Abstract

We demonstrate experimentally the generation of square and hexagonal lattices of optical vortices and reveal their propagation in a saturable nonlinear medium. If the topological charges of the vortices have identical signs, the lattice exhibits rotation, whereas if their signs alternate between being the same and being opposite to each other, we observe stable propagation of the structures. In the nonlinear medium the lattices induce periodic modulation of the refractive index. Diffraction of a probe beam by this nonlinearity-induced periodic structure is observed.

© 2002 Optical Society of America

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    [Crossref]
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  8. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristiasen, and J. P. Woerdman, “Helical-wave-front laser-beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
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  15. G. Molina-Terriza, E. M. Wright, and L. Torner, “Propagation and control of noncanonical optical vortices,” Opt. Lett. 26, 163–165 (2001).
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    [Crossref]
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    [Crossref]
  18. D. Rozas, Z. S. Sacks, and G. A. Swartzlander, “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. 79, 3399–3402 (1997).
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  25. G. A. Swartzlander and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503–2506 (1992).
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    [Crossref]
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    [Crossref] [PubMed]
  28. Z. Chen and M. Segev, “Self-trapping of an optical vortex by use of the bulk photovoltaic effect,” Phys. Rev. Lett. 78, 2948–2951 (1997).
    [Crossref]
  29. P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000).
    [Crossref] [PubMed]
  30. A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Decay of high order optical vortices in anisotropic nonlinear optical media,” Phys. Rev. Lett. 78, 2108–2111 (1997).
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  31. I. Velchev, A. Dreischuh, D. Neshev, and S. Dinev, “Multiple-charged optical vortex solitons in bulk Kerr media,” Opt. Commun. 140, 77–82 (1997).
    [Crossref]
  32. A. Dreischuh, G. G. Paulus, F. Zacher, F. Grabson, D. Neshev, and H. Walther, “Modulational instability of multiple-charged optical vortex solitons under saturation of the nonlinearity,” Phys. Rev. E 60, 7518–7524 (1999).
    [Crossref]
  33. D. Neshev, A. Dreischuh, M. Assa, and S. Dinev, “Motion control of ensembles of ordered optical vortices generated on finite extent background,” Opt. Commun. 151, 413–421 (1998).
    [Crossref]
  34. B. Luther-Davies, R. Powles, and V. Tikhonenko, “Nonlinear rotation of three-dimensional dark spatial solitons in a Gaussian laser beam,” Opt. Lett. 19, 1816–1818 (1994).
    [Crossref] [PubMed]
  35. D. Rozas and G. A. Swartzlander, “Observed rotational enhancement of nonlinear optical vortices,” Opt. Lett. 25, 126–128 (2000).
    [Crossref]
  36. G.-H. Kim, J.-H. Jeon, Y.-Ch. Noh, K.-H. Ko, H.-J. Moon, J.-H. Lee, and J.-S. Chang, “An array of phase singularities in a self-defocusing medium,” Opt. Commun. 147, 131–137 (1998).
    [Crossref]
  37. G. M. McDonald, K. S. Syed, and W. J. Firth, “Dark spatial soliton break-up in the transverse plane,” Opt. Commun. 95, 281–288 (1993).
    [Crossref]
  38. V. Tikhonenko, J. Christou, B. Luther-Davies, and Yu. S. Kivshar, “Observation of vortex solitons created by the instability of dark soliton stripes,” Opt. Lett. 21, 1129–1131 (1996).
    [Crossref] [PubMed]
  39. A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Propagation of dark stripe beams in nonlinear media: snake instability and creation of optical vortices,” Phys. Rev. Lett. 76, 2262–2265 (1996).
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    [Crossref]
  41. G. Molina-Terriza and L. Torner, “Reconfigurable dynamic beam shaping in seeded frequency doubling,” Opt. Lett. 26, 154–156 (2001).
    [Crossref]
  42. G. S. McDonald, K. S. Syed, and W. J. Firth, “Optical vortices in beam propagation through a self-defocusing medium,” Opt. Commun. 94, 469–476 (1992).
    [Crossref]
  43. S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, and V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389–R2392 (2000).
    [Crossref]
  44. K. T. Gahagan and G. A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. 21, 827–829 (1996).
    [Crossref] [PubMed]
  45. Ch. Slinger, P. Brett, V. Hui, G. Monnington, D. Pain, and I. Sage, “Electrically controllable multiple, active, computer-generated hologram,” Opt. Lett. 22, 1113–1115 (1997).
    [Crossref] [PubMed]
  46. K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, “Vortex formation in a stirred Bose–Einstein condensate,” Phys. Rev. Lett. 84, 806–809 (2000).
    [Crossref] [PubMed]
  47. B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, “Watching dark solitons decay into vortex rings in a Bose–Einstein condensate,” Phys. Rev. Lett. 86, 2926–2929 (2001).
    [Crossref] [PubMed]
  48. J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, “Observation of vortex lattices in Bose–Einstein condensates,” Science 292, 476–479 (2001).
    [Crossref] [PubMed]
  49. A. Dreischuh, D. Neshev, G. G. Paulus, and H. Walther, “Experimental generation of steering odd dark beams of finite length,” J. Opt. Soc. Am. B 17, 2011–2017 (2000).
    [Crossref]
  50. A. M. Deykoon and G. A. Swartzlander, “Pinched optical-vortex soliton,” J. Opt. Soc. Am. B 18, 804–810 (2001).
    [Crossref]

