Abstract

The problem of solitonlike pulse propagation in optical fiber devices with two-component fields is studied. Two examples, viz., propagation of pulses in birefringent optical fibers and in nonlinear couplers, are considered. It is shown that radiation processes play an essential role in the soliton dynamics in two-component field fiber devices. Radiation influences the transformation of the main pulses in different ways in the two cases considered. The influence of radiation is stronger when the soliton is closer to the point of bifurcation or the separatrix and weaker when it is far from it.

© 1995 Optical Society of America

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References

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  1. M. Romagnoli, S. Trillo, and S. Wabnitz, "Soliton switching in nonlinear couplers" Opt. Quantum Electron. 24, S1237–S1267 (1992).
    [CrossRef]
  2. K. J. Blow, N. J. Doran, and D. Wood, "Polarization instabilities for solitons in birefringent fibers," Opt. Lett. 12, 202–204 (1987).
    [CrossRef] [PubMed]
  3. C. R. Menyuk, "Pulse propagation in an elliptically birefringent Kerr medium," IEEE J. Quantum Electron. 25, 2674–2682 (1989).
    [CrossRef]
  4. 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]
  5. S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, "Soliton switching in fiber nonlinear directional couplers," Opt. Lett. 13, 672–674 (1988).
    [CrossRef] [PubMed]
  6. M. V. Tratnik and J. E. Sipe, "Nonlinear polarization dynamics. I. The single-pulse equations," Phys. Rev. A 35, 2965–2975 (1987).
    [CrossRef] [PubMed]
  7. F. Kh. Abdullaev, R. M. Abrarov, and S. A. Darmanyan, "Dynamics of solitons in coupled optical fibers," Opt. Lett. 14, 131–133 (1989).
    [CrossRef] [PubMed]
  8. J. M. Soto-Crespo and E. M. Wright, "All-optical switching of solitons in two- and three-core nonlinear fiber couplers," J. Appl. Phys. 70, 7240–7243 (1991).
    [CrossRef]
  9. P. A. Bélanger and C. Paré, "Soliton switching and energy coupling in two-mode fibers: analytical results," Phys. Rev. A 41, 5254–5256 (1990).
    [CrossRef] [PubMed]
  10. P. L. Chu, G. D. Peng, and B. Malomed, "Analytical solution to soliton switching in nonlinear twin-core fibers," Opt. Lett. 18, 328–330 (1993).
    [CrossRef] [PubMed]
  11. E. M. Wright, G. I. Stegeman, and S. Wabnitz, "Solitarywave decay and symmetry-breaking instabilities in twomode fibers," Phys. Rev. A 40, 4455–4466 (1989).
    [CrossRef] [PubMed]
  12. C. R. Menyuk, "Soliton robustness in optical fibers," J. Opt. Soc. Am. B 10, 1585–1591 (1993).
    [CrossRef]
  13. N. N. Akhmediev and J. M. Soto-Crespo, "Dynamics of solitonlike pulse propagation in birefringent optical fibers," Phys. Rev. E 49, 5742–5754 (1994).
    [CrossRef]
  14. N. N. Akhmediev and J. M. Soto-Crespo, "Propagation dynamics of ultrashort pulses in nonlinear fiber couplers," Phys. Rev. E 49, 4519–4529 (1994).
    [CrossRef]
  15. A. V. Buryak and N. N. Akhmediev, "Influence of radiation on soliton dynamics in nonlinear fibre couplers," Opt. Commun. 110, 287–292 (1994).
    [CrossRef]
  16. G. Gregory and S. Wabnitz, "New exact solutions and bifurcations in the spatial distribution of polarization in thirdorder nonlinear optical interactions," Phys. Rev. Lett. 56, 600–603 (1986).
    [CrossRef]
  17. B. Daino, G. Gergory, and S. Wabnitz, "Stability analysis of nonlinear coherent coupling," J. Appl. phys. 58, 4512–4514 (1985).
    [CrossRef]
  18. N. N. Akhmediev and A. Ankiewicz, "Novel soliton states and bifurcation phenomena in nonlinear fiber couplers," Phys. Rev. Lett. 70, 2395–2398 (1993).
    [CrossRef] [PubMed]
  19. N. N. Akhmediev, A. V. Buryak, and J. M. Soto-Crespo, "Elliptically polarized solitons in birefringent optical fibers," Opt. Commun. 112, 278–282 (1994).
    [CrossRef]
  20. L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer-Verlag, Berlin, 1987).
    [CrossRef]
  21. H. G. Winful, "Self-induced polarization changes in birefringent optical fibers," Appl. Phys. Lett. 47, 213–215 (1985).
    [CrossRef]
  22. J. M. Soto-Crespo and N. N. Akhmediev, "Stability of soliton states in a nonlinear fiber coupler," Phys. Rev. E 48, 4710–4715 (1993).
    [CrossRef]
  23. K. J. Blow, N. J. Doran, and S. J. D. Phoenix, "The soliton phase," Opt. Commun. 88, 137–140 (1992).
    [CrossRef]
  24. S. V. Manakov, "On the theory of two-dimensional stationary self-focusing of electromagnetic waves," Zh. Eksp. Teor. Fiz. 65, 505–516 (1973) [Sov. Phys. JETP 38, 248–253 (1974)].

