Abstract

We consider femtosecond soliton-pulse propagation in a birefringent optical fiber where rapidly oscillating terms, the difference in polarization dispersions, and the difference in group velocities of the two polarization components have to be taken into account. We demonstrate the existence of a novel class of linearly polarized soliton states (with the linear polarization ranging from 0 to 2π). We also find the elliptically polarized soliton states, which do not appear to be acceptable to the coupled nonlinear Schrödinger equations describing the pulse evolution in the birefringent fiber when the different dispersions between the two polarizations are ignored and the group-velocity difference is taken into account. More importantly, the corresponding stability analysis reveals that within certain operating regions the fast soliton can be stable and the slow soliton can be unstable, whereas in the others the fast soliton is unstable and the slow soliton is stable, in contrast to those reported earlier by neglecting different polarization dispersions. On the other hand, both the linearly polarized soliton states and the elliptically polarized soliton states are found to be unstable. This indicates that for high-capacity coherent soliton communication in the femtosecond regime, the pulse must be launched along either the slow or the fast axis of a practical polarization-maintaining fiber. Finally, the potential applications of weakly unstable linearly polarized soliton states for ultrafast soliton switching are discussed.

© 1997 Optical Society of America

Full Article  |  PDF Article

Corrections

Yijiang Chen and Javid Atai, "Femtosecond soliton pulses in birefringent optical fibers: errata," J. Opt. Soc. Am. B 15, 1246-1246 (1998)
https://www.osapublishing.org/josab/abstract.cfm?uri=josab-15-3-1246