2001 (8)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

G. Molina-Terriza, E. M. Wright, and L. Torner, “Propagation and control of noncanonical optical vortices,” Opt. Lett. 26, 163–165 (2001).
[Crossref]

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[Crossref]

D. Neshev, A. Nepomnyashchy, and Yu. S. Kivshar, “Nonlinear Aharonov–Bohm scattering for optical vortices,” Phys. Rev. Lett. 87, 043901 (2001).
[Crossref]

G. Molina-Terriza and L. Torner, “Reconfigurable dynamic beam shaping in seeded frequency doubling,” Opt. Lett. 26, 154–156 (2001).
[Crossref]

B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, “Watching dark solitons decay into vortex rings in a Bose–Einstein condensate,” Phys. Rev. Lett. 86, 2926–2929 (2001).
[Crossref] [PubMed]

J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, “Observation of vortex lattices in Bose–Einstein condensates,” Science 292, 476–479 (2001).
[Crossref] [PubMed]

A. M. Deykoon and G. A. Swartzlander, “Pinched optical-vortex soliton,” J. Opt. Soc. Am. B 18, 804–810 (2001).
[Crossref]

2000 (7)

K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, “Vortex formation in a stirred Bose–Einstein condensate,” Phys. Rev. Lett. 84, 806–809 (2000).
[Crossref] [PubMed]

A. Dreischuh, D. Neshev, G. G. Paulus, and H. Walther, “Experimental generation of steering odd dark beams of finite length,” J. Opt. Soc. Am. B 17, 2011–2017 (2000).
[Crossref]

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, and V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389–R2392 (2000).
[Crossref]

A. H. Carlsson, J. N. Malmberg, D. Anderson, M. Lisak, E. A. Ostrovskaya, T. Alexander, and Yu. S. Kivshar, “Linear and nonlinear waveguides induced by optical vortex solitons,” Opt. Lett. 25, 660–662 (2000).
[Crossref]

C. T. Law, X. Zhang, and G. A. Swartzlander, “Waveguiding properties of optical vortex solitons,” Opt. Lett. 25, 55–57 (2000).
[Crossref]

P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000).
[Crossref] [PubMed]

D. Rozas and G. A. Swartzlander, “Observed rotational enhancement of nonlinear optical vortices,” Opt. Lett. 25, 126–128 (2000).
[Crossref]

1999 (4)

A. Dreischuh, G. G. Paulus, F. Zacher, F. Grabson, D. Neshev, and H. Walther, “Modulational instability of multiple-charged optical vortex solitons under saturation of the nonlinearity,” Phys. Rev. E 60, 7518–7524 (1999).
[Crossref]

I. Freund, “Critical point explosions in two-dimensional wave fields,” Opt. Commun. 159, 99–117 (1999).
[Crossref]

J. E. Williams and M. J. Holland, “Preparing topological states of a Bose–Einstein condensate,” Nature 401, 568–572 (1999).
[Crossref]

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, “Vortices in a Bose–Einstein condensate,” Phys. Rev. Lett. 83, 2498–2501 (1999).
[Crossref]

1998 (5)

D. Neshev, A. Dreischuh, M. Assa, and S. Dinev, “Motion control of ensembles of ordered optical vortices generated on finite extent background,” Opt. Commun. 151, 413–421 (1998).
[Crossref]

G.-H. Kim, J.-H. Jeon, Y.-Ch. Noh, K.-H. Ko, H.-J. Moon, J.-H. Lee, and J.-S. Chang, “An array of phase singularities in a self-defocusing medium,” Opt. Commun. 147, 131–137 (1998).
[Crossref]

V. Tikhonenko, Yu. S. Kivshar, V. V. Steblina, and A. A. Zozulya, “Vortex solitons in a saturable optical medium,” J. Opt. Soc. Am. B 15, 79–86 (1998).
[Crossref]

Yu. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies, and L. M. Pismen, “Dynamics of optical vortex solitons,” Opt. Commun. 152, 198–206 (1998).
[Crossref]

Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep.,  298, 172–185 (1998).
[Crossref]

1997 (7)

