Abstract

In a recent paper [Opt. Lett. 25, 290 (2000)] and a personal communication (MIT, Cambridge, Mass. 02139) Y. Chen and H. A. Haus analytically described the relative displacement among the orthogonally polarized components of a soliton that undergoes propagation in a fiber affected by polarization-mode dispersion. We show that the same results may also be derived by use of an approach alternative to that of Y. Chen and H. A. Haus.

© 2002 Optical Society of America

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References

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  1. F. Heismann, D. A. Fishman, and D. L. Wilson, “Automatic compensation of first order polarization mode dispersion in a 10 Gb/s transmission system,” Proceedings of the 24th European Conference on Optical Communication (Telefonica, Madrid, Spain, 1998), pp. 529–530.
  2. H. Bulow, D. Schlump, J. Weber, B. Wedding, and R. Heidemann, “Electronic equalization of fiber PMD-induced distortion at 10 Gbit/s,” Optical Fiber Communication Conference, Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 151–152.
  3. A. Sahara, H. Kubota, and M. Nakazawa, “Ultra-high speed soliton transmission in presence of polarisation mode dispersion using in-line synchronous modulation,” Electron. Lett. 35, 76–78 (1999).
    [CrossRef]
  4. H. Bulow, “PMD mitigation techniques and their effectiveness in installed fiber,” Optical Fiber Communication Conference, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 110–112.
  5. C. Francia, F. Bruyere, J. P. Thiery, and D. Penninckx, “Simple dynamic polarisation mode dispersion compensator,” Electron. Lett. 35, 414–415 (1999).
    [CrossRef]
  6. R. Noè, D. Sandel, M. Yoshida-Dierolf, S. Hinz, V. Mirvoda, A. Schopflin, C. Gungener, E. Gottwald, C. Scheerer, G. Fischer, T. Weyrauch, and W. Haase, “Polarization mode dispersion compensation at 10, 20, and 40 Gb/s with various optical equalizers,” J. Lightwave Technol. 17, 1602–1616 (1999).
    [CrossRef]
  7. M. Romagnoli, P. Franco, R. Corsini, A. Schiffini, and M. Midrio, “Time-domain Fourier optics for polarization-mode dispersion compensation,” Opt. Lett. 24, 1197–1199 (1999).
    [CrossRef]
  8. S. Lee, R. Khosravani, J. Peng, V. Grubsky, D. S. Starodubov, A. E. Willner, and J. Feinberg, “Adjustable compensation of polarization mode dispersion using a high-birefringence nonlinearly chirped fiber Bragg grating,” IEEE Photonics Technol. Lett. 11, 1277–1279 (1999).
    [CrossRef]
  9. L. F. Mollenauer, K. Smith, J. P. Gordon, and C. R. Menyuk, “Resistance of solitons to the effect of polarization dispersion in optical fibers,” Opt. Lett. 14, 1219–1221 (1989).
    [CrossRef] [PubMed]
  10. 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]
  11. Y. Chen and H. A. Haus, “Solitons and polarization mode dispersion,” Opt. Lett. 25, 290–292 (2000).
    [CrossRef]
  12. H. A. Haus and Y. Chen, “Perturbation of Manakov solitons,” MIT, Cambridge, Mass. 02139 (personal communication, 2000).
  13. T. I. Lakoba and D. J. Kaup, “Perturbation theory for the Manakov soliton and its application to pulse propagation in birefringent fibers,” Phys. Rev. E 56, 6147–6165 (1997).
    [CrossRef]
  14. G. J. Foschini and C. D. Poole, “Statistical theory of polarization dispersion in single mode fibers,” J. Lightwave Technol. 9, 1439–1456 (1991).
    [CrossRef]
  15. 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]
  16. M. Midrio, “Nonlinear principal states of polarization in optical fiber with randomly varying birefringence,” J. Opt. Soc. Am. B 17, 169–177 (2000).
    [CrossRef]
  17. D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov–PMD equations to the study of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 15, 1735–1745 (1997).
    [CrossRef]
  18. S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).
  19. C. R. Menyuk, “Stability of solitons in birefringent optical fibers. II. Arbitrary amplitudes,” J. Opt. Soc. Am. B 5, 392–402 (1988).
    [CrossRef]
  20. D. J. Kaup and B. A. Malomed, “Soliton trapping and daughter waves in the Manakov model,” Phys. Rev. A 48, 599–604 (1993).
    [CrossRef] [PubMed]
  21. H. A. Haus, W. S. Wong, and F. J. Khatri, “Continuum generation by perturbation of soliton,” J. Opt. Soc. Am. B 14, 304–313 (1997).
    [CrossRef]

