Abstract

We present, to the best of our knowledge, the first exact analytical solitons of a nonlinear Helmholtz equation with a saturable refractive-index model. These new two-dimensional spatial solitons have a bistable characteristic in some parameter regimes, and they capture oblique (arbitrary-angle) beam propagation in both the forward and backward directions. New conservation laws are reported, and the classic paraxial solution is recovered in an appropriate multiple limit. Analysis and simulations examine the stability of both solution branches, and stationary Helmholtz solitons are found to emerge from a range of perturbed input beams.

© 2009 Optical Society of America

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    [CrossRef]

2007 (6)

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, “Bistable Helmholtz solitons in cubic-quintic materials,” Phys. Rev. A 76, 033833 (2007).
[CrossRef]

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, “Helmholtz bright and boundary solitons,” J. Phys. A: Math. Theor. 40, 1545-1560 (2007).
[CrossRef]

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, “Helmholtz bright and boundary solitons--corrigendum,”J. Phys. A: Math. Theor. 40, 8601 (2007).
[CrossRef]

J. M. Christian, G. S. McDonald, R. J. Potton, and P. Chamorro-Posada, “Helmholtz solitons in power-law optical materials,” Phys. Rev. A 76, 033834 (2007).
[CrossRef]

G. Baruch, G. Fibich, and S. Tsynkov, “High-order numerical solution of the nonlinear Helmholtz equation with axial symmetry,” J. Comput. Appl. Math. 204, 477-492 (2007).
[CrossRef]

J. Sánchez-Curto, P. Chamorro-Posada, and G. S. McDonald, “Helmholtz solitons at nonlinear interfaces,” Opt. Lett. 32, 1126-1128 (2007).
[CrossRef] [PubMed]

2006 (2)

Y.-D. Wu, “New all-optical switch based on the spatial soliton repulsion,” Opt. Express 14, 4005-4012 (2006).
[CrossRef] [PubMed]

P. Chamorro-Posada and G. S. McDonald, “Spatial Kerr soliton collisions at arbitrary angles,” Phys. Rev. E 74, 036609 (2006).
[CrossRef]

2005 (2)

2004 (2)

2002 (2)

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Exact soliton solutions of the nonlinear Helmholtz equation: communication,” J. Opt. Soc. Am. B 19, 1216-1217 (2002).
[CrossRef]

O. Cohen, R. Uzdin, T. Carmon, J. Fleischer, M. Segev, and S. Odouov, “Collisions between optical spatial solitons propagating in opposite directions,” Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

2001 (1)

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Nonparaxial 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 nonparaxial spatial solitons,” J. Mod. Opt. 47, 1877-1886 (2000).

1999 (5)

S. Bian, M. Martinelli, and R. J. Horowicz, “Z-scan formula for saturable Kerr media,” Opt. Commun. 172, 347-353 (1999).
[CrossRef]

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518-1523 (1999).
[CrossRef] [PubMed]

C. Anastassiou, M. Segev, K. Steiglitz, J. A. Giordmaine, M. Mitchell, M. Shih, S. Lan, and J. Martin, “Energy-exchange interactions between colliding vector solitons,” Phys. Rev. Lett. 83, 2332-2335 (1999).
[CrossRef]

S. Blair and K. Wagner, “Spatial soliton angular deflection logic gates,” Appl. Opt. 38, 6749-6772 (1999).
[CrossRef]

J. Scheuer and M. Orenstein, “Interactions and switching of spatial soliton pairs in the vicinity of a nonlinear interface,” Opt. Lett. 24, 1735-1737 (1999).
[CrossRef]

1998 (4)

1997 (2)

1996 (1)

1995 (2)

1992 (2)

W. Krolikowski and B. Luther-Davies, “Analytic solution for soliton propagation in a nonlinear saturable medium,” Opt. Lett. 17, 1414-1416 (1992).
[CrossRef] [PubMed]

S. Gatz and J. Herrmann, “Soliton collision and soliton fusion in dispersive materials with a linear and quadratic intensity depending refraction index change,” IEEE J. Quantum Electron. 28, 1732-1738 (1992).
[CrossRef]

1991 (3)

1990 (1)

1989 (3)

D. Mihalache, M. Bertolotti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” Prog. Opt. 27, 229-313 (1989).

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. I. Equivalent-particle theory for a single interface,” Phys. Rev. A 39, 1809-1827 (1989).
[CrossRef] [PubMed]

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. II. Multiple-particle and multiple-interface extensions,” Phys. Rev. A 39, 1828-1840 (1989).
[CrossRef] [PubMed]

1988 (3)

D. Mihalache, D. Mazilu, M. Bertolotti, and C. Sibilia, “Exact solution for transverse electric polarized nonlinear guided waves in saturable media,” J. Mod. Opt. 35, 1017-1027 (1988).
[CrossRef]

V. E. Wood, E. D. Evans, and R. P. Kenan, “Soluble saturable refractive-index nonlinearity model,” Opt. Commun. 69, 156-160 (1988).
[CrossRef]

M. D. Feit and J. A. Fleck, “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633-640 (1988).
[CrossRef]

1987 (3)

