Abstract

Steering properties of bright-soliton Y-junctions excited from second-order Hermite–Gaussian beams in a nonlinear Kerr medium are investigated numerically. The numerical study is supplemented by a variational approach of two different types. The first one solves the Zakharov–Shabat scattering problem, and the moment method is applied to determine the trial eigensolution. The second approach utilizes the trial function ansatz of the second-order Hermite–Gaussian form and solves the nonlinear Schrödinger equation. Good agreement is found between numerical calculations and variational approximations.

© 1996 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Maneuf, R. Desailly, and C. Froehly, “Stable self-trapping of laser beams: observation in nonlinear planar waveguide,” Opt. Commun. 65, 193–198 (1988).
    [CrossRef]
  2. J. S. Aitchison, Y. Silberberg, A. M. Weiner, D. E. Learid, M. K. Oliver, J. L. Jackel, E. M. Vogel, and P. W. E. Smith, “Spatial optical solitons in planar glass waveguides,” J. Opt. Soc. Am. B 8, 1290–1297 (1991).
    [CrossRef]
  3. T.-T. Shi and S. Chi, “Nonlinear photonic switching by using the spatial soliton collision,” Opt. Lett. 15, 1123–1125 (1990).
    [CrossRef] [PubMed]
  4. R. De La Fuente and A. Barthelemy, “Spatial solitons pairing by cross phase modulation,” Opt. Commun. 88, 419–423 (1992).
    [CrossRef]
  5. B. Luther-Davies and X. P. Yang, “Waveguides and Y-junctions formed in bulk media by using dark spatial solitons,” Opt. Lett. 17, 496–498 (1992); “Steerable optical waveguides formed in self-defocusing media by using dark spatial solitons,” Opt. Lett. 17, 1755–1757 (1992).
    [CrossRef] [PubMed]
  6. G. A. Swartzlander, D. R. Anderson, J. J. Regan, H. Yin, and A. E. Kaplan, “Spatial dark-soliton stripes and grids in self-defocusing materials,” Phys. Rev. Lett. 66, 1583–1586 (1991); G. R. Allan, S. R. Skinner, D. R. Andersen, and A. L. Smirl, “Observation of fundamental dark spatial solitons in semiconductors using picosecond pulses,” Opt. Lett. 16, 156–158 (1991).
    [CrossRef] [PubMed]
  7. G. Khitrova, H. M. Gibbs, Y. Kawamura, H. Iwamura, T. Ikegami, J. E. Sipe, and L. Ming, “Spatial solitons in a self-focusing semiconductor gain medium,” Phys. Rev. Lett. 70, 920–923 (1993).
    [CrossRef] [PubMed]
  8. A. W. Snyder, S. J. Hewlett, and D. J. Mitchell, “Dynamic spatial solitons,” Phys. Rev. Lett. 72, 1012–1015 (1994).
    [CrossRef] [PubMed]
  9. M. Sagev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923–926 (1992); G. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, and M. C. Bashaw, “Dark and bright photovoltaic spatial solitons,” Phys. Rev. A 50, 4457–4460 (1994).
    [CrossRef]
  10. C. R. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to χ(2):χ(2) cascading,” J. Opt. Soc. Am. B 11, 2434–2443 (1994).
    [CrossRef]
  11. E. M. Wright, ed., “All-optical switching using solitons” (special issue), Opt. Quantum Electron. 24, 1215–1336 (1992).
    [CrossRef]
  12. P. V. Mamyshev, A. Villeneuve, G. I. Stegeman, and J. S. Aitchison, “Steerable optical waveguides formed by bright spatial solitons in AlGaAs,” Electron. Lett. 30, 726–727 (1994).
    [CrossRef]
  13. N. Akhmediev and A. Ankiewicz, “Spatial soliton X-junctions and couplers,” Opt. Commun. 100, 186–192 (1993).
    [CrossRef]
  14. X. Yang, B. Luther-Davies, and W. Krolikowski, “On the properties of waveguide X-junctions written by spatial solitons,” Int. J. Nonlinear Opt. Phys. 2, 339–352 (1993).
    [CrossRef]
  15. W. Krolikowski, X. Yang, B. Luther-Davies, and J. Breslin, “Dark soliton steering in saturable nonlinear medium,” Opt. Commun. 105, 219–225 (1994).
    [CrossRef]
  16. A. P. Sheppard, “Devices written by colliding spatial solitons: a coupled mode theory approach,” Opt. Commun. 102, 317–323 (1993).
    [CrossRef]
  17. J. P. Gordon, “Interaction forces among solitons in optical fibers,” Opt. Lett. 8, 596–598 (1983).
    [CrossRef] [PubMed]
  18. J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, and P. W. E. Smith, “Experimental observation of spatial soliton interactions,” Opt. Lett. 16, 15–17 (1991); M. Shalby and A. Barthelemy, “Experimental spatial soliton trapping and switching,” Opt. Lett. 16, 1472–1474 (1991).
    [CrossRef] [PubMed]
  19. G. A. Swartzlander and A. E. Kaplan, “Self-deflection of laser beams in a thin nonlinear film,” J. Opt. Soc. Am. B 5, 765–768 (1988); G. A. Swartzlander, H. Yin, and A. E. Kaplan, “Continuous-wave self-deflection effect in sodium vapor,” J. Opt. Soc. Am. B 6, 1317–1325 (1989).
    [CrossRef]
  20. R.-P. Pan, S.-M. Chen, and C.-L. Pan, “Self-bending and asymmetric spatial self-phase modulation effects in nematic liquid-crystal films,” J. Opt. Soc. Am. B 8, 1065–1071 (1991).
    [CrossRef]
  21. X. D. Cao, D. D. Meyerhofer, and G. P. Agrawal, “Optimization of optical beam steering in nonlinear Kerr media by spatial phase modulation,” J. Opt. Soc. Am. B 11, 2224–2231 (1994).
    [CrossRef]
  22. D. Burak and W. Nasalski, “Spatial solitons excited by the second-order Hermite–Gaussian beams,” Opt. Appl. 24, 209–212 (1994); “Selective excitation of higher-order optical solitons by Hermite–Gaussian beams in Kerr medium,” submitted to Int. J. Nonlinear Opt. Phys.
  23. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118–134 (1971).
  24. W. Nasalski, “Nonspecular bistability versus diffraction at nonlinear hybrid interfaces,” Opt. Commun. 77, 443–450 (1990); “Ray analysis of Gaussian beam nonspecular scattering,” Opt. Commun. 92, 307–314 (1992).
    [CrossRef]
  25. H. A. Haus and M. N. Islam, “Theory of the soliton laser,” IEEE J. Quantum Electron. 21, 1172–1188 (1985).
    [CrossRef]
  26. J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Suppl. Prog. Theor. Phys. 55, 284–306 (1974).
    [CrossRef]
  27. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989), pp. 44–48.
  28. D. Burak, “Steering properties of bright soliton pairs excited by symmetric and real initial profiles,” Phys. Rev. A 52, 4054–4058 (1995).
    [CrossRef] [PubMed]
  29. A. E. Siegman, “Hermite–Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63, 1093–1094 (1973).
    [CrossRef]
  30. D. Burak, “Gaussian beam propagation in nonlinear Kerr medium,” Opt. Appl. 21, 3–8 (1991); D. Burak and W. Nasalski, “Gaussian beam to spatial soliton formation in Kerr medium,” Appl. Opt. 33, 6393–6401 (1994).
    [CrossRef] [PubMed]
  31. M. Desaix, D. Anderson, and M. Lisak, “Variational approach to the Zakharov–Shabat scattering problem,” Phys. Rev. E 50, 2253–2256 (1994).
    [CrossRef]
  32. D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
    [CrossRef]
  33. W. Nasalski, “Linear formulation of nonlinear propagation of optical beams and pulses,” Opt. Appl. 24, 205–208 (1994); “Complex ray tracing of nonlinear propagation,” Opt. Commun. 119, 218–227 (1995).
    [CrossRef]
  34. O. Bogolyavlensky, Methods in the Theory of Dynamical Systems in Astrophysics and Gas Dynamics (Springer Verlag, Berlin, 1985), Chap. 1, pp. 1–23.
    [CrossRef]
  35. T. Ueda and W. L. Kath, “Dynamics of coupled solitons in nonlinear optical fibers,” Phys. Rev. A 42, 563–571 (1990).
    [CrossRef] [PubMed]
  36. M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1972), pp. 773–792.
  37. A. Wünsche, “Generalized Gaussian beam solutions of paraxial optics and their connection to a hidden symmetry,” J. Opt. Soc. Am. A 6, 1320–1329 (1989).
    [CrossRef]

