Abstract

We investigate the existence and form of (2+1)-dimensional ground-state counterpropagating solitons in photorefractive media with saturable nonlinearity. General conditions for the existence of fundamental solitons in a local isotropic model that includes an intensity-dependent saturable nonlinearity are identified. We confirm our theoretical findings numerically and determine the ground-state profiles. We check their stability in propagation and identify the coupling constant threshold for their existence. Critical exponents of the power and beam width are determined as functions of the propagation constant at the threshold. We finally formulate a variational approach to the same problem, introduce an approximate fundamental Gaussian solution, and verify that this method leads to the same threshold and similar critical exponents as the theoretical and numerical methods.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  22. D. Jović, R. Jovanović, S. Prvanović, M. Petrović, and M. Belić, “Counterpropagating beams in rotationally symmetric photonic lattices,” Opt. Mater. 30, 1173–1176 (2008).
    [CrossRef]
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    [CrossRef]
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  29. N. Aleksić, M. Petrović, A. Strinić, and M. Belić, “Solitons in highly nonlocal nematic liquid crystals: variational approach,” Phys. Rev. A 85, 033826 (2012).
    [CrossRef]
  30. K. Motzek, P. Jander, A. Desyatnikov, M. Belić, C. Denz, and F. Kaiser, “Dynamic counterpropagating vector solitons in saturable self-focusing media,” Phys. Rev. E 68, 066611(2003).
    [CrossRef]
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    [CrossRef]

2012 (1)

N. Aleksić, M. Petrović, A. Strinić, and M. Belić, “Solitons in highly nonlocal nematic liquid crystals: variational approach,” Phys. Rev. A 85, 033826 (2012).
[CrossRef]

2011 (1)

M. S. Petrović, M. R. Belić, C. Denz, and Y. S. Kivshar, “Counterpropagating optical beams and solitons,” Laser Photon. Rev. 5, 214–233 (2011).
[CrossRef]

2009 (1)

2008 (1)

D. Jović, R. Jovanović, S. Prvanović, M. Petrović, and M. Belić, “Counterpropagating beams in rotationally symmetric photonic lattices,” Opt. Mater. 30, 1173–1176 (2008).
[CrossRef]

2006 (1)

2005 (2)

K. Motzek, M. Belić, T. Richter, C. Denz, A. Desyatnikov, P. Jander, and F. Kaiser, “Counterpropagating beams in biased photorefractive crystals: anisotropic theory,” Phys. Rev. E 71, 016610 (2005).
[CrossRef]

E. DelRe, A. D’Ercole, and E. Palange, “Mechanisms supporting long propagation regimes of photorefractive solitons,” Phys. Rev. E 71, 036610 (2005).
[CrossRef]

2004 (5)

E. DelRe, G. De Masi, A. Ciattoni, and E. Palange, “Pairing space-charge field conditions with self-guiding for the attainment of circular symmetry in photorefractive solitons,” Appl. Phys. Lett. 85, 5499–5501 (2004).
[CrossRef]

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717–754 (2004).
[CrossRef]

M. Belić, P. Jander, K. Motzek, A. Desyatnikov, D. Jović, A. Strinić, M. Petrović, C. Denz, and F. Kaiser, “Counterpropagating self-trapped beams in photorefractive crystals,” J. Opt. B Quantum. Semiclass. Opt. 6, S190–S196 (2004).
[CrossRef]

J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, “Dipole and quadrupole solitons in optically induced two-dimensional photonic lattices: theory and experiment,” Stud. Appl. Math. 113, 389–412 (2004).
[CrossRef]

C. Rotschild, O. Cohen, O. Manela, T. Carmon, and M. Segev, “Interactions between spatial screening solitons propagating in opposite directions,” J. Opt. Soc. Am. B 21, 1354–1357(2004).
[CrossRef]

2003 (1)

K. Motzek, P. Jander, A. Desyatnikov, M. Belić, C. Denz, and F. Kaiser, “Dynamic counterpropagating vector solitons in saturable self-focusing media,” Phys. Rev. E 68, 066611(2003).
[CrossRef]

2002 (3)

D. Kip, C. Herden, and M. Wesner, “All-optical signal routing using interaction of mutually incoherent spatial solitons,” Ferroelectrics 274, 135–142 (2002).

