Abstract

The fast contraction, or pinching, of optical vortices in both thermal and Kerr self-defocusing media is investigated by numerical techniques. For the thermal case, heat diffusion across the vortex core is described, and the heretofore unexplained stability of optical-vortex solitons in thermal media is explained.

© 2001 Optical Society of America

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1998 (1)

1996 (2)

1995 (2)

G. A. Swartzlander, Jr., D. L. Drugan, N. Hallak, M. O. Freeman, and C. T. Law, “Optical transistor effect using an optical vortex soliton,” Laser Phys. 5, 704–709 (1995).

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 (1995).
[CrossRef] [PubMed]

1994 (2)

F. B. de Colstoun, G. Khitrova, A. V. Fedorov, T. R. Nelson, C. Lowry, T. M. Brennan, B. G. Hammons, and P. D. Maker, “Transverse modes, vortices and vertical-cavity surface-emitting lasers,” Chaos, Solitons Fractals 4, 1575–1596 (1994).
[CrossRef]

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Pratti, A. J. Kent, G. L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. D’Angelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[CrossRef] [PubMed]

1993 (2)

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123 (1993).
[CrossRef]

G. A. Swartzlander, Jr., B. L. Justus, A. L. Huston, A. J. Campillo, and C. T. Law, “Characteristics of a low f-number broadband visible thermal optical limiter,” Int. J. Nonlinear Opt. Phys. 2, 577–611 (1993).
[CrossRef]

1992 (4)

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]

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

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

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wave fronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

1991 (1)

G. A. Swartzlander, Jr., D. R. Anderson, J. J. Regan, H. Yin, and A. Kaplan, “Spatial dark-soliton stripes and grids in self-defocusing materials,” Phys. Rev. Lett. 66, 1583–1586 (1991).
[CrossRef] [PubMed]

1989 (2)

1988 (1)

1983 (1)

1978 (1)

1974 (1)

J. E. Bjorkholm and A. Ashkin, “CW self-focusing and self-trapping of light in sodium vapor,” Phys. Rev. Lett. 32, 129–132 (1974).
[CrossRef]

1968 (1)

J. R. Whinnery, D. T. Miller, and F. Dabby, “Thermal convection and spherical aberration distortion of laser beams in low-loss liquids,” IEEE J. Quantum Electron. QE-3, 382–383 (1968).

1965 (1)

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–7 (1965).
[CrossRef]

1961 (1)

G. Goubau and F. Schwering, “On the propagation of electromagnetic wave beams,” IRE Trans. Antennas Propag. 9, 248–256 (1961).
[CrossRef]

Allen, L.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123 (1993).
[CrossRef]

Anderson, D. R.

G. A. Swartzlander, Jr., D. R. Anderson, J. J. Regan, H. Yin, and A. Kaplan, “Spatial dark-soliton stripes and grids in self-defocusing materials,” Phys. Rev. Lett. 66, 1583–1586 (1991).
[CrossRef] [PubMed]

Ashkin, A.

J. E. Bjorkholm and A. Ashkin, “CW self-focusing and self-trapping of light in sodium vapor,” Phys. Rev. Lett. 32, 129–132 (1974).
[CrossRef]

Bazhenov, V. Yu.

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wave fronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123 (1993).
[CrossRef]

Bjorkholm, J. E.

J. E. Bjorkholm and A. Ashkin, “CW self-focusing and self-trapping of light in sodium vapor,” Phys. Rev. Lett. 32, 129–132 (1974).
[CrossRef]

Brambilla, M.

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Pratti, A. J. Kent, G. L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. D’Angelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[CrossRef] [PubMed]

Brennan, T. M.

F. B. de Colstoun, G. Khitrova, A. V. Fedorov, T. R. Nelson, C. Lowry, T. M. Brennan, B. G. Hammons, and P. D. Maker, “Transverse modes, vortices and vertical-cavity surface-emitting lasers,” Chaos, Solitons Fractals 4, 1575–1596 (1994).
[CrossRef]

Campillo, A. J.

G. A. Swartzlander, Jr., B. L. Justus, A. L. Huston, A. J. Campillo, and C. T. Law, “Characteristics of a low f-number broadband visible thermal optical limiter,” Int. J. Nonlinear Opt. Phys. 2, 577–611 (1993).
[CrossRef]

Cattaneo, M.

