Abstract

We report on the stabilization of inherently unstable counterpropagating photorefractive spatial solitons by the use of one- and two-dimensional photonic lattices. We numerically investigate the dependence of the instability dynamics on period and amplitude of the lattice and present experimental verification for the dynamic stabilization of the bi-directional soliton state.

© 2007 Optical Society of America

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  41. D. Träger, R. Fischer, D. N. Neshev, A. A. Sukhorukov, C. Denz, W. Krolikowski, and Y. S. Kivshar, "Nonlinear Bloch modes in two-dimensional photonic lattices," Opt. Express 14(5), 1913-1923 (2006). URL http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-5-1913
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2007

2006

D. Träger, R. Fischer, D. N. Neshev, A. A. Sukhorukov, C. Denz, W. Krolikowski, and Y. S. Kivshar, "Nonlinear Bloch modes in two-dimensional photonic lattices," Opt. Express 14(5), 1913-1923 (2006). URL http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-5-1913
[CrossRef] [PubMed]

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, C. Denz, and Y. S. Kivshar, "Reducedsymmetry two-dimensional solitons in photonic lattices," Phys. Rev. Lett. 96(2), 023,905-4 (2006). URL http://link.aps.org/abstract/PRL/v96/p023905.
[CrossRef]

D. Träger, N. Sagemerten, and C. Denz, "Guiding of DynamicallyModulated Signals in Arrays of Photorefractive Spatial Solitons," IEEE J. Sel. Top. Quantum Electron. 12, 383-387 (2006).
[CrossRef]

P. Jander, J. Schröder, T. Richter, K. Motzek, F. Kaiser, M. R. Belic, and C. Denz, "Dynamic instability of counterpropagating self-trapped beams in photorefractive media," Proc. SPIE 6255, 62,550A (2006).

2005

2004

M. Belic, P. Jander, K. Motzek, A. Desyatnikov, D. Jovic, A. Strinic, M. Petrovic, C. Denz, and F. Kaiser, "Counterpropagating self-trapped beams in photorefractive crystals," J. Opt. B 6, S190-S196 (2004).
[CrossRef]

2003

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003).
[CrossRef] [PubMed]

J.W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003).
[CrossRef] [PubMed]

W. Krolikowski, B. Luther-Davies, and C. Denz, "Photorefractive Solitons," IEEE J. Sel. Top. Quantum Electron. 39, 3-12 (2003).
[CrossRef]

K. Motzek, P. Jander, A. Desyatnikov, M. Belic, C. Denz, and F. Kaiser, "Dynamic counterpropagating vector solitons in saturable self-focusing media," Phys. Rev. E 68, 066,611-4 (2003).
[CrossRef]

M. R. Belic, P. Jander, A. Strinic, A. Desyatnikov, and C. Denz, "Self-trapped bidirectional waveguides in a saturable photorefractive medium," Phys. Rev. E 68, R025,601-4 (2003).
[CrossRef]

D. Neshev, E. A. Ostrovskaya, Y. Kivshar, and W. Krolikowski, "Spatial solitons in optically induced gratings," Opt. Lett. 28, 710-712 (2003).
[CrossRef] [PubMed]

2002

O. Cohen, S. Lan, T. Carmon, J. A. Giordmaine, and M. Segev, "Spatial vector solitons consisting of counterpropagating fields," Opt. Lett. 27(22), 2013 (2002).
[CrossRef]

C. Weilnau, M. Ahles, J. Petter, D. Träger, J. Schröder, and C. Denz, "Spatial optical (2+1)-dimensional scalarand vector-solitons in saturable nonlinear media," Ann. Phys. 11, 573-629 (2002).
[CrossRef]

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, "Collisions between Optical Spatial Solitons Propagating in Opposite Directions," Phys. Rev. Lett. 89, 133,901-4 (2002).

D. Kip, C. Herden, and M. Wesner, "All-Optical Signal Routing Using Interaction ofMutually Incoherent Spatial Solitons," Ferroelectrics 274, 135-142 (2002).

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, "Discrete solitons in photorefractive optically induced photonic lattices," Phys. Rev. E 66, 046,602-5 (2002).
[CrossRef]

1999

F. T. Arecchi, S. Boccaletti, and P. Ramazza, "Pattern formation and competition in nonlinear optics," Phys. Rep. 318, 1-83 (1999).
[CrossRef]

G. I. Stegeman and M. Segev, "Optical Spatial Solitons and Their Interactions: Universality and Diversity," Science 286, 1518-1523 (1999).
[CrossRef] [PubMed]

1998

W. Krolikowski, M. Saffman, B. Luther-Davies, and C. Denz, "Anomalous Interaction of Spatial Solitons in Photorefractive Media," Phys. Rev. Lett. 80, 3240-3243 (1998).
[CrossRef]

