Abstract

We investigate some of the theoretical notions underlying the observation of spatial solitons in nonlinear planar waveguides. We find that, when the beam is confined in one dimension principally by the action of a linear refractive-index profile, the nonlinear behavior of the beam in the orthogonal dimension is governed by the usual nonlinear Schrödinger equation with parameters modified by the linear waveguide modal properties. When either the power or the nonlinearity of the material is high, nonlinearity affects both dimensions, and a form of three-dimensional self-trapping begins to occur. A simple variational approximation gives an accurate picture of what is happening in this regime.

© 1993 Optical Society of America

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References

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  1. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).
  2. A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, Redwood City, Calif., 1992).
  3. G. I. Stegeman and R. H. Stolen, “Waveguides and fibers for nonlinear optics,” J. Opt. Soc. Am. B 6, 652–662 (1989).
    [Crossref]
  4. G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
    [Crossref]
  5. D. Mihalache, M. Bertolotti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. XXVII, pp. 227–313.
    [Crossref]
  6. A. D. Boardman, P. Egan, F. Lederer, U. Langbein, and D. Mihalache, “Third-order nonlinear electromagnetic TE and TM guided waves,” in Nonlinear Surface Electromagnetic Phenomena, H.-E. Ponath and G. I. Stegeman, eds. (North-Holland, Amsterdam, 1991).
    [Crossref]
  7. A. B. Aceves, P. Varatharajah, A. C. Newell, E. M. Wright, G. I. Stegeman, D. R. Heatley, J. V. Moloney, and H. Adachihara, “Particle aspects of collimated light channel propagation at nonlinear interfaces and in waveguides,” J. Opt. Soc. Am. B 7, 963–974 (1990).
    [Crossref]
  8. Y. R. Shen, “Self-focusing: experimental,” Prog. Quantum Electron. 4, 1–34 (1975).
    [Crossref]
  9. J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
    [Crossref]
  10. A. Barthelemy, S. Maneuf, and C. Froehly, “Propagation soliton et autoconfinement de faisceaux laser par non-linéarité optique de Kerr,” Opt. Commun. 55, 201–206 (1985).
    [Crossref]
  11. S. Maneuf and F. Reynaud, “Quasi-steady state self-trapping of first, second and third order subnanosecond soliton beams,” Opt. Commun. 66, 325–328 (1988).
    [Crossref]
  12. F. Reynaud and A. Barthelemy, “Optically controlled interaction between two fundamental soliton beams,” Europhys. Lett. 12, 401–405 (1990).
    [Crossref]
  13. M. Shalaby, F. Reynaud, and A. Barthelemy, “Experimental observation of spatial soliton interactions with a π/2 relative phase difference,” Opt. Lett. 17, 778–780 (1992).
    [Crossref] [PubMed]
  14. A. Barthelemy, C. Froehly, S. Maneuf, and F. Reynaud, “Experimental observation of beams’ self-deflection appearing with two-dimensional spatial soliton propagation in bulk Kerr material,” Opt. Lett. 17, 884–886 (1992).
    [Crossref]
  15. F. Reynaud and A. Barthelemy, “Soliton beam propagation in nonlinear Kerr media,” in Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, eds. (Kluwer, Dordrecht, The Netherlands, 1992).
    [Crossref]
  16. G. R. Allan, S. R. Skinner, D. R. Anderson, and A. L. Smirl, “Observation of fundamental dark spatial solitons in semiconductors using picosecond pulses,” Opt. Lett. 16, 156–159 (1991).
    [PubMed]
  17. B. Luther-Davies and Y. Xiaoping, “Waveguides and Y junctions formed in bulk media by using dark spatial solitons,” Opt. Lett. 17, 496–498 (1992).
    [Crossref] [PubMed]
  18. S. Maneuf, R. Desailly, and C. Froehly, “Stable self-trapping of laser beams: observation in a nonlinear planar waveguide,” Opt. Commun. 65, 193–198 (1988).
    [Crossref]
  19. J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. M. Vogel, and P. W. E. Smith, “Observation of spatial optical solitons in a nonlinear glass waveguide,” Opt. Lett. 15, 471–473 (1990).
    [Crossref] [PubMed]
  20. J. S. Aitchison, Y. Silberberg, A. M. Weiner, D. E. Leaird, 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]
  21. 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).
    [Crossref] [PubMed]
  22. D. H. Reitze, A. M. Weiner, and D. E. Leaird, “High-power femtosecond optical pulse compression by using spatial solitons,” Opt. Lett. 16, 1409–1411 (1991).
    [Crossref] [PubMed]
  23. Q. Y. Li, C. Pask, and R. A. Sammut, “Simple model for spatial optical solitons in planar waveguides,” Opt. Lett. 16, 1083–1085 (1991).
    [Crossref] [PubMed]
  24. N. N. Akhmediev, R. F. Nabiev, and Yu. M. Popov, “Three-dimensional modes of a symmetric nonlinear plane waveguide,” Opt. Commun. 69, 247–252 (1989).
    [Crossref]
  25. N. N. Akhmediev, R. F. Nabiev, and Yu. M. Popov, “Stripe nonlinear waves in a symmetrical planar structure,” Opt. Commun. 72, 190–194 (1989).
    [Crossref]
  26. Q. Y. Li, R. A. Sammut, and C. Pask, “Variational and finite element analyses of nonlinear strip optical waveguides,” Opt. Commun. 94, 37–43 (1992).
    [Crossref]
  27. R. A. Sammut and C. Pask, “Gaussian and equivalent-step-index approximations for nonlinear waveguides,” J. Opt. Soc. Am. B 8, 395–402 (1991).
    [Crossref]
  28. A. W. Snyder, Y. Chen, L. Poladian, and D. J. Mitchell, “Fundamental mode of highly nonlinear fibres,” Electron. Lett. 26, 643–644 (1990).
    [Crossref]
  29. R. Weinstock, Calculus of Variations (Dover, New York, 1974).
  30. R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley Interscience, New York, 1989), Vol. 1.

