Abstract

We present analytical results on the polarization-mode dispersion characteristics of single-mode birefringent fibers based on statistical properties of the backscattered signal. In particular, we calculate exactly the relationship between polarization-mode dispersion characteristics in forward propagation with respect to round-trip propagation in terms of dynamical equations, differential group delays, correlation lengths, and second-order effects. The theory applies to fibers affected by superposition of linear and circular birefringence in both the short-length and the long-length regimes.

© 1999 Optical Society of America

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References

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  1. A. J. Rogers, “Polarization-optical time domain reflectometry: a technique for the measurement of field distributions,” Appl. Opt. 20, 1060–1074 (1981).
    [CrossRef] [PubMed]
  2. B. Y. Kim, S. S. Choi, “Analysis and measurement of birefringence in single-mode fiber using the backscattering method,” Opt. Lett. 6, 578–580 (1981).
    [CrossRef] [PubMed]
  3. E. Brinkmeyer, “Forward–backward transmission in birefringent single-mode fibers: interpretation of polarization-sensitive measurements,” Opt. Lett. 11, 575–577 (1981).
    [CrossRef]
  4. J. N. Ross, “Birefringence measurement in optical fibers by polarization optical time-domain reflectometry,” Appl. Opt. 21, 3489–3495 (1982).
    [CrossRef] [PubMed]
  5. A. Galtarossa, P. Pistolato, M. Schiano, “Measurements of stress birefringence in optical cables by polarization otdr,” presented at the European Fibre Optic Communications and Networks Conference, Brighton, UK, June 27–30, 1995.
  6. E. Collett, Polarized Light, Fundamentals and Applications (Dekker, New York, 1993).
  7. D. H. O. Beddinghton, T. G. Ellison, X. Shon, A. S. Siddiqui, S. D. Walker, “Fully polarimetric optical time domain reflectometer with one meter spatial resolution,” in Digest of 1997 Optical Fiber Conference (Optical Society of America, Washington, D.C., 1997), paper WL24.
  8. A. Galtarossa, M. Schiano, “Complete characterization of polarization mode dispersion in erbium doped optical amplifiers,” Electron. Lett. 28, 2143–2144 (1992).
    [CrossRef]
  9. A. Galtarossa, B. A. Schrefler, M. Schiano, C. G. Someda, A. Tommasini, G. Zavarise, “Stress distribution in optical-fiber ribbons,” IEEE Photonics Technol. Lett. 9, 354–356 (1997).
    [CrossRef]
  10. C. D. Poole, J. Nagel, “Polarization effects in lightwave systems,” in Optical Fiber Telecommunications, I. P. Kaminow, T. Koch, eds. (Academic, San Diego, Calif., 1997), pp. 114–155.
  11. F. Curti, B. Daino, G. De Marchis, 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]
  12. F. Corsi, A. Galtarossa, L. Palmieri, “Experimental investigation of polarization mode dispersion properties in single-mode fibers using a new backscattering technique,” in Digest of the 1998 Optical Fiber Conference (Optical Society of America, Washington, D.C., 1998), paper ThR4.
  13. F. Corsi, A. Galtarossa, L. Palmieri, “Polarization mode dispersion characterization of single-mode optical fiber using backscattering technique,” J. Lightwave Technol. 16, 1832–1843 (1998).
    [CrossRef]
  14. R. Ulrich, A. Simon, “Polarization optics of twisted single-mode fibers,” Appl. Opt. 18, 2241–2251 (1979).
    [CrossRef] [PubMed]
  15. G. J. Foschini, C. D. Poole, “Statistical theory of polarization dispersion in single mode fibers,” J. Lightwave Technol. 9, 1439–1456 (1991).
    [CrossRef]
  16. C. D. Poole, J. H. Winters, J. A. Nagel, “Dynamical equation for polarization dispersion,” Opt. Lett. 6, 372–374 (1991).
    [CrossRef]
  17. R. E. Schuh, E. S. R. Sikora, N. G. Walker, A. S. Siddiqui, L. M. Gleeson, D. H. O. Bebbington, “Theoretical analysis and measurement of effects of fiber twist on polarization mode dispersion of optical fibers,” Electron. Lett. 28, 1772–1773 (1995).
    [CrossRef]
  18. M. O. Van Deventer, “Probability density functions of optical polarization state: theory and applications,” J. Lightwave Technol. 12, 2147–2152 (1994).
    [CrossRef]
  19. S. C. Rashleigh, “Origins and control of polarization effect in single-mode fibers,” J. Lightwave Technol. 1, 312–331 (1983).
    [CrossRef]
  20. P. K. A. Wai, C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (1996).
    [CrossRef]
  21. P. K. A. Wai, C. R. Menyuk, “Polarization decorrelation in optical fibers with randomly varying birefringence,” Opt. Lett. 19, 1517–1519 (1994).
    [CrossRef] [PubMed]
  22. P. Ciprut, B. Gisin, N. Gisin, R. Passy, J. P. Von der Weid, F. Prieto, C. W. Zimmer, “Second-order polarization mode dispersion: impact on analog and digital transmissions,” J. Lightwave Technol. 16, 757–771 (1998).
    [CrossRef]

