Abstract

A general dark-soliton solution of the Helmholtz equation (with defocusing Kerr nonlinearity) that has on- and off-axis, gray and black, paraxial and Helmholtz solitons as particular solutions, is reported. Modifications to soliton transverse velocity, width, phase period, and existence conditions are derived and explained in geometrical terms. Simulations verify analytical predictions and also demonstrate spontaneous formation of Helmholtz solitons and transparency of their interactions.

© 2003 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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  7. P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, J. Mod. Opt. 47, 1877 (2000).
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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]
  16. K. J. Blow and N. J. Doran, Phys. Lett. 107A, 55 (1985).
    [CrossRef]

2002 (1)

2001 (1)

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, Opt. Commun. 19, 1 (2001).
[CrossRef]

2000 (5)

S. Blair, Chaos 10, 570 (2000).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, J. Mod. Opt. 47, 1877 (2000).
[CrossRef]

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, Opt. Commun. 178, 431 (2000).
[CrossRef]

T. A. Laine and A. T. Friberg, J. Opt. Soc. Am. B 17, 751 (2000).
[CrossRef]

A. Ciattoni, P. Di Porto, B. Crosignani, and A. Yariv, J. Opt. Soc. Am. B 17, 809 (2000).
[CrossRef]

1999 (1)

G. I. Stegeman and M. Segev, Science 286, 1518 (1999).
[CrossRef] [PubMed]

1998 (2)

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, J. Mod. Opt. 45, 1111 (1998).
[CrossRef]

Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
[CrossRef]

1995 (1)

1993 (1)

Y. S. Kivshar, IEEE J. Quantum Electron. 29, 250 (1993).
[CrossRef]

1991 (2)

G. A. Swartzlander, D. R. Anderson, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef]

G. R. Allan, S. K. Skinner, D. R. Andersen, and A. L. Smirl, Opt. Lett. 16, 156 (1991).
[PubMed]

1985 (1)

K. J. Blow and N. J. Doran, Phys. Lett. 107A, 55 (1985).
[CrossRef]

1973 (1)

V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 64, 1627 (1973).

Allan, G. R.

Andersen, D. R.

Anderson, D. R.

G. A. Swartzlander, D. R. Anderson, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef]

Blair, S.

S. Blair, Chaos 10, 570 (2000).
[CrossRef]

Blow, K. J.

K. J. Blow and N. J. Doran, Phys. Lett. 107A, 55 (1985).
[CrossRef]

Chamorro-Posada, P.

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, J. Opt. Soc. Am. B 19, 1216 (2002).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, Opt. Commun. 19, 1 (2001).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, J. Mod. Opt. 47, 1877 (2000).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, J. Mod. Opt. 45, 1111 (1998).
[CrossRef]

Chi, S.

Ciattoni, A.

Crosignani, B.

Di Porto, P.

Doran, N. J.

K. J. Blow and N. J. Doran, Phys. Lett. 107A, 55 (1985).
[CrossRef]

Friberg, A. T.

Granot, E.

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, Opt. Commun. 178, 431 (2000).
[CrossRef]

Guo, Q.

Isbi, Y.

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, Opt. Commun. 178, 431 (2000).
[CrossRef]

Kaplan, A. E.

G. A. Swartzlander, D. R. Anderson, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef]

Kivshar, Y. S.

Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
[CrossRef]

Y. S. Kivshar, IEEE J. Quantum Electron. 29, 250 (1993).
[CrossRef]

Laine, T. A.

Lewis, A.

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, Opt. Commun. 178, 431 (2000).
[CrossRef]

Luther-Davies, B.

Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
[CrossRef]

Malomed, B.

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, Opt. Commun. 178, 431 (2000).
[CrossRef]

McDonald, G. S.

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, J. Opt. Soc. Am. B 19, 1216 (2002).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, Opt. Commun. 19, 1 (2001).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, J. Mod. Opt. 47, 1877 (2000).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, J. Mod. Opt. 45, 1111 (1998).
[CrossRef]

New, G. H. C.