1995 (1)

D. Burak, “Steering properties of bright soliton pairs excited by symmetric and real initial profiles,” Phys. Rev. A 52, 4054–4058 (1995).
[CrossRef] [PubMed]

1994 (8)

X. D. Cao, D. D. Meyerhofer, and G. P. Agrawal, “Optimization of optical beam steering in nonlinear Kerr media by spatial phase modulation,” J. Opt. Soc. Am. B 11, 2224–2231 (1994).
[CrossRef]

D. Burak and W. Nasalski, “Spatial solitons excited by the second-order Hermite–Gaussian beams,” Opt. Appl. 24, 209–212 (1994); “Selective excitation of higher-order optical solitons by Hermite–Gaussian beams in Kerr medium,” submitted to Int. J. Nonlinear Opt. Phys.

M. Desaix, D. Anderson, and M. Lisak, “Variational approach to the Zakharov–Shabat scattering problem,” Phys. Rev. E 50, 2253–2256 (1994).
[CrossRef]

W. Nasalski, “Linear formulation of nonlinear propagation of optical beams and pulses,” Opt. Appl. 24, 205–208 (1994); “Complex ray tracing of nonlinear propagation,” Opt. Commun. 119, 218–227 (1995).
[CrossRef]

A. W. Snyder, S. J. Hewlett, and D. J. Mitchell, “Dynamic spatial solitons,” Phys. Rev. Lett. 72, 1012–1015 (1994).
[CrossRef] [PubMed]

C. R. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to χ(2):χ(2) cascading,” J. Opt. Soc. Am. B 11, 2434–2443 (1994).
[CrossRef]

P. V. Mamyshev, A. Villeneuve, G. I. Stegeman, and J. S. Aitchison, “Steerable optical waveguides formed by bright spatial solitons in AlGaAs,” Electron. Lett. 30, 726–727 (1994).
[CrossRef]

W. Krolikowski, X. Yang, B. Luther-Davies, and J. Breslin, “Dark soliton steering in saturable nonlinear medium,” Opt. Commun. 105, 219–225 (1994).
[CrossRef]

1993 (4)

A. P. Sheppard, “Devices written by colliding spatial solitons: a coupled mode theory approach,” Opt. Commun. 102, 317–323 (1993).
[CrossRef]

