Abstract

A procedure is described that allows one to solve the evolution equations in birefringent optical fibers by using repeated diagonalization. With this approach several practical problems are solved in a unified evolution in twisted fibers, sinusoidally rocked fibers, and fibers with randomly varying way. Included are last case it is shown that a phenomenological model described by Poole and others birefringence. In the applies to fibers whose axes of birefringence can take on any orientation.

© 1994 Optical Society of America

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  1. R. H. Stolen, V. Ramaswamy, P. Kaiser, and W. Pleibel, “Linear polarization in birefringent single-mode fibers,” Appl. Phys. Lett. 33, 699–701 (1978).
    [CrossRef]
  2. I. P. Kaminow, “Polarization in fibers,” Laser Focus 16(6), 80–84 (1980).
  3. I. P. Kaminow, “Polarization in optical fibers,” IEEE J. Quantum Electron. QE-17, 15-22 (1981).
    [CrossRef]
  4. L. Li, J. R. Qian, and D. N. Payne, “Current sensors using highly birefringent bow-tie fibers,” Electron. Lett. 22, 1142–1144 (1986).
    [CrossRef]
  5. R. I. Laming and D. N. Payne, “Electric current sensors employing spun highly birefringent optical fibers,” J. Lightwave Technol. 7, 2084–2094 (1989).
    [CrossRef]
  6. A. D. Kersey, A. Dandridge, and A. B. Tveten, “Dependence of visibility on input polarization in interferometric fiber-optic sensors,” Opt. Lett. 13, 288–290 (1988).
    [CrossRef] [PubMed]
  7. A. D. Kersey, M. J. Marone, and A. Dandridge, “Observation of input-polarization-induced phase noise in interferometric fiber-optics sensors,” Opt. Lett. 13, 847–849 (1988).
    [CrossRef] [PubMed]
  8. W. K. Burns, C.-L. Chen, and R. P. Moeller, “Fiber-optic gyroscopes with broad-band sources,” J. Lightwave Technol. 1, 98–105 (1983).
    [CrossRef]
  9. W. K. Burns, “Phase error bounds of fiber gyro with polarization holding fiber,” J. Lightwave Technol. 4, 8–14 (1986).
    [CrossRef]
  10. M. N. Islam, Ultrafast Fiber Switching Devices and Systems (Cambridge U. Press, Cambridge, 1992).
  11. K. Kitayama, Y. Kimura, and S. Seikai, “Fiber-optic logic gate,” Appl. Phys. Lett. 46, 317–319 (1985).
    [CrossRef]
  12. H. G. Winful, “Polarization instabilities in birefringent nonlinear media: application to fiber-optic devices,” Opt. Lett. 11, 33–35 (1986).
    [CrossRef] [PubMed]
  13. T. Morioka, M. Saruwatari, and A. Takada, “Ultrafast optical multi/demultiplexer utilizing optical Kerr effect in polarisation-maintaining single-mode fibers,” Electron. Lett. 23, 453–454 (1987).
    [CrossRef]
  14. M. J. Marrone and C. A. Villaruel, “Fiber in-line polarization rotator and mode interchanger,” Appl. Opt. 26, 3194–3195 (1987).
    [CrossRef] [PubMed]
  15. R. H. Stolen, A. Ashkin, W. Pleibel, and J. M. Dziedzic, “Inline fiber-polarization-rocking rotator and filter,” Opt. Lett. 9, 300–302 (1984).
    [CrossRef] [PubMed]
  16. B. Daino, G. Gregori, and S. Wabnitz, “New all-optical devices based on third-order nonlinearity of birefringent fibers,” Opt. Lett. 11, 42–44 (1986).
    [CrossRef] [PubMed]
  17. S. C. Rashleigh, W. K. Burns, R. P. Moeller, and R. Ulrich, “Polarization holding in birefringent single-mode fibers,” Opt. Lett. 7, 40–42 (1982).
    [CrossRef] [PubMed]
  18. W. K. Burns and R. P. Moeller, “Measurement of polarization dispersion in high-birefringence fibers,” Opt. Lett. 8, 195–197 (1983).
    [CrossRef] [PubMed]
  19. A. Simon and R. Ulrich, “Evolution of polarization along a single-mode fiber,” Appl. Phys. Lett. 31, 517–520 (1977).
    [CrossRef]
  20. V. Ramaswamy, R. D. Standley, D. Sze, and W. G. French, “Polarization effects in short length, single mode fibers,” Bell Syst. Tech. J. 57, 635–651 (1978).
    [CrossRef]
  21. A. J. Barlow, J. J. Ramskov-Hansen, and D. N. Payne, “Birefringence and polarization mode-dispersion in spun single-mode fibers,” Appl. Opt. 20, 2962–2968 (1981).
    [CrossRef] [PubMed]
  22. S. C. Rashleigh, “Origins and control of polarization effects in single-mode fibers,” J. Lightwave Technol. 1, 312–331 (1983).
    [CrossRef]
  23. M. J. Marrone, C. A. Villaruel, N. J. Frigo, and A. Dandridge, “Internal rotation of the birefringence axes in polarization-holding fibers,” Opt. Lett. 12, 60–62 (1987).
    [CrossRef] [PubMed]
  24. R. Calvani, R. Caponi, and F. Cisternino, “Polarization measurements on single-mode fibers,” J. Lightwave Technol. 7, 1187–1196 (1989).
    [CrossRef]
  25. N. S. Bergano, “Undersea lightwave transmission systems using Er-doped fiber amplifiers,” Opt. Photon. News 4(1), 8–14 (1993).
    [CrossRef]
  26. L. F. Mollenauer, K. Smith, J. P. Gordon, and C. R. Menyuk, “Resistance of solitons to the effects of polarization dispersion in optical fibers,” Opt. Lett. 14, 1219–1221 (1989).
    [CrossRef] [PubMed]
  27. S. G. Evangelides, L. F. Mollenauer, and J. P. Gordon, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–34 (1992).
    [CrossRef]
  28. P. K. A. Wai, C. R. Menyuk, and H. H. Chen, “Stability of solitons in randomly varying birefringent fibers,” Opt. Lett. 16, 1231–1233 (1991).
    [CrossRef] [PubMed]
  29. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, Boston, 1991).
  30. N. S. Bergano, C. D. Poole, and R. E. Wagner, “Investigation of polarization dispersion in long lengths of single-mode fiber using multilongitudinal mode lasers,” J. Lightwave Technol. 5, 1618–1622 (1987).
    [CrossRef]
  31. C. D. Poole and R. E. Wagner, “Phenomenological approach to polarisation dispersion in long single-mode fibres,” Electron. Lett. 22, 1029–1030 (1986).
    [CrossRef]
  32. C. D. Poole, “Statistical treatment of polarization dispersion in single-mode fiber,” Opt. Lett. 13, 687–689 (1988).
    [CrossRef] [PubMed]
  33. C. D. Poole, “Measurement of polarization-mode dispersion in single-mode fibers with random coupling,” Opt. Lett. 14, 523–525 (1989).
    [CrossRef] [PubMed]
  34. C. D. Poole, J. H. Winters, and J. A. Nagel, “Dynamical equation for polarization dispersion,” Opt. Lett. 16, 372–374 (1991).
    [CrossRef] [PubMed]
  35. G. J. Foschini and C. D. Poole, “Statistical theory of polarization dispersion in single mode fibers,” J. Lightwave Technol. 9, 1439–1456 (1991).
    [CrossRef]
  36. D. Andresciani, F. Curti, F. Matera, and B. Daino, “Measurement of the group-delay difference between principal states of polarization on a low-birefringence terrestrial fiber cable,” Opt. Lett. 12, 844–846 (1987).
    [CrossRef] [PubMed]
  37. F. Curti, D. Daino, Q. Mao, F. Matera, and C. G. Someda, “Concatenation of polarisation dispersion in single-mode fibres,” Electron. Lett. 25, 290–292 (1989).
    [CrossRef]
  38. F. Curti, B. Daino, G. De Marchis, and F. Maternola, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Tech. 8, 1162–1166 (1990).
    [CrossRef]
  39. S. Betti, F. Curti, B. Daino, G. De Marchis, E. Iannone, and F. Matera, “Evolution of the bandwidth of the principal states of polarization in single-mode fibers,” Opt. Lett. 16, 467–469 (1991).
    [CrossRef] [PubMed]
  40. See, e.g., the discussion at the beginning of Chap. 1 of A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984).
  41. H. Poincaré, Théorie Mathématique de la Lumière (Georges Carré, Paris, 1892), Vol. 2, Chap. 12; see also M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light (Pergamon, Oxford, 1984), Chap. 1.
  42. N. J. Frigo, “A generalized geometrical representation of coupled mode theory,” IEEE J. Quantum Electron. QE-22, 2131–2140 (1986).
    [CrossRef]
  43. D. David, D. D. Holm, and M. V. Tratnik, “Hamiltonian chaos in nonlinear optical polarization dynamics,” Phys. Rep. 187, 218–367 (1990).
    [CrossRef]
  44. G. B. Arfken, Mathematical Methods for Physicists (Academic, New York, 1985), Chap. 4.10.
  45. See, e.g., H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1980).
  46. C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr medium,” IEEE J. Quantum Electron. 25, 2674–2682 (1989).
    [CrossRef]
  47. C. R. Menyuk and P. K. A. Wai, “Elimination of nonlinear polarization rotation in twisted fibers,” J. Opt. Soc. Am. B 11, 1305–1309 (1994).
    [CrossRef]
  48. C. Kittel, Introduction to Solid State Phyics (Wiley, New York, 1986), Chap. 7. In the mathematical community, this result is typically referred to as Floquet’s theorem. See, e.g., G. Blanch, “Mathieu functions,” in Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, eds. (U.S. Government Printing Office, Washington, D.C., 1964). Note that in contrast to the usual result, the Bloch–Floquet exponent can only be imaginary, because the motion is confined to a sphere. Mathematically, D is unitary, so its eigenvalues can have only imaginary exponents.
  49. G. Keiser, Optical Fiber Communications (McGraw-Hill, New York, 1991).
  50. T. Ueda and W. Kath, “Dynamics of optical pulses in randomly birefringent fibers,” Physica (The Hague) D55, 166–181 (1992).

