Abstract

We study the formation and propagation of two-dimensional vortex solitons, i.e. solitons with a phase singularity, in optical materials with a nonlocal focusing nonlinearity. We show that nonlocality stabilizes the dynamics of an otherwise unstable vortex beam. This occurs for either single or higher charge fundamental vortices as well as higher order (multiple ring) vortex solitons. Our results pave the way for experimental observation of stable vortex rings in other nonlocal nonlinear systems including Bose-Einstein condensates with pronounced long-range interparticle interaction.

© 2005 Optical Society of America

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References

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    [Crossref]
  4. L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L.D. Carr, Y. Castin, and C. Salomon, “Formation of a Matter-Wave Bright Soliton,” Science 296, 1290–1293 (2002).
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    [Crossref] [PubMed]
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    [Crossref]
  28. M. Bertolotti, R. Li Voti, S. Marchetti, and C. Sibili, “Interaction of soliton-like beam in a diffusive nonlinear planar waveguide,” Opt. Commun. 133, 578–586 (1997).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  35. W. Królikowski and O. Bang, “Solitons in nonlocal nonlinear media: Exact solutions,” Phys. Rev. E 63, 016610 1–6 (2001).
  36. N.I. Nikolov, D. Neshev, O. Bang, and W. Królikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E 68, 036614 1–5 (2003).
    [Crossref]
  37. D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
    [Crossref]
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  40. A. Yakimenko, Y. Zaliznyak, and Y. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” http://au.arxiv.org/abs/nlin.PS/0411024.

2004 (3)

C.-C. Jeng, M.-F. Shih, K. Motzek, and Y.S. Kivshar, “Partially incoherent optical vortices in self-focusing monlinear media,” Phys. Rev. Lett. 92, 043904 1–4 (2004).
[Crossref]

W. Królikowski, O. Bang, N.I. Nikolov, D. Neshev, J. Wyller, J.J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B. 6, S288–S294 (2004).
[Crossref]

P.I. Krepostnov, V.O. Popov, and N.N. Rozanov, “Internal modes of Solitons in a Bose-Einstein Condensate,” JETP 99, 279–285 (2004).
[Crossref]

2003 (5)

W. Królikowski, O. Bang, J. Wyller, and J.J. Rasmussen, “Optical Beams in Nonlocal Nonlinear Media,” Acta Physica Polonica A 103, 133–147 (2003).

G. Assanto and M. Peccianti, “Spatial solitons in nematic liquid crystals,” IEEE J. Quantum Electron. 39, 13–21 (2003).
[Crossref]

N.I. Nikolov, D. Neshev, O. Bang, and W. Królikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E 68, 036614 1–5 (2003).
[Crossref]

S.K. Adhikari, “Mean-field model of interaction between bright vortex solitons in Bose-Einstein condensate,” New J. Phys. 5, 137.1–137.13 (2003).
[Crossref]

J. Yang and D. Pelinovsky, “Stable vortex and dipole vector solitons in a saturable nonlinear medium,” Phys. Rev. E 67, 016608 1–12 (2003).
[Crossref]

2002 (4)

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A.V. Buryak, B.A. Malomed, L. Torner, J.P. Torres, and F. Lederer, “Stable spinning optical solitons in three dimensions,” Phys. Rev. Lett. 88, 073902 1–4 (2002).
[Crossref]

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L.D. Carr, Y. Castin, and C. Salomon, “Formation of a Matter-Wave Bright Soliton,” Science 296, 1290–1293 (2002).
[Crossref] [PubMed]

J. Wyller, W. Królikowski, O. Bang, and J.J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E 66, 066615 1–13 (2002).
[Crossref]

O. Bang, W. Królikowski, J. Wyller, and J.J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619 1–5 (2002).
[Crossref]

2001 (2)

W. Królikowski and O. Bang, “Solitons in nonlocal nonlinear media: Exact solutions,” Phys. Rev. E 63, 016610 1–6 (2001).

W. Królikowski, O. Bang, J.J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E 64, 016612 1–8 (2001).
[Crossref]

2000 (3)

K. Goral, K. Rzazewski, and T. Pfau, “Bose-Einstein condensation with magnetic dipole-dipole forces,” Phys. Rev. A 61, 051601 1–4 (2000).
[Crossref]

J. Denschlag, J.E. Simsarian, D.L. Feder, C.W. Clark, L.A. Collins, J. Cubizolles, L. Deng, E.W. Hagley, K. Helmerson, W.P. Reinhardt, S.L. Rolston, B.I. Schneider, and W.D. Phillips, “Generating Solitons by Phase Engineering of a Bose-Einstein Condensate,” Science 287, 97–101 (2000).
[Crossref]

Z.H. Musslimani, M. Segev, D.N. Christodoulides, and M. Soljacic, “Composite multihump vector solitons carrying topological charge,” Phys. Rev. Lett. 84, 1164–1167 (2000).
[Crossref] [PubMed]

1999 (2)

G.I. Stegeman and M. Segev, “Optical Spatial Solitons and Their Interactions: Universality and Diversity,” Science 286, 1518–1523 (1999).
[Crossref] [PubMed]

A. Snyder and F. Ladouceur, “Light guiding light: letting light be the master of its own destiny,” Optics & Photonics News 10, 35–39 (1999).
[Crossref]

1998 (3)

D.V. Petrov, L. Torner, J. Nartorell, D.W. Wilson, R.E. Muller, and P.D. Maker, “Observation of azimuthal modulational instability and formation of patterns of optical solitons in a quadratic nonlinear crystal,” Opt. Lett. 23, 1444–1446 (1998).
[Crossref]

