Abstract

Recent generalizations of the standard nonlinear Schroedinger equation (NLSE), aimed at describing nonparaxial prop-agation in Kerr media are examined. An analysis of their limitations, based on available exact results for transverse electric (TE) and trans-verse magnetic (TM) (1+1)-D spatial solitons, is presented. Numerical stability analysis reveals that nonparaxial TM soltions are unstable to perturbations and tend to catastrophically collapse while TE solitons are stable even in the extreme nonparaxial limit.

©2006 Optical Society of America

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References

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  1. R. de la Fuente, O. Varela, and H. Michenel, “Fourier analysis of non-paraxial self-focusing,” Opt.Commun. 173, 403–410 (2000).
    [Crossref]
  2. A. Ciattoni et al., “Polarization and energy dynamics in ultra focused optical Kerr propagation,” Opt.Lett. 27, 734–736 (2002).
    [Crossref]
  3. G. Fibich and B. Ilan, “Vectorial and random effects in self-focusing and in multiple filamenta-tions,” Physica D 157, 112–146 (2001).
    [Crossref]
  4. M. Matuszewski et al., “Self-consistent treatment of the full nonlinear opical pulse propagation equation in an isotropic medium,” Opt.Commun. 221, 337–351 (2003).
    [Crossref]
  5. Timo A. Laine and Ari T. Friberg, “Self-guided waves and exact solutions of the nonlinear Schroedinger equation,” J.Opt.Soc.Am. B 17,751–757 (2000).
    [Crossref]
  6. Ciattoni et al., “Perfect optical solitons: spatial Kerr solitons as exact solutions of Maxwell’s equations,” J.Opt.Soc.Am. B 22, 1384–1394 (2005).
    [Crossref]
  7. A. Ciattoni et al., “Nonparaxial dark solitons in optical Kerr media,” Opt. Lett. 30, 516–518 (2005).
    [Crossref] [PubMed]
  8. M. Lax, W.H. Louisell, and W.B. McKnight, “From Maxwell to paraxial wave optics,” Phys.Rev. A 11, 1365–1370 (1975).
    [Crossref]
  9. A. Taflove and S. C. Hagness, Computational electrodynamics : the finite-difference time-domain method, Second Edition (Artech House, Boston, 2000).

2005 (2)

Ciattoni et al., “Perfect optical solitons: spatial Kerr solitons as exact solutions of Maxwell’s equations,” J.Opt.Soc.Am. B 22, 1384–1394 (2005).
[Crossref]

A. Ciattoni et al., “Nonparaxial dark solitons in optical Kerr media,” Opt. Lett. 30, 516–518 (2005).
[Crossref] [PubMed]

2003 (1)

M. Matuszewski et al., “Self-consistent treatment of the full nonlinear opical pulse propagation equation in an isotropic medium,” Opt.Commun. 221, 337–351 (2003).
[Crossref]

2002 (1)

A. Ciattoni et al., “Polarization and energy dynamics in ultra focused optical Kerr propagation,” Opt.Lett. 27, 734–736 (2002).
[Crossref]

2001 (1)

G. Fibich and B. Ilan, “Vectorial and random effects in self-focusing and in multiple filamenta-tions,” Physica D 157, 112–146 (2001).
[Crossref]

2000 (2)

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

R. de la Fuente, O. Varela, and H. Michenel, “Fourier analysis of non-paraxial self-focusing,” Opt.Commun. 173, 403–410 (2000).
[Crossref]

1975 (1)

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

Ciattoni,

Ciattoni et al., “Perfect optical solitons: spatial Kerr solitons as exact solutions of Maxwell’s equations,” J.Opt.Soc.Am. B 22, 1384–1394 (2005).
[Crossref]

Ciattoni, A.

A. Ciattoni et al., “Nonparaxial dark solitons in optical Kerr media,” Opt. Lett. 30, 516–518 (2005).
[Crossref] [PubMed]

A. Ciattoni et al., “Polarization and energy dynamics in ultra focused optical Kerr propagation,” Opt.Lett. 27, 734–736 (2002).
[Crossref]

de la Fuente, R.

R. de la Fuente, O. Varela, and H. Michenel, “Fourier analysis of non-paraxial self-focusing,” Opt.Commun. 173, 403–410 (2000).
[Crossref]

Fibich, G.

G. Fibich and B. Ilan, “Vectorial and random effects in self-focusing and in multiple filamenta-tions,” Physica D 157, 112–146 (2001).
[Crossref]

Friberg, Ari T.

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

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational electrodynamics : the finite-difference time-domain method, Second Edition (Artech House, Boston, 2000).

Ilan, B.

G. Fibich and B. Ilan, “Vectorial and random effects in self-focusing and in multiple filamenta-tions,” Physica D 157, 112–146 (2001).
[Crossref]

Laine, Timo A.

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

Lax, M.

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

Louisell, W.H.

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

Matuszewski, M.

M. Matuszewski et al., “Self-consistent treatment of the full nonlinear opical pulse propagation equation in an isotropic medium,” Opt.Commun. 221, 337–351 (2003).
[Crossref]

McKnight, W.B.

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

Michenel, H.

R. de la Fuente, O. Varela, and H. Michenel, “Fourier analysis of non-paraxial self-focusing,” Opt.Commun. 173, 403–410 (2000).
[Crossref]

Taflove, A.