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Decay of high order optical vortices in anisotropic nonlinear optical media,” Phys. Rev. Lett. 78, 2108–2111 (1997).
[Crossref]

I. Velchev, A. Dreischuh, D. Neshev, and S. Dinev, “Multiple-charged optical vortex solitons in bulk Kerr media,” Opt. Commun. 140, 77–82 (1997).
[Crossref]

D. V. Petrov, F. Canal, and L. Torner, “A simple method to generate optical beams with a screw phase dislocation,” Opt. Commun. 143, 265–267 (1997).
[Crossref]

D. Rozas, C. T. Law, and G. A. Swartzlander, “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B 14, 3054–3065 (1997).
[Crossref]

D. Rozas, Z. S. Sacks, and G. A. Swartzlander, “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. 79, 3399–3402 (1997).
[Crossref]

Z. Chen and M. Segev, “Self-trapping of an optical vortex by use of the bulk photovoltaic effect,” Phys. Rev. Lett. 78, 2948–2951 (1997).
[Crossref]

Ch. Slinger, P. Brett, V. Hui, G. Monnington, D. Pain, and I. Sage, “Electrically controllable multiple, active, computer-generated hologram,” Opt. Lett. 22, 1113–1115 (1997).
[Crossref] [PubMed]

1996 (4)

1995 (1)

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, “Dark photorefractive spatial solitons and photorefractive vortex solitons,” Phys. Rev. Lett. 74, 1978–1981 (1995).
[Crossref] [PubMed]

1994 (3)

B. Luther-Davies, R. Powles, and V. Tikhonenko, “Nonlinear rotation of three-dimensional dark spatial solitons in a Gaussian laser beam,” Opt. Lett. 19, 1816–1818 (1994).
[Crossref] [PubMed]

K. Staliunas, “Vortices and dark solitons in two-dimensional nonlinear Schrödinger equation,” Chaos, Solitons Fractals 4, 1783–1796 (1994).
[Crossref]

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristiasen, and J. P. Woerdman, “Helical-wave-front laser-beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[Crossref]

1993 (2)

I. V. Basisty, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[Crossref]

G. M. McDonald, K. S. Syed, and W. J. Firth, “Dark spatial soliton break-up in the transverse plane,” Opt. Commun. 95, 281–288 (1993).
[Crossref]

1992 (4)

G. A. Swartzlander and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503–2506 (1992).
[Crossref] [PubMed]

A. W. Snyder, L. Poladian, and D. J. Mitchell, “Stable dark self-guided beams of circular symmetry in a bulk Kerr medium,” Opt. Lett. 17, 789–791 (1992).
[Crossref] [PubMed]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[Crossref] [PubMed]

G. S. McDonald, K. S. Syed, and W. J. Firth, “Optical vortices in beam propagation through a self-defocusing medium,” Opt. Commun. 94, 469–476 (1992).
[Crossref]

1990 (1)

1989 (1)

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

Abo-Shaeer, J. R.

J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, “Observation of vortex lattices in Bose–Einstein condensates,” Science 292, 476–479 (2001).
[Crossref] [PubMed]

Alexander, T.

Anderson, B. P.

B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, “Watching dark solitons decay into vortex rings in a Bose–Einstein condensate,” Phys. Rev. Lett. 86, 2926–2929 (2001).
[Crossref] [PubMed]

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, “Vortices in a Bose–Einstein condensate,” Phys. Rev. Lett. 83, 2498–2501 (1999).
[Crossref]

Anderson, D.

Assa, M.

D. Neshev, A. Dreischuh, M. Assa, and S. Dinev, “Motion control of ensembles of ordered optical vortices generated on finite extent background,” Opt. Commun. 151, 413–421 (1998).
[Crossref]

Basisty, I. V.

I. V. Basisty, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[Crossref]

Bazhenov, V. Y.

I. V. Basisty, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[Crossref]

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristiasen, and J. P. Woerdman, “Helical-wave-front laser-beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[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]

Birner, A.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, and V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389–R2392 (2000).
[Crossref]

Brett, P.

Busch, K.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, and V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389–R2392 (2000).
[Crossref]

Canal, F.

D. V. Petrov, F. Canal, and L. Torner, “A simple method to generate optical beams with a screw phase dislocation,” Opt. Commun. 143, 265–267 (1997).
[Crossref]

Carlsson, A. H.

Chang, J.-S.

G.-H. Kim, J.-H. Jeon, Y.-Ch. Noh, K.-H. Ko, H.-J. Moon, J.-H. Lee, and J.-S. Chang, “An array of phase singularities in a self-defocusing medium,” Opt. Commun. 147, 131–137 (1998).
[Crossref]

Chen, Z.

Z. Chen and M. Segev, “Self-trapping of an optical vortex by use of the bulk photovoltaic effect,” Phys. Rev. Lett. 78, 2948–2951 (1997).
[Crossref]

Chevy, F.