D.V. Skryabin and W.J. Firth, “Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media,” Phys. Rev. E 58, 3916–3930 (1998).
[Crossref]

A. Parola, L. Salasnich, and L. Reatto, “Structure and stability of bosonic clouds: Alkali-metal atoms with negative scattering length,” Phys. Rev. A 57, R3180–R3183 (1998).
[Crossref]

1997 (3)

A. Snyder and J. Mitchell, “Accessible Solitons,” Science 276, 1538–1541 (1997).
[Crossref]

M. Bertolotti, R. Li Voti, S. Marchetti, and C. Sibili, “Interaction of soliton-like beam in a diffusive nonlinear planar waveguide,” Opt. Commun. 133, 578–586 (1997).
[Crossref]

M. Quiroga-Teixeiro and H. Michinel, “Stable azimuthal stationary state in quintic nonlinear optical media,” J. Opt. Soc. Am. B 14, 2004–2009 (1997).
[Crossref]

1995 (2)

T.A. Davydova and A.I. Fishchuk, “Upper hybrid nonlinear wave structures,” Ukr. J. Phys. 40, 487–494 (1995).

D.W. McLaughlin, D.J. Muraki, M.J. Shelley, and W. Xiao, “A paraxial model for optical self-focussing in a nematic liquid crystal,” Physica D 88, 55–81 (1995).
[Crossref]

1993 (1)

D. Suter and T. Blasberg, “Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium,” Phys. Rev. A 48, 4583–4587 (1993).
[Crossref] [PubMed]

1992 (1)

1991 (1)

J.M. Soto-Crespo, D.R. Heatley, E.M. Wright, and N.N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A 44, 636–644 (1991).
[Crossref] [PubMed]

1989 (1)

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[Crossref]

1985 (2)

V.I. Kruglov and R.A. Vlasov, “Spiral self-trapping propagation of optical beams in media with cubic nonlinearity,” Phys. Lett. A 111, 401–404 (1985).
[Crossref]

S.K. Turitsyn, “Spatial dispersion of nonlinearity and stability of multidimensional solitons,” Theor. Math. Phys. 64, 226–232 (1985).
[Crossref]

1983 (1)

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

1980 (1)

H.L. Pecseli and J.J. Rasmussen, “Nonlinear electron waves in strongly magnetized plasmas,” Plasma Phys. 22, 421–438 (1980).
[Crossref]

1975 (1)

A.G. Litvak, V.A. Mironov, G.M. Fraiman, and A.D. Yunakovskii, “Thermal self-effect of wave beams in plasma with a nonlocal nonlinearity,” Sov. J. Plasma Phys. 1, 31–37 (1975).

1968 (1)

S. Akhmanov, D. Krindach, A. Migulin, A. Sukhorukov, and R. Khokhlov, “Thermal self-actions of laser beams,” IEEE J. Quantum Electron. 4, 568–575 (1968).
[Crossref]

Adhikari, S.K.

S.K. Adhikari, “Mean-field model of interaction between bright vortex solitons in Bose-Einstein condensate,” New J. Phys. 5, 137.1–137.13 (2003).
[Crossref]

Agrawal, G.

Y. Kivshar and G. AgrawalOptical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, Calif., 2003).

Akhmanov, S.

S. Akhmanov, D. Krindach, A. Migulin, A. Sukhorukov, and R. Khokhlov, “Thermal self-actions of laser beams,” IEEE J. Quantum Electron. 4, 568–575 (1968).
[Crossref]

Akhmediev, N.N.

J.M. Soto-Crespo, D.R. Heatley, E.M. Wright, and N.N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A 44, 636–644 (1991).
[Crossref] [PubMed]

Anderson, D.

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

Assanto, G.

G. Assanto and M. Peccianti, “Spatial solitons in nematic liquid crystals,” IEEE J. Quantum Electron. 39, 13–21 (2003).
[Crossref]

Bang, O.

W. Królikowski, O. Bang, N.I. Nikolov, D. Neshev, J. Wyller, J.J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B. 6, S288–S294 (2004).
[Crossref]

W. Królikowski, O. Bang, J. Wyller, and J.J. Rasmussen, “Optical Beams in Nonlocal Nonlinear Media,” Acta Physica Polonica A 103, 133–147 (2003).

N.I. Nikolov, D. Neshev, O. Bang, and W. Królikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E 68, 036614 1–5 (2003).
[Crossref]

J. Wyller, W. Królikowski, O. Bang, and J.J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E 66, 066615 1–13 (2002).
[Crossref]

O. Bang, W. Królikowski, J. Wyller, and J.J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619 1–5 (2002).
[Crossref]

W. Królikowski, O. Bang, J.J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E 64, 016612 1–8 (2001).
[Crossref]

W. Królikowski and O. Bang, “Solitons in nonlocal nonlinear media: Exact solutions,” Phys. Rev. E 63, 016610 1–6 (2001).

Bertolotti, M.

M. Bertolotti, R. Li Voti, S. Marchetti, and C. Sibili, “Interaction of soliton-like beam in a diffusive nonlinear planar waveguide,” Opt. Commun. 133, 578–586 (1997).
[Crossref]

Blasberg, T.