1992 (4)

M. Shalaby, F. Reynaud, and A. Barthelemy, “Experimental observation of spatial soliton interactions with a π/2 relative phase difference,” Opt. Lett. 17, 778–780 (1992).
[Crossref] [PubMed]

A. Barthelemy, C. Froehly, S. Maneuf, and F. Reynaud, “Experimental observation of beams’ self-deflection appearing with two-dimensional spatial soliton propagation in bulk Kerr material,” Opt. Lett. 17, 884–886 (1992).
[Crossref]

B. Luther-Davies and Y. Xiaoping, “Waveguides and Y junctions formed in bulk media by using dark spatial solitons,” Opt. Lett. 17, 496–498 (1992).
[Crossref] [PubMed]

Q. Y. Li, R. A. Sammut, and C. Pask, “Variational and finite element analyses of nonlinear strip optical waveguides,” Opt. Commun. 94, 37–43 (1992).
[Crossref]

1991 (6)

1990 (5)

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
[Crossref]

A. B. Aceves, P. Varatharajah, A. C. Newell, E. M. Wright, G. I. Stegeman, D. R. Heatley, J. V. Moloney, and H. Adachihara, “Particle aspects of collimated light channel propagation at nonlinear interfaces and in waveguides,” J. Opt. Soc. Am. B 7, 963–974 (1990).
[Crossref]

F. Reynaud and A. Barthelemy, “Optically controlled interaction between two fundamental soliton beams,” Europhys. Lett. 12, 401–405 (1990).
[Crossref]

A. W. Snyder, Y. Chen, L. Poladian, and D. J. Mitchell, “Fundamental mode of highly nonlinear fibres,” Electron. Lett. 26, 643–644 (1990).
[Crossref]

J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. M. Vogel, and P. W. E. Smith, “Observation of spatial optical solitons in a nonlinear glass waveguide,” Opt. Lett. 15, 471–473 (1990).
[Crossref] [PubMed]

1989 (3)

G. I. Stegeman and R. H. Stolen, “Waveguides and fibers for nonlinear optics,” J. Opt. Soc. Am. B 6, 652–662 (1989).
[Crossref]

N. N. Akhmediev, R. F. Nabiev, and Yu. M. Popov, “Three-dimensional modes of a symmetric nonlinear plane waveguide,” Opt. Commun. 69, 247–252 (1989).
[Crossref]

N. N. Akhmediev, R. F. Nabiev, and Yu. M. Popov, “Stripe nonlinear waves in a symmetrical planar structure,” Opt. Commun. 72, 190–194 (1989).
[Crossref]