P. Roussignol, D. Ricard, J. Lukasik, and C. Flytzanis, “New results on optical phase conjugation in semiconductor-doped glasses,” J. Opt. Soc. Am. B 4, 5-13 (1987).
[CrossRef]

D. Mihalache and D. Mazilu, “Stability and instability of nonlinear guided waves in saturable media,” Solid State Commun. 63, 215-217 (1987).
[CrossRef]

D. Mihalache and D. Mazilu, “Stability of nonlinear slab-guided waves in saturable media: a numerical analysis,” Phys. Lett. A 122, 381-384 (1987).
[CrossRef]

1986 (1)

D. Mihalache and D. Mazilu, “TM-polarized nonlinear slab-guided waves in saturable media,” Solid State Commun. 60, 397-399 (1986).
[CrossRef]

1985 (2)

A. E. Kaplan, “Bistable solitons,” Phys. Rev. Lett. 55, 1291-1294 (1985).
[CrossRef] [PubMed]

A. E. Kaplan, “Multistable self-trapping of light and multistable soliton pulse propagation,” IEEE J. Quantum Electron. 21, 1538-1543 (1985).
[CrossRef]

1983 (1)

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]

1973 (1)

M. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in the medium with nonlinearity saturation,” Radiophys. Quantum Electron. 16, 783-789 (1973).
[CrossRef]

Aceves, A. B.

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. I. Equivalent-particle theory for a single interface,” Phys. Rev. A 39, 1809-1827 (1989).
[CrossRef] [PubMed]

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. II. Multiple-particle and multiple-interface extensions,” Phys. Rev. A 39, 1828-1840 (1989).
[CrossRef] [PubMed]

Aitchison, J. S.

Anastassiou, C.

C. Anastassiou, M. Segev, K. Steiglitz, J. A. Giordmaine, M. Mitchell, M. Shih, S. Lan, and J. Martin, “Energy-exchange interactions between colliding vector solitons,” Phys. Rev. Lett. 83, 2332-2335 (1999).
[CrossRef]

Baruch, G.

G. Baruch, G. Fibich, and S. Tsynkov, “High-order numerical solution of the nonlinear Helmholtz equation with axial symmetry,” J. Comput. Appl. Math. 204, 477-492 (2007).
[CrossRef]

Bertolotti, M.

D. Mihalache, M. Bertolotti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” Prog. Opt. 27, 229-313 (1989).

D. Mihalache, D. Mazilu, M. Bertolotti, and C. Sibilia, “Exact solution for transverse electric polarized nonlinear guided waves in saturable media,” J. Mod. Opt. 35, 1017-1027 (1988).
[CrossRef]

Bian, S.

S. Bian, M. Martinelli, and R. J. Horowicz, “Z-scan formula for saturable Kerr media,” Opt. Commun. 172, 347-353 (1999).
[CrossRef]

S. Bian, J. Frejlich, and K. H. Ringhofer, “Photorefractive saturable Kerr-type nonlinearity in photovoltaic crystals,” Phys. Rev. Lett. 78, 4035-4038 (1997).
[CrossRef]

Birge, R. R.

Blair, S.

Carcalho, M. I.

Carmon, T.

O. Cohen, R. Uzdin, T. Carmon, J. Fleischer, M. Segev, and S. Odouov, “Collisions between optical spatial solitons propagating in opposite directions,” Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

Catunda, T.

Chamorro-Posada, P.

J. M. Christian, G. S. McDonald, R. J. Potton, and P. Chamorro-Posada, “Helmholtz solitons in power-law optical materials,” Phys. Rev. A 76, 033834 (2007).
[CrossRef]

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, “Helmholtz bright and boundary solitons,” J. Phys. A: Math. Theor. 40, 1545-1560 (2007).
[CrossRef]

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, “Bistable Helmholtz solitons in cubic-quintic materials,” Phys. Rev. A 76, 033833 (2007).
[CrossRef]

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, “Helmholtz bright and boundary solitons--corrigendum,”J. Phys. A: Math. Theor. 40, 8601 (2007).
[CrossRef]

J. Sánchez-Curto, P. Chamorro-Posada, and G. S. McDonald, “Helmholtz solitons at nonlinear interfaces,” Opt. Lett. 32, 1126-1128 (2007).
[CrossRef] [PubMed]

P. Chamorro-Posada and G. S. McDonald, “Spatial Kerr soliton collisions at arbitrary angles,” Phys. Rev. E 74, 036609 (2006).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Exact soliton solutions of the nonlinear Helmholtz equation: communication,” J. Opt. Soc. Am. B 19, 1216-1217 (2002).
[CrossRef]

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

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

Chi, S.

Christian, J. M.

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, “Bistable Helmholtz solitons in cubic-quintic materials,” Phys. Rev. A 76, 033833 (2007).
[CrossRef]

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, “Helmholtz bright and boundary solitons--corrigendum,”J. Phys. A: Math. Theor. 40, 8601 (2007).
[CrossRef]

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, “Helmholtz bright and boundary solitons,” J. Phys. A: Math. Theor. 40, 1545-1560 (2007).
[CrossRef]

J. M. Christian, G. S. McDonald, R. J. Potton, and P. Chamorro-Posada, “Helmholtz solitons in power-law optical materials,” Phys. Rev. A 76, 033834 (2007).
[CrossRef]

Christodoulides, D. N.

