Abstract

Optical vortices in linear and nonlinear media may exhibit propagation dynamics similar to hydrodynamic vortex phenomena. Analytical and numerical methods are used to describe and investigate the interaction between vortices and the background field. We demonstrate that optical vortices that have quasi-point core functions, such as optical vortex solitons, may orbit one another at rates that are orders of magnitude larger than those with nonlocalized cores.

© 1997 Optical Society of America

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1996 (9)

N. R. Heckenberg, M. Vaupel, J. T. Malos, and C. O. Weiss, “Optical-vortex pair creation and annihilation and helical astigmatism of a nonplanar ring resonator,” Phys. Rev. A 54, 1–10 (1996).

W. J. Firth and A. Lord, “Two-dimensional solitons in a Kerr cavity,” J. Mod. Opt. 43, 1071–1077 (1996).
[CrossRef]

A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetry breaking to spatial turbulence,” Phys. Rev. A 54, 870–879 (1996).
[CrossRef] [PubMed]

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Break-up of two-dimensional bright spatial solitons due to transverse modulation instability,” Europhys. Lett. 35, 25–30 (1996).
[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]

I. Velchev, A. Dreischuh, D. Neshev, and S. Dinev, “Interactions of optical vortex solitons superimposed on different background beams,” Opt. Commun. 130, 385–392 (1996).
[CrossRef]

K. T. Gahagan and G. A. Swartzlander, Jr., “Optical vortex trapping of particles,” Opt. Lett. 21, 827–829 (1996).
[CrossRef] [PubMed]

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]

J. Christou, V. Tikhonenko, Yu. S. Kivshar, and B. Luther-Davies, “Vortex soliton motion and steering,” Opt. Lett. 21, 1649–1651 (1996).
[CrossRef] [PubMed]

1995 (10)

K. Staliunas and C. O. Weiss, “Nonstationary vortex lattices in large-aperture class B lasers,” J. Opt. Soc. Am. B 12, 1142–1149 (1995).
[CrossRef]

F. S. Roux, “Dynamical behavior of optical vortices,” J. Opt. Soc. Am. B 12, 1215–1221 (1995).
[CrossRef]

M. Morin, G. Duree, and M. Segev, “Waveguides formed by quasi-steady-state photorefractive spatial solitons,” Opt. Lett. 20, 2066–2068 (1995).
[CrossRef] [PubMed]

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

T. Ackemann, E. Kriege, and W. Lange, “Phase singularities via nonlinear beam propagation in sodium vapor,” Opt. Commun. 115, 339–346 (1995).
[CrossRef]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high-efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

D. Y. Tang, N. R. Heckenberg, and C. O. Weiss, “Phase-dependent helical pattern formation in a laser,” Opt. Commun. 114, 95–100 (1995).
[CrossRef]

G. Slekys, K. Staliunas, and C. O. Weiss, “Motion of phase singularities in a class-B laser,” Opt. Commun. 119, 433–446 (1995).
[CrossRef]

K. Staliunas, A. Berzanskis, and V. Jarutis, “Vortex statistics in optical speckle fields,” Opt. Commun. 120, 23–28 (1995).
[CrossRef]

1994 (10)

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

C. T. Law and G. A. Swartzlander, Jr., “Polarized optical vortex solitons: instabilities and dynamics in Kerr nonlinear media,” Chaos Solitons Fractals 4, 1759–1766 (1994).
[CrossRef]

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

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]

G. Indebetouw and D. R. Korwan, “Model of vortices nucleation in a photorefractive phase-conjugate resonator,” J. Mod. Opt. 41, 941–950 (1994).
[CrossRef]

I. Freund, “Optical vortices in Gaussian random wave fields: statistical probability densities,” J. Opt. Soc. Am. A 11, 1644–1652 (1994).
[CrossRef]

Y. Chen and J. Atai, “Dynamics of optical-vortex solitons in a perturbed nonlinear medium,” J. Opt. Soc. Am. B 11, 2000–2003 (1994).
[CrossRef]

E. M. Wright, R. Y. Chiao, and J. C. Garrison, “Optical anyons: atoms trapped on electromagnetic vortices,” Chaos Solitons Fractals 4, 1797–1803 (1994).
[CrossRef]

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]

1993 (10)

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, 123–132 (1993).
[CrossRef]

C. Nore, M. E. Brachet, and S. Fauve, “Numerical study of hydrodynamics using the nonlinear Schrödinger (NLS) equation,” Physica D 65, 154–162 (1993).
[CrossRef]

C. T. Law and G. A. Swartzlander, Jr., “Optical vortex solitons and the stability of dark soliton stripes,” Opt. Lett. 18, 586–588 (1993).
[CrossRef]

G. A. Swartzlander, Jr., and C. T. Law, “The optical vortex soliton,” Opt. Photonics News 4 (12), 10 (1993).
[CrossRef]

G. L. Lippi, T. Ackemann, L. M. Hoffer, A. Gahl, and W. Lange, “Interplay of linear and nonlinear effects in the formation of optical vortices in a nonlinear resonator,” Phys. Rev. A 48, R4043–R4046 (1993).
[CrossRef]

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

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

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

C. O. Weiss, H. R. Telle, and K. Staliunas, “Restless optical vortex,” Phys. Rev. A 47, R1616–R1619 (1993).
[CrossRef] [PubMed]

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

1992 (11)

K. Staliunas, “Dynamics of optical vortices in a laser beam,” Opt. Commun. 90, 123–127 (1992).
[CrossRef]

K. Staliunas, “Optical vortices during three-way nonlinear coupling,” Opt. Commun. 91, 82–86 (1992).
[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]

L. M. Pismen and A. A. Nepomnyashchy, “On interaction of spiral waves,” Physica D 54, 183–193 (1992).
[CrossRef]

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

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[CrossRef]

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]

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. 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]

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in array optic regime,” Biophys. J. 61, 569–582 (1992).
[CrossRef] [PubMed]

1991 (2)

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinsztein-Dunlop, R. McDuff, C. O. Weiss, and Chr. Tamm, “Interferometric measurements of phase singularities in the output of a visible laser,” J. Mod. Opt. 38, 2531–2541 (1991).
[CrossRef]

E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[CrossRef]

1990 (1)

V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis’ma Zh. Eksp. Teor. Fiz. 52, 1037–1039 (1990) [ JETP Lett. 52, 429–431 (1990)].