2000 (2)

1999 (5)

A. Sahara, H. Kubota, and M. Nakazawa, “Ultra-high speed soliton transmission in presence of polarisation mode dispersion using in-line synchronous modulation,” Electron. Lett. 35, 76–78 (1999).
[CrossRef]

C. Francia, F. Bruyere, J. P. Thiery, and D. Penninckx, “Simple dynamic polarisation mode dispersion compensator,” Electron. Lett. 35, 414–415 (1999).
[CrossRef]

R. Noè, D. Sandel, M. Yoshida-Dierolf, S. Hinz, V. Mirvoda, A. Schopflin, C. Gungener, E. Gottwald, C. Scheerer, G. Fischer, T. Weyrauch, and W. Haase, “Polarization mode dispersion compensation at 10, 20, and 40 Gb/s with various optical equalizers,” J. Lightwave Technol. 17, 1602–1616 (1999).
[CrossRef]

M. Romagnoli, P. Franco, R. Corsini, A. Schiffini, and M. Midrio, “Time-domain Fourier optics for polarization-mode dispersion compensation,” Opt. Lett. 24, 1197–1199 (1999).
[CrossRef]

S. Lee, R. Khosravani, J. Peng, V. Grubsky, D. S. Starodubov, A. E. Willner, and J. Feinberg, “Adjustable compensation of polarization mode dispersion using a high-birefringence nonlinearly chirped fiber Bragg grating,” IEEE Photonics Technol. Lett. 11, 1277–1279 (1999).
[CrossRef]

1997 (3)

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

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov–PMD equations to the study of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 15, 1735–1745 (1997).
[CrossRef]

H. A. Haus, W. S. Wong, and F. J. Khatri, “Continuum generation by perturbation of soliton,” J. Opt. Soc. Am. B 14, 304–313 (1997).
[CrossRef]

1996 (1)

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

1993 (1)

D. J. Kaup and B. A. Malomed, “Soliton trapping and daughter waves in the Manakov model,” Phys. Rev. A 48, 599–604 (1993).
[CrossRef] [PubMed]

1991 (2)

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]

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

1989 (1)

1988 (1)

1974 (1)

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

Bruyere, F.

C. Francia, F. Bruyere, J. P. Thiery, and D. Penninckx, “Simple dynamic polarisation mode dispersion compensator,” Electron. Lett. 35, 414–415 (1999).
[CrossRef]

Chen, H. H.

Chen, Y.

Corsini, R.

Feinberg, J.

S. Lee, R. Khosravani, J. Peng, V. Grubsky, D. S. Starodubov, A. E. Willner, and J. Feinberg, “Adjustable compensation of polarization mode dispersion using a high-birefringence nonlinearly chirped fiber Bragg grating,” IEEE Photonics Technol. Lett. 11, 1277–1279 (1999).
[CrossRef]

Fischer, G.

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]

Francia, C.

C. Francia, F. Bruyere, J. P. Thiery, and D. Penninckx, “Simple dynamic polarisation mode dispersion compensator,” Electron. Lett. 35, 414–415 (1999).
[CrossRef]

Franco, P.

Gordon, J. P.

Gottwald, E.

Grubsky, V.

S. Lee, R. Khosravani, J. Peng, V. Grubsky, D. S. Starodubov, A. E. Willner, and J. Feinberg, “Adjustable compensation of polarization mode dispersion using a high-birefringence nonlinearly chirped fiber Bragg grating,” IEEE Photonics Technol. Lett. 11, 1277–1279 (1999).
[CrossRef]

Gungener, C.

Haase, W.

Haus, H. A.

Hinz, S.

Kaup, D. J.