References

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  1. K. C. Kao and G. A. Hockman, “Dielectric fibre surface waveguides for optical frequencies,” IEE Proc. Optoelectron. 113, 1151–1158 (1966); D. Gloge, “Dispersion in weakly guiding fibers,” Appl. Opt. 10, 2442–2445 (1971).
    [CrossRef] [PubMed]
  2. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
    [CrossRef]
  3. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observations of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
    [CrossRef]
  4. I. P. Kaminow, “Polarization in optical fibers,” IEEE J. Quantum Electron. 17, 15–22 (1981).
    [CrossRef]
  5. 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]
  6. C. R. Menyuk, “Stability of solitons in birefringent optical fibers. I: Equal propagation amplitudes,” Opt. Lett. 12, 614–616 (1987).
    [CrossRef] [PubMed]
  7. C. R. Menyuk, “Stability of solitons in birefringent optical fibers. II: Arbitrary amplitudes,” J. Opt. Soc. Am. B 5, 392–402 (1988).
    [CrossRef]
  8. M. N. Ialam, C. D. Poole, and J. P. Gordon, “Soliton trapping in birefringent optical fibers,” Opt. Lett. 18, 1011–1013 (1989).
  9. H. G. Unger, Planar Optical Waveguides and Fibres (Oxford, New York, 1977), p. 43.
  10. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995), p. 244.
  11. Y. Silberberg and Y. Barad, “Rotating vector solitary waves in isotropic fibers,” Opt. Lett. 20, 246–248 (1995).
    [CrossRef] [PubMed]
  12. V. M. Eleonskii, V. G. Korolev, N. E. Kulagin, and L. P. Shil’nikov, “Branching bifurcations of vector envelope solitons and integrability of Hamiltonian systems,” Sov. Phys. JETP 72, 619–623 (1991).
  13. M. Haelterman and A. P. Sheppard, “The elliptically polarized fundamental vector soliton of isotropic Kerr media,” Phys. Lett. A 194, 191–196 (1994).
    [CrossRef]
  14. J. M. Soto-Crespo, N. N. Akhmediev, and A. Ankiewicz, “Stationary solitonlike pulses in birefringent optical fibers,” Phys. Rev. E 51, 3547–3555 (1995).
    [CrossRef]
  15. S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
    [CrossRef]
  16. C. D. Poole and C. R. Giles, “Polarization-dependent pulse compression and broadening due to polarization dispersion in dispersion-shift fiber,” Opt. Lett. 13, 155–157 (1988).
    [CrossRef]
  17. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), p. 264.
  18. D. N. Christodoulides and R. I. Joseph, “Vector solitons in birefringent nonlinear dispersive media,” Opt. Lett. 13, 53–55 (1988).
    [CrossRef] [PubMed]
  19. M. V. Tratnik and J. E. Sipe, “Bound solitary waves in a birefringent optical fiber,” Phys. Rev. A 38, 2011–2017 (1988).
    [CrossRef] [PubMed]
  20. N. N. Akhmediev, A. V. Buryak, J. M. Soto-Crespo, and D. R. Andersen, “Phase-locked stationary soliton states in birefringent nonlinear optical fibers,” J. Opt. Soc. Am. B 12, 434–439 (1995).
    [CrossRef]
  21. In K. J. Blow, N. J. Doran, and D. Wood, “Polarization instabilities for solitons in birefringent fibers,” Opt. Lett. 12, 202–204 (1987) and in E. M. Wright, G. I. Stegeman, and S. Wabnitz, “Solitary-wave decay and symmetry-breaking instabilities in two-mode fibers,” Phys. Rev. A 40, 4455–4466 (1989), it was shown that for rd=1, the slow soliton is always stable and the fast soliton is unstable (in the region where the slow soliton is stable). In addition to confirming the result of the above two references for rd=1,Ref. 14 indicated that both the slow soliton and the fast soliton can be unstable for rd=1 when δ¯=δ/Δ/4=2δ/R=B/(2Dλc)>1, which is possible only for very small dispersion D<0.1 ps/nm/km with the birefringence B=10-4 or D<0.01 ps/nm/km with B=10-5. This conclusion requires further examination by considering rd≠1, which is a prominent effect for very small D<0.1 ps/nm/km.
    [CrossRef] [PubMed]
  22. At this value κ=1.01(<κulb<κvlb), the slow soliton is unstable.
  23. F. M. Mitschke and L. F. Mollenauer, “Discovery of the soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986).
    [CrossRef] [PubMed]
  24. J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986).
    [CrossRef] [PubMed]
  25. K. J. Blow, N. J. Doran, and D. Wood, “Suppression of the soliton self-frequency shift by bandwidth-limited amplification,” J. Opt. Soc. Am. B 5, 1301–1304 (1988); M. Nakazawa, K. Kurokawa, H. Kubota, and E. Yamada, “Observation of the trapping of an optical soliton by adiabatic gain narrowing and its escape,” Phys. Rev. Lett. 65, 1881–1884 (1990).
    [CrossRef] [PubMed]
  26. M. Nakazawa, H. Kubota, K. Kurokawa, and E. Yamada, “Femtosecond optical soliton transmission over long distances using adiabatic trapping and soliton standardization,” J. Opt. Soc. Am. B 8, 1811–1817 (1991).
    [CrossRef]
  27. S. Liu and W. Wang, “Complete compensation for the soliton self-frequency shift and third-order dispersion of a fiber,” Opt. Lett. 18, 1911–1912 (1993).
    [CrossRef] [PubMed]

1995 (3)

1994 (1)

M. Haelterman and A. P. Sheppard, “The elliptically polarized fundamental vector soliton of isotropic Kerr media,” Phys. Lett. A 194, 191–196 (1994).
[CrossRef]

1993 (1)

1992 (1)

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[CrossRef]

1991 (2)

M. Nakazawa, H. Kubota, K. Kurokawa, and E. Yamada, “Femtosecond optical soliton transmission over long distances using adiabatic trapping and soliton standardization,” J. Opt. Soc. Am. B 8, 1811–1817 (1991).
[CrossRef]

V. M. Eleonskii, V. G. Korolev, N. E. Kulagin, and L. P. Shil’nikov, “Branching bifurcations of vector envelope solitons and integrability of Hamiltonian systems,” Sov. Phys. JETP 72, 619–623 (1991).

1989 (1)

M. N. Ialam, C. D. Poole, and J. P. Gordon, “Soliton trapping in birefringent optical fibers,” Opt. Lett. 18, 1011–1013 (1989).