1988 (2)

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

S. Maneuf and F. Reynaud, “Quasi-steady state self-trapping of first, second and third order subnanosecond soliton beams,” Opt. Commun. 66, 325–328 (1988).
[Crossref]

1985 (1)

A. Barthelemy, S. Maneuf, and C. Froehly, “Propagation soliton et autoconfinement de faisceaux laser par non-linéarité optique de Kerr,” Opt. Commun. 55, 201–206 (1985).
[Crossref]

1975 (2)

Y. R. Shen, “Self-focusing: experimental,” Prog. Quantum Electron. 4, 1–34 (1975).
[Crossref]

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[Crossref]

Aceves, A. B.

Adachihara, H.

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).

Aitchison, J. S.

Akhmediev, N. N.

N. N. Akhmediev, R. F. Nabiev, and Yu. M. Popov, “Three-dimensional modes of a symmetric nonlinear plane waveguide,” Opt. Commun. 69, 247–252 (1989).
[Crossref]

N. N. Akhmediev, R. F. Nabiev, and Yu. M. Popov, “Stripe nonlinear waves in a symmetrical planar structure,” Opt. Commun. 72, 190–194 (1989).
[Crossref]

Allan, G. R.

Anderson, D. R.

Barthelemy, A.

M. Shalaby, F. Reynaud, and A. Barthelemy, “Experimental observation of spatial soliton interactions with a π/2 relative phase difference,” Opt. Lett. 17, 778–780 (1992).
[Crossref] [PubMed]

A. Barthelemy, C. Froehly, S. Maneuf, and F. Reynaud, “Experimental observation of beams’ self-deflection appearing with two-dimensional spatial soliton propagation in bulk Kerr material,” Opt. Lett. 17, 884–886 (1992).
[Crossref]

F. Reynaud and A. Barthelemy, “Optically controlled interaction between two fundamental soliton beams,” Europhys. Lett. 12, 401–405 (1990).
[Crossref]

A. Barthelemy, S. Maneuf, and C. Froehly, “Propagation soliton et autoconfinement de faisceaux laser par non-linéarité optique de Kerr,” Opt. Commun. 55, 201–206 (1985).
[Crossref]

F. Reynaud and A. Barthelemy, “Soliton beam propagation in nonlinear Kerr media,” in Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, eds. (Kluwer, Dordrecht, The Netherlands, 1992).
[Crossref]

Bertolotti, M.

D. Mihalache, M. Bertolotti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. XXVII, pp. 227–313.
[Crossref]

Boardman, A. D.

A. D. Boardman, P. Egan, F. Lederer, U. Langbein, and D. Mihalache, “Third-order nonlinear electromagnetic TE and TM guided waves,” in Nonlinear Surface Electromagnetic Phenomena, H.-E. Ponath and G. I. Stegeman, eds. (North-Holland, Amsterdam, 1991).
[Crossref]

Chen, Y.

A. W. Snyder, Y. Chen, L. Poladian, and D. J. Mitchell, “Fundamental mode of highly nonlinear fibres,” Electron. Lett. 26, 643–644 (1990).
[Crossref]

Courant, R.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley Interscience, New York, 1989), Vol. 1.

Desailly, R.

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

Egan, P.

A. D. Boardman, P. Egan, F. Lederer, U. Langbein, and D. Mihalache, “Third-order nonlinear electromagnetic TE and TM guided waves,” in Nonlinear Surface Electromagnetic Phenomena, H.-E. Ponath and G. I. Stegeman, eds. (North-Holland, Amsterdam, 1991).
[Crossref]

Froehly, C.

A. Barthelemy, C. Froehly, S. Maneuf, and F. Reynaud, “Experimental observation of beams’ self-deflection appearing with two-dimensional spatial soliton propagation in bulk Kerr material,” Opt. Lett. 17, 884–886 (1992).
[Crossref]

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

A. Barthelemy, S. Maneuf, and C. Froehly, “Propagation soliton et autoconfinement de faisceaux laser par non-linéarité optique de Kerr,” Opt. Commun. 55, 201–206 (1985).
[Crossref]

Heatley, D. R.

Hilbert, D.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley Interscience, New York, 1989), Vol. 1.

Jackel, J. L.

Langbein, U.