K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, “Vortex formation in a stirred Bose–Einstein condensate,” Phys. Rev. Lett. 84, 806–809 (2000).
[Crossref] [PubMed]

Chinaglia, W.

P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000).
[Crossref] [PubMed]

Christou, J.

Clark, C. W.

B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, “Watching dark solitons decay into vortex rings in a Bose–Einstein condensate,” Phys. Rev. Lett. 86, 2926–2929 (2001).
[Crossref] [PubMed]

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristiasen, and J. P. Woerdman, “Helical-wave-front laser-beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[Crossref]

Collins, L. A.

B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, “Watching dark solitons decay into vortex rings in a Bose–Einstein condensate,” Phys. Rev. Lett. 86, 2926–2929 (2001).
[Crossref] [PubMed]

Cornell, E. A.

B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, “Watching dark solitons decay into vortex rings in a Bose–Einstein condensate,” Phys. Rev. Lett. 86, 2926–2929 (2001).
[Crossref] [PubMed]

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, “Vortices in a Bose–Einstein condensate,” Phys. Rev. Lett. 83, 2498–2501 (1999).
[Crossref]

Coullet, P.

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

Crosignani, B.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, “Dark photorefractive spatial solitons and photorefractive vortex solitons,” Phys. Rev. Lett. 74, 1978–1981 (1995).
[Crossref] [PubMed]

Dalibard, J.

K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, “Vortex formation in a stirred Bose–Einstein condensate,” Phys. Rev. Lett. 84, 806–809 (2000).
[Crossref] [PubMed]

Deykoon, A. M.

Di Porto, P.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, “Dark photorefractive spatial solitons and photorefractive vortex solitons,” Phys. Rev. Lett. 74, 1978–1981 (1995).
[Crossref] [PubMed]

Di Trapani, P.

P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000).
[Crossref] [PubMed]

Dinev, S.

D. Neshev, A. Dreischuh, M. Assa, and S. Dinev, “Motion control of ensembles of ordered optical vortices generated on finite extent background,” Opt. Commun. 151, 413–421 (1998).
[Crossref]

I. Velchev, A. Dreischuh, D. Neshev, and S. Dinev, “Multiple-charged optical vortex solitons in bulk Kerr media,” Opt. Commun. 140, 77–82 (1997).
[Crossref]

Donnelly, R. J.

See, e.g., R. J. Donnelly, Quantized Vortices in Helium II (Cambridge U. Press, Cambridge, 1991).

Dreischuh, A.

A. Dreischuh, D. Neshev, G. G. Paulus, and H. Walther, “Experimental generation of steering odd dark beams of finite length,” J. Opt. Soc. Am. B 17, 2011–2017 (2000).
[Crossref]

A. Dreischuh, G. G. Paulus, F. Zacher, F. Grabson, D. Neshev, and H. Walther, “Modulational instability of multiple-charged optical vortex solitons under saturation of the nonlinearity,” Phys. Rev. E 60, 7518–7524 (1999).
[Crossref]

D. Neshev, A. Dreischuh, M. Assa, and S. Dinev, “Motion control of ensembles of ordered optical vortices generated on finite extent background,” Opt. Commun. 151, 413–421 (1998).
[Crossref]

I. Velchev, A. Dreischuh, D. Neshev, and S. Dinev, “Multiple-charged optical vortex solitons in bulk Kerr media,” Opt. Commun. 140, 77–82 (1997).
[Crossref]

Duree, G.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, “Dark photorefractive spatial solitons and photorefractive vortex solitons,” Phys. Rev. Lett. 74, 1978–1981 (1995).
[Crossref] [PubMed]

Feder, D. L.

B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, “Watching dark solitons decay into vortex rings in a Bose–Einstein condensate,” Phys. Rev. Lett. 86, 2926–2929 (2001).
[Crossref] [PubMed]

Firth, W. J.

G. M. McDonald, K. S. Syed, and W. J. Firth, “Dark spatial soliton break-up in the transverse plane,” Opt. Commun. 95, 281–288 (1993).
[Crossref]

G. S. McDonald, K. S. Syed, and W. J. Firth, “Optical vortices in beam propagation through a self-defocusing medium,” Opt. Commun. 94, 469–476 (1992).
[Crossref]

Freund, I.

I. Freund, “Critical point explosions in two-dimensional wave fields,” Opt. Commun. 159, 99–117 (1999).
[Crossref]

Gahagan, K. T.

Gil, L.

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

Gösele, U.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, and V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389–R2392 (2000).
[Crossref]

Grabson, F.