M. R. Belić, D. Vujić, A. Stepken, F. Kaiser, G. F. Calvo, F. Agullo-Lopez, and M. Carrascosa, “Isotropic versus anisotropic modeling of photorefractive solitons,” Phys. Rev. E 65, 066610 (2002).
[CrossRef]

O. Cohen, S. Lan, T. Carmon, J. A. Giordmaine, and M. Segev, “Spatial vector solitons consisting of counterpropagating fields,” Opt. Lett. 27, 2013–2015 (2002).
[CrossRef]

2001 (1)

1999 (1)

E. DelRe, A. Ciattoni, B. Crosignani, and P. Di Porto, “Nonlinear optical propagation phenomena in near-transition centrosymmetric photorefractive crystals,” J. Nonlinear Opt. Phys. Mater. 8, 1–20 (1999).
[CrossRef]

1997 (1)

1988 (1)

M. V. Tratnik and J. E. Sipe, “Bound solitary waves in a birefringent optical fibre,” Phys. Rev. A 38, 2011–2017 (1988).
[CrossRef]

1983 (1)

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

1976 (1)

V. I. Petviashvili, “Equation of an extraordinary soliton,” Fiz. Plazmy 2, 469–471 (1976).

Ablowitz, M. J.

M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations, and Inverse Scattering (Cambridge University, 1991).

Agranat, A. J.

Agrawal, G. P.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Agullo-Lopez, F.

M. R. Belić, D. Vujić, A. Stepken, F. Kaiser, G. F. Calvo, F. Agullo-Lopez, and M. Carrascosa, “Isotropic versus anisotropic modeling of photorefractive solitons,” Phys. Rev. E 65, 066610 (2002).
[CrossRef]

Akhmediev, N. N.

N. N. Akhmediev and A. A. Ankiewicz, Solitons (Chapman & Hall, 1997).

Aleksic, N.

N. Aleksić, M. Petrović, A. Strinić, and M. Belić, “Solitons in highly nonlocal nematic liquid crystals: variational approach,” Phys. Rev. A 85, 033826 (2012).
[CrossRef]

Aleksic, N. B.

M. S. Petrović, A. I. Strinić, N. B. Aleksić, and M. R. Belić, “Do shape invariant solitons in highly nonlocal nematic liquid crystals really exist?,” arXiv:1110.5053v1 [physics.optics] (2011).

Anderson, D.

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

Ankiewicz, A. A.

N. N. Akhmediev and A. A. Ankiewicz, Solitons (Chapman & Hall, 1997).

Arsenovic, D.

Belic, M.

N. Aleksić, M. Petrović, A. Strinić, and M. Belić, “Solitons in highly nonlocal nematic liquid crystals: variational approach,” Phys. Rev. A 85, 033826 (2012).
[CrossRef]

D. Jović, R. Jovanović, S. Prvanović, M. Petrović, and M. Belić, “Counterpropagating beams in rotationally symmetric photonic lattices,” Opt. Mater. 30, 1173–1176 (2008).
[CrossRef]

M. Belić, D. Jović, S. Prvanović, D. Arsenović, and M. Petrović, “Counterpropagating self-trapped beams in optical photonic lattices,” Opt. Express 14, 794–799 (2006).
[CrossRef]

K. Motzek, M. Belić, T. Richter, C. Denz, A. Desyatnikov, P. Jander, and F. Kaiser, “Counterpropagating beams in biased photorefractive crystals: anisotropic theory,” Phys. Rev. E 71, 016610 (2005).
[CrossRef]

M. Belić, P. Jander, K. Motzek, A. Desyatnikov, D. Jović, A. Strinić, M. Petrović, C. Denz, and F. Kaiser, “Counterpropagating self-trapped beams in photorefractive crystals,” J. Opt. B Quantum. Semiclass. Opt. 6, S190–S196 (2004).
[CrossRef]

K. Motzek, P. Jander, A. Desyatnikov, M. Belić, C. Denz, and F. Kaiser, “Dynamic counterpropagating vector solitons in saturable self-focusing media,” Phys. Rev. E 68, 066611(2003).
[CrossRef]

Belic, M. R.

M. S. Petrović, M. R. Belić, C. Denz, and Y. S. Kivshar, “Counterpropagating optical beams and solitons,” Laser Photon. Rev. 5, 214–233 (2011).
[CrossRef]

M. R. Belić, D. Vujić, A. Stepken, F. Kaiser, G. F. Calvo, F. Agullo-Lopez, and M. Carrascosa, “Isotropic versus anisotropic modeling of photorefractive solitons,” Phys. Rev. E 65, 066610 (2002).
[CrossRef]

M. S. Petrović, A. I. Strinić, N. B. Aleksić, and M. R. Belić, “Do shape invariant solitons in highly nonlocal nematic liquid crystals really exist?,” arXiv:1110.5053v1 [physics.optics] (2011).