N. Akhmediev and A. Ankiewicz, “Spatial soliton X-junctions and couplers,” Opt. Commun. 100, 186–192 (1993).
[CrossRef]

X. Yang, B. Luther-Davies, and W. Krolikowski, “On the properties of waveguide X-junctions written by spatial solitons,” Int. J. Nonlinear Opt. Phys. 2, 339–352 (1993).
[CrossRef]

G. Khitrova, H. M. Gibbs, Y. Kawamura, H. Iwamura, T. Ikegami, J. E. Sipe, and L. Ming, “Spatial solitons in a self-focusing semiconductor gain medium,” Phys. Rev. Lett. 70, 920–923 (1993).
[CrossRef] [PubMed]

1992 (4)

M. Sagev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923–926 (1992); G. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, and M. C. Bashaw, “Dark and bright photovoltaic spatial solitons,” Phys. Rev. A 50, 4457–4460 (1994).
[CrossRef]

R. De La Fuente and A. Barthelemy, “Spatial solitons pairing by cross phase modulation,” Opt. Commun. 88, 419–423 (1992).
[CrossRef]

B. Luther-Davies and X. P. Yang, “Waveguides and Y-junctions formed in bulk media by using dark spatial solitons,” Opt. Lett. 17, 496–498 (1992); “Steerable optical waveguides formed in self-defocusing media by using dark spatial solitons,” Opt. Lett. 17, 1755–1757 (1992).
[CrossRef] [PubMed]

E. M. Wright, ed., “All-optical switching using solitons” (special issue), Opt. Quantum Electron. 24, 1215–1336 (1992).
[CrossRef]

1991 (5)

J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, and P. W. E. Smith, “Experimental observation of spatial soliton interactions,” Opt. Lett. 16, 15–17 (1991); M. Shalby and A. Barthelemy, “Experimental spatial soliton trapping and switching,” Opt. Lett. 16, 1472–1474 (1991).
[CrossRef] [PubMed]

G. A. Swartzlander, D. R. Anderson, J. J. Regan, H. Yin, and A. E. Kaplan, “Spatial dark-soliton stripes and grids in self-defocusing materials,” Phys. Rev. Lett. 66, 1583–1586 (1991); G. R. Allan, S. R. Skinner, D. R. Andersen, and A. L. Smirl, “Observation of fundamental dark spatial solitons in semiconductors using picosecond pulses,” Opt. Lett. 16, 156–158 (1991).
[CrossRef] [PubMed]

J. S. Aitchison, Y. Silberberg, A. M. Weiner, D. E. Learid, M. K. Oliver, J. L. Jackel, E. M. Vogel, and P. W. E. Smith, “Spatial optical solitons in planar glass waveguides,” J. Opt. Soc. Am. B 8, 1290–1297 (1991).
[CrossRef]

R.-P. Pan, S.-M. Chen, and C.-L. Pan, “Self-bending and asymmetric spatial self-phase modulation effects in nematic liquid-crystal films,” J. Opt. Soc. Am. B 8, 1065–1071 (1991).
[CrossRef]

D. Burak, “Gaussian beam propagation in nonlinear Kerr medium,” Opt. Appl. 21, 3–8 (1991); D. Burak and W. Nasalski, “Gaussian beam to spatial soliton formation in Kerr medium,” Appl. Opt. 33, 6393–6401 (1994).
[CrossRef] [PubMed]

1990 (3)

W. Nasalski, “Nonspecular bistability versus diffraction at nonlinear hybrid interfaces,” Opt. Commun. 77, 443–450 (1990); “Ray analysis of Gaussian beam nonspecular scattering,” Opt. Commun. 92, 307–314 (1992).
[CrossRef]

T. Ueda and W. L. Kath, “Dynamics of coupled solitons in nonlinear optical fibers,” Phys. Rev. A 42, 563–571 (1990).
[CrossRef] [PubMed]

T.-T. Shi and S. Chi, “Nonlinear photonic switching by using the spatial soliton collision,” Opt. Lett. 15, 1123–1125 (1990).
[CrossRef] [PubMed]

1989 (1)

1988 (2)

1985 (1)

H. A. Haus and M. N. Islam, “Theory of the soliton laser,” IEEE J. Quantum Electron. 21, 1172–1188 (1985).
[CrossRef]

1983 (2)

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

J. P. Gordon, “Interaction forces among solitons in optical fibers,” Opt. Lett. 8, 596–598 (1983).
[CrossRef] [PubMed]

1974 (1)

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Suppl. Prog. Theor. Phys. 55, 284–306 (1974).
[CrossRef]

1973 (1)

1971 (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118–134 (1971).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1972), pp. 773–792.

Agrawal, G. P.

Aitchison, J. S.

Akhmediev, N.

N. Akhmediev and A. Ankiewicz, “Spatial soliton X-junctions and couplers,” Opt. Commun. 100, 186–192 (1993).
[CrossRef]

Anderson, D.

M. Desaix, D. Anderson, and M. Lisak, “Variational approach to the Zakharov–Shabat scattering problem,” Phys. Rev. E 50, 2253–2256 (1994).
[CrossRef]

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

Anderson, D. R.

G. A. Swartzlander, D. R. Anderson, J. J. Regan, H. Yin, and A. E. Kaplan, “Spatial dark-soliton stripes and grids in self-defocusing materials,” Phys. Rev. Lett. 66, 1583–1586 (1991); G. R. Allan, S. R. Skinner, D. R. Andersen, and A. L. Smirl, “Observation of fundamental dark spatial solitons in semiconductors using picosecond pulses,” Opt. Lett. 16, 156–158 (1991).
[CrossRef] [PubMed]

Ankiewicz, A.

N. Akhmediev and A. Ankiewicz, “Spatial soliton X-junctions and couplers,” Opt. Commun. 100, 186–192 (1993).
[CrossRef]

Barthelemy, A.