1998 (2)

1997 (1)

A. Galtarossa, B. A. Schrefler, M. Schiano, C. G. Someda, A. Tommasini, G. Zavarise, “Stress distribution in optical-fiber ribbons,” IEEE Photonics Technol. Lett. 9, 354–356 (1997).
[CrossRef]

1996 (1)

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

1995 (1)

R. E. Schuh, E. S. R. Sikora, N. G. Walker, A. S. Siddiqui, L. M. Gleeson, D. H. O. Bebbington, “Theoretical analysis and measurement of effects of fiber twist on polarization mode dispersion of optical fibers,” Electron. Lett. 28, 1772–1773 (1995).
[CrossRef]

1994 (2)

M. O. Van Deventer, “Probability density functions of optical polarization state: theory and applications,” J. Lightwave Technol. 12, 2147–2152 (1994).
[CrossRef]

P. K. A. Wai, C. R. Menyuk, “Polarization decorrelation in optical fibers with randomly varying birefringence,” Opt. Lett. 19, 1517–1519 (1994).
[CrossRef] [PubMed]

1992 (1)

A. Galtarossa, M. Schiano, “Complete characterization of polarization mode dispersion in erbium doped optical amplifiers,” Electron. Lett. 28, 2143–2144 (1992).
[CrossRef]

1991 (2)

G. J. Foschini, 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, J. A. Nagel, “Dynamical equation for polarization dispersion,” Opt. Lett. 6, 372–374 (1991).
[CrossRef]

1990 (1)

F. Curti, B. Daino, G. De Marchis, 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]

1983 (1)

S. C. Rashleigh, “Origins and control of polarization effect in single-mode fibers,” J. Lightwave Technol. 1, 312–331 (1983).
[CrossRef]

1982 (1)

1981 (3)

1979 (1)

Bebbington, D. H. O.

R. E. Schuh, E. S. R. Sikora, N. G. Walker, A. S. Siddiqui, L. M. Gleeson, D. H. O. Bebbington, “Theoretical analysis and measurement of effects of fiber twist on polarization mode dispersion of optical fibers,” Electron. Lett. 28, 1772–1773 (1995).
[CrossRef]

Beddinghton, D. H. O.

D. H. O. Beddinghton, T. G. Ellison, X. Shon, A. S. Siddiqui, S. D. Walker, “Fully polarimetric optical time domain reflectometer with one meter spatial resolution,” in Digest of 1997 Optical Fiber Conference (Optical Society of America, Washington, D.C., 1997), paper WL24.

Brinkmeyer, E.

Choi, S. S.

Ciprut, P.

Collett, E.

E. Collett, Polarized Light, Fundamentals and Applications (Dekker, New York, 1993).

Corsi, F.

F. Corsi, A. Galtarossa, L. Palmieri, “Polarization mode dispersion characterization of single-mode optical fiber using backscattering technique,” J. Lightwave Technol. 16, 1832–1843 (1998).
[CrossRef]

F. Corsi, A. Galtarossa, L. Palmieri, “Experimental investigation of polarization mode dispersion properties in single-mode fibers using a new backscattering technique,” in Digest of the 1998 Optical Fiber Conference (Optical Society of America, Washington, D.C., 1998), paper ThR4.