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Pratti, A. J. Kent, G. L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. D’Angelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[CrossRef] [PubMed]

Christou, J.

Coates, A. B.

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Pratti, A. J. Kent, G. L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. D’Angelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[CrossRef] [PubMed]

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 (1995).
[CrossRef] [PubMed]

D’Angelo, E. J.

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Pratti, A. J. Kent, G. L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. D’Angelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[CrossRef] [PubMed]

Dabby, F.

J. R. Whinnery, D. T. Miller, and F. Dabby, “Thermal convection and spherical aberration distortion of laser beams in low-loss liquids,” IEEE J. Quantum Electron. QE-3, 382–383 (1968).

de Colstoun, F. B.

F. B. de Colstoun, G. Khitrova, A. V. Fedorov, T. R. Nelson, C. Lowry, T. M. Brennan, B. G. Hammons, and P. D. Maker, “Transverse modes, vortices and vertical-cavity surface-emitting lasers,” Chaos, Solitons Fractals 4, 1575–1596 (1994).
[CrossRef]

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 (1995).
[CrossRef] [PubMed]

Drugan, D. L.

G. A. Swartzlander, Jr., D. L. Drugan, N. Hallak, M. O. Freeman, and C. T. Law, “Optical transistor effect using an optical vortex soliton,” Laser Phys. 5, 704–709 (1995).

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 (1995).
[CrossRef] [PubMed]

Fedorov, A. V.

F. B. de Colstoun, G. Khitrova, A. V. Fedorov, T. R. Nelson, C. Lowry, T. M. Brennan, B. G. Hammons, and P. D. Maker, “Transverse modes, vortices and vertical-cavity surface-emitting lasers,” Chaos, Solitons Fractals 4, 1575–1596 (1994).
[CrossRef]

Feit, M. D.

Fleck Jr., J. A.

Freeman, M. O.

G. A. Swartzlander, Jr., D. L. Drugan, N. Hallak, M. O. Freeman, and C. T. Law, “Optical transistor effect using an optical vortex soliton,” Laser Phys. 5, 704–709 (1995).

Gil, L.

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

Gordon, J. P.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–7 (1965).
[CrossRef]

Goubau, G.

G. Goubau and F. Schwering, “On the propagation of electromagnetic wave beams,” IRE Trans. Antennas Propag. 9, 248–256 (1961).
[CrossRef]

Green, C.

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Pratti, A. J. Kent, G. L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. D’Angelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[CrossRef] [PubMed]

Hallak, N.

G. A. Swartzlander, Jr., D. L. Drugan, N. Hallak, M. O. Freeman, and C. T. Law, “Optical transistor effect using an optical vortex soliton,” Laser Phys. 5, 704–709 (1995).

Hammons, B. G.

F. B. de Colstoun, G. Khitrova, A. V. Fedorov, T. R. Nelson, C. Lowry, T. M. Brennan, B. G. Hammons, and P. D. Maker, “Transverse modes, vortices and vertical-cavity surface-emitting lasers,” Chaos, Solitons Fractals 4, 1575–1596 (1994).
[CrossRef]

Heckenberg, N. R.

Huston, A. L.

G. A. Swartzlander, Jr., B. L. Justus, A. L. Huston, A. J. Campillo, and C. T. Law, “Characteristics of a low f-number broadband visible thermal optical limiter,” Int. J. Nonlinear Opt. Phys. 2, 577–611 (1993).
[CrossRef]

Iturbe Castillo, M. D.

Justus, B. L.

G. A. Swartzlander, Jr., B. L. Justus, A. L. Huston, A. J. Campillo, and C. T. Law, “Characteristics of a low f-number broadband visible thermal optical limiter,” Int. J. Nonlinear Opt. Phys. 2, 577–611 (1993).
[CrossRef]

Kaplan, A.

G. A. Swartzlander, Jr., D. R. Anderson, J. J. Regan, H. Yin, and A. Kaplan, “Spatial dark-soliton stripes and grids in self-defocusing materials,” Phys. Rev. Lett. 66, 1583–1586 (1991).
[CrossRef] [PubMed]

Kaplan, A. E.