T. C. Lin, M. R. Belić, M. S. Petrović, and G. Chen, “Ground state of nonlinear Schrödinger systems with saturable nonlinearity,” arXiv:1208.6259v1 [math-ph] (2012).

Bezryadina, A.

J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, “Dipole and quadrupole solitons in optically induced two-dimensional photonic lattices: theory and experiment,” Stud. Appl. Math. 113, 389–412 (2004).
[CrossRef]

Calvo, G. F.

M. R. Belić, D. Vujić, A. Stepken, F. Kaiser, G. F. Calvo, F. Agullo-Lopez, and M. Carrascosa, “Isotropic versus anisotropic modeling of photorefractive solitons,” Phys. Rev. E 65, 066610 (2002).
[CrossRef]

Carmon, T.

Carrascosa, M.

M. R. Belić, D. Vujić, A. Stepken, F. Kaiser, G. F. Calvo, F. Agullo-Lopez, and M. Carrascosa, “Isotropic versus anisotropic modeling of photorefractive solitons,” Phys. Rev. E 65, 066610 (2002).
[CrossRef]

Chen, G.

T. C. Lin, M. R. Belić, M. S. Petrović, and G. Chen, “Ground state of nonlinear Schrödinger systems with saturable nonlinearity,” arXiv:1208.6259v1 [math-ph] (2012).

Chen, Z.

J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, “Dipole and quadrupole solitons in optically induced two-dimensional photonic lattices: theory and experiment,” Stud. Appl. Math. 113, 389–412 (2004).
[CrossRef]

Ciattoni, A.

A. Ciattoni, A. Marini, C. Rizza, and E. DelRe, “Collision and fusion of counterpropagating micrometer-sized optical beams in periodically biased photorefractive crystals,” Opt. Lett. 34, 911–913 (2009).
[CrossRef]

E. DelRe, G. De Masi, A. Ciattoni, and E. Palange, “Pairing space-charge field conditions with self-guiding for the attainment of circular symmetry in photorefractive solitons,” Appl. Phys. Lett. 85, 5499–5501 (2004).
[CrossRef]

E. DelRe, A. Ciattoni, and A. J. Agranat, “Anisotropic charge displacement supporting isolated photorefractive optical needles,” Opt. Lett. 26, 908–910 (2001).
[CrossRef]

E. DelRe, A. Ciattoni, B. Crosignani, and P. Di Porto, “Nonlinear optical propagation phenomena in near-transition centrosymmetric photorefractive crystals,” J. Nonlinear Opt. Phys. Mater. 8, 1–20 (1999).
[CrossRef]

Clarkson, P. A.

M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations, and Inverse Scattering (Cambridge University, 1991).

Cohen, O.

Crosignani, B.

E. DelRe, A. Ciattoni, B. Crosignani, and P. Di Porto, “Nonlinear optical propagation phenomena in near-transition centrosymmetric photorefractive crystals,” J. Nonlinear Opt. Phys. Mater. 8, 1–20 (1999).
[CrossRef]

D’Ercole, A.

E. DelRe, A. D’Ercole, and E. Palange, “Mechanisms supporting long propagation regimes of photorefractive solitons,” Phys. Rev. E 71, 036610 (2005).
[CrossRef]

De Masi, G.

E. DelRe, G. De Masi, A. Ciattoni, and E. Palange, “Pairing space-charge field conditions with self-guiding for the attainment of circular symmetry in photorefractive solitons,” Appl. Phys. Lett. 85, 5499–5501 (2004).
[CrossRef]

DelRe, E.

A. Ciattoni, A. Marini, C. Rizza, and E. DelRe, “Collision and fusion of counterpropagating micrometer-sized optical beams in periodically biased photorefractive crystals,” Opt. Lett. 34, 911–913 (2009).
[CrossRef]

E. DelRe, A. D’Ercole, and E. Palange, “Mechanisms supporting long propagation regimes of photorefractive solitons,” Phys. Rev. E 71, 036610 (2005).
[CrossRef]

E. DelRe, G. De Masi, A. Ciattoni, and E. Palange, “Pairing space-charge field conditions with self-guiding for the attainment of circular symmetry in photorefractive solitons,” Appl. Phys. Lett. 85, 5499–5501 (2004).
[CrossRef]

E. DelRe, A. Ciattoni, and A. J. Agranat, “Anisotropic charge displacement supporting isolated photorefractive optical needles,” Opt. Lett. 26, 908–910 (2001).
[CrossRef]

E. DelRe, A. Ciattoni, B. Crosignani, and P. Di Porto, “Nonlinear optical propagation phenomena in near-transition centrosymmetric photorefractive crystals,” J. Nonlinear Opt. Phys. Mater. 8, 1–20 (1999).
[CrossRef]

Denz, C.