R. De La Fuente and A. Barthelemy, “Spatial solitons pairing by cross phase modulation,” Opt. Commun. 88, 419–423 (1992).
[CrossRef]

Bogolyavlensky, O.

O. Bogolyavlensky, Methods in the Theory of Dynamical Systems in Astrophysics and Gas Dynamics (Springer Verlag, Berlin, 1985), Chap. 1, pp. 1–23.
[CrossRef]

Breslin, J.

W. Krolikowski, X. Yang, B. Luther-Davies, and J. Breslin, “Dark soliton steering in saturable nonlinear medium,” Opt. Commun. 105, 219–225 (1994).
[CrossRef]

Burak, D.

D. Burak, “Steering properties of bright soliton pairs excited by symmetric and real initial profiles,” Phys. Rev. A 52, 4054–4058 (1995).
[CrossRef] [PubMed]

D. Burak and W. Nasalski, “Spatial solitons excited by the second-order Hermite–Gaussian beams,” Opt. Appl. 24, 209–212 (1994); “Selective excitation of higher-order optical solitons by Hermite–Gaussian beams in Kerr medium,” submitted to Int. J. Nonlinear Opt. Phys.

D. Burak, “Gaussian beam propagation in nonlinear Kerr medium,” Opt. Appl. 21, 3–8 (1991); D. Burak and W. Nasalski, “Gaussian beam to spatial soliton formation in Kerr medium,” Appl. Opt. 33, 6393–6401 (1994).
[CrossRef] [PubMed]

Cao, X. D.

Chen, S.-M.

Chi, S.

Crosignani, B.

M. Sagev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923–926 (1992); G. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, and M. C. Bashaw, “Dark and bright photovoltaic spatial solitons,” Phys. Rev. A 50, 4457–4460 (1994).
[CrossRef]

De La Fuente, R.

R. De La Fuente and A. Barthelemy, “Spatial solitons pairing by cross phase modulation,” Opt. Commun. 88, 419–423 (1992).
[CrossRef]

Desailly, R.

S. Maneuf, R. Desailly, and C. Froehly, “Stable self-trapping of laser beams: observation in nonlinear planar waveguide,” Opt. Commun. 65, 193–198 (1988).
[CrossRef]

Desaix, M.

M. Desaix, D. Anderson, and M. Lisak, “Variational approach to the Zakharov–Shabat scattering problem,” Phys. Rev. E 50, 2253–2256 (1994).
[CrossRef]

Fischer, B.

M. Sagev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923–926 (1992); G. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, and M. C. Bashaw, “Dark and bright photovoltaic spatial solitons,” Phys. Rev. A 50, 4457–4460 (1994).
[CrossRef]

Froehly, C.

S. Maneuf, R. Desailly, and C. Froehly, “Stable self-trapping of laser beams: observation in nonlinear planar waveguide,” Opt. Commun. 65, 193–198 (1988).
[CrossRef]

Gibbs, H. M.

G. Khitrova, H. M. Gibbs, Y. Kawamura, H. Iwamura, T. Ikegami, J. E. Sipe, and L. Ming, “Spatial solitons in a self-focusing semiconductor gain medium,” Phys. Rev. Lett. 70, 920–923 (1993).
[CrossRef] [PubMed]

Gordon, J. P.

Haus, H. A.

H. A. Haus and M. N. Islam, “Theory of the soliton laser,” IEEE J. Quantum Electron. 21, 1172–1188 (1985).
[CrossRef]

Hewlett, S. J.

A. W. Snyder, S. J. Hewlett, and D. J. Mitchell, “Dynamic spatial solitons,” Phys. Rev. Lett. 72, 1012–1015 (1994).
[CrossRef] [PubMed]

Ikegami, T.

G. Khitrova, H. M. Gibbs, Y. Kawamura, H. Iwamura, T. Ikegami, J. E. Sipe, and L. Ming, “Spatial solitons in a self-focusing semiconductor gain medium,” Phys. Rev. Lett. 70, 920–923 (1993).
[CrossRef] [PubMed]

Islam, M. N.

H. A. Haus and M. N. Islam, “Theory of the soliton laser,” IEEE J. Quantum Electron. 21, 1172–1188 (1985).
[CrossRef]

Iwamura, H.

G. Khitrova, H. M. Gibbs, Y. Kawamura, H. Iwamura, T. Ikegami, J. E. Sipe, and L. Ming, “Spatial solitons in a self-focusing semiconductor gain medium,” Phys. Rev. Lett. 70, 920–923 (1993).
[CrossRef] [PubMed]

Jackel, J. L.

Kaplan, A. E.

G. A. Swartzlander, D. R. Anderson, J. J. Regan, H. Yin, and A. E. Kaplan, “Spatial dark-soliton stripes and grids in self-defocusing materials,” Phys. Rev. Lett. 66, 1583–1586 (1991); G. R. Allan, S. R. Skinner, D. R. Andersen, and A. L. Smirl, “Observation of fundamental dark spatial solitons in semiconductors using picosecond pulses,” Opt. Lett. 16, 156–158 (1991).
[CrossRef] [PubMed]

G. A. Swartzlander and A. E. Kaplan, “Self-deflection of laser beams in a thin nonlinear film,” J. Opt. Soc. Am. B 5, 765–768 (1988); G. A. Swartzlander, H. Yin, and A. E. Kaplan, “Continuous-wave self-deflection effect in sodium vapor,” J. Opt. Soc. Am. B 6, 1317–1325 (1989).
[CrossRef]

Kath, W. L.

