Abstract

We study spatially localized optical vortices created by self-trapping of partially incoherent light with a phase dislocation in a biased photorefractive crystal. In contrast to a decay of coherent self-trapped vortex beams due to the azimuthal modulational instability, the incoherent vortices are stabilized for large values of the spatial incoherence; this was confirmed by experiment. We analyze the spatial coherence properties of the incoherent optical vortices and reveal the existence of ringlike singularities in the spatial coherence function of a vortex field that can characterize the stable propagation of vortices through nonlinear media.

© 2005 Optical Society of America

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  1. J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
    [CrossRef]
  2. See for example a comprehensive review paper by M. S. Soskin and M. V. Vasnetsov, "Optical vortices," in Progress in Optics, Vol. 42, E.Wolf, ed. (Elsevier, Amsterdam, 2001).
  3. See an extensive list of references on optical vortices in G. A. Swartzlander, Jr., Singular Optics/Optical Vortex References, http://www.u.arizona.edu/∼grovers/SO/so.html.
  4. Such self-trapped singular beams were first suggested in V. I. Kruglov and R. A. Vlasov, "Spiral self-trapping propagation of optical beams in media with cubic nonlinearity," Phys. Lett. 111, 401-404 (1985).
    [CrossRef]
  5. See, for example Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, San Diego, Calif., 2003), Chap. 8.
  6. W. J. Firth and D. V. Skryabin, "Optical solitons carrying orbital angular momentum," Phys. Rev. Lett. 79, 2450-2453 (1997).
    [CrossRef]
  7. L. Torner and D. V. Petrov, "Azimuthal instabilities and self-breaking of beams into sets of solitons in bulk second-harmonic generation," Electron. Lett. 33, 608-610 (1997).
    [CrossRef]
  8. D. V. Skryabin and W. Firth, "Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media," Phys. Rev. E 58, 3916-3930 (1998).
    [CrossRef]
  9. V. Tikhonenko, J. Christou, and B. Luther-Davies, "Spiraling bright spatial solitons formed by the breakup of an optical vortex in a saturable self-focusing medium," J. Opt. Soc. Am. B 12, 2046-2052 (1995).
    [CrossRef]
  10. V. Tikhonenko, J. Christou, and B. Luther-Davies, "Three-dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium," Phys. Rev. Lett. 76, 2698-2701 (1996).
    [CrossRef] [PubMed]
  11. Z. Chen, M. Shih, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, "Steady-state vortex-screening solitons formed in biased photorefractive media," Opt. Lett. 22, 1751-1753 (1997).
    [CrossRef]
  12. D. V. Petrov, L. Torner, J. Nartorell, R. Vilaseca, J. P. Torres, and C. Cojocaru, "Observation of azimuthal modulational instability and formation of patterns of optical solitons in a quadratic nonlinear crystal," Opt. Lett. 23, 1444-1446 (1998).
    [CrossRef]
  13. H. Saito and M. Ueda, "Split instability of a vortex in an attractive Bose-Einstein condensate," Phys. Rev. Lett. 89, 190402(4) (2002).
    [CrossRef]
  14. M. Quiroga-Teìxeìro and H. Michinel, "Stable azimuthal stationary state in quintic nonlinear optical media," J. Opt. Soc. Am. B 14, 2004-2008 (1997).
    [CrossRef]
  15. A. Desyatnikov, A. Maimistov, and B. A. Malomed, "Three-dimensional spinning solitons in dispersive media with the cubic-quintic nonlinearity," Phys. Rev. E 61, 3107-3113 (2000).
    [CrossRef]
  16. H. Michinel, J. Campo-Táboas, M. L. Quiroga-Teixeiro, J. R. Salqueiro, and R. Gracía-Fernández, "Excitation of stable vortex solitons in nonlinear cubic-quintic materials," J. Opt. B: Quantum Semiclassical Opt. 3, 314-317 (2001).
    [CrossRef]
  17. V. Skarka, N. B. Aleksic, and V. I. Berezhiani, "Dynamics of electromagnetic beam with phase dislocation in saturable nonlinear media," Phys. Lett. A 291, 124-127 (2001).
    [CrossRef]
  18. B. A. Malomed, L.-C. Crasovan, and D. Mihalache, "Stability of vortex solitons in the cubic-quintic model," Physica D 161, 187-201 (2002).
    [CrossRef]
  19. T. A. Davydova, A. I. Yakimenko, and Yu. A. Zaliznyak, "Two-dimensional solitons and vortices in normal and anomalous dispersive media," Phys. Rev. E 67, 026402(5) (2003).
    [CrossRef]
  20. I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L.-C. Crasovan, and D. Mihalache, "Stability of spinning ring solitons of the cubic-quintic nonlinear Schrödinger equation," Phys. Lett. A 288, 292-298 (2001).
    [CrossRef]
  21. 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 (2002).
    [CrossRef] [PubMed]
  22. C.-C. Jeng, M.-F. Shih, K. Motzek, and Yu. S. Kivshar, "Partially incoherent optical vortices in self-focusing nonlinear media," Phys. Rev. Lett. 92, 043904 (2004).
    [CrossRef] [PubMed]
  23. C. Anastassiou, M. Soljacic, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, "Eliminating the transverse instabilities of Kerr solitons," Phys. Rev. Lett. 85, 4888-4891 (2000).
    [CrossRef] [PubMed]
  24. J. P. Torres, C. Anastassiou, M. Segev, M. Soljacic, and D. N. Christodoulides, "Transverse instability of incoherent solitons in Kerr media," Phys. Rev. E 65, 015601(R) (2002).
    [CrossRef]
  25. M. Soljacic, M. Segev, T. Coskun, D. Christodoulides, and A. Vishwanath, "Modulation instability of incoherent beams in noninstantaneous nonlinear media," Phys. Rev. Lett. 84, 467-470 (2000).
    [CrossRef] [PubMed]
  26. D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. Christodoulides, "Modulation instability and pattern formation in spatially incoherent light beams," Science 290, 495-498 (2000).
    [CrossRef] [PubMed]
  27. H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, "Phase singularities of the coherence functions in Young's interference pattern," Opt. Lett. 28, 968-970 (2003).
    [CrossRef] [PubMed]
  28. D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., "Spatial correlation singularity of a vortex field," Phys. Rev. Lett. 92, 143905(4) (2004).
    [CrossRef] [PubMed]
  29. D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, "Theory of incoherent self-focusing in biased photorefractive media," Phys. Rev. Lett. 78, 646-649 (1997).
    [CrossRef]
  30. D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, "Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media," Phys. Rev. E 63, 035601(R) (2001).
    [CrossRef]
  31. C. Anastassiou, C. Pigier, M. Segev, D. Kip, E. Eugenieva, and D. Christodoulides, "Self-trapping of bright rings," Opt. Lett. 26, 911-913 (2001).
    [CrossRef]
  32. M. Mitchell, M. Segev, T. Coskun, and D. N. Christodoulides, "Theory of self-trapped spatially incoherent light beams," Phys. Rev. Lett. 79, 4990-4993 (1997).
    [CrossRef]
  33. K. Motzek, A. Stepken, F. Kaiser, M. R. Belic, M. Ahles, C. Weilnau, and C. Denz, "Dipole-mode vector solitons in anisotropic photorefractive media," Opt. Commun. 197, 161-168 (2001).
    [CrossRef]