Curti, F.

F. Curti, B. Daino, G. De Marchis, 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, 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, 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]

Ellison, T. G.

D. H. O. Beddinghton, T. G. Ellison, X. Shon, A. S. Siddiqui, S. D. Walker, “Fully polarimetric optical time domain reflectometer with one meter spatial resolution,” in Digest of 1997 Optical Fiber Conference (Optical Society of America, Washington, D.C., 1997), paper WL24.

Foschini, G. J.

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

Galtarossa, A.

F. Corsi, A. Galtarossa, L. Palmieri, “Polarization mode dispersion characterization of single-mode optical fiber using backscattering technique,” J. Lightwave Technol. 16, 1832–1843 (1998).
[CrossRef]

A. Galtarossa, B. A. Schrefler, M. Schiano, C. G. Someda, A. Tommasini, G. Zavarise, “Stress distribution in optical-fiber ribbons,” IEEE Photonics Technol. Lett. 9, 354–356 (1997).
[CrossRef]

A. Galtarossa, M. Schiano, “Complete characterization of polarization mode dispersion in erbium doped optical amplifiers,” Electron. Lett. 28, 2143–2144 (1992).
[CrossRef]

A. Galtarossa, P. Pistolato, M. Schiano, “Measurements of stress birefringence in optical cables by polarization otdr,” presented at the European Fibre Optic Communications and Networks Conference, Brighton, UK, June 27–30, 1995.

F. Corsi, A. Galtarossa, L. Palmieri, “Experimental investigation of polarization mode dispersion properties in single-mode fibers using a new backscattering technique,” in Digest of the 1998 Optical Fiber Conference (Optical Society of America, Washington, D.C., 1998), paper ThR4.

Gisin, B.

Gisin, N.

Gleeson, L. M.

R. E. Schuh, E. S. R. Sikora, N. G. Walker, A. S. Siddiqui, L. M. Gleeson, D. H. O. Bebbington, “Theoretical analysis and measurement of effects of fiber twist on polarization mode dispersion of optical fibers,” Electron. Lett. 28, 1772–1773 (1995).
[CrossRef]

Kim, B. Y.

Matera, F.

F. Curti, B. Daino, G. De Marchis, 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]

Menyuk, C. R.

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

P. K. A. Wai, C. R. Menyuk, “Polarization decorrelation in optical fibers with randomly varying birefringence,” Opt. Lett. 19, 1517–1519 (1994).
[CrossRef] [PubMed]

Nagel, J.

C. D. Poole, J. Nagel, “Polarization effects in lightwave systems,” in Optical Fiber Telecommunications, I. P. Kaminow, T. Koch, eds. (Academic, San Diego, Calif., 1997), pp. 114–155.

Nagel, J. A.

Palmieri, L.

F. Corsi, A. Galtarossa, L. Palmieri, “Polarization mode dispersion characterization of single-mode optical fiber using backscattering technique,” J. Lightwave Technol. 16, 1832–1843 (1998).
[CrossRef]

F. Corsi, A. Galtarossa, L. Palmieri, “Experimental investigation of polarization mode dispersion properties in single-mode fibers using a new backscattering technique,” in Digest of the 1998 Optical Fiber Conference (Optical Society of America, Washington, D.C., 1998), paper ThR4.

Passy, R.

Pistolato, P.

A. Galtarossa, P. Pistolato, M. Schiano, “Measurements of stress birefringence in optical cables by polarization otdr,” presented at the European Fibre Optic Communications and Networks Conference, Brighton, UK, June 27–30, 1995.

Poole, C. D.

G. J. Foschini, 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, J. A. Nagel, “Dynamical equation for polarization dispersion,” Opt. Lett. 6, 372–374 (1991).
[CrossRef]

C. D. Poole, J. Nagel, “Polarization effects in lightwave systems,” in Optical Fiber Telecommunications, I. P. Kaminow, T. Koch, eds. (Academic, San Diego, Calif., 1997), pp. 114–155.

Prieto, F.

Rashleigh, S. C.