1994 (1)

1993 (1)

N. S. Bergano, “Undersea lightwave transmission systems using Er-doped fiber amplifiers,” Opt. Photon. News 4(1), 8–14 (1993).
[CrossRef]

1992 (2)

S. G. Evangelides, L. F. Mollenauer, and J. P. Gordon, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–34 (1992).
[CrossRef]

T. Ueda and W. Kath, “Dynamics of optical pulses in randomly birefringent fibers,” Physica (The Hague) D55, 166–181 (1992).

1991 (4)

1990 (2)

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

D. David, D. D. Holm, and M. V. Tratnik, “Hamiltonian chaos in nonlinear optical polarization dynamics,” Phys. Rep. 187, 218–367 (1990).
[CrossRef]

1989 (6)

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

F. Curti, D. Daino, Q. Mao, F. Matera, and C. G. Someda, “Concatenation of polarisation dispersion in single-mode fibres,” Electron. Lett. 25, 290–292 (1989).
[CrossRef]

C. D. Poole, “Measurement of polarization-mode dispersion in single-mode fibers with random coupling,” Opt. Lett. 14, 523–525 (1989).
[CrossRef] [PubMed]

L. F. Mollenauer, K. Smith, J. P. Gordon, and C. R. Menyuk, “Resistance of solitons to the effects of polarization dispersion in optical fibers,” Opt. Lett. 14, 1219–1221 (1989).
[CrossRef] [PubMed]

R. Calvani, R. Caponi, and F. Cisternino, “Polarization measurements on single-mode fibers,” J. Lightwave Technol. 7, 1187–1196 (1989).
[CrossRef]

R. I. Laming and D. N. Payne, “Electric current sensors employing spun highly birefringent optical fibers,” J. Lightwave Technol. 7, 2084–2094 (1989).
[CrossRef]

1988 (3)

1987 (5)