Kent, A. J.

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Pratti, A. J. Kent, G. L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. D’Angelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[CrossRef] [PubMed]

Khitrova, G.

F. B. de Colstoun, G. Khitrova, A. V. Fedorov, T. R. Nelson, C. Lowry, T. M. Brennan, B. G. Hammons, and P. D. Maker, “Transverse modes, vortices and vertical-cavity surface-emitting lasers,” Chaos, Solitons Fractals 4, 1575–1596 (1994).
[CrossRef]

Kivshar, Yu. S.

Law, C. T.

G. A. Swartzlander, Jr., D. L. Drugan, N. Hallak, M. O. Freeman, and C. T. Law, “Optical transistor effect using an optical vortex soliton,” Laser Phys. 5, 704–709 (1995).

G. A. Swartzlander, Jr., B. L. Justus, A. L. Huston, A. J. Campillo, and C. T. Law, “Characteristics of a low f-number broadband visible thermal optical limiter,” Int. J. Nonlinear Opt. Phys. 2, 577–611 (1993).
[CrossRef]

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

Leite, R. C. C.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–7 (1965).
[CrossRef]

Lowry, C.

F. B. de Colstoun, G. Khitrova, A. V. Fedorov, T. R. Nelson, C. Lowry, T. M. Brennan, B. G. Hammons, and P. D. Maker, “Transverse modes, vortices and vertical-cavity surface-emitting lasers,” Chaos, Solitons Fractals 4, 1575–1596 (1994).
[CrossRef]

Lugiato, L. A.

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Pratti, A. J. Kent, G. L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. D’Angelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[CrossRef] [PubMed]

Luther-Davies, B.

Maker, P. D.

F. B. de Colstoun, G. Khitrova, A. V. Fedorov, T. R. Nelson, C. Lowry, T. M. Brennan, B. G. Hammons, and P. D. Maker, “Transverse modes, vortices and vertical-cavity surface-emitting lasers,” Chaos, Solitons Fractals 4, 1575–1596 (1994).
[CrossRef]

McDuff, R.

Miller, D. T.

J. R. Whinnery, D. T. Miller, and F. Dabby, “Thermal convection and spherical aberration distortion of laser beams in low-loss liquids,” IEEE J. Quantum Electron. QE-3, 382–383 (1968).

Mitchell, D. J.

Moore, R. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–7 (1965).
[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 (1995).
[CrossRef] [PubMed]

Nelson, T. R.

F. B. de Colstoun, G. Khitrova, A. V. Fedorov, T. R. Nelson, C. Lowry, T. M. Brennan, B. G. Hammons, and P. D. Maker, “Transverse modes, vortices and vertical-cavity surface-emitting lasers,” Chaos, Solitons Fractals 4, 1575–1596 (1994).
[CrossRef]

Oppo, G. L.

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Pratti, A. J. Kent, G. L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. D’Angelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[CrossRef] [PubMed]

Pirovano, R.

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Pratti, A. J. Kent, G. L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. D’Angelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[CrossRef] [PubMed]

Poladian, L.

Porto, S. P.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–7 (1965).
[CrossRef]

Pratti, F.

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Pratti, A. J. Kent, G. L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. D’Angelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[CrossRef] [PubMed]

Regan, J. J.

G. A. Swartzlander, Jr., D. R. Anderson, J. J. Regan, H. Yin, and A. Kaplan, “Spatial dark-soliton stripes and grids in self-defocusing materials,” Phys. Rev. Lett. 66, 1583–1586 (1991).
[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.

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 (1995).
[CrossRef] [PubMed]

Sánchez-Mondragón, J. J.

Schwering, F.

G. Goubau and F. Schwering, “On the propagation of electromagnetic wave beams,” IRE Trans. Antennas Propag. 9, 248–256 (1961).
[CrossRef]

Segev, 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 (1995).
[CrossRef] [PubMed]

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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 (1995).
[CrossRef] [PubMed]

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Snyder, A. W.

Soskin, M. S.