S. C. Rashleigh, “Origins and control of polarization effect in single-mode fibers,” J. Lightwave Technol. 1, 312–331 (1983).
[CrossRef]

Rogers, A. J.

Ross, J. N.

Schiano, M.

A. Galtarossa, B. A. Schrefler, M. Schiano, C. G. Someda, A. Tommasini, G. Zavarise, “Stress distribution in optical-fiber ribbons,” IEEE Photonics Technol. Lett. 9, 354–356 (1997).
[CrossRef]

A. Galtarossa, M. Schiano, “Complete characterization of polarization mode dispersion in erbium doped optical amplifiers,” Electron. Lett. 28, 2143–2144 (1992).
[CrossRef]

A. Galtarossa, P. Pistolato, M. Schiano, “Measurements of stress birefringence in optical cables by polarization otdr,” presented at the European Fibre Optic Communications and Networks Conference, Brighton, UK, June 27–30, 1995.

Schrefler, B. A.

A. Galtarossa, B. A. Schrefler, M. Schiano, C. G. Someda, A. Tommasini, G. Zavarise, “Stress distribution in optical-fiber ribbons,” IEEE Photonics Technol. Lett. 9, 354–356 (1997).
[CrossRef]

Schuh, R. E.

R. E. Schuh, E. S. R. Sikora, N. G. Walker, A. S. Siddiqui, L. M. Gleeson, D. H. O. Bebbington, “Theoretical analysis and measurement of effects of fiber twist on polarization mode dispersion of optical fibers,” Electron. Lett. 28, 1772–1773 (1995).
[CrossRef]

Shon, X.

D. H. O. Beddinghton, T. G. Ellison, X. Shon, A. S. Siddiqui, S. D. Walker, “Fully polarimetric optical time domain reflectometer with one meter spatial resolution,” in Digest of 1997 Optical Fiber Conference (Optical Society of America, Washington, D.C., 1997), paper WL24.

Siddiqui, A. S.

R. E. Schuh, E. S. R. Sikora, N. G. Walker, A. S. Siddiqui, L. M. Gleeson, D. H. O. Bebbington, “Theoretical analysis and measurement of effects of fiber twist on polarization mode dispersion of optical fibers,” Electron. Lett. 28, 1772–1773 (1995).
[CrossRef]

D. H. O. Beddinghton, T. G. Ellison, X. Shon, A. S. Siddiqui, S. D. Walker, “Fully polarimetric optical time domain reflectometer with one meter spatial resolution,” in Digest of 1997 Optical Fiber Conference (Optical Society of America, Washington, D.C., 1997), paper WL24.

Sikora, E. S. R.

R. E. Schuh, E. S. R. Sikora, N. G. Walker, A. S. Siddiqui, L. M. Gleeson, D. H. O. Bebbington, “Theoretical analysis and measurement of effects of fiber twist on polarization mode dispersion of optical fibers,” Electron. Lett. 28, 1772–1773 (1995).
[CrossRef]

Simon, A.

Someda, C. G.

A. Galtarossa, B. A. Schrefler, M. Schiano, C. G. Someda, A. Tommasini, G. Zavarise, “Stress distribution in optical-fiber ribbons,” IEEE Photonics Technol. Lett. 9, 354–356 (1997).
[CrossRef]

Tommasini, A.

A. Galtarossa, B. A. Schrefler, M. Schiano, C. G. Someda, A. Tommasini, G. Zavarise, “Stress distribution in optical-fiber ribbons,” IEEE Photonics Technol. Lett. 9, 354–356 (1997).
[CrossRef]

Ulrich, R.

Van Deventer, M. O.

M. O. Van Deventer, “Probability density functions of optical polarization state: theory and applications,” J. Lightwave Technol. 12, 2147–2152 (1994).
[CrossRef]

Von der Weid, J. P.

Wai, P. K. A.

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

P. K. A. Wai, C. R. Menyuk, “Polarization decorrelation in optical fibers with randomly varying birefringence,” Opt. Lett. 19, 1517–1519 (1994).
[CrossRef] [PubMed]

Walker, N. G.