2004 (2)

C.-C. Jeng, M.-F. Shih, K. Motzek, and Yu. S. Kivshar, "Partially incoherent optical vortices in self-focusing nonlinear media," Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef] [PubMed]

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., "Spatial correlation singularity of a vortex field," Phys. Rev. Lett. 92, 143905(4) (2004).
[CrossRef] [PubMed]

2003 (2)

H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, "Phase singularities of the coherence functions in Young's interference pattern," Opt. Lett. 28, 968-970 (2003).
[CrossRef] [PubMed]

T. A. Davydova, A. I. Yakimenko, and Yu. A. Zaliznyak, "Two-dimensional solitons and vortices in normal and anomalous dispersive media," Phys. Rev. E 67, 026402(5) (2003).
[CrossRef]

2002 (4)

H. Saito and M. Ueda, "Split instability of a vortex in an attractive Bose-Einstein condensate," Phys. Rev. Lett. 89, 190402(4) (2002).
[CrossRef]

J. P. Torres, C. Anastassiou, M. Segev, M. Soljacic, and D. N. Christodoulides, "Transverse instability of incoherent solitons in Kerr media," Phys. Rev. E 65, 015601(R) (2002).
[CrossRef]

B. A. Malomed, L.-C. Crasovan, and D. Mihalache, "Stability of vortex solitons in the cubic-quintic model," Physica D 161, 187-201 (2002).
[CrossRef]

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 (2002).
[CrossRef] [PubMed]

2001 (6)

K. Motzek, A. Stepken, F. Kaiser, M. R. Belic, M. Ahles, C. Weilnau, and C. Denz, "Dipole-mode vector solitons in anisotropic photorefractive media," Opt. Commun. 197, 161-168 (2001).
[CrossRef]

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, "Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media," Phys. Rev. E 63, 035601(R) (2001).
[CrossRef]

C. Anastassiou, C. Pigier, M. Segev, D. Kip, E. Eugenieva, and D. Christodoulides, "Self-trapping of bright rings," Opt. Lett. 26, 911-913 (2001).
[CrossRef]

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L.-C. Crasovan, and D. Mihalache, "Stability of spinning ring solitons of the cubic-quintic nonlinear Schrödinger equation," Phys. Lett. A 288, 292-298 (2001).
[CrossRef]

H. Michinel, J. Campo-Táboas, M. L. Quiroga-Teixeiro, J. R. Salqueiro, and R. Gracía-Fernández, "Excitation of stable vortex solitons in nonlinear cubic-quintic materials," J. Opt. B: Quantum Semiclassical Opt. 3, 314-317 (2001).
[CrossRef]

V. Skarka, N. B. Aleksic, and V. I. Berezhiani, "Dynamics of electromagnetic beam with phase dislocation in saturable nonlinear media," Phys. Lett. A 291, 124-127 (2001).
[CrossRef]

2000 (4)

A. Desyatnikov, A. Maimistov, and B. A. Malomed, "Three-dimensional spinning solitons in dispersive media with the cubic-quintic nonlinearity," Phys. Rev. E 61, 3107-3113 (2000).
[CrossRef]

C. Anastassiou, M. Soljacic, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, "Eliminating the transverse instabilities of Kerr solitons," Phys. Rev. Lett. 85, 4888-4891 (2000).
[CrossRef] [PubMed]

M. Soljacic, M. Segev, T. Coskun, D. Christodoulides, and A. Vishwanath, "Modulation instability of incoherent beams in noninstantaneous nonlinear media," Phys. Rev. Lett. 84, 467-470 (2000).
[CrossRef] [PubMed]

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. Christodoulides, "Modulation instability and pattern formation in spatially incoherent light beams," Science 290, 495-498 (2000).
[CrossRef] [PubMed]