D. Suter and T. Blasberg, “Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium,” Phys. Rev. A 48, 4583–4587 (1993).
[Crossref] [PubMed]

Bourdel, T.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L.D. Carr, Y. Castin, and C. Salomon, “Formation of a Matter-Wave Bright Soliton,” Science 296, 1290–1293 (2002).
[Crossref] [PubMed]

Buryak, A.V.

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A.V. Buryak, B.A. Malomed, L. Torner, J.P. Torres, and F. Lederer, “Stable spinning optical solitons in three dimensions,” Phys. Rev. Lett. 88, 073902 1–4 (2002).
[Crossref]

Carr, L.D.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L.D. Carr, Y. Castin, and C. Salomon, “Formation of a Matter-Wave Bright Soliton,” Science 296, 1290–1293 (2002).
[Crossref] [PubMed]

Castin, Y.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L.D. Carr, Y. Castin, and C. Salomon, “Formation of a Matter-Wave Bright Soliton,” Science 296, 1290–1293 (2002).
[Crossref] [PubMed]

Cerullo, G.

Christodoulides, D.N.

Z.H. Musslimani, M. Segev, D.N. Christodoulides, and M. Soljacic, “Composite multihump vector solitons carrying topological charge,” Phys. Rev. Lett. 84, 1164–1167 (2000).
[Crossref] [PubMed]

Clark, C.W.

J. Denschlag, J.E. Simsarian, D.L. Feder, C.W. Clark, L.A. Collins, J. Cubizolles, L. Deng, E.W. Hagley, K. Helmerson, W.P. Reinhardt, S.L. Rolston, B.I. Schneider, and W.D. Phillips, “Generating Solitons by Phase Engineering of a Bose-Einstein Condensate,” Science 287, 97–101 (2000).
[Crossref]

Collins, L.A.

J. Denschlag, J.E. Simsarian, D.L. Feder, C.W. Clark, L.A. Collins, J. Cubizolles, L. Deng, E.W. Hagley, K. Helmerson, W.P. Reinhardt, S.L. Rolston, B.I. Schneider, and W.D. Phillips, “Generating Solitons by Phase Engineering of a Bose-Einstein Condensate,” Science 287, 97–101 (2000).
[Crossref]

Coullet, P.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[Crossref]

Crasovan, L.-C.

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A.V. Buryak, B.A. Malomed, L. Torner, J.P. Torres, and F. Lederer, “Stable spinning optical solitons in three dimensions,” Phys. Rev. Lett. 88, 073902 1–4 (2002).
[Crossref]

Cubizolles, J.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L.D. Carr, Y. Castin, and C. Salomon, “Formation of a Matter-Wave Bright Soliton,” Science 296, 1290–1293 (2002).
[Crossref] [PubMed]

J. Denschlag, J.E. Simsarian, D.L. Feder, C.W. Clark, L.A. Collins, J. Cubizolles, L. Deng, E.W. Hagley, K. Helmerson, W.P. Reinhardt, S.L. Rolston, B.I. Schneider, and W.D. Phillips, “Generating Solitons by Phase Engineering of a Bose-Einstein Condensate,” Science 287, 97–101 (2000).
[Crossref]

Davydova, T.A.

T.A. Davydova and A.I. Fishchuk, “Upper hybrid nonlinear wave structures,” Ukr. J. Phys. 40, 487–494 (1995).

De Silvestri, S.

Deng, L.

J. Denschlag, J.E. Simsarian, D.L. Feder, C.W. Clark, L.A. Collins, J. Cubizolles, L. Deng, E.W. Hagley, K. Helmerson, W.P. Reinhardt, S.L. Rolston, B.I. Schneider, and W.D. Phillips, “Generating Solitons by Phase Engineering of a Bose-Einstein Condensate,” Science 287, 97–101 (2000).
[Crossref]

Denschlag, J.

J. Denschlag, J.E. Simsarian, D.L. Feder, C.W. Clark, L.A. Collins, J. Cubizolles, L. Deng, E.W. Hagley, K. Helmerson, W.P. Reinhardt, S.L. Rolston, B.I. Schneider, and W.D. Phillips, “Generating Solitons by Phase Engineering of a Bose-Einstein Condensate,” Science 287, 97–101 (2000).
[Crossref]

Edmundson, D.

W. Królikowski, O. Bang, N.I. Nikolov, D. Neshev, J. Wyller, J.J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B. 6, S288–S294 (2004).
[Crossref]

Feder, D.L.

J. Denschlag, J.E. Simsarian, D.L. Feder, C.W. Clark, L.A. Collins, J. Cubizolles, L. Deng, E.W. Hagley, K. Helmerson, W.P. Reinhardt, S.L. Rolston, B.I. Schneider, and W.D. Phillips, “Generating Solitons by Phase Engineering of a Bose-Einstein Condensate,” Science 287, 97–101 (2000).
[Crossref]

Ferrari, G.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L.D. Carr, Y. Castin, and C. Salomon, “Formation of a Matter-Wave Bright Soliton,” Science 296, 1290–1293 (2002).
[Crossref] [PubMed]

Firth, W.J.

D.V. Skryabin and W.J. Firth, “Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media,” Phys. Rev. E 58, 3916–3930 (1998).
[Crossref]

Fishchuk, A.I.

T.A. Davydova and A.I. Fishchuk, “Upper hybrid nonlinear wave structures,” Ukr. J. Phys. 40, 487–494 (1995).

Fraiman, G.M.

A.G. Litvak, V.A. Mironov, G.M. Fraiman, and A.D. Yunakovskii, “Thermal self-effect of wave beams in plasma with a nonlocal nonlinearity,” Sov. J. Plasma Phys. 1, 31–37 (1975).

