Abstract

Propagation of short pulses in birefringent single-mode fibers is considered. Initial pulses are assumed to be linearly polarized at an arbitrary angle with respect to the polarization axes. The Kerr nonlinearity leads to a substantial interaction between the partial pulses in each of the two polarizations. When the amplitudes of the partial pulses are equal, it is found that above a certain amplitude threshold, whose size increases with birefringence, the two partial pulses lock together and travel as one unit. This unit can be a single soliton or, at higher amplitudes, a breather. At the same time, the central frequencies of both polarizations shift just far enough so that, if a rapid oscillation is ignored, their group velocities become identical. When the initial amplitudes are unequal, it is found as before that above a certain threshold one or more solitons emerge from the initial pulse. However, the breathers that appeared when the amplitudes were equal are unstable; they break up into two distinct solitons moving at different velocities when the amplitudes become slightly unequal. It is further shown that realistic fiber attenuation has little effect on these results. The numerical method used to obtain these results is described in detail.

© 1988 Optical Society of America

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References

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  1. I. P. Kaminow, “Polarization in optical fibers,” IEEE J. Quantum Electron. QE-17, 15–22 (1981).
    [CrossRef]
  2. A. Hasegawa, F. Tappert, “Transmission of stationary nonlinear pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
    [CrossRef]
  3. L. F. Mollenauer, R. H. Stolen, J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
    [CrossRef]
  4. C. R. Menyuk, “Stability of solitons in birefringent optical fibers. I: Equal propagation amplitudes,” Opt. Lett. 12, 614–616 (1987).
    [CrossRef] [PubMed]
  5. A. Hasegawa, “Self-confinement of multimode optical pulse in a glass fiber,” Opt. Lett. 5, 416–417 (1980).
    [CrossRef] [PubMed]
  6. B. Crosignani, P. Di Porto, “Soliton propagation in multimode fibers,” Opt. Lett. 6, 329–330 (1981).
    [CrossRef] [PubMed]
  7. C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. QE-23, 174–176 (1987).
    [CrossRef]
  8. A. Hasegawa, Y. Kodama, “Signal transmission by optical solitons in monomode fibers,” Proc. IEEE 69, 1145–1150 (1981).
    [CrossRef]
  9. L. F. Mollenauer, J. P. Gordon, M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. QE-22, 157–173 (1986).
    [CrossRef]
  10. C. R. Menyuk, “Origin of solitons in the ‘real’ world,” Phys. Rev. A 33, 4367–4374 (1986).
    [CrossRef] [PubMed]
  11. C. R. Menyuk, P. K. A. Wai, H. H. Chen, Y. C. Lee, “Hamiltonian deformations of integrable, nonlinear field equations (with applications to optical fibers),” in Published Proceedings of the 4th Conference on Applied Mathematics and Computing, F. Dressel, ed., (U.S. Government Printing Office, Washington, D.C., 1987), pp. 373–386.
  12. S. V. Manakov, “On the theory of two-dimensional self-focusing of electromagnetic waves,” Zh. Eksp. Teor. Fiz. 65, 505–516 (1973) [Sov. Phys. JETP 38, 248–253 (1974)].
  13. P. K. A. Wai, C. R. Menyuk, H. H. Chen, Y. C. Lee, “Soliton propagation at the zero dispersion wavelength of single mode optical fibers,” Opt. Lett. 12, 628–630 (1987).
    [CrossRef] [PubMed]
  14. L. F. Mollenauer, R. H. Stolen, “The soliton laser,” Opt. Lett. 9, 13–15 (1984).
    [CrossRef] [PubMed]
  15. M. N. Islam, L. F. Mollenauer, R. H. Stolen, “Fiber Raman amplification soliton laser (FRASL),” in Ultrafast Phenomena V, G. R. Fleming, A. E. Siegman, eds. (Springer-Verlag, Berlin, 1986), pp. 46–50.
    [CrossRef]
  16. B. Zysset, P. Beaud, W. Hodel, H. P. Weber, “80-fs soliton-like pulses from an optical nonlinear fiber resonator,” in Ultrafast Phenomena V, G. R. Fleming, A. E. Siegman, eds. (Springer-Verlag, Berlin, 1986), pp. 54–57.
    [CrossRef]
  17. J. D. Kafka, T. Baer, “Fiber soliton Raman laser pumped by a Nd:YAG laser,” Opt. Lett. 12, 181–183 (1987).
    [CrossRef] [PubMed]
  18. A. S. Gouveia-Neto, A. S. L. Gomes, J. R. Taylor, “Higher-order soliton pulse compression and splitting at 1.32 μ m in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-23, 1193–1198 (1987).
    [CrossRef]
  19. R. Stolen, AT&T Bell Laboratories, Holmdel, New Jersey 07733 (personal communication).
  20. R. H. Stolen, J. Botineau, A. Ashkin, “Intensity discrimination of optical pulses with birefringent fibers,” Opt. Lett. 7, 512–514 (1982).
    [CrossRef] [PubMed]
  21. R. H. Stolen, L. F. Mollenauer, W. J. Tomlinson, “Observation of pulse restoration at the soliton period in optical fibers,” Opt. Lett. 8, 186–188 (1983).
    [CrossRef] [PubMed]
  22. K. J. Blow, N. J. Doran, D. Wood, “Polarization instabilities for solitons in birefringent fibers,” Opt. Lett. 12, 202–204 (1987).
    [CrossRef] [PubMed]
  23. J. Botineau, R. H. Stolen, “Effect of polarization on spectral broadening in optical fibers,” J. Opt. Soc. Am. 72, 1592–1596 (1982).
    [CrossRef]
  24. See, e.g., P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), pp. 437–440.
  25. See, e.g., T. R. Taha, M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation,” J. Comp. Phys. 55, 203–230 (1984).
    [CrossRef]
  26. L. F. Mollenauer, R. H. Stolen, J. P. Gordon, W. J. Tomlinson, “Extreme picosecond pulse narrowing by means of soliton effect in single-mode fibers,” Opt. Lett. 8, 289–291 (1983).
    [CrossRef] [PubMed]