Ciattoni, A.

Cohen, O.

O. Cohen, R. Uzdin, T. Carmon, J. Fleischer, M. Segev, and S. Odouov, “Collisions between optical spatial solitons propagating in opposite directions,” Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

Coutaz, J. -L.

Crosignani, B.

Cury, L. A.

de Araújo, C. B.

de Melo, C. P.

Demenicis, L.

dos Santos, C. G.

Downie, J. D.

Evans, E. D.

V. E. Wood, E. D. Evans, and R. P. Kenan, “Soluble saturable refractive-index nonlinearity model,” Opt. Commun. 69, 156-160 (1988).
[CrossRef]

Feit, M. D.

Fibich, G.

G. Baruch, G. Fibich, and S. Tsynkov, “High-order numerical solution of the nonlinear Helmholtz equation with axial symmetry,” J. Comput. Appl. Math. 204, 477-492 (2007).
[CrossRef]

Fleck, J. A.

Fleischer, J.

O. Cohen, R. Uzdin, T. Carmon, J. Fleischer, M. Segev, and S. Odouov, “Collisions between optical spatial solitons propagating in opposite directions,” Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

Flytzanis, C.

Frejlich, J.

S. Bian, J. Frejlich, and K. H. Ringhofer, “Photorefractive saturable Kerr-type nonlinearity in photovoltaic crystals,” Phys. Rev. Lett. 78, 4035-4038 (1997).
[CrossRef]

Friberg, A. T.

Gary, C.

Gatz, S.

S. Gatz and J. Herrmann, “Soliton collision and soliton fusion in dispersive materials with a linear and quadratic intensity depending refraction index change,” IEEE J. Quantum Electron. 28, 1732-1738 (1992).
[CrossRef]

S. Gatz and J. Herrmann, “Soliton propagation in materials with saturable nonlinearity,” J. Opt. Soc. Am. B 8, 2296-2302 (1991).
[CrossRef]

Giordmaine, J. A.

C. Anastassiou, M. Segev, K. Steiglitz, J. A. Giordmaine, M. Mitchell, M. Shih, S. Lan, and J. Martin, “Energy-exchange interactions between colliding vector solitons,” Phys. Rev. Lett. 83, 2332-2335 (1999).
[CrossRef]

Goldstein, H.

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, 1980).

Gomes, A. S. L.

Gordon, J. P.

Guo, Q.

Haelterman, M.

Herrmann, J.

Horowicz, R. J.

S. Bian, M. Martinelli, and R. J. Horowicz, “Z-scan formula for saturable Kerr media,” Opt. Commun. 172, 347-353 (1999).
[CrossRef]

Kang, J. U.

Kaplan, A. E.

A. E. Kaplan, “Bistable solitons,” Phys. Rev. Lett. 55, 1291-1294 (1985).
[CrossRef] [PubMed]

A. E. Kaplan, “Multistable self-trapping of light and multistable soliton pulse propagation,” IEEE J. Quantum Electron. 21, 1538-1543 (1985).
[CrossRef]

Kartashov, Y. V.

Kenan, R. P.

V. E. Wood, E. D. Evans, and R. P. Kenan, “Soluble saturable refractive-index nonlinearity model,” Opt. Commun. 69, 156-160 (1988).
[CrossRef]

Kivshar, Y. S.

Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81-197 (1998).
[CrossRef]

Y. S. Kivshar, “Bright and dark spatial solitons in non-Kerr media,” Opt. Quantum Electron. 30, 571-614 (1998).
[CrossRef]

Kolokolov, A. A.

M. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in the medium with nonlinearity saturation,” Radiophys. Quantum Electron. 16, 783-789 (1973).
[CrossRef]

Krolikowski, W.

Kull, M.

Laine, T. A.

Lan, S.

C. Anastassiou, M. Segev, K. Steiglitz, J. A. Giordmaine, M. Mitchell, M. Shih, S. Lan, and J. Martin, “Energy-exchange interactions between colliding vector solitons,” Phys. Rev. Lett. 83, 2332-2335 (1999).
[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]

Lukasik, J.

Luther-Davies, B.

Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81-197 (1998).
[CrossRef]

W. Krolikowski and B. Luther-Davies, “Analytic solution for soliton propagation in a nonlinear saturable medium,” Opt. Lett. 17, 1414-1416 (1992).
[CrossRef] [PubMed]

Martin, J.

C. Anastassiou, M. Segev, K. Steiglitz, J. A. Giordmaine, M. Mitchell, M. Shih, S. Lan, and J. Martin, “Energy-exchange interactions between colliding vector solitons,” Phys. Rev. Lett. 83, 2332-2335 (1999).
[CrossRef]

Martinelli, M.