1995

M. Carvalho, S. Singh, and D. Christodoulides, "Self-deflection of steady-state bright spatial solitons in biased photorefractive crystals," Opt. Commun. 120(5), 311-315 (1995).
[CrossRef]

M. Shih, M. Segev, G. Valley, G. Salomono, B. Crosignani, and P. DiPorto, "Observation of two-dimensional steady-state photorefractive screening solitons," Electron. Lett. 31(10), 826-828 (1995).
[CrossRef]

1994

J. B. Geddes, R. A. Indik, J. V. Moloney, and W. J. Firth, "Hexagons and squares in a passive nonlinear optical system," Phys. Rev. A 50, 3471-3485, (1994).
[CrossRef] [PubMed]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, "Steady-State Spatial Screening Solitons in Photorefractive Materials with External Applied Field," Phys. Rev. Lett. 73, 3211-3214 (1994).
[CrossRef] [PubMed]

1993

M. Haelterman, A. P. Sheppard, and A. W. Snyder, "Bimodal counterpropagating spatial solitary-waves," Opt. Commun. 103, 145-152 (1993).
[CrossRef]

1990

1988

1982

Y. Silberberg and I. B. Joseph, "Instabilities, Self-Oscillation, and Chaos in a Simple Nonlinear Optical Interaction," Phys. Rev. Lett. 48, 1541-1543 (1982). URL http://link.aps.org/abstract/PRL/v48/p1541.
[CrossRef]

1973

S. Somekh, E. Gamire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-48 (1973).
[CrossRef]

1965

Ahles, M.

C. Weilnau, M. Ahles, J. Petter, D. Träger, J. Schröder, and C. Denz, "Spatial optical (2+1)-dimensional scalarand vector-solitons in saturable nonlinear media," Ann. Phys. 11, 573-629 (2002).
[CrossRef]

Arecchi, F. T.

F. T. Arecchi, S. Boccaletti, and P. Ramazza, "Pattern formation and competition in nonlinear optics," Phys. Rep. 318, 1-83 (1999).
[CrossRef]

Belic, M.

M. Belic, P. Jander, K. Motzek, A. Desyatnikov, D. Jovic, A. Strinic, M. Petrovic, C. Denz, and F. Kaiser, "Counterpropagating self-trapped beams in photorefractive crystals," J. Opt. B 6, S190-S196 (2004).
[CrossRef]

K. Motzek, P. Jander, A. Desyatnikov, M. Belic, C. Denz, and F. Kaiser, "Dynamic counterpropagating vector solitons in saturable self-focusing media," Phys. Rev. E 68, 066,611-4 (2003).
[CrossRef]

Belic, M. R.

P. Jander, J. Schröder, T. Richter, K. Motzek, F. Kaiser, M. R. Belic, and C. Denz, "Dynamic instability of counterpropagating self-trapped beams in photorefractive media," Proc. SPIE 6255, 62,550A (2006).

P. Jander, J. Schröder, C. Denz, M. Petrovic, and M. R. Belic, "Dynamic instability of self-induced bidirectional waveguides in photorefractive media," Opt. Lett. 30, 750-752 (2005).
[CrossRef] [PubMed]

M. R. Belic, P. Jander, A. Strinic, A. Desyatnikov, and C. Denz, "Self-trapped bidirectional waveguides in a saturable photorefractive medium," Phys. Rev. E 68, R025,601-4 (2003).
[CrossRef]

Boccaletti, S.

F. T. Arecchi, S. Boccaletti, and P. Ramazza, "Pattern formation and competition in nonlinear optics," Phys. Rep. 318, 1-83 (1999).
[CrossRef]

Carmon, T.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, "Collisions between Optical Spatial Solitons Propagating in Opposite Directions," Phys. Rev. Lett. 89, 133,901-4 (2002).

O. Cohen, S. Lan, T. Carmon, J. A. Giordmaine, and M. Segev, "Spatial vector solitons consisting of counterpropagating fields," Opt. Lett. 27(22), 2013 (2002).
[CrossRef]

Carvalho, M.

M. Carvalho, S. Singh, and D. Christodoulides, "Self-deflection of steady-state bright spatial solitons in biased photorefractive crystals," Opt. Commun. 120(5), 311-315 (1995).
[CrossRef]

Christodoulides, D.

M. Carvalho, S. Singh, and D. Christodoulides, "Self-deflection of steady-state bright spatial solitons in biased photorefractive crystals," Opt. Commun. 120(5), 311-315 (1995).
[CrossRef]

Christodoulides, D. N.

J.W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003).
[CrossRef] [PubMed]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, "Discrete solitons in photorefractive optically induced photonic lattices," Phys. Rev. E 66, 046,602-5 (2002).
[CrossRef]

D. N. Christodoulides and R. I. Joseph, "Discrete self-focusing in nonlinear arrays of coupled waveguides," Opt. Lett. 13, 794-796 (1988).
[CrossRef] [PubMed]

Cohen, O.

