Abstract

We present a detailed analysis of the generation and propagation of bright spatial soliton beams in nonlinear Kerr media, in which an input beam is assumed to be of a Gaussian or hyperbolic secant form. The problem is solved by the use of the inverse-scattering transform (IST). The analysis of the discrete spectrum obtained from the direct-scattering problem gives exact information about the parameters of the generated soliton. A condition of soliton appearance in the spectrum as a function of the complex width of the initial Gaussian beam is given numerically. The similarities and differences between the hyperbolic secant and Gaussian beams entering the Kerr medium are analyzed in detail. A case is found in which almost all (approximately 99.5%) the total intensity of the Gaussian beam entering the Kerr medium is transformed into the soliton beam. However, this analogy to the self-trapping of soliton beams occurs for higher total-intensity values than in the case of the soliton input profile. The evolution from the Gaussian to the soliton envelope is studied and the condition of self-trapping in the near field is found. The numerical method based on the IST of the solution to the nonlinear Schrödinger equation is refined.

© 1994 Optical Society of America

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  1. Y. S. Shen, Principles of Modern Optics (Wiley-Interscience, New York, 1984), pp. 303–333.
  2. A. W. Snyder, D. J. Mitchell, L. Poladian, F. Ladouceur, “Self-induced optical fibers—spatial solitary waves,” Opt. Lett. 16, 21–23 (1991).
    [Crossref] [PubMed]
  3. J. S. Aitchison, Y. Silberberg, A. M. Weiner, D. E. Learid, M. K. Oliver, J. L. Jackel, E. M. Vogel, P. W. E. Smith, “Spatial optical solitons in planar glass waveguides,” J. Opt. Soc. Am. B 8, 1290–1297 (1991).
    [Crossref]
  4. S. Maneuf, R. Desailly, C. Froehly, “Stable self-trapping of laser beams: observation in nonlinear planar waveguide,” Opt. Commun. 65, 193–198 (1988).
    [Crossref]
  5. G. R. Allan, S. R. Skinner, D. R. Andersen, A. L. Smirl, “Observation of fundamental dark spatial solitons in semiconductors using picosecond pulses,” Opt. Lett. 16, 156–158 (1991); S. R. Skinner, G. R. Allan, D. R. Andersen, A. L. Smirl, “Dark spatial soliton propagation in bulk ZnSe,” J. Quantum Electron. 27, 2211–2219 (1991).
    [Crossref] [PubMed]
  6. G. Khitrova, H. M. Gibbs, Y. Kawamura, H. Iwamura, T. Ikegami, J. E. Sipe, L. Ming, “Spatial solitons in a self-focusing semiconductor gain medium,” Phys. Rev. Lett. 70, 920–923 (1993).
    [Crossref] [PubMed]
  7. T.-T. Shi, Sien Chi, “Nonlinear photonic switching by using the spatial soliton collision,” Opt. Lett. 15, 1123–1125 (1990).
    [Crossref] [PubMed]
  8. R. De La Fuente, A. Barthelemy, “Spatial solitons pairing by cross phase modulation,” Opt. Commun. 88, 419–423 (1992).
    [Crossref]
  9. J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, P. W. E. Smith, “Experimental observation of spatial solitons interactions,” Opt. Lett. 16, 15–17 (1991).
    [Crossref] [PubMed]
  10. B. Luther-Davies, X. P. Yang, “Waveguides and Y-junctions formed in bulk media by using dark spatial solitons,” Opt. Lett. 17, 496–498 (1992).
    [Crossref] [PubMed]
  11. E. M. Wright, ed., “All-optical switching using solitons,” Opt. Quantum Electr.24, 1215–1337 (1992).
  12. G. S. McDonald, W. J. Firth, “Spatial solitary-wave optical memory,” J. Opt. Soc. Am. B 7, 1328–1335 (1990).
    [Crossref]