1998 (2)

1997 (6)

Z. Chen, M. Shih, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, "Steady-state vortex-screening solitons formed in biased photorefractive media," Opt. Lett. 22, 1751-1753 (1997).
[CrossRef]

W. J. Firth and D. V. Skryabin, "Optical solitons carrying orbital angular momentum," Phys. Rev. Lett. 79, 2450-2453 (1997).
[CrossRef]

L. Torner and D. V. Petrov, "Azimuthal instabilities and self-breaking of beams into sets of solitons in bulk second-harmonic generation," Electron. Lett. 33, 608-610 (1997).
[CrossRef]

M. Quiroga-Teìxeìro and H. Michinel, "Stable azimuthal stationary state in quintic nonlinear optical media," J. Opt. Soc. Am. B 14, 2004-2008 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, "Theory of incoherent self-focusing in biased photorefractive media," Phys. Rev. Lett. 78, 646-649 (1997).
[CrossRef]

M. Mitchell, M. Segev, T. Coskun, and D. N. Christodoulides, "Theory of self-trapped spatially incoherent light beams," Phys. Rev. Lett. 79, 4990-4993 (1997).
[CrossRef]

1996 (1)

V. Tikhonenko, J. Christou, and B. Luther-Davies, "Three-dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium," Phys. Rev. Lett. 76, 2698-2701 (1996).
[CrossRef] [PubMed]

1995 (1)

1985 (1)

Such self-trapped singular beams were first suggested in V. I. Kruglov and R. A. Vlasov, "Spiral self-trapping propagation of optical beams in media with cubic nonlinearity," Phys. Lett. 111, 401-404 (1985).
[CrossRef]

1974 (1)

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

Agrawal, G. P.

See, for example Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, San Diego, Calif., 2003), Chap. 8.

Ahles, M.

K. Motzek, A. Stepken, F. Kaiser, M. R. Belic, M. Ahles, C. Weilnau, and C. Denz, "Dipole-mode vector solitons in anisotropic photorefractive media," Opt. Commun. 197, 161-168 (2001).
[CrossRef]

Aleksic, N. B.

V. Skarka, N. B. Aleksic, and V. I. Berezhiani, "Dynamics of electromagnetic beam with phase dislocation in saturable nonlinear media," Phys. Lett. A 291, 124-127 (2001).
[CrossRef]

Anastassiou, C.

J. P. Torres, C. Anastassiou, M. Segev, M. Soljacic, and D. N. Christodoulides, "Transverse instability of incoherent solitons in Kerr media," Phys. Rev. E 65, 015601(R) (2002).
[CrossRef]

C. Anastassiou, C. Pigier, M. Segev, D. Kip, E. Eugenieva, and D. Christodoulides, "Self-trapping of bright rings," Opt. Lett. 26, 911-913 (2001).
[CrossRef]

C. Anastassiou, M. Soljacic, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, "Eliminating the transverse instabilities of Kerr solitons," Phys. Rev. Lett. 85, 4888-4891 (2000).
[CrossRef] [PubMed]

Belic, M. R.

K. Motzek, A. Stepken, F. Kaiser, M. R. Belic, M. Ahles, C. Weilnau, and C. Denz, "Dipole-mode vector solitons in anisotropic photorefractive media," Opt. Commun. 197, 161-168 (2001).
[CrossRef]

Berezhiani, V. I.

V. Skarka, N. B. Aleksic, and V. I. Berezhiani, "Dynamics of electromagnetic beam with phase dislocation in saturable nonlinear media," Phys. Lett. A 291, 124-127 (2001).
[CrossRef]

Berry, M. V.

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

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 (2002).
[CrossRef] [PubMed]

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L.-C. Crasovan, and D. Mihalache, "Stability of spinning ring solitons of the cubic-quintic nonlinear Schrödinger equation," Phys. Lett. A 288, 292-298 (2001).
[CrossRef]

Campo-Táboas, J.

H. Michinel, J. Campo-Táboas, M. L. Quiroga-Teixeiro, J. R. Salqueiro, and R. Gracía-Fernández, "Excitation of stable vortex solitons in nonlinear cubic-quintic materials," J. Opt. B: Quantum Semiclassical Opt. 3, 314-317 (2001).
[CrossRef]

Chen, Z.

Christodoulides, D.

C. Anastassiou, C. Pigier, M. Segev, D. Kip, E. Eugenieva, and D. Christodoulides, "Self-trapping of bright rings," Opt. Lett. 26, 911-913 (2001).
[CrossRef]

M. Soljacic, M. Segev, T. Coskun, D. Christodoulides, and A. Vishwanath, "Modulation instability of incoherent beams in noninstantaneous nonlinear media," Phys. Rev. Lett. 84, 467-470 (2000).
[CrossRef] [PubMed]

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. Christodoulides, "Modulation instability and pattern formation in spatially incoherent light beams," Science 290, 495-498 (2000).
[CrossRef] [PubMed]

Christodoulides, D. N.

J. P. Torres, C. Anastassiou, M. Segev, M. Soljacic, and D. N. Christodoulides, "Transverse instability of incoherent solitons in Kerr media," Phys. Rev. E 65, 015601(R) (2002).
[CrossRef]

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, "Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media," Phys. Rev. E 63, 035601(R) (2001).
[CrossRef]

C. Anastassiou, M. Soljacic, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, "Eliminating the transverse instabilities of Kerr solitons," Phys. Rev. Lett. 85, 4888-4891 (2000).
[CrossRef] [PubMed]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, "Theory of incoherent self-focusing in biased photorefractive media," Phys. Rev. Lett. 78, 646-649 (1997).
[CrossRef]

M. Mitchell, M. Segev, T. Coskun, and D. N. Christodoulides, "Theory of self-trapped spatially incoherent light beams," Phys. Rev. Lett. 79, 4990-4993 (1997).
[CrossRef]

Christou, J.