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, J. Opt. Soc. Am. B 19, 1216 (2002).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, Opt. Commun. 19, 1 (2001).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, J. Mod. Opt. 47, 1877 (2000).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, J. Mod. Opt. 45, 1111 (1998).
[CrossRef]

Regan, J. J.

G. A. Swartzlander, D. R. Anderson, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef]

Segev, M.

G. I. Stegeman and M. Segev, Science 286, 1518 (1999).
[CrossRef] [PubMed]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 64, 1627 (1973).

Skinner, S. K.

Smirl, A. L.

Stegeman, G. I.

G. I. Stegeman and M. Segev, Science 286, 1518 (1999).
[CrossRef] [PubMed]

Sternklar, S.

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, Opt. Commun. 178, 431 (2000).
[CrossRef]

Swartzlander, G. A.

G. A. Swartzlander, D. R. Anderson, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef]

Yariv, A.

Yin, H.

G. A. Swartzlander, D. R. Anderson, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 64, 1627 (1973).

Chaos (1)

S. Blair, Chaos 10, 570 (2000).
[CrossRef]

IEEE J. Quantum Electron. (1)

Y. S. Kivshar, IEEE J. Quantum Electron. 29, 250 (1993).
[CrossRef]

J. Mod. Opt. (2)

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, J. Mod. Opt. 45, 1111 (1998).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, J. Mod. Opt. 47, 1877 (2000).
[CrossRef]

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

Opt. Commun. (2)

E. Granot, S. Sternklar, Y. Isbi, B. Malomed, and A. Lewis, Opt. Commun. 178, 431 (2000).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, Opt. Commun. 19, 1 (2001).
[CrossRef]

Opt. Lett. (2)

Phys. Lett. (1)

K. J. Blow and N. J. Doran, Phys. Lett. 107A, 55 (1985).
[CrossRef]

Phys. Rep. (1)

Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

G. A. Swartzlander, D. R. Anderson, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef]

Science (1)

G. I. Stegeman and M. Segev, Science 286, 1518 (1999).
[CrossRef] [PubMed]

Zh. Eksp. Teor. Fiz. (1)

V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 64, 1627 (1973).

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Figures (4)

Fig. 1
Fig. 1

Comparison of transverse profiles and geometries of paraxial and Helmholtz solitons (κ=0.001): black solitons (top) with V=25 and u0=1. F=0.8 gray solitons (bottom) with V=10 and u0=10. wθ and wθ-θ0 are the unscaled x widths.

Fig. 2
Fig. 2

Spontaneous generation and subsequent interactions of paraxial (left) and Helmholtz (right) dark solitons.

Fig. 3
Fig. 3

Normalized transverse velocities of simulated gray solitons (symbols) and the corresponding analytical predictions (curves).

Fig. 4
Fig. 4

Evolution of normalized beam widths (toward ξ0) for initially perturbed off-axis Helmholtz black solitons (horizontal lines denote analytical predictions for their asymptotic values).

Equations (11)

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

κ2uζ2+iuζ+122uξ2-u2u=0,
uξ,ζ=u0A tanh Θ+iF×expi1-4κu021+2κV21/2-Vξ+ζ2κ×exp-iζ2κ,
Θ=u0Aξ+Wζ1+2κW21/2
W=V-V01+2κVV0
V0=u0F1-2+F22κu021/2.
uξ,ζ=u0A tanhΘ+iFexp-iVξ×exp-iu02ζ-i12V2ζ,
Θ=u0Aξ+V-Fu0ζ.
ξ=ξ+Vζ1+2κV21/2,    ζ=-2κVξ+ζ1+2κV21/2,uξ,ζ=expiVξ1+2κV21/2+12κ1-11+2κV21/2ζuξ,ζ,
ξ0-1=u0A1+2κW21/2.
uξ,0=u0 tanhu0aξ.
uξ,0=u0 tanhu0ξexp-iS0ξ.

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