Abstract

The refraction of dark solitons at a planar boundary separating two defocusing Kerr media is simulated and analyzed, for the first time (to our knowledge). Analysis is based on the nonlinear Helmholtz equation and is thus valid for any angle of incidence. A new law, governing refraction of black solitons, is combined with one describing bright soliton refraction to yield a generalized Snell’s law whose validity is verified numerically. The complexity of gray soliton refraction is also analyzed, and illustrated by a change from external to internal refraction on varying the soliton contrast parameter.

© 2010 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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2009 (1)

J. Sánchez-Curto, P. Chamorro-Posada, and G. S. McDonald, J. Opt. A 11, 054015 (2009).
[CrossRef]

2007 (2)

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, Phys. Rev. A 76, 033833 (2007).
[CrossRef]

J. Sánchez-Curto, P. Chamorro-Posada, and G. S. McDonald, Opt. Lett. 32, 1126 (2007).
[CrossRef] [PubMed]

2006 (2)

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, Phys. Rev. E 74, 066612 (2006).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Express 14, 12365 (2006).
[CrossRef] [PubMed]

2005 (2)

2003 (2)

2002 (2)

2001 (1)

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, Opt. Commun. 192, 1 (2001).
[CrossRef]

2000 (1)

A. D. Boardman, P. Bontemps, W. Ilecki, and A. A. Zharov, J. Mod. Opt. 47, 1941 (2000).

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]

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

1996 (2)

G. Fibich, Phys. Rev. Lett. 76, 4356 (1996).
[CrossRef] [PubMed]

P. J. Bradley and C. De Angelis, Opt. Commun. 130, 205 (1996).
[CrossRef]

1995 (2)

G. Cancellieri, F. Chiaraluce, E. Gambi, and P. Pierleoni, J. Opt. Soc. Am. B 12, 1300 (1995).
[CrossRef]

M. D. Iturbe-Castillo, J. J. Sánchez-Mondragón, S. I. Stepanov, M. B. Klein, and B. A. Wechsler, Opt. Commun. 118, 515 (1995).
[CrossRef]

1993 (1)

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

1992 (2)

1991 (2)

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

S. R. Skinner and D. R. Andersen, J. Opt. Soc. Am. B 8, 759 (1991).
[CrossRef]

1990 (1)

1989 (3)

D. Mihalache, M. Bertolotti, and C. Sibilia, Prog. Opt. 27, 227 (1989).
[CrossRef]

A. B. Aceves, J. V. Moloney, and A. C. Newell, Phys. Rev. A 39, 1828 (1989).
[CrossRef] [PubMed]

A. B. Aceves, J. V. Moloney, and A. C. Newell, Phys. Rev. A 39, 1809 (1989).
[CrossRef] [PubMed]

Aceves, A. B.

A. B. Aceves, J. V. Moloney, and A. C. Newell, Phys. Rev. A 39, 1828 (1989).
[CrossRef] [PubMed]

A. B. Aceves, J. V. Moloney, and A. C. Newell, Phys. Rev. A 39, 1809 (1989).
[CrossRef] [PubMed]

Andersen, D. R.

Bertolotti, M.

D. Mihalache, M. Bertolotti, and C. Sibilia, Prog. Opt. 27, 227 (1989).
[CrossRef]

Boardman, A. D.

A. D. Boardman, P. Bontemps, W. Ilecki, and A. A. Zharov, J. Mod. Opt. 47, 1941 (2000).

Bontemps, P.

A. D. Boardman, P. Bontemps, W. Ilecki, and A. A. Zharov, J. Mod. Opt. 47, 1941 (2000).

Bradley, P. J.

P. J. Bradley and C. De Angelis, Opt. Commun. 130, 205 (1996).
[CrossRef]

Cancellieri, G.

Chamorro-Posada, P.

J. Sánchez-Curto, P. Chamorro-Posada, and G. S. McDonald, J. Opt. A 11, 054015 (2009).
[CrossRef]

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, Phys. Rev. A 76, 033833 (2007).
[CrossRef]

J. Sánchez-Curto, P. Chamorro-Posada, and G. S. McDonald, Opt. Lett. 32, 1126 (2007).
[CrossRef] [PubMed]

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, Phys. Rev. E 74, 066612 (2006).
[CrossRef]

P. Chamorro-Posada and G. S. McDonald, Opt. Lett. 28, 825 (2003).
[CrossRef] [PubMed]

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. 192, 1 (2001).
[CrossRef]

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

Chen, Y.

Y. Chen, Phys. Rev. A 45, 4974 (1992).
[CrossRef] [PubMed]

Chiaraluce, F.

Christian, J. M.

