Abstract

The linear propagation of pulses driven by random polarization-mode dispersion is considered. Analytical expressions are derived for the probability-density functions of the pulse width, timing displacement, and degree of polarization. The study is performed in Stokes space, and frequency correlation between modes is shown to play an important role in it.

© 2002 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. B. Bahkshi, J. Handryd, P. A. Andrekson, J. Brentel, E. Kollveit, B. E. Olsson, and M. Karlsson, “Experimental observation of soliton robustness to polarization dispersion pulse broadening,” Electron. Lett. 35, 65–66 (1999).
    [CrossRef]
  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]
  3. C. R. Menyuk and P. K. A. Wai, “Polarization evolution and dispersion in fibers with spatially varying birefringence,” J. Opt. Soc. Am. B 11, 1288–1296 (1994).
    [CrossRef]
  4. 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]
  5. M. Karlsson and J. Brentel, “Autocorrelation function of the polarization-mode dispersion vector,” Opt. Lett. 24, 939–941 (1999).
    [CrossRef]
  6. M. Shtaif and A. Mecozzi, “Study of the frequency autocorrelation of the differential group delay in fibers with polarization mode dispersion,” Opt. Lett. 25, 707–709 (2000).
    [CrossRef]
  7. F. Bruyere, “Impact of first- and second-order PMD in optical digital transmission systems,” Opt. Fiber Technol. Mater., Devices Syst. 2, 269–280 (1996).
    [CrossRef]
  8. C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr medium,” IEEE J. Quantum Electron. 25, 2674–2682 (1989).
    [CrossRef]
  9. M. Matsumoto, Y. Akagi, and A. Hasegawa, “Propagation of solitons in fibers with randomly varying birefringence: effects of soliton transmission control,” J. Lightwave Technol. 15, 584–589 (1997).
    [CrossRef]
  10. C. D. Poole and R. E. Wagner, “Phenomenological approach to polarization dispersion in long single-mode fibers,” Electron. Lett. 22, 1029–1030 (1986).
    [CrossRef]
  11. N. Gisin, J. P. Pellaux, and J. P. Von der Weid, “Polarization mode dispersion for short and long single-mode fibers,” J. Lightwave Technol. 9, 821–827 (1991).
    [CrossRef]
  12. T. I. Lakoba and D. J. Kaup, “Perturbation theory for the Manakov soliton and its application to pulse propagation in randomly birefringent fibers,” Phys. Rev. E 56, 6147–6165 (1997).
    [CrossRef]
  13. M. C. de Lignie, H. G. J. Nagel, and M. O. van Deventer, “Large polarization mode dispersion in fiber optic cables,” J. Lightwave Technol. 12, 1325–1329 (1994).
    [CrossRef]
  14. A. Galtarossa, G. Gianello, C. G. Someda, and M. Schiano, “In-field comparison among polarization mode dispersion measurement techniques,” J. Lightwave Technol. 14, 42–49 (1996).
    [CrossRef]
  15. C. Xie, M. Karlsson, and P. A. Andrekson, “Soliton robustness to the polarization-mode dispersion in optical fibers,” IEEE Photonics Technol. Lett. 12, 801–803 (2000).
    [CrossRef]
  16. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).
  17. M. Karlsson, “Polarization mode dispersion-induced pulse broadening in optical fibers,” Opt. Lett. 23, 688–690 (1998).
    [CrossRef]
  18. L. Arnold, Stochastic Differential Equations: Theory and Applications (Wiley, New York, 1974).
  19. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New-York, 1965).
  20. Y. Li and A. Yariv, “Solutions to the dynamical equation of polarization-mode dispersion and polarization-dependent losses,” J. Opt. Soc. Am. B 17, 1821–1827 (2000).
    [CrossRef]

2000 (3)

1999 (2)

M. Karlsson and J. Brentel, “Autocorrelation function of the polarization-mode dispersion vector,” Opt. Lett. 24, 939–941 (1999).
[CrossRef]