1994 (4)

N. N. Akhmediev and J. M. Soto-Crespo, "Dynamics of solitonlike pulse propagation in birefringent optical fibers," Phys. Rev. E 49, 5742–5754 (1994).
[CrossRef]

N. N. Akhmediev and J. M. Soto-Crespo, "Propagation dynamics of ultrashort pulses in nonlinear fiber couplers," Phys. Rev. E 49, 4519–4529 (1994).
[CrossRef]

A. V. Buryak and N. N. Akhmediev, "Influence of radiation on soliton dynamics in nonlinear fibre couplers," Opt. Commun. 110, 287–292 (1994).
[CrossRef]

N. N. Akhmediev, A. V. Buryak, and J. M. Soto-Crespo, "Elliptically polarized solitons in birefringent optical fibers," Opt. Commun. 112, 278–282 (1994).
[CrossRef]

1993 (4)

J. M. Soto-Crespo and N. N. Akhmediev, "Stability of soliton states in a nonlinear fiber coupler," Phys. Rev. E 48, 4710–4715 (1993).
[CrossRef]

P. L. Chu, G. D. Peng, and B. Malomed, "Analytical solution to soliton switching in nonlinear twin-core fibers," Opt. Lett. 18, 328–330 (1993).
[CrossRef] [PubMed]

C. R. Menyuk, "Soliton robustness in optical fibers," J. Opt. Soc. Am. B 10, 1585–1591 (1993).
[CrossRef]

N. N. Akhmediev and A. Ankiewicz, "Novel soliton states and bifurcation phenomena in nonlinear fiber couplers," Phys. Rev. Lett. 70, 2395–2398 (1993).
[CrossRef] [PubMed]

1992 (3)

M. Romagnoli, S. Trillo, and S. Wabnitz, "Soliton switching in nonlinear couplers" Opt. Quantum Electron. 24, S1237–S1267 (1992).
[CrossRef]

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]

K. J. Blow, N. J. Doran, and S. J. D. Phoenix, "The soliton phase," Opt. Commun. 88, 137–140 (1992).
[CrossRef]

1991 (1)

J. M. Soto-Crespo and E. M. Wright, "All-optical switching of solitons in two- and three-core nonlinear fiber couplers," J. Appl. Phys. 70, 7240–7243 (1991).
[CrossRef]

1990 (1)

P. A. Bélanger and C. Paré, "Soliton switching and energy coupling in two-mode fibers: analytical results," Phys. Rev. A 41, 5254–5256 (1990).
[CrossRef] [PubMed]

1989 (3)

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

F. Kh. Abdullaev, R. M. Abrarov, and S. A. Darmanyan, "Dynamics of solitons in coupled optical fibers," Opt. Lett. 14, 131–133 (1989).
[CrossRef] [PubMed]

E. M. Wright, G. I. Stegeman, and S. Wabnitz, "Solitarywave decay and symmetry-breaking instabilities in twomode fibers," Phys. Rev. A 40, 4455–4466 (1989).
[CrossRef] [PubMed]