D. Andresciani, F. Curti, F. Matera, and B. Daino, “Measurement of the group-delay difference between principal states of polarization on a low-birefringence terrestrial fiber cable,” Opt. Lett. 12, 844–846 (1987).
[CrossRef] [PubMed]

N. S. Bergano, C. D. Poole, and R. E. Wagner, “Investigation of polarization dispersion in long lengths of single-mode fiber using multilongitudinal mode lasers,” J. Lightwave Technol. 5, 1618–1622 (1987).
[CrossRef]

M. J. Marrone, C. A. Villaruel, N. J. Frigo, and A. Dandridge, “Internal rotation of the birefringence axes in polarization-holding fibers,” Opt. Lett. 12, 60–62 (1987).
[CrossRef] [PubMed]

T. Morioka, M. Saruwatari, and A. Takada, “Ultrafast optical multi/demultiplexer utilizing optical Kerr effect in polarisation-maintaining single-mode fibers,” Electron. Lett. 23, 453–454 (1987).
[CrossRef]

M. J. Marrone and C. A. Villaruel, “Fiber in-line polarization rotator and mode interchanger,” Appl. Opt. 26, 3194–3195 (1987).
[CrossRef] [PubMed]

1986 (6)

W. K. Burns, “Phase error bounds of fiber gyro with polarization holding fiber,” J. Lightwave Technol. 4, 8–14 (1986).
[CrossRef]

B. Daino, G. Gregori, and S. Wabnitz, “New all-optical devices based on third-order nonlinearity of birefringent fibers,” Opt. Lett. 11, 42–44 (1986).
[CrossRef] [PubMed]

L. Li, J. R. Qian, and D. N. Payne, “Current sensors using highly birefringent bow-tie fibers,” Electron. Lett. 22, 1142–1144 (1986).
[CrossRef]

H. G. Winful, “Polarization instabilities in birefringent nonlinear media: application to fiber-optic devices,” Opt. Lett. 11, 33–35 (1986).
[CrossRef] [PubMed]

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

N. J. Frigo, “A generalized geometrical representation of coupled mode theory,” IEEE J. Quantum Electron. QE-22, 2131–2140 (1986).
[CrossRef]

1985 (1)

K. Kitayama, Y. Kimura, and S. Seikai, “Fiber-optic logic gate,” Appl. Phys. Lett. 46, 317–319 (1985).
[CrossRef]

1984 (1)

1983 (3)

W. K. Burns and R. P. Moeller, “Measurement of polarization dispersion in high-birefringence fibers,” Opt. Lett. 8, 195–197 (1983).
[CrossRef] [PubMed]

W. K. Burns, C.-L. Chen, and R. P. Moeller, “Fiber-optic gyroscopes with broad-band sources,” J. Lightwave Technol. 1, 98–105 (1983).
[CrossRef]

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

1982 (1)

1981 (2)

1980 (1)

I. P. Kaminow, “Polarization in fibers,” Laser Focus 16(6), 80–84 (1980).

1978 (2)

R. H. Stolen, V. Ramaswamy, P. Kaiser, and W. Pleibel, “Linear polarization in birefringent single-mode fibers,” Appl. Phys. Lett. 33, 699–701 (1978).
[CrossRef]

V. Ramaswamy, R. D. Standley, D. Sze, and W. G. French, “Polarization effects in short length, single mode fibers,” Bell Syst. Tech. J. 57, 635–651 (1978).
[CrossRef]

1977 (1)

A. Simon and R. Ulrich, “Evolution of polarization along a single-mode fiber,” Appl. Phys. Lett. 31, 517–520 (1977).
[CrossRef]

Andresciani, D.

Arfken, G. B.

G. B. Arfken, Mathematical Methods for Physicists (Academic, New York, 1985), Chap. 4.10.

Ashkin, A.

Barlow, A. J.

Bergano, N. S.

N. S. Bergano, “Undersea lightwave transmission systems using Er-doped fiber amplifiers,” Opt. Photon. News 4(1), 8–14 (1993).
[CrossRef]

N. S. Bergano, C. D. Poole, and R. E. Wagner, “Investigation of polarization dispersion in long lengths of single-mode fiber using multilongitudinal mode lasers,” J. Lightwave Technol. 5, 1618–1622 (1987).
[CrossRef]

Betti, S.

Burns, W. K.

W. K. Burns, “Phase error bounds of fiber gyro with polarization holding fiber,” J. Lightwave Technol. 4, 8–14 (1986).
[CrossRef]

W. K. Burns, C.-L. Chen, and R. P. Moeller, “Fiber-optic gyroscopes with broad-band sources,” J. Lightwave Technol. 1, 98–105 (1983).
[CrossRef]

W. K. Burns and R. P. Moeller, “Measurement of polarization dispersion in high-birefringence fibers,” Opt. Lett. 8, 195–197 (1983).
[CrossRef] [PubMed]

S. C. Rashleigh, W. K. Burns, R. P. Moeller, and R. Ulrich, “Polarization holding in birefringent single-mode fibers,” Opt. Lett. 7, 40–42 (1982).
[CrossRef] [PubMed]

Calvani, R.

R. Calvani, R. Caponi, and F. Cisternino, “Polarization measurements on single-mode fibers,” J. Lightwave Technol. 7, 1187–1196 (1989).
[CrossRef]

Caponi, R.

R. Calvani, R. Caponi, and F. Cisternino, “Polarization measurements on single-mode fibers,” J. Lightwave Technol. 7, 1187–1196 (1989).
[CrossRef]

Chen, C.-L.

W. K. Burns, C.-L. Chen, and R. P. Moeller, “Fiber-optic gyroscopes with broad-band sources,” J. Lightwave Technol. 1, 98–105 (1983).
[CrossRef]

Chen, H. H.

Cisternino, F.

R. Calvani, R. Caponi, and F. Cisternino, “Polarization measurements on single-mode fibers,” J. Lightwave Technol. 7, 1187–1196 (1989).
[CrossRef]

Curti, F.