1987 (6)

1986 (2)

L. F. Mollenauer, J. P. Gordon, M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. QE-22, 157–173 (1986).
[CrossRef]

C. R. Menyuk, “Origin of solitons in the ‘real’ world,” Phys. Rev. A 33, 4367–4374 (1986).
[CrossRef] [PubMed]

1984 (2)

L. F. Mollenauer, R. H. Stolen, “The soliton laser,” Opt. Lett. 9, 13–15 (1984).
[CrossRef] [PubMed]

See, e.g., T. R. Taha, M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation,” J. Comp. Phys. 55, 203–230 (1984).
[CrossRef]

1983 (2)

1982 (2)

1981 (3)

A. Hasegawa, Y. Kodama, “Signal transmission by optical solitons in monomode fibers,” Proc. IEEE 69, 1145–1150 (1981).
[CrossRef]

B. Crosignani, P. Di Porto, “Soliton propagation in multimode fibers,” Opt. Lett. 6, 329–330 (1981).
[CrossRef] [PubMed]

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

1980 (2)

A. Hasegawa, “Self-confinement of multimode optical pulse in a glass fiber,” Opt. Lett. 5, 416–417 (1980).
[CrossRef] [PubMed]

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

1973 (2)

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

S. V. Manakov, “On the theory of two-dimensional self-focusing of electromagnetic waves,” Zh. Eksp. Teor. Fiz. 65, 505–516 (1973) [Sov. Phys. JETP 38, 248–253 (1974)].

Ablowitz, M. J.

See, e.g., T. R. Taha, M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation,” J. Comp. Phys. 55, 203–230 (1984).
[CrossRef]

Ashkin, A.

Baer, T.

Beaud, P.

