Abstract

We show that, in (2+1)-D case, both bright and dark solitons can exist in optical Kerr media when the optical beams are cylindrically symmetric and almost circularly polarized. We characterize the dependence of the properties associated with these solitons, such as their spatial width and intensity profiles, on their normalized intensity and the non-paraxial degree.

©2005 Optical Society of America

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References

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  1. Yu. S. Kivshar and G. P. Agrawal, “Optical solitons: From Fiber to Photonic Crystals”, Academic Press, San Diego, 2003 and references therein.
  2. M. D. Feit and J. A. Fleck, “Beam non-paraxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988).
    [Crossref]
  3. S. Chi and Q. Guo, “Vector theory of self-focusing of an optical beam in Kerr media,” Opt. Lett. 20, 1598 (1995).
    [Crossref] [PubMed]
  4. S. Blair, “Non-paraxial one-dimensional spatial solitons,” Chaos 10, 570(2000).
    [Crossref]
  5. P. Chamorro-Posada and G. S. McDonald, “Helmholtz dark solitons,” Opt. Lett. 28, 825 (2003).
    [Crossref] [PubMed]
  6. R. de la Fuente, R. Varela, and H. Michinel, “Fourier analysis of non-paraxial self-focusing,” Opt. Commun. 173, 403(2000).
    [Crossref]
  7. A. Ciattoni, P. Di. Porto, B. Crosignani, and A. Yariv, “Vectorial non-paraxial propagation equation in the presence of a tensorial refractive-index perturbation,” J. Opt. Soc. Am. B 17, 809 (2000).
    [Crossref]
  8. A. Ferrando, M. Zacarés, P. F. de Córdoba, D. Binosi, and J. A. Monsoriu, “Vortex solitons in photonic crystal fibers,” Opt. Express 12, 817 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-5-817.
    [Crossref] [PubMed]
  9. B. Crosignani, A. Yariv, and S. Mookherjea, “Non-paraxial spatial solitons and propagation-invariant pattern solutions in optical Kerr media,” Opt. Lett. 29, 1254 (2004).
    [Crossref] [PubMed]
  10. A. Ciattoni, B. Crosignani, S. Mookherjea, and A. Yariv, “Non-paraxial dark solitons in optical Kerr media,” Opt. Lett. 30, 516 (2005).
    [Crossref] [PubMed]
  11. H. Wang and W. She, “Modulation Instability and Interaction of Non-paraxial Beams in Self-focusing Kerr Media,” Opt. Commun. (2005, to be published).
    [Crossref]
  12. Y. Chen and J. Atai, “Solitary Waves of Maxwell’s Equations in Nonlinear Anisotropic Media,” J. Mod. Opt. 42, 1649 (1995).
    [Crossref]
  13. A. W. Snyder, D. J. Mitchell, and Y. Chen, “Spatial solitons of Maxwell’s equation,” Opt. Lett. 19, 524 (1994).
    [Crossref] [PubMed]
  14. A. Ciattoni, B. Crosignani, P. Di. Porto, and A. Yariv, “Azimuthally Polarized Spatial Dark Solitons: Exact Solutions of Maxwell’s Equations in a Kerr Medium,” Phys. Rev. Lett. 94, 073902 (2005).
    [Crossref] [PubMed]
  15. B. Crosignani, A. Cutolo, and P. Di Porto, “Coupled-mode theory of nonlinear propagation in multimode and singlemode fibers: envelope solitons and self-confinement,” J. Opt. Soc. Am. 72, 1136–1144 (1982).
    [Crossref]
  16. G. Fibich and B. Ilan, “Multiple Filamentation of Circularly Polarized Beams,” Phys. Rev. Lett. 89, 013901 (2002).
    [Crossref] [PubMed]

2005 (2)

A. Ciattoni, B. Crosignani, S. Mookherjea, and A. Yariv, “Non-paraxial dark solitons in optical Kerr media,” Opt. Lett. 30, 516 (2005).
[Crossref] [PubMed]

A. Ciattoni, B. Crosignani, P. Di. Porto, and A. Yariv, “Azimuthally Polarized Spatial Dark Solitons: Exact Solutions of Maxwell’s Equations in a Kerr Medium,” Phys. Rev. Lett. 94, 073902 (2005).
[Crossref] [PubMed]

2004 (2)

2003 (1)

2002 (1)

G. Fibich and B. Ilan, “Multiple Filamentation of Circularly Polarized Beams,” Phys. Rev. Lett. 89, 013901 (2002).
[Crossref] [PubMed]

2000 (3)

S. Blair, “Non-paraxial one-dimensional spatial solitons,” Chaos 10, 570(2000).
[Crossref]

R. de la Fuente, R. Varela, and H. Michinel, “Fourier analysis of non-paraxial self-focusing,” Opt. Commun. 173, 403(2000).
[Crossref]

A. Ciattoni, P. Di. Porto, B. Crosignani, and A. Yariv, “Vectorial non-paraxial propagation equation in the presence of a tensorial refractive-index perturbation,” J. Opt. Soc. Am. B 17, 809 (2000).
[Crossref]

1995 (2)

S. Chi and Q. Guo, “Vector theory of self-focusing of an optical beam in Kerr media,” Opt. Lett. 20, 1598 (1995).
[Crossref] [PubMed]

Y. Chen and J. Atai, “Solitary Waves of Maxwell’s Equations in Nonlinear Anisotropic Media,” J. Mod. Opt. 42, 1649 (1995).
[Crossref]

1994 (1)

1988 (1)

1982 (1)

Agrawal, G. P.