A. Taflove and S. C. Hagness, Computational electrodynamics : the finite-difference time-domain method, Second Edition (Artech House, Boston, 2000).

Varela, O.

R. de la Fuente, O. Varela, and H. Michenel, “Fourier analysis of non-paraxial self-focusing,” Opt.Commun. 173, 403–410 (2000).
[Crossref]

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

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

Ciattoni et al., “Perfect optical solitons: spatial Kerr solitons as exact solutions of Maxwell’s equations,” J.Opt.Soc.Am. B 22, 1384–1394 (2005).
[Crossref]

Opt. Lett. (1)

Opt.Commun. (2)

M. Matuszewski et al., “Self-consistent treatment of the full nonlinear opical pulse propagation equation in an isotropic medium,” Opt.Commun. 221, 337–351 (2003).
[Crossref]

R. de la Fuente, O. Varela, and H. Michenel, “Fourier analysis of non-paraxial self-focusing,” Opt.Commun. 173, 403–410 (2000).
[Crossref]

Opt.Lett. (1)

A. Ciattoni et al., “Polarization and energy dynamics in ultra focused optical Kerr propagation,” Opt.Lett. 27, 734–736 (2002).
[Crossref]

Phys.Rev. A (1)

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

Physica D (1)

G. Fibich and B. Ilan, “Vectorial and random effects in self-focusing and in multiple filamenta-tions,” Physica D 157, 112–146 (2001).
[Crossref]

Other (1)

A. Taflove and S. C. Hagness, Computational electrodynamics : the finite-difference time-domain method, Second Edition (Artech House, Boston, 2000).

Supplementary Material (2)

» Media 1: AVI (2100 KB)     
» Media 2: AVI (2429 KB)     

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Figures (2)

Fig. 1.
Fig. 1.

BPM simulations of the solitons solutions of a) TE-equation (12); b) TE-equation (13); c) TM-equation (14); d) TM-equation (15).

Fig. 2.
Fig. 2.

(a) (2.04 MB) Movie of FDTD simulation showing the stable propagation of a TE polarized nonparaxial soliton with initial amplitude of A=0.4. (b) (2.37 MB) Movie of FDTD simulation showing the unstable propagation of a TM polarized nonparaxial soliton with initial amplitude of A=0.4

Equations (22)

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

E ( x , y , z , t ) = A ( x , y , z ) exp ( ikz i ω t ) ,
( i z + 1 2k 2 ) A = 2k 3 n 2 n 0 | A | 2 A 1 3k n 2 n 0 ( A A ) A * ,
( i ζ + 1 2 2 ξ 2 ) U = γ U 2 U ,
( i ζ + 1 2 2 ξ 2 ) U = γ [ U 2 U + a U 2 2 U ξ 2 + b U 2 2 U * ξ 2
+ c U ξ 2 U + d ( U ξ ) 2 U * ] ,
β u + 1 2 u = γ ( u 3 + Au 2 u + Buu 2 ) ,
df du + 4 γ Bu 1 + 2 γ Au 2 f = 4 u β γ u 2 1 + 2 γ Au 2 .
f ( u ) = γ B [ β + 1 2 ( α + 1 ) A ] u 2 ( α + 1 ) A
+ { f ( u 0 ) γ B [ β + 1 2 ( α + 1 ) A ] + u 0 2 ( α + 1 ) A } ( 1 + 2 γ Au 0 2 ) α ( 1 + 2 γ Au 2 ) α + 1
u = u ( α + 1 ) A 2 γ Bu [ f ( u 0 )
γ B ( β + 1 2 ( α + 1 ) A ) + u 0 2 ( α + 1 ) A ] ( 1 + 2 γ A u 0 2 ) α ( 1 + 2 γ Au 2 ) α + 1
β = α α + 1 [ u 0 2 ( 1 + 2 Au 0 2 ) α ( 1 + 2 Au 0 2 ) α 1 1 2 B ] 1 2 u 0 2 + 1 6 ( B A ) u 0 4 +
β = u 2 ,
( i ζ + 1 2 2 ξ 2 ) u = γ [ u 2 u + 1 2 2 ξ 2 ( u 2 u ) ] ,
i u ζ + 1 2 u = γ ( u 2 u + u 2 u + 1 2 u 2 u * + 2 u u 2 + u * u 2 ) .
i u ζ + 1 2 u = γ ( u 2 u + u 2 u + u u 2 + 1 2 u * u 2 ) .
i u ζ + 1 2 u = γ ( u 2 u + 1 3 u 2 u + 5 6 u 2 u * + 8 3 u u 2 ) ,
i u ζ + 1 2 u = γ ( u 2 u + 13 12 u 2 u + 4 3 u 2 u * + 11 3 u u 2 + 1 2 u * u 2 ) .
β = 1 + 1 + u 0 2 1 2 u 0 2 1 8 u 0 4 + ,
β = 1 + 1 2 u 2 u 2 1 2 u 4 + ,
β = 1 + 2 u 0 2 1 + 3 u 0 2 1 = 1 2 + 3 8 u 0 4 + ,
β = 1 + 1 2 u 2 = u 2 1 2 u 4 +

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