1988 (1)

1987 (2)

M. V. Tratnik and J. E. Sipe, "Nonlinear polarization dynamics. I. The single-pulse equations," Phys. Rev. A 35, 2965–2975 (1987).
[CrossRef] [PubMed]

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

1986 (1)

G. Gregory and S. Wabnitz, "New exact solutions and bifurcations in the spatial distribution of polarization in thirdorder nonlinear optical interactions," Phys. Rev. Lett. 56, 600–603 (1986).
[CrossRef]

1985 (2)

B. Daino, G. Gergory, and S. Wabnitz, "Stability analysis of nonlinear coherent coupling," J. Appl. phys. 58, 4512–4514 (1985).
[CrossRef]

H. G. Winful, "Self-induced polarization changes in birefringent optical fibers," Appl. Phys. Lett. 47, 213–215 (1985).
[CrossRef]

1973 (1)

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

Abdullaev, F. Kh.

Abrarov, R. M.

Akhmediev, N. N.

N. N. Akhmediev and J. M. Soto-Crespo, "Dynamics of solitonlike pulse propagation in birefringent optical fibers," Phys. Rev. E 49, 5742–5754 (1994).
[CrossRef]

N. N. Akhmediev and J. M. Soto-Crespo, "Propagation dynamics of ultrashort pulses in nonlinear fiber couplers," Phys. Rev. E 49, 4519–4529 (1994).
[CrossRef]

A. V. Buryak and N. N. Akhmediev, "Influence of radiation on soliton dynamics in nonlinear fibre couplers," Opt. Commun. 110, 287–292 (1994).
[CrossRef]

N. N. Akhmediev, A. V. Buryak, and J. M. Soto-Crespo, "Elliptically polarized solitons in birefringent optical fibers," Opt. Commun. 112, 278–282 (1994).
[CrossRef]

N. N. Akhmediev and A. Ankiewicz, "Novel soliton states and bifurcation phenomena in nonlinear fiber couplers," Phys. Rev. Lett. 70, 2395–2398 (1993).
[CrossRef] [PubMed]

J. M. Soto-Crespo and N. N. Akhmediev, "Stability of soliton states in a nonlinear fiber coupler," Phys. Rev. E 48, 4710–4715 (1993).
[CrossRef]

Ankiewicz, A.

N. N. Akhmediev and A. Ankiewicz, "Novel soliton states and bifurcation phenomena in nonlinear fiber couplers," Phys. Rev. Lett. 70, 2395–2398 (1993).
[CrossRef] [PubMed]

Bélanger, P. A.

P. A. Bélanger and C. Paré, "Soliton switching and energy coupling in two-mode fibers: analytical results," Phys. Rev. A 41, 5254–5256 (1990).
[CrossRef] [PubMed]

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.

A. V. Buryak and N. N. Akhmediev, "Influence of radiation on soliton dynamics in nonlinear fibre couplers," Opt. Commun. 110, 287–292 (1994).
[CrossRef]

N. N. Akhmediev, A. V. Buryak, and J. M. Soto-Crespo, "Elliptically polarized solitons in birefringent optical fibers," Opt. Commun. 112, 278–282 (1994).
[CrossRef]

Chu, P. L.

Daino, B.

B. Daino, G. Gergory, and S. Wabnitz, "Stability analysis of nonlinear coherent coupling," J. Appl. phys. 58, 4512–4514 (1985).
[CrossRef]

Darmanyan, S. A.

Doran, N. J.

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]

Faddeev, L. D.

L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer-Verlag, Berlin, 1987).
[CrossRef]

Gergory, G.

B. Daino, G. Gergory, and S. Wabnitz, "Stability analysis of nonlinear coherent coupling," J. Appl. phys. 58, 4512–4514 (1985).
[CrossRef]

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]

Gregory, G.

G. Gregory and S. Wabnitz, "New exact solutions and bifurcations in the spatial distribution of polarization in thirdorder nonlinear optical interactions," Phys. Rev. Lett. 56, 600–603 (1986).
[CrossRef]

Malomed, B.

Manakov, S. V.