A. D. Boardman, P. Egan, F. Lederer, U. Langbein, and D. Mihalache, “Third-order nonlinear electromagnetic TE and TM guided waves,” in Nonlinear Surface Electromagnetic Phenomena, H.-E. Ponath and G. I. Stegeman, eds. (North-Holland, Amsterdam, 1991).
[Crossref]

Leaird, D. E.

Lederer, F.

A. D. Boardman, P. Egan, F. Lederer, U. Langbein, and D. Mihalache, “Third-order nonlinear electromagnetic TE and TM guided waves,” in Nonlinear Surface Electromagnetic Phenomena, H.-E. Ponath and G. I. Stegeman, eds. (North-Holland, Amsterdam, 1991).
[Crossref]

Li, Q. Y.

Q. Y. Li, R. A. Sammut, and C. Pask, “Variational and finite element analyses of nonlinear strip optical waveguides,” Opt. Commun. 94, 37–43 (1992).
[Crossref]

Q. Y. Li, C. Pask, and R. A. Sammut, “Simple model for spatial optical solitons in planar waveguides,” Opt. Lett. 16, 1083–1085 (1991).
[Crossref] [PubMed]

Luther-Davies, B.

Maneuf, S.

A. Barthelemy, C. Froehly, S. Maneuf, and F. Reynaud, “Experimental observation of beams’ self-deflection appearing with two-dimensional spatial soliton propagation in bulk Kerr material,” Opt. Lett. 17, 884–886 (1992).
[Crossref]

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

S. Maneuf and F. Reynaud, “Quasi-steady state self-trapping of first, second and third order subnanosecond soliton beams,” Opt. Commun. 66, 325–328 (1988).
[Crossref]

A. Barthelemy, S. Maneuf, and C. Froehly, “Propagation soliton et autoconfinement de faisceaux laser par non-linéarité optique de Kerr,” Opt. Commun. 55, 201–206 (1985).
[Crossref]

Marburger, J. H.

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[Crossref]

Mihalache, D.

D. Mihalache, M. Bertolotti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. XXVII, pp. 227–313.
[Crossref]

A. D. Boardman, P. Egan, F. Lederer, U. Langbein, and D. Mihalache, “Third-order nonlinear electromagnetic TE and TM guided waves,” in Nonlinear Surface Electromagnetic Phenomena, H.-E. Ponath and G. I. Stegeman, eds. (North-Holland, Amsterdam, 1991).
[Crossref]

Mitchell, D. J.

A. W. Snyder, Y. Chen, L. Poladian, and D. J. Mitchell, “Fundamental mode of highly nonlinear fibres,” Electron. Lett. 26, 643–644 (1990).
[Crossref]

Moloney, J. V.

Nabiev, R. F.

N. N. Akhmediev, R. F. Nabiev, and Yu. M. Popov, “Stripe nonlinear waves in a symmetrical planar structure,” Opt. Commun. 72, 190–194 (1989).
[Crossref]

N. N. Akhmediev, R. F. Nabiev, and Yu. M. Popov, “Three-dimensional modes of a symmetric nonlinear plane waveguide,” Opt. Commun. 69, 247–252 (1989).
[Crossref]

Newell, A. C.

Oliver, M. K.

Pask, C.

Poladian, L.

A. W. Snyder, Y. Chen, L. Poladian, and D. J. Mitchell, “Fundamental mode of highly nonlinear fibres,” Electron. Lett. 26, 643–644 (1990).
[Crossref]

Popov, Yu. M.

N. N. Akhmediev, R. F. Nabiev, and Yu. M. Popov, “Three-dimensional modes of a symmetric nonlinear plane waveguide,” Opt. Commun. 69, 247–252 (1989).
[Crossref]

N. N. Akhmediev, R. F. Nabiev, and Yu. M. Popov, “Stripe nonlinear waves in a symmetrical planar structure,” Opt. Commun. 72, 190–194 (1989).
[Crossref]

Reitze, D. H.

Reynaud, F.