1989 (1)

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

1983 (2)

1982 (1)

1981 (1)

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetskii, and V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” Pis’ma Zh. Eksp. Teor. Fiz. 33, 206–210 (1981) [ JETP Lett. 33, 195–199 (1981)].

1978 (1)

1974 (1)

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

1968 (1)

W. G. Wagner, H. A. Haus, and J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
[CrossRef]

1965 (1)

A. L. Fetter, “Vortices in an imperfect Bose gas,” Phys. Rev. 138, A429–A431 (1965).
[CrossRef]

1962 (1)

G. A. Askar’yan, “Effects of the gradient of a strong electromagnetic beam on electrons and atoms,” Sov. Phys. JETP 15, 1088–1090 (1962).

1961 (2)

L. P. Pitaevskii, “Vortex lines in an imperfect Bose gas,” Zh. Eksp. Teor. Fiz. 40, 646–651 (1961) [ Sov. Phys. JETP 13, 451–454 (1961)].

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

1958 (1)

V. L. Ginzburg and L. P. Pitaevskii, “On the theory of superfluidity,” Zh. Eksp. Teor. Fiz. 34, 1240–1245 (1958) [ Sov. Phys. JETP 7, 858–861 (1958)].

Abramochkin, E.

E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[CrossRef]

Ackemann, T.

T. Ackemann, E. Kriege, and W. Lange, “Phase singularities via nonlinear beam propagation in sodium vapor,” Opt. Commun. 115, 339–346 (1995).
[CrossRef]

G. L. Lippi, T. Ackemann, L. M. Hoffer, A. Gahl, and W. Lange, “Interplay of linear and nonlinear effects in the formation of optical vortices in a nonlinear resonator,” Phys. Rev. A 48, R4043–R4046 (1993).
[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, 123–132 (1993).
[CrossRef]

Anderson, D. Z.

A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetry breaking to spatial turbulence,” Phys. Rev. A 54, 870–879 (1996).
[CrossRef] [PubMed]

Ashkin, A.

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in array optic regime,” Biophys. J. 61, 569–582 (1992).
[CrossRef] [PubMed]

Askar’yan, G. A.

G. A. Askar’yan, “Effects of the gradient of a strong electromagnetic beam on electrons and atoms,” Sov. Phys. JETP 15, 1088–1090 (1962).

Atai, J.

Baranova, N. B.

N. B. Baranova, A. V. Mamaev, N. F. Pilipetskii, V. V. Shkukov, and B. Ya. Zel’dovich, “Wavefront dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. 73, 525–528 (1983).
[CrossRef]

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetskii, and V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” Pis’ma Zh. Eksp. Teor. Fiz. 33, 206–210 (1981) [ JETP Lett. 33, 195–199 (1981)].

Basistiy, I. V.

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

Bazhenov, V. Yu.

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

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

V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis’ma Zh. Eksp. Teor. Fiz. 52, 1037–1039 (1990) [ JETP Lett. 52, 429–431 (1990)].

Beijersbergen, M. W.

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

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, 123–132 (1993).
[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]

Berzanskis, A.

K. Staliunas, A. Berzanskis, and V. Jarutis, “Vortex statistics in optical speckle fields,” Opt. Commun. 120, 23–28 (1995).
[CrossRef]

Brachet, M. E.

C. Nore, M. E. Brachet, and S. Fauve, “Numerical study of hydrodynamics using the nonlinear Schrödinger (NLS) equation,” Physica D 65, 154–162 (1993).
[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]

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]

Chen, Y.

Chiao, R. Y.

E. M. Wright, R. Y. Chiao, and J. C. Garrison, “Optical anyons: atoms trapped on electromagnetic vortices,” Chaos Solitons Fractals 4, 1797–1803 (1994).
[CrossRef]

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]

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[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. DiPorto, E. Sharp, and A. Yariv, “Dark photorefractive spatial solitons and photorefractive vortex solitons,” Phys. Rev. Lett. 74, 1978–1981 (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]

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]

Dinev, S.

I. Velchev, A. Dreischuh, D. Neshev, and S. Dinev, “Interactions of optical vortex solitons superimposed on different background beams,” Opt. Commun. 130, 385–392 (1996).
[CrossRef]

DiPorto, P.

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

Dreischuh, A.

I. Velchev, A. Dreischuh, D. Neshev, and S. Dinev, “Interactions of optical vortex solitons superimposed on different background beams,” Opt. Commun. 130, 385–392 (1996).
[CrossRef]

Duree, G.

M. Morin, G. Duree, and M. Segev, “Waveguides formed by quasi-steady-state photorefractive spatial solitons,” Opt. Lett. 20, 2066–2068 (1995).
[CrossRef] [PubMed]

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

Fauve, S.

C. Nore, M. E. Brachet, and S. Fauve, “Numerical study of hydrodynamics using the nonlinear Schrödinger (NLS) equation,” Physica D 65, 154–162 (1