A. Dreischuh, G. G. Paulus, F. Zacher, F. Grabson, D. Neshev, and H. Walther, “Modulational instability of multiple-charged optical vortex solitons under saturation of the nonlinearity,” Phys. Rev. E 60, 7518–7524 (1999).
[Crossref]

Haljan, P. C.

B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, “Watching dark solitons decay into vortex rings in a Bose–Einstein condensate,” Phys. Rev. Lett. 86, 2926–2929 (2001).
[Crossref] [PubMed]

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, “Vortices in a Bose–Einstein condensate,” Phys. Rev. Lett. 83, 2498–2501 (1999).
[Crossref]

Hall, D. S.

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, “Vortices in a Bose–Einstein condensate,” Phys. Rev. Lett. 83, 2498–2501 (1999).
[Crossref]

Heckenberg, N. R.

Holland, M. J.

J. E. Williams and M. J. Holland, “Preparing topological states of a Bose–Einstein condensate,” Nature 401, 568–572 (1999).
[Crossref]

Hui, V.

Jeon, J.-H.

G.-H. Kim, J.-H. Jeon, Y.-Ch. Noh, K.-H. Ko, H.-J. Moon, J.-H. Lee, and J.-S. Chang, “An array of phase singularities in a self-defocusing medium,” Opt. Commun. 147, 131–137 (1998).
[Crossref]

John, S.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, and V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389–R2392 (2000).
[Crossref]

Ketterle, W.

J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, “Observation of vortex lattices in Bose–Einstein condensates,” Science 292, 476–479 (2001).
[Crossref] [PubMed]

Kim, G.-H.

G.-H. Kim, J.-H. Jeon, Y.-Ch. Noh, K.-H. Ko, H.-J. Moon, J.-H. Lee, and J.-S. Chang, “An array of phase singularities in a self-defocusing medium,” Opt. Commun. 147, 131–137 (1998).
[Crossref]

Kivshar, Y. S.

Kivshar, Yu. S.

D. Neshev, A. Nepomnyashchy, and Yu. S. Kivshar, “Nonlinear Aharonov–Bohm scattering for optical vortices,” Phys. Rev. Lett. 87, 043901 (2001).
[Crossref]

A. H. Carlsson, J. N. Malmberg, D. Anderson, M. Lisak, E. A. Ostrovskaya, T. Alexander, and Yu. S. Kivshar, “Linear and nonlinear waveguides induced by optical vortex solitons,” Opt. Lett. 25, 660–662 (2000).
[Crossref]

Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep.,  298, 172–185 (1998).
[Crossref]

Yu. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies, and L. M. Pismen, “Dynamics of optical vortex solitons,” Opt. Commun. 152, 198–206 (1998).
[Crossref]

V. Tikhonenko, Yu. S. Kivshar, V. V. Steblina, and A. A. Zozulya, “Vortex solitons in a saturable optical medium,” J. Opt. Soc. Am. B 15, 79–86 (1998).
[Crossref]

V. Tikhonenko, J. Christou, B. Luther-Davies, and Yu. S. Kivshar, “Observation of vortex solitons created by the instability of dark soliton stripes,” Opt. Lett. 21, 1129–1131 (1996).
[Crossref] [PubMed]

Ko, K.-H.

G.-H. Kim, J.-H. Jeon, Y.-Ch. Noh, K.-H. Ko, H.-J. Moon, J.-H. Lee, and J.-S. Chang, “An array of phase singularities in a self-defocusing medium,” Opt. Commun. 147, 131–137 (1998).
[Crossref]

Kristiasen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristiasen, and J. P. Woerdman, “Helical-wave-front laser-beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[Crossref]

Law, C. T.

Lee, J.-H.

G.-H. Kim, J.-H. Jeon, Y.-Ch. Noh, K.-H. Ko, H.-J. Moon, J.-H. Lee, and J.-S. Chang, “An array of phase singularities in a self-defocusing medium,” Opt. Commun. 147, 131–137 (1998).
[Crossref]

Lehmann, V.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, and V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389–R2392 (2000).
[Crossref]

Leonard, S. W.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, and V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389–R2392 (2000).
[Crossref]

Lisak, M.

Luther-Davies, B.

Madison, K. W.

K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, “Vortex formation in a stirred Bose–Einstein condensate,” Phys. Rev. Lett. 84, 806–809 (2000).
[Crossref] [PubMed]

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Malmberg, J. N.

Mamaev, A. V.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Decay of high order optical vortices in anisotropic nonlinear optical media,” Phys. Rev. Lett. 78, 2108–2111 (1997).
[Crossref]

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Propagation of dark stripe beams in nonlinear media: snake instability and creation of optical vortices,” Phys. Rev. Lett. 76, 2262–2265 (1996).
[Crossref] [PubMed]

Matthews, M. R.

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, “Vortices in a Bose–Einstein condensate,” Phys. Rev. Lett. 83, 2498–2501 (1999).
[Crossref]

McDonald, G. M.