B. Bahkshi, J. Handryd, P. A. Andrekson, J. Brentel, E. Kollveit, B. E. Olsson, and M. Karlsson, “Experimental observation of soliton robustness to polarization dispersion pulse broadening,” Electron. Lett. 35, 65–66 (1999).
[CrossRef]

1998 (1)

1997 (2)

T. I. Lakoba and D. J. Kaup, “Perturbation theory for the Manakov soliton and its application to pulse propagation in randomly birefringent fibers,” Phys. Rev. E 56, 6147–6165 (1997).
[CrossRef]

M. Matsumoto, Y. Akagi, and A. Hasegawa, “Propagation of solitons in fibers with randomly varying birefringence: effects of soliton transmission control,” J. Lightwave Technol. 15, 584–589 (1997).
[CrossRef]

1996 (3)

F. Bruyere, “Impact of first- and second-order PMD in optical digital transmission systems,” Opt. Fiber Technol. Mater., Devices Syst. 2, 269–280 (1996).
[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]

A. Galtarossa, G. Gianello, C. G. Someda, and M. Schiano, “In-field comparison among polarization mode dispersion measurement techniques,” J. Lightwave Technol. 14, 42–49 (1996).
[CrossRef]

1994 (2)

M. C. de Lignie, H. G. J. Nagel, and M. O. van Deventer, “Large polarization mode dispersion in fiber optic cables,” J. Lightwave Technol. 12, 1325–1329 (1994).
[CrossRef]

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

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]

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

1989 (1)

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr medium,” IEEE J. Quantum Electron. 25, 2674–2682 (1989).
[CrossRef]

1986 (1)

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

Akagi, Y.

M. Matsumoto, Y. Akagi, and A. Hasegawa, “Propagation of solitons in fibers with randomly varying birefringence: effects of soliton transmission control,” J. Lightwave Technol. 15, 584–589 (1997).
[CrossRef]

Andrekson, P. A.

C. Xie, M. Karlsson, and P. A. Andrekson, “Soliton robustness to the polarization-mode dispersion in optical fibers,” IEEE Photonics Technol. Lett. 12, 801–803 (2000).
[CrossRef]

B. Bahkshi, J. Handryd, P. A. Andrekson, J. Brentel, E. Kollveit, B. E. Olsson, and M. Karlsson, “Experimental observation of soliton robustness to polarization dispersion pulse broadening,” Electron. Lett. 35, 65–66 (1999).
[CrossRef]

Bahkshi, B.

B. Bahkshi, J. Handryd, P. A. Andrekson, J. Brentel, E. Kollveit, B. E. Olsson, and M. Karlsson, “Experimental observation of soliton robustness to polarization dispersion pulse broadening,” Electron. Lett. 35, 65–66 (1999).
[CrossRef]

Brentel, J.

B. Bahkshi, J. Handryd, P. A. Andrekson, J. Brentel, E. Kollveit, B. E. Olsson, and M. Karlsson, “Experimental observation of soliton robustness to polarization dispersion pulse broadening,” Electron. Lett. 35, 65–66 (1999).
[CrossRef]

M. Karlsson and J. Brentel, “Autocorrelation function of the polarization-mode dispersion vector,” Opt. Lett. 24, 939–941 (1999).
[CrossRef]

Bruyere, F.

F. Bruyere, “Impact of first- and second-order PMD in optical digital transmission systems,” Opt. Fiber Technol. Mater., Devices Syst. 2, 269–280 (1996).
[CrossRef]

de Lignie, M. C.

M. C. de Lignie, H. G. J. Nagel, and M. O. van Deventer, “Large polarization mode dispersion in fiber optic cables,” J. Lightwave Technol. 12, 1325–1329 (1994).
[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]

Galtarossa, A.

A. Galtarossa, G. Gianello, C. G. Someda, and M. Schiano, “In-field comparison among polarization mode dispersion measurement techniques,” J. Lightwave Technol. 14, 42–49 (1996).
[CrossRef]

Gianello, G.