O. Cohen, S. Lan, T. Carmon, J. A. Giordmaine, and M. Segev, "Spatial vector solitons consisting of counterpropagating fields," Opt. Lett. 27(22), 2013 (2002).
[CrossRef]

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, "Collisions between Optical Spatial Solitons Propagating in Opposite Directions," Phys. Rev. Lett. 89, 133,901-4 (2002).

Crosignani, B.

M. Shih, M. Segev, G. Valley, G. Salomono, B. Crosignani, and P. DiPorto, "Observation of two-dimensional steady-state photorefractive screening solitons," Electron. Lett. 31(10), 826-828 (1995).
[CrossRef]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, "Steady-State Spatial Screening Solitons in Photorefractive Materials with External Applied Field," Phys. Rev. Lett. 73, 3211-3214 (1994).
[CrossRef] [PubMed]

Denz, C.

D. Träger, N. Sagemerten, and C. Denz, "Guiding of DynamicallyModulated Signals in Arrays of Photorefractive Spatial Solitons," IEEE J. Sel. Top. Quantum Electron. 12, 383-387 (2006).
[CrossRef]

P. Jander, J. Schröder, T. Richter, K. Motzek, F. Kaiser, M. R. Belic, and C. Denz, "Dynamic instability of counterpropagating self-trapped beams in photorefractive media," Proc. SPIE 6255, 62,550A (2006).

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, C. Denz, and Y. S. Kivshar, "Reducedsymmetry two-dimensional solitons in photonic lattices," Phys. Rev. Lett. 96(2), 023,905-4 (2006). URL http://link.aps.org/abstract/PRL/v96/p023905.
[CrossRef]

D. Träger, R. Fischer, D. N. Neshev, A. A. Sukhorukov, C. Denz, W. Krolikowski, and Y. S. Kivshar, "Nonlinear Bloch modes in two-dimensional photonic lattices," Opt. Express 14(5), 1913-1923 (2006). URL http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-5-1913
[CrossRef] [PubMed]

P. Jander, J. Schröder, C. Denz, M. Petrovic, and M. R. Belic, "Dynamic instability of self-induced bidirectional waveguides in photorefractive media," Opt. Lett. 30, 750-752 (2005).
[CrossRef] [PubMed]

M. Belic, P. Jander, K. Motzek, A. Desyatnikov, D. Jovic, A. Strinic, M. Petrovic, C. Denz, and F. Kaiser, "Counterpropagating self-trapped beams in photorefractive crystals," J. Opt. B 6, S190-S196 (2004).
[CrossRef]

M. R. Belic, P. Jander, A. Strinic, A. Desyatnikov, and C. Denz, "Self-trapped bidirectional waveguides in a saturable photorefractive medium," Phys. Rev. E 68, R025,601-4 (2003).
[CrossRef]

K. Motzek, P. Jander, A. Desyatnikov, M. Belic, C. Denz, and F. Kaiser, "Dynamic counterpropagating vector solitons in saturable self-focusing media," Phys. Rev. E 68, 066,611-4 (2003).
[CrossRef]

W. Krolikowski, B. Luther-Davies, and C. Denz, "Photorefractive Solitons," IEEE J. Sel. Top. Quantum Electron. 39, 3-12 (2003).
[CrossRef]

C. Weilnau, M. Ahles, J. Petter, D. Träger, J. Schröder, and C. Denz, "Spatial optical (2+1)-dimensional scalarand vector-solitons in saturable nonlinear media," Ann. Phys. 11, 573-629 (2002).
[CrossRef]

W. Krolikowski, M. Saffman, B. Luther-Davies, and C. Denz, "Anomalous Interaction of Spatial Solitons in Photorefractive Media," Phys. Rev. Lett. 80, 3240-3243 (1998).
[CrossRef]

Desyatnikov, A.

M. Belic, P. Jander, K. Motzek, A. Desyatnikov, D. Jovic, A. Strinic, M. Petrovic, C. Denz, and F. Kaiser, "Counterpropagating self-trapped beams in photorefractive crystals," J. Opt. B 6, S190-S196 (2004).
[CrossRef]

K. Motzek, P. Jander, A. Desyatnikov, M. Belic, C. Denz, and F. Kaiser, "Dynamic counterpropagating vector solitons in saturable self-focusing media," Phys. Rev. E 68, 066,611-4 (2003).
[CrossRef]

M. R. Belic, P. Jander, A. Strinic, A. Desyatnikov, and C. Denz, "Self-trapped bidirectional waveguides in a saturable photorefractive medium," Phys. Rev. E 68, R025,601-4 (2003).
[CrossRef]

Desyatnikov, A. S.

DiPorto, P.

M. Shih, M. Segev, G. Valley, G. Salomono, B. Crosignani, and P. DiPorto, "Observation of two-dimensional steady-state photorefractive screening solitons," Electron. Lett. 31(10), 826-828 (1995).
[CrossRef]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, "Steady-State Spatial Screening Solitons in Photorefractive Materials with External Applied Field," Phys. Rev. Lett. 73, 3211-3214 (1994).
[CrossRef] [PubMed]

Efremidis, N. K.