S. Betti, F. Curti, B. Daino, G. De Marchis, E. Iannone, and F. Matera, “Evolution of the bandwidth of the principal states of polarization in single-mode fibers,” Opt. Lett. 16, 467–469 (1991).
[CrossRef] [PubMed]

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

F. Curti, D. Daino, Q. Mao, F. Matera, and C. G. Someda, “Concatenation of polarisation dispersion in single-mode fibres,” Electron. Lett. 25, 290–292 (1989).
[CrossRef]

D. Andresciani, F. Curti, F. Matera, and B. Daino, “Measurement of the group-delay difference between principal states of polarization on a low-birefringence terrestrial fiber cable,” Opt. Lett. 12, 844–846 (1987).
[CrossRef] [PubMed]

Daino, B.

Daino, D.

F. Curti, D. Daino, Q. Mao, F. Matera, and C. G. Someda, “Concatenation of polarisation dispersion in single-mode fibres,” Electron. Lett. 25, 290–292 (1989).
[CrossRef]

Dandridge, A.

David, D.

D. David, D. D. Holm, and M. V. Tratnik, “Hamiltonian chaos in nonlinear optical polarization dynamics,” Phys. Rep. 187, 218–367 (1990).
[CrossRef]

De Marchis, G.

S. Betti, F. Curti, B. Daino, G. De Marchis, E. Iannone, and F. Matera, “Evolution of the bandwidth of the principal states of polarization in single-mode fibers,” Opt. Lett. 16, 467–469 (1991).
[CrossRef] [PubMed]

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

Dziedzic, J. M.

Evangelides, S. G.

S. G. Evangelides, L. F. Mollenauer, and J. P. Gordon, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–34 (1992).
[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]

French, W. G.

V. Ramaswamy, R. D. Standley, D. Sze, and W. G. French, “Polarization effects in short length, single mode fibers,” Bell Syst. Tech. J. 57, 635–651 (1978).
[CrossRef]

Frigo, N. J.

M. J. Marrone, C. A. Villaruel, N. J. Frigo, and A. Dandridge, “Internal rotation of the birefringence axes in polarization-holding fibers,” Opt. Lett. 12, 60–62 (1987).
[CrossRef] [PubMed]

N. J. Frigo, “A generalized geometrical representation of coupled mode theory,” IEEE J. Quantum Electron. QE-22, 2131–2140 (1986).
[CrossRef]

Goldstein, H.

See, e.g., H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1980).

Gordon, J. P.

S. G. Evangelides, L. F. Mollenauer, and J. P. Gordon, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–34 (1992).
[CrossRef]

L. F. Mollenauer, K. Smith, J. P. Gordon, and C. R. Menyuk, “Resistance of solitons to the effects of polarization dispersion in optical fibers,” Opt. Lett. 14, 1219–1221 (1989).
[CrossRef] [PubMed]

Gregori, G.

Holm, D. D.

D. David, D. D. Holm, and M. V. Tratnik, “Hamiltonian chaos in nonlinear optical polarization dynamics,” Phys. Rep. 187, 218–367 (1990).
[CrossRef]

Iannone, E.

Islam, M. N.

M. N. Islam, Ultrafast Fiber Switching Devices and Systems (Cambridge U. Press, Cambridge, 1992).

Kaiser, P.

R. H. Stolen, V. Ramaswamy, P. Kaiser, and W. Pleibel, “Linear polarization in birefringent single-mode fibers,” Appl. Phys. Lett. 33, 699–701 (1978).
[CrossRef]

Kaminow, I. P.

I. P. Kaminow, “Polarization in optical fibers,” IEEE J. Quantum Electron. QE-17, 15-22 (1981).
[CrossRef]

I. P. Kaminow, “Polarization in fibers,” Laser Focus 16(6), 80–84 (1980).

Kath, W.

T. Ueda and W. Kath, “Dynamics of optical pulses in randomly birefringent fibers,” Physica (The Hague) D55, 166–181 (1992).

Keiser, G.

G. Keiser, Optical Fiber Communications (McGraw-Hill, New York, 1991).

Kersey, A. D.

Kimura, Y.

K. Kitayama, Y. Kimura, and S. Seikai, “Fiber-optic logic gate,” Appl. Phys. Lett. 46, 317–319 (1985).
[CrossRef]

Kitayama, K.

K. Kitayama, Y. Kimura, and S. Seikai, “Fiber-optic logic gate,” Appl. Phys. Lett. 46, 317–319 (1985).
[CrossRef]

Kittel, C.

C. Kittel, Introduction to Solid State Phyics (Wiley, New York, 1986), Chap. 7. In the mathematical community, this result is typically referred to as Floquet’s theorem. See, e.g., G. Blanch, “Mathieu functions,” in Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, eds. (U.S. Government Printing Office, Washington, D.C., 1964). Note that in contrast to the usual result, the Bloch–Floquet exponent can only be imaginary, because the motion is confined to a sphere. Mathematically, D is unitary, so its eigenvalues can have only imaginary exponents.

Laming, R. I.

R. I. Laming and D. N. Payne, “Electric current sensors employing spun highly birefringent optical fibers,” J. Lightwave Technol. 7, 2084–2094 (1989).
[CrossRef]

Li, L.

L. Li, J. R. Qian, and D. N. Payne, “Current sensors using highly birefringent bow-tie fibers,” Electron. Lett. 22, 1142–1144 (1986).
[CrossRef]

Mao, Q.

F. Curti, D. Daino, Q. Mao, F. Matera, and C. G. Someda, “Concatenation of polarisation dispersion in single-mode fibres,” Electron. Lett. 25, 290–292 (1989).
[CrossRef]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, Boston, 1991).

Marone, M. J.

Marrone, M. J.

Matera, F.

Maternola, F.

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

Menyuk, C. R.

Moeller, R. P.

Mollenauer, L. F.

S. G. Evangelides, L. F. Mollenauer, and J. P. Gordon, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–34 (1992).
[CrossRef]

L. F. Mollenauer, K. Smith, J. P. Gordon, and C. R. Menyuk, “Resistance of solitons to the effects of polarization dispersion in optical fibers,” Opt. Lett. 14, 1219–1221 (1989).
[CrossRef] [PubMed]

Morioka, T.

T. Morioka, M. Saruwatari, and A. Takada, “Ultrafast optical multi/demultiplexer utilizing optical Kerr effect in polarisation-maintaining single-mode fibers,” Electron. Lett. 23, 453–454 (1987).
[CrossRef]

Nagel, J. A.