S. Bian, M. Martinelli, and R. J. Horowicz, “Z-scan formula for saturable Kerr media,” Opt. Commun. 172, 347-353 (1999).
[CrossRef]

Mazilu, D.

D. Mihalache, D. Mazilu, M. Bertolotti, and C. Sibilia, “Exact solution for transverse electric polarized nonlinear guided waves in saturable media,” J. Mod. Opt. 35, 1017-1027 (1988).
[CrossRef]

D. Mihalache and D. Mazilu, “Stability of nonlinear slab-guided waves in saturable media: a numerical analysis,” Phys. Lett. A 122, 381-384 (1987).
[CrossRef]

D. Mihalache and D. Mazilu, “Stability and instability of nonlinear guided waves in saturable media,” Solid State Commun. 63, 215-217 (1987).
[CrossRef]

D. Mihalache and D. Mazilu, “TM-polarized nonlinear slab-guided waves in saturable media,” Solid State Commun. 60, 397-399 (1986).
[CrossRef]

McDonald, G. S.

J. M. Christian, G. S. McDonald, R. J. Potton, and P. Chamorro-Posada, “Helmholtz solitons in power-law optical materials,” Phys. Rev. A 76, 033834 (2007).
[CrossRef]

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, “Helmholtz bright and boundary solitons,” J. Phys. A: Math. Theor. 40, 1545-1560 (2007).
[CrossRef]

J. Sánchez-Curto, P. Chamorro-Posada, and G. S. McDonald, “Helmholtz solitons at nonlinear interfaces,” Opt. Lett. 32, 1126-1128 (2007).
[CrossRef] [PubMed]

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, “Helmholtz bright and boundary solitons--corrigendum,”J. Phys. A: Math. Theor. 40, 8601 (2007).
[CrossRef]

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, “Bistable Helmholtz solitons in cubic-quintic materials,” Phys. Rev. A 76, 033833 (2007).
[CrossRef]

P. Chamorro-Posada and G. S. McDonald, “Spatial Kerr soliton collisions at arbitrary angles,” Phys. Rev. E 74, 036609 (2006).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Exact soliton solutions of the nonlinear Helmholtz equation: communication,” J. Opt. Soc. Am. B 19, 1216-1217 (2002).
[CrossRef]

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

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

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]

Mihalache, D.

D. Mihalache, M. Bertolotti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” Prog. Opt. 27, 229-313 (1989).

D. Mihalache, D. Mazilu, M. Bertolotti, and C. Sibilia, “Exact solution for transverse electric polarized nonlinear guided waves in saturable media,” J. Mod. Opt. 35, 1017-1027 (1988).
[CrossRef]

D. Mihalache and D. Mazilu, “Stability of nonlinear slab-guided waves in saturable media: a numerical analysis,” Phys. Lett. A 122, 381-384 (1987).
[CrossRef]

D. Mihalache and D. Mazilu, “Stability and instability of nonlinear guided waves in saturable media,” Solid State Commun. 63, 215-217 (1987).
[CrossRef]

D. Mihalache and D. Mazilu, “TM-polarized nonlinear slab-guided waves in saturable media,” Solid State Commun. 60, 397-399 (1986).
[CrossRef]

Mitchell, M.

C. Anastassiou, M. Segev, K. Steiglitz, J. A. Giordmaine, M. Mitchell, M. Shih, S. Lan, and J. Martin, “Energy-exchange interactions between colliding vector solitons,” Phys. Rev. Lett. 83, 2332-2335 (1999).
[CrossRef]

Moloney, J. V.

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. II. Multiple-particle and multiple-interface extensions,” Phys. Rev. A 39, 1828-1840 (1989).
[CrossRef] [PubMed]

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. I. Equivalent-particle theory for a single interface,” Phys. Rev. A 39, 1809-1827 (1989).
[CrossRef] [PubMed]

Mookherjea, S.

New, G. H. C.

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Exact soliton solutions of the nonlinear Helmholtz equation: communication,” J. Opt. Soc. Am. B 19, 1216-1217 (2002).
[CrossRef]

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

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

Newell, A. C.

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. I. Equivalent-particle theory for a single interface,” Phys. Rev. A 39, 1809-1827 (1989).
[CrossRef] [PubMed]

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. II. Multiple-particle and multiple-interface extensions,” Phys. Rev. A 39, 1828-1840 (1989).
[CrossRef] [PubMed]

Odouov, S.

O. Cohen, R. Uzdin, T. Carmon, J. Fleischer, M. Segev, and S. Odouov, “Collisions between optical spatial solitons propagating in opposite directions,” Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

Orenstein, M.

Petrov, D. V.

Potton, R. J.

J. M. Christian, G. S. McDonald, R. J. Potton, and P. Chamorro-Posada, “Helmholtz solitons in power-law optical materials,” Phys. Rev. A 76, 033834 (2007).
[CrossRef]

Ricard, D.

Ringhofer, K. H.

S. Bian, J. Frejlich, and K. H. Ringhofer, “Photorefractive saturable Kerr-type nonlinearity in photovoltaic crystals,” Phys. Rev. Lett. 78, 4035-4038 (1997).
[CrossRef]

Roussignol, P.