Gil, L.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[Crossref]

Goral, K.

K. Goral, K. Rzazewski, and T. Pfau, “Bose-Einstein condensation with magnetic dipole-dipole forces,” Phys. Rev. A 61, 051601 1–4 (2000).
[Crossref]

Hagley, E.W.

J. Denschlag, J.E. Simsarian, D.L. Feder, C.W. Clark, L.A. Collins, J. Cubizolles, L. Deng, E.W. Hagley, K. Helmerson, W.P. Reinhardt, S.L. Rolston, B.I. Schneider, and W.D. Phillips, “Generating Solitons by Phase Engineering of a Bose-Einstein Condensate,” Science 287, 97–101 (2000).
[Crossref]

Heatley, D.R.

J.M. Soto-Crespo, D.R. Heatley, E.M. Wright, and N.N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A 44, 636–644 (1991).
[Crossref] [PubMed]

Helmerson, K.

J. Denschlag, J.E. Simsarian, D.L. Feder, C.W. Clark, L.A. Collins, J. Cubizolles, L. Deng, E.W. Hagley, K. Helmerson, W.P. Reinhardt, S.L. Rolston, B.I. Schneider, and W.D. Phillips, “Generating Solitons by Phase Engineering of a Bose-Einstein Condensate,” Science 287, 97–101 (2000).
[Crossref]

Jeng, C.-C.

C.-C. Jeng, M.-F. Shih, K. Motzek, and Y.S. Kivshar, “Partially incoherent optical vortices in self-focusing monlinear media,” Phys. Rev. Lett. 92, 043904 1–4 (2004).
[Crossref]

Khaykovich, L.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L.D. Carr, Y. Castin, and C. Salomon, “Formation of a Matter-Wave Bright Soliton,” Science 296, 1290–1293 (2002).
[Crossref] [PubMed]

Khokhlov, R.

S. Akhmanov, D. Krindach, A. Migulin, A. Sukhorukov, and R. Khokhlov, “Thermal self-actions of laser beams,” IEEE J. Quantum Electron. 4, 568–575 (1968).
[Crossref]

Kivshar, Y.

Y. Kivshar and G. AgrawalOptical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, Calif., 2003).

A. Yakimenko, Y. Zaliznyak, and Y. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” http://au.arxiv.org/abs/nlin.PS/0411024.

Kivshar, Y.S.

C.-C. Jeng, M.-F. Shih, K. Motzek, and Y.S. Kivshar, “Partially incoherent optical vortices in self-focusing monlinear media,” Phys. Rev. Lett. 92, 043904 1–4 (2004).
[Crossref]

Krepostnov, P.I.

P.I. Krepostnov, V.O. Popov, and N.N. Rozanov, “Internal modes of Solitons in a Bose-Einstein Condensate,” JETP 99, 279–285 (2004).
[Crossref]

Krindach, D.

S. Akhmanov, D. Krindach, A. Migulin, A. Sukhorukov, and R. Khokhlov, “Thermal self-actions of laser beams,” IEEE J. Quantum Electron. 4, 568–575 (1968).
[Crossref]

Królikowski, W.

W. Królikowski, O. Bang, N.I. Nikolov, D. Neshev, J. Wyller, J.J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B. 6, S288–S294 (2004).
[Crossref]

W. Królikowski, O. Bang, J. Wyller, and J.J. Rasmussen, “Optical Beams in Nonlocal Nonlinear Media,” Acta Physica Polonica A 103, 133–147 (2003).

N.I. Nikolov, D. Neshev, O. Bang, and W. Królikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E 68, 036614 1–5 (2003).
[Crossref]

J. Wyller, W. Królikowski, O. Bang, and J.J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E 66, 066615 1–13 (2002).
[Crossref]

O. Bang, W. Królikowski, J. Wyller, and J.J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619 1–5 (2002).
[Crossref]

W. Królikowski and O. Bang, “Solitons in nonlocal nonlinear media: Exact solutions,” Phys. Rev. E 63, 016610 1–6 (2001).

W. Królikowski, O. Bang, J.J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E 64, 016612 1–8 (2001).
[Crossref]

Kruglov, V.I.

V.I. Kruglov and R.A. Vlasov, “Spiral self-trapping propagation of optical beams in media with cubic nonlinearity,” Phys. Lett. A 111, 401–404 (1985).
[Crossref]

Ladouceur, F.

A. Snyder and F. Ladouceur, “Light guiding light: letting light be the master of its own destiny,” Optics & Photonics News 10, 35–39 (1999).
[Crossref]

Lederer, F.

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A.V. Buryak, B.A. Malomed, L. Torner, J.P. Torres, and F. Lederer, “Stable spinning optical solitons in three dimensions,” Phys. Rev. Lett. 88, 073902 1–4 (2002).
[Crossref]

Li Voti, R.

M. Bertolotti, R. Li Voti, S. Marchetti, and C. Sibili, “Interaction of soliton-like beam in a diffusive nonlinear planar waveguide,” Opt. Commun. 133, 578–586 (1997).
[Crossref]

Litvak, A.G.

A.G. Litvak, V.A. Mironov, G.M. Fraiman, and A.D. Yunakovskii, “Thermal self-effect of wave beams in plasma with a nonlocal nonlinearity,” Sov. J. Plasma Phys. 1, 31–37 (1975).

Magni, V.

Maker, P.D.

Malomed, B.A.