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

D. J. Kaup and B. A. Malomed, “Soliton trapping and daughter waves in the Manakov model,” Phys. Rev. A 48, 599–604 (1993).
[CrossRef] [PubMed]

Khatri, F. J.

Khosravani, R.

S. Lee, R. Khosravani, J. Peng, V. Grubsky, D. S. Starodubov, A. E. Willner, and J. Feinberg, “Adjustable compensation of polarization mode dispersion using a high-birefringence nonlinearly chirped fiber Bragg grating,” IEEE Photonics Technol. Lett. 11, 1277–1279 (1999).
[CrossRef]

Kubota, H.

A. Sahara, H. Kubota, and M. Nakazawa, “Ultra-high speed soliton transmission in presence of polarisation mode dispersion using in-line synchronous modulation,” Electron. Lett. 35, 76–78 (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 birefringent fibers,” Phys. Rev. E 56, 6147–6165 (1997).
[CrossRef]

Lee, S.

S. Lee, R. Khosravani, J. Peng, V. Grubsky, D. S. Starodubov, A. E. Willner, and J. Feinberg, “Adjustable compensation of polarization mode dispersion using a high-birefringence nonlinearly chirped fiber Bragg grating,” IEEE Photonics Technol. Lett. 11, 1277–1279 (1999).
[CrossRef]

Malomed, B. A.

D. J. Kaup and B. A. Malomed, “Soliton trapping and daughter waves in the Manakov model,” Phys. Rev. A 48, 599–604 (1993).
[CrossRef] [PubMed]

Manakov, S. V.

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

Marcuse, D.

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov–PMD equations to the study of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 15, 1735–1745 (1997).
[CrossRef]

Menyuk, C. R.

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov–PMD equations to the study of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 15, 1735–1745 (1997).
[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]

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]

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

C. R. Menyuk, “Stability of solitons in birefringent optical fibers. II. Arbitrary amplitudes,” J. Opt. Soc. Am. B 5, 392–402 (1988).
[CrossRef]

Midrio, M.

Mirvoda, V.

Mollenauer, L. F.

Nakazawa, M.

A. Sahara, H. Kubota, and M. Nakazawa, “Ultra-high speed soliton transmission in presence of polarisation mode dispersion using in-line synchronous modulation,” Electron. Lett. 35, 76–78 (1999).
[CrossRef]

Noè, R.

Peng, J.

S. Lee, R. Khosravani, J. Peng, V. Grubsky, D. S. Starodubov, A. E. Willner, and J. Feinberg, “Adjustable compensation of polarization mode dispersion using a high-birefringence nonlinearly chirped fiber Bragg grating,” IEEE Photonics Technol. Lett. 11, 1277–1279 (1999).
[CrossRef]

Penninckx, D.

C. Francia, F. Bruyere, J. P. Thiery, and D. Penninckx, “Simple dynamic polarisation mode dispersion compensator,” Electron. Lett. 35, 414–415 (1999).
[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]

Romagnoli, M.

Sahara, A.

A. Sahara, H. Kubota, and M. Nakazawa, “Ultra-high speed soliton transmission in presence of polarisation mode dispersion using in-line synchronous modulation,” Electron. Lett. 35, 76–78 (1999).
[CrossRef]

Sandel, D.

Scheerer, C.

Schiffini, A.

Schopflin, A.

Smith, K.

Starodubov, D. S.

S. Lee, R. Khosravani, J. Peng, V. Grubsky, D. S. Starodubov, A. E. Willner, and J. Feinberg, “Adjustable compensation of polarization mode dispersion using a high-birefringence nonlinearly chirped fiber Bragg grating,” IEEE Photonics Technol. Lett. 11, 1277–1279 (1999).
[CrossRef]

Thiery, J. P.

C. Francia, F. Bruyere, J. P. Thiery, and D. Penninckx, “Simple dynamic polarisation mode dispersion compensator,” Electron. Lett. 35, 414–415 (1999).
[CrossRef]

Wai, P. K. A.

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov–PMD equations to the study of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 15, 1735–1745 (1997).
[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]

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]

Weyrauch, T.

Willner, A. E.