Yu. S. Kivshar and G. P. Agrawal, “Optical solitons: From Fiber to Photonic Crystals”, Academic Press, San Diego, 2003 and references therein.

Atai, J.

Y. Chen and J. Atai, “Solitary Waves of Maxwell’s Equations in Nonlinear Anisotropic Media,” J. Mod. Opt. 42, 1649 (1995).
[Crossref]

Binosi, D.

Blair, S.

S. Blair, “Non-paraxial one-dimensional spatial solitons,” Chaos 10, 570(2000).
[Crossref]

Chamorro-Posada, P.

Chen, Y.

Y. Chen and J. Atai, “Solitary Waves of Maxwell’s Equations in Nonlinear Anisotropic Media,” J. Mod. Opt. 42, 1649 (1995).
[Crossref]

A. W. Snyder, D. J. Mitchell, and Y. Chen, “Spatial solitons of Maxwell’s equation,” Opt. Lett. 19, 524 (1994).
[Crossref] [PubMed]

Chi, S.

Ciattoni, A.

Crosignani, B.

Cutolo, A.

de Córdoba, P. F.

de la Fuente, R.

R. de la Fuente, R. Varela, and H. Michinel, “Fourier analysis of non-paraxial self-focusing,” Opt. Commun. 173, 403(2000).
[Crossref]

Di Porto, P.

Feit, M. D.

Ferrando, A.

Fibich, G.

G. Fibich and B. Ilan, “Multiple Filamentation of Circularly Polarized Beams,” Phys. Rev. Lett. 89, 013901 (2002).
[Crossref] [PubMed]

Fleck, J. A.

Guo, Q.

Ilan, B.

G. Fibich and B. Ilan, “Multiple Filamentation of Circularly Polarized Beams,” Phys. Rev. Lett. 89, 013901 (2002).
[Crossref] [PubMed]

Kivshar, Yu. S.

Yu. S. Kivshar and G. P. Agrawal, “Optical solitons: From Fiber to Photonic Crystals”, Academic Press, San Diego, 2003 and references therein.

McDonald, G. S.

Michinel, H.

R. de la Fuente, R. Varela, and H. Michinel, “Fourier analysis of non-paraxial self-focusing,” Opt. Commun. 173, 403(2000).
[Crossref]

Mitchell, D. J.

Monsoriu, J. A.

Mookherjea, S.

Porto, P. Di.

A. Ciattoni, B. Crosignani, P. Di. Porto, and A. Yariv, “Azimuthally Polarized Spatial Dark Solitons: Exact Solutions of Maxwell’s Equations in a Kerr Medium,” Phys. Rev. Lett. 94, 073902 (2005).
[Crossref] [PubMed]

A. Ciattoni, P. Di. Porto, B. Crosignani, and A. Yariv, “Vectorial non-paraxial propagation equation in the presence of a tensorial refractive-index perturbation,” J. Opt. Soc. Am. B 17, 809 (2000).
[Crossref]

She, W.

H. Wang and W. She, “Modulation Instability and Interaction of Non-paraxial Beams in Self-focusing Kerr Media,” Opt. Commun. (2005, to be published).
[Crossref]

Snyder, A. W.

Varela, R.

R. de la Fuente, R. Varela, and H. Michinel, “Fourier analysis of non-paraxial self-focusing,” Opt. Commun. 173, 403(2000).
[Crossref]

Wang, H.

H. Wang and W. She, “Modulation Instability and Interaction of Non-paraxial Beams in Self-focusing Kerr Media,” Opt. Commun. (2005, to be published).
[Crossref]

Yariv, A.

Zacarés, M.