V. Tikhonenko, J. Christou, and B. Luther-Davies, "Three-dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium," Phys. Rev. Lett. 76, 2698-2701 (1996).
[CrossRef] [PubMed]

V. Tikhonenko, J. Christou, and B. Luther-Davies, "Spiraling bright spatial solitons formed by the breakup of an optical vortex in a saturable self-focusing medium," J. Opt. Soc. Am. B 12, 2046-2052 (1995).
[CrossRef]

Cojocaru, C.

Coskun, T.

M. Soljacic, M. Segev, T. Coskun, D. Christodoulides, and A. Vishwanath, "Modulation instability of incoherent beams in noninstantaneous nonlinear media," Phys. Rev. Lett. 84, 467-470 (2000).
[CrossRef] [PubMed]

M. Mitchell, M. Segev, T. Coskun, and D. N. Christodoulides, "Theory of self-trapped spatially incoherent light beams," Phys. Rev. Lett. 79, 4990-4993 (1997).
[CrossRef]

Coskun, T. H.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, "Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media," Phys. Rev. E 63, 035601(R) (2001).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, "Theory of incoherent self-focusing in biased photorefractive media," Phys. Rev. Lett. 78, 646-649 (1997).
[CrossRef]

Crasovan, L.-C.

B. A. Malomed, L.-C. Crasovan, and D. Mihalache, "Stability of vortex solitons in the cubic-quintic model," Physica D 161, 187-201 (2002).
[CrossRef]

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 (2002).
[CrossRef] [PubMed]

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L.-C. Crasovan, and D. Mihalache, "Stability of spinning ring solitons of the cubic-quintic nonlinear Schrödinger equation," Phys. Lett. A 288, 292-298 (2001).
[CrossRef]

Davydova, T. A.

T. A. Davydova, A. I. Yakimenko, and Yu. A. Zaliznyak, "Two-dimensional solitons and vortices in normal and anomalous dispersive media," Phys. Rev. E 67, 026402(5) (2003).
[CrossRef]

Denz, C.

K. Motzek, A. Stepken, F. Kaiser, M. R. Belic, M. Ahles, C. Weilnau, and C. Denz, "Dipole-mode vector solitons in anisotropic photorefractive media," Opt. Commun. 197, 161-168 (2001).
[CrossRef]

Desyatnikov, A.

A. Desyatnikov, A. Maimistov, and B. A. Malomed, "Three-dimensional spinning solitons in dispersive media with the cubic-quintic nonlinearity," Phys. Rev. E 61, 3107-3113 (2000).
[CrossRef]

Eugenieva, E.

C. Anastassiou, C. Pigier, M. Segev, D. Kip, E. Eugenieva, and D. Christodoulides, "Self-trapping of bright rings," Opt. Lett. 26, 911-913 (2001).
[CrossRef]

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. Christodoulides, "Modulation instability and pattern formation in spatially incoherent light beams," Science 290, 495-498 (2000).
[CrossRef] [PubMed]

Eugenieva, E. D.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, "Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media," Phys. Rev. E 63, 035601(R) (2001).
[CrossRef]

C. Anastassiou, M. Soljacic, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, "Eliminating the transverse instabilities of Kerr solitons," Phys. Rev. Lett. 85, 4888-4891 (2000).
[CrossRef] [PubMed]

Firth, W.

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

Firth, W. J.

W. J. Firth and D. V. Skryabin, "Optical solitons carrying orbital angular momentum," Phys. Rev. Lett. 79, 2450-2453 (1997).
[CrossRef]

Gbur, G.

Gracía-Fernández, R.

H. Michinel, J. Campo-Táboas, M. L. Quiroga-Teixeiro, J. R. Salqueiro, and R. Gracía-Fernández, "Excitation of stable vortex solitons in nonlinear cubic-quintic materials," J. Opt. B: Quantum Semiclassical Opt. 3, 314-317 (2001).
[CrossRef]

Jeng, C.-C.

C.-C. Jeng, M.-F. Shih, K. Motzek, and Yu. S. Kivshar, "Partially incoherent optical vortices in self-focusing nonlinear media," Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef] [PubMed]

Kaiser, F.

K. Motzek, A. Stepken, F. Kaiser, M. R. Belic, M. Ahles, C. Weilnau, and C. Denz, "Dipole-mode vector solitons in anisotropic photorefractive media," Opt. Commun. 197, 161-168 (2001).
[CrossRef]

Kip, D.

C. Anastassiou, C. Pigier, M. Segev, D. Kip, E. Eugenieva, and D. Christodoulides, "Self-trapping of bright rings," Opt. Lett. 26, 911-913 (2001).
[CrossRef]

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. Christodoulides, "Modulation instability and pattern formation in spatially incoherent light beams," Science 290, 495-498 (2000).
[CrossRef] [PubMed]

C. Anastassiou, M. Soljacic, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, "Eliminating the transverse instabilities of Kerr solitons," Phys. Rev. Lett. 85, 4888-4891 (2000).
[CrossRef] [PubMed]

Kivshar, Yu. S.