M. Shalaby, F. Reynaud, and A. Barthelemy, “Experimental observation of spatial soliton interactions with a π/2 relative phase difference,” Opt. Lett. 17, 778–780 (1992).
[Crossref] [PubMed]

A. Barthelemy, C. Froehly, S. Maneuf, and F. Reynaud, “Experimental observation of beams’ self-deflection appearing with two-dimensional spatial soliton propagation in bulk Kerr material,” Opt. Lett. 17, 884–886 (1992).
[Crossref]

F. Reynaud and A. Barthelemy, “Optically controlled interaction between two fundamental soliton beams,” Europhys. Lett. 12, 401–405 (1990).
[Crossref]

S. Maneuf and F. Reynaud, “Quasi-steady state self-trapping of first, second and third order subnanosecond soliton beams,” Opt. Commun. 66, 325–328 (1988).
[Crossref]

F. Reynaud and A. Barthelemy, “Soliton beam propagation in nonlinear Kerr media,” in Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, eds. (Kluwer, Dordrecht, The Netherlands, 1992).
[Crossref]

Sammut, R. A.

Shalaby, M.

Shen, Y. R.

Y. R. Shen, “Self-focusing: experimental,” Prog. Quantum Electron. 4, 1–34 (1975).
[Crossref]

Sibilia, C.

D. Mihalache, M. Bertolotti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. XXVII, pp. 227–313.
[Crossref]

Silberberg, Y.

Skinner, S. R.

Smirl, A. L.

Smith, P. W. E.

Snyder, A. W.

A. W. Snyder, Y. Chen, L. Poladian, and D. J. Mitchell, “Fundamental mode of highly nonlinear fibres,” Electron. Lett. 26, 643–644 (1990).
[Crossref]

Stegeman, G. I.

Stolen, R. H.

Varatharajah, P.

Vogel, E. M.

Weiner, A. M.

Weinstock, R.

R. Weinstock, Calculus of Variations (Dover, New York, 1974).

Wright, E. M.

Xiaoping, Y.

Electron. Lett. (1)

A. W. Snyder, Y. Chen, L. Poladian, and D. J. Mitchell, “Fundamental mode of highly nonlinear fibres,” Electron. Lett. 26, 643–644 (1990).
[Crossref]

Europhys. Lett. (1)

F. Reynaud and A. Barthelemy, “Optically controlled interaction between two fundamental soliton beams,” Europhys. Lett. 12, 401–405 (1990).
[Crossref]

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

Opt. Commun. (6)

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

N. N. Akhmediev, R. F. Nabiev, and Yu. M. Popov, “Three-dimensional modes of a symmetric nonlinear plane waveguide,” Opt. Commun. 69, 247–252 (1989).
[Crossref]

N. N. Akhmediev, R. F. Nabiev, and Yu. M. Popov, “Stripe nonlinear waves in a symmetrical planar structure,” Opt. Commun. 72, 190–194 (1989).
[Crossref]

Q. Y. Li, R. A. Sammut, and C. Pask, “Variational and finite element analyses of nonlinear strip optical waveguides,” Opt. Commun. 94, 37–43 (1992).
[Crossref]

A. Barthelemy, S. Maneuf, and C. Froehly, “Propagation soliton et autoconfinement de faisceaux laser par non-linéarité optique de Kerr,” Opt. Commun. 55, 201–206 (1985).
[Crossref]

S. Maneuf and F. Reynaud, “Quasi-steady state self-trapping of first, second and third order subnanosecond soliton beams,” Opt. Commun. 66, 325–328 (1988).
[Crossref]

Opt. Lett. (8)

G. R. Allan, S. R. Skinner, D. R. Anderson, and A. L. Smirl, “Observation of fundamental dark spatial solitons in semiconductors using picosecond pulses,” Opt. Lett. 16, 156–159 (1991).
[PubMed]

B. Luther-Davies and Y. Xiaoping, “Waveguides and Y junctions formed in bulk media by using dark spatial solitons,” Opt. Lett. 17, 496–498 (1992).
[Crossref] [PubMed]

M. Shalaby, F. Reynaud, and A. Barthelemy, “Experimental observation of spatial soliton interactions with a π/2 relative phase difference,” Opt. Lett. 17, 778–780 (1992).
[Crossref] [PubMed]

A. Barthelemy, C. Froehly, S. Maneuf, and F. Reynaud, “Experimental observation of beams’ self-deflection appearing with two-dimensional spatial soliton propagation in bulk Kerr material,” Opt. Lett. 17, 884–886 (1992).
[Crossref]

J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. M. Vogel, and P. W. E. Smith, “Observation of spatial optical solitons in a nonlinear glass waveguide,” Opt. Lett. 15, 471–473 (1990).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