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A.V. Buryak, B.A. Malomed, L. Torner, J.P. Torres, and F. Lederer, “Stable spinning optical solitons in three dimensions,” Phys. Rev. Lett. 88, 073902 1–4 (2002).
[Crossref]

Marchetti, S.

M. Bertolotti, R. Li Voti, S. Marchetti, and C. Sibili, “Interaction of soliton-like beam in a diffusive nonlinear planar waveguide,” Opt. Commun. 133, 578–586 (1997).
[Crossref]

Mazilu, D.

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A.V. Buryak, B.A. Malomed, L. Torner, J.P. Torres, and F. Lederer, “Stable spinning optical solitons in three dimensions,” Phys. Rev. Lett. 88, 073902 1–4 (2002).
[Crossref]

McLaughlin, D.W.

D.W. McLaughlin, D.J. Muraki, M.J. Shelley, and W. Xiao, “A paraxial model for optical self-focussing in a nematic liquid crystal,” Physica D 88, 55–81 (1995).
[Crossref]

Michinel, H.

Migulin, A.

S. Akhmanov, D. Krindach, A. Migulin, A. Sukhorukov, and R. Khokhlov, “Thermal self-actions of laser beams,” IEEE J. Quantum Electron. 4, 568–575 (1968).
[Crossref]

Mihalache, D.

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A.V. Buryak, B.A. Malomed, L. Torner, J.P. Torres, and F. Lederer, “Stable spinning optical solitons in three dimensions,” Phys. Rev. Lett. 88, 073902 1–4 (2002).
[Crossref]

Mironov, V.A.

A.G. Litvak, V.A. Mironov, G.M. Fraiman, and A.D. Yunakovskii, “Thermal self-effect of wave beams in plasma with a nonlocal nonlinearity,” Sov. J. Plasma Phys. 1, 31–37 (1975).

Mitchell, J.

A. Snyder and J. Mitchell, “Accessible Solitons,” Science 276, 1538–1541 (1997).
[Crossref]

Motzek, K.

C.-C. Jeng, M.-F. Shih, K. Motzek, and Y.S. Kivshar, “Partially incoherent optical vortices in self-focusing monlinear media,” Phys. Rev. Lett. 92, 043904 1–4 (2004).
[Crossref]

Muller, R.E.

Muraki, D.J.

D.W. McLaughlin, D.J. Muraki, M.J. Shelley, and W. Xiao, “A paraxial model for optical self-focussing in a nematic liquid crystal,” Physica D 88, 55–81 (1995).
[Crossref]

Musslimani, Z.H.

Z.H. Musslimani, M. Segev, D.N. Christodoulides, and M. Soljacic, “Composite multihump vector solitons carrying topological charge,” Phys. Rev. Lett. 84, 1164–1167 (2000).
[Crossref] [PubMed]

Nartorell, J.

Neshev, D.

W. Królikowski, O. Bang, N.I. Nikolov, D. Neshev, J. Wyller, J.J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B. 6, S288–S294 (2004).
[Crossref]

N.I. Nikolov, D. Neshev, O. Bang, and W. Królikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E 68, 036614 1–5 (2003).
[Crossref]

Nikolov, N.I.

W. Królikowski, O. Bang, N.I. Nikolov, D. Neshev, J. Wyller, J.J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B. 6, S288–S294 (2004).
[Crossref]

N.I. Nikolov, D. Neshev, O. Bang, and W. Królikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E 68, 036614 1–5 (2003).
[Crossref]

Parola, A.

A. Parola, L. Salasnich, and L. Reatto, “Structure and stability of bosonic clouds: Alkali-metal atoms with negative scattering length,” Phys. Rev. A 57, R3180–R3183 (1998).
[Crossref]

Peccianti, M.

G. Assanto and M. Peccianti, “Spatial solitons in nematic liquid crystals,” IEEE J. Quantum Electron. 39, 13–21 (2003).
[Crossref]

Pecseli, H.L.

H.L. Pecseli and J.J. Rasmussen, “Nonlinear electron waves in strongly magnetized plasmas,” Plasma Phys. 22, 421–438 (1980).
[Crossref]

Pelinovsky, D.

J. Yang and D. Pelinovsky, “Stable vortex and dipole vector solitons in a saturable nonlinear medium,” Phys. Rev. E 67, 016608 1–12 (2003).
[Crossref]

Petrov, D.V.

Pfau, T.

K. Goral, K. Rzazewski, and T. Pfau, “Bose-Einstein condensation with magnetic dipole-dipole forces,” Phys. Rev. A 61, 051601 1–4 (2000).
[Crossref]

Phillips, W.D.

J. Denschlag, J.E. Simsarian, D.L. Feder, C.W. Clark, L.A. Collins, J. Cubizolles, L. Deng, E.W. Hagley, K. Helmerson, W.P. Reinhardt, S.L. Rolston, B.I. Schneider, and W.D. Phillips, “Generating Solitons by Phase Engineering of a Bose-Einstein Condensate,” Science 287, 97–101 (2000).
[Crossref]

Popov, V.O.

P.I. Krepostnov, V.O. Popov, and N.N. Rozanov, “Internal modes of Solitons in a Bose-Einstein Condensate,” JETP 99, 279–285 (2004).
[Crossref]

Quiroga-Teixeiro, M.

Rasmussen, J.J.