B. Zysset, P. Beaud, W. Hodel, H. P. Weber, “80-fs soliton-like pulses from an optical nonlinear fiber resonator,” in Ultrafast Phenomena V, G. R. Fleming, A. E. Siegman, eds. (Springer-Verlag, Berlin, 1986), pp. 54–57.
[CrossRef]

Blow, K. J.

Botineau, J.

Chen, H. H.

P. K. A. Wai, C. R. Menyuk, H. H. Chen, Y. C. Lee, “Soliton propagation at the zero dispersion wavelength of single mode optical fibers,” Opt. Lett. 12, 628–630 (1987).
[CrossRef] [PubMed]

C. R. Menyuk, P. K. A. Wai, H. H. Chen, Y. C. Lee, “Hamiltonian deformations of integrable, nonlinear field equations (with applications to optical fibers),” in Published Proceedings of the 4th Conference on Applied Mathematics and Computing, F. Dressel, ed., (U.S. Government Printing Office, Washington, D.C., 1987), pp. 373–386.

Crosignani, B.

Di Porto, P.

Doran, N. J.

Feshbach, H.

See, e.g., P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), pp. 437–440.

Gomes, A. S. L.

A. S. Gouveia-Neto, A. S. L. Gomes, J. R. Taylor, “Higher-order soliton pulse compression and splitting at 1.32 μ m in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-23, 1193–1198 (1987).
[CrossRef]

Gordon, J. P.

L. F. Mollenauer, J. P. Gordon, M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. QE-22, 157–173 (1986).
[CrossRef]

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, W. J. Tomlinson, “Extreme picosecond pulse narrowing by means of soliton effect in single-mode fibers,” Opt. Lett. 8, 289–291 (1983).
[CrossRef] [PubMed]

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

Gouveia-Neto, A. S.

A. S. Gouveia-Neto, A. S. L. Gomes, J. R. Taylor, “Higher-order soliton pulse compression and splitting at 1.32 μ m in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-23, 1193–1198 (1987).
[CrossRef]

Hasegawa, A.

A. Hasegawa, Y. Kodama, “Signal transmission by optical solitons in monomode fibers,” Proc. IEEE 69, 1145–1150 (1981).
[CrossRef]

A. Hasegawa, “Self-confinement of multimode optical pulse in a glass fiber,” Opt. Lett. 5, 416–417 (1980).
[CrossRef] [PubMed]

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

Hodel, W.

B. Zysset, P. Beaud, W. Hodel, H. P. Weber, “80-fs soliton-like pulses from an optical nonlinear fiber resonator,” in Ultrafast Phenomena V, G. R. Fleming, A. E. Siegman, eds. (Springer-Verlag, Berlin, 1986), pp. 54–57.
[CrossRef]

Islam, M. N.

L. F. Mollenauer, J. P. Gordon, M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. QE-22, 157–173 (1986).
[CrossRef]

M. N. Islam, L. F. Mollenauer, R. H. Stolen, “Fiber Raman amplification soliton laser (FRASL),” in Ultrafast Phenomena V, G. R. Fleming, A. E. Siegman, eds. (Springer-Verlag, Berlin, 1986), pp. 46–50.
[CrossRef]

Kafka, J. D.

Kaminow, I. P.

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

Kodama, Y.

A. Hasegawa, Y. Kodama, “Signal transmission by optical solitons in monomode fibers,” Proc. IEEE 69, 1145–1150 (1981).
[CrossRef]

Lee, Y. C.

P. K. A. Wai, C. R. Menyuk, H. H. Chen, Y. C. Lee, “Soliton propagation at the zero dispersion wavelength of single mode optical fibers,” Opt. Lett. 12, 628–630 (1987).
[CrossRef] [PubMed]

C. R. Menyuk, P. K. A. Wai, H. H. Chen, Y. C. Lee, “Hamiltonian deformations of integrable, nonlinear field equations (with applications to optical fibers),” in Published Proceedings of the 4th Conference on Applied Mathematics and Computing, F. Dressel, ed., (U.S. Government Printing Office, Washington, D.C., 1987), pp. 373–386.

Manakov, S. V.