A. Galtarossa, G. Gianello, C. G. Someda, and M. Schiano, “In-field comparison among polarization mode dispersion measurement techniques,” J. Lightwave Technol. 14, 42–49 (1996).
[CrossRef]

Gisin, N.

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

Handryd, J.

B. Bahkshi, J. Handryd, P. A. Andrekson, J. Brentel, E. Kollveit, B. E. Olsson, and M. Karlsson, “Experimental observation of soliton robustness to polarization dispersion pulse broadening,” Electron. Lett. 35, 65–66 (1999).
[CrossRef]

Hasegawa, A.

M. Matsumoto, Y. Akagi, and A. Hasegawa, “Propagation of solitons in fibers with randomly varying birefringence: effects of soliton transmission control,” J. Lightwave Technol. 15, 584–589 (1997).
[CrossRef]

Karlsson, M.

C. Xie, M. Karlsson, and P. A. Andrekson, “Soliton robustness to the polarization-mode dispersion in optical fibers,” IEEE Photonics Technol. Lett. 12, 801–803 (2000).
[CrossRef]

B. Bahkshi, J. Handryd, P. A. Andrekson, J. Brentel, E. Kollveit, B. E. Olsson, and M. Karlsson, “Experimental observation of soliton robustness to polarization dispersion pulse broadening,” Electron. Lett. 35, 65–66 (1999).
[CrossRef]

M. Karlsson and J. Brentel, “Autocorrelation function of the polarization-mode dispersion vector,” Opt. Lett. 24, 939–941 (1999).
[CrossRef]

M. Karlsson, “Polarization mode dispersion-induced pulse broadening in optical fibers,” Opt. Lett. 23, 688–690 (1998).
[CrossRef]

Kaup, D. J.

T. I. Lakoba and D. J. Kaup, “Perturbation theory for the Manakov soliton and its application to pulse propagation in randomly birefringent fibers,” Phys. Rev. E 56, 6147–6165 (1997).
[CrossRef]

Kollveit, E.

B. Bahkshi, J. Handryd, P. A. Andrekson, J. Brentel, E. Kollveit, B. E. Olsson, and M. Karlsson, “Experimental observation of soliton robustness to polarization dispersion pulse broadening,” Electron. Lett. 35, 65–66 (1999).
[CrossRef]

Lakoba, T. I.

T. I. Lakoba and D. J. Kaup, “Perturbation theory for the Manakov soliton and its application to pulse propagation in randomly birefringent fibers,” Phys. Rev. E 56, 6147–6165 (1997).
[CrossRef]

Li, Y.

Matsumoto, M.

M. Matsumoto, Y. Akagi, and A. Hasegawa, “Propagation of solitons in fibers with randomly varying birefringence: effects of soliton transmission control,” J. Lightwave Technol. 15, 584–589 (1997).
[CrossRef]

Mecozzi, A.

Menyuk, C. R.

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

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

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr medium,” IEEE J. Quantum Electron. 25, 2674–2682 (1989).
[CrossRef]

Nagel, H. G. J.

M. C. de Lignie, H. G. J. Nagel, and M. O. van Deventer, “Large polarization mode dispersion in fiber optic cables,” J. Lightwave Technol. 12, 1325–1329 (1994).
[CrossRef]

Olsson, B. E.

B. Bahkshi, J. Handryd, P. A. Andrekson, J. Brentel, E. Kollveit, B. E. Olsson, and M. Karlsson, “Experimental observation of soliton robustness to polarization dispersion pulse broadening,” Electron. Lett. 35, 65–66 (1999).
[CrossRef]

Pellaux, J. P.

N. Gisin, J. P. Pellaux, and J. P. Von der Weid, “Polarization mode dispersion for short and long single-mode fibers,” J. Lightwave Technol. 9, 821–827 (1991).
[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 and R. E. Wagner, “Phenomenological approach to polarization dispersion in long single-mode fibers,” Electron. Lett. 22, 1029–1030 (1986).
[CrossRef]

Schiano, M.