1988 (5)

1987 (2)

1986 (2)

1981 (1)

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

1980 (1)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observations of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[CrossRef]

1978 (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]

1973 (1)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

1966 (1)

K. C. Kao and G. A. Hockman, “Dielectric fibre surface waveguides for optical frequencies,” IEE Proc. Optoelectron. 113, 1151–1158 (1966); D. Gloge, “Dispersion in weakly guiding fibers,” Appl. Opt. 10, 2442–2445 (1971).
[CrossRef] [PubMed]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995), p. 244.

Akhmediev, N. N.

J. M. Soto-Crespo, N. N. Akhmediev, and A. Ankiewicz, “Stationary solitonlike pulses in birefringent optical fibers,” Phys. Rev. E 51, 3547–3555 (1995).
[CrossRef]

N. N. Akhmediev, A. V. Buryak, J. M. Soto-Crespo, and D. R. Andersen, “Phase-locked stationary soliton states in birefringent nonlinear optical fibers,” J. Opt. Soc. Am. B 12, 434–439 (1995).
[CrossRef]

Andersen, D. R.

Ankiewicz, A.

J. M. Soto-Crespo, N. N. Akhmediev, and A. Ankiewicz, “Stationary solitonlike pulses in birefringent optical fibers,” Phys. Rev. E 51, 3547–3555 (1995).
[CrossRef]

Barad, Y.

Bergano, N. S.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[CrossRef]

Blow, K. J.

Buryak, A. V.

Christodoulides, D. N.

Doran, N. J.

Eleonskii, V. M.

V. M. Eleonskii, V. G. Korolev, N. E. Kulagin, and L. P. Shil’nikov, “Branching bifurcations of vector envelope solitons and integrability of Hamiltonian systems,” Sov. Phys. JETP 72, 619–623 (1991).

Evangelides, S. G.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[CrossRef]

Giles, C. R.

Gordon, J. P.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[CrossRef]

M. N. Ialam, C. D. Poole, and J. P. Gordon, “Soliton trapping in birefringent optical fibers,” Opt. Lett. 18, 1011–1013 (1989).

J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986).
[CrossRef] [PubMed]

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observations of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[CrossRef]

Haelterman, M.

M. Haelterman and A. P. Sheppard, “The elliptically polarized fundamental vector soliton of isotropic Kerr media,” Phys. Lett. A 194, 191–196 (1994).
[CrossRef]

Hasegawa, A.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Hockman, G. A.

K. C. Kao and G. A. Hockman, “Dielectric fibre surface waveguides for optical frequencies,” IEE Proc. Optoelectron. 113, 1151–1158 (1966); D. Gloge, “Dispersion in weakly guiding fibers,” Appl. Opt. 10, 2442–2445 (1971).
[CrossRef] [PubMed]

Ialam, M. N.

M. N. Ialam, C. D. Poole, and J. P. Gordon, “Soliton trapping in birefringent optical fibers,” Opt. Lett. 18, 1011–1013 (1989).

Joseph, R. I.

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. 17, 15–22 (1981).
[CrossRef]

Kao, K. C.

K. C. Kao and G. A. Hockman, “Dielectric fibre surface waveguides for optical frequencies,” IEE Proc. Optoelectron. 113, 1151–1158 (1966); D. Gloge, “Dispersion in weakly guiding fibers,” Appl. Opt. 10, 2442–2445 (1971).
[CrossRef] [PubMed]

Korolev, V. G.

V. M. Eleonskii, V. G. Korolev, N. E. Kulagin, and L. P. Shil’nikov, “Branching bifurcations of vector envelope solitons and integrability of Hamiltonian systems,” Sov. Phys. JETP 72, 619–623 (1991).

Kubota, H.

Kulagin, N. E.

V. M. Eleonskii, V. G. Korolev, N. E. Kulagin, and L. P. Shil’nikov, “Branching bifurcations of vector envelope solitons and integrability of Hamiltonian systems,” Sov. Phys. JETP 72, 619–623 (1991).

Kurokawa, K.

Liu, S.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), p. 264.

Menyuk, C. R.

Mitschke, F. M.