  13. L. Gagnon, C. Pare, “Nonlinear radiation modes connected to parabolic graded-index profiles by the lens transformation,” J. Opt. Soc. Am. A 8, 601–607 (1991).
    [Crossref]
  14. A. Yariv, A. Yech, “The application of Gaussian beam formalism to optical propagation in nonlinear media,” Opt. Commun. 27, 295–298 (1978).
    [Crossref]
  15. A. W. Snyder, D. J. Mitchell, L. Poladin, “Linear approach for approximating spatial solitons and nonlinear guided modes,” J. Opt. Soc. Am. B 8, 1618–1620 (1991).
    [Crossref]
  16. Q. Y. Li, C. Pask, R. A. Sammut, “Simple model for spatial optical solitons in planar waveguides,” Opt. Lett. 16, 1083–1085 (1991); R. A. Sammut, C. Pask, Q. Y. Li, “Theoretical study of spatial solitons in planar waveguides,” J. Opt. Soc. Am. B 10, 485–491 (1993).
    [Crossref] [PubMed]
  17. V. E. Zakharov, 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).
  18. J. Satsuma, N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–306 (1974).
    [Crossref]
  19. H. Segur, M. J. Ablowitz, “Asymptotic solutions and conservation laws for the nonlinear Schrbdinger equations. I,” J. Math. Phys. 17, 710–713 (1976).
    [Crossref]
  20. H. Segur, “Asymptotic solutions and conservation laws for the nonlinear Schrödinger equation. II,” J. Math. Phys. 17, 714–716 (1976).
    [Crossref]
  21. D. Burak, “Gaussian beam propagation in nonlinear Kerr medium,” Opt. Appl. 21, 3–8 (1991).
  22. 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]
  23. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paralaxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [Crossref]
  24. G. J. Lamb, Elements of Soliton Theory (Mir, Moscow, 1983), pp.32–121.
  25. M. J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform (Mir, Moscow, 1987), pp. 11–111; M. J. Ablowitz, P. A. Clarcson, Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge U. Press, Cambridge, 1991), pp. 70–162.
    [Crossref]
  26. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 16, p. 722.
  27. G. Boffetta, A. R. Osborne, “Computation of the direct scattering transform for the nonlinear Schrödinger equation,” J. Computat. Phys. 102, 252–264 (1992).
    [Crossref]
  28. P. V. Frangos, D. J. Frantzeskakis, C. N. Capsalis, “Pulse propagation in an optical fibre of parabolic index profile by direct numerical solution of the Gelfand–Levitan integral equations,” Proc. Inst. Electr. Eng. 140, 141–149 (1993).
  29. P. Frangos, D. Jaggard, “A numerical solution to the Zakharov–Schabat inverse scattering problem,” IEEE Trans. Antennas Propag. 39, 74–79 (1991).
    [Crossref]
  30. T. R. Taha, M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations,” J. Computat. Phys. 55, 192–230 (1984).
    [Crossref]
  31. M. J. Ablowitz, J. F. Ladik, “Nonlinear differential-difference equations,” J. Math. Phys. 16, 598–603 (1975); “On the solution of a class of partial differential equations,” Stud. Appl. Math. 57, 1–12 (1977).
    [Crossref]
  32. M. J. Ablowitz, J. F. Ladik, “Nonlinear differential-difference equations and Fourier analysis,” J. Math. Phys. 17, 1011–1018 (1976); “A nonlinear difference scheme and inverse scattering,” Stud. Appl. Math. 55, 213–229 (1976).
    [Crossref]

1993 (2)

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

P. V. Frangos, D. J. Frantzeskakis, C. N. Capsalis, “Pulse propagation in an optical fibre of parabolic index profile by direct numerical solution of the Gelfand–Levitan integral equations,” Proc. Inst. Electr. Eng. 140, 141–149 (1993).

