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] [PubMed]
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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).
    [CrossRef] [PubMed]
  5. Y. Kivshar and G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, Calif., 2003).
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    [CrossRef]
  7. 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]
  8. 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).
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  10. M. Quiroga-Teixeiro and H. Michinel, �??Stable azimuthal stationary state in quintic nonlinear optical media,�?? J. Opt. Soc. Am. B 14, 2004-2009 (1997).
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  11. D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A.V. Buryak, B.A. Malomed, L. Torner, J.P. Torres, F. Lederer, �??Stable spinning optical solitons in three dimensions,�?? Phys. Rev. Lett. 88, 073902 1-4 (2002).
    [CrossRef]
  12. 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]
  13. 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]
  14. 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]
  15. 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]
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    [CrossRef]
  17. Our first verification of stable vortex ring propagation was presented in: O.Bang, Nonlocal solitons, talk at theWorkshop on Mathematical Ideas in Nonlinear Optics: GuidedWaves in Inhomogenous Nonlinear Media, 19-23 July 2004, Edinburgh , UK.
  18. 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).
  19. H.L. Pecseli and J.J. Rasmussen, �??Nonlinear electron waves in strongly magnetized plasmas,�?? Plasma Phys. 22, 421-438 (1980).
    [CrossRef]
  20. T.A. Davydova and A.I. Fishchuk, �??Upper hybrid nonlinear wave structures,�?? Ukr. J. Phys. 40, 487-494 (1995).
  21. 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]
  22. 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]
  23. 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]
  24. G. Assanto and M. Peccianti, �??Spatial solitons in nematic liquid crystals,�?? IEEE J. Quantum Electron. 39, 13-21 (2003).
    [CrossRef]
  25. 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]
  26. K. Goral, K. Rzazewski, and T. Pfau, �??Bose-Einstein condensation with magnetic dipole-dipole forces,�?? Phys. Rev. A 61, 051601 1-4 (2000).
    [CrossRef]
  27. A. Snyder and J. Mitchell, �??Accessible Solitons,�?? Science 276, 1538-1541 (1997).
    [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]
  29. 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]
  30. W. Królikowski, O. Bang, J. Wyller, and J.J. Rasmussen, �??Optical Beams in Nonlocal Nonlinear Media,�?? Acta Physica Polonica A 103, 133-147 (2003).
  31. 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]
  32. J. Wyller, W. Królikowski, O. Bang, J.J. Rasmussen, �??Generic features of modulational instability in nonlocal Kerr media,�?? Phys. Rev. E 66, 066615 1-13 (2002).
    [CrossRef]
  33. S.K. Turitsyn, �??Spatial dispersion of nonlinearity and stability of multidimensional solitons,�?? Theor. Math. Phys. 64, 226-232 (1985).
    [CrossRef]
  34. 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]
  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, 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]
  38. V. Magni, G. Cerullo, and S. De Silvestri, �??High-accuracy fast Hankel transform for optical beam propagation,�?? J. Opt. Soc. Am. A 9, 2031-2033 (1992).
    [CrossRef]
  39. 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]
  40. A. Yakimenko, Y. Zaliznyak, Y. Kivshar, �??Stable vortex solitons in nonlocal self-focusing nonlinear media,�?? <a href="http://au.arxiv.org/abs/nlin.PS/0411024">http://au.arxiv.org/abs/nlin.PS/0411024</a>

Acta Physica Polonica A

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.

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.

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

J. Opt. Soc. Am. B.

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]

JETP

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.

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.

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.

Optics & Photonics News

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

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

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]

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]

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

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]

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

Phys. Rev. E

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).

N.I. Nikolov, D. Neshev, O. Bang, W. Królikowski, �??Quadratic solitons as nonlocal solitons,�?? Phys. Rev. E 68, 036614 1-5 (2003).
[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]

J. Wyller, W. Królikowski, O. Bang, J.J. Rasmussen, �??Generic features of modulational instability in nonlocal Kerr media,�?? Phys. Rev. E 66, 066615 1-13 (2002).
[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]

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]

Phys. Rev. Lett.

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]

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]

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

Physica D

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]

Plasma Phys.

H.L. Pecseli and J.J. Rasmussen, �??Nonlinear electron waves in strongly magnetized plasmas,�?? Plasma Phys. 22, 421-438 (1980).
[CrossRef]

Science

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

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]

A. Snyder and J. Mitchell, �??Accessible Solitons,�?? Science 276, 1538-1541 (1997).
[CrossRef]

Sov. J. Plasma Phys.

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).

Theor. Math. Phys.

S.K. Turitsyn, �??Spatial dispersion of nonlinearity and stability of multidimensional solitons,�?? Theor. Math. Phys. 64, 226-232 (1985).
[CrossRef]

Ukr. J. Phys.

T.A. Davydova and A.I. Fishchuk, �??Upper hybrid nonlinear wave structures,�?? Ukr. J. Phys. 40, 487-494 (1995).

Other

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

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

A. Yakimenko, Y. Zaliznyak, Y. Kivshar, �??Stable vortex solitons in nonlocal self-focusing nonlinear media,�?? <a href="http://au.arxiv.org/abs/nlin.PS/0411024">http://au.arxiv.org/abs/nlin.PS/0411024</a>

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)

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

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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