Sánchez-Curto, J.

Scheuer, J.

Segev, M.

O. Cohen, R. Uzdin, T. Carmon, J. Fleischer, M. Segev, and S. Odouov, “Collisions between optical spatial solitons propagating in opposite directions,” Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518-1523 (1999).
[CrossRef] [PubMed]

C. Anastassiou, M. Segev, K. Steiglitz, J. A. Giordmaine, M. Mitchell, M. Shih, S. Lan, and J. Martin, “Energy-exchange interactions between colliding vector solitons,” Phys. Rev. Lett. 83, 2332-2335 (1999).
[CrossRef]

Sheppard, A. P.

Shih, M.

C. Anastassiou, M. Segev, K. Steiglitz, J. A. Giordmaine, M. Mitchell, M. Shih, S. Lan, and J. Martin, “Energy-exchange interactions between colliding vector solitons,” Phys. Rev. Lett. 83, 2332-2335 (1999).
[CrossRef]

Sibilia, C.

D. Mihalache, M. Bertolotti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” Prog. Opt. 27, 229-313 (1989).

D. Mihalache, D. Mazilu, M. Bertolotti, and C. Sibilia, “Exact solution for transverse electric polarized nonlinear guided waves in saturable media,” J. Mod. Opt. 35, 1017-1027 (1988).
[CrossRef]

Souto-Maior, R.

Stegeman, G. I.

Steiglitz, K.

C. Anastassiou, M. Segev, K. Steiglitz, J. A. Giordmaine, M. Mitchell, M. Shih, S. Lan, and J. Martin, “Energy-exchange interactions between colliding vector solitons,” Phys. Rev. Lett. 83, 2332-2335 (1999).
[CrossRef]

Timucin, D.

Torner, L.

Tsynkov, S.

G. Baruch, G. Fibich, and S. Tsynkov, “High-order numerical solution of the nonlinear Helmholtz equation with axial symmetry,” J. Comput. Appl. Math. 204, 477-492 (2007).
[CrossRef]

Uzdin, R.

O. Cohen, R. Uzdin, T. Carmon, J. Fleischer, M. Segev, and S. Odouov, “Collisions between optical spatial solitons propagating in opposite directions,” Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

Vakhitov, M. G.

M. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in the medium with nonlinearity saturation,” Radiophys. Quantum Electron. 16, 783-789 (1973).
[CrossRef]

Vysloukh, V. A.

Wagner, K.

Wang, X.

Wang Song, Q.

Wood, V. E.

V. E. Wood, E. D. Evans, and R. P. Kenan, “Soluble saturable refractive-index nonlinearity model,” Opt. Commun. 69, 156-160 (1988).
[CrossRef]

Wu, Y. -D.

Yariv, A.

Zelenina, A. S.

Appl. Opt. (2)

IEEE J. Quantum Electron. (2)

A. E. Kaplan, “Multistable self-trapping of light and multistable soliton pulse propagation,” IEEE J. Quantum Electron. 21, 1538-1543 (1985).
[CrossRef]

S. Gatz and J. Herrmann, “Soliton collision and soliton fusion in dispersive materials with a linear and quadratic intensity depending refraction index change,” IEEE J. Quantum Electron. 28, 1732-1738 (1992).
[CrossRef]

J. Comput. Appl. Math. (1)

G. Baruch, G. Fibich, and S. Tsynkov, “High-order numerical solution of the nonlinear Helmholtz equation with axial symmetry,” J. Comput. Appl. Math. 204, 477-492 (2007).
[CrossRef]

J. Mod. Opt. (2)

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

D. Mihalache, D. Mazilu, M. Bertolotti, and C. Sibilia, “Exact solution for transverse electric polarized nonlinear guided waves in saturable media,” J. Mod. Opt. 35, 1017-1027 (1988).
[CrossRef]

J. Opt. Soc. Am. B (11)

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. Roussignol, D. Ricard, J. Lukasik, and C. Flytzanis, “New results on optical phase conjugation in semiconductor-doped glasses,” J. Opt. Soc. Am. B 4, 5-13 (1987).
[CrossRef]

M. D. Feit and J. A. Fleck, “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633-640 (1988).
[CrossRef]

T. Catunda and L. A. Cury, “Transverse self-phase modulation in ruby and GdAlO3:Cr+3 crystals,” J. Opt. Soc. Am. B 7, 1445-1455 (1990).
[CrossRef]

J.-L. Coutaz and M. Kull, “Saturation of the nonlinear index of refraction in semiconductor-doped glass,” J. Opt. Soc. Am. B 8, 95-98 (1991).
[CrossRef]

J. Herrmann, “Propagation of ultrashort light pulses in fibers with saturable nonlinearity in the normal-dispersion region,” J. Opt. Soc. Am. B 8, 1507-1511 (1991).
[CrossRef]

S. Gatz and J. Herrmann, “Soliton propagation in materials with saturable nonlinearity,” J. Opt. Soc. Am. B 8, 2296-2302 (1991).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Exact soliton solutions of the nonlinear Helmholtz equation: communication,” J. Opt. Soc. Am. B 19, 1216-1217 (2002).
[CrossRef]