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wave fronts,” J. Mod. Opt. 39, 985–990 (1992).
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Z. S. Sacks, D. Rozas, and G. A. Swartzlander, Jr., “Holographic formation of optical vortex filaments,” J. Opt. Soc. Am. B 15, 2226–2234 (1998).
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[CrossRef]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef]

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M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Pratti, A. J. Kent, G. L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. D’Angelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
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[CrossRef]

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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 (1995).
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Appl. Opt. (1)

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

G. A. Swartzlander, Jr., B. L. Justus, A. L. Huston, A. J. Campillo, and C. T. Law, “Characteristics of a low f-number broadband visible thermal optical limiter,” Int. J. Nonlinear Opt. Phys. 2, 577–611 (1993).
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[CrossRef]

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[CrossRef]

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

Laser Phys. (1)

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Opt. Commun. (2)

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[CrossRef] [PubMed]

Phys. Rev. Lett. (4)

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

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Figures (8)

Fig. 1
Fig. 1

Formation of an overshooting ring around the vortex core. (a) The intensity profile at the input is transformed into an OVS with a bright ring around it at (b) z=152 mm and (c) z=200 mm. Parameters of the beam are wg=5 mm, wv,0=0.4 mm, P=1.4 W, and wavelength λ=0.514 µm. Nonlinear coefficient n2=-2×10-6 cm2/W, refractive-index change Δn=7×10-6, the corresponding size of the OVS is wovs=34 µm, and Zpinch=150 mm.

Fig. 2
Fig. 2

Initially large vortex core contracts to a subsoliton size of wvmin=23 µm in a Kerr material (parameters are identical to those in Fig. 1). The vortex remains contracted to subsoliton size over ∼60 characteristic nonlinear distances. As expected, the collapse distance, Zc, is found to agree with the theoretical value of Zpinch.

Fig. 3
Fig. 3

Pinch distance Zc as a function of the input refractive-index change Δn0=n2P/(πwg2/2) in a Kerr nonlinear medium. Symbols represent numerically determined values of Zc; solid lines represent Zpinch=wv,0/Δn0.

Fig. 4
Fig. 4

Temperature profiles in a thin layer of absorbing liquid heated by an annular beam (dashed) with parameters listed in Table 1 at t=0.01 s and 4 s. The temperature difference across the core, δT=Tb-Tc, varies with time.

Fig. 5
Fig. 5

Time evolution of the temperature difference δT for two input vortex sizes. The maximum of δT occurs at topt=3.2 s when wv,0=0.4 mm, and topt=1.7 s when wv,0=0.2 mm. Beam and material parameters are listed in Table 1.

Fig. 6
Fig. 6

Vortex size wv,L at the output face contracts to a minimum size wv,min=65 µm at time twv,min=3.1 s. Beam and material parameters are listed in Table 1. The gray curve is drawn to aid the eye.

Fig. 7
Fig. 7

Intensity profiles at the output face of a thermal medium of length L=200 mm at time (a) t=0.002 s, (b) t=0.40 s, (c) t=3.1 s, and (d) t=8.0 s. Beam and material parameters are listed in Table 1.

Fig. 8
Fig. 8

Output vortex size as a function of incident power. Triangles represent the Kerr medium (n2=2×10-6 cm2/W), and squares represent the thermal medium (α=3.6 m-1, t=1.5 s). The OVS size wv=1.27/kΔn/n0 is shown for comparison by a solid curve. Gray curves are drawn to aid the eye.

Tables (1)

Tables Icon

Table 1 Parameters of the Heat Source and Material Used to Produce Fig. 4

Equations (9)

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

E(ρ, θ)=E0 tanh(ρ/wv,0)exp(-ρ2/wg2)exp(iθ),
-2ik(E/z)+2E+2k2ΔnE/n0=0,
wovs1.27/k|Δn|/n0,
Zpinch=wv,0/Δn0,
Δn(x, y, z, t)=(n/T)T(x, y, z, t),
T(x, y, z, t)/t=D2T+α|E(x, y, z)|2/(cpρ),
toptwv,0wg/7D,
T(x, y, z)t=(n+1)Δt
=T(x, y, z)t=nΔt+Δt(D2T(x, y, z)t=nΔt+α|E(x, y, z)|t=nΔt2/cpρ),

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