S. Lee, R. Khosravani, J. Peng, V. Grubsky, D. S. Starodubov, A. E. Willner, and J. Feinberg, “Adjustable compensation of polarization mode dispersion using a high-birefringence nonlinearly chirped fiber Bragg grating,” IEEE Photonics Technol. Lett. 11, 1277–1279 (1999).
[CrossRef]

Wong, W. S.

Yoshida-Dierolf, M.

Electron. Lett. (2)

A. Sahara, H. Kubota, and M. Nakazawa, “Ultra-high speed soliton transmission in presence of polarisation mode dispersion using in-line synchronous modulation,” Electron. Lett. 35, 76–78 (1999).
[CrossRef]

C. Francia, F. Bruyere, J. P. Thiery, and D. Penninckx, “Simple dynamic polarisation mode dispersion compensator,” Electron. Lett. 35, 414–415 (1999).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

S. Lee, R. Khosravani, J. Peng, V. Grubsky, D. S. Starodubov, A. E. Willner, and J. Feinberg, “Adjustable compensation of polarization mode dispersion using a high-birefringence nonlinearly chirped fiber Bragg grating,” IEEE Photonics Technol. Lett. 11, 1277–1279 (1999).
[CrossRef]

J. Lightwave Technol. (4)

R. Noè, D. Sandel, M. Yoshida-Dierolf, S. Hinz, V. Mirvoda, A. Schopflin, C. Gungener, E. Gottwald, C. Scheerer, G. Fischer, T. Weyrauch, and W. Haase, “Polarization mode dispersion compensation at 10, 20, and 40 Gb/s with various optical equalizers,” J. Lightwave Technol. 17, 1602–1616 (1999).
[CrossRef]

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

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]

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov–PMD equations to the study of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 15, 1735–1745 (1997).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Lett. (4)

Phys. Rev. A (1)

D. J. Kaup and B. A. Malomed, “Soliton trapping and daughter waves in the Manakov model,” Phys. Rev. A 48, 599–604 (1993).
[CrossRef] [PubMed]

Phys. Rev. E (1)

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

Sov. Phys. JETP (1)

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

Other (4)

H. A. Haus and Y. Chen, “Perturbation of Manakov solitons,” MIT, Cambridge, Mass. 02139 (personal communication, 2000).

H. Bulow, “PMD mitigation techniques and their effectiveness in installed fiber,” Optical Fiber Communication Conference, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 110–112.

F. Heismann, D. A. Fishman, and D. L. Wilson, “Automatic compensation of first order polarization mode dispersion in a 10 Gb/s transmission system,” Proceedings of the 24th European Conference on Optical Communication (Telefonica, Madrid, Spain, 1998), pp. 529–530.

H. Bulow, D. Schlump, J. Weber, B. Wedding, and R. Heidemann, “Electronic equalization of fiber PMD-induced distortion at 10 Gbit/s,” Optical Fiber Communication Conference, Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 151–152.

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Figures (3)

Fig. 1
Fig. 1

Relative time displacement versus normalized distance for a fiber with λ=0.01. The polarization angle is θ=π/4. The solid curve is theoretical [Eq. (12)]; the filled dots are numerical results.

Fig. 2
Fig. 2

Soliton plus dispersive wave on u¯ polarization for propagation after the normalized distance z=15 in a fiber with λ=0.01. The solid curve is theoretical; the dots are numerical results.

Fig. 3
Fig. 3

Relative time displacement for a soliton traveling in a two-trunk fiber with the characteristics as in the main text. The shaded region denotes the second fiber.

Equations (45)