A. Galtarossa, G. Gianello, C. G. Someda, and M. Schiano, “In-field comparison among polarization mode dispersion measurement techniques,” J. Lightwave Technol. 14, 42–49 (1996).
[CrossRef]

Shtaif, M.

Someda, C. G.

A. Galtarossa, G. Gianello, C. G. Someda, and M. Schiano, “In-field comparison among polarization mode dispersion measurement techniques,” J. Lightwave Technol. 14, 42–49 (1996).
[CrossRef]

van Deventer, M. O.

M. C. de Lignie, H. G. J. Nagel, and M. O. van Deventer, “Large polarization mode dispersion in fiber optic cables,” J. Lightwave Technol. 12, 1325–1329 (1994).
[CrossRef]

Von der Weid, J. P.

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

Wagner, R. E.

C. D. Poole and R. E. Wagner, “Phenomenological approach to polarization 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]

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

Xie, C.

C. Xie, M. Karlsson, and P. A. Andrekson, “Soliton robustness to the polarization-mode dispersion in optical fibers,” IEEE Photonics Technol. Lett. 12, 801–803 (2000).
[CrossRef]

Yariv, A.

Electron. Lett. (2)

B. Bahkshi, J. Handryd, P. A. Andrekson, J. Brentel, E. Kollveit, B. E. Olsson, and M. Karlsson, “Experimental observation of soliton robustness to polarization dispersion pulse broadening,” Electron. Lett. 35, 65–66 (1999).
[CrossRef]

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

IEEE J. Quantum Electron. (1)

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr medium,” IEEE J. Quantum Electron. 25, 2674–2682 (1989).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

C. Xie, M. Karlsson, and P. A. Andrekson, “Soliton robustness to the polarization-mode dispersion in optical fibers,” IEEE Photonics Technol. Lett. 12, 801–803 (2000).
[CrossRef]

J. Lightwave Technol. (6)

M. C. de Lignie, H. G. J. Nagel, and M. O. van Deventer, “Large polarization mode dispersion in fiber optic cables,” J. Lightwave Technol. 12, 1325–1329 (1994).
[CrossRef]

A. Galtarossa, G. Gianello, C. G. Someda, and M. Schiano, “In-field comparison among polarization mode dispersion measurement techniques,” J. Lightwave Technol. 14, 42–49 (1996).
[CrossRef]

M. Matsumoto, Y. Akagi, and A. Hasegawa, “Propagation of solitons in fibers with randomly varying birefringence: effects of soliton transmission control,” J. Lightwave Technol. 15, 584–589 (1997).
[CrossRef]

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

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

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

J. Opt. Soc. Am. B (2)

Opt. Fiber Technol. Mater., Devices Syst. (1)

F. Bruyere, “Impact of first- and second-order PMD in optical digital transmission systems,” Opt. Fiber Technol. Mater., Devices Syst. 2, 269–280 (1996).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. E (1)

T. I. Lakoba and D. J. Kaup, “Perturbation theory for the Manakov soliton and its application to pulse propagation in randomly birefringent fibers,” Phys. Rev. E 56, 6147–6165 (1997).
[CrossRef]

Other (3)

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

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New-York, 1965).

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

PDF of correlation degree C(ω1, ω2) for σ2Δω2=1.

Fig. 2
Fig. 2

Square rms widths of pulses with Gaussian (gaus) or sech shapes. Curves, theoretical (theo) values; crosses and circles, results averaged over 104 numerical (num) simulations. (a) Mean values; (b) variances.

Fig. 3
Fig. 3

Rotations of the polarization of pulses with Gaussian or sech shapes. Curves, theoretical values; crosses and circles, results averaged over 104 numerical simulations. (a) Mean values; (b) variances. Abbreviations as for Fig. 2.