1992 (3)

G. Boffetta, A. R. Osborne, “Computation of the direct scattering transform for the nonlinear Schrödinger equation,” J. Computat. Phys. 102, 252–264 (1992).
[Crossref]

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

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

1991 (9)

J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, P. W. E. Smith, “Experimental observation of spatial solitons interactions,” Opt. Lett. 16, 15–17 (1991).
[Crossref] [PubMed]

A. W. Snyder, D. J. Mitchell, L. Poladian, F. Ladouceur, “Self-induced optical fibers—spatial solitary waves,” Opt. Lett. 16, 21–23 (1991).
[Crossref] [PubMed]

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

L. Gagnon, C. Pare, “Nonlinear radiation modes connected to parabolic graded-index profiles by the lens transformation,” J. Opt. Soc. Am. A 8, 601–607 (1991).
[Crossref]

A. W. Snyder, D. J. Mitchell, L. Poladin, “Linear approach for approximating spatial solitons and nonlinear guided modes,” J. Opt. Soc. Am. B 8, 1618–1620 (1991).
[Crossref]

Q. Y. Li, C. Pask, R. A. Sammut, “Simple model for spatial optical solitons in planar waveguides,” Opt. Lett. 16, 1083–1085 (1991); R. A. Sammut, C. Pask, Q. Y. Li, “Theoretical study of spatial solitons in planar waveguides,” J. Opt. Soc. Am. B 10, 485–491 (1993).
[Crossref] [PubMed]

G. R. Allan, S. R. Skinner, D. R. Andersen, A. L. Smirl, “Observation of fundamental dark spatial solitons in semiconductors using picosecond pulses,” Opt. Lett. 16, 156–158 (1991); S. R. Skinner, G. R. Allan, D. R. Andersen, A. L. Smirl, “Dark spatial soliton propagation in bulk ZnSe,” J. Quantum Electron. 27, 2211–2219 (1991).
[Crossref] [PubMed]

D. Burak, “Gaussian beam propagation in nonlinear Kerr medium,” Opt. Appl. 21, 3–8 (1991).

P. Frangos, D. Jaggard, “A numerical solution to the Zakharov–Schabat inverse scattering problem,” IEEE Trans. Antennas Propag. 39, 74–79 (1991).
[Crossref]

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

G. S. McDonald, W. J. Firth, “Spatial solitary-wave optical memory,” J. Opt. Soc. Am. B 7, 1328–1335 (1990).
[Crossref]

1988 (1)

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

1984 (1)

T. R. Taha, M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations,” J. Computat. Phys. 55, 192–230 (1984).
[Crossref]

1978 (1)

A. Yariv, A. Yech, “The application of Gaussian beam formalism to optical propagation in nonlinear media,” Opt. Commun. 27, 295–298 (1978).
[Crossref]

1976 (3)

H. Segur, M. J. Ablowitz, “Asymptotic solutions and conservation laws for the nonlinear Schrbdinger equations. I,” J. Math. Phys. 17, 710–713 (1976).
[Crossref]

H. Segur, “Asymptotic solutions and conservation laws for the nonlinear Schrödinger equation. II,” J. Math. Phys. 17, 714–716 (1976).
[Crossref]

M. J. Ablowitz, J. F. Ladik, “Nonlinear differential-difference equations and Fourier analysis,” J. Math. Phys. 17, 1011–1018 (1976); “A nonlinear difference scheme and inverse scattering,” Stud. Appl. Math. 55, 213–229 (1976).
[Crossref]

1975 (2)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paralaxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

M. J. Ablowitz, J. F. Ladik, “Nonlinear differential-difference equations,” J. Math. Phys. 16, 598–603 (1975); “On the solution of a class of partial differential equations,” Stud. Appl. Math. 57, 1–12 (1977).
[Crossref]

1974 (1)

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

1971 (1)

V. E. Zakharov, 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).

Ablowitz, M. J.

