Abstract

We study the evolution of optical signals in single-mode optical fibers in the presence of polarization-mode dispersion and polarization-dependent losses. Two geometric vectors on the Poincaré sphere are defined to characterize the effects of polarization-mode dispersion and polarization-dependent losses on the optical field in the fiber. By solving the dynamical equation for these two vectors, several general statistical results are obtained. The practically important weak polarization-dependent-loss situation is discussed in detail.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Foschini and C. Poole, “Statistical theory of polarization dispersion in single mode fibers,” J. Lightwave Technol. 9, 1439–1456 (1991).
    [CrossRef]
  2. N. Gisin, “Solutions of the dynamical equation for polarization dispersion,” Opt. Commun. 86, 371–373 (1991).
    [CrossRef]
  3. P. Wai and C. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (1996).
    [CrossRef]
  4. P. Ciprut, B. Gisin, N. Gisin, R. Passy, J. Von der Weid, F. Prieto, and C. Zimmer, “Second order polarization mode dispersion: impact on analog and digital transmissions,” J. Lightwave Technol. 16, 757–771 (1998).
    [CrossRef]
  5. G. Foschini, R. Jopson, L. Nelson, and H. Kogelnik, “The statistics of PMD-induced chromatic fiber dispersion,” J. Lightwave Technol. 17, 1560–1565 (1999).
    [CrossRef]
  6. M. Karlsson and J. Brentel, “Autocorrelation function of the polarization-mode dispersion vector,” Opt. Lett. 24, 939–941 (1999).
    [CrossRef]
  7. M. Shtaif, A. Mecozzi, and J. Nagel, “Mean-square magnitude of all orders of polarization mode dispersion and the relation with the bandwidth of the principal states,” IEEE Photonics Technol. Lett. 12, 53–55 (2000).
    [CrossRef]
  8. N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers,” Opt. Commun. 142, 119–125 (1997).
    [CrossRef]
  9. B. Huttner and N. Gisin, “Anomalous pulse spreading in birefringent optical fibers with polarization-dependent losses,” Opt. Lett. 22, 504–506 (1997).
    [CrossRef] [PubMed]
  10. B. Huttner, C. De Barros, B. Gisin, and N. Gisin, “Polarization-induced pulse spreading in birefringent optical fibers with zero differential group delay,” Opt. Lett. 24, 370–372 (1999).
    [CrossRef]
  11. L. Chen, J. Cameron, and X. Bao, “Statistics of polarization mode dispersion in presence of the polarization dependent loss in single mode fibers,” Opt. Commun. 169, 69–73 (1999).
    [CrossRef]
  12. A. Yariv, Optical Electronics in Modern Communications (Oxford University, New York, 1997).
  13. P. Dirac, The Principles of Quantum Mechanics (Oxford University, New York, 1958).
  14. L. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1968).
  15. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, Calif., 1995).
  16. A. Eyal and M. Tur, “A modified Poincaré sphere technique for the determination of polarization mode dispersion in the presence of differential loss/gain,” In Optical Fiber Communications Conference, Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 340–341.
  17. C. D. Poole and R. E. Wagner, “Phenomenological approach to polarization dispersion in long single-mode fibers,” Electron. Lett. 22, 1029–1030 (1986).
    [CrossRef]
  18. W. Horsthemke and R. Lefever, Noise Induced Transitions (Springer, Berlin, 1984).
  19. L. Arnold, Stochastic Differential Equations: Theory and Applications (Wiley, New York, 1974).
  20. B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, 5th ed. (Springer, Berlin, 1998).

2000 (1)

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

1999 (4)

1998 (1)

1997 (2)

N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers,” Opt. Commun. 142, 119–125 (1997).
[CrossRef]

B. Huttner and N. Gisin, “Anomalous pulse spreading in birefringent optical fibers with polarization-dependent losses,” Opt. Lett. 22, 504–506 (1997).
[CrossRef] [PubMed]

1996 (1)

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

1991 (2)

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

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

Bao, X.

L. Chen, J. Cameron, and X. Bao, “Statistics of polarization mode dispersion in presence of the polarization dependent loss in single mode fibers,” Opt. Commun. 169, 69–73 (1999).
[CrossRef]

Brentel, J.

Cameron, J.

L. Chen, J. Cameron, and X. Bao, “Statistics of polarization mode dispersion in presence of the polarization dependent loss in single mode fibers,” Opt. Commun. 169, 69–73 (1999).
[CrossRef]

Chen, L.

L. Chen, J. Cameron, and X. Bao, “Statistics of polarization mode dispersion in presence of the polarization dependent loss in single mode fibers,” Opt. Commun. 169, 69–73 (1999).
[CrossRef]

Ciprut, P.

De Barros, C.

Foschini, G.

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

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

Gisin, B.

Gisin, N.

Huttner, B.

Jopson, R.

Karlsson, M.

Kogelnik, H.

Mecozzi, A.

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

Menyuk, C.

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

Nagel, J.

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

Nelson, L.

Passy, R.

Poole, C.

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

Poole, C. D.

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

Prieto, F.

Shtaif, M.

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

Von der Weid, J.

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.

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

Zimmer, C.

Electron. Lett. (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]

IEEE Photonics Technol. Lett. (1)

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

J. Lightwave Technol. (4)

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

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

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

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

Opt. Commun. (3)

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

N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers,” Opt. Commun. 142, 119–125 (1997).
[CrossRef]

L. Chen, J. Cameron, and X. Bao, “Statistics of polarization mode dispersion in presence of the polarization dependent loss in single mode fibers,” Opt. Commun. 169, 69–73 (1999).
[CrossRef]

Opt. Lett. (3)

Other (8)

A. Yariv, Optical Electronics in Modern Communications (Oxford University, New York, 1997).

P. Dirac, The Principles of Quantum Mechanics (Oxford University, New York, 1958).

L. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1968).