M. S. Petrović, M. R. Belić, C. Denz, and Y. S. Kivshar, “Counterpropagating optical beams and solitons,” Laser Photon. Rev. 5, 214–233 (2011).
[CrossRef]

K. Motzek, M. Belić, T. Richter, C. Denz, A. Desyatnikov, P. Jander, and F. Kaiser, “Counterpropagating beams in biased photorefractive crystals: anisotropic theory,” Phys. Rev. E 71, 016610 (2005).
[CrossRef]

M. Belić, P. Jander, K. Motzek, A. Desyatnikov, D. Jović, A. Strinić, M. Petrović, C. Denz, and F. Kaiser, “Counterpropagating self-trapped beams in photorefractive crystals,” J. Opt. B Quantum. Semiclass. Opt. 6, S190–S196 (2004).
[CrossRef]

K. Motzek, P. Jander, A. Desyatnikov, M. Belić, C. Denz, and F. Kaiser, “Dynamic counterpropagating vector solitons in saturable self-focusing media,” Phys. Rev. E 68, 066611(2003).
[CrossRef]

Desyatnikov, A.

K. Motzek, M. Belić, T. Richter, C. Denz, A. Desyatnikov, P. Jander, and F. Kaiser, “Counterpropagating beams in biased photorefractive crystals: anisotropic theory,” Phys. Rev. E 71, 016610 (2005).
[CrossRef]

M. Belić, P. Jander, K. Motzek, A. Desyatnikov, D. Jović, A. Strinić, M. Petrović, C. Denz, and F. Kaiser, “Counterpropagating self-trapped beams in photorefractive crystals,” J. Opt. B Quantum. Semiclass. Opt. 6, S190–S196 (2004).
[CrossRef]

K. Motzek, P. Jander, A. Desyatnikov, M. Belić, C. Denz, and F. Kaiser, “Dynamic counterpropagating vector solitons in saturable self-focusing media,” Phys. Rev. E 68, 066611(2003).
[CrossRef]

Di Porto, P.

E. DelRe, A. Ciattoni, B. Crosignani, and P. Di Porto, “Nonlinear optical propagation phenomena in near-transition centrosymmetric photorefractive crystals,” J. Nonlinear Opt. Phys. Mater. 8, 1–20 (1999).
[CrossRef]

Gatz, S.

Giordmaine, J. A.

Herden, C.

D. Kip, C. Herden, and M. Wesner, “All-optical signal routing using interaction of mutually incoherent spatial solitons,” Ferroelectrics 274, 135–142 (2002).

Herrmann, J.

Jander, P.

K. Motzek, M. Belić, T. Richter, C. Denz, A. Desyatnikov, P. Jander, and F. Kaiser, “Counterpropagating beams in biased photorefractive crystals: anisotropic theory,” Phys. Rev. E 71, 016610 (2005).
[CrossRef]

M. Belić, P. Jander, K. Motzek, A. Desyatnikov, D. Jović, A. Strinić, M. Petrović, C. Denz, and F. Kaiser, “Counterpropagating self-trapped beams in photorefractive crystals,” J. Opt. B Quantum. Semiclass. Opt. 6, S190–S196 (2004).
[CrossRef]

K. Motzek, P. Jander, A. Desyatnikov, M. Belić, C. Denz, and F. Kaiser, “Dynamic counterpropagating vector solitons in saturable self-focusing media,” Phys. Rev. E 68, 066611(2003).
[CrossRef]

Jovanovic, R.

D. Jović, R. Jovanović, S. Prvanović, M. Petrović, and M. Belić, “Counterpropagating beams in rotationally symmetric photonic lattices,” Opt. Mater. 30, 1173–1176 (2008).
[CrossRef]

Jovic, D.