R. E. Schuh, E. S. R. Sikora, N. G. Walker, A. S. Siddiqui, L. M. Gleeson, D. H. O. Bebbington, “Theoretical analysis and measurement of effects of fiber twist on polarization mode dispersion of optical fibers,” Electron. Lett. 28, 1772–1773 (1995).
[CrossRef]

Walker, S. D.

D. H. O. Beddinghton, T. G. Ellison, X. Shon, A. S. Siddiqui, S. D. Walker, “Fully polarimetric optical time domain reflectometer with one meter spatial resolution,” in Digest of 1997 Optical Fiber Conference (Optical Society of America, Washington, D.C., 1997), paper WL24.

Winters, J. H.

Zavarise, G.

A. Galtarossa, B. A. Schrefler, M. Schiano, C. G. Someda, A. Tommasini, G. Zavarise, “Stress distribution in optical-fiber ribbons,” IEEE Photonics Technol. Lett. 9, 354–356 (1997).
[CrossRef]

Zimmer, C. W.

Appl. Opt. (3)

Electron. Lett. (2)

R. E. Schuh, E. S. R. Sikora, N. G. Walker, A. S. Siddiqui, L. M. Gleeson, D. H. O. Bebbington, “Theoretical analysis and measurement of effects of fiber twist on polarization mode dispersion of optical fibers,” Electron. Lett. 28, 1772–1773 (1995).
[CrossRef]

A. Galtarossa, M. Schiano, “Complete characterization of polarization mode dispersion in erbium doped optical amplifiers,” Electron. Lett. 28, 2143–2144 (1992).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

A. Galtarossa, B. A. Schrefler, M. Schiano, C. G. Someda, A. Tommasini, G. Zavarise, “Stress distribution in optical-fiber ribbons,” IEEE Photonics Technol. Lett. 9, 354–356 (1997).
[CrossRef]

J. Lightwave Technol. (7)

M. O. Van Deventer, “Probability density functions of optical polarization state: theory and applications,” J. Lightwave Technol. 12, 2147–2152 (1994).
[CrossRef]

S. C. Rashleigh, “Origins and control of polarization effect in single-mode fibers,” J. Lightwave Technol. 1, 312–331 (1983).
[CrossRef]

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

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

F. Curti, B. Daino, G. De Marchis, 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]

F. Corsi, A. Galtarossa, L. Palmieri, “Polarization mode dispersion characterization of single-mode optical fiber using backscattering technique,” J. Lightwave Technol. 16, 1832–1843 (1998).
[CrossRef]

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

Opt. Lett. (4)

Other (5)

C. D. Poole, J. Nagel, “Polarization effects in lightwave systems,” in Optical Fiber Telecommunications, I. P. Kaminow, T. Koch, eds. (Academic, San Diego, Calif., 1997), pp. 114–155.

A. Galtarossa, P. Pistolato, M. Schiano, “Measurements of stress birefringence in optical cables by polarization otdr,” presented at the European Fibre Optic Communications and Networks Conference, Brighton, UK, June 27–30, 1995.

E. Collett, Polarized Light, Fundamentals and Applications (Dekker, New York, 1993).

D. H. O. Beddinghton, T. G. Ellison, X. Shon, A. S. Siddiqui, S. D. Walker, “Fully polarimetric optical time domain reflectometer with one meter spatial resolution,” in Digest of 1997 Optical Fiber Conference (Optical Society of America, Washington, D.C., 1997), paper WL24.

F. Corsi, A. Galtarossa, L. Palmieri, “Experimental investigation of polarization mode dispersion properties in single-mode fibers using a new backscattering technique,” in Digest of the 1998 Optical Fiber Conference (Optical Society of America, Washington, D.C., 1998), paper ThR4.

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Figures (6)

Fig. 1
Fig. 1

Evolution of Δτ (solid curve) and ΔτB (dashed curve) as a function of twisting rate for a fiber with fixed length.

Fig. 2
Fig. 2

Evolution of LC/LB (solid curve) and LCB/LB (dashed curve) as a function of LR/LB.

Fig. 3
Fig. 3

Evolution of LC/LCB as a function of LR/LB.

Fig. 4
Fig. 4

Histogram of the simulated ΔτB; the solid curve is the Rayleigh PDF.