Mollenauer, L. F.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[CrossRef]

F. M. Mitschke and L. F. Mollenauer, “Discovery of the soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986).
[CrossRef] [PubMed]

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observations of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[CrossRef]

Nakazawa, M.

Pleibel, W.

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]

Poole, C. D.

M. N. Ialam, C. D. Poole, and J. P. Gordon, “Soliton trapping in birefringent optical fibers,” Opt. Lett. 18, 1011–1013 (1989).

C. D. Poole and C. R. Giles, “Polarization-dependent pulse compression and broadening due to polarization dispersion in dispersion-shift fiber,” Opt. Lett. 13, 155–157 (1988).
[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]

Sheppard, A. P.

M. Haelterman and A. P. Sheppard, “The elliptically polarized fundamental vector soliton of isotropic Kerr media,” Phys. Lett. A 194, 191–196 (1994).
[CrossRef]

Shil’nikov, L. P.

V. M. Eleonskii, V. G. Korolev, N. E. Kulagin, and L. P. Shil’nikov, “Branching bifurcations of vector envelope solitons and integrability of Hamiltonian systems,” Sov. Phys. JETP 72, 619–623 (1991).

Silberberg, Y.

Sipe, J. E.

M. V. Tratnik and J. E. Sipe, “Bound solitary waves in a birefringent optical fiber,” Phys. Rev. A 38, 2011–2017 (1988).
[CrossRef] [PubMed]

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), p. 264.

Soto-Crespo, J. M.

N. N. Akhmediev, A. V. Buryak, J. M. Soto-Crespo, and D. R. Andersen, “Phase-locked stationary soliton states in birefringent nonlinear optical fibers,” J. Opt. Soc. Am. B 12, 434–439 (1995).
[CrossRef]

J. M. Soto-Crespo, N. N. Akhmediev, and A. Ankiewicz, “Stationary solitonlike pulses in birefringent optical fibers,” Phys. Rev. E 51, 3547–3555 (1995).
[CrossRef]

Stolen, R. H.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observations of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[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]

Tappert, F.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Tratnik, M. V.

M. V. Tratnik and J. E. Sipe, “Bound solitary waves in a birefringent optical fiber,” Phys. Rev. A 38, 2011–2017 (1988).
[CrossRef] [PubMed]

Unger, H. G.

H. G. Unger, Planar Optical Waveguides and Fibres (Oxford, New York, 1977), p. 43.

Wang, W.

Wood, D.

Yamada, E.

Appl. Phys. Lett. (2)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[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]

IEE Proc. Optoelectron. (1)

K. C. Kao and G. A. Hockman, “Dielectric fibre surface waveguides for optical frequencies,” IEE Proc. Optoelectron. 113, 1151–1158 (1966); D. Gloge, “Dispersion in weakly guiding fibers,” Appl. Opt. 10, 2442–2445 (1971).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

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

J. Lightwave Technol. (1)

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[CrossRef]

J. Opt. Soc. Am. B (4)

Opt. Lett. (9)

S. Liu and W. Wang, “Complete compensation for the soliton self-frequency shift and third-order dispersion of a fiber,” Opt. Lett. 18, 1911–1912 (1993).
[CrossRef] [PubMed]

F. M. Mitschke and L. F. Mollenauer, “Discovery of the soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986).
[CrossRef] [PubMed]

J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986).
[CrossRef] [PubMed]

In K. J. Blow, N. J. Doran, and D. Wood, “Polarization instabilities for solitons in birefringent fibers,” Opt. Lett. 12, 202–204 (1987) and in E. M. Wright, G. I. Stegeman, and S. Wabnitz, “Solitary-wave decay and symmetry-breaking instabilities in two-mode fibers,” Phys. Rev. A 40, 4455–4466 (1989), it was shown that for rd=1, the slow soliton is always stable and the fast soliton is unstable (in the region where the slow soliton is stable). In addition to confirming the result of the above two references for rd=1,Ref. 14 indicated that both the slow soliton and the fast soliton can be unstable for rd=1 when δ¯=δ/Δ/4=2δ/R=B/(2Dλc)>1, which is possible only for very small dispersion D<0.1 ps/nm/km with the birefringence B=10-4 or D<0.01 ps/nm/km with B=10-5. This conclusion requires further examination by considering rd≠1, which is a prominent effect for very small D<0.1 ps/nm/km.
[CrossRef] [PubMed]