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

Menyuk, C. R.

C. R. Menyuk, "Soliton robustness in optical fibers," J. Opt. Soc. Am. B 10, 1585–1591 (1993).
[CrossRef]

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

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]

Paré, C.

P. A. Bélanger and C. Paré, "Soliton switching and energy coupling in two-mode fibers: analytical results," Phys. Rev. A 41, 5254–5256 (1990).
[CrossRef] [PubMed]

Peng, G. D.

Phoenix, S. J. D.

K. J. Blow, N. J. Doran, and S. J. D. Phoenix, "The soliton phase," Opt. Commun. 88, 137–140 (1992).
[CrossRef]

Romagnoli, M.

M. Romagnoli, S. Trillo, and S. Wabnitz, "Soliton switching in nonlinear couplers" Opt. Quantum Electron. 24, S1237–S1267 (1992).
[CrossRef]

Sipe, J. E.

M. V. Tratnik and J. E. Sipe, "Nonlinear polarization dynamics. I. The single-pulse equations," Phys. Rev. A 35, 2965–2975 (1987).
[CrossRef] [PubMed]

Soto-Crespo, J. M.

N. N. Akhmediev and J. M. Soto-Crespo, "Propagation dynamics of ultrashort pulses in nonlinear fiber couplers," Phys. Rev. E 49, 4519–4529 (1994).
[CrossRef]

N. N. Akhmediev and J. M. Soto-Crespo, "Dynamics of solitonlike pulse propagation in birefringent optical fibers," Phys. Rev. E 49, 5742–5754 (1994).
[CrossRef]

N. N. Akhmediev, A. V. Buryak, and J. M. Soto-Crespo, "Elliptically polarized solitons in birefringent optical fibers," Opt. Commun. 112, 278–282 (1994).
[CrossRef]

J. M. Soto-Crespo and N. N. Akhmediev, "Stability of soliton states in a nonlinear fiber coupler," Phys. Rev. E 48, 4710–4715 (1993).
[CrossRef]

J. M. Soto-Crespo and E. M. Wright, "All-optical switching of solitons in two- and three-core nonlinear fiber couplers," J. Appl. Phys. 70, 7240–7243 (1991).
[CrossRef]

Stegeman, G. I.

E. M. Wright, G. I. Stegeman, and S. Wabnitz, "Solitarywave decay and symmetry-breaking instabilities in twomode fibers," Phys. Rev. A 40, 4455–4466 (1989).
[CrossRef] [PubMed]

S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, "Soliton switching in fiber nonlinear directional couplers," Opt. Lett. 13, 672–674 (1988).
[CrossRef] [PubMed]

Takhtajan, L. A.

L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer-Verlag, Berlin, 1987).
[CrossRef]

Tratnik, M. V.

M. V. Tratnik and J. E. Sipe, "Nonlinear polarization dynamics. I. The single-pulse equations," Phys. Rev. A 35, 2965–2975 (1987).
[CrossRef] [PubMed]

Trillo, S.

M. Romagnoli, S. Trillo, and S. Wabnitz, "Soliton switching in nonlinear couplers" Opt. Quantum Electron. 24, S1237–S1267 (1992).
[CrossRef]

S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, "Soliton switching in fiber nonlinear directional couplers," Opt. Lett. 13, 672–674 (1988).
[CrossRef] [PubMed]

Wabnitz, S.

M. Romagnoli, S. Trillo, and S. Wabnitz, "Soliton switching in nonlinear couplers" Opt. Quantum Electron. 24, S1237–S1267 (1992).
[CrossRef]

E. M. Wright, G. I. Stegeman, and S. Wabnitz, "Solitarywave decay and symmetry-breaking instabilities in twomode fibers," Phys. Rev. A 40, 4455–4466 (1989).
[CrossRef] [PubMed]

S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, "Soliton switching in fiber nonlinear directional couplers," Opt. Lett. 13, 672–674 (1988).
[CrossRef] [PubMed]

G. Gregory and S. Wabnitz, "New exact solutions and bifurcations in the spatial distribution of polarization in thirdorder nonlinear optical interactions," Phys. Rev. Lett. 56, 600–603 (1986).
[CrossRef]

B. Daino, G. Gergory, and S. Wabnitz, "Stability analysis of nonlinear coherent coupling," J. Appl. phys. 58, 4512–4514 (1985).
[CrossRef]

Winful, H. G.