Fig. 4
Fig. 4

(a) Mean-square width and (b) PDF of the pulse width of pulses with Gaussian shapes. The numerical values were computed from 3000 runs. The PDF is plotted at z=2000 km. T02=16.4 ps2 and Dp=0.1 ps/km. The PDF is plotted at z=2000 km, which corresponds to σ2z/T02=0.12. The theoretical PDF is Eq. (46).

Fig. 5
Fig. 5

(a) Degree of polarization and (b) PDF of the pulse width of pulses with Gaussian shapes. The numerical values were computed from 3000 runs. T02=3.6 ps2 and Dp=0.2 ps/km. The PDF is plotted at z=2000 km, which corresponds to σ2z/T02=2.2. The theoretical PDF is Eq. (47).

Equations (125)

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

iAz+K0A+iK1At-β2Att=0,
iUz-β2Utt=iRUt,
R(z)=m1(z)Σ1+m2(z)Σ2+m3(z)Σ3,
Σ1=0110,Σ2=0-ii0,Σ3=100-1,
mj(z)mj(z)=σ2δ(z-z),σ2=Δβ2Δz12.
Tc=t(|u|2+|v|2)dt(|u|2+|v|2)dt.
Pr=|u|2dt(|u|2+|v|2)dt.
Pd=(s12+s22+s33)1/2
E0=(|u|2+|v|2(t)dt,
s1=(|u|2-|v|2)(t)dt/E0,
s2=2 Re(u*v)(t)dt/E0,
s3=2 Im(u*v)(t)dt/E0.
uθ,ξ(t)=cos(θ)u(t)+sin(θ)eiξv(t),
vθ,ξ(t)=-sin(θ)u(t)+cos(θ)eiξv(t).
Pd1=|uθ,ξ|2dt(|u|2+|v|2)dt,
Tw12=(t-Tc)2(|u|2+|v|2)dt(|u|2+|v|2)dt.
Tw22= t2(|u|2+|v|2)dt(|u|2+|v|2)dt=Tw12+Tc2.
uˆ(ω)= u(t)exp(iωt)dt,
vˆ(ω)= v(t)exp(iωt)dt,
Uˆz=iωR(z)Uˆ,
sˆ1(ω)=(|uˆ|2-|vˆ|2)(ω)/Eˆ0(ω),
sˆ2(ω)=2 Re(uˆ*vˆ)(ω)/Eˆ0(ω),
sˆ3(ω)=2 Im(uˆ*vˆ)(ω)/Eˆ0(ω),
Eˆ0(ω)=|uˆ|2(ω)+|vˆ|2(ω)= u0(t)exp(iωt)dt2,
sˆz=2σωW˙(z)×sˆ,
dsˆ1(ω)=2σω(sˆ2°dW3-sˆ3°dW2),
dsˆ2(ω)=2σω(sˆ3°dW1-sˆ1°dW3),
dsˆ3(ω)=2σω(sˆ1°dW2-sˆ2°dW1),
C(ω1, ω2)sˆ(ω1)·sˆ(ω2).
L=2σ2Δω2C(1-C2)C,
C(ω1, ω2)=exp(-4Δω2σ2z).
p(z, C)=12n=0Pn(C)exp[-2n(n+1)σ2Δω2z]
Tc= tˆ(ω)Eˆ0(ω)dω Eˆ0(ω)dω,