D. N. Christodoulides and R. I. Joseph, “Vector solitons in birefringent nonlinear dispersive media,” Opt. Lett. 13, 53–55 (1988).
[CrossRef] [PubMed]

M. N. Ialam, C. D. Poole, and J. P. Gordon, “Soliton trapping in birefringent optical fibers,” Opt. Lett. 18, 1011–1013 (1989).

C. D. Poole and C. R. Giles, “Polarization-dependent pulse compression and broadening due to polarization dispersion in dispersion-shift fiber,” Opt. Lett. 13, 155–157 (1988).
[CrossRef]

C. R. Menyuk, “Stability of solitons in birefringent optical fibers. I: Equal propagation amplitudes,” Opt. Lett. 12, 614–616 (1987).
[CrossRef] [PubMed]

Y. Silberberg and Y. Barad, “Rotating vector solitary waves in isotropic fibers,” Opt. Lett. 20, 246–248 (1995).
[CrossRef] [PubMed]

Phys. Lett. A (1)

M. Haelterman and A. P. Sheppard, “The elliptically polarized fundamental vector soliton of isotropic Kerr media,” Phys. Lett. A 194, 191–196 (1994).
[CrossRef]

Phys. Rev. A (1)

M. V. Tratnik and J. E. Sipe, “Bound solitary waves in a birefringent optical fiber,” Phys. Rev. A 38, 2011–2017 (1988).
[CrossRef] [PubMed]

Phys. Rev. E (1)

J. M. Soto-Crespo, N. N. Akhmediev, and A. Ankiewicz, “Stationary solitonlike pulses in birefringent optical fibers,” Phys. Rev. E 51, 3547–3555 (1995).
[CrossRef]

Phys. Rev. Lett. (1)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observations of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[CrossRef]

Sov. Phys. JETP (1)

V. M. Eleonskii, V. G. Korolev, N. E. Kulagin, and L. P. Shil’nikov, “Branching bifurcations of vector envelope solitons and integrability of Hamiltonian systems,” Sov. Phys. JETP 72, 619–623 (1991).

Other (4)

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), p. 264.

H. G. Unger, Planar Optical Waveguides and Fibres (Oxford, New York, 1977), p. 43.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995), p. 244.

At this value κ=1.01(<κulb<κvlb), the slow soliton is unstable.

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Figures (9)

Fig. 1
Fig. 1

(a) Field profiles of the linearly polarized soliton state at κ=1.017158 (corresponding to the polarization angle θ=32°) and rd=0.95 with the solid curves from numerical solution and the dotted curves from the analytical approximation. (b) The paired relationships between peak amplitudes f(0) and g(0), κ and f(0), and κ and θ of the linearly polarized soliton states at rd=0.95.

Fig. 2
Fig. 2

Bifurcation values κulb and κvlb of the linearly polarized soliton states versus rd with the shaded area indicating the existence region of the linearly polarized soliton states.

Fig. 3
Fig. 3

Power of the linearly polarized soliton states together with those of the slow and the fast solitons versus κ for rd=0.95. Note that by assuming rd=1, power Pv of the fast soliton is always greater than Pu of the slow soliton for κ>1 except at κ=1 when Pu=Pv. The dashed lines represent unstable branches and the solid lines represent the stable branches.

Fig. 4
Fig. 4

(a) Power of the elliptically polarized soliton states versus κ for rd=0.95, where the dashed curves represent unstable branches and the solid line represents the stable branches. (b) Bifurcation values κveb of the elliptically polarized soliton states (α=1/3 or cp=2/3) versus rd with the existence region of the soliton states identified by the shaded area.