H. G. Winful, "Self-induced polarization changes in birefringent optical fibers," Appl. Phys. Lett. 47, 213–215 (1985).
[CrossRef]

Wood, D.

Wright, E. M.

J. M. Soto-Crespo and E. M. Wright, "All-optical switching of solitons in two- and three-core nonlinear fiber couplers," J. Appl. Phys. 70, 7240–7243 (1991).
[CrossRef]

E. M. Wright, G. I. Stegeman, and S. Wabnitz, "Solitarywave decay and symmetry-breaking instabilities in twomode fibers," Phys. Rev. A 40, 4455–4466 (1989).
[CrossRef] [PubMed]

S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, "Soliton switching in fiber nonlinear directional couplers," Opt. Lett. 13, 672–674 (1988).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

H. G. Winful, "Self-induced polarization changes in birefringent optical fibers," Appl. Phys. Lett. 47, 213–215 (1985).
[CrossRef]

IEEE J. Quantum Electron. (1)

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

J. Appl. Phys. (1)

J. M. Soto-Crespo and E. M. Wright, "All-optical switching of solitons in two- and three-core nonlinear fiber couplers," J. Appl. Phys. 70, 7240–7243 (1991).
[CrossRef]

B. Daino, G. Gergory, and S. Wabnitz, "Stability analysis of nonlinear coherent coupling," J. Appl. phys. 58, 4512–4514 (1985).
[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 (1)

Opt. Commun. (3)

A. V. Buryak and N. N. Akhmediev, "Influence of radiation on soliton dynamics in nonlinear fibre couplers," Opt. Commun. 110, 287–292 (1994).
[CrossRef]

N. N. Akhmediev, A. V. Buryak, and J. M. Soto-Crespo, "Elliptically polarized solitons in birefringent optical fibers," Opt. Commun. 112, 278–282 (1994).
[CrossRef]

K. J. Blow, N. J. Doran, and S. J. D. Phoenix, "The soliton phase," Opt. Commun. 88, 137–140 (1992).
[CrossRef]

Opt. Lett. (4)

Opt. Quantum Electron. (1)

M. Romagnoli, S. Trillo, and S. Wabnitz, "Soliton switching in nonlinear couplers" Opt. Quantum Electron. 24, S1237–S1267 (1992).
[CrossRef]

Phys. Rev. A (3)

P. A. Bélanger and C. Paré, "Soliton switching and energy coupling in two-mode fibers: analytical results," Phys. Rev. A 41, 5254–5256 (1990).
[CrossRef] [PubMed]

M. V. Tratnik and J. E. Sipe, "Nonlinear polarization dynamics. I. The single-pulse equations," Phys. Rev. A 35, 2965–2975 (1987).
[CrossRef] [PubMed]

E. M. Wright, G. I. Stegeman, and S. Wabnitz, "Solitarywave decay and symmetry-breaking instabilities in twomode fibers," Phys. Rev. A 40, 4455–4466 (1989).
[CrossRef] [PubMed]

Phys. Rev. E (3)

N. N. Akhmediev and J. M. Soto-Crespo, "Dynamics of solitonlike pulse propagation in birefringent optical fibers," Phys. Rev. E 49, 5742–5754 (1994).
[CrossRef]

N. N. Akhmediev and J. M. Soto-Crespo, "Propagation dynamics of ultrashort pulses in nonlinear fiber couplers," Phys. Rev. E 49, 4519–4529 (1994).
[CrossRef]

J. M. Soto-Crespo and N. N. Akhmediev, "Stability of soliton states in a nonlinear fiber coupler," Phys. Rev. E 48, 4710–4715 (1993).
[CrossRef]

Phys. Rev. Lett. (2)

G. Gregory and S. Wabnitz, "New exact solutions and bifurcations in the spatial distribution of polarization in thirdorder nonlinear optical interactions," Phys. Rev. Lett. 56, 600–603 (1986).
[CrossRef]