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, Calif., 1995).

A. Eyal and M. Tur, “A modified Poincaré sphere technique for the determination of polarization mode dispersion in the presence of differential loss/gain,” In Optical Fiber Communications Conference, Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 340–341.

W. Horsthemke and R. Lefever, Noise Induced Transitions (Springer, Berlin, 1984).

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

B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, 5th ed. (Springer, Berlin, 1998).

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 (7)

Fig. 1
Fig. 1

Ensemble average of Ω·Λ and Ω2-Λ2.

Fig. 2
Fig. 2

Ensemble average of Λ2.

Fig. 3
Fig. 3

Ensemble average of (Ω·Λ)2.

Fig. 4
Fig. 4

Simulation of Ω1,2,32 in the rotating coordinate system. Simulation parameters: h=20 (m),α=10-3 (m-1), and β=1.67×10-13 (sec m-1).

Fig. 5
Fig. 5

Simulation of Λ1,2,32 in the rotating coordinate system. Simulation parameters: h=20 (m),α=10-3 (m-1), and β=1.67×10-13 (sec m-1).

Fig. 6
Fig. 6

Simulation of PDL effect on the mean-square DGD for zΓ. The solid line is a theoretical prediction of Eq. (44), and discrete circles are numerical data. The numerical value of Γ for the simulation is 1875 (km).

Fig. 7
Fig. 7

Simulation of τ4 for zΓ. The solid line is a theoretical prediction of Eq. (45), and circles are numerical data. The Γ value for this simulation is 20 (km).

Equations (48)

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

|φ=T(ω)|φ0.
ω|φ=TωT-1|φ.
TωT-1=-i2 W·σ,
σ1=0110,σ2=0-ii0,σ3=100-1.
Wz=β(z)+[ωβ(z)+iα(z)]×W,
ω|φ=-i2(Ω+iΛ)·σ|φ.
Sφ|σ|φφ|φ.
Sω=Ω×S-(Λ×S)×S.
σmσn=δmnI+il=13mnlσl,
S/ω=ΩPMD×S,
Ω-Λ×Sp=λSp,
Ωz=β+ωβ×Ω-α×Λ,
Λz=ωβ×Λ+α×Ω.
ξz=0,ξzξz=2hδ(z-z),ξzdθ/dz.
z Ω1Ω2Ω3Λ1Λ2Λ3=Ω2-Ω10Λ2-Λ10ξz+β-ωβΩ3+αΛ3ωβΩ2-αΛ20-αΩ3-ωβΛ3αΩ2+ωβΛ2.
zf=Gˆf.
Gˆ=β Ω1-(ωβΩ3-αΛ3) Ω2+(ωβΩ2-αΛ2) Ω3-(αΩ3+ωβΛ3) Λ2+(αΩ2+ωβΛ2) Λ3+1h Ω22 2Ω12+Ω12 2Ω22+Λ22 2Λ12+Λ12 2Λ22-2Ω1Ω2 2Ω1Ω2+2Ω2Λ2 2Ω1Λ1-2Ω2Λ1 2Ω1Λ2-2Ω1Λ2 2Ω2Λ1+2Ω1Λ1 2Ω2Λ2-2Λ1Λ2 2Λ1Λ2-Ω1 Ω1-Ω2 Ω2-Λ1 Λ1-Λ2 Λ2.
z(Ω·Λ=zΩ1Λ1+Ω2Λ2+Ω3Λ3=βΛ1.
zΛ1=-1hΛ1.
Ω·Λ=0.
zΩ2=z(Ω12+Ω22+Ω32=2βΩ1+2αg,
zΛ2)=zΛ12+Λ22+Λ32=2αg,
zΩ2-Λ2=2βΩ1.
zΩ1=β-1hΩ1.
Ω1=βh[1-exp(-z/h)],
Ω2-Λ2=2β2h2zh-1+exp(-z/h).
Ω2PMD=2β2h2zh-1+exp(-z/h).
zg=-1hg+αΩ22+Ω32+Λ22+Λ32.
Ω12Ω22Ω32,Λ12Λ22Λ32.
zg-1hg+2α3Ω2+Λ2.
d2gdz2+1h dgdz-8α23g=43β2αh[1-exp(-z/h)].
g=Ω2Λ3-Ω3Λ2β2h2α[exp(z/Γ)-1],
Γ8α2h3-1.
Ω2β2hΓexp(z/Γ)-1+zΓ,
Λ2β2hΓexp(z/Γ)-1-zΓ.
z(Ω·Λ)2=2βΩ1Λ12+Ω2Λ1Λ2+Ω3Λ1Λ3.
z(Ω·Λ)22βΩ1Λ122β2h Λ23.
(Ω·Λ)22β4h2Γ23 exp(z/Γ)-1-zΓ-z22Γ2.
zΛ12=2hΛ22-Λ12.
h2 zΛ12=Λ22-Λ12Λ2,
zΛ21hΛ2.
h2 Λ2zΛ2=h2Γ exp(z/Γ)-1exp(z/Γ)-1-z/Γ
h/z,hzΓ.h/(2Γ)O(1),zΓorzΓ
τ2=Ω2-Λ22+(Ω2-Λ2)24+(Ω·Λ)2.
(Ω·Λ)2(Ω2-Λ2)2136 zΓ1,zΓO(1),zΓΓ26z2 expzΓ1,zΓ.
τ2Ω2-Λ2+(Ω·Λ)2Ω2-Λ2Ω2-Λ2+(Ω·Λ)2Ω2-Λ2,zΓ.
τ2-Ω2-Λ2β2h18Γz2,zΓ.
τ4(Ω·Λ)2,zΓ.

Metrics