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, Phys. Rev. A 76, 033833 (2007).
[CrossRef]

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, Phys. Rev. E 74, 066612 (2006).
[CrossRef]

Ciattoni, A.

Crosignani, B.

De Angelis, C.

P. J. Bradley and C. De Angelis, Opt. Commun. 130, 205 (1996).
[CrossRef]

Fibich, G.

G. Fibich, Phys. Rev. Lett. 76, 4356 (1996).
[CrossRef] [PubMed]

Gambi, E.

Hooton, D. E.

Ilecki, W.

A. D. Boardman, P. Bontemps, W. Ilecki, and A. A. Zharov, J. Mod. Opt. 47, 1941 (2000).

Iturbe-Castillo, M. D.

M. D. Iturbe-Castillo, J. J. Sánchez-Mondragón, S. I. Stepanov, M. B. Klein, and B. A. Wechsler, Opt. Commun. 118, 515 (1995).
[CrossRef]

Kaplan, A. E.

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

D. R. Andersen, D. E. Hooton, G. A. Swartzlander, Jr., and A. E. Kaplan, Opt. Lett. 15, 783 (1990).
[CrossRef] [PubMed]

Kartashov, Y. V.

Kivshar, Yu. S.

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

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

Klein, M. B.

M. D. Iturbe-Castillo, J. J. Sánchez-Mondragón, S. I. Stepanov, M. B. Klein, and B. A. Wechsler, Opt. Commun. 118, 515 (1995).
[CrossRef]

Luther-Davies, B.

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

B. Luther-Davies and Y. Xiaoping, Opt. Lett. 17, 496 (1992).
[CrossRef] [PubMed]

McDonald, G. S.

J. Sánchez-Curto, P. Chamorro-Posada, and G. S. McDonald, J. Opt. A 11, 054015 (2009).
[CrossRef]

J. Sánchez-Curto, P. Chamorro-Posada, and G. S. McDonald, Opt. Lett. 32, 1126 (2007).
[CrossRef] [PubMed]

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, Phys. Rev. A 76, 033833 (2007).
[CrossRef]

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, Phys. Rev. E 74, 066612 (2006).
[CrossRef]

P. Chamorro-Posada and G. S. McDonald, Opt. Lett. 28, 825 (2003).
[CrossRef] [PubMed]

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. 192, 1 (2001).
[CrossRef]

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

Mihalache, D.

D. Mihalache, M. Bertolotti, and C. Sibilia, Prog. Opt. 27, 227 (1989).
[CrossRef]

Moloney, J. V.

A. B. Aceves, J. V. Moloney, and A. C. Newell, Phys. Rev. A 39, 1828 (1989).
[CrossRef] [PubMed]

A. B. Aceves, J. V. Moloney, and A. C. Newell, Phys. Rev. A 39, 1809 (1989).
[CrossRef] [PubMed]

Mookherjea, S.

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. 192, 1 (2001).
[CrossRef]

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

Newell, A. C.

A. B. Aceves, J. V. Moloney, and A. C. Newell, Phys. Rev. A 39, 1809 (1989).
[CrossRef] [PubMed]

A. B. Aceves, J. V. Moloney, and A. C. Newell, Phys. Rev. A 39, 1828 (1989).
[CrossRef] [PubMed]

Orenstein, M.

Pierleoni, P.

Regan, J. J.

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

Sánchez-Curto, J.

J. Sánchez-Curto, P. Chamorro-Posada, and G. S. McDonald, J. Opt. A 11, 054015 (2009).
[CrossRef]

J. Sánchez-Curto, P. Chamorro-Posada, and G. S. McDonald, Opt. Lett. 32, 1126 (2007).
[CrossRef] [PubMed]

Sánchez-Mondragón, J. J.

M. D. Iturbe-Castillo, J. J. Sánchez-Mondragón, S. I. Stepanov, M. B. Klein, and B. A. Wechsler, Opt. Commun. 118, 515 (1995).
[CrossRef]

Scheuer, J.

Segev, M.

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

Shadrivov, I. V.

I. V. Shadrivov and A. A. Zharov, Opt. Commun. 216, 47 (2003).
[CrossRef]

I. V. Shadrivov and A. A. Zharov, J. Opt. Soc. Am. B 19, 596 (2002).
[CrossRef]

Sibilia, C.

D. Mihalache, M. Bertolotti, and C. Sibilia, Prog. Opt. 27, 227 (1989).
[CrossRef]

Skinner, S. R.

Stegeman, G. I.

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

Stepanov, S. I.

M. D. Iturbe-Castillo, J. J. Sánchez-Mondragón, S. I. Stepanov, M. B. Klein, and B. A. Wechsler, Opt. Commun. 118, 515 (1995).
[CrossRef]

Swartzlander, G. A.