N. N. Akhmediev and A. Ankiewicz, "Novel soliton states and bifurcation phenomena in nonlinear fiber couplers," Phys. Rev. Lett. 70, 2395–2398 (1993).
[CrossRef] [PubMed]

Zh. Eksp. Teor. Fiz. (1)

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

Other (1)

L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer-Verlag, Berlin, 1987).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

(a) Energy-dispersion diagram for linearly polarized solitons in birefringent optical fibers. (b) Energy-dispersion diagram for stationary soliton states in nonlinear fiber couplers. (c) Hamiltonian H versus energy Q for linearly polarized solitons in birefringent optical fibers. The dotted curve corresponds to the initial conditions given by Eqs. (33) below. (d) H versus Q for stationary soliton states in nonlinear fiber couplers. The dotted curve represents the initial conditions given by Eqs. (34) below. The trajectories shown by the solid curves correspond to those in Figs. 4 and 5 below.

Fig. 2
Fig. 2

Trajectories on the Poincaré sphere for solutions in the birefringent fiber: (a) β > gS0, (b) β < gS0.

Fig. 3
Fig. 3

Trajectories on the Poincaré sphere for the nonlinear coupler: (a) K >|g|S0, (b) K < |g|S0.

Fig. 4
Fig. 4

Evolution of the integrated Stokes parameters for pulses in a birefringent fiber. The initial condition is given by Eqs. (33) with Q = (a) 2.4, (b) 4.0, (c) 6.0, (d) 7.2, (e) 8.0, and (f) 12.0.

Fig. 5
Fig. 5

Evolution of the integrated Stokes parameters for pulses in a nonlinear coupler. The initial condition is a soliton of an individual core as given by Eqs. (34) with Q = (a) 3.2, (b) 4.0, (c) 4.8, (d) 5.2, (e) 5.6 and (f) 7.2. All the simulations have K = 1.

Fig. 6
Fig. 6

(a) Evolution of the integrated Stokes parameters for a soliton pair in a nonlinear coupler with K = 0.59. The initial condition is the unstable A-type asymmetric soliton state with q = 1, plus a small perturbation. The perturbation produces oscillations around a symmetric or a stable asymmetric state. (b) Evolution of the component energies in each core: E P , G = | P , G | 2 d τ.

Fig. 7
Fig. 7

(a) Evolution of the integrated Stokes parameters for a soliton pair in a nonlinear coupler having K = 0.1. The initial condition is a symmetric soliton state with q = 1, plus a small perturbation. (b) Evolution of S 0 (total energy in the pulse). (c) Evolution of the component energies in each core: E P , G = | P , G | 2 d τ.

Fig. 8
Fig. 8

Evolution of the integrated Stokes parameters for a soliton pair in a nonlinear coupler with K = 1. The initial condition is the antisymmetric soliton state with q = 0.6, plus a small perturbation.

Equations (34)