S. V. Manakov, “On the theory of two-dimensional self-focusing of electromagnetic waves,” Zh. Eksp. Teor. Fiz. 65, 505–516 (1973) [Sov. Phys. JETP 38, 248–253 (1974)].

Menyuk, C. R.

P. K. A. Wai, C. R. Menyuk, H. H. Chen, Y. C. Lee, “Soliton propagation at the zero dispersion wavelength of single mode optical fibers,” Opt. Lett. 12, 628–630 (1987).
[CrossRef] [PubMed]

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. QE-23, 174–176 (1987).
[CrossRef]

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

C. R. Menyuk, “Origin of solitons in the ‘real’ world,” Phys. Rev. A 33, 4367–4374 (1986).
[CrossRef] [PubMed]

C. R. Menyuk, P. K. A. Wai, H. H. Chen, Y. C. Lee, “Hamiltonian deformations of integrable, nonlinear field equations (with applications to optical fibers),” in Published Proceedings of the 4th Conference on Applied Mathematics and Computing, F. Dressel, ed., (U.S. Government Printing Office, Washington, D.C., 1987), pp. 373–386.

Mollenauer, L. F.

L. F. Mollenauer, J. P. Gordon, M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. QE-22, 157–173 (1986).
[CrossRef]

L. F. Mollenauer, R. H. Stolen, “The soliton laser,” Opt. Lett. 9, 13–15 (1984).
[CrossRef] [PubMed]

R. H. Stolen, L. F. Mollenauer, W. J. Tomlinson, “Observation of pulse restoration at the soliton period in optical fibers,” Opt. Lett. 8, 186–188 (1983).
[CrossRef] [PubMed]

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, W. J. Tomlinson, “Extreme picosecond pulse narrowing by means of soliton effect in single-mode fibers,” Opt. Lett. 8, 289–291 (1983).
[CrossRef] [PubMed]

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

M. N. Islam, L. F. Mollenauer, R. H. Stolen, “Fiber Raman amplification soliton laser (FRASL),” in Ultrafast Phenomena V, G. R. Fleming, A. E. Siegman, eds. (Springer-Verlag, Berlin, 1986), pp. 46–50.
[CrossRef]

Morse, P. M.

See, e.g., P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), pp. 437–440.

Stolen, R.

R. Stolen, AT&T Bell Laboratories, Holmdel, New Jersey 07733 (personal communication).

Stolen, R. H.

Taha, T. R.

See, e.g., T. R. Taha, M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation,” J. Comp. Phys. 55, 203–230 (1984).
[CrossRef]

Tappert, F.

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

Taylor, J. R.

A. S. Gouveia-Neto, A. S. L. Gomes, J. R. Taylor, “Higher-order soliton pulse compression and splitting at 1.32 μ m in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-23, 1193–1198 (1987).
[CrossRef]

Tomlinson, W. J.

Wai, P. K. A.

P. K. A. Wai, C. R. Menyuk, H. H. Chen, Y. C. Lee, “Soliton propagation at the zero dispersion wavelength of single mode optical fibers,” Opt. Lett. 12, 628–630 (1987).
[CrossRef] [PubMed]

C. R. Menyuk, P. K. A. Wai, H. H. Chen, Y. C. Lee, “Hamiltonian deformations of integrable, nonlinear field equations (with applications to optical fibers),” in Published Proceedings of the 4th Conference on Applied Mathematics and Computing, F. Dressel, ed., (U.S. Government Printing Office, Washington, D.C., 1987), pp. 373–386.

Weber, H. P.

B. Zysset, P. Beaud, W. Hodel, H. P. Weber, “80-fs soliton-like pulses from an optical nonlinear fiber resonator,” in Ultrafast Phenomena V, G. R. Fleming, A. E. Siegman, eds. (Springer-Verlag, Berlin, 1986), pp. 54–57.
[CrossRef]

Wood, D.

Zysset, B.