D. H. Reitze, A. M. Weiner, and D. E. Leaird, “High-power femtosecond optical pulse compression by using spatial solitons,” Opt. Lett. 16, 1409–1411 (1991).
[Crossref] [PubMed]

Q. Y. Li, C. Pask, and R. A. Sammut, “Simple model for spatial optical solitons in planar waveguides,” Opt. Lett. 16, 1083–1085 (1991).
[Crossref] [PubMed]

Opt. Quantum Electron. (1)

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
[Crossref]

Prog. Quantum Electron. (2)

Y. R. Shen, “Self-focusing: experimental,” Prog. Quantum Electron. 4, 1–34 (1975).
[Crossref]

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[Crossref]

Other (7)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).

A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, Redwood City, Calif., 1992).

D. Mihalache, M. Bertolotti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. XXVII, pp. 227–313.
[Crossref]

A. D. Boardman, P. Egan, F. Lederer, U. Langbein, and D. Mihalache, “Third-order nonlinear electromagnetic TE and TM guided waves,” in Nonlinear Surface Electromagnetic Phenomena, H.-E. Ponath and G. I. Stegeman, eds. (North-Holland, Amsterdam, 1991).
[Crossref]

F. Reynaud and A. Barthelemy, “Soliton beam propagation in nonlinear Kerr media,” in Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, eds. (Kluwer, Dordrecht, The Netherlands, 1992).
[Crossref]

R. Weinstock, Calculus of Variations (Dover, New York, 1974).

R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley Interscience, New York, 1989), Vol. 1.

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Figures (7)

Fig. 1
Fig. 1

Diagram showing an elliptical beam launched into a planar, nonlinear waveguide. The linear refractive-index profile is also shown.

Fig. 2
Fig. 2

Field distribution at a power of 0.01 mW calculated by using the finite-element method. Waveguide parameters are nf = 1.57, nc = ns = 1.55, n2 = 10−9 m2/W, λ = 0.515 μm, and film thickness 2 μm.

Fig. 3
Fig. 3

Percentage difference between the exact (finite-element) solution and an approximate solution obtained by multiplying solutions along the x and y axes, calculated by using Eq. (3): (a) variation with power for films of thickness 1, 2, and 4 μm; (b) variation with film thickness at a power of 0.025 mW Other waveguide parameters are as in Fig. 2.

Fig. 4
Fig. 4

Effective index as a function of power calculated by using the finite-element method (solid curve) and the sech-linear approximation (dashed curve). Waveguide parameters are as in Fig. 2.

Fig. 5
Fig. 5

Field profiles at powers corresponding to the three points A, B, and C in Fig. 4. (a) Field variation in the y direction calculated by using the finite-element method. The curves labeled A and B are virtually indistinguishable and coincide with the field of the linear mode of a planar waveguide. (b) Field variation in the x direction calculated by using the finite-element method (solid curves) and the sech-function approximation (dashed curves).

Fig. 6
Fig. 6

As in Fig. 4 but showing the improvement obtained at high power when several different trial functions in the variational solution are used: FE, finite-element solution; SL, sech–linear; SS, sech–sech; SG, sech–Gaussian; and SN, sech–nonlinear approximation.

Fig. 7
Fig. 7

Inverse effective length as a function of film width.

Equations (44)