C.-C. Jeng, M.-F. Shih, K. Motzek, and Yu. S. Kivshar, "Partially incoherent optical vortices in self-focusing nonlinear media," Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef] [PubMed]

See, for example Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, San Diego, Calif., 2003), Chap. 8.

Kruglov, V. I.

Such self-trapped singular beams were first suggested in V. I. Kruglov and R. A. Vlasov, "Spiral self-trapping propagation of optical beams in media with cubic nonlinearity," Phys. Lett. 111, 401-404 (1985).
[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 (2002).
[CrossRef] [PubMed]

Luther-Davies, B.

V. Tikhonenko, J. Christou, and B. Luther-Davies, "Three-dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium," Phys. Rev. Lett. 76, 2698-2701 (1996).
[CrossRef] [PubMed]

V. Tikhonenko, J. Christou, and B. Luther-Davies, "Spiraling bright spatial solitons formed by the breakup of an optical vortex in a saturable self-focusing medium," J. Opt. Soc. Am. B 12, 2046-2052 (1995).
[CrossRef]

Maimistov, A.

A. Desyatnikov, A. Maimistov, and B. A. Malomed, "Three-dimensional spinning solitons in dispersive media with the cubic-quintic nonlinearity," Phys. Rev. E 61, 3107-3113 (2000).
[CrossRef]

Maker, P. D.

Maleev, I. D.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., "Spatial correlation singularity of a vortex field," Phys. Rev. Lett. 92, 143905(4) (2004).
[CrossRef] [PubMed]

Malomed, B. A.

B. A. Malomed, L.-C. Crasovan, and D. Mihalache, "Stability of vortex solitons in the cubic-quintic model," Physica D 161, 187-201 (2002).
[CrossRef]

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 (2002).
[CrossRef] [PubMed]

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L.-C. Crasovan, and D. Mihalache, "Stability of spinning ring solitons of the cubic-quintic nonlinear Schrödinger equation," Phys. Lett. A 288, 292-298 (2001).
[CrossRef]

A. Desyatnikov, A. Maimistov, and B. A. Malomed, "Three-dimensional spinning solitons in dispersive media with the cubic-quintic nonlinearity," Phys. Rev. E 61, 3107-3113 (2000).
[CrossRef]

Marathay, A. S.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., "Spatial correlation singularity of a vortex field," Phys. Rev. Lett. 92, 143905(4) (2004).
[CrossRef] [PubMed]

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 (2002).
[CrossRef] [PubMed]

Michinel, H.

H. Michinel, J. Campo-Táboas, M. L. Quiroga-Teixeiro, J. R. Salqueiro, and R. Gracía-Fernández, "Excitation of stable vortex solitons in nonlinear cubic-quintic materials," J. Opt. B: Quantum Semiclassical Opt. 3, 314-317 (2001).
[CrossRef]

M. Quiroga-Teìxeìro and H. Michinel, "Stable azimuthal stationary state in quintic nonlinear optical media," J. Opt. Soc. Am. B 14, 2004-2008 (1997).
[CrossRef]

Mihalache, D.

B. A. Malomed, L.-C. Crasovan, and D. Mihalache, "Stability of vortex solitons in the cubic-quintic model," Physica D 161, 187-201 (2002).
[CrossRef]

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 (2002).
[CrossRef] [PubMed]

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L.-C. Crasovan, and D. Mihalache, "Stability of spinning ring solitons of the cubic-quintic nonlinear Schrödinger equation," Phys. Lett. A 288, 292-298 (2001).
[CrossRef]

Mitchell, M.

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, "Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media," Phys. Rev. E 63, 035601(R) (2001).
[CrossRef]

M. Mitchell, M. Segev, T. Coskun, and D. N. Christodoulides, "Theory of self-trapped spatially incoherent light beams," Phys. Rev. Lett. 79, 4990-4993 (1997).
[CrossRef]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, "Theory of incoherent self-focusing in biased photorefractive media," Phys. Rev. Lett. 78, 646-649 (1997).
[CrossRef]

Motzek, K.

C.-C. Jeng, M.-F. Shih, K. Motzek, and Yu. S. Kivshar, "Partially incoherent optical vortices in self-focusing nonlinear media," Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef] [PubMed]

K. Motzek, A. Stepken, F. Kaiser, M. R. Belic, M. Ahles, C. Weilnau, and C. Denz, "Dipole-mode vector solitons in anisotropic photorefractive media," Opt. Commun. 197, 161-168 (2001).
[CrossRef]

Muller, R. E.

Musslimani, Z. H.

C. Anastassiou, M. Soljacic, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, "Eliminating the transverse instabilities of Kerr solitons," Phys. Rev. Lett. 85, 4888-4891 (2000).
[CrossRef] [PubMed]

Nartorell, J.

Nye, J. F.

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

Palacios, D. M.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., "Spatial correlation singularity of a vortex field," Phys. Rev. Lett. 92, 143905(4) (2004).
[CrossRef] [PubMed]

Petrov, D. V.

Pigier, C.

Quiroga-Teixeiro, M. L.

H. Michinel, J. Campo-Táboas, M. L. Quiroga-Teixeiro, J. R. Salqueiro, and R. Gracía-Fernández, "Excitation of stable vortex solitons in nonlinear cubic-quintic materials," J. Opt. B: Quantum Semiclassical Opt. 3, 314-317 (2001).
[CrossRef]

Quiroga-Teìxeìro, M.

Saito, H.

H. Saito and M. Ueda, "Split instability of a vortex in an attractive Bose-Einstein condensate," Phys. Rev. Lett. 89, 190402(4) (2002).
[CrossRef]

Salqueiro, J. R.