B. Zysset, P. Beaud, W. Hodel, H. P. Weber, “80-fs soliton-like pulses from an optical nonlinear fiber resonator,” in Ultrafast Phenomena V, G. R. Fleming, A. E. Siegman, eds. (Springer-Verlag, Berlin, 1986), pp. 54–57.
[CrossRef]

Appl. Phys. Lett. (1)

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

IEEE J. Quantum Electron. (4)

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. QE-23, 174–176 (1987).
[CrossRef]

L. F. Mollenauer, J. P. Gordon, M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. QE-22, 157–173 (1986).
[CrossRef]

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

A. S. Gouveia-Neto, A. S. L. Gomes, J. R. Taylor, “Higher-order soliton pulse compression and splitting at 1.32 μ m in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-23, 1193–1198 (1987).
[CrossRef]

J. Comp. Phys. (1)

See, e.g., T. R. Taha, M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation,” J. Comp. Phys. 55, 203–230 (1984).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Lett. (10)

R. H. Stolen, J. Botineau, A. Ashkin, “Intensity discrimination of optical pulses with birefringent fibers,” Opt. Lett. 7, 512–514 (1982).
[CrossRef] [PubMed]

R. H. Stolen, L. F. Mollenauer, W. J. Tomlinson, “Observation of pulse restoration at the soliton period in optical fibers,” Opt. Lett. 8, 186–188 (1983).
[CrossRef] [PubMed]

K. J. Blow, N. J. Doran, D. Wood, “Polarization instabilities for solitons in birefringent fibers,” Opt. Lett. 12, 202–204 (1987).
[CrossRef] [PubMed]

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, W. J. Tomlinson, “Extreme picosecond pulse narrowing by means of soliton effect in single-mode fibers,” Opt. Lett. 8, 289–291 (1983).
[CrossRef] [PubMed]

J. D. Kafka, T. Baer, “Fiber soliton Raman laser pumped by a Nd:YAG laser,” Opt. Lett. 12, 181–183 (1987).
[CrossRef] [PubMed]

P. K. A. Wai, C. R. Menyuk, H. H. Chen, Y. C. Lee, “Soliton propagation at the zero dispersion wavelength of single mode optical fibers,” Opt. Lett. 12, 628–630 (1987).
[CrossRef] [PubMed]

L. F. Mollenauer, R. H. Stolen, “The soliton laser,” Opt. Lett. 9, 13–15 (1984).
[CrossRef] [PubMed]

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

A. Hasegawa, “Self-confinement of multimode optical pulse in a glass fiber,” Opt. Lett. 5, 416–417 (1980).
[CrossRef] [PubMed]

B. Crosignani, P. Di Porto, “Soliton propagation in multimode fibers,” Opt. Lett. 6, 329–330 (1981).
[CrossRef] [PubMed]

Phys. Rev. A (1)

C. R. Menyuk, “Origin of solitons in the ‘real’ world,” Phys. Rev. A 33, 4367–4374 (1986).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

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

Proc. IEEE (1)

A. Hasegawa, Y. Kodama, “Signal transmission by optical solitons in monomode fibers,” Proc. IEEE 69, 1145–1150 (1981).
[CrossRef]

Zh. Eksp. Teor. Fiz. (1)

S. V. Manakov, “On the theory of two-dimensional self-focusing of electromagnetic waves,” Zh. Eksp. Teor. Fiz. 65, 505–516 (1973) [Sov. Phys. JETP 38, 248–253 (1974)].

Other (5)

R. Stolen, AT&T Bell Laboratories, Holmdel, New Jersey 07733 (personal communication).

M. N. Islam, L. F. Mollenauer, R. H. Stolen, “Fiber Raman amplification soliton laser (FRASL),” in Ultrafast Phenomena V, G. R. Fleming, A. E. Siegman, eds. (Springer-Verlag, Berlin, 1986), pp. 46–50.
[CrossRef]

B. Zysset, P. Beaud, W. Hodel, H. P. Weber, “80-fs soliton-like pulses from an optical nonlinear fiber resonator,” in Ultrafast Phenomena V, G. R. Fleming, A. E. Siegman, eds. (Springer-Verlag, Berlin, 1986), pp. 54–57.
[CrossRef]

C. R. Menyuk, P. K. A. Wai, H. H. Chen, Y. C. Lee, “Hamiltonian deformations of integrable, nonlinear field equations (with applications to optical fibers),” in Published Proceedings of the 4th Conference on Applied Mathematics and Computing, F. Dressel, ed., (U.S. Government Printing Office, Washington, D.C., 1987), pp. 373–386.