W. Królikowski, O. Bang, N.I. Nikolov, D. Neshev, J. Wyller, J.J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B. 6, S288–S294 (2004).
[Crossref]

W. Królikowski, O. Bang, J. Wyller, and J.J. Rasmussen, “Optical Beams in Nonlocal Nonlinear Media,” Acta Physica Polonica A 103, 133–147 (2003).

O. Bang, W. Królikowski, J. Wyller, and J.J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619 1–5 (2002).
[Crossref]

J. Wyller, W. Królikowski, O. Bang, and J.J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E 66, 066615 1–13 (2002).
[Crossref]

W. Królikowski, O. Bang, J.J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E 64, 016612 1–8 (2001).
[Crossref]

H.L. Pecseli and J.J. Rasmussen, “Nonlinear electron waves in strongly magnetized plasmas,” Plasma Phys. 22, 421–438 (1980).
[Crossref]

Reatto, L.

A. Parola, L. Salasnich, and L. Reatto, “Structure and stability of bosonic clouds: Alkali-metal atoms with negative scattering length,” Phys. Rev. A 57, R3180–R3183 (1998).
[Crossref]

Reinhardt, W.P.

J. Denschlag, J.E. Simsarian, D.L. Feder, C.W. Clark, L.A. Collins, J. Cubizolles, L. Deng, E.W. Hagley, K. Helmerson, W.P. Reinhardt, S.L. Rolston, B.I. Schneider, and W.D. Phillips, “Generating Solitons by Phase Engineering of a Bose-Einstein Condensate,” Science 287, 97–101 (2000).
[Crossref]

Rocca, F.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[Crossref]

Rolston, S.L.

J. Denschlag, J.E. Simsarian, D.L. Feder, C.W. Clark, L.A. Collins, J. Cubizolles, L. Deng, E.W. Hagley, K. Helmerson, W.P. Reinhardt, S.L. Rolston, B.I. Schneider, and W.D. Phillips, “Generating Solitons by Phase Engineering of a Bose-Einstein Condensate,” Science 287, 97–101 (2000).
[Crossref]

Rozanov, N.N.

P.I. Krepostnov, V.O. Popov, and N.N. Rozanov, “Internal modes of Solitons in a Bose-Einstein Condensate,” JETP 99, 279–285 (2004).
[Crossref]

Rzazewski, K.

K. Goral, K. Rzazewski, and T. Pfau, “Bose-Einstein condensation with magnetic dipole-dipole forces,” Phys. Rev. A 61, 051601 1–4 (2000).
[Crossref]

Salasnich, L.

A. Parola, L. Salasnich, and L. Reatto, “Structure and stability of bosonic clouds: Alkali-metal atoms with negative scattering length,” Phys. Rev. A 57, R3180–R3183 (1998).
[Crossref]

Salomon, C.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L.D. Carr, Y. Castin, and C. Salomon, “Formation of a Matter-Wave Bright Soliton,” Science 296, 1290–1293 (2002).
[Crossref] [PubMed]

Schneider, B.I.

J. Denschlag, J.E. Simsarian, D.L. Feder, C.W. Clark, L.A. Collins, J. Cubizolles, L. Deng, E.W. Hagley, K. Helmerson, W.P. Reinhardt, S.L. Rolston, B.I. Schneider, and W.D. Phillips, “Generating Solitons by Phase Engineering of a Bose-Einstein Condensate,” Science 287, 97–101 (2000).
[Crossref]

Schreck, F.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L.D. Carr, Y. Castin, and C. Salomon, “Formation of a Matter-Wave Bright Soliton,” Science 296, 1290–1293 (2002).
[Crossref] [PubMed]

Segev, M.

Z.H. Musslimani, M. Segev, D.N. Christodoulides, and M. Soljacic, “Composite multihump vector solitons carrying topological charge,” Phys. Rev. Lett. 84, 1164–1167 (2000).
[Crossref] [PubMed]

G.I. Stegeman and M. Segev, “Optical Spatial Solitons and Their Interactions: Universality and Diversity,” Science 286, 1518–1523 (1999).
[Crossref] [PubMed]

Shelley, M.J.

D.W. McLaughlin, D.J. Muraki, M.J. Shelley, and W. Xiao, “A paraxial model for optical self-focussing in a nematic liquid crystal,” Physica D 88, 55–81 (1995).
[Crossref]

Shih, M.-F.

C.-C. Jeng, M.-F. Shih, K. Motzek, and Y.S. Kivshar, “Partially incoherent optical vortices in self-focusing monlinear media,” Phys. Rev. Lett. 92, 043904 1–4 (2004).
[Crossref]

Sibili, C.

M. Bertolotti, R. Li Voti, S. Marchetti, and C. Sibili, “Interaction of soliton-like beam in a diffusive nonlinear planar waveguide,” Opt. Commun. 133, 578–586 (1997).
[Crossref]

Simsarian, J.E.

J. Denschlag, J.E. Simsarian, D.L. Feder, C.W. Clark, L.A. Collins, J. Cubizolles, L. Deng, E.W. Hagley, K. Helmerson, W.P. Reinhardt, S.L. Rolston, B.I. Schneider, and W.D. Phillips, “Generating Solitons by Phase Engineering of a Bose-Einstein Condensate,” Science 287, 97–101 (2000).
[Crossref]

Skryabin, D.V.

D.V. Skryabin and W.J. Firth, “Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media,” Phys. Rev. E 58, 3916–3930 (1998).
[Crossref]

Snyder, A.