H. Michinel, J. Campo-Táboas, M. L. Quiroga-Teixeiro, J. R. Salqueiro, and R. Gracía-Fernández, "Excitation of stable vortex solitons in nonlinear cubic-quintic materials," J. Opt. B: Quantum Semiclassical Opt. 3, 314-317 (2001).
[CrossRef]

Sammut, R. A.

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L.-C. Crasovan, and D. Mihalache, "Stability of spinning ring solitons of the cubic-quintic nonlinear Schrödinger equation," Phys. Lett. A 288, 292-298 (2001).
[CrossRef]

Schouten, H. F.

Segev, M.

J. P. Torres, C. Anastassiou, M. Segev, M. Soljacic, and D. N. Christodoulides, "Transverse instability of incoherent solitons in Kerr media," Phys. Rev. E 65, 015601(R) (2002).
[CrossRef]

C. Anastassiou, C. Pigier, M. Segev, D. Kip, E. Eugenieva, and D. Christodoulides, "Self-trapping of bright rings," Opt. Lett. 26, 911-913 (2001).
[CrossRef]

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, "Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media," Phys. Rev. E 63, 035601(R) (2001).
[CrossRef]

C. Anastassiou, M. Soljacic, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, "Eliminating the transverse instabilities of Kerr solitons," Phys. Rev. Lett. 85, 4888-4891 (2000).
[CrossRef] [PubMed]

M. Soljacic, M. Segev, T. Coskun, D. Christodoulides, and A. Vishwanath, "Modulation instability of incoherent beams in noninstantaneous nonlinear media," Phys. Rev. Lett. 84, 467-470 (2000).
[CrossRef] [PubMed]

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. Christodoulides, "Modulation instability and pattern formation in spatially incoherent light beams," Science 290, 495-498 (2000).
[CrossRef] [PubMed]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, "Theory of incoherent self-focusing in biased photorefractive media," Phys. Rev. Lett. 78, 646-649 (1997).
[CrossRef]

M. Mitchell, M. Segev, T. Coskun, and D. N. Christodoulides, "Theory of self-trapped spatially incoherent light beams," Phys. Rev. Lett. 79, 4990-4993 (1997).
[CrossRef]

Z. Chen, M. Shih, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, "Steady-state vortex-screening solitons formed in biased photorefractive media," Opt. Lett. 22, 1751-1753 (1997).
[CrossRef]

Shih, M.

Shih, M.-F.

C.-C. Jeng, M.-F. Shih, K. Motzek, and Yu. S. Kivshar, "Partially incoherent optical vortices in self-focusing nonlinear media," Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef] [PubMed]

Skarka, V.

V. Skarka, N. B. Aleksic, and V. I. Berezhiani, "Dynamics of electromagnetic beam with phase dislocation in saturable nonlinear media," Phys. Lett. A 291, 124-127 (2001).
[CrossRef]

Skryabin, D. V.

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

W. J. Firth and D. V. Skryabin, "Optical solitons carrying orbital angular momentum," Phys. Rev. Lett. 79, 2450-2453 (1997).
[CrossRef]

Soljacic, M.

J. P. Torres, C. Anastassiou, M. Segev, M. Soljacic, and D. N. Christodoulides, "Transverse instability of incoherent solitons in Kerr media," Phys. Rev. E 65, 015601(R) (2002).
[CrossRef]

M. Soljacic, M. Segev, T. Coskun, D. Christodoulides, and A. Vishwanath, "Modulation instability of incoherent beams in noninstantaneous nonlinear media," Phys. Rev. Lett. 84, 467-470 (2000).
[CrossRef] [PubMed]

C. Anastassiou, M. Soljacic, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, "Eliminating the transverse instabilities of Kerr solitons," Phys. Rev. Lett. 85, 4888-4891 (2000).
[CrossRef] [PubMed]

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. Christodoulides, "Modulation instability and pattern formation in spatially incoherent light beams," Science 290, 495-498 (2000).
[CrossRef] [PubMed]

Soskin, M. S.

See for example a comprehensive review paper by M. S. Soskin and M. V. Vasnetsov, "Optical vortices," in Progress in Optics, Vol. 42, E.Wolf, ed. (Elsevier, Amsterdam, 2001).

Stepken, A.

K. Motzek, A. Stepken, F. Kaiser, M. R. Belic, M. Ahles, C. Weilnau, and C. Denz, "Dipole-mode vector solitons in anisotropic photorefractive media," Opt. Commun. 197, 161-168 (2001).
[CrossRef]

Swartzlander, G. A.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., "Spatial correlation singularity of a vortex field," Phys. Rev. Lett. 92, 143905(4) (2004).
[CrossRef] [PubMed]

See an extensive list of references on optical vortices in G. A. Swartzlander, Jr., Singular Optics/Optical Vortex References, http://www.u.arizona.edu/∼grovers/SO/so.html.

Tikhonenko, V.