See, e.g., P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953), pp. 437–440.

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Figures (18)

Fig. 1
Fig. 1

Variation of (a) the maximum location s max u and (b) the frequency centroid ω cent u with distance along the fiber measured in soliton periods (δ = 0.15, γ = 0.0).

Fig. 2
Fig. 2

Variation of (a) the maximum location s max u and (b) the frequency centroid ω cent u with distance along the fiber (δ = 0.5, γ = 0.0).

Fig. 3
Fig. 3

Details of the pulse evolution (A = 0.1, δ = 0.5, γ = 0.0). Solid lines indicate u and ũ; dashed lines indicate v and v ˜. (A) u(s) and v(s), ξ = 0; (B) ũ(ω) and v ˜(ω), ξ = 0; (C) u(s) and v(s), ξ = 5π (10 soliton periods); (D) ũ(ω) and v ˜(ω), ξ = 5π.

Fig. 4
Fig. 4

Details of the pulse evolution (A = 1.0, δ = 0.5, γ = 0.0). Solid lines indicate u and ũ; dashed lines indicate v and v ˜. (A) u(s) and v(s), ξ = 0; (B) ũ(ω) and v ˜(ω), ξ = 0; (C) u(s) and v(s), ξ = 5π (10 soliton periods); (D) ũ(ω) and v ˜(ω), ξ = 5π.

Fig. 5
Fig. 5

Variation of (a) the maximum location s max u and (b) the frequency centroid ω cent u with distance along the fiber (δ = 1.0, γ = 0.0).

Fig. 6
Fig. 6

Details of the pulse structure at 10 soliton periods, ξ = 5π (δ = 1.0). Only the u polarization is shown. (a) A = 2.0, (b) A = 2.05.

Fig. 7
Fig. 7

Pulse evolution (A = 1.0, δ = 0.5). Solid lines indicate γ = 0.0; dashed lines indicate γ = 0.0105. (A) ξ = 0, (B) ξ = 10π (20 soliton periods), (C) ξ = 25π.

Fig. 8
Fig. 8

Variation of the maximum location s max u(A = 1.0, δ = 0.5).

Fig. 9
Fig. 9

Pulse evolution (A = 0.8, δ = 0.15, γ = 0.0, α = 30°). Solid lines indicate the u polarization; dashed lines indicate the v polarization. (A) ξ = 0, (B) ξ = 5π (10 soliton periods), (C) ξ = 10π.

Fig. 10
Fig. 10

Parameter variation with distance along the fiber (A = 0.8, δ = 0.15, γ = 0.0, α = 30°). (a) The maximum locations s max u and s max v, (b) the pulse widths wu and wv, (c) the frequency centroids ω cent u and ω cent v. Solid lines indicate the u polarization; dotted lines indicate the v polarization.

Fig. 11
Fig. 11

Pulse evolution (A = 1.1, δ = 0.5, γ = 0.0, α = 30°). Solid lines indicate the u polarization; dashed lines indicate the v polarization. (A) ξ = 0, (B) ξ = 5π (10 soliton periods), (C) ξ = 10π.

Fig. 12
Fig. 12

Parameter variation with distance along the fiber (A = 1.1, δ = 0.5, γ = 0.0, α = 30°). (a) The maximum locations s max u and s max v, (b) the pulse widths wu and wv, and (c) the frequency centroids ω cent u and ω cent v, Solid lines indicate the u polarization; dotted lines indicate the v polarization.

Fig. 13
Fig. 13

Pulse evolution (A = 0.8, δ = 0.15, γ = 0.0, α = 15°). Solid lines indicate the u polarization; dashed lines indicate the v polarization. (A) ξ = 0, (B) ξ = 10π (20 soliton periods).