T. Ueda and W. L. Kath, “Dynamics of coupled solitons in nonlinear optical fibers,” Phys. Rev. A 42, 563–571 (1990).
[CrossRef] [PubMed]

Kawamura, Y.

G. Khitrova, H. M. Gibbs, Y. Kawamura, H. Iwamura, T. Ikegami, J. E. Sipe, and L. Ming, “Spatial solitons in a self-focusing semiconductor gain medium,” Phys. Rev. Lett. 70, 920–923 (1993).
[CrossRef] [PubMed]

Khitrova, G.

G. Khitrova, H. M. Gibbs, Y. Kawamura, H. Iwamura, T. Ikegami, J. E. Sipe, and L. Ming, “Spatial solitons in a self-focusing semiconductor gain medium,” Phys. Rev. Lett. 70, 920–923 (1993).
[CrossRef] [PubMed]

Krolikowski, W.

W. Krolikowski, X. Yang, B. Luther-Davies, and J. Breslin, “Dark soliton steering in saturable nonlinear medium,” Opt. Commun. 105, 219–225 (1994).
[CrossRef]

X. Yang, B. Luther-Davies, and W. Krolikowski, “On the properties of waveguide X-junctions written by spatial solitons,” Int. J. Nonlinear Opt. Phys. 2, 339–352 (1993).
[CrossRef]

Leaird, D. E.

Learid, D. E.

Lisak, M.

M. Desaix, D. Anderson, and M. Lisak, “Variational approach to the Zakharov–Shabat scattering problem,” Phys. Rev. E 50, 2253–2256 (1994).
[CrossRef]

Luther-Davies, B.

W. Krolikowski, X. Yang, B. Luther-Davies, and J. Breslin, “Dark soliton steering in saturable nonlinear medium,” Opt. Commun. 105, 219–225 (1994).
[CrossRef]

X. Yang, B. Luther-Davies, and W. Krolikowski, “On the properties of waveguide X-junctions written by spatial solitons,” Int. J. Nonlinear Opt. Phys. 2, 339–352 (1993).
[CrossRef]

B. Luther-Davies and X. P. Yang, “Waveguides and Y-junctions formed in bulk media by using dark spatial solitons,” Opt. Lett. 17, 496–498 (1992); “Steerable optical waveguides formed in self-defocusing media by using dark spatial solitons,” Opt. Lett. 17, 1755–1757 (1992).
[CrossRef] [PubMed]

Mamyshev, P. V.

P. V. Mamyshev, A. Villeneuve, G. I. Stegeman, and J. S. Aitchison, “Steerable optical waveguides formed by bright spatial solitons in AlGaAs,” Electron. Lett. 30, 726–727 (1994).
[CrossRef]

Maneuf, S.

S. Maneuf, R. Desailly, and C. Froehly, “Stable self-trapping of laser beams: observation in nonlinear planar waveguide,” Opt. Commun. 65, 193–198 (1988).
[CrossRef]

Menyuk, C. R.

Meyerhofer, D. D.

Ming, L.

G. Khitrova, H. M. Gibbs, Y. Kawamura, H. Iwamura, T. Ikegami, J. E. Sipe, and L. Ming, “Spatial solitons in a self-focusing semiconductor gain medium,” Phys. Rev. Lett. 70, 920–923 (1993).
[CrossRef] [PubMed]

Mitchell, D. J.

A. W. Snyder, S. J. Hewlett, and D. J. Mitchell, “Dynamic spatial solitons,” Phys. Rev. Lett. 72, 1012–1015 (1994).
[CrossRef] [PubMed]

Nasalski, W.

D. Burak and W. Nasalski, “Spatial solitons excited by the second-order Hermite–Gaussian beams,” Opt. Appl. 24, 209–212 (1994); “Selective excitation of higher-order optical solitons by Hermite–Gaussian beams in Kerr medium,” submitted to Int. J. Nonlinear Opt. Phys.

W. Nasalski, “Linear formulation of nonlinear propagation of optical beams and pulses,” Opt. Appl. 24, 205–208 (1994); “Complex ray tracing of nonlinear propagation,” Opt. Commun. 119, 218–227 (1995).
[CrossRef]

W. Nasalski, “Nonspecular bistability versus diffraction at nonlinear hybrid interfaces,” Opt. Commun. 77, 443–450 (1990); “Ray analysis of Gaussian beam nonspecular scattering,” Opt. Commun. 92, 307–314 (1992).
[CrossRef]

Oliver, M. K.

Pan, C.-L.

Pan, R.-P.

Regan, J. J.

G. A. Swartzlander, D. R. Anderson, J. J. Regan, H. Yin, and A. E. Kaplan, “Spatial dark-soliton stripes and grids in self-defocusing materials,” Phys. Rev. Lett. 66, 1583–1586 (1991); G. R. Allan, S. R. Skinner, D. R. Andersen, and A. L. Smirl, “Observation of fundamental dark spatial solitons in semiconductors using picosecond pulses,” Opt. Lett. 16, 156–158 (1991).
[CrossRef] [PubMed]

Sagev, M.

M. Sagev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923–926 (1992); G. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, and M. C. Bashaw, “Dark and bright photovoltaic spatial solitons,” Phys. Rev. A 50, 4457–4460 (1994).
[CrossRef]

Satsuma, J.

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Suppl. Prog. Theor. Phys. 55, 284–306 (1974).
[CrossRef]

Schiek, R.

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118–134 (1971).

Sheppard, A. P.

A. P. Sheppard, “Devices written by colliding spatial solitons: a coupled mode theory approach,” Opt. Commun. 102, 317–323 (1993).
[CrossRef]

Shi, T.-T.