Fig. 5
Fig. 5

(a) Paired relationships between peak amplitudes f(0), and g(0), f(0) and κ, and κ and R=f(0)/g(0) of the elliptically polarized soliton states at rd=0.95. (b) Field profiles of the elliptically polarized soliton states at κ=2 and rd=0.95.

Fig. 6
Fig. 6

(a) Growth rates χ of the linearly polarized soliton states versus the polarization angle θ=tan-1[g(0)/f(0)]. (b) Growth rates χ of the linearly polarized soliton states, the slow soliton, and part of the fast soliton versus κ for rd=0.95.

Fig. 7
Fig. 7

Growth rates χe of the elliptically polarized soliton states and χv of the fast soliton versus κ for rd=0.95, where Re refers to the real part of the complex value χ.

Fig. 8
Fig. 8

Stable evolution of the femtosecond fast soliton at κ=1.01 and rd=0.95: (a) |u|/N, (b) |v|/N; s=(T-Z/vp)N, X=N2Z, and initial input is u=u0+0.2 max×{|u0|}sech[4(s+1.5)] and v=v0+0.2 max{|v0|}sech[4(s+1.5)] with (u0, v0) the soliton state and max{ } referring to the maximum value.

Fig. 9
Fig. 9

Unstable evolution of the linearly polarized soliton state at κ=1.017158 (θ=32°) and rd=0.95: (a), (c) |u|/N; (b), (d) |v|/N. In (a) and (b) no initial perturbation is imposed, and in (c) and (d) initial excitation is u=u0+0.0055 max{|u0|}=sech(4s) and v=v0+0.0055 max{|v0|}sech(4s).

Equations (33)

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iuZ+δ uT+122uT2+(|u|2+cp|v|2)u
+(1-cp)v2u* exp(-iΔZ)=0,
ivZ-δ vT+rd22vT2+(cp|u|2+|v|2)v
+(1-cp)u2v* exp(iΔZ)=0,
u=(1+cp)-1/2 exp[i0.5(1+δ2)Z-iδT]sech T,
v=u exp(i2δT),
u(T, Z)=Nf[(T-Z/vp)N]exp[i(γ1Z+η1T)],
v(T, Z)=μNg[(T-Z/vp)N]exp[i(γ2Z+η2T)],
η1=η2=η=-2δ/(1-rd),
γ1=γ-Δ/4,γ2=γ+Δ/4,
vp=-(1-rd)/[δ(1+rd)],
12d2fds2-f+(f2+αg2)f=0,
rd2d2gds2-κg+(αf2+g2)g=0,
κ=[(rd/2)η2-δη+γ+Δ/4]/(1/2η2+δη+γ-Δ/4)=[2δ2/(1-rd)2+γ+Δ/4]/[2δ2rd/(1-rd)2+γ-Δ/4],
N2=1/2η2+δη+γ-Δ/4=2δ2rd/(1-rd)2+γ-Δ/4
f(s)=2 sech 2s,g=0,
f=0,g(s)=2κ sech 2κ/rds.
g2(0)=κ-αf2(0)+{[κ-αf2(0)]2-f4(0)+2f2(0)}1/2,
12dfds2+rd2dgds2-f2-κg2+12f4+12g4+αf2g2=const.
rd2d2gpds2-κgp+2α sech2(2s)gp=0,
q-m=(κ/rb)1/2
q=[(8α/rd+1)1/2-1]/2.
κu=rd48αrd+11/2-12.
q-m=(rb/κ)1/2,
q=[(8αrd+1)1/2-1]/2,
κv=4rd/[(8αrd+1)1/2-1]2
H=-12dfds2+rd2dgds2+f2+κg2-12f4-12g4-αf2g2ds
f=A1 sech(s/s0),
g=A2 sech(s/s0),
A22=-A12+κ+[κ2-2(κ-1)A12]1/2.
κ=4(1-rd)2A12/3-rd(1+2rd)-2(1-rd)[rd2-2(2rd2-3rd+1)A12/3+4(1-rd)2A14/9]1/21-4rd
A12=3(κ-rd)[κ-(4κ-3)rd]8(κ-1)(1-rd)2,
s02=0.5(A12+rdA22)/(A12+κA22).

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