D. Jović, R. Jovanović, S. Prvanović, M. Petrović, and M. Belić, “Counterpropagating beams in rotationally symmetric photonic lattices,” Opt. Mater. 30, 1173–1176 (2008).
[CrossRef]

M. Belić, D. Jović, S. Prvanović, D. Arsenović, and M. Petrović, “Counterpropagating self-trapped beams in optical photonic lattices,” Opt. Express 14, 794–799 (2006).
[CrossRef]

M. Belić, P. Jander, K. Motzek, A. Desyatnikov, D. Jović, A. Strinić, M. Petrović, C. Denz, and F. Kaiser, “Counterpropagating self-trapped beams in photorefractive crystals,” J. Opt. B Quantum. Semiclass. Opt. 6, S190–S196 (2004).
[CrossRef]

Kaiser, F.

K. Motzek, M. Belić, T. Richter, C. Denz, A. Desyatnikov, P. Jander, and F. Kaiser, “Counterpropagating beams in biased photorefractive crystals: anisotropic theory,” Phys. Rev. E 71, 016610 (2005).
[CrossRef]

M. Belić, P. Jander, K. Motzek, A. Desyatnikov, D. Jović, A. Strinić, M. Petrović, C. Denz, and F. Kaiser, “Counterpropagating self-trapped beams in photorefractive crystals,” J. Opt. B Quantum. Semiclass. Opt. 6, S190–S196 (2004).
[CrossRef]

K. Motzek, P. Jander, A. Desyatnikov, M. Belić, C. Denz, and F. Kaiser, “Dynamic counterpropagating vector solitons in saturable self-focusing media,” Phys. Rev. E 68, 066611(2003).
[CrossRef]

M. R. Belić, D. Vujić, A. Stepken, F. Kaiser, G. F. Calvo, F. Agullo-Lopez, and M. Carrascosa, “Isotropic versus anisotropic modeling of photorefractive solitons,” Phys. Rev. E 65, 066610 (2002).
[CrossRef]

Kip, D.

D. Kip, C. Herden, and M. Wesner, “All-optical signal routing using interaction of mutually incoherent spatial solitons,” Ferroelectrics 274, 135–142 (2002).

Kivshar, Y. S.

M. S. Petrović, M. R. Belić, C. Denz, and Y. S. Kivshar, “Counterpropagating optical beams and solitons,” Laser Photon. Rev. 5, 214–233 (2011).
[CrossRef]

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Lamb, G. L.

G. L. Lamb, Elements of Soliton Theory (Wiley, 1980).

Lan, S.

Lieb, E. H.

E. H. Lieb and M. Loss, Analysis, 2nd ed. (American Mathematical Society, 2001).

Lin, T. C.

T. C. Lin, M. R. Belić, M. S. Petrović, and G. Chen, “Ground state of nonlinear Schrödinger systems with saturable nonlinearity,” arXiv:1208.6259v1 [math-ph] (2012).

Loss, M.

E. H. Lieb and M. Loss, Analysis, 2nd ed. (American Mathematical Society, 2001).

Makasyuk, I.

J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, “Dipole and quadrupole solitons in optically induced two-dimensional photonic lattices: theory and experiment,” Stud. Appl. Math. 113, 389–412 (2004).
[CrossRef]

Manakov, S. V.

S. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method (Plenum, 1984).

Manela, O.

Marini, A.

Motzek, K.

K. Motzek, M. Belić, T. Richter, C. Denz, A. Desyatnikov, P. Jander, and F. Kaiser, “Counterpropagating beams in biased photorefractive crystals: anisotropic theory,” Phys. Rev. E 71, 016610 (2005).
[CrossRef]

M. Belić, P. Jander, K. Motzek, A. Desyatnikov, D. Jović, A. Strinić, M. Petrović, C. Denz, and F. Kaiser, “Counterpropagating self-trapped beams in photorefractive crystals,” J. Opt. B Quantum. Semiclass. Opt. 6, S190–S196 (2004).
[CrossRef]

K. Motzek, P. Jander, A. Desyatnikov, M. Belić, C. Denz, and F. Kaiser, “Dynamic counterpropagating vector solitons in saturable self-focusing media,” Phys. Rev. E 68, 066611(2003).
[CrossRef]

Novikov, S.

S. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method (Plenum, 1984).

Palange, E.

E. DelRe, A. D’Ercole, and E. Palange, “Mechanisms supporting long propagation regimes of photorefractive solitons,” Phys. Rev. E 71, 036610 (2005).
[CrossRef]

E. DelRe, G. De Masi, A. Ciattoni, and E. Palange, “Pairing space-charge field conditions with self-guiding for the attainment of circular symmetry in photorefractive solitons,” Appl. Phys. Lett. 85, 5499–5501 (2004).
[CrossRef]

Petrovic, M.