Siegman, A. E.

Silberberg, Y.

Sipe, J. E.

G. Khitrova, H. M. Gibbs, Y. Kawamura, H. Iwamura, T. Ikegami, J. E. Sipe, and L. Ming, “Spatial solitons in a self-focusing semiconductor gain medium,” Phys. Rev. Lett. 70, 920–923 (1993).
[CrossRef] [PubMed]

Smith, P. W. E.

Snyder, A. W.

A. W. Snyder, S. J. Hewlett, and D. J. Mitchell, “Dynamic spatial solitons,” Phys. Rev. Lett. 72, 1012–1015 (1994).
[CrossRef] [PubMed]

Stegeman, G. I.

P. V. Mamyshev, A. Villeneuve, G. I. Stegeman, and J. S. Aitchison, “Steerable optical waveguides formed by bright spatial solitons in AlGaAs,” Electron. Lett. 30, 726–727 (1994).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1972), pp. 773–792.

Swartzlander, G. A.

G. A. Swartzlander, D. R. Anderson, J. J. Regan, H. Yin, and A. E. Kaplan, “Spatial dark-soliton stripes and grids in self-defocusing materials,” Phys. Rev. Lett. 66, 1583–1586 (1991); G. R. Allan, S. R. Skinner, D. R. Andersen, and A. L. Smirl, “Observation of fundamental dark spatial solitons in semiconductors using picosecond pulses,” Opt. Lett. 16, 156–158 (1991).
[CrossRef] [PubMed]

G. A. Swartzlander and A. E. Kaplan, “Self-deflection of laser beams in a thin nonlinear film,” J. Opt. Soc. Am. B 5, 765–768 (1988); G. A. Swartzlander, H. Yin, and A. E. Kaplan, “Continuous-wave self-deflection effect in sodium vapor,” J. Opt. Soc. Am. B 6, 1317–1325 (1989).
[CrossRef]

Torner, L.

Ueda, T.

T. Ueda and W. L. Kath, “Dynamics of coupled solitons in nonlinear optical fibers,” Phys. Rev. A 42, 563–571 (1990).
[CrossRef] [PubMed]

Villeneuve, A.

P. V. Mamyshev, A. Villeneuve, G. I. Stegeman, and J. S. Aitchison, “Steerable optical waveguides formed by bright spatial solitons in AlGaAs,” Electron. Lett. 30, 726–727 (1994).
[CrossRef]

Vogel, E. M.

Weiner, A. M.

Wünsche, A.

Yajima, N.

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Suppl. Prog. Theor. Phys. 55, 284–306 (1974).
[CrossRef]

Yang, X.

W. Krolikowski, X. Yang, B. Luther-Davies, and J. Breslin, “Dark soliton steering in saturable nonlinear medium,” Opt. Commun. 105, 219–225 (1994).
[CrossRef]

X. Yang, B. Luther-Davies, and W. Krolikowski, “On the properties of waveguide X-junctions written by spatial solitons,” Int. J. Nonlinear Opt. Phys. 2, 339–352 (1993).
[CrossRef]

Yang, X. P.

Yariv, A.

M. Sagev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923–926 (1992); G. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, and M. C. Bashaw, “Dark and bright photovoltaic spatial solitons,” Phys. Rev. A 50, 4457–4460 (1994).
[CrossRef]

Yin, H.

G. A. Swartzlander, D. R. Anderson, J. J. Regan, H. Yin, and A. E. Kaplan, “Spatial dark-soliton stripes and grids in self-defocusing materials,” Phys. Rev. Lett. 66, 1583–1586 (1991); G. R. Allan, S. R. Skinner, D. R. Andersen, and A. L. Smirl, “Observation of fundamental dark spatial solitons in semiconductors using picosecond pulses,” Opt. Lett. 16, 156–158 (1991).
[CrossRef] [PubMed]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118–134 (1971).

Electron. Lett. (1)

P. V. Mamyshev, A. Villeneuve, G. I. Stegeman, and J. S. Aitchison, “Steerable optical waveguides formed by bright spatial solitons in AlGaAs,” Electron. Lett. 30, 726–727 (1994).
[CrossRef]

IEEE J. Quantum Electron. (1)

H. A. Haus and M. N. Islam, “Theory of the soliton laser,” IEEE J. Quantum Electron. 21, 1172–1188 (1985).
[CrossRef]

Int. J. Nonlinear Opt. Phys. (1)

X. Yang, B. Luther-Davies, and W. Krolikowski, “On the properties of waveguide X-junctions written by spatial solitons,” Int. J. Nonlinear Opt. Phys. 2, 339–352 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Opt. Soc. Am. B (5)

Opt. Appl. (3)

D. Burak and W. Nasalski, “Spatial solitons excited by the second-order Hermite–Gaussian beams,” Opt. Appl. 24, 209–212 (1994); “Selective excitation of higher-order optical solitons by Hermite–Gaussian beams in Kerr medium,” submitted to Int. J. Nonlinear Opt. Phys.