V. Tikhonenko, J. Christou, and B. Luther-Davies, "Three-dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium," Phys. Rev. Lett. 76, 2698-2701 (1996).
[CrossRef] [PubMed]

V. Tikhonenko, J. Christou, and B. Luther-Davies, "Spiraling bright spatial solitons formed by the breakup of an optical vortex in a saturable self-focusing medium," J. Opt. Soc. Am. B 12, 2046-2052 (1995).
[CrossRef]

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 (2002).
[CrossRef] [PubMed]

D. V. Petrov, L. Torner, J. Nartorell, R. Vilaseca, J. P. Torres, and C. Cojocaru, "Observation of azimuthal modulational instability and formation of patterns of optical solitons in a quadratic nonlinear crystal," Opt. Lett. 23, 1444-1446 (1998).
[CrossRef]

L. Torner and D. V. Petrov, "Azimuthal instabilities and self-breaking of beams into sets of solitons in bulk second-harmonic generation," Electron. Lett. 33, 608-610 (1997).
[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 (2002).
[CrossRef] [PubMed]

J. P. Torres, C. Anastassiou, M. Segev, M. Soljacic, and D. N. Christodoulides, "Transverse instability of incoherent solitons in Kerr media," Phys. Rev. E 65, 015601(R) (2002).
[CrossRef]

C. Anastassiou, M. Soljacic, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, "Eliminating the transverse instabilities of Kerr solitons," Phys. Rev. Lett. 85, 4888-4891 (2000).
[CrossRef] [PubMed]

D. V. Petrov, L. Torner, J. Nartorell, R. Vilaseca, J. P. Torres, and C. Cojocaru, "Observation of azimuthal modulational instability and formation of patterns of optical solitons in a quadratic nonlinear crystal," Opt. Lett. 23, 1444-1446 (1998).
[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 (2002).
[CrossRef] [PubMed]

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L.-C. Crasovan, and D. Mihalache, "Stability of spinning ring solitons of the cubic-quintic nonlinear Schrödinger equation," Phys. Lett. A 288, 292-298 (2001).
[CrossRef]

Ueda, M.

H. Saito and M. Ueda, "Split instability of a vortex in an attractive Bose-Einstein condensate," Phys. Rev. Lett. 89, 190402(4) (2002).
[CrossRef]

Vasnetsov, M. V.

See for example a comprehensive review paper by M. S. Soskin and M. V. Vasnetsov, "Optical vortices," in Progress in Optics, Vol. 42, E.Wolf, ed. (Elsevier, Amsterdam, 2001).

Vilaseca, R.

Vishwanath, A.

M. Soljacic, M. Segev, T. Coskun, D. Christodoulides, and A. Vishwanath, "Modulation instability of incoherent beams in noninstantaneous nonlinear media," Phys. Rev. Lett. 84, 467-470 (2000).
[CrossRef] [PubMed]

Visser, T. D.

Vlasov, R. A.

Such self-trapped singular beams were first suggested in V. I. Kruglov and R. A. Vlasov, "Spiral self-trapping propagation of optical beams in media with cubic nonlinearity," Phys. Lett. 111, 401-404 (1985).
[CrossRef]

Weilnau, C.

K. Motzek, A. Stepken, F. Kaiser, M. R. Belic, M. Ahles, C. Weilnau, and C. Denz, "Dipole-mode vector solitons in anisotropic photorefractive media," Opt. Commun. 197, 161-168 (2001).
[CrossRef]

Wilson, D. W.

Wolf, E.

Yakimenko, A. I.

T. A. Davydova, A. I. Yakimenko, and Yu. A. Zaliznyak, "Two-dimensional solitons and vortices in normal and anomalous dispersive media," Phys. Rev. E 67, 026402(5) (2003).
[CrossRef]

Zaliznyak, Yu. A.

T. A. Davydova, A. I. Yakimenko, and Yu. A. Zaliznyak, "Two-dimensional solitons and vortices in normal and anomalous dispersive media," Phys. Rev. E 67, 026402(5) (2003).
[CrossRef]

Electron. Lett. (1)

L. Torner and D. V. Petrov, "Azimuthal instabilities and self-breaking of beams into sets of solitons in bulk second-harmonic generation," Electron. Lett. 33, 608-610 (1997).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt. (1)

H. Michinel, J. Campo-Táboas, M. L. Quiroga-Teixeiro, J. R. Salqueiro, and R. Gracía-Fernández, "Excitation of stable vortex solitons in nonlinear cubic-quintic materials," J. Opt. B: Quantum Semiclassical Opt. 3, 314-317 (2001).
[CrossRef]

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

Opt. Commun. (1)

K. Motzek, A. Stepken, F. Kaiser, M. R. Belic, M. Ahles, C. Weilnau, and C. Denz, "Dipole-mode vector solitons in anisotropic photorefractive media," Opt. Commun. 197, 161-168 (2001).
[CrossRef]

Opt. Lett. (4)

Phys. Lett. (1)

Such self-trapped singular beams were first suggested in V. I. Kruglov and R. A. Vlasov, "Spiral self-trapping propagation of optical beams in media with cubic nonlinearity," Phys. Lett. 111, 401-404 (1985).
[CrossRef]

Phys. Lett. A (2)

V. Skarka, N. B. Aleksic, and V. I. Berezhiani, "Dynamics of electromagnetic beam with phase dislocation in saturable nonlinear media," Phys. Lett. A 291, 124-127 (2001).
[CrossRef]

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L.-C. Crasovan, and D. Mihalache, "Stability of spinning ring solitons of the cubic-quintic nonlinear Schrödinger equation," Phys. Lett. A 288, 292-298 (2001).
[CrossRef]

Phys. Rev. E (5)

J. P. Torres, C. Anastassiou, M. Segev, M. Soljacic, and D. N. Christodoulides, "Transverse instability of incoherent solitons in Kerr media," Phys. Rev. E 65, 015601(R) (2002).
[CrossRef]

D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, "Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media," Phys. Rev. E 63, 035601(R) (2001).
[CrossRef]

T. A. Davydova, A. I. Yakimenko, and Yu. A. Zaliznyak, "Two-dimensional solitons and vortices in normal and anomalous dispersive media," Phys. Rev. E 67, 026402(5) (2003).
[CrossRef]