Papoulis, A.

See, e.g., the discussion at the beginning of Chap. 1 of A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984).

Payne, D. N.

R. I. Laming and D. N. Payne, “Electric current sensors employing spun highly birefringent optical fibers,” J. Lightwave Technol. 7, 2084–2094 (1989).
[CrossRef]

L. Li, J. R. Qian, and D. N. Payne, “Current sensors using highly birefringent bow-tie fibers,” Electron. Lett. 22, 1142–1144 (1986).
[CrossRef]

A. J. Barlow, J. J. Ramskov-Hansen, and D. N. Payne, “Birefringence and polarization mode-dispersion in spun single-mode fibers,” Appl. Opt. 20, 2962–2968 (1981).
[CrossRef] [PubMed]

Pleibel, W.

R. H. Stolen, A. Ashkin, W. Pleibel, and J. M. Dziedzic, “Inline fiber-polarization-rocking rotator and filter,” Opt. Lett. 9, 300–302 (1984).
[CrossRef] [PubMed]

R. H. Stolen, V. Ramaswamy, P. Kaiser, and W. Pleibel, “Linear polarization in birefringent single-mode fibers,” Appl. Phys. Lett. 33, 699–701 (1978).
[CrossRef]

Poincaré, H.

H. Poincaré, Théorie Mathématique de la Lumière (Georges Carré, Paris, 1892), Vol. 2, Chap. 12; see also M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light (Pergamon, Oxford, 1984), Chap. 1.

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, “Measurement of polarization-mode dispersion in single-mode fibers with random coupling,” Opt. Lett. 14, 523–525 (1989).
[CrossRef] [PubMed]

C. D. Poole, “Statistical treatment of polarization dispersion in single-mode fiber,” Opt. Lett. 13, 687–689 (1988).
[CrossRef] [PubMed]

N. S. Bergano, C. D. Poole, and R. E. Wagner, “Investigation of polarization dispersion in long lengths of single-mode fiber using multilongitudinal mode lasers,” J. Lightwave Technol. 5, 1618–1622 (1987).
[CrossRef]

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

Qian, J. R.

L. Li, J. R. Qian, and D. N. Payne, “Current sensors using highly birefringent bow-tie fibers,” Electron. Lett. 22, 1142–1144 (1986).
[CrossRef]

Ramaswamy, V.

R. H. Stolen, V. Ramaswamy, P. Kaiser, and W. Pleibel, “Linear polarization in birefringent single-mode fibers,” Appl. Phys. Lett. 33, 699–701 (1978).
[CrossRef]

V. Ramaswamy, R. D. Standley, D. Sze, and W. G. French, “Polarization effects in short length, single mode fibers,” Bell Syst. Tech. J. 57, 635–651 (1978).
[CrossRef]

Ramskov-Hansen, J. J.

Rashleigh, S. C.

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

S. C. Rashleigh, W. K. Burns, R. P. Moeller, and R. Ulrich, “Polarization holding in birefringent single-mode fibers,” Opt. Lett. 7, 40–42 (1982).
[CrossRef] [PubMed]

Saruwatari, M.

T. Morioka, M. Saruwatari, and A. Takada, “Ultrafast optical multi/demultiplexer utilizing optical Kerr effect in polarisation-maintaining single-mode fibers,” Electron. Lett. 23, 453–454 (1987).
[CrossRef]

Seikai, S.

K. Kitayama, Y. Kimura, and S. Seikai, “Fiber-optic logic gate,” Appl. Phys. Lett. 46, 317–319 (1985).
[CrossRef]

Simon, A.

A. Simon and R. Ulrich, “Evolution of polarization along a single-mode fiber,” Appl. Phys. Lett. 31, 517–520 (1977).
[CrossRef]

Smith, K.

Someda, C. G.

F. Curti, D. Daino, Q. Mao, F. Matera, and C. G. Someda, “Concatenation of polarisation dispersion in single-mode fibres,” Electron. Lett. 25, 290–292 (1989).
[CrossRef]

Standley, R. D.

V. Ramaswamy, R. D. Standley, D. Sze, and W. G. French, “Polarization effects in short length, single mode fibers,” Bell Syst. Tech. J. 57, 635–651 (1978).
[CrossRef]

Stolen, R. H.

R. H. Stolen, A. Ashkin, W. Pleibel, and J. M. Dziedzic, “Inline fiber-polarization-rocking rotator and filter,” Opt. Lett. 9, 300–302 (1984).
[CrossRef] [PubMed]

R. H. Stolen, V. Ramaswamy, P. Kaiser, and W. Pleibel, “Linear polarization in birefringent single-mode fibers,” Appl. Phys. Lett. 33, 699–701 (1978).
[CrossRef]

Sze, D.

V. Ramaswamy, R. D. Standley, D. Sze, and W. G. French, “Polarization effects in short length, single mode fibers,” Bell Syst. Tech. J. 57, 635–651 (1978).
[CrossRef]

Takada, A.

T. Morioka, M. Saruwatari, and A. Takada, “Ultrafast optical multi/demultiplexer utilizing optical Kerr effect in polarisation-maintaining single-mode fibers,” Electron. Lett. 23, 453–454 (1987).
[CrossRef]

Tratnik, M. V.

D. David, D. D. Holm, and M. V. Tratnik, “Hamiltonian chaos in nonlinear optical polarization dynamics,” Phys. Rep. 187, 218–367 (1990).
[CrossRef]

Tveten, A. B.

Ueda, T.

T. Ueda and W. Kath, “Dynamics of optical pulses in randomly birefringent fibers,” Physica (The Hague) D55, 166–181 (1992).

Ulrich, R.

S. C. Rashleigh, W. K. Burns, R. P. Moeller, and R. Ulrich, “Polarization holding in birefringent single-mode fibers,” Opt. Lett. 7, 40–42 (1982).
[CrossRef] [PubMed]

A. Simon and R. Ulrich, “Evolution of polarization along a single-mode fiber,” Appl. Phys. Lett. 31, 517–520 (1977).
[CrossRef]

Villaruel, C. A.