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n 2 ( x , y ) = n 0 2 ( y ) + α ψ 2 ( x , y ) ,
t 2 ψ + k 2 ( n 0 2 ψ + α ψ 3 ) = β 2 ψ ,
= [ ψ ex ( x , y ) - ψ ex ( x , 0 ) ψ ex ( 0 , y ) ] 2 d x d y ψ ex 2 ( x , y ) d x d y .
J = - - ( - t ψ 2 + k 2 n 0 2 ψ 2 + ½ k 2 α ψ 4 ) d x d y
- - ψ 2 d x d y = C = 2 k P β c 0 ,
ψ ( x , y ) = A X ( x / ω x ) Y ( y / ω y ) ,
J = - C X 2 ω x 2 X 2 - C Y 2 ω y 2 Y 2 + k 2 n c 2 C + α k 2 C 2 2 ω x ω y X 4 Y 4 X 2 2 Y 2 2 + 2 C k 2 ( n f 2 - n c 2 ) Y 2 0 α ω 2 y Y 2 ( u ) d u ,
A 2 = C ω x ω y X 2 Y 2 .
ω x = 4 ω y α k 2 C X 2 X 2 Y 2 2 X 4 Y 4 ,
1 - k 2 ( n f 2 - n c 2 ) a ω y Y 2 Y 2 ( a ω y ) - ( α k 2 C 4 ) 2 X 4 2 Y 4 2 X 2 3 Y 2 3 X 2 Y 2 = 0.
A ω x = ( 4 X 2 Y 2 α k 2 X 4 Y 4 ) 1 / 2 = B k ( 2 α ) 1 / 2 ,
ψ ( x , y ) = ψ L ( y ) ψ S ( x ) ,
d 2 ψ L d y 2 + k 2 n 0 2 ψ L = β L 2 ψ L .
J = - [ N 2 β L 2 X 2 - N 2 ( d X d x ) 2 + 1 2 k 2 α N 4 X 4 ] d x ,
- X 2 d x = C N 2 ,
d 2 X d x 2 + k 2 α N 4 N 2 X 3 + ( β L 2 - β 2 ) X = 0 ,
X = [ 2 ( β 2 - β L 2 ) N 2 k 2 α N 4 ] 1 / 2 sech [ ( β 2 - β L 2 ) 1 / 2 x ] ,
E = A sech ( x / ω x ) ψ L ( y ) exp [ i ( β L 2 + ω x - 2 ) 1 / 2 z - ω t ]
A sech ( x / ω x ) exp ( i z / 2 β L ω x 2 ) ψ L ( y ) exp [ i ( β L z - ω t ) ] ,
ω x = 4 N 2 2 / α C k 2 N 4
B = ( N 2 / N 4 ) 1 / 2 .
ψ ( x , y ) = A sech ( x ω x ) exp ( - y 2 2 ω y 2 ) .
ω x = 4 ω y α k 2 C 2 π ,
A ω x = 2 1 / 4 k ( 2 α ) 1 / 2 ,
1 - k 2 ( n f 2 - n c 2 ) 2 a ω y π exp [ - ( a 2 / ω y 2 ) ] - ( α k 2 C 4 3 π ) 2 = 0.
ψ ( x , y ) = A sech ( x ω x ) sech ( y ω y ) .
ω x = 12 ω y α k 2 C ,
A ω x = 1 k ( 3 α ) 1 / 2 ,
1 - 3 2 k 2 ( n f 2 - n c 2 ) a ω y sech ( a ω y ) - ( α k 2 C 12 ) 2 = 0.
ψ ( x , y ) = sech ( x ω x ) ϕ ( y )
J = 2 ω x - [ - ( d ϕ d y ) 2 + k 2 [ n 0 2 ( y ) - 1 3 k 2 ω x 2 ] ϕ 2 + 1 3 k 2 α ϕ 4 ] d y ,
C = 2 ω x - ϕ 2 d y .
d 2 ϕ d y 2 + k 2 [ n 0 2 ( y ) - 1 3 k 2 ω x 2 - β 2 + 2 α 3 ϕ 2 ] ϕ = 0.
J = β 2 C - 2 ω x k 2 3 - α ϕ 4 d y .
Ψ = ψ L ( y ) exp ( i β L z ) Φ ( x , z )
2 Ψ + k 2 [ n 0 2 ( y ) Ψ + α Ψ 2 Ψ ] = 0.
2 Φ z 2 + 2 i β L Φ z + 2 Φ x 2 + k 2 α ( N 4 N 2 ) Φ 2 Φ = 0.
2 i β L Φ z + 2 Φ x 2 + k 2 α ( N 4 N 2 ) Φ 2 Φ = 0 ,
Ψ ( x , y , z ) = η L ( y ) Φ ( x , z ) exp ( i β L z ) ,
η L ( y ) = ψ L ( y ) / N 2 1 / 2 .
2 Φ z 2 + 2 i β L Φ z + 2 Φ x 2 + k 2 α L eff - 1 Φ 2 Φ = 0
2 i β L Φ z + 2 Φ x 2 + k 2 α L eff - 1 Φ 2 Φ = 0 ,
L eff = N 2 2 N 4
P = β c 0 2 k - Φ 2 d x .

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