G. M. McDonald, K. S. Syed, and W. J. Firth, “Dark spatial soliton break-up in the transverse plane,” Opt. Commun. 95, 281–288 (1993).
[Crossref]

McDonald, G. S.

G. S. McDonald, K. S. Syed, and W. J. Firth, “Optical vortices in beam propagation through a self-defocusing medium,” Opt. Commun. 94, 469–476 (1992).
[Crossref]

McDuff, R.

Minardi, S.

P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000).
[Crossref] [PubMed]

Mitchell, D. J.

Molina-Terriza, G.

Mondia, J. P.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, and V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389–R2392 (2000).
[Crossref]

Monnington, G.

Moon, H.-J.

G.-H. Kim, J.-H. Jeon, Y.-Ch. Noh, K.-H. Ko, H.-J. Moon, J.-H. Lee, and J.-S. Chang, “An array of phase singularities in a self-defocusing medium,” Opt. Commun. 147, 131–137 (1998).
[Crossref]

Morin, M.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, “Dark photorefractive spatial solitons and photorefractive vortex solitons,” Phys. Rev. Lett. 74, 1978–1981 (1995).
[Crossref] [PubMed]

Nepomnyashchy, A.

D. Neshev, A. Nepomnyashchy, and Yu. S. Kivshar, “Nonlinear Aharonov–Bohm scattering for optical vortices,” Phys. Rev. Lett. 87, 043901 (2001).
[Crossref]

Neshev, D.

D. Neshev, A. Nepomnyashchy, and Yu. S. Kivshar, “Nonlinear Aharonov–Bohm scattering for optical vortices,” Phys. Rev. Lett. 87, 043901 (2001).
[Crossref]

A. Dreischuh, D. Neshev, G. G. Paulus, and H. Walther, “Experimental generation of steering odd dark beams of finite length,” J. Opt. Soc. Am. B 17, 2011–2017 (2000).
[Crossref]

A. Dreischuh, G. G. Paulus, F. Zacher, F. Grabson, D. Neshev, and H. Walther, “Modulational instability of multiple-charged optical vortex solitons under saturation of the nonlinearity,” Phys. Rev. E 60, 7518–7524 (1999).
[Crossref]

D. Neshev, A. Dreischuh, M. Assa, and S. Dinev, “Motion control of ensembles of ordered optical vortices generated on finite extent background,” Opt. Commun. 151, 413–421 (1998).
[Crossref]

I. Velchev, A. Dreischuh, D. Neshev, and S. Dinev, “Multiple-charged optical vortex solitons in bulk Kerr media,” Opt. Commun. 140, 77–82 (1997).
[Crossref]

Noh, Y.-Ch.

G.-H. Kim, J.-H. Jeon, Y.-Ch. Noh, K.-H. Ko, H.-J. Moon, J.-H. Lee, and J.-S. Chang, “An array of phase singularities in a self-defocusing medium,” Opt. Commun. 147, 131–137 (1998).
[Crossref]

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]

Ostrovskaya, E. A.

Pain, D.

Paulus, G. G.

A. Dreischuh, D. Neshev, G. G. Paulus, and H. Walther, “Experimental generation of steering odd dark beams of finite length,” J. Opt. Soc. Am. B 17, 2011–2017 (2000).
[Crossref]

A. Dreischuh, G. G. Paulus, F. Zacher, F. Grabson, D. Neshev, and H. Walther, “Modulational instability of multiple-charged optical vortex solitons under saturation of the nonlinearity,” Phys. Rev. E 60, 7518–7524 (1999).
[Crossref]

Petrov, D. V.

D. V. Petrov, F. Canal, and L. Torner, “A simple method to generate optical beams with a screw phase dislocation,” Opt. Commun. 143, 265–267 (1997).
[Crossref]

Piskarskas, A.

P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000).
[Crossref] [PubMed]

Pismen, L. M.

Yu. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies, and L. M. Pismen, “Dynamics of optical vortex solitons,” Opt. Commun. 152, 198–206 (1998).
[Crossref]

Poladian, L.

Powles, R.

Raman, C.

J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, “Observation of vortex lattices in Bose–Einstein condensates,” Science 292, 476–479 (2001).
[Crossref] [PubMed]

Recolons, J.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[Crossref]

Regal, C. A.

B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, “Watching dark solitons decay into vortex rings in a Bose–Einstein condensate,” Phys. Rev. Lett. 86, 2926–2929 (2001).
[Crossref] [PubMed]

Rocca, F.

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

Rozas, D.

Sacks, Z. S.