D. N. Christodoulides and M. I. Carcalho, “Bright, dark, and gray spatial soliton states in photorefractive media,” J. Opt. Soc. Am. B 12, 1628-1633 (1995).
[CrossRef]

L. Demenicis, A. S. L. Gomes, D. V. Petrov, C. B. de Araújo, C. P. de Melo, C. G. dos Santos, and R. Souto-Maior, “Saturation effects in the nonlinear-optical susceptibility of poly(3-hexadecylthiophene),” J. Opt. Soc. Am. B 14, 609-614 (1997).
[CrossRef]

Q. Wang Song, X. Wang, R. R. Birge, J. D. Downie, D. Timucin, and C. Gary, “Propagation of a Gaussian beam in a bacteriorhodopsin film,” J. Opt. Soc. Am. B 15, 1602-1609 (1998).
[CrossRef]

J. Phys. A: Math. Theor. (2)

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, “Helmholtz bright and boundary solitons,” J. Phys. A: Math. Theor. 40, 1545-1560 (2007).
[CrossRef]

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, “Helmholtz bright and boundary solitons--corrigendum,”J. Phys. A: Math. Theor. 40, 8601 (2007).
[CrossRef]

Opt. Commun. (3)

S. Bian, M. Martinelli, and R. J. Horowicz, “Z-scan formula for saturable Kerr media,” Opt. Commun. 172, 347-353 (1999).
[CrossRef]

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

V. E. Wood, E. D. Evans, and R. P. Kenan, “Soluble saturable refractive-index nonlinearity model,” Opt. Commun. 69, 156-160 (1988).
[CrossRef]

Opt. Express (1)

Opt. Lett. (10)

J. Sánchez-Curto, P. Chamorro-Posada, and G. S. McDonald, “Helmholtz solitons at nonlinear interfaces,” Opt. Lett. 32, 1126-1128 (2007).
[CrossRef] [PubMed]

Y. V. Kartashov, A. S. Zelenina, L. Torner, and V. A. Vysloukh, “Spatial soliton switching in quasi-continuous optical arrays,” Opt. Lett. 29, 766-768 (2004).
[CrossRef] [PubMed]

B. Crosignani, A. Yariv, and S. Mookherjea, “Nonparaxial spatial solitons and propagation-invariant pattern solutions in optical Kerr media,” Opt. Lett. 29, 1254-1256 (2004).
[CrossRef] [PubMed]

A. Ciattoni, B. Crosignani, S. Mookherjea, and A. Yariv, “Nonparaxial dark solitons in optical Kerr media,” Opt. Lett. 30, 516-518 (2005).
[CrossRef] [PubMed]

S. Chi and Q. Guo, “Vector theory of self-focusing of an optical beam in Kerr media,” Opt. Lett. 20, 1598-1600 (1995).
[CrossRef] [PubMed]

A. P. Sheppard and M. Haelterman, “Nonparaxiality stabilizes three dimensional soliton beams in Kerr media,” Opt. Lett. 23, 1820-1822 (1998).
[CrossRef]

J. Scheuer and M. Orenstein, “Interactions and switching of spatial soliton pairs in the vicinity of a nonlinear interface,” Opt. Lett. 24, 1735-1737 (1999).
[CrossRef]

J. U. Kang, G. I. Stegeman, and J. S. Aitchison, “One-dimensional spatial soliton dragging, trapping, and all-optical switching in AlGaAs waveguides,” Opt. Lett. 21, 189-191 (1996).
[CrossRef] [PubMed]

J. P. Gordon, “Interaction forces among solitons in optical fibers,” Opt. Lett. 8, 596-598 (1983).
[CrossRef] [PubMed]

W. Krolikowski and B. Luther-Davies, “Analytic solution for soliton propagation in a nonlinear saturable medium,” Opt. Lett. 17, 1414-1416 (1992).
[CrossRef] [PubMed]

Opt. Quantum Electron. (1)

Y. S. Kivshar, “Bright and dark spatial solitons in non-Kerr media,” Opt. Quantum Electron. 30, 571-614 (1998).
[CrossRef]

Phys. Lett. A (1)

D. Mihalache and D. Mazilu, “Stability of nonlinear slab-guided waves in saturable media: a numerical analysis,” Phys. Lett. A 122, 381-384 (1987).
[CrossRef]

Phys. Rep. (1)

Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81-197 (1998).
[CrossRef]

Phys. Rev. A (5)

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, “Bistable Helmholtz solitons in cubic-quintic materials,” Phys. Rev. A 76, 033833 (2007).
[CrossRef]

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. I. Equivalent-particle theory for a single interface,” Phys. Rev. A 39, 1809-1827 (1989).
[CrossRef] [PubMed]

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. II. Multiple-particle and multiple-interface extensions,” Phys. Rev. A 39, 1828-1840 (1989).
[CrossRef] [PubMed]

J. M. Christian, G. S. McDonald, R. J. Potton, and P. Chamorro-Posada, “Helmholtz solitons in power-law optical materials,” Phys. Rev. A 76, 033834 (2007).
[CrossRef]