J.W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003).
[CrossRef] [PubMed]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, "Discrete solitons in photorefractive optically induced photonic lattices," Phys. Rev. E 66, 046,602-5 (2002).
[CrossRef]

Firth, W. J.

Fischer, R.

Fitzgerald, A.

Fleischer, J. W.

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, "Discrete solitons in photorefractive optically induced photonic lattices," Phys. Rev. E 66, 046,602-5 (2002).
[CrossRef]

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, "Collisions between Optical Spatial Solitons Propagating in Opposite Directions," Phys. Rev. Lett. 89, 133,901-4 (2002).

Fleischer, J.W.

J.W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003).
[CrossRef] [PubMed]

Gamire, E.

S. Somekh, E. Gamire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-48 (1973).
[CrossRef]

Garvin, H. L.

S. Somekh, E. Gamire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-48 (1973).
[CrossRef]

Geddes, J. B.

J. B. Geddes, R. A. Indik, J. V. Moloney, and W. J. Firth, "Hexagons and squares in a passive nonlinear optical system," Phys. Rev. A 50, 3471-3485, (1994).
[CrossRef] [PubMed]

Giordmaine, J. A.

Haelterman, M.

M. Haelterman, A. P. Sheppard, and A. W. Snyder, "Bimodal counterpropagating spatial solitary-waves," Opt. Commun. 103, 145-152 (1993).
[CrossRef]

Herden, C.

D. Kip, C. Herden, and M. Wesner, "All-Optical Signal Routing Using Interaction ofMutually Incoherent Spatial Solitons," Ferroelectrics 274, 135-142 (2002).

Hunsperger, R. G.

S. Somekh, E. Gamire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-48 (1973).
[CrossRef]

Indik, R. A.

J. B. Geddes, R. A. Indik, J. V. Moloney, and W. J. Firth, "Hexagons and squares in a passive nonlinear optical system," Phys. Rev. A 50, 3471-3485, (1994).
[CrossRef] [PubMed]

Jander, P.

P. Jander, J. Schröder, T. Richter, K. Motzek, F. Kaiser, M. R. Belic, and C. Denz, "Dynamic instability of counterpropagating self-trapped beams in photorefractive media," Proc. SPIE 6255, 62,550A (2006).

P. Jander, J. Schröder, C. Denz, M. Petrovic, and M. R. Belic, "Dynamic instability of self-induced bidirectional waveguides in photorefractive media," Opt. Lett. 30, 750-752 (2005).
[CrossRef] [PubMed]

M. Belic, P. Jander, K. Motzek, A. Desyatnikov, D. Jovic, A. Strinic, M. Petrovic, C. Denz, and F. Kaiser, "Counterpropagating self-trapped beams in photorefractive crystals," J. Opt. B 6, S190-S196 (2004).
[CrossRef]

M. R. Belic, P. Jander, A. Strinic, A. Desyatnikov, and C. Denz, "Self-trapped bidirectional waveguides in a saturable photorefractive medium," Phys. Rev. E 68, R025,601-4 (2003).
[CrossRef]

K. Motzek, P. Jander, A. Desyatnikov, M. Belic, C. Denz, and F. Kaiser, "Dynamic counterpropagating vector solitons in saturable self-focusing media," Phys. Rev. E 68, 066,611-4 (2003).
[CrossRef]

Jones, A. L.

Joseph, I. B.

Y. Silberberg and I. B. Joseph, "Instabilities, Self-Oscillation, and Chaos in a Simple Nonlinear Optical Interaction," Phys. Rev. Lett. 48, 1541-1543 (1982). URL http://link.aps.org/abstract/PRL/v48/p1541.
[CrossRef]

Joseph, R. I.

Jovic, D.

M. Belic, P. Jander, K. Motzek, A. Desyatnikov, D. Jovic, A. Strinic, M. Petrovic, C. Denz, and F. Kaiser, "Counterpropagating self-trapped beams in photorefractive crystals," J. Opt. B 6, S190-S196 (2004).
[CrossRef]

Kaiser, F.

P. Jander, J. Schröder, T. Richter, K. Motzek, F. Kaiser, M. R. Belic, and C. Denz, "Dynamic instability of counterpropagating self-trapped beams in photorefractive media," Proc. SPIE 6255, 62,550A (2006).

M. Belic, P. Jander, K. Motzek, A. Desyatnikov, D. Jovic, A. Strinic, M. Petrovic, C. Denz, and F. Kaiser, "Counterpropagating self-trapped beams in photorefractive crystals," J. Opt. B 6, S190-S196 (2004).
[CrossRef]

K. Motzek, P. Jander, A. Desyatnikov, M. Belic, C. Denz, and F. Kaiser, "Dynamic counterpropagating vector solitons in saturable self-focusing media," Phys. Rev. E 68, 066,611-4 (2003).
[CrossRef]

Kip, D.

