Abstract

Exact analytical soliton solutions of the nonlinear Helmholtz equation are reported. A lucid generalization of paraxial soliton theory that incorporates nonparaxial effects is found.

© 2002 Optical Society of America

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References

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  1. P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Non-paraxial solitons,” J. Mod. Opt. 45, 1111–1121 (1998).
    [CrossRef]
  2. P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Propagation properties of non-paraxial spatial solitons,” J. Mod. Opt. 47, 1877–1886 (2000).
    [CrossRef]
  3. P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Non-paraxial beam propagation methods,” Opt. Commun. 192, 1–12 (2001).
    [CrossRef]
  4. P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Nonparaxial solitons,” presented at the UK National Conference of Quantum Electronics, Cardiff, UK, 1997.
  5. P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Exact analytical nonparaxial optical solitons,” presented at the 1998 International Conference on Quantum Electronics, San Francisco, Calif.
  6. P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Nonparaxial solitons: accurate numerical methods and new results,” presented at the 1998 European Conference on Quantum Electronics, Glasgow, UK.
  7. P. Chamorro-Posada, G. S. McDonald, G. H. C. New, J. Noon, and S. Chávez-Cerda, “Fundamental properties of (1+1)D and (2+1)D nonparaxial optical solitons,” presented at the UK National Conference of Quantum Electronics, Manchester, UK, 1999.
  8. T. A. Laine and A. T. Friberg, “Self-guided waves and exact solutions of the nonlinear Helmholtz equation,” J. Opt. Soc. Am. B 17, 751–757 (2000).
    [CrossRef]
  9. 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]
  10. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Does the nonlinear Schrödinger equation correctly describe beam propagation?” Opt. Lett. 18, 411–413 (1993).
    [CrossRef] [PubMed]
  11. J. M. Soto-Crespo and N. Akhmediev, “Description of the self-focusing and collapse effects by a modified nonlinear Schrodinger equation,” Opt. Commun. 101, 223–230 (1993).
    [CrossRef]

2001 (1)

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Non-paraxial beam propagation methods,” Opt. Commun. 192, 1–12 (2001).
[CrossRef]

2000 (2)

T. A. Laine and A. T. Friberg, “Self-guided waves and exact solutions of the nonlinear Helmholtz equation,” J. Opt. Soc. Am. B 17, 751–757 (2000).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Propagation properties of non-paraxial spatial solitons,” J. Mod. Opt. 47, 1877–1886 (2000).
[CrossRef]

1998 (1)

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Non-paraxial solitons,” J. Mod. Opt. 45, 1111–1121 (1998).
[CrossRef]

1993 (2)

N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Does the nonlinear Schrödinger equation correctly describe beam propagation?” Opt. Lett. 18, 411–413 (1993).
[CrossRef] [PubMed]

J. M. Soto-Crespo and N. Akhmediev, “Description of the self-focusing and collapse effects by a modified nonlinear Schrodinger equation,” Opt. Commun. 101, 223–230 (1993).
[CrossRef]

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]

Akhmediev, N.

N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Does the nonlinear Schrödinger equation correctly describe beam propagation?” Opt. Lett. 18, 411–413 (1993).
[CrossRef] [PubMed]

J. M. Soto-Crespo and N. Akhmediev, “Description of the self-focusing and collapse effects by a modified nonlinear Schrodinger equation,” Opt. Commun. 101, 223–230 (1993).
[CrossRef]

Ankiewicz, A.

Chamorro-Posada, P.

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Non-paraxial beam propagation methods,” Opt. Commun. 192, 1–12 (2001).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Propagation properties of non-paraxial spatial solitons,” J. Mod. Opt. 47, 1877–1886 (2000).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Non-paraxial solitons,” J. Mod. Opt. 45, 1111–1121 (1998).
[CrossRef]

Friberg, A. T.

Laine, T. A.

McDonald, G. S.

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Non-paraxial beam propagation methods,” Opt. Commun. 192, 1–12 (2001).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Propagation properties of non-paraxial spatial solitons,” J. Mod. Opt. 47, 1877–1886 (2000).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Non-paraxial solitons,” J. Mod. Opt. 45, 1111–1121 (1998).
[CrossRef]

New, G. H. C.

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Non-paraxial beam propagation methods,” Opt. Commun. 192, 1–12 (2001).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Propagation properties of non-paraxial spatial solitons,” J. Mod. Opt. 47, 1877–1886 (2000).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Non-paraxial solitons,” J. Mod. Opt. 45, 1111–1121 (1998).
[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]

Soto-Crespo, J. M.

N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Does the nonlinear Schrödinger equation correctly describe beam propagation?” Opt. Lett. 18, 411–413 (1993).
[CrossRef] [PubMed]

J. M. Soto-Crespo and N. Akhmediev, “Description of the self-focusing and collapse effects by a modified nonlinear Schrodinger equation,” Opt. Commun. 101, 223–230 (1993).
[CrossRef]

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]

J. Mod. Opt. (2)

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Non-paraxial solitons,” J. Mod. Opt. 45, 1111–1121 (1998).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Propagation properties of non-paraxial spatial solitons,” J. Mod. Opt. 47, 1877–1886 (2000).
[CrossRef]

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

Opt. Commun. (2)

J. M. Soto-Crespo and N. Akhmediev, “Description of the self-focusing and collapse effects by a modified nonlinear Schrodinger equation,” Opt. Commun. 101, 223–230 (1993).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Non-paraxial beam propagation methods,” Opt. Commun. 192, 1–12 (2001).
[CrossRef]

Opt. Lett. (1)

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]

Other (4)

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Nonparaxial solitons,” presented at the UK National Conference of Quantum Electronics, Cardiff, UK, 1997.

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Exact analytical nonparaxial optical solitons,” presented at the 1998 International Conference on Quantum Electronics, San Francisco, Calif.

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Nonparaxial solitons: accurate numerical methods and new results,” presented at the 1998 European Conference on Quantum Electronics, Glasgow, UK.

P. Chamorro-Posada, G. S. McDonald, G. H. C. New, J. Noon, and S. Chávez-Cerda, “Fundamental properties of (1+1)D and (2+1)D nonparaxial optical solitons,” presented at the UK National Conference of Quantum Electronics, Manchester, UK, 1999.

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Equations (6)

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

2Ez2+2Ex2+k2E+γ|E|2E=0,
κ2uζ2+iuζ+122uξ2+|u|2u=0,
u(ξ, ζ)=η sechη(ξ+Vζ)1+2κV2×expi1+2κη21+2κV21/2-Vξ+ζ2κ×exp-iζ2κ,
ξ=ξ+Vζ(1+2κV2)1/2,ζ=-2κVξ+ζ(1+2κV2)1/2,
u(ξ, ζ)=expiVξ(1+2κV2)1/2+12κ1-1(1+2κV2)1/2ζu(ξ, ζ),
-+12κ+ϕ(ξ, ζ)ζ|u(ξ, ζ)|2dξ=C,

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