A. Snyder and F. Ladouceur, “Light guiding light: letting light be the master of its own destiny,” Optics & Photonics News 10, 35–39 (1999).
[Crossref]

A. Snyder and J. Mitchell, “Accessible Solitons,” Science 276, 1538–1541 (1997).
[Crossref]

Soljacic, M.

Z.H. Musslimani, M. Segev, D.N. Christodoulides, and M. Soljacic, “Composite multihump vector solitons carrying topological charge,” Phys. Rev. Lett. 84, 1164–1167 (2000).
[Crossref] [PubMed]

Soto-Crespo, J.M.

J.M. Soto-Crespo, D.R. Heatley, E.M. Wright, and N.N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A 44, 636–644 (1991).
[Crossref] [PubMed]

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]

Sukhorukov, A.

S. Akhmanov, D. Krindach, A. Migulin, A. Sukhorukov, and R. Khokhlov, “Thermal self-actions of laser beams,” IEEE J. Quantum Electron. 4, 568–575 (1968).
[Crossref]

Suter, D.

D. Suter and T. Blasberg, “Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium,” Phys. Rev. A 48, 4583–4587 (1993).
[Crossref] [PubMed]

Torner, L.

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A.V. Buryak, B.A. Malomed, L. Torner, J.P. Torres, and F. Lederer, “Stable spinning optical solitons in three dimensions,” Phys. Rev. Lett. 88, 073902 1–4 (2002).
[Crossref]

D.V. Petrov, L. Torner, J. Nartorell, D.W. Wilson, R.E. Muller, and P.D. Maker, “Observation of azimuthal modulational instability and formation of patterns of optical solitons in a quadratic nonlinear crystal,” Opt. Lett. 23, 1444–1446 (1998).
[Crossref]

Torres, J.P.

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A.V. Buryak, B.A. Malomed, L. Torner, J.P. Torres, and F. Lederer, “Stable spinning optical solitons in three dimensions,” Phys. Rev. Lett. 88, 073902 1–4 (2002).
[Crossref]

Towers, I.

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A.V. Buryak, B.A. Malomed, L. Torner, J.P. Torres, and F. Lederer, “Stable spinning optical solitons in three dimensions,” Phys. Rev. Lett. 88, 073902 1–4 (2002).
[Crossref]

Turitsyn, S.K.

S.K. Turitsyn, “Spatial dispersion of nonlinearity and stability of multidimensional solitons,” Theor. Math. Phys. 64, 226–232 (1985).
[Crossref]

Vlasov, R.A.

V.I. Kruglov and R.A. Vlasov, “Spiral self-trapping propagation of optical beams in media with cubic nonlinearity,” Phys. Lett. A 111, 401–404 (1985).
[Crossref]

Wilson, D.W.

Wright, E.M.

J.M. Soto-Crespo, D.R. Heatley, E.M. Wright, and N.N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A 44, 636–644 (1991).
[Crossref] [PubMed]

Wyller, J.

W. Królikowski, O. Bang, N.I. Nikolov, D. Neshev, J. Wyller, J.J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B. 6, S288–S294 (2004).
[Crossref]

W. Królikowski, O. Bang, J. Wyller, and J.J. Rasmussen, “Optical Beams in Nonlocal Nonlinear Media,” Acta Physica Polonica A 103, 133–147 (2003).

O. Bang, W. Królikowski, J. Wyller, and J.J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619 1–5 (2002).
[Crossref]

J. Wyller, W. Królikowski, O. Bang, and J.J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E 66, 066615 1–13 (2002).
[Crossref]

W. Królikowski, O. Bang, J.J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E 64, 016612 1–8 (2001).
[Crossref]

Xiao, W.

D.W. McLaughlin, D.J. Muraki, M.J. Shelley, and W. Xiao, “A paraxial model for optical self-focussing in a nematic liquid crystal,” Physica D 88, 55–81 (1995).
[Crossref]

Yakimenko, A.

A. Yakimenko, Y. Zaliznyak, and Y. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” http://au.arxiv.org/abs/nlin.PS/0411024.

Yang, J.

J. Yang and D. Pelinovsky, “Stable vortex and dipole vector solitons in a saturable nonlinear medium,” Phys. Rev. E 67, 016608 1–12 (2003).
[Crossref]

Yunakovskii, A.D.

A.G. Litvak, V.A. Mironov, G.M. Fraiman, and A.D. Yunakovskii, “Thermal self-effect of wave beams in plasma with a nonlocal nonlinearity,” Sov. J. Plasma Phys. 1, 31–37 (1975).

Zaliznyak, Y.

A. Yakimenko, Y. Zaliznyak, and Y. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” http://au.arxiv.org/abs/nlin.PS/0411024.

Acta Physica Polonica A (1)

W. Królikowski, O. Bang, J. Wyller, and J.J. Rasmussen, “Optical Beams in Nonlocal Nonlinear Media,” Acta Physica Polonica A 103, 133–147 (2003).