W. Nasalski, “Linear formulation of nonlinear propagation of optical beams and pulses,” Opt. Appl. 24, 205–208 (1994); “Complex ray tracing of nonlinear propagation,” Opt. Commun. 119, 218–227 (1995).
[CrossRef]

D. Burak, “Gaussian beam propagation in nonlinear Kerr medium,” Opt. Appl. 21, 3–8 (1991); D. Burak and W. Nasalski, “Gaussian beam to spatial soliton formation in Kerr medium,” Appl. Opt. 33, 6393–6401 (1994).
[CrossRef] [PubMed]

Opt. Commun. (6)

N. Akhmediev and A. Ankiewicz, “Spatial soliton X-junctions and couplers,” Opt. Commun. 100, 186–192 (1993).
[CrossRef]

S. Maneuf, R. Desailly, and C. Froehly, “Stable self-trapping of laser beams: observation in nonlinear planar waveguide,” Opt. Commun. 65, 193–198 (1988).
[CrossRef]

W. Nasalski, “Nonspecular bistability versus diffraction at nonlinear hybrid interfaces,” Opt. Commun. 77, 443–450 (1990); “Ray analysis of Gaussian beam nonspecular scattering,” Opt. Commun. 92, 307–314 (1992).
[CrossRef]

W. Krolikowski, X. Yang, B. Luther-Davies, and J. Breslin, “Dark soliton steering in saturable nonlinear medium,” Opt. Commun. 105, 219–225 (1994).
[CrossRef]

A. P. Sheppard, “Devices written by colliding spatial solitons: a coupled mode theory approach,” Opt. Commun. 102, 317–323 (1993).
[CrossRef]

R. De La Fuente and A. Barthelemy, “Spatial solitons pairing by cross phase modulation,” Opt. Commun. 88, 419–423 (1992).
[CrossRef]

Opt. Lett. (4)

Opt. Quantum Electron. (1)

E. M. Wright, ed., “All-optical switching using solitons” (special issue), Opt. Quantum Electron. 24, 1215–1336 (1992).
[CrossRef]

Phys. Rev. A (3)

D. Burak, “Steering properties of bright soliton pairs excited by symmetric and real initial profiles,” Phys. Rev. A 52, 4054–4058 (1995).
[CrossRef] [PubMed]

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

T. Ueda and W. L. Kath, “Dynamics of coupled solitons in nonlinear optical fibers,” Phys. Rev. A 42, 563–571 (1990).
[CrossRef] [PubMed]

Phys. Rev. E (1)

M. Desaix, D. Anderson, and M. Lisak, “Variational approach to the Zakharov–Shabat scattering problem,” Phys. Rev. E 50, 2253–2256 (1994).
[CrossRef]

Phys. Rev. Lett. (4)

G. A. Swartzlander, D. R. Anderson, J. J. Regan, H. Yin, and A. E. Kaplan, “Spatial dark-soliton stripes and grids in self-defocusing materials,” Phys. Rev. Lett. 66, 1583–1586 (1991); G. R. Allan, S. R. Skinner, D. R. Andersen, and A. L. Smirl, “Observation of fundamental dark spatial solitons in semiconductors using picosecond pulses,” Opt. Lett. 16, 156–158 (1991).
[CrossRef] [PubMed]

G. Khitrova, H. M. Gibbs, Y. Kawamura, H. Iwamura, T. Ikegami, J. E. Sipe, and L. Ming, “Spatial solitons in a self-focusing semiconductor gain medium,” Phys. Rev. Lett. 70, 920–923 (1993).
[CrossRef] [PubMed]

A. W. Snyder, S. J. Hewlett, and D. J. Mitchell, “Dynamic spatial solitons,” Phys. Rev. Lett. 72, 1012–1015 (1994).
[CrossRef] [PubMed]

M. Sagev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923–926 (1992); G. Valley, M. Segev, B. Crosignani, A. Yariv, M. M. Fejer, and M. C. Bashaw, “Dark and bright photovoltaic spatial solitons,” Phys. Rev. A 50, 4457–4460 (1994).
[CrossRef]

Suppl. Prog. Theor. Phys. (1)

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Suppl. Prog. Theor. Phys. 55, 284–306 (1974).
[CrossRef]

Zh. Eksp. Teor. Fiz. (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118–134 (1971).

Other (3)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989), pp. 44–48.

M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1972), pp. 773–792.

O. Bogolyavlensky, Methods in the Theory of Dynamical Systems in Astrophysics and Gas Dynamics (Springer Verlag, Berlin, 1985), Chap. 1, pp. 1–23.
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Soliton parameters excited by the second-order Hermite–Gaussian beam [Eq. (7)]. The imaginary parts (solid curves, the left-hand-side axis) and the real parts (dotted curves, the right-hand-side axis) of eigenvalues λi, i = 1 … 4 (corresponding to the amplitudes and the deflection angles, λi = −arctan 2ξi, of solitons, respectively), are plotted as a function of the incident-beam amplitude q0.

Fig. 2
Fig. 2

Evolution of initial HG2 profiles [Eq. (7)] in a Kerr medium for the incident-beam amplitudes (a) q0 ≈ 1.67, (b) q0 ≈ 2.22, (c) q0 ≈ 2.24.

Fig. 3
Fig. 3

Amplitudes (the left-hand-side axis) and transverse velocities (the right-hand-side axis) of excited solitons as obtained from numerical and variational solutions of Eqs. (5). Solid and dashed lines: numerically obtained values of 2η and 2ξ, respectively. Solid diamonds and circles: variational results for 2η and 2ξ, respectively. The subscripts IST and v in η and ξ refer to IST and variational approaches, respectively.

Fig. 4
Fig. 4

Left-hand-side axis: peak amplitude of the beams |U(xmax, ζ0)| in numerical (solid curve) and variational (solid diamonds) approaches evaluated at ζ0 = 10zF versus the amplitude of the incident beam q0. Right-hand-side axis: the deflection angles ϑ ( n , v ) = x max ( n , v ) / ζ 0 versus q0 in numerical (dashed curve) and variational (solid circles) approaches.