tˆ(ω)Im[uˆ*(ω)uˆ(ω)]+Im[vˆ*(ω)vˆ(ω)]|uˆ|2(ω)+|vˆ|2(ω),
dtˆ=σ(sˆ1°dW1+sˆ2°dW2+sˆ3°dW3).
L=12σ22tˆ2.
p(tˆ)=12πσzexp-tˆ22σ2z.
L=12σ22tˆ12+12σ22tˆ22+σ2C2tˆ1tˆ2+2σ2Δω2C(1-C2)C.
tˆ(ω1)tˆ(ω2)=1-exp[-4(ω1-ω2)2σ2z]4(ω1-ω2)2.
Tc2= Eˆ0(ω1)Eˆ0(ω2)tˆ(ω1)tˆ(ω2)dω1dω2 Eˆ0(ω)dω2.
Tc2=T0221+4σ2zT021/2-1.
Tc2σ2zT02σ2z-4(c2-c42)σ4z2T02
σ2zT02πc1σ2z,
c1= Eˆ0(ω)2dω Eˆ0(ω)dω2=12π|u0u0|2(t)dt|u0|2(t)dt2,
c2= ω2Eˆ0(ω)dω Eˆ0(ω)dω=|u0t|2dt|u0|2dt,
c4=ωEˆ0(ω)dω Eˆ0(ω)dω=i  u0tu0*dt|u0|2dt,
Tc2n=(2n+1)!!Tc2n1+Oσ6z3T06,
p(t)=12πTc21/2exp-t22Tc2,
Tw22= Rˆ(ω)Eˆ0(ω)dω Eˆ0(ω)dω,
Rˆ(ω)|uˆ|2(ω)+|vˆ|2(ω)|uˆ|2(ω)+|vˆ|2(ω).
dRˆ=σ(rˆ1°dW1+rˆ2°dW2+rˆ3°dW3),
rˆ1(ω)=2 Im(uˆuˆ*-vˆvˆ*)/Eˆ0(ω),
rˆ2(ω)=2 Im(uˆvˆ*+vˆuˆ*)/Eˆ0(ω),
rˆ3(ω)=2 Re(uˆvˆ*-vˆuˆ*)/Eˆ0(ω),
rˆz=2σωW˙(z)×rˆ+2σW˙(z).
L=8σ2τ2τ2+12σ2τ,
p(τ)=τ1/22π(4σ2z)3/2exp-τ8σ2z,τ0.
L=8σ2τ12τ12+12σ2τ1+8σ2τ22τ22+12σ2τ2+16σ2Cp2τ1τ2+σ2(12-4Δω2Cp)Cp+2σ2[Δω2(τ1τ2-Cp2)+τ1+τ2+2Cp]2Cp2+8σ2(Cp+τ1)2Cpτ1+8σ2(Cp+τ2)2Cpτ2,
Cp=3Δω2[1-exp(-4σ2Δω2z)],
Cov(τ1, τ2)=12Δω4[4Δω2σ2z-1+exp(-4Δω2σ2z)],
Cp2=124Δω2[4+24Δω2σ2z+72Δω4σ4z2-9 exp(-4Δω2σ2z)+5 exp(-12Δω2σ2z)].
Tw22=T02+3σ2z,
Var(Tw22)= Eˆ0(ω1)Eˆ0(ω2)Cov[τ(ω1),τ(ω2)]dω1dω216 Eˆ0(ω)dω2.
Var(Tw22)=T0(T02+4σ2z)3/2-6T02σ2z-T04.
d Var(Tw22)dz=12σ2Tc2.
Var(Tw22)σ2zT026σ4z2
σ2zT028πc1(σ2z)3/2.
t2 p(u)du=(Tw22t2).
p(t2)=σ2zT02t2-T022π(σ2z)3/2exp-t2-T022σ2z,t2T02.
p(t2)=σ2zT0212π Var(Tw22)exp-(t2-Tw22)22 Var(Tw22).
p(t2)σ2zT02t2-T222πγ23/2exp-t2-T222γ2,t2T22,
γ2=[Var(Tw22)/6]1/2,
T22=Tw22-[3 Var(Tw22)/2]1/2.
Var(Tw12)=Var(Tw22)+Var(Tc2)-2 Cov(Tw22, Tc2).
Tw12=T02+3σ2z-T0221+4σ2zT021/2-1.
Var(Tw12)=T041+4σ2zT021/21+4σ2zT021/2-12.