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

Phys. Rev. E (1)

P. Chamorro-Posada and G. S. McDonald, “Spatial Kerr soliton collisions at arbitrary angles,” Phys. Rev. E 74, 036609 (2006).
[CrossRef]

Phys. Rev. Lett. (4)

S. Bian, J. Frejlich, and K. H. Ringhofer, “Photorefractive saturable Kerr-type nonlinearity in photovoltaic crystals,” Phys. Rev. Lett. 78, 4035-4038 (1997).
[CrossRef]

O. Cohen, R. Uzdin, T. Carmon, J. Fleischer, M. Segev, and S. Odouov, “Collisions between optical spatial solitons propagating in opposite directions,” Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

A. E. Kaplan, “Bistable solitons,” Phys. Rev. Lett. 55, 1291-1294 (1985).
[CrossRef] [PubMed]

C. Anastassiou, M. Segev, K. Steiglitz, J. A. Giordmaine, M. Mitchell, M. Shih, S. Lan, and J. Martin, “Energy-exchange interactions between colliding vector solitons,” Phys. Rev. Lett. 83, 2332-2335 (1999).
[CrossRef]

Prog. Opt. (1)

D. Mihalache, M. Bertolotti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” Prog. Opt. 27, 229-313 (1989).

Radiophys. Quantum Electron. (1)

M. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in the medium with nonlinearity saturation,” Radiophys. Quantum Electron. 16, 783-789 (1973).
[CrossRef]

Science (1)

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518-1523 (1999).
[CrossRef] [PubMed]

Solid State Commun. (2)

D. Mihalache and D. Mazilu, “TM-polarized nonlinear slab-guided waves in saturable media,” Solid State Commun. 60, 397-399 (1986).
[CrossRef]

D. Mihalache and D. Mazilu, “Stability and instability of nonlinear guided waves in saturable media,” Solid State Commun. 63, 215-217 (1987).
[CrossRef]

Other (1)

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, 1980).

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Figures (6)

Fig. 1
Fig. 1

Geometry of Helmholtz solitons in the ( ξ , ζ ) plane. The forward solutions have K ζ > 0 , while the direction of K ξ is defined by V [(a) K ξ < 0 when V > 0 ; (b) K ξ > 0 when V < 0 ]. The backward solutions have K ζ < 0 [(c) K ξ > 0 when V > 0 ; (d) K ξ < 0 when V < 0 ]. The red (dashed) lines mark ξ + V ζ = 0 , where | V | is when the beam coincides with the ξ axis. The propagation angle of the beam with respect to the ζ axis is Θ, where tan   Θ K ζ / K ξ = V K ζ / V K ξ = V and V / V .

Fig. 2
Fig. 2

Propagation domains for (a) forward and (b) backward Helmholtz solitons. Each beam is restricted by the condition V + , which corresponds to 90 ° θ + 90 ° in the ( x , z ) frame since tan   θ = ( 2 κ ) 1 / 2 V . The gray region denotes forbidden directions. Since x and z are scaled by different factors ( L D w 0 ) , the propagation angle θ in the ( x , z ) frame is represented by the angle Θ in the ( ξ , ζ ) frame (see Fig. 1), where θ and Θ are related by tan   θ = ( 2 κ ) 1 / 2 tan   Θ .

Fig. 3
Fig. 3

Angular beam-broadening effect for bistable Helmholtz solitons [Eq. (9)] when γ = 0.25 for (a) lower- ( ρ 0 2.32 ) and (b) upper- ( ρ 0 8.68 ) branch solutions. Geometrical broadening is absent for the paraxial solution ( | θ | = 0 ° ) . For a launching angle of | θ | = 60 ° , the perceived width of the beam has doubled relative to the paraxial profile.

Fig. 4
Fig. 4

Curves defining nondegenerate bistable solution families for four different values of the width parameter ν. These plots are obtained by solving Eq. (11) numerically.

Fig. 5
Fig. 5

Paraxial beam power P as a function of β for four different values of the saturation parameter γ. The curve for γ = 0.00 corresponds to a Kerr nonlinearity, where β = ρ 0 / 2 [from Eq. (8)]. One finds that the slope d P / d β is always positive (solitons predicted to be stable) and P ( β ) is single valued (no Kaplan-type degenerate bistability).

Fig. 6
Fig. 6

Beam reshaping simulations for (a) lower- and (b) upper-branch canonical solitons. Solid curve: | θ | = 10 ° ; dashed curve: | θ | = 30 ° ; dot-dashed curve: | θ | = 50 ° . Perturbed lower-branch beams exhibit decaying oscillations in the amplitude, width, and area. These curves are universal and hold for any combination of κ V 2 = 1 2 tan 2 θ (see text for the specific numerical values of κ and V used in these simulations).