D. Rozas, Z. S. Sacks, and G. A. Swartzlander, “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. 79, 3399–3402 (1997).
[Crossref]

Saffman, M.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Decay of high order optical vortices in anisotropic nonlinear optical media,” Phys. Rev. Lett. 78, 2108–2111 (1997).
[Crossref]

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Propagation of dark stripe beams in nonlinear media: snake instability and creation of optical vortices,” Phys. Rev. Lett. 76, 2262–2265 (1996).
[Crossref] [PubMed]

Sage, I.

Salamo, G.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, “Dark photorefractive spatial solitons and photorefractive vortex solitons,” Phys. Rev. Lett. 74, 1978–1981 (1995).
[Crossref] [PubMed]

Segev, M.

Z. Chen and M. Segev, “Self-trapping of an optical vortex by use of the bulk photovoltaic effect,” Phys. Rev. Lett. 78, 2948–2951 (1997).
[Crossref]

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, “Dark photorefractive spatial solitons and photorefractive vortex solitons,” Phys. Rev. Lett. 74, 1978–1981 (1995).
[Crossref] [PubMed]

Sharp, E.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, “Dark photorefractive spatial solitons and photorefractive vortex solitons,” Phys. Rev. Lett. 74, 1978–1981 (1995).
[Crossref] [PubMed]

Slinger, Ch.

Smith, C. P.

Snyder, A. W.

Soskin, M. S.

I. V. Basisty, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[Crossref]

Staliunas, K.

K. Staliunas, “Vortices and dark solitons in two-dimensional nonlinear Schrödinger equation,” Chaos, Solitons Fractals 4, 1783–1796 (1994).
[Crossref]

Steblina, V. V.

Swartzlander, G. A.

Syed, K. S.

G. M. McDonald, K. S. Syed, and W. J. Firth, “Dark spatial soliton break-up in the transverse plane,” Opt. Commun. 95, 281–288 (1993).
[Crossref]

G. S. McDonald, K. S. Syed, and W. J. Firth, “Optical vortices in beam propagation through a self-defocusing medium,” Opt. Commun. 94, 469–476 (1992).
[Crossref]

Tamm, Chr.

Tikhonenko, V.

Toader, O.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, and V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389–R2392 (2000).
[Crossref]

Torner, L.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[Crossref]

G. Molina-Terriza and L. Torner, “Reconfigurable dynamic beam shaping in seeded frequency doubling,” Opt. Lett. 26, 154–156 (2001).
[Crossref]

G. Molina-Terriza, E. M. Wright, and L. Torner, “Propagation and control of noncanonical optical vortices,” Opt. Lett. 26, 163–165 (2001).
[Crossref]

D. V. Petrov, F. Canal, and L. Torner, “A simple method to generate optical beams with a screw phase dislocation,” Opt. Commun. 143, 265–267 (1997).
[Crossref]

Torres, J. P.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[Crossref]

Valiulis, G.

P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000).
[Crossref] [PubMed]

van Driel, H. M.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, and V. Lehmann, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61, R2389–R2392 (2000).
[Crossref]

Vasnetsov, M. V.

I. V. Basisty, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[Crossref]

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Velchev, I.

I. Velchev, A. Dreischuh, D. Neshev, and S. Dinev, “Multiple-charged optical vortex solitons in bulk Kerr media,” Opt. Commun. 140, 77–82 (1997).
[Crossref]

Vogels, J. M.

J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, “Observation of vortex lattices in Bose–Einstein condensates,” Science 292, 476–479 (2001).
[Crossref] [PubMed]

Walther, H.

A. Dreischuh, D. Neshev, G. G. Paulus, and H. Walther, “Experimental generation of steering odd dark beams of finite length,” J. Opt. Soc. Am. B 17, 2011–2017 (2000).
[Crossref]

A. Dreischuh, G. G. Paulus, F. Zacher, F. Grabson, D. Neshev, and H. Walther, “Modulational instability of multiple-charged optical vortex solitons under saturation of the nonlinearity,” Phys. Rev. E 60, 7518–7524 (1999).
[Crossref]

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Weiss, C. O.

White, A. G.

Wieman, C. E.

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, “Vortices in a Bose–Einstein condensate,” Phys. Rev. Lett. 83, 2498–2501 (1999).
[Crossref]

Williams, J. E.

J. E. Williams and M. J. Holland, “Preparing topological states of a Bose–Einstein condensate,” Nature 401, 568–572 (1999).
[Crossref]

Woerdman, J. P.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristiasen, and J. P. Woerdman, “Helical-wave-front laser-beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[Crossref]

Wohlleben, W.

K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, “Vortex formation in a stirred Bose–Einstein condensate,” Phys. Rev. Lett. 84, 806–809 (2000).
[Crossref] [PubMed]

Wright, E. M.