E. Smirnov, M. Stepic, C. Ruter, V. Shandarov, and D. Kip, "Interaction of counterpropagating discrete solitons in a nonlinear one-dimensional waveguide array," Opt. Lett. 32, 512-514 (2007).
[CrossRef] [PubMed]

D. Kip, C. Herden, and M. Wesner, "All-Optical Signal Routing Using Interaction ofMutually Incoherent Spatial Solitons," Ferroelectrics 274, 135-142 (2002).

Kivshar, Y.

Kivshar, Y. S.

Krolikowski, W.

D. Träger, R. Fischer, D. N. Neshev, A. A. Sukhorukov, C. Denz, W. Krolikowski, and Y. S. Kivshar, "Nonlinear Bloch modes in two-dimensional photonic lattices," Opt. Express 14(5), 1913-1923 (2006). URL http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-5-1913
[CrossRef] [PubMed]

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, C. Denz, and Y. S. Kivshar, "Reducedsymmetry two-dimensional solitons in photonic lattices," Phys. Rev. Lett. 96(2), 023,905-4 (2006). URL http://link.aps.org/abstract/PRL/v96/p023905.
[CrossRef]

D. Neshev, E. A. Ostrovskaya, Y. Kivshar, and W. Krolikowski, "Spatial solitons in optically induced gratings," Opt. Lett. 28, 710-712 (2003).
[CrossRef] [PubMed]

W. Krolikowski, B. Luther-Davies, and C. Denz, "Photorefractive Solitons," IEEE J. Sel. Top. Quantum Electron. 39, 3-12 (2003).
[CrossRef]

W. Krolikowski, M. Saffman, B. Luther-Davies, and C. Denz, "Anomalous Interaction of Spatial Solitons in Photorefractive Media," Phys. Rev. Lett. 80, 3240-3243 (1998).
[CrossRef]

Lan, S.

Lederer, F.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003).
[CrossRef] [PubMed]

Luther-Davies, B.

W. Krolikowski, B. Luther-Davies, and C. Denz, "Photorefractive Solitons," IEEE J. Sel. Top. Quantum Electron. 39, 3-12 (2003).
[CrossRef]

W. Krolikowski, M. Saffman, B. Luther-Davies, and C. Denz, "Anomalous Interaction of Spatial Solitons in Photorefractive Media," Phys. Rev. Lett. 80, 3240-3243 (1998).
[CrossRef]

Moloney, J. V.

J. B. Geddes, R. A. Indik, J. V. Moloney, and W. J. Firth, "Hexagons and squares in a passive nonlinear optical system," Phys. Rev. A 50, 3471-3485, (1994).
[CrossRef] [PubMed]

Motzek, K.

P. Jander, J. Schröder, T. Richter, K. Motzek, F. Kaiser, M. R. Belic, and C. Denz, "Dynamic instability of counterpropagating self-trapped beams in photorefractive media," Proc. SPIE 6255, 62,550A (2006).

M. Belic, P. Jander, K. Motzek, A. Desyatnikov, D. Jovic, A. Strinic, M. Petrovic, C. Denz, and F. Kaiser, "Counterpropagating self-trapped beams in photorefractive crystals," J. Opt. B 6, S190-S196 (2004).
[CrossRef]

K. Motzek, P. Jander, A. Desyatnikov, M. Belic, C. Denz, and F. Kaiser, "Dynamic counterpropagating vector solitons in saturable self-focusing media," Phys. Rev. E 68, 066,611-4 (2003).
[CrossRef]

Neshev, D.

Neshev, D. N.

Odoulov, S.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, "Collisions between Optical Spatial Solitons Propagating in Opposite Directions," Phys. Rev. Lett. 89, 133,901-4 (2002).

Ostrovskaya, E. A.

Pare, C.

Petrovic, M.

P. Jander, J. Schröder, C. Denz, M. Petrovic, and M. R. Belic, "Dynamic instability of self-induced bidirectional waveguides in photorefractive media," Opt. Lett. 30, 750-752 (2005).
[CrossRef] [PubMed]

M. Belic, P. Jander, K. Motzek, A. Desyatnikov, D. Jovic, A. Strinic, M. Petrovic, C. Denz, and F. Kaiser, "Counterpropagating self-trapped beams in photorefractive crystals," J. Opt. B 6, S190-S196 (2004).
[CrossRef]

Petter, J.

C. Weilnau, M. Ahles, J. Petter, D. Träger, J. Schröder, and C. Denz, "Spatial optical (2+1)-dimensional scalarand vector-solitons in saturable nonlinear media," Ann. Phys. 11, 573-629 (2002).
[CrossRef]

Ramazza, P.