T. R. Taha, M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations,” J. Computat. Phys. 55, 192–230 (1984).
[Crossref]

H. Segur, M. J. Ablowitz, “Asymptotic solutions and conservation laws for the nonlinear Schrbdinger equations. I,” J. Math. Phys. 17, 710–713 (1976).
[Crossref]

M. J. Ablowitz, J. F. Ladik, “Nonlinear differential-difference equations and Fourier analysis,” J. Math. Phys. 17, 1011–1018 (1976); “A nonlinear difference scheme and inverse scattering,” Stud. Appl. Math. 55, 213–229 (1976).
[Crossref]

M. J. Ablowitz, J. F. Ladik, “Nonlinear differential-difference equations,” J. Math. Phys. 16, 598–603 (1975); “On the solution of a class of partial differential equations,” Stud. Appl. Math. 57, 1–12 (1977).
[Crossref]

M. J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform (Mir, Moscow, 1987), pp. 11–111; M. J. Ablowitz, P. A. Clarcson, Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge U. Press, Cambridge, 1991), pp. 70–162.
[Crossref]

Aitchison, J. S.

Allan, G. R.

Andersen, D. R.

Barthelemy, A.

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

Boffetta, G.

G. Boffetta, A. R. Osborne, “Computation of the direct scattering transform for the nonlinear Schrödinger equation,” J. Computat. Phys. 102, 252–264 (1992).
[Crossref]

Burak, D.

D. Burak, “Gaussian beam propagation in nonlinear Kerr medium,” Opt. Appl. 21, 3–8 (1991).

Capsalis, C. N.

P. V. Frangos, D. J. Frantzeskakis, C. N. Capsalis, “Pulse propagation in an optical fibre of parabolic index profile by direct numerical solution of the Gelfand–Levitan integral equations,” Proc. Inst. Electr. Eng. 140, 141–149 (1993).

Chi, Sien

De La Fuente, R.

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

Desailly, R.

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

Firth, W. J.

Flannery, B. P.

H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 16, p. 722.

Frangos, P.

P. Frangos, D. Jaggard, “A numerical solution to the Zakharov–Schabat inverse scattering problem,” IEEE Trans. Antennas Propag. 39, 74–79 (1991).
[Crossref]

Frangos, P. V.

P. V. Frangos, D. J. Frantzeskakis, C. N. Capsalis, “Pulse propagation in an optical fibre of parabolic index profile by direct numerical solution of the Gelfand–Levitan integral equations,” Proc. Inst. Electr. Eng. 140, 141–149 (1993).

Frantzeskakis, D. J.

P. V. Frangos, D. J. Frantzeskakis, C. N. Capsalis, “Pulse propagation in an optical fibre of parabolic index profile by direct numerical solution of the Gelfand–Levitan integral equations,” Proc. Inst. Electr. Eng. 140, 141–149 (1993).

Froehly, C.

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

Gagnon, L.

Gibbs, H. M.

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

Ikegami, T.

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

Iwamura, H.

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

Jackel, J. L.

Jaggard, D.

P. Frangos, D. Jaggard, “A numerical solution to the Zakharov–Schabat inverse scattering problem,” IEEE Trans. Antennas Propag. 39, 74–79 (1991).
[Crossref]

Kawamura, Y.

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

Ladik, J. F.

M. J. Ablowitz, J. F. Ladik, “Nonlinear differential-difference equations and Fourier analysis,” J. Math. Phys. 17, 1011–1018 (1976); “A nonlinear difference scheme and inverse scattering,” Stud. Appl. Math. 55, 213–229 (1976).
[Crossref]

M. J. Ablowitz, J. F. Ladik, “Nonlinear differential-difference equations,” J. Math. Phys. 16, 598–603 (1975); “On the solution of a class of partial differential equations,” Stud. Appl. Math. 57, 1–12 (1977).
[Crossref]

Ladouceur, F.

Lamb, G. J.

G. J. Lamb, Elements of Soliton Theory (Mir, Moscow, 1983), pp.32–121.

Lax, M.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paralaxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

Leaird, D. E.

Learid, D. E.

Li, Q. Y.

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paralaxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

Luther-Davies, B.

Maneuf, S.

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

McDonald, G. S.