Fig. 5
Fig. 5

Evolution of 〈Δτ〉 (dashed curve) and ΔτB (solid curve) as a function of z. Black dots are obtained by multiplying 〈Δτ〉 by π/2.

Fig. 6
Fig. 6

Parametric plot of τω2 versus τωB2 as z increases. The black dots are numerical data; the solid line is plotted with Eq. (28).

Equations (90)

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

α¯(γ, ϕ, θ)=γcos ϕ cos θcos ϕ sin θsin ϕ=γαˆ(ϕ, θ),
R=I+sin γ(αˆ)+(1-cos γ)(αˆ)2,
R(γ, ϕ, θ)=R(γ,-ϕ, θ)=MRT(γ, ϕ, θ)M,
sˆB(z)=R(z)R(z)sˆ0=MRT(z)MR(z)sˆ0=RB(z)sˆ0,
sˆ(z)z=R(z)zRT(z)sˆ(z)=β¯(z)×sˆ(z),
sˆB(z)z=RB(z)zRBT(z)sˆB(z)=β¯B(z)×sˆB(z),
β¯B(z)=2MRT(z)β1(z)β2(z)0.
sˆ(ω)ω=R(ω)ωRT(ω)sˆ(ω)=Ω¯(ω)×sˆ(ω),
Ω¯(z, ω)z=β¯(z, ω)ω+β¯(z, ω)×Ω¯(z, ω).
sˆB(ω)ω=RB(ω)ωRBT(ω)sˆB(ω)=Ω¯B(ω)×sˆB(ω),
Ω¯B(ω)=2MRT(ω)Ω1(ω)Ω2(ω)0.
ΔτB=|Ω¯B|=2Ω12+Ω22.
Ω¯Bω=2MRTωΩ1Ω20+RT ωΩ1Ω20.
R RTω=(Ω¯×)T=-(Ω¯×).
Ω¯Bω=2MRTΩ2-Ω10Ω3+ωΩ1Ω20.
sˆn=RnRn-1R1sˆ0,
sˆBn=MR1TR2TRnTMRnR2R1sˆ0.
Ω¯z=βL0βC+βL0βC×Ω¯,
Δτ(z)1β|βLβL+βCβC|z.
ρc=±βLω(2-g) dgdω1/2
ΔτB2(z)=4β2(βLβC-βCβL)sin2 βz22+2βCβ3(βLβC-βCβL)sin βz+2βLβ2(βLβL+βCβC)z2.
ΔτB(z)2βLβ2|βLβL+βCβC|z.
ΔτBΔτ2βL(βL2+βC2)1/2.
limρ ΔτB(z=L)=2gkβLω|g-2|L.
|sˆB|=sˆBω=ΔτB|sin ξ|,
sˆB(z, ω)=-2 sin γ(1-cos γ)sin ϕ cos2 ϕcos2 γ-sin2 γ cos 2ϕ2 sin γ cos ϕ(cos γ cos2 ϕ+sin2 ϕ),
sˆBω=z 2βLβ2|βLβL+βCβC|βC2β2cos2 βz+βL2β21/2,
sˆBω=ΔτB(z)βC2β2cos2 βz+βL2β21/2.
Fn(i, j)=0,ij,
Fn(1, 1)=[1-14(3-c2ϕ)(1-cγ)]n,
Fn(2, 2)=Fn(1, 1),
Fn(3, 3)=[1-12(1+c2ϕ)(1-cγ)]n,
Fn(i, j)=k=13rikfkj,
Fn(i, j)=k=13rikfkj.
limnFn=[Ø].
Bn=[uij][bij][rij]=l=13k=13uikbklrlj,
Bn(i, j)=l=13k=13uikbklrlj=l=13k=13uikrljbkl,
Bn(i, j)=ui1r1jb11+ui2r2jb22+ui3r3jb33,
Bn(i, j)=0,ij,
Bn(1, 1)=13(1-Υ)+ΥBn-1(1, 1),
Bn(2, 2)=13(1-Υ)+ΥBn-1(2, 2),