F. T. Arecchi, S. Boccaletti, and P. Ramazza, "Pattern formation and competition in nonlinear optics," Phys. Rep. 318, 1-83 (1999).
[CrossRef]

Richter, T.

P. Jander, J. Schröder, T. Richter, K. Motzek, F. Kaiser, M. R. Belic, and C. Denz, "Dynamic instability of counterpropagating self-trapped beams in photorefractive media," Proc. SPIE 6255, 62,550A (2006).

Ruter, C.

Saffman, M.

W. Krolikowski, M. Saffman, B. Luther-Davies, and C. Denz, "Anomalous Interaction of Spatial Solitons in Photorefractive Media," Phys. Rev. Lett. 80, 3240-3243 (1998).
[CrossRef]

Sagemerten, N.

Salomono, G.

M. Shih, M. Segev, G. Valley, G. Salomono, B. Crosignani, and P. DiPorto, "Observation of two-dimensional steady-state photorefractive screening solitons," Electron. Lett. 31(10), 826-828 (1995).
[CrossRef]

Schröder, J.

P. Jander, J. Schröder, T. Richter, K. Motzek, F. Kaiser, M. R. Belic, and C. Denz, "Dynamic instability of counterpropagating self-trapped beams in photorefractive media," Proc. SPIE 6255, 62,550A (2006).

P. Jander, J. Schröder, C. Denz, M. Petrovic, and M. R. Belic, "Dynamic instability of self-induced bidirectional waveguides in photorefractive media," Opt. Lett. 30, 750-752 (2005).
[CrossRef] [PubMed]

C. Weilnau, M. Ahles, J. Petter, D. Träger, J. Schröder, and C. Denz, "Spatial optical (2+1)-dimensional scalarand vector-solitons in saturable nonlinear media," Ann. Phys. 11, 573-629 (2002).
[CrossRef]

Sears, S.

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, "Discrete solitons in photorefractive optically induced photonic lattices," Phys. Rev. E 66, 046,602-5 (2002).
[CrossRef]

Segev, M.

J.W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003).
[CrossRef] [PubMed]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, "Discrete solitons in photorefractive optically induced photonic lattices," Phys. Rev. E 66, 046,602-5 (2002).
[CrossRef]

O. Cohen, S. Lan, T. Carmon, J. A. Giordmaine, and M. Segev, "Spatial vector solitons consisting of counterpropagating fields," Opt. Lett. 27(22), 2013 (2002).
[CrossRef]

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, "Collisions between Optical Spatial Solitons Propagating in Opposite Directions," Phys. Rev. Lett. 89, 133,901-4 (2002).

G. I. Stegeman and M. Segev, "Optical Spatial Solitons and Their Interactions: Universality and Diversity," Science 286, 1518-1523 (1999).
[CrossRef] [PubMed]

M. Shih, M. Segev, G. Valley, G. Salomono, B. Crosignani, and P. DiPorto, "Observation of two-dimensional steady-state photorefractive screening solitons," Electron. Lett. 31(10), 826-828 (1995).
[CrossRef]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, "Steady-State Spatial Screening Solitons in Photorefractive Materials with External Applied Field," Phys. Rev. Lett. 73, 3211-3214 (1994).
[CrossRef] [PubMed]

Shandarov, V.

Sheppard, A. P.

M. Haelterman, A. P. Sheppard, and A. W. Snyder, "Bimodal counterpropagating spatial solitary-waves," Opt. Commun. 103, 145-152 (1993).
[CrossRef]

Shih, M.

M. Shih, M. Segev, G. Valley, G. Salomono, B. Crosignani, and P. DiPorto, "Observation of two-dimensional steady-state photorefractive screening solitons," Electron. Lett. 31(10), 826-828 (1995).
[CrossRef]

Silberberg, Y.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003).
[CrossRef] [PubMed]

Y. Silberberg and I. B. Joseph, "Instabilities, Self-Oscillation, and Chaos in a Simple Nonlinear Optical Interaction," Phys. Rev. Lett. 48, 1541-1543 (1982). URL http://link.aps.org/abstract/PRL/v48/p1541.
[CrossRef]

Singh, S.

M. Carvalho, S. Singh, and D. Christodoulides, "Self-deflection of steady-state bright spatial solitons in biased photorefractive crystals," Opt. Commun. 120(5), 311-315 (1995).
[CrossRef]

Smirnov, E.

Snyder, A. W.

M. Haelterman, A. P. Sheppard, and A. W. Snyder, "Bimodal counterpropagating spatial solitary-waves," Opt. Commun. 103, 145-152 (1993).
[CrossRef]

Somekh, S.

S. Somekh, E. Gamire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-48 (1973).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman and M. Segev, "Optical Spatial Solitons and Their Interactions: Universality and Diversity," Science 286, 1518-1523 (1999).
[CrossRef] [PubMed]

Stepic, M.

Strinic, A.

