(ΩS)4}=μ40+116 (Ω0y2+Ω0z2)(4Ω0x2+Ω0y2+Ω0z2).
P(μ4)=2π3Ω23/204μ4-μ40×dy exp{-3[16(μ4-μ40)+3y2]/(8yΩ2)}y[16(μ4-μ40)-y2].
μ4=σ0427x216+21x2-(2x+15)x+1+3100 (5x-4)10x+4+163tan-1(3x+3)+45625-16π33,
dRidz=Qig+Ui,i=1,2 ,, n,
g(z)=0,g(z1)g(z2)=η2Iδ(z2-z1).
Ψ(z+δz)-Ψ(z)=dΨ(z)dz δz+12d2Ψ(z)dz2 (δz)2+O[(δz)3],
dΨ(z)dz=i=1ndRidziΨ(z),
d2Ψ(z)dz2=i=1ndRidzidRidziΨ(z)+zdRidziΨ(z)+i,j=1ndRjdzdRidz:ijΨ(z),
dΨ(z)dz=limδz0Ψ(z+δz)g-Ψ(z)δz,
dΨ(z)dz=G{Ψ(z)},
G=limδz0i=1ndRidzi+δz2i=1ndRidzidRidzi+zdRidzi+i,j=1ndRjdzdRidz:ijg.
Qi=-f(ωi)Si×0-f(ωi)Ωi×+fω(ωi)I0.
G=-η2i=1nfi2(SiSi+ΩiΩi)+η22i,j=1n{fifj[(SiSj)I-SiSj]:SiSj+fifj[(ΩiΩj)I-ΩiΩj]:ΩiΩj+fiωfjωΩiΩj+[fifjωΩi-fjfiωΩj](Ωi×Ωj)+fifj[(SiΩj)I-SiΩj]:SiΩj+fifj[(ΩiSj)I-ΩiSj]:ΩiSj+fifjωSi(Si×Ωj)-fjfiωSj(Ωi×Sj)},
Tω=-i2 [Ω(z, ω)β]T,
Tωω=14 {-Ω2(z, ω)-2iΩω(z, ω)β}T.
(Ω1S1)(Ω2×Ω2ω)S2
=limω3ω2ω3 (Ω1S1)(Ω2×Ω3)S2.
Ω4¯=-+Ω4(z, ω)a2(ω)dω=15σ04x2,
Ωω2¯=-+dω1a2(ω1)limω2ω12Ω1Ω2/ω1ω2=3σ04x2,
(ΩS¯)(Ω×Ωω)S¯=-+-+dω1dω2a2(ω1)a2(ω2)(Ω1S1)(Ω2×Ω2ω)S2=2σ04xx+1 (2x+1-x-2),
Ω2(ΩS)¯(ΩS¯)=-+-+dω1dω2a2(ω1)a2(ω2)Ω12(Ω1S1)(Ω2S2)=10σ04x(x+1-1),
Ω2¯(ΩS¯)2=-+-+-+dω1dω2dω3a2(ω1)a2(ω2)a2(ω3)Ω12(Ω2S2)(Ω3S3)=σ04294875-128π93+10x+4x-1163x+1+225 (5x-4)10x+4+12833tan-1[3(x+1)].
-+Ω2[(aω)2-2aaωω]dω=9σ04x,
(ΩS¯)2-+(aω)2dω=2σ04(x+1-1),
(ΩS¯)-+(ΩS)[(aω)2-2aaωω]dω
=σ047x+8x+1-8.

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