Chaos (1)

S. Blair, “Non-paraxial one-dimensional spatial solitons,” Chaos 10, 570(2000).
[Crossref]

J. Mod. Opt. (1)

Y. Chen and J. Atai, “Solitary Waves of Maxwell’s Equations in Nonlinear Anisotropic Media,” J. Mod. Opt. 42, 1649 (1995).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

R. de la Fuente, R. Varela, and H. Michinel, “Fourier analysis of non-paraxial self-focusing,” Opt. Commun. 173, 403(2000).
[Crossref]

Opt. Express (1)

Opt. Lett. (5)

Phys. Rev. Lett. (2)

A. Ciattoni, B. Crosignani, P. Di. Porto, and A. Yariv, “Azimuthally Polarized Spatial Dark Solitons: Exact Solutions of Maxwell’s Equations in a Kerr Medium,” Phys. Rev. Lett. 94, 073902 (2005).
[Crossref] [PubMed]

G. Fibich and B. Ilan, “Multiple Filamentation of Circularly Polarized Beams,” Phys. Rev. Lett. 89, 013901 (2002).
[Crossref] [PubMed]

Other (2)

Yu. S. Kivshar and G. P. Agrawal, “Optical solitons: From Fiber to Photonic Crystals”, Academic Press, San Diego, 2003 and references therein.

H. Wang and W. She, “Modulation Instability and Interaction of Non-paraxial Beams in Self-focusing Kerr Media,” Opt. Commun. (2005, to be published).
[Crossref]

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Figures (4)

Fig. 1.
Fig. 1. Normalized soliton profile u(ρ)/u 0 of a non-paraxial bright soliton for various values of u 0 at a 2=0.015.

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Fig. 2.
Fig. 2. Existence curve of nonpraxial bright solitons for a 2=0.015.

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Fig. 3.
Fig. 3. Non-paraxial vortex soliton solutions for various values of u ∞: (a) m=1; (b) m=2.

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Fig 4.
Fig 4. Existence curves of 2-D non-paraxial vortex solitons for m=1 (dashed curve) and m=2 (solid curve) when a 2=0.015.

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Equations (18)

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∇ × E = iω B , ∇ × B = − i ω c 2 n 0 2 E − iω μ 0 P nl .
P nl = 4 3 ε 0 n 0 n 2 [ ∣ E ∣ 2 E + 1 2 ( E · E ) E * ] ,
∇ × ∇ × E = k 2 E + 4 3 k 2 n 2 n 0 [ ∣ E ∣ 2 E + 1 2 ( E · E ) E * ] ,
∇ · E = − 4 3 n 2 n 0 ∇ · [ ∣ E ∣ 2 E + 1 2 ( E · E ) E * ] .
s = x r 0 , t = y r 0 , ξ = z ( k x 0 2 ) , U = k r 0 ∣ n 2 ∣ n 0 E ,
a 2 ∂ 2 U + ∂ ξ 2 + 2 i ∂ U + ∂ ξ + Δ ⊥ U + + 4 γ 3 ∣ U + ∣ 2 U +
= − 2 γ 3 a 2 [ 4 ( ∣ ∂ U + ∂ s ∣ 2 + ∣ ∂ U + ∂ t ∣ 2 ) U + + [ ( ∂ U + ∂ s ) 2 + ( ∂ U + ∂ t ) 2 ] U + * + ∣ U + ∣ 2 ∇ ⊥ U + + U + 2 ∇ ⊥ U + * ] ,
2 i ∂ U − ∂ ξ + Δ ⊥ U − + 8 γ 3 ∣ U + ∣ 2 U − = − 2 γ 3 a 2 ( ∂ ∂ s − i ∂ ∂ t ) [ U + ( ∂ ∂ s − i ∂ ∂ t ) ∣ U + ∣ 2 ] ,
U 3 = a 2 ( i ∂ U + ∂ s + ∂ U + ∂ t ) ,
U + ( ρ , φ , ξ ) = u ( ρ ) exp ⁡ ( iβξ ) ,
− ( 2 β + β 2 a 2 ) u + u ″ + u ′ ρ + 4 u 3 3 + 2 a 2 3 [ 5 a 2 u u ′ 2 + 2 u 2 ρ u ′ + 2 u 2 u ″ ] = 0 ,
u ( ρ ) = u 0 + 3 ρ 2 u 0 2 ( 3 + 4 a 2 u 0 2 ) ( 2 β + β 2 a 2 − 4 3 u 0 2 ) , as ρ → 0 .
U + ( ρ , φ , ξ ) = u ( ρ ) exp ⁡ ( imφ ) exp ⁡ ( iβξ ) ,
− ( 2 β + β 2 a 2 ) u + u ″ + u ′ ρ − m 2 ρ 2 u − 4 u 3 3 − 2 a 2 3 [ 5 u u ′ 2 + 2 u 2 ρ u ′ + 2 m 2 u 2 ρ 2 + 2 u 2 u ″ ] = 0 ,
u ( 0 ) = 0 , u ( ∞ ) = u ∞ , u ′ ( ∞ ) = 0 , u ″ ( ∞ ) = 0 .
u ( ρ ) = δ ρ m + O ( ρ m + 2 ) .
β = a − 2 ( − 1 ± 1 − 4 a 2 u ∞ 2 3 ) ,
u ∞ < 3 ( 4 a 2 ) ,

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