Bn(3, 3)=-13(1-Υ)+ΥBn-1(3, 3),
Bn(1, 1)=Bn(2, 2)=13(1+2Υn) Bn(3, 3)=-13(1-4Υn).
limnBn=13M.
Ψ(z)=Ψ(nLR)=Py(z)Px(z)+Py(z)=12[1-exp(-2hnLR)],
Py(nLR)=12[1+sˆy·(Fnsˆx)],
Ψ(nLR)=12{1-[1-12(1-cγ)]n}.
cγ=cos γ=cos(βLR)=1-2πlr exp(-4πlr2)erfi(lr4π),
erfi(u)=1ierf(iu)=2π0u exp(t2)dt.
lc=LCLB=-2lrln[(1+cγ)/2].
ΨB(z)=Py(z)Px(z)+Py(z)=13[1-exp(-2hBz)],
ΨB(nLR)=13(1-|Υ|n),
lcB=LCBLB=-2lrln(14|1+3c2γ|),
c2γ=cos 2γ=1-4πlr exp(-16πlr2)erfi(2lr4π).
Δτ=8π dτzLC,
fΔτB(a)=a4σ2exp-a28σ2,a[0,+[.
ΔτB=σ2π=dτ2πzLC=π2Δτ.
|Ω¯B|2=4[Ω22Ω32+2Ω2Ω3Ω1+(Ω1)2+Ω12Ω32-2Ω1Ω3Ω2+(Ω2)2],
E[Ωi2Ωj2]=E[Ωi2]E[Ωj2]=σ4,ij, E[(Ωi)2]=E[Ωi2]2=σ4.
E[τωB2]=16σ4+8(E[Ω2Ω3Ω1]-E[Ω1Ω3Ω2]).
E[τωB2]=π24E[Δτ]4.
E[τω2]=13E[Δτ2]2=3π264E[Δτ]4,
τωB2=163τω2.
Rx(ψ)=1000cos ψ-sin ψ0sin ψcos ψ,
Ry(ψ)=cos ψ0sin ψ010-sin ψ0cos ψ,
Rz(ψ)=cos ψ-sin ψ0sin ψcos ψ0001,
R(γ, ϕ, θ)=Rz(θ)Ry(-ϕ)Rx(γ)Ry(ϕ)Rz(-θ).
Rh(ψ)=I+Ih sin ψ+Ih2(1-cos ψ),
Ix=00000-1010,Iy=001000-100, Iz=0-10100000.
R(γ, ϕ, θ)=I+A sin γ+A2(1-cos γ),
A=0-sin ϕcos ϕ sin θsin ϕ0-cos ϕ cos θ-cos ϕ sin θcos ϕ cos θ0.
αˆ=cos ϕ cos θcos ϕ sin θsin ϕ.
R(α¯)=I+sin γ(αˆ×)+(1-cos γ)(αˆ×)2,
RT(Γ¯×)R=(RTΓ¯)×,
QT(Γ¯×)Q=-(QTΓ¯)×.
sˆz=RzR-1sˆ=Bsˆ=β¯×sˆ,
B=0-β3β2β30-β1-β2β10.
sˆBz=RBzRB-1sˆB=β¯B×sˆB.
β¯B×=RBzRB-1=MRTzMR+RTM RzRTMRM=M RTzRM+(MRT)MBM(RM).
BT=R RTz=-B,
β¯B×=MRT(-B+MBM)RM=-2MRTβ1β20×RM.
β¯B=2MRTβ1β20.
Ω¯B=2MRTΩ1Ω20.
c2ϕ=cos 2ϕ,c4ϕ=cos 4ϕ, cγ=cos γ,c2γ=cos 2γ.
Υ=332(3+4c2ϕ+c4ϕ)c2γ+38(1-c4ϕ)cγ+132(11-12c2ϕ+9c4ϕ);
Bn(1, 1)=Υn-1B1(1, 1)+13(1-Υ)j=0n-2Υj=Υn-1B1(1, 1)+13(1-Υn-1).
B1(1, 1)=13(1+2Υ),
Bn(1, 1)=13(1+2Υn).
B(1, 1)=B(2, 2)=13,B(3, 3)=-13
B(1, 1)=B(2, 2)=B(3, 3)=1.

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