N. Aleksić, M. Petrović, A. Strinić, and M. Belić, “Solitons in highly nonlocal nematic liquid crystals: variational approach,” Phys. Rev. A 85, 033826 (2012).
[CrossRef]

D. Jović, R. Jovanović, S. Prvanović, M. Petrović, and M. Belić, “Counterpropagating beams in rotationally symmetric photonic lattices,” Opt. Mater. 30, 1173–1176 (2008).
[CrossRef]

M. Belić, D. Jović, S. Prvanović, D. Arsenović, and M. Petrović, “Counterpropagating self-trapped beams in optical photonic lattices,” Opt. Express 14, 794–799 (2006).
[CrossRef]

M. Belić, P. Jander, K. Motzek, A. Desyatnikov, D. Jović, A. Strinić, M. Petrović, C. Denz, and F. Kaiser, “Counterpropagating self-trapped beams in photorefractive crystals,” J. Opt. B Quantum. Semiclass. Opt. 6, S190–S196 (2004).
[CrossRef]

Petrovic, M. S.

M. S. Petrović, M. R. Belić, C. Denz, and Y. S. Kivshar, “Counterpropagating optical beams and solitons,” Laser Photon. Rev. 5, 214–233 (2011).
[CrossRef]

M. S. Petrović, A. I. Strinić, N. B. Aleksić, and M. R. Belić, “Do shape invariant solitons in highly nonlocal nematic liquid crystals really exist?,” arXiv:1110.5053v1 [physics.optics] (2011).

T. C. Lin, M. R. Belić, M. S. Petrović, and G. Chen, “Ground state of nonlinear Schrödinger systems with saturable nonlinearity,” arXiv:1208.6259v1 [math-ph] (2012).

Petviashvili, V. I.

V. I. Petviashvili, “Equation of an extraordinary soliton,” Fiz. Plazmy 2, 469–471 (1976).

Pitaevskii, L. P.

S. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method (Plenum, 1984).

Potton, R. J.

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717–754 (2004).
[CrossRef]

Prvanovic, S.

D. Jović, R. Jovanović, S. Prvanović, M. Petrović, and M. Belić, “Counterpropagating beams in rotationally symmetric photonic lattices,” Opt. Mater. 30, 1173–1176 (2008).
[CrossRef]

M. Belić, D. Jović, S. Prvanović, D. Arsenović, and M. Petrović, “Counterpropagating self-trapped beams in optical photonic lattices,” Opt. Express 14, 794–799 (2006).
[CrossRef]

Richter, T.

K. Motzek, M. Belić, T. Richter, C. Denz, A. Desyatnikov, P. Jander, and F. Kaiser, “Counterpropagating beams in biased photorefractive crystals: anisotropic theory,” Phys. Rev. E 71, 016610 (2005).
[CrossRef]

Rizza, C.

Rotschild, C.

Segev, M.

Sipe, J. E.

M. V. Tratnik and J. E. Sipe, “Bound solitary waves in a birefringent optical fibre,” Phys. Rev. A 38, 2011–2017 (1988).
[CrossRef]

Stepken, A.

M. R. Belić, D. Vujić, A. Stepken, F. Kaiser, G. F. Calvo, F. Agullo-Lopez, and M. Carrascosa, “Isotropic versus anisotropic modeling of photorefractive solitons,” Phys. Rev. E 65, 066610 (2002).
[CrossRef]

Strinic, A.

N. Aleksić, M. Petrović, A. Strinić, and M. Belić, “Solitons in highly nonlocal nematic liquid crystals: variational approach,” Phys. Rev. A 85, 033826 (2012).
[CrossRef]

M. Belić, P. Jander, K. Motzek, A. Desyatnikov, D. Jović, A. Strinić, M. Petrović, C. Denz, and F. Kaiser, “Counterpropagating self-trapped beams in photorefractive crystals,” J. Opt. B Quantum. Semiclass. Opt. 6, S190–S196 (2004).
[CrossRef]

Strinic, A. I.

M. S. Petrović, A. I. Strinić, N. B. Aleksić, and M. R. Belić, “Do shape invariant solitons in highly nonlocal nematic liquid crystals really exist?,” arXiv:1110.5053v1 [physics.optics] (2011).

Sulem, C.

C. Sulem and P. Sulem, The Nonlinear Schrödinger Equation: Self-focusing and Wave Collapse (Springer-Verlag, 1999).

Sulem, P.

C. Sulem and P. Sulem, The Nonlinear Schrödinger Equation: Self-focusing and Wave Collapse (Springer-Verlag, 1999).

Tratnik, M. V.