M. Belic, P. Jander, K. Motzek, A. Desyatnikov, D. Jovic, A. Strinic, M. Petrovic, C. Denz, and F. Kaiser, "Counterpropagating self-trapped beams in photorefractive crystals," J. Opt. B 6, S190-S196 (2004).
[CrossRef]

M. R. Belic, P. Jander, A. Strinic, A. Desyatnikov, and C. Denz, "Self-trapped bidirectional waveguides in a saturable photorefractive medium," Phys. Rev. E 68, R025,601-4 (2003).
[CrossRef]

Sukhorukov, A. A.

Träger, D.

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, C. Denz, and Y. S. Kivshar, "Reducedsymmetry two-dimensional solitons in photonic lattices," Phys. Rev. Lett. 96(2), 023,905-4 (2006). URL http://link.aps.org/abstract/PRL/v96/p023905.
[CrossRef]

D. Träger, R. Fischer, D. N. Neshev, A. A. Sukhorukov, C. Denz, W. Krolikowski, and Y. S. Kivshar, "Nonlinear Bloch modes in two-dimensional photonic lattices," Opt. Express 14(5), 1913-1923 (2006). URL http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-5-1913
[CrossRef] [PubMed]

D. Träger, N. Sagemerten, and C. Denz, "Guiding of DynamicallyModulated Signals in Arrays of Photorefractive Spatial Solitons," IEEE J. Sel. Top. Quantum Electron. 12, 383-387 (2006).
[CrossRef]

A. S. Desyatnikov, D. N. Neshev, Y. S. Kivshar, N. Sagemerten, D. Träger, J. Jagers, C. Denz, and Y. V. Kartashov, "Nonlinear photonic lattices in anisotropic nonlocal self-focusing media," Opt. Lett. 30(8), 869-871 (2005). URL http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-8-869
[CrossRef] [PubMed]

C. Weilnau, M. Ahles, J. Petter, D. Träger, J. Schröder, and C. Denz, "Spatial optical (2+1)-dimensional scalarand vector-solitons in saturable nonlinear media," Ann. Phys. 11, 573-629 (2002).
[CrossRef]

Uzdin, R.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, "Collisions between Optical Spatial Solitons Propagating in Opposite Directions," Phys. Rev. Lett. 89, 133,901-4 (2002).

Valley, G.

M. Shih, M. Segev, G. Valley, G. Salomono, B. Crosignani, and P. DiPorto, "Observation of two-dimensional steady-state photorefractive screening solitons," Electron. Lett. 31(10), 826-828 (1995).
[CrossRef]

Valley, G. C.

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, "Steady-State Spatial Screening Solitons in Photorefractive Materials with External Applied Field," Phys. Rev. Lett. 73, 3211-3214 (1994).
[CrossRef] [PubMed]

Weilnau, C.

C. Weilnau, M. Ahles, J. Petter, D. Träger, J. Schröder, and C. Denz, "Spatial optical (2+1)-dimensional scalarand vector-solitons in saturable nonlinear media," Ann. Phys. 11, 573-629 (2002).
[CrossRef]

Wesner, M.

D. Kip, C. Herden, and M. Wesner, "All-Optical Signal Routing Using Interaction ofMutually Incoherent Spatial Solitons," Ferroelectrics 274, 135-142 (2002).

Yariv, A.

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, "Steady-State Spatial Screening Solitons in Photorefractive Materials with External Applied Field," Phys. Rev. Lett. 73, 3211-3214 (1994).
[CrossRef] [PubMed]

S. Somekh, E. Gamire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-48 (1973).
[CrossRef]

Ann. Phys.

C. Weilnau, M. Ahles, J. Petter, D. Träger, J. Schröder, and C. Denz, "Spatial optical (2+1)-dimensional scalarand vector-solitons in saturable nonlinear media," Ann. Phys. 11, 573-629 (2002).
[CrossRef]

Appl. Phys. Lett.

S. Somekh, E. Gamire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-48 (1973).
[CrossRef]

Electron. Lett.

M. Shih, M. Segev, G. Valley, G. Salomono, B. Crosignani, and P. DiPorto, "Observation of two-dimensional steady-state photorefractive screening solitons," Electron. Lett. 31(10), 826-828 (1995).
[CrossRef]

Ferroelectrics

D. Kip, C. Herden, and M. Wesner, "All-Optical Signal Routing Using Interaction ofMutually Incoherent Spatial Solitons," Ferroelectrics 274, 135-142 (2002).

IEEE J. Sel. Top. Quantum Electron.

W. Krolikowski, B. Luther-Davies, and C. Denz, "Photorefractive Solitons," IEEE J. Sel. Top. Quantum Electron. 39, 3-12 (2003).
[CrossRef]

D. Träger, N. Sagemerten, and C. Denz, "Guiding of DynamicallyModulated Signals in Arrays of Photorefractive Spatial Solitons," IEEE J. Sel. Top. Quantum Electron. 12, 383-387 (2006).
[CrossRef]

J. Opt. B

M. Belic, P. Jander, K. Motzek, A. Desyatnikov, D. Jovic, A. Strinic, M. Petrovic, C. Denz, and F. Kaiser, "Counterpropagating self-trapped beams in photorefractive crystals," J. Opt. B 6, S190-S196 (2004).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

Nature

J.W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003).
[CrossRef] [PubMed]

Opt. Commun.