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paralaxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

Ming, L.

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

Mitchell, D. J.

Nasalski, W.

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.

Osborne, A. R.

G. Boffetta, A. R. Osborne, “Computation of the direct scattering transform for the nonlinear Schrödinger equation,” J. Computat. Phys. 102, 252–264 (1992).
[Crossref]

Pare, C.

Pask, C.

Poladian, L.

Poladin, L.

Press, H.

H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 16, p. 722.

Sammut, R. A.

Satsuma, J.

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

Segur, H.

H. Segur, M. J. Ablowitz, “Asymptotic solutions and conservation laws for the nonlinear Schrbdinger equations. I,” J. Math. Phys. 17, 710–713 (1976).
[Crossref]

H. Segur, “Asymptotic solutions and conservation laws for the nonlinear Schrödinger equation. II,” J. Math. Phys. 17, 714–716 (1976).
[Crossref]

M. J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform (Mir, Moscow, 1987), pp. 11–111; M. J. Ablowitz, P. A. Clarcson, Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge U. Press, Cambridge, 1991), pp. 70–162.
[Crossref]

Shabat, A. B.

V. E. Zakharov, 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).

Shen, Y. S.

Y. S. Shen, Principles of Modern Optics (Wiley-Interscience, New York, 1984), pp. 303–333.

Shi, T.-T.

Silberberg, Y.

Sipe, J. E.

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

Skinner, S. R.

Smirl, A. L.

Smith, P. W. E.

Snyder, A. W.

Taha, T. R.

T. R. Taha, M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations,” J. Computat. Phys. 55, 192–230 (1984).
[Crossref]

Teukolsky, S. A.

H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 16, p. 722.

Vetterling, W. T.

H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 16, p. 722.

Vogel, E. M.

Weiner, A. M.

Yajima, N.

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

Yang, X. P.

Yariv, A.

A. Yariv, A. Yech, “The application of Gaussian beam formalism to optical propagation in nonlinear media,” Opt. Commun. 27, 295–298 (1978).
[Crossref]

Yech, A.

A. Yariv, A. Yech, “The application of Gaussian beam formalism to optical propagation in nonlinear media,” Opt. Commun. 27, 295–298 (1978).
[Crossref]

Zakharov, V. E.

V. E. Zakharov, 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).

IEEE Trans. Antennas Propag. (1)

P. Frangos, D. Jaggard, “A numerical solution to the Zakharov–Schabat inverse scattering problem,” IEEE Trans. Antennas Propag. 39, 74–79 (1991).
[Crossref]

J. Computat. Phys. (2)

T. R. Taha, M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations,” J. Computat. Phys. 55, 192–230 (1984).
[Crossref]

G. Boffetta, A. R. Osborne, “Computation of the direct scattering transform for the nonlinear Schrödinger equation,” J. Computat. Phys. 102, 252–264 (1992).
[Crossref]

J. Math. Phys. (4)

M. J. Ablowitz, J. F. Ladik, “Nonlinear differential-difference equations,” J. Math. Phys. 16, 598–603 (1975); “On the solution of a class of partial differential equations,” Stud. Appl. Math. 57, 1–12 (1977).
[Crossref]

M. J. Ablowitz, J. F. Ladik, “Nonlinear differential-difference equations and Fourier analysis,” J. Math. Phys. 17, 1011–1018 (1976); “A nonlinear difference scheme and inverse scattering,” Stud. Appl. Math. 55, 213–229 (1976).
[Crossref]

H. Segur, M. J. Ablowitz, “Asymptotic solutions and conservation laws for the nonlinear Schrbdinger equations. I,” J. Math. Phys. 17, 710–713 (1976).
[Crossref]

H. Segur, “Asymptotic solutions and conservation laws for the nonlinear Schrödinger equation. II,” J. Math. Phys. 17, 714–716 (1976).
[Crossref]

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

J. Opt. Soc. Am. B (3)

Opt. Appl. (1)

D. Burak, “Gaussian beam propagation in nonlinear Kerr medium,” Opt. Appl. 21, 3–8 (1991).

Opt. Commun. (4)

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]

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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

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Figures (5)

Fig. 1
Fig. 1

Total intensities thresholds for soliton generation (2η = 0) versus ζ i for the incident Gaussian beam (solid curve), hyperbolic-secant beam (short-dashed curve), and ray-potential theory predictions (long-dashed curve).