G. Molina-Terriza, J. Recolons, J. P. Torres, L. Torner, and E. M. Wright, “Observation of the dynamical inversion of the topological charge of an optical vortex,” Phys. Rev. Lett. 87, 023902 (2001).
[Crossref]

G. Molina-Terriza, E. M. Wright, and L. Torner, “Propagation and control of noncanonical optical vortices,” Opt. Lett. 26, 163–165 (2001).
[Crossref]

Yariv, A.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, “Dark photorefractive spatial solitons and photorefractive vortex solitons,” Phys. Rev. Lett. 74, 1978–1981 (1995).
[Crossref] [PubMed]

Zacher, F.

A. Dreischuh, G. G. Paulus, F. Zacher, F. Grabson, D. Neshev, and H. Walther, “Modulational instability of multiple-charged optical vortex solitons under saturation of the nonlinearity,” Phys. Rev. E 60, 7518–7524 (1999).
[Crossref]

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Zhang, X.

Zozulya, A. A.

V. Tikhonenko, Yu. S. Kivshar, V. V. Steblina, and A. A. Zozulya, “Vortex solitons in a saturable optical medium,” J. Opt. Soc. Am. B 15, 79–86 (1998).
[Crossref]

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Decay of high order optical vortices in anisotropic nonlinear optical media,” Phys. Rev. Lett. 78, 2108–2111 (1997).
[Crossref]

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Propagation of dark stripe beams in nonlinear media: snake instability and creation of optical vortices,” Phys. Rev. Lett. 76, 2262–2265 (1996).
[Crossref] [PubMed]

Chaos, Solitons Fractals (1)

K. Staliunas, “Vortices and dark solitons in two-dimensional nonlinear Schrödinger equation,” Chaos, Solitons Fractals 4, 1783–1796 (1994).
[Crossref]

J. Opt. Soc. Am. B (5)

Nature (2)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

J. E. Williams and M. J. Holland, “Preparing topological states of a Bose–Einstein condensate,” Nature 401, 568–572 (1999).
[Crossref]

Opt. Commun. (11)

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

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Other (2)

See, e.g., M. Vasnetsov and K. Staliunas, eds., Optical Vortices (Nova Science, New York, 1999).

See, e.g., R. J. Donnelly, Quantized Vortices in Helium II (Cambridge U. Press, Cambridge, 1991).

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Figures (6)

Fig. 1
Fig. 1

Background beams containing vortex lattices of different geometries: (a) at the input of the NLM and (b), (c) at z=10 for lattices with equal and alternating TCs, respectively. Images of square-shaped and hexagonal lattices are shown. Bottom, transverse slices of hexagonal lattices. For comparison, a transverse slice of the background beam (without vortices imposed) is shown by dashed curves. In all cases Δ=5.0, I0=1, and s=0.4. The images are gray-scale coded; white corresponds to maximal intensity.

Fig. 2
Fig. 2

Angles of rotation of vortex lattices of equal TCs versus propagation distance for square (open squares) and hexagonal (filled circles) geometry. Solid curves are guides for the eye. The lattice parameters are the same as in Fig. 1.

Fig. 3
Fig. 3

Experimental setup: D, iris diaphragm; L1, L2, lenses of focal lengths 7.0 and 8.0 cm, respectively: M, mirror; E, screen; F, neutral-density filters; CCD, camera. The characteristic distances between elements are shown.

Fig. 4
Fig. 4

Interferograms of three experimentally generated lattices. Two neighboring vortices in each image are marked with arrows.

Fig. 5
Fig. 5

Experimental images of the vortex lattices after 10-cm propagation in a NLM. (a) Hexagonal lattice with alternating TCs for powers of 10 and 50 mW. (b) Hexagonal lattice with equal TCs for the same powers. The inset in each image represents the size and the orientation of the elementary cell of each lattice.

Fig. 6
Fig. 6

Images of the probe He–Ne laser beam on screen E (Fig. 3). (a) Input He–Ne laser beam profile; (b) intensity profiles of the probe beam at low pump power (10 mW) for different vertical displacements with respect to the pump beam; (c), (d) diffraction of the He–Ne beam from periodic phase gratings induced in the NLM by square and hexagonal lattices, respectively (pump power, 80 mW); (e) diffraction pattern when a single vortex is superimposed upon the pump beam (pump power, 30 mW).

Equations (5)

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iEz+12ΔE-|E|2E(1+s|E|2)γ=0,
E(r, z=0)=j,k=-sq(r-rjk)hex(r-rjk),
sq(r-rjk)=tanh(|r-rjk|)×expi sgnj+k arctan(r-rjk)y(r-rjk)x,
hex(r-rjk)=tanh(|r-rjk-r1|)tanh(|r-rjk-r2|)×expi arctan(r-rjk-r1)y(r-rjk-r1)x×expi sgn arctan(r-rjk-r2)y(r-rjk-r2)x.
B(x, y, z=0)=I0 exp-x2+y2w14,

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