Equations (33)

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

n NL ( I ) = n 2 I sat 2 [ 1 1 ( 1 + I / I sat ) 2 ] .
( 2 z 2 + 2 x 2 ) E ( x , z ) + ω 2 n 2 c 2 E ( x , z ) = 0.
κ 2 u ζ 2 + i u ζ + 1 2 2 u ξ 2 + 1 2 2 + γ | u | 2 ( 1 + γ | u | 2 ) 2 | u | 2 u = 0.
2 ρ ( 2 ρ ξ 2 + 2 κ 2 ρ ζ 2 ) 1 ρ 2 [ ( ρ ξ ) 2 + 2 κ ( ρ ζ ) 2 ] = 8 β 4 ρ 2 + γ ρ ( 1 + γ ρ ) 2 ,
2 κ K ζ 1 ρ ρ ζ + K ξ 1 ρ ρ ξ = 0 ,
κ K ζ 2 1 4 κ + 1 2 K ξ 2 β .
d d ρ [ 1 ρ ( d ρ d s ) 2 ] = 8 β 4 ρ ( 2 + γ ρ ) ( 1 + γ ρ ) 2 ,
K ( K ξ , K ζ ) = ± 1 + 4 κ β 1 + 2 κ V 2 ( V , + 1 2 κ ) .
( d ρ d s ) 2 = 4 1 + γ ρ 0 ( ρ 0 ρ 1 + γ ρ ) ρ 2 ,
β β ( ρ 0 , γ ) = 1 1 + γ ρ 0 ( ρ 0 2 ) .
u ( ξ , ζ ) = ρ 1 / 2 ( ξ , ζ ) exp [ ± i 1 + 4 κ β 1 + 2 κ V 2 ( V ξ + ζ 2 κ ) ] exp ( i ζ 2 κ ) ,
where       2 tan 1 ( Ψ ) + 1 γ ρ 0 ln ( Ψ + γ ρ 0 Ψ γ ρ 0 )             = 1 γ 2 1 + γ ρ 0 ( ξ + V ζ 1 + 2 κ V 2 ) ,
Ψ ( γ ρ 0 ρ 1 + γ ρ 0 ) 1 / 2 .
u ( ξ , ζ ) = ρ 1 / 2 ( ξ , ζ ) exp [ i 1 + 4 κ β 2 κ ( ξ   sin   θ + ζ 2 κ cos   θ ) ] exp ( i ζ 2 κ ) ,
2 tan 1 ( Ψ ) + 1 γ ρ 0 ln ( Ψ + γ ρ 0 Ψ γ ρ 0 ) = 1 γ 2 1 + γ ρ 0 ( ξ   cos   θ + ζ 2 κ sin   θ ) ,
2 tan 1 [ ( γ ρ 0 2 + γ ρ 0 ) 1 / 2 ] + 1 γ ρ 0 ln ( 2 + γ ρ 0 + 1 2 + γ ρ 0 1 ) = 2 ν Δ γ 1 + γ ρ 0 .
L = i 2 ( u u ζ u u ζ ) κ u ζ u ζ 1 2 u ξ u ξ + G ( u , u ) ,
G ( u , u * ) 0 u * u d Y 1 2 ( 2 + γ Y ) Y ( 1 + γ Y ) 2 = 1 2 ( u u ) 2 ( 1 + γ u u ) ,
π L u ζ = ( i 2 κ ζ ) u ,     π ̃ L u ζ = ( i 2 + κ ζ ) u ,
W = + d ξ [ | u | 2 i κ ( u u ζ u u ζ ) ] ,
M = + d ξ [ i 2 ( u u ξ u u ξ ) κ ( u ξ u ζ + u ζ u ξ ) ] ,
H = + d ξ [ 1 2 u ξ u ξ κ u ζ u ζ G ( u , u ) ] .
W = ± ( 1 + 4 κ β ) 1 / 2 P ,
M = V 1 + 2 κ V 2 [ ( 1 + 4 κ β ) P 2 κ Q ] ,
H = W 2 κ 1 1 + 2 κ V 2 ( 1 2 κ ) [ ( 1 + 4 κ β ) P 2 κ Q ] ,
P + d s ρ ( s ) ,
Q 1 4 + d s 1 ρ ( s ) [ d ρ ( s ) d s ] 2 .
u ( ξ , ζ ) ρ 1 / 2 ( ξ , ζ ) exp [ i V ξ + i ( β V 2 2 ) ζ ] ,
2 tan 1 ( Ψ ) + 1 γ ρ 0 ln ( Ψ + γ ρ 0 Ψ γ ρ 0 ) 2 γ ξ + V ζ 1 + γ ρ 0 ,
u ( ξ , ζ ) ρ 1 / 2 ( ξ , ζ ) exp [ i V ξ i ( β V 2 2 ) ζ ] exp ( i 2 ζ 2 κ ) .
d P d β = d d β + d ξ | u ( ξ , ζ ; β ) | 2 > 0.
P ( β ) = 1 2 0 ρ 0 ( β ) d ρ 0 [ β 1 1 + γ ρ 0 ( ρ 0 2 ) ] 1 / 2 .
u ( ξ , 0 ) = ρ 1 / 2 ( ξ , 0 ) exp ( i V 1 + 4 κ β 1 + 2 κ V 2 ξ ) ,

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