Fig. 2
Fig. 2

The field distributions in the incident plane ζ = 0 of Gaussian (solid line) and hyperbolic-secant (short-dashed curve) beams and the soliton field distribution in the far-field plane (long-dashed curve) for different soliton amplitudes: (a) 2η = 1/2 (q 0 = 1.1, A = 3/4), (b) 2η = 1 (q 0 = 1.3, A = 1), (c) 2η = 1.77 (q 0 = 1.69, A = 1.38). Plots (c), (d), and (e) show the total intensities I (tot) of Gaussian and hyperbolic-secant beams necessary for soliton excitation with the certain amplitude 2η.

Fig. 3
Fig. 3

Field amplitude distribution during Gaussian to soliton beam formation; parameters are the same as in Fig. 2(c).

Fig. 4
Fig. 4

(a) Ratio 4η/I (inc) versus I ( inc ) = π / 2 q 0 2 for Gaussian (solid curve) and I (inc) = 2A 2 for hyperbolic-secant (dashed curve) input profiles. (b) Soliton amplitude 2η versus the incident Gaussian beam amplitude q 0 (solid); for ζ i = 0, 2η ≈ 2.13q 0 − 1.84, for ζ i = ±1, 2η ≈ 1.89q 0 − 1.70, for ζ i = ±2, 2η ≈ 1.67q 0 − 1.66. The short-dashed curve is the approximate value of 2η as calculated from Eq. (3). Long-dashed curve is from relation (22).

Fig. 5
Fig. 5

Propagation of the central part of the beam |q(0, ζ) in a nonlinear Kerr medium for (a) different amplitudes of the collimated (ζ i = 0) incident Gaussian beam: curve (1) for q 0 = 2.0 and 2η = 2.4, curve (2) for q 0 = 1.609 and 2η = 1.609, curve (3) for q 0 = 1.13 and 2η = 0.94, and curve (4) for q 0 = 1.0 and 2η = 0.26. (b) the same for different ζ i of an incident Gaussian beam with amplitude q 0 = 1.609: solid curves represent ζ i = 0.5 and ζ i = 1, short-dashed curves represent ζ i = −0.5 and ζ i = −1, and the long-dashed curve represents ζ i = 0.

Equations (37)