Fig. 14
Fig. 14

Pulse evolution (A = 1.0, δ = 0.5, γ = 0.0, α = 15°). Solid lines indicate the u polarization; dashed lines indicate the v polarization. (A) ξ = 0, (B) ξ = 10π (20 soliton periods).

Fig. 15
Fig. 15

Details of the pulse structure at 20 soliton periods, ξ = 10π (A = 0.9, δ = 0.5). Solid lines indicate the u polarization; dashed lines indicate the v polarization. (a) α = 30°, (b) α = 35°, (c) α = 40°.

Fig. 16
Fig. 16

Details of the pulse structure at 10 soliton periods, ξ = 5π (A = 2.1, δ = 1.0). (a) u polarization, α = 44.5°; (b) v polarization, α = 44.5°; (c) u polarization, α = 44.8°; (d) v polarization, α = 44.8°; (e) u polarization, α = 45°; (f) v polarization, α = 45°.

Fig. 17
Fig. 17

Details of the pulse structure at 10 soliton periods, ξ = 5π (A = 2.0, δ = 1.0). Solid lines indicate the u polarization; dashed lines indicate the v polarization. (a) α = 25°; (b) α = 30°.

Fig. 18
Fig. 18

Details of the pulse structure at 10 soliton periods, ξ = 5π (δ = 1.0, α = 40°). (a) u polarization, A = 2.2; (b) v polarization, A = 2.2; (c) u polarization, A = 2.3; (d) v polarization, A = 2.3.

Tables (1)

Tables Icon

Table 1 Threshold Values of A at Which the Kerr Nonlinearity Is Sufficient to Compensate for Linear Birefringence at α = 45°

Equations (31)