IEEE J. Quantum Electron. (2)

S. Akhmanov, D. Krindach, A. Migulin, A. Sukhorukov, and R. Khokhlov, “Thermal self-actions of laser beams,” IEEE J. Quantum Electron. 4, 568–575 (1968).
[Crossref]

G. Assanto and M. Peccianti, “Spatial solitons in nematic liquid crystals,” IEEE J. Quantum Electron. 39, 13–21 (2003).
[Crossref]

J. Opt. B. (1)

W. Królikowski, O. Bang, N.I. Nikolov, D. Neshev, J. Wyller, J.J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B. 6, S288–S294 (2004).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

JETP (1)

P.I. Krepostnov, V.O. Popov, and N.N. Rozanov, “Internal modes of Solitons in a Bose-Einstein Condensate,” JETP 99, 279–285 (2004).
[Crossref]

New J. Phys. (1)

S.K. Adhikari, “Mean-field model of interaction between bright vortex solitons in Bose-Einstein condensate,” New J. Phys. 5, 137.1–137.13 (2003).
[Crossref]

Opt. Commun. (2)

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[Crossref]

M. Bertolotti, R. Li Voti, S. Marchetti, and C. Sibili, “Interaction of soliton-like beam in a diffusive nonlinear planar waveguide,” Opt. Commun. 133, 578–586 (1997).
[Crossref]

Opt. Lett. (1)

Optics & Photonics News (1)

A. Snyder and F. Ladouceur, “Light guiding light: letting light be the master of its own destiny,” Optics & Photonics News 10, 35–39 (1999).
[Crossref]

Phys. Lett. A (1)

V.I. Kruglov and R.A. Vlasov, “Spiral self-trapping propagation of optical beams in media with cubic nonlinearity,” Phys. Lett. A 111, 401–404 (1985).
[Crossref]

Phys. Rev. A (5)

J.M. Soto-Crespo, D.R. Heatley, E.M. Wright, and N.N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A 44, 636–644 (1991).
[Crossref] [PubMed]

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Other (3)

Our first verification of stable vortex ring propagation was presented in: O.Bang, Nonlocal solitons, talk at the Workshop on Mathematical Ideas in Nonlinear Optics: GuidedWaves in Inhomogenous Nonlinear Media, 19–23 July 2004, Edinburgh, UK.

Y. Kivshar and G. AgrawalOptical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, Calif., 2003).

A. Yakimenko, Y. Zaliznyak, and Y. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” http://au.arxiv.org/abs/nlin.PS/0411024.

Supplementary Material (4)

» Media 1: MOV (3312 KB)     
» Media 2: MOV (1513 KB)     
» Media 3: MOV (15520 KB)     
» Media 4: MOV (10462 KB)     

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Figures (4)

Fig. 1.
Fig. 1.

Examples of ring vortex solitons in a nonlocal medium with charge m=1 (a) and m=3 (b). Solid line - normalized amplitude profile, dotted line - refractive index profile, dashed line - profile of the nonlocal response function. Top row - strongly nonlocal regime (σ0=5, σ=1,Λ=26.0783). Bottom row - weak nonlocality (σ0=0.1, σ=1,Λ=2.0198).

Fig. 2.
Fig. 2.

Propagation of nonlocal charge m=1 ring vortex solitons. (a) unstable propagation in the weakly nonlocal case with σ0/σ=0.1 and σ=1, Λ=2.0198, (x,y)∊[-10,10]×[-10,10] and z∊[0,5]. (b) stable propagation in the highly nonlocal case with σ0/σ=10, Λ=101.0199 and σ=1, (x,y)∊[-30,30]×[-30,30] and z∊[0,50].

Fig. 3.
Fig. 3.

Transverse structure of the nonlinearity-induced potential nesting the ring vortex soliton from Fig. 2. (a) Movie (3.2MB) depicting spatial evolution of the potential in a weakly nonlocal case (unstable propagation) (http://www.opticsexpress.org/view_media.cfm?umid=11676 for high resolution movie - 15MB); (b) Transverse structure of potential for strong nonlocality (stable propagation of the vortex soliton). Simulation parameters are the same as in Fig. 2.

Fig. 4.
Fig. 4.

Stable propagation of a double ring nonlocal vortex soliton with charge m=1, σ0=9, σ=1, (x,y)∊[-25,25]×[-25,25] and z∊[0,25]. (a) Transverse intensity structure. (b) Movie (1.5MB) illustrating dynamics of the vortex nested in the self-induced nonlocal potential. Note the difference in oscillations of the inner and outer rings. (http://www.opticsexpress.org/view_media.cfm?umid=11677 for high resolution movie - 10MB).

Equations (10)

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N ( I ) = R ( r r ) I ( r , z ) d 2 r ,
i z ψ + ( x 2 + y 2 ) ψ + N ( I ) ψ = 0 .
R ( r ) = 1 π σ 0 2 exp ( r 2 σ 0 2 ) .
ψ ( r , z ) = ψ ( r , ϕ , z ) = u ( r ) exp ( im ϕ ) exp ( i Λ z )
r 2 u ( r ) + 1 r r u ( r ) ( m 2 r 2 + Λ ) u ( r ) + u ( r ) 0 2 π 0 R ( r r ) u ( r ) 2 r d r d ϕ = 0
𝓛 = Λ u ( r ) 2 u ( r ) 2 + u ( r ) 2 R ( r r ) u ( r ) 2 d 2 r .
u ( r ) = Ar exp ( r 2 ( 2 σ 2 ) ) .
Λ u n + 1 ( r ) r 2 u n + 1 ( r ) 1 r r u n + 1 ( r ) + m 2 r 2 u n + 1 ( r ) = u n ( r ) N ( u n ( r ) 2 )
ψ ( r , ϕ ) = 2 σ 0 2 σ 4 r m exp ( r 2 ( 2 σ 2 ) ) exp ( im ϕ ) ,
ψ ( r , ϕ ) = 4 2 σ 0 2 σ 4 r ( 1 r 2 ( 2 σ 2 ) ) exp ( r 2 ( 2 σ 2 ) ) exp ( i ϕ ) .

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