Wabnitz, S.

Wagner, R. E.

N. S. Bergano, C. D. Poole, and R. E. Wagner, “Investigation of polarization dispersion in long lengths of single-mode fiber using multilongitudinal mode lasers,” J. Lightwave Technol. 5, 1618–1622 (1987).
[CrossRef]

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

Wai, P. K. A.

Winful, H. G.

Winters, J. H.

Appl. Opt. (2)

Appl. Phys. Lett. (3)

A. Simon and R. Ulrich, “Evolution of polarization along a single-mode fiber,” Appl. Phys. Lett. 31, 517–520 (1977).
[CrossRef]

K. Kitayama, Y. Kimura, and S. Seikai, “Fiber-optic logic gate,” Appl. Phys. Lett. 46, 317–319 (1985).
[CrossRef]

R. H. Stolen, V. Ramaswamy, P. Kaiser, and W. Pleibel, “Linear polarization in birefringent single-mode fibers,” Appl. Phys. Lett. 33, 699–701 (1978).
[CrossRef]

Bell Syst. Tech. J. (1)

V. Ramaswamy, R. D. Standley, D. Sze, and W. G. French, “Polarization effects in short length, single mode fibers,” Bell Syst. Tech. J. 57, 635–651 (1978).
[CrossRef]

Electron. Lett. (4)

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

L. Li, J. R. Qian, and D. N. Payne, “Current sensors using highly birefringent bow-tie fibers,” Electron. Lett. 22, 1142–1144 (1986).
[CrossRef]

T. Morioka, M. Saruwatari, and A. Takada, “Ultrafast optical multi/demultiplexer utilizing optical Kerr effect in polarisation-maintaining single-mode fibers,” Electron. Lett. 23, 453–454 (1987).
[CrossRef]

F. Curti, D. Daino, Q. Mao, F. Matera, and C. G. Someda, “Concatenation of polarisation dispersion in single-mode fibres,” Electron. Lett. 25, 290–292 (1989).
[CrossRef]

IEEE J. Quantum Electron. (3)

N. J. Frigo, “A generalized geometrical representation of coupled mode theory,” IEEE J. Quantum Electron. QE-22, 2131–2140 (1986).
[CrossRef]

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

I. P. Kaminow, “Polarization in optical fibers,” IEEE J. Quantum Electron. QE-17, 15-22 (1981).
[CrossRef]

J. Lightwave Tech. (1)

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

J. Lightwave Technol. (8)

R. I. Laming and D. N. Payne, “Electric current sensors employing spun highly birefringent optical fibers,” J. Lightwave Technol. 7, 2084–2094 (1989).
[CrossRef]

W. K. Burns, C.-L. Chen, and R. P. Moeller, “Fiber-optic gyroscopes with broad-band sources,” J. Lightwave Technol. 1, 98–105 (1983).
[CrossRef]

W. K. Burns, “Phase error bounds of fiber gyro with polarization holding fiber,” J. Lightwave Technol. 4, 8–14 (1986).
[CrossRef]

N. S. Bergano, C. D. Poole, and R. E. Wagner, “Investigation of polarization dispersion in long lengths of single-mode fiber using multilongitudinal mode lasers,” J. Lightwave Technol. 5, 1618–1622 (1987).
[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]

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

R. Calvani, R. Caponi, and F. Cisternino, “Polarization measurements on single-mode fibers,” J. Lightwave Technol. 7, 1187–1196 (1989).
[CrossRef]

S. G. Evangelides, L. F. Mollenauer, and J. P. Gordon, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–34 (1992).
[CrossRef]

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

Laser Focus (1)

I. P. Kaminow, “Polarization in fibers,” Laser Focus 16(6), 80–84 (1980).

Opt. Lett. (15)

A. D. Kersey, A. Dandridge, and A. B. Tveten, “Dependence of visibility on input polarization in interferometric fiber-optic sensors,” Opt. Lett. 13, 288–290 (1988).
[CrossRef] [PubMed]

A. D. Kersey, M. J. Marone, and A. Dandridge, “Observation of input-polarization-induced phase noise in interferometric fiber-optics sensors,” Opt. Lett. 13, 847–849 (1988).
[CrossRef] [PubMed]

H. G. Winful, “Polarization instabilities in birefringent nonlinear media: application to fiber-optic devices,” Opt. Lett. 11, 33–35 (1986).
[CrossRef] [PubMed]

R. H. Stolen, A. Ashkin, W. Pleibel, and J. M. Dziedzic, “Inline fiber-polarization-rocking rotator and filter,” Opt. Lett. 9, 300–302 (1984).
[CrossRef] [PubMed]

B. Daino, G. Gregori, and S. Wabnitz, “New all-optical devices based on third-order nonlinearity of birefringent fibers,” Opt. Lett. 11, 42–44 (1986).
[CrossRef] [PubMed]

S. C. Rashleigh, W. K. Burns, R. P. Moeller, and R. Ulrich, “Polarization holding in birefringent single-mode fibers,” Opt. Lett. 7, 40–42 (1982).
[CrossRef] [PubMed]

W. K. Burns and R. P. Moeller, “Measurement of polarization dispersion in high-birefringence fibers,” Opt. Lett. 8, 195–197 (1983).
[CrossRef] [PubMed]

P. K. A. Wai, C. R. Menyuk, and H. H. Chen, “Stability of solitons in randomly varying birefringent fibers,” Opt. Lett. 16, 1231–1233 (1991).
[CrossRef] [PubMed]

M. J. Marrone, C. A. Villaruel, N. J. Frigo, and A. Dandridge, “Internal rotation of the birefringence axes in polarization-holding fibers,” Opt. Lett. 12, 60–62 (1987).
[CrossRef] [PubMed]

D. Andresciani, F. Curti, F. Matera, and B. Daino, “Measurement of the group-delay difference between principal states of polarization on a low-birefringence terrestrial fiber cable,” Opt. Lett. 12, 844–846 (1987).
[CrossRef] [PubMed]

L. F. Mollenauer, K. Smith, J. P. Gordon, and C. R. Menyuk, “Resistance of solitons to the effects of polarization dispersion in optical fibers,” Opt. Lett. 14, 1219–1221 (1989).
[CrossRef] [PubMed]