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q ( x , ζ ) = 2 η exp ( i η 2 ζ ) sech ( 2 η x ) ,
v ( x ) = v 0 exp ( - x 2 w 0 2 ) ,
2 η = - 2 π cot [ δ ( ω 0 , v 0 ) ] w 0 δ ( w 0 , v 0 ) ,
δ ( w 0 , v 0 ) = π w 0 v 0 ,
2 E z 2 + 2 E x 2 + k 2 E = 0 ,
2 i β V z + 2 V x 2 + ( k 2 - β 2 ) V = 0 ,
U ( x , ζ ) = u 0 v ( ζ ) exp [ - x 2 v 2 ( ζ ) ] ,
4 i q ζ + 2 q x 2 + 2 q 2 q = 0 ,
χ = ( n 0 k 0 ) 2 n 2 / n 0 w 0 ,
q ( x , ζ ) = - 2 K ( x , x , ζ ) ,
K ( x , y , ζ ) + x + K ( x , z , ζ ) x + F ( z + s , ζ ) × F * ( s + y , ζ ) d s d z = F * ( x + y , ζ ) ,
F ( x , ζ ) = 1 2 π - + r ( λ , 0 ) exp ( i λ x + i λ 2 ζ ) d λ - i c ( λ 1 , 0 ) exp ( i λ 1 x + i λ 1 2 ζ ) .
Ψ 1 x + i λ Ψ 1 = q 0 ( x ) Ψ 2 , Ψ 2 x - i λ Ψ 2 = - q 0 * ( x ) Ψ 1 .
[ Ψ 1 ( x , λ ) Ψ 2 ( x , λ ) ] = ( 0 1 ) exp ( i λ x ) ,
[ Ψ 1 ( x , λ ) Ψ 2 ( x , λ ) ] = [ b * ( λ * ) exp ( - i λ * x ) a ( λ ) exp ( i λ x ) ] ,
c ( η , 0 ) = - 1 2 i - + Ψ 1 ( x , η ) , Ψ 2 ( x , η ) d x .
q 0 ( x , ζ i ) = χ U ( x , ζ i ) = q 0 v ( ζ i ) exp [ - x 2 v 2 ( ζ i ) ] ,
K i j ( m ) ( ζ ) + l = i N ( m ) - 1 [ x l + 1 ( m ) - x l ( m ) ] f l j ( ζ ) K i l ( m ) ( ζ ) = F * [ x i ( m ) + x j ( m ) , ζ ] , i = 1 , 2 , , N ( m ) - 1 , K N ( m ) N ( m ) ( m ) ( ζ ) = F * [ 2 x N ( m ) ( m ) , ζ ] ,
f l j ( ζ ) = x i ( m ) + F [ x l ( m ) + s , ζ ] F * [ s + x j ( m ) , ζ ] d s ,
η ( q 0 , ζ i ) = 0 ,
I ( th ) = - + q ( x , ζ i ) 2 d x = ( π ) 1 / 2 ( 2 ) 1 / 2 q 0 ( th ) ( ζ i ) 2 .
I ( rp ) ( th ) = [ π 2 ( ζ i 2 + 1 ) ] 1 / 2 ,
q ( x ) = A sech ( x ) ,
2 η = 2 A - 1 ,             A 1 / 2 ,
I ( sech ) ( th ) = 2 [ A ( th ) ] 2 = 1 / 2.
Ψ 1 x * - i λ * Ψ 1 * = q 0 * ( x , ζ i ) Ψ 2 * , Ψ 2 x * + i λ * Ψ 2 * = - q 0 ( x , ζ i ) Ψ 1 * ,
[ φ 1 ( x , λ ) φ 2 ( x , λ ) ] = n = 0 λ n [ exp ( - i λ x ) f n ( x ) exp ( i λ x ) g n ( x ) ] ,
[ φ 1 ( x , λ ) φ 2 ( x , λ ) ] = ( 1 0 ) exp ( - i λ x ) .
[ φ 1 ( x , λ ) φ 2 ( x , λ ) ] = [ a ( λ ) exp ( i λ x ) b ( λ ) exp ( i λ x ) ] .
f 0 ( x ) = cos δ ( x ) , g 0 ( x ) = - sin δ ( x ) , [ f n ( x ) g n ( x ) ] = W ( x ) - x W - 1 ( s ) b n ( s ) d s ,             n 1 ,
b n ( x ) = k = 1 n ( i ) k γ k ( x ) [ g n - k ( x ) ( - 1 ) k + 1 f n - k ( x ) ] ,             n 1 ,
W ( x ) = [ cos δ ( x ) sin δ ( x ) - sin δ ( x ) cos δ ( x ) ] ,
γ k ( x ) = q 0 ( x ) ( 2 x ) k k ! ,
δ ( x ) = - x q 0 ( s ) d s ,
η = - cos δ A cos δ - B sin δ ,
A = - + 2 x q 0 ( x ) sin [ 2 δ ( x ) ] d x , B = - + 2 x q 0 ( x ) cos [ 2 δ ( x ) ] d x , δ δ ( ) = π w 0 v 0 .
- x exp [ - ( s / w 0 ) 2 ] d s 2 - x exp [ - 2 ( s / w 0 ) 2 ] d s ,

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