M. V. Tratnik and J. E. Sipe, “Bound solitary waves in a birefringent optical fibre,” Phys. Rev. A 38, 2011–2017 (1988).
[CrossRef]

Vujic, D.

M. R. Belić, D. Vujić, A. Stepken, F. Kaiser, G. F. Calvo, F. Agullo-Lopez, and M. Carrascosa, “Isotropic versus anisotropic modeling of photorefractive solitons,” Phys. Rev. E 65, 066610 (2002).
[CrossRef]

Wesner, M.

D. Kip, C. Herden, and M. Wesner, “All-optical signal routing using interaction of mutually incoherent spatial solitons,” Ferroelectrics 274, 135–142 (2002).

Yang, J.

J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, “Dipole and quadrupole solitons in optically induced two-dimensional photonic lattices: theory and experiment,” Stud. Appl. Math. 113, 389–412 (2004).
[CrossRef]

Zakharov, V. E.

S. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method (Plenum, 1984).

Appl. Phys. Lett. (1)

E. DelRe, G. De Masi, A. Ciattoni, and E. Palange, “Pairing space-charge field conditions with self-guiding for the attainment of circular symmetry in photorefractive solitons,” Appl. Phys. Lett. 85, 5499–5501 (2004).
[CrossRef]

Ferroelectrics (1)

D. Kip, C. Herden, and M. Wesner, “All-optical signal routing using interaction of mutually incoherent spatial solitons,” Ferroelectrics 274, 135–142 (2002).

Fiz. Plazmy (1)

V. I. Petviashvili, “Equation of an extraordinary soliton,” Fiz. Plazmy 2, 469–471 (1976).

J. Nonlinear Opt. Phys. Mater. (1)

E. DelRe, A. Ciattoni, B. Crosignani, and P. Di Porto, “Nonlinear optical propagation phenomena in near-transition centrosymmetric photorefractive crystals,” J. Nonlinear Opt. Phys. Mater. 8, 1–20 (1999).
[CrossRef]

J. Opt. B Quantum. Semiclass. Opt. (1)

M. Belić, P. Jander, K. Motzek, A. Desyatnikov, D. Jović, A. Strinić, M. Petrović, C. Denz, and F. Kaiser, “Counterpropagating self-trapped beams in photorefractive crystals,” J. Opt. B Quantum. Semiclass. Opt. 6, S190–S196 (2004).
[CrossRef]

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

Laser Photon. Rev. (1)

M. S. Petrović, M. R. Belić, C. Denz, and Y. S. Kivshar, “Counterpropagating optical beams and solitons,” Laser Photon. Rev. 5, 214–233 (2011).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Opt. Mater. (1)

D. Jović, R. Jovanović, S. Prvanović, M. Petrović, and M. Belić, “Counterpropagating beams in rotationally symmetric photonic lattices,” Opt. Mater. 30, 1173–1176 (2008).
[CrossRef]

Phys. Rev. A (3)

M. V. Tratnik and J. E. Sipe, “Bound solitary waves in a birefringent optical fibre,” Phys. Rev. A 38, 2011–2017 (1988).
[CrossRef]

N. Aleksić, M. Petrović, A. Strinić, and M. Belić, “Solitons in highly nonlocal nematic liquid crystals: variational approach,” Phys. Rev. A 85, 033826 (2012).
[CrossRef]

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

Phys. Rev. E (4)

M. R. Belić, D. Vujić, A. Stepken, F. Kaiser, G. F. Calvo, F. Agullo-Lopez, and M. Carrascosa, “Isotropic versus anisotropic modeling of photorefractive solitons,” Phys. Rev. E 65, 066610 (2002).
[CrossRef]

K. Motzek, P. Jander, A. Desyatnikov, M. Belić, C. Denz, and F. Kaiser, “Dynamic counterpropagating vector solitons in saturable self-focusing media,” Phys. Rev. E 68, 066611(2003).
[CrossRef]

K. Motzek, M. Belić, T. Richter, C. Denz, A. Desyatnikov, P. Jander, and F. Kaiser, “Counterpropagating beams in biased photorefractive crystals: anisotropic theory,” Phys. Rev. E 71, 016610 (2005).
[CrossRef]

E. DelRe, A. D’Ercole, and E. Palange, “Mechanisms supporting long propagation regimes of photorefractive solitons,” Phys. Rev. E 71, 036610 (2005).
[CrossRef]

Rep. Prog. Phys. (1)

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717–754 (2004).
[CrossRef]

Stud. Appl. Math. (1)

J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, “Dipole and quadrupole solitons in optically induced two-dimensional photonic lattices: theory and experiment,” Stud. Appl. Math. 113, 389–412 (2004).
[CrossRef]

Other (9)

M. S. Petrović, A. I. Strinić, N. B. Aleksić, and M. R. Belić, “Do shape invariant solitons in highly nonlocal nematic liquid crystals really exist?,” arXiv:1110.5053v1 [physics.optics] (2011).