C. D. Poole, “Statistical treatment of polarization dispersion in single-mode fiber,” Opt. Lett. 13, 687–689 (1988).
[CrossRef] [PubMed]

C. D. Poole, “Measurement of polarization-mode dispersion in single-mode fibers with random coupling,” Opt. Lett. 14, 523–525 (1989).
[CrossRef] [PubMed]

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

S. Betti, F. Curti, B. Daino, G. De Marchis, E. Iannone, and F. Matera, “Evolution of the bandwidth of the principal states of polarization in single-mode fibers,” Opt. Lett. 16, 467–469 (1991).
[CrossRef] [PubMed]

Opt. Photon. News (1)

N. S. Bergano, “Undersea lightwave transmission systems using Er-doped fiber amplifiers,” Opt. Photon. News 4(1), 8–14 (1993).
[CrossRef]

Phys. Rep. (1)

D. David, D. D. Holm, and M. V. Tratnik, “Hamiltonian chaos in nonlinear optical polarization dynamics,” Phys. Rep. 187, 218–367 (1990).
[CrossRef]

Physica (The Hague) (1)

T. Ueda and W. Kath, “Dynamics of optical pulses in randomly birefringent fibers,” Physica (The Hague) D55, 166–181 (1992).

Other (8)

C. Kittel, Introduction to Solid State Phyics (Wiley, New York, 1986), Chap. 7. In the mathematical community, this result is typically referred to as Floquet’s theorem. See, e.g., G. Blanch, “Mathieu functions,” in Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, eds. (U.S. Government Printing Office, Washington, D.C., 1964). Note that in contrast to the usual result, the Bloch–Floquet exponent can only be imaginary, because the motion is confined to a sphere. Mathematically, D is unitary, so its eigenvalues can have only imaginary exponents.

G. Keiser, Optical Fiber Communications (McGraw-Hill, New York, 1991).

G. B. Arfken, Mathematical Methods for Physicists (Academic, New York, 1985), Chap. 4.10.

See, e.g., H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1980).

See, e.g., the discussion at the beginning of Chap. 1 of A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984).

H. Poincaré, Théorie Mathématique de la Lumière (Georges Carré, Paris, 1892), Vol. 2, Chap. 12; see also M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light (Pergamon, Oxford, 1984), Chap. 1.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, Boston, 1991).

M. N. Islam, Ultrafast Fiber Switching Devices and Systems (Cambridge U. Press, Cambridge, 1992).

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Equations (62)