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

D. R. Andersen, D. E. Hooton, G. A. Swartzlander, Jr., and A. E. Kaplan, Opt. Lett. 15, 783 (1990).
[CrossRef] [PubMed]

Torner, L.

Torruellas, W.

S. Trillo and W. Torruellas, Spatial Solitons (Springer Verlag, 2000).

Trillo, S.

S. Trillo and W. Torruellas, Spatial Solitons (Springer Verlag, 2000).

Vysloukh, V. A.

Wechsler, B. A.

M. D. Iturbe-Castillo, J. J. Sánchez-Mondragón, S. I. Stepanov, M. B. Klein, and B. A. Wechsler, Opt. Commun. 118, 515 (1995).
[CrossRef]

Xiaoping, Y.

Yariv, A.

Yin, H.

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

Zharov, A. A.

I. V. Shadrivov and A. A. Zharov, Opt. Commun. 216, 47 (2003).
[CrossRef]

I. V. Shadrivov and A. A. Zharov, J. Opt. Soc. Am. B 19, 596 (2002).
[CrossRef]

A. D. Boardman, P. Bontemps, W. Ilecki, and A. A. Zharov, J. Mod. Opt. 47, 1941 (2000).

IEEE J. Quantum Electron. (1)

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

J. Mod. Opt. (2)

A. D. Boardman, P. Bontemps, W. Ilecki, and A. A. Zharov, J. Mod. Opt. 47, 1941 (2000).

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

J. Opt. A (1)

J. Sánchez-Curto, P. Chamorro-Posada, and G. S. McDonald, J. Opt. A 11, 054015 (2009).
[CrossRef]

J. Opt. Soc. Am. B (5)

Opt. Commun. (4)

I. V. Shadrivov and A. A. Zharov, Opt. Commun. 216, 47 (2003).
[CrossRef]

P. J. Bradley and C. De Angelis, Opt. Commun. 130, 205 (1996).
[CrossRef]

M. D. Iturbe-Castillo, J. J. Sánchez-Mondragón, S. I. Stepanov, M. B. Klein, and B. A. Wechsler, Opt. Commun. 118, 515 (1995).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, Opt. Commun. 192, 1 (2001).
[CrossRef]

Opt. Express (1)

Opt. Lett. (5)

Phys. Rep. (1)

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

Phys. Rev. A (4)

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, Phys. Rev. A 76, 033833 (2007).
[CrossRef]

Y. Chen, Phys. Rev. A 45, 4974 (1992).
[CrossRef] [PubMed]

A. B. Aceves, J. V. Moloney, and A. C. Newell, Phys. Rev. A 39, 1809 (1989).
[CrossRef] [PubMed]

A. B. Aceves, J. V. Moloney, and A. C. Newell, Phys. Rev. A 39, 1828 (1989).
[CrossRef] [PubMed]

Phys. Rev. E (1)

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, Phys. Rev. E 74, 066612 (2006).
[CrossRef]

Phys. Rev. Lett. (2)

G. Fibich, Phys. Rev. Lett. 76, 4356 (1996).
[CrossRef] [PubMed]

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

Prog. Opt. (1)

D. Mihalache, M. Bertolotti, and C. Sibilia, Prog. Opt. 27, 227 (1989).
[CrossRef]

Science (1)

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

Other (1)

S. Trillo and W. Torruellas, Spatial Solitons (Springer Verlag, 2000).

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Figures (2)

Fig. 1
Fig. 1

(a) Refraction geometry: a black soliton experiencing (b) internal refraction, (c) transparency, and (d) external refraction. Here, u 0 = 1 and κ = 10 3 , giving θ t = 24.851 ° , 25°, and 25.147°, respectively. In scaled units, one obtains W = 10.356 , 10.427, and 10.497.

Fig. 2
Fig. 2

External (left) and internal refraction (right).

Equations (5)

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

κ 2 u ζ 2 + j u ζ + 1 2 2 u ξ 2 | u | 2 u = [ Δ 4 κ ( 1 α ) | u | 2 ] H ( ξ ) u .
u ( ξ , ζ ) = u 0 ( A tanh Θ + j F ) exp ( j ζ 2 κ ) × exp [ j 1 Δ 4 κ u 0 2 α 1 + 2 κ V 2 ( V ξ + ζ 2 κ ) ] ,
Θ = u 0 A α 1 2 ( ξ + W ζ ) 1 + 2 κ W 2 , W = V V 0 1 + 2 κ V V 0 .
V 0 = u 0 F α 1 2 [ 1 Δ ( 2 + F 2 ) 2 κ u 0 2 α ] 1 2 .
γ ± n 01 cos ( θ i ) = n 02 cos ( θ t ) ,

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