Equations (41)

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

2 E z 2 + 2 E x 2 + k 0 2 n 2 E = 0 ,
2 i β V z + 2 V x 2 + 2 k 0 n 0 n 2 | V | 2 V = 0
i U ζ + 1 2 2 U x 2 + | U | 2 U = 0 ,
U ( x , ζ ) = i = 1 N 2 η i exp [ 2 i ( n i 2 ξ i 2 ) ζ 2 i ξ i x ] × sech ( 2 η i x + 4 η i ξ i ζ ),
φ 1 x = i λ i φ 1 + i U 0 ( x ) φ 2 ,
φ 2 x = i λ i φ 2 + i U 0 * ( x ) φ 1 ,
+ U 0 ( x ) d x = 0 .
U 0 ( x ) = q 0 ( 1 + x 2 ) exp ( x 2 2 ) ,
q 0 th 0.86
q 0 ( b ) 2.22
δ + L d x = 0 ,
L = 1 2 ( φ 2 d φ 1 d x φ 1 d φ 2 d x ) + i λ φ 1 φ 2 + i 2 [ U 0 * ( x ) φ 1 2 U 0 ( x ) φ 2 2 ] .
φ 1 ( x , λ ˜ ) = B exp ( i λ ˜ x ) { cos β ( x ) η ˜ [ B ( x ) sin β ( x ) A ( x ) cos β ( x ) ] } ,
φ 2 ( x , λ ˜ ) = i φ 1 ( x , λ ˜ ) ,
β ( x ) = x U ˜ 0 ( y ) d y ,
A ( x ) = x 2 y β ( y ) sin [ 2 β ( y ) ] d y ,
B ( x ) = x 2 y β ( y ) cos [ 2 β ( y ) ] d y .
U ˜ 0 ( x ) = exp ( 2 i ξ ˜ x ) U 0 ( x ) ,
tan ϑ def = P M = 2 ξ ˜ ,
M = + | U ˜ 0 ( x ) | 2 d x ,
P = 1 2 i + ( U ˜ 0 * U ˜ 0 x U ˜ 0 U ˜ 0 * x ) d x .
η ˜ = cos β B sin β A cos β ,
L = i 2 ( U * U ζ U U * ζ ) 1 2 | U x | 2 + 1 2 | U | 4 .
U ( x , ζ ) = A ( ζ ) v 3 ( ζ ) { 1 + [ x v ( ζ ) ] 2 } exp [ x 2 2 v 2 ( ζ ) ] ,
d v d ζ = i 2 v i 8 q 0 2 ( 1 + v 2 v * 2 ) 3 / 2 ( 43 320 + 7 80 v 2 v * 2 + 5 80 v 4 v * 4 1 80 v * 2 v 2 5 80 v * 4 v 4 4 80 v * 6 v 6 ) ,
d A d ζ = i 2 q 0 2 1 v 2 + 1 v * 2 ( 9 16 27 256 v 2 v * 2 + 9 64 v * 2 v 2 3 64 v 4 v * 4 + 1 8 v * 4 v 4 5 64 v 6 v * 6 ) A .
- | U ( x , ζ ) | 2 d x = 3 2 π | A | 2 v 2 + v * 2 5 / 2 = const .= 3 π q 0 2 4 ,
H = 3 π | A | 2 ( v * 2 + v 2 ) 9 / 2 { 5 2 Re [ v 2 ] | A | 2 4 | v | 10 [ Re [ v 8 ] + | v | 4 ( Re [ v 4 ] + 35 8 | v 4 | ) ] } ,
r ˙ = 1 2 sin ( 2 φ ) [ 1 r + q 0 2 2 cos 3 / 2 ( 2 φ ) ] × [ 47 320 + 12 80 sin ( 6 φ ) sin ( 2 φ ) + 9 80 sin ( 10 φ ) sin ( 2 φ ) ] ,
φ ˙ = 1 2 r cos ( 2 φ ) [ 1 r + q 0 2 2 cos 3 / 2 ( 2 φ ) ] × [ 39 320 + 1 40 cos ( 6 φ ) cos ( 2 φ ) + 1 80 cos ( 10 φ ) cos ( 2 φ ) ] .
r c 1 = 320 2 51 q 0 2 , φ c 1 = 0 ,
1 r c 2 = 0 , φ c 2 = π 4 .
q 0 ( v ) = 8 5 51 2 2.11 ,
x max ( v ) = ± [ | v | 4 + ( Re [ v 2 ] ) 2 + | v | 8 | v | 4 ( Re [ v 2 ] ) 2 + ( Re [ v 2 ] ) 4 Re [ v 2 ] ] 1 / 2 ,
i V ζ + 1 2 2 V x 2 = 0 .
V 0 ( x , ζ ) = q 0 v ( ζ ) exp [ x 2 2 v 2 ( ζ ) ] ,
V n ( x , ζ ) = q 0 v n + 1 ( ζ ) He n [ x v ( ζ ) ] exp [ x 2 2 v 2 ( ζ ) ] ,
d 2 He n ( x / v ) d ( x / v ) 2 x v d He n ( x / v ) d ( x / v ) + n He n ( x / v ) = 0 .
He n [ x v ( ζ ) ] = ( 1 ) n exp [ ( x / v ) 2 / 2 ] d n d ( x / v ) n exp [ ( x / v ) 2 / 2 ] .
V 2 ( x , ζ ) = q 0 v 3 ( ζ ) { 1 + [ x v ( ζ ) ] 2 } exp [ x 2 2 v 2 ( ζ ) ] .
+ V n ( x , 0 ) d x = 0 , n 1 .

Metrics