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i ( U z + k U t ) - 1 2 k 2 U t 2 + χ 2 ( U 2 + 2 3 V 2 ) U + χ 6 V 2 U * exp [ - 2 i ( k 0 - l 0 ) z ] = - i Γ U ,
i ( V z + l V t ) - 1 2 l 2 V t 2 + χ 2 ( 2 3 U 2 + V 2 ) V + χ 6 U 2 V * exp [ 2 i ( k 0 - l 0 ) z ] = - i Γ V ,
U ¯ = U cos θ + V sin θ , V ¯ = V cos θ - U sin θ .
k = l = λ 0 2 π c 2 D ( λ ) ,
ξ = π z 2 z 0 ,             z 0 = π 2 c 2 t 0 2 D ( λ ) λ 0 ,             t 0 = 0.568 τ , s = 1 t 0 ( t - z v ¯ g ) ,             v ¯ g = 2 k + l , u = ( χ 2 ) 1 / 2 U ,             v = ( χ 2 ) 1 / 2 V , δ = k - l 2 k t 0 = π c Δ n D ( λ ) λ 0 t 0 ,             R = 8 π c λ 0 t 0 ,             γ = 2 z 0 Γ π ,
i ( u ξ + δ u s ) + 1 2 2 u s 2 + ( u 2 + 2 3 v 2 ) u + 1 3 v 2 u * exp ( - i R δ ξ ) = - i γ u ,
i ( v ξ - δ v s ) + 1 2 2 v s 2 + ( 2 3 u 2 + v 2 ) v + 1 3 u 2 v * exp ( i R δ ξ ) = - i γ v .
i ( u ξ + δ u s ) + 1 2 2 u s 2 + ( u 2 + 2 3 v 2 ) u = - i γ u ,
i ( v ξ - δ v s ) + 1 2 2 v s 2 + ( 2 3 u 2 + v 2 ) v = - i γ v .
u ( ξ = 0 ) = A cos α sech s , v ( ξ = 0 ) = A sin α sech s .
u ( ξ , s ) = exp [ i 2 ( 1 + δ 2 ) ξ - i δ s ] sech s , v ( ξ , s ) = 0.
u ( ξ , s ) = ( 3 5 ) 1 / 2 exp [ i 2 ( 1 + δ 2 ) ξ - i δ s ] sech s , v ( ξ , s ) = ( 3 5 ) 1 / 2 exp [ i 2 ( 1 + δ 2 ) ξ + i δ s ] sech s .
i ( u ξ + δ u s ) + 1 2 2 u s 2 = 0 , i ( v ξ - δ v s ) + 1 2 2 v s 2 = 0.
u ( ξ , s ) = ( π 2 ξ ) 1 / 2 ( 1 - i ) A 2 exp [ i ( s - δ ξ ) 2 ] sech [ π 2 ξ ( s - δ ξ ) ] , v ( ξ , s ) = ( π 2 ξ ) 1 / 2 ( 1 - i ) A 2 exp [ i ( s + δ ξ ) 2 ] sech [ π 2 ξ ( s + δ ξ ) ] .
u ˜ ( ω , ξ ) = 1 2 π - d s e i ω s u ( s , ξ ) , v ˜ ( ω , ξ ) = 1 2 π - d s e i ω s v ( s , ξ ) .
u ˜ ( ω , ξ + Δ ξ ) = u ˜ ( ω , ξ ) exp [ i ω ( δ - ω / 2 ) Δ ξ - γ Δ ξ ] , v ˜ ( ω , ξ + Δ ξ ) = v ˜ ( ω , ξ ) exp [ - i ω ( δ + ω / 2 ) Δ ξ - γ Δ ξ ] .
u ( s , ξ + Δ ξ ) = u ( s , ξ ) exp [ i ( u 2 + v 2 ) Δ ξ ] , v ( s , ξ + Δ ξ ) = v ( s , ξ ) exp [ i ( u 2 + v 2 ) Δ ξ ] .
N u = exp [ i ( u 2 + v 2 ) Δ ξ ] , N v = exp [ i ( u 2 + v 2 ) Δ ξ ] ,
N u = [ 1 + i ( u 2 + v 2 ) Δ ξ / 4 1 - i ( u 2 + v 2 ) Δ ξ / 4 ] 2 , N v = [ 1 + i ( u 2 + v 2 ) Δ ξ / 4 1 - i ( u 2 + v 2 ) Δ ξ / 4 ] 2 .
u ( new ) = Γ u ( old ) ,             v ( new ) = Γ v ( old )
Γ = 1 - sech [ β ( s - s edge ) ]
v ( s , ξ ) = u ( - s , ξ )
v ˜ ( ω , ξ ) = u ˜ ( - ω , ξ ) .
s max v ( ξ ) = - s max u ( ξ ) .
ω cent u ( ξ ) = - d ω ω u ˜ ( ω , ξ ) 2 - d ω u ˜ ( ω , ξ ) 2 , ω cent v ( ξ ) = - d ω ω v ˜ ( ω , ξ ) 2 - d ω v ˜ ( ω , ξ ) 2 .
ω cent u ( ξ ) = - ω cent v ( ξ ) .
s avg u = - d s s u ( s , ξ ) 2 - d s u ( s , ξ ) 2 ,             ( s 2 ) avg u = - d s s 2 u ( s , ξ ) 2 - d s u ( s , ξ ) 2 , s avg u = - d s s v ( s , ξ ) 2 - d s v ( s , ξ ) 2 ,             ( s 2 ) avg v = - d s s 2 v ( s , ξ ) 2 - d s v ( s , ξ ) 2 ,
w u = [ ( s 2 ) avg u - ( s avg u ) 2 ] 1 / 2 , w v = [ ( s 2 ) avg v - ( s avg v ) 2 ] 1 / 2 .
ω cent u ( ξ ) = - ( cot 2 α ) ω cent u ( ξ ) .
ξ - d ω u ˜ ( ω , ξ ) 2 = ξ - d ω v ˜ ( ω , ξ ) 2 = 0
ξ - d ω ω ( u ˜ ( ω , ξ ) 2 + v ˜ ( ω , ξ ) 2 ) = 0 ,

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