E. H. Lieb and M. Loss, Analysis, 2nd ed. (American Mathematical Society, 2001).

T. C. Lin, M. R. Belić, M. S. Petrović, and G. Chen, “Ground state of nonlinear Schrödinger systems with saturable nonlinearity,” arXiv:1208.6259v1 [math-ph] (2012).

G. L. Lamb, Elements of Soliton Theory (Wiley, 1980).

N. N. Akhmediev and A. A. Ankiewicz, Solitons (Chapman & Hall, 1997).

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

S. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method (Plenum, 1984).

M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations, and Inverse Scattering (Cambridge University, 1991).

C. Sulem and P. Sulem, The Nonlinear Schrödinger Equation: Self-focusing and Wave Collapse (Springer-Verlag, 1999).

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Figures (4)

Fig. 1.
Fig. 1.

Soliton intensity u2 profile obtained numerically (dots), fitted with a Gaussian (line), presented as a function of the transverse x distance. Inset: the same plot but presented on the logarithmic scale. Parameters: coupling constant Γ=10, propagation constant μ=5.

Fig. 2.
Fig. 2.

(a) Product of the soliton power and coupling constant, as a function of μ/Γ. Squares represent numerical results; black solid line represents the results of the variational approach. (b) Same dependence plotted on a log–log scale in the region Γ/2<μ<Γ. The red (light) curve is a linear fit through the numerical points, with the slope 2.33±0.07. The black (dark) curve is the corresponding prediction of the variational study, with an approximate slope of 2.41.

Fig. 3.
Fig. 3.

Same as Fig. 2(b) but presenting the width R of the soliton multiplied by the square root of the coupling constant, as a function of μ/Γ. The slopes here read 0.470 and 0.455, respectively.

Fig. 4.
Fig. 4.

(a) F beam intensity in the transverse plane, presented at different propagation lengths, for propagation in the medium where Γ<μ (above threshold; Γ=4, μ=5). The size of the transverse windows is 4 input beam widths. (b) Stable soliton propagation below the threshold, Γ=8 and μ=5. (c) Same as (b) but displaying the actual soliton profiles. Arrows indicate the direction of propagation.

Equations (27)

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

izF=ΔF+ΓE0F,izB=ΔB+ΓE0B,
τtE0+E0=I1+I,
F=u(x,y)eiμz,B=v(x,y)eiλz,
Δu+Γu2+v21+u2+v2u=μu,
Δv+Γu2+v21+u2+v2v=λv.
μΓ=PR2ln(1+I)P.
eΓ=inf{E[u,v]: u,vH1(R2),P[u,v]=1},
E[u,v]=R2(|u|2+|v|2Γ[Iln(1+I)])
T=infwH1(R2)w2=1R2|w|2R2[w2ln(1+w2)].
MinimizeH[ρ]overρH1(R2),R2ρ2=1,
H[ρ]=R2[|ρ|2Γ[ρ2ln(1+ρ2)]].
PΓ|1μ/Γ|α,
RΓ1/2|1μ/Γ|β,
U=Aexp[r22R2]exp[iCr2+iψ].
L=i2(U*zUUzU*)+|U|2Γ[|U|2ln(1+|U|2)],
L(A,R,C,ψ)=πA2[R2ψ+R4(C+4C2)+1]ΓπR2[A2+Li2(A2)],
ddzA=4CA,
ddzR=4CR,
dCdz=4C2+1R4+ΓR2A2(ln[1+A2]+Li2[A2]),
dψdz=2R2+ΓΓA2(2ln[1+A2]+Li2[A2]).
R=AΓLi2(A2)ln(1+A2),
μdψdz=Γ[1+Li2(A2)/A2],
P=(π/Γ)A4[Li2(A2)ln(1+A2)],
μΓ[12ln2(A)/A2],
R1/2ΓA/ln(A),
PΓ2πln2(A)(1μ/Γ)2,
RΓ1/2(1μ/Γ)1/2,

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