Tw12σ2zT02T02+2σ2z
σ2zT023σ2z,
Var(Tw12)σ2zT024σ4z2
σ2zT028πc1(σ2z)3/2.
p(t2)=σ2zT0212σ2zexp-t2-T022σ2z,t2>T02.
d1=(Tw12-T02)/(σ2z),
p(t2)=(t2-T12)d1/2-12d1/2Γ(d1/2)γ1d1/2exp-t2-T122γ1,
γ1=[Var(Tw12)/(2d1)]1/2,
T12=Tw12-[d1 Var(Tw12)/2]1/2.
Pr=12+12 sˆ1(ω)Eˆ0(ω)dω Eˆ0(ω)dω.
sˆ1(ω)=exp(-4ω2σ2z),
Cov[sˆ1(ω1)sˆ1(ω2)]
= exp[-4(ω1-ω2)2σ2z]+ exp[-4(ω12+ω22+ω1ω2)σ2z]-exp[-4(ω12+ω22)σ2z].
Pr=12+12 exp(-4ω2σ2z)Eˆ0(ω)dω Eˆ0(ω)dω,
Pr=12+121+2σ2zT02-1/2.
Prσ2zT021-2σ2zc2
σ2zT0212+c3π4σ2z,
c3=Eˆ0(0)Eˆ0(ω)dω=12π u0dt|2|u0|2dt.
Var(Pr)= Cov[sˆ1(ω1),sˆ1(ω2)]Eˆ0(ω1)Eˆ0(ω2)dω1dω24 Eˆ0(ω)dω2.
Var(Pr)=1121+4σ2zT02-1/2-141+2σ2zT02-1+161+3σ2zT02-1/21+σ2zT02-1/2.
Var(Pr)σ2zT0216σ4z2c22
σ2zT02πc16σ2z.
Pd= C(ω1, ω2)Eˆ0(ω1)Eˆ0(ω2)dω1dω21/2 Eˆ0(ω)dω.
dTc2dz=σ2Pd2.
Pd2=1+4σ2zT02-1/2.
Pd2σ2zT021-8σ2z(c2-c42),
Pd2σ2zT02πc12σz,
p(x)=14σ2(c2-c42)zexp-x4σ2(c2-c42)z.
p(x)=2πα3/2x2 exp-x22α,
Uˆz=iωR(z)Uˆ+iβ2ω2Uˆ.
Eˆ0(ω)=Eˇ0(ω),
Rˆ(ω)=Rˇ(ω)+β2ω2z2+2βωztˇ(ω),
tˆ(ω)=tˇ(ω)+βωz,
Tc=βc4z,Var(Tc)=Var(Tc)|β=0.
Tw22=T02+3σ2z+c2β2z2,
Var(Tw22)=Var(Tw22)|β=0+4β2z2× ω1ω2tˇ(ω1)tˇ(ω2)Eˇ0(ω1)Eˇ0(ω2)dω1dω2 Eˇ0(ω)dω2,
Tw12=T02+3σ2z+(c2+c42)β2z2+Var(Tc)|β=0,
Var(Tw12)=Var(Tw22)-2 Var(Tc)2.
Tw22=T02+3σ2z+β2z24T02
Var(Tw22)=T041+4σ2zT023/2-6 σ2zT02-1+β2z24{[1+(4σ2z/T02)]1/2-1}2[1+(4σ2z/T02)]1/2
Tw12=T02+3σ2z+β2z24T02-T0221+4σ2zT021/2-1
Var(Tw12)=T041+4σ2zT021/21+4σ2zT021/2-12×1+β2z24T04[1+(4σ2z/T02)].
dXi(z)dz=j=1d Mij[X(z)]W˙j(z),i=1,, n,
L=i=1n bi(x)xi+i,j=1n aij(x)2xixj,
aij(x)=k=1d Mik(x)Mjk(x),
bi(x)=j=1nk=1dMik(x)xjMjk(x).
f(X)z=Lf(X),
pz=L*p,
L*pk=-i=1nxi[bi(x)p]+i,j=1n2xixj[aij(x)p].

Metrics