A. Desyatnikov, A. Maimistov, and B. A. Malomed, "Three-dimensional spinning solitons in dispersive media with the cubic-quintic nonlinearity," Phys. Rev. E 61, 3107-3113 (2000).
[CrossRef]

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

Phys. Rev. Lett. (10)

V. Tikhonenko, J. Christou, and B. Luther-Davies, "Three-dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium," Phys. Rev. Lett. 76, 2698-2701 (1996).
[CrossRef] [PubMed]

H. Saito and M. Ueda, "Split instability of a vortex in an attractive Bose-Einstein condensate," Phys. Rev. Lett. 89, 190402(4) (2002).
[CrossRef]

W. J. Firth and D. V. Skryabin, "Optical solitons carrying orbital angular momentum," Phys. Rev. Lett. 79, 2450-2453 (1997).
[CrossRef]

M. Mitchell, M. Segev, T. Coskun, and D. N. Christodoulides, "Theory of self-trapped spatially incoherent light beams," Phys. Rev. Lett. 79, 4990-4993 (1997).
[CrossRef]

M. Soljacic, M. Segev, T. Coskun, D. Christodoulides, and A. Vishwanath, "Modulation instability of incoherent beams in noninstantaneous nonlinear media," Phys. Rev. Lett. 84, 467-470 (2000).
[CrossRef] [PubMed]

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., "Spatial correlation singularity of a vortex field," Phys. Rev. Lett. 92, 143905(4) (2004).
[CrossRef] [PubMed]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, "Theory of incoherent self-focusing in biased photorefractive media," Phys. Rev. Lett. 78, 646-649 (1997).
[CrossRef]

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 (2002).
[CrossRef] [PubMed]

C.-C. Jeng, M.-F. Shih, K. Motzek, and Yu. S. Kivshar, "Partially incoherent optical vortices in self-focusing nonlinear media," Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef] [PubMed]

C. Anastassiou, M. Soljacic, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, "Eliminating the transverse instabilities of Kerr solitons," Phys. Rev. Lett. 85, 4888-4891 (2000).
[CrossRef] [PubMed]

Physica D (1)

B. A. Malomed, L.-C. Crasovan, and D. Mihalache, "Stability of vortex solitons in the cubic-quintic model," Physica D 161, 187-201 (2002).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

Science (1)

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. Christodoulides, "Modulation instability and pattern formation in spatially incoherent light beams," Science 290, 495-498 (2000).
[CrossRef] [PubMed]

Other (3)

See for example a comprehensive review paper by M. S. Soskin and M. V. Vasnetsov, "Optical vortices," in Progress in Optics, Vol. 42, E.Wolf, ed. (Elsevier, Amsterdam, 2001).

See an extensive list of references on optical vortices in G. A. Swartzlander, Jr., Singular Optics/Optical Vortex References, http://www.u.arizona.edu/∼grovers/SO/so.html.

See, for example Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, San Diego, Calif., 2003), Chap. 8.

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Figures (5)

Fig. 1
Fig. 1

Comparison between numerical (upper row) and experimental (lower row) results for the vortex stabilization effect. Numerical results are shown for the vortex after 9 mm of propagation for (from left to right) the coherent case and for three partially incoherent cases at θ 0 = 0.14 ° , θ 0 = 0.29 ° , θ 0 = 0.38 ° , respectively.

Fig. 2
Fig. 2

Images of the intensity (left column) and the modulus of the cross correlation (right column) of an incoherent vortex with θ 0 = 0.64 ° (strong incoherence). Contrary to the case of the linear propagation, there is a local intensity minimum in the beam’s center. The cross correlation, however, shows the same ring of phase singularities as predicted in the linear theory.

Fig. 3
Fig. 3

Images of the intensity (upper row) and the modulus of the cross correlation (lower row) for the breakup of an incoherent vortex at θ 0 = 0.37 ° (weak incoherence), when the ring is not preserved.

Fig. 4
Fig. 4

The intensity (left column) and the cross correlation (right column) of the far field. The effects of the nonlinearity on the intensity distribution can be clearly seen, whereas the cross correlation maintains more or less the structure one would expect in the case of linear propagation.

Fig. 5
Fig. 5

A composite soliton calculated by using the three modes with the topological charges m = 0 , 1 , 2 . (a) Profiles of the three components, (b) total intensity of the vortex soliton, (c) vortex cross correlation Γ ( r , r ) .

Equations (9)

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

E ( r , t ) = j E j ( r ) exp { i k j r } exp [ i γ j ( t ) ] ,
E j ( r ) = { 1 π θ 0 exp [ ( α j 2 + β j 2 ) θ 0 2 ] } 1 2 A ( r ) ,
A ( r ) = ( r w 0 ) 2 exp ( i m ϕ ) exp ( r 2 σ 2 )
i E j ( r , z ) z + 2 E j ( r , z ) + η ( r , z ) E j ( r , z ) = 0 ,
η ( r , z ) = I ( r , z ) 1 + s I ( r , z ) ,
Γ ( r 1 , r 2 ; z ) = E * ( r 2 , z , t ) E ( r 1 , z , t ) ,
E ( r , t ) = m = 0 2 E m ( r ) exp ( i m φ ) exp [ i γ m ( t ) ] .
m tot = Im { E * ( r × E ) d r } e z I d r ,
Γ ( r , r ) = m , m = 0 2 E m * ( r ) E m ( r ) = m = 0 2 E m * ( r ) E m ( r )

Metrics