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E ( z , r t ) = E 1 ( z ) Ψ 1 ( r t ) + E 2 ( z ) Ψ 2 ( r t ) ,
d d z ( E 1 E 2 ) = i [ k 11 k 12 k 21 k 22 ] ( E 1 E 2 ) ,
d E d z = i KE .
I = [ 1 0 0 1 ] ,             σ 1 = [ 0 1 1 0 ] ,             σ 2 = [ 0 - i i 0 ] ,             σ 3 = [ 1 0 0 - 1 ] ,
K = k 0 I + κ 1 σ 1 + κ 2 σ 2 + κ 3 σ 3 ,
A = E exp [ - i 0 z k 0 ( z ) d z ]
d A d z = i Θ A ,
κ 3 = b sin θ ,             κ 1 = b sin θ ,
Θ = b [ cos θ sin θ sin θ - cos θ ] ,
Θ = [ b γ γ - b ] ,
U 1 Θ U 1 - 1 = b σ 3 ,
U 1 = cos ( θ / 2 ) I + i sin ( θ / 2 ) σ 2 , U 1 - 1 = cos ( θ / 2 ) I + i sin ( θ / 2 ) σ 2 .
d B 1 d z = ( i U 1 Θ U 1 - 1 - U 1 d U 1 - 1 d z ) B 1 = i ( b σ 3 + θ z 2 σ 2 ) i Ψ 1 B 1 .
b = c 2 cos ϕ 2 ,             θ z / 2 = c 2 sin ϕ 2 ,
d B 1 d z = i Ψ 2 B 1 ,
ϕ 2 , z = b 2 b 2 + θ z 2 / 4 d d z θ z 2 b ,
V = U 3 U 2 U 1
d B d z = i Ψ B
B , 1 ( z ) = B , 1 ( 0 ) exp [ i 0 z c ( z ) d z ] , B , 2 ( z ) = B , 2 ( 0 ) exp [ - i 0 z c ( z ) d z ] .
X = A σ 3 A ,             Y = A σ 1 A ,             Z = A σ 1 A
A ( z ) = T ( z ) A 0 ,
j = 1 3 R j σ j = T j = 1 3 R j , 0 σ j T ,
A A = 1 2 I + 1 2 j = 1 3 σ j A σ j A .
d B 1 d z = i Ψ 1 B 1 ,
B 1 ( z ) = [ cos ( b z ) I + i sin ( b z ) σ 3 ] B 1 ( 0 ) .
T = U 1 - 1 [ cos ( b z ) I + i sin ( b z ) σ 3 ] U 1 = cos ( b z ) I + i cos θ 0 sin ( b z ) σ 3 + i sin θ 0 sin ( b z ) σ 1 .
X = [ cos 2 ( b z ) + cos ( 2 θ 0 ) sin 2 ( b z ) ] X 0 + sin ( 2 θ 0 ) sin 2 ( b z ) Y 0 - sin θ 0 sin ( 2 b z ) Z 0 , Y = sin ( 2 θ 0 ) sin 2 ( b z ) X 0 + [ cos 2 ( b z ) - cos ( 2 θ 0 ) sin 2 ( b z ) ] Y 0 + cos θ 0 sin ( 2 b z ) Z 0 , Z = sin θ 0 sin ( 2 b z ) X 0 - cos θ 0 sin ( 2 b z ) Y 0 + cos ( 2 b z ) Z 0 .
X = X 0 , Y = cos ( 2 b z ) Y 0 + sin ( 2 b z ) Z 0 , Z = - sin ( 2 b z ) Y 0 + cos ( 2 b z ) Z 0 .
d B 1 d z = i ( b σ 3 + Ω 3 σ 2 ) B 1 , d B 2 d z = i c 2 σ 3 B 2 ,
B 2 ( z ) = T 2 B 2 , 0 ,
T 1 = cos ( c 2 z ) I + i cos ϕ 2 sin ( c 2 z ) σ 3 + i sin ϕ 2 sin ( c 2 z ) σ 2 .
X 1 = [ cos 2 ( c 2 z ) + cos ( 2 ϕ 2 ) sin 2 ( c 2 z ) ] X 0 + sin ϕ 2 sin ( 2 c 2 z ) Y 0 + sin ( 2 ϕ 2 ) sin 2 ( c 2 z ) Z 0 , Y 1 = - sin ϕ 2 sin ( 2 c 2 z ) X 0 + cos ( 2 c 2 z ) Y 0 + cos ϕ 2 sin ( 2 c 2 z ) Z 0 , Z 1 = sin ( 2 ϕ 2 ) sin 2 ( c 2 z ) X 0 - cos ϕ 2 sin ( 2 c 2 z ) Y 0 + [ cos 2 ( c 2 z ) - cos ( 2 ϕ 2 ) sin 2 ( c 2 z ) ] Z 0 .
X 0 = cos ϕ 2 = b ( b 2 + Ω 2 / 4 ) 1 / 2 ,             Y 0 = 0 , Z 0 = sin ϕ 2 = Ω / 2 ( b 2 + Ω 2 / 4 ) 1 / 2 ,
d A d z = i [ b cos ( sin Ω z ) σ 3 + b sin ( sin Ω z ) σ 1 ] A , d B 1 d z = i [ b σ 3 + 1 2 Ω cos ( Ω z ) σ 2 ] B 1 , d B 2 d z = i [ c 2 σ 3 + Ω 2 4 b sin Ω z 1 + ( 2 Ω 2 / 4 b 2 ) cos 2 Ω z σ 1 ] B 2 ,
B 1 ( z ) = T 1 ( z ) B 1 , 0 = T 1 ( s ) T 1 m ( 2 π / Ω ) B 1 , 0 ,
T 1 ( 2 π / Ω ) = R - 1 DR ,
B 1 ( z ) = T 1 ( s ) R - 1 D - 1 ( s ) D ( z ) R B 1 , 0 , = R - 1 S ( z ) D ( z ) R B 1 , 0 ,
d B 1 d z = i Ψ 1 B 1 ,
Ψ 1 = b σ 3 + θ z 2 σ 2 = [ b - i θ z / 2 i θ z / 2 - b ] .
d B 1 d ω = i F 1 B 1 ,
F 1 z = Ψ 1 ω + i [ Ψ 1 , F 1 ] .
Ψ 1 ω = b σ 3 .
F 1 = f X σ 3 + f Y σ 1 + f Z σ 2 ,
d τ d d z = 2 b cos ξ ,
F 1 T 1 = - i T 1 ω ,
τ d 2 = - 4 det F 1 = 4 det ( T 1 / ω ) ,
S = exp [ - i φ ( z ) σ 3 ] T 1 = { cos [ φ ( z ) ] I - i sin [ φ ( z ) ] σ 3 } T 1 ,
φ ( z ) = 0 z b ( z ) d z .
d S d z = [ 0 1 2 θ z exp ( - 2 i φ ) - 1 2 θ z exp ( 2 i φ ) 0 ] S .
τ d 2 = 4 det ( S + i σ 3 φ S ) ,
S = [ s 1 s 2 - s 2 * s 1 * ]
d τ d 2 d z = 8 b φ + 8 i b ( s 1 s 1 * + s 2 s 2 * ) ,
d 2 τ d 2 d z 2 = 8 ( b ) 2 + 8 φ d b d z - 4 i b θ z [ s 1 s 2 exp ( 2 i φ ) - s 1 * s 2 * exp ( - 2 i φ ) - s 1 s 2 exp ( 2 i φ ) + s 1 * s 2 * exp ( - 2 i φ ) ] .
s 1 ( z ) = s 1 ( ζ ) - ζ z θ z ( z ) 2 exp [ - 2 i φ ( z ) ] s 2 * ( z ) d z s 1 ( ζ ) - s 2 * ( ζ ) ζ z θ z ( z ) 2 exp [ - 2 i φ ( z ) ] d z ,
s 2 ( z ) s 2 ( ζ ) + s 1 * ( ζ ) ζ z θ z ( z ) 2 exp [ - 2 i φ ( z ) ] d z ,
s 1 ( z ) s 1 ( ζ ) - [ s 2 * ( ζ ) - 2 i φ ( ζ ) s 2 * ( ζ ) ] × ζ z θ z ( z ) 2 exp [ - 2 i φ ( z ) ] d z , s 2 ( z ) s 2 ( ζ ) + [ s 1 * ( ζ ) - 2 i φ ( ζ ) s 1 * ( ζ ) ] × ζ z θ z ( z ) 2 exp [ - 2 i φ ( z ) ] d z .
H ( z ) = Re [ ζ z θ z ( z ) θ z ( z ) 4 exp [ 2 i φ ( z ) - 2 i φ ( z ) ] d z ] 1 2 - θ z ( z ) θ z ( z ) 4 exp [ 2 i φ ( z ) - 2 i φ ( z ) ] d z ,
d 2 τ d 2 d z 2 = 8 ( b ) 2 + 8 φ d b d z - 32 b φ H ( z ) - 32 i b H ( z ) ( s 1 s 1 * + s 2 s 2 * ) ,
X ( z ) = 1 L z - L z X ( z ) d z ,
d τ d 2 d z = 8 b φ + 8 i b s 1 s 1 * + s 2 s 2 * , d 2 τ d 2 d z 2 = 8 ( b ) 2 - 32 φ b H - 32 i b H s 1 s 1 * + s 2 s 2 * ,
d 2 τ d 2 d z 2 = - 4 r H d 2 τ d 2 d z + 8 ( b ) 2 ,
τ d 2 = ( b ) 2 2 r 2 H 2 [ exp ( - 4 r H z ) - 1 + 4 r H z ] ,

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