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i U ξ + β U + ½ U τ τ + ( | U | 2 + A | V | 2 ) U + ( 1 A ) V 2 U * = 0 , i V ξ β V + ½ V τ τ + ( A | U | 2 + | V | 2 ) V + ( 1 A ) U 2 V * = 0 ,
Q = ( | U | 2 + | V | 2 ) d τ
H = [ ½ ( | U τ | 2 + | V τ | 2 ) β ( | U | 2 | V | 2 ) ½ ( | U | 2 + | V | 2 ) 2 ½ ( 1 A ) ( U V * U * V ) 2 ] d τ .
i U ξ = δ H δ U * , i V ξ = δ H δ V * .
s 0 = | U | 2 + | V | 2 , s 1 = | U | 2 | V | 2 , s 2 = U * V + U V * , s 3 = i ( U * V U V * ) .
d d ξ s 0 d τ = 0 , d d ξ s 1 d τ = 2 ( 1 A ) s 2 s 3 d τ , d d ξ s 2 d τ = 2 β s 3 d τ + 2 ( 1 A ) s 1 s 3 d τ , d d ξ s 3 d τ = 2 β s 2 d τ ,
U = U ( ξ , τ , q ) exp ( i β ξ ) , V = V ( ξ , τ , q ) exp ( i β ξ ) ,
i U ξ + ½ U τ τ + ( | U | 2 + A | V | 2 ) U + ( 1 A ) V 2 U * × exp ( i 4 β ξ ) = 0 , i V ξ + ½ V τ τ + ( A | U | 2 + | V | 2 ) V + ( 1 A ) U 2 V * × exp ( i 4 β ξ ) = 0 .
s 0 = ( | U | 2 + | V | 2 ) = s 0 , s 1 = ( | U | 2 | V | 2 ) = s 1 , s 2 = ( U * V + U V * ) = s 2 cos ( 2 β ξ ) s 3 sin ( 2 β ξ ) , s 3 = ( U * V U V * ) = s 2 sin ( 2 β ξ ) + s 3 cos ( 2 β ξ ) .
P = U + i V , G = U i V .
i P ξ + K G + ½ P τ τ + ( A 2 | P | 2 + 2 A 2 | G | 2 ) P = 0 , i G ξ + K P + ½ G τ τ + ( 2 A 2 | P | 2 + A 2 | G | 2 ) G = 0 ,
H = [ ½ ( | P τ | 2 + | G τ | 2 ) ½ ( | P | 4 + | G | 4 ) K ( P G * + P * G ) ] d τ ,
Q = ( | P | 2 + | G | 2 ) d τ .
2 s 0 = ( | P | 2 + | G | 2 ) = s 0 , 2 s 1 = ( P G * + P * G ) = s 2 , 2 s 2 = i ( P * G P G * ) = s 3 , 2 s 3 = ( | P | 2 | G | 2 ) = s 1 .
U = u ( ξ , τ , q ) exp ( i q ξ ) , V = υ ( ξ , τ , q ) exp ( i q ξ ) ,
δ ( H q Q ) = 0 .
½ u τ τ ( q β ) u + ( | u | 2 + A | υ | 2 ) u + ( 1 A ) υ 2 u * = 0 , ½ υ τ τ ( q + β ) υ + ( A | u | 2 + | υ | 2 ) υ + ( 1 A ) u 2 υ * = 0 ,
u = 2 ( q β ) cosh ( 2 ( q β ) τ ) , υ = 0 ,
u = 0 , υ = 2 ( q + β ) cosh ( 2 ( q + β ) τ ) .
P = G = 2 ( q K ) cosh ( 2 ( q K ) τ ) exp ( i q ξ ) ,
P = G = i 2 ( q + K ) cosh ( 2 ( q + K ) τ ) exp ( i q ξ ) .
| P | = | G | = u 2 + υ 2 = 2 q cosh ( 2 q τ ) .
u = X ( ξ ) f ( τ ) , υ = Y ( ξ ) f ( τ ) ,
d d ξ S 0 = 0 , d d ξ S 1 = 2 g S 2 S 3 , d d ξ S 2 = 2 β S 3 + 2 g S 1 S 3 , d d ξ S 3 = 2 β S 2 ,
S 0 = ( | X | 2 + | Y | 2 ) , S 1 = ( | X | 2 | Y | 2 ) , S 2 = ( X * Y + X Y * ) , S 3 = i ( X * Y X Y * ) ,
g = ( 1 A ) f 4 d τ f 2 d τ .
S 1 2 + S 2 2 + S 3 2 = S 0 2 ,
W = g 2 β S 3 2 S 1 ,
W = H β I + g S 0 2 2 β ( 1 A ) ρ S 0 2 β , ρ = f τ 2 d τ f 2 d τ , I = f 2 d τ .
2 S 0 = S 0 , 2 S 1 = S 2 , 2 S 2 = S 3 , 2 S 3 = S 1 .
d d ξ S 0 = 0 , d d ξ S 2 = 2 g S 1 S 3 , d d ξ S 3 = 2 K S 1 + 2 g S 1 S 2 , d d ξ S 1 = 2 K S 3 ,
g = 1 A 2 f 4 d τ f 2 d τ = g 2 .
u = Q 4 sech ( Q 4 τ ) , υ = i u .
P = 0 , G = Q 2 sech ( Q 2 τ ) .

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