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

i UZ+12 2UT2+(|U|2+|V|2)U=-iλ UTiRu,
i VZ+12 2VT2+(|U|2+|V|2)V=iλ VTiRv,
U(T, Z)=η cos(θ)sech[η(T-ξU)]exp[iκU(T-ξU)],
V(T, Z)=η sin(θ)sech[η(T-ξV)]exp[iκV(T-ξV)],
ddZ ξU+ξV2=Re [cos(θ)RU+sin(θ)RV] x exp(-iZ/2)cosh(x) dx=λ cos(2θ),
i uz+12 2ut2+(|u|2+|v|2)u=-2iλ sin2(θ) ut,
i vz+12 2vt2+(|u|2+|v|2)v=2iλ cos2(θ) vt,
u¯(t, z)=u(t, z)exp[2iλ sin2(θ)t]exp[-2iλ2 sin4(θ)z],
v¯(t, z)=v(t, z)exp[-2iλ cos2(θ)t]exp[-2iλ2 cos4(θ)z].
u¯(t, z=0)=cos(θ)sech(t)exp[2iλ sin2 (θ)t],
v¯(t, z=0)=sin(θ)sech(t)exp[-2iλ cos2(θ)t].
Δu¯(t, z=0)=2iλ cos(θ)sin2(θ)t sech(t),
Δv¯(t, z=0)=-2iλ sin(θ)cos2(θ)t sech(t),
es=u¯(t, z=0)e1+v¯(t, z=0)e2|u¯(t, z=0)|2+|v¯(t, z=0)|2=cos(θ)e1+sin(θ)e2,
e=-sin(θ)e1+cos(θ)e2,
u¯e1+v¯e2=ψes+ϕe.
i Δψz+12 2Δψt2+2|ψs|2Δψ+ψs2Δψ*=0,
i Δϕz+12 2Δϕt2+|ψs|2Δϕ=0,
Δψ(t, z=0)=[Δu¯(t, 0)e1+Δv¯(t, 0)e2]·es=0,
Δϕ(t, z=0)=[Δu¯(t, 0)e1+Δv¯(t,0)e2]·e=-iλ sin(2θ)t sech(t),
Δϕ(t, z)=-λ cos(θ)sin(θ)h(t, z),
h(t, z)=-+ sechπ2Ω [Ω+i tanh(t)]Ω2+1×exp[iΩt-i(Ω2/2)z]dΩ,
Δu¯(t, z)Δv¯(t, z)=-sin(θ)cos(θ)Δϕ=cos(θ)sin2(θ)-cos2(θ)sin(θ)λh(t, z).
Δξu cos2(θ)=Re -+Δu¯(t, z)u¯s*(t, z)tdt=λ cos2(θ)sin2(θ)Re -+h(t ,z)sech(t)×exp(-iz/2)tdt,
Δξv sin2(θ)=λ cos2(θ)sin2(θ)Re -+h(t, z)sech(t)×exp(-iz/2)tdt,
Δξ=Δξu-Δξv=λF(z),
F(z)=π-+ sech2π2Ω1+Ω2 sin[(1+Ω2)z/2]dΩ.
i u¯z+12 2u¯t2+(|u¯|2+|v¯|2)u¯
=-2i(λ2-λ1)sin2(θ) u¯t,
i v¯z+12 2v¯t2+(|u¯|2+|v¯|2)v¯
=2i(λ2-λ1)cos2(θ) v¯t.
i Δϕkz+12 2Δϕkt2+|ψs|2Δϕk=0,
Δϕk(t, z)=Δϕ(t, z)|zSk,
k=1,2,
Δϕ1(t, 0)=-iλ1 sin(2θ)t sech(t),
Δϕ2(t, L1)=Δϕ1(t, L1)-i(λ2-λ1)sin(2θ)×exp(iL1/2)t sech(t),
Δϕ1(t, z)=-λ1 sin(θ)cos(θ)h(t, z),zS1,
Δϕ2(t, z)=-sin(θ)cos(θ)[λ1h(t, z)+(λ2-λ1)×exp(iL1/2)h(t, z-L1)],zS2,
Δξ(z)=λ1F(z)zS1λ1F(z)+(λ2-λ1)F(z-L1)zS2.
Δξ(LTOT)=λ1F(LTOT)+k=2N(λk-λk-1)×F(LTOT-sk-1),
sk=i=1kLk,LTOT=i=1NLk.
Δξ(z)=0zλ(s)f(z-s)ds,
f(z)=dF(z)dz=π2  sech2π2Ωcos[(1+Ω2)z/2]dΩ.
Δξ(LTOT)=k=1kλkF(LTOT-sk-1)-k=2Nλk-1F(LTOT-sk-1)=k=1Nλk[F(LTOT-sk-1)-F(LTOT-sk)].
Δξ(LTOT)k=1Nλ(ck)f(LTOT-ck)(sk-sk-1),

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