M. Haelterman, A. P. Sheppard, and A. W. Snyder, "Bimodal counterpropagating spatial solitary-waves," Opt. Commun. 103, 145-152 (1993).
[CrossRef]

M. Carvalho, S. Singh, and D. Christodoulides, "Self-deflection of steady-state bright spatial solitons in biased photorefractive crystals," Opt. Commun. 120(5), 311-315 (1995).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rep.

F. T. Arecchi, S. Boccaletti, and P. Ramazza, "Pattern formation and competition in nonlinear optics," Phys. Rep. 318, 1-83 (1999).
[CrossRef]

Phys. Rev. A

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Phys. Rev. E

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Phys. Rev. Lett.

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Figures (11)

Fig. 1.
Fig. 1.

Steady state modulation of the refractive index (bottom) resulting from lattices with different peak intensities (top) in the applied isotropic model.

Fig. 2.
Fig. 2.

lodN for counterpropagating solitons launched on-site as a function of lattice intensity A for different lattice periods p. (a) period smaller than the beam diameter; (b) lattice period larger than the beam diameter. Non-monotonous decrease of lodN for p = 1d and A = 0.4 - 0.8, is due to long lasting transient dynamics.

Fig. 3.
Fig. 3.

Temporal evolution of the intensity distribution at one crystal face with increasing lattice peak intensity at the plane of incidence of F. The constant line (red) marks the input of F while the intensity distribution of B oscillates (green) since it is the output plane of beam B. Depicted is the evolution along the curve for periodicity equal to beam diameter. Qualitatively similar results are obtained at the plane of incidence of B for the transmitted beam F.

Fig. 4.
Fig. 4.

Illustration of the intensity distribution for the stabilized states shown for the same lattice peak intensity of A = 2.4 and periodicity of p = 0.5d in (a), p = 1.0d in (b) and p = 1.5d in (c). The blurred nature in (a) results from the mutual interaction of the solitons.

Fig. 5.
Fig. 5.

Illustration of the dynamic states for p = 0.25d and A = 2.2. The temporal evolution is depicted in (a) where as (b-d) show snapshots of the oscillation comprised of sequences of focussing and deflection.

Fig. 6.
Fig. 6.

Decrease in the dynamics of counterpropagating solitons launched off-site as a function of peak intensity and periodicity. (a) period smaller than the beam diameter; (b) period larger than the beam diameter.

Fig. 7.
Fig. 7.

Illustration of the increase of the short time scale dynamics for beams launched off-site on the lattice of periodicity p= 1d.

Fig. 8.
Fig. 8.

Experimental setup for the observation of counterpropagating solitons on a one-dimensional lattice. MO: microscope objective, PH: pinhole, L: lens, HWP: half wave plate, M: mirror, PBS: polarizing beam splitter, PM: piezo-mounted mirror, BS: beam splitter

Fig. 9.
Fig. 9.

Comparison between the modulation of the interference pattern (top) and the induced refractive index modulation (bottom) measured in terms of the intensity modulation of a guided plane wave. Shown are the results for lattice wave powers of 0.5mW (dashed line), 1.5mW (dotted line) and 2.5mW (dash-dot line).

Fig. 10.
Fig. 10.

Temporal evolution of the intensity distribution projected onto the x-axis (parallel to the c-axis) for different lattice powers of (a) 250μ W, (b) 1.0mW, (c) 1.5mW, (d) 2.0mW, and 2.5mW. The faint horizontal lines at x = 0μm mark the reflected beam at this crystal face which acts as a reference. Again the results for the other crystal face show similar behaviour.

Fig. 11.
Fig. 11.

Stabilization of the instability in two transverse directions: (a) 2D square lattice of 6μm period. (b) Input intensity distribution for the forward (F) and backwards (B) propagating beams, respectively. (c,d) Digitally combined beam profiles at the left and right face of the crystal when each soliton propagates independently. (e,f) Stabilization on the lattice for both faces of the crystal, respectively. (g) Position of the output beams center without the lattice (solid line) and with the lattice (dotted line) as a function of time.

Equations (4)

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i z F + x 2 F = Γ E sc F
i z B + x 2 B = Γ E sc B
τ ( I ) t E sc + E sc = I 1 + I ,
lod N = t = 1 N x [ F ( x , L , t ) F ( x , L , t 1 ) 2 F ( x , L , t ) 2 + B ( x , 0 , t ) B ( x , 0 , t 1 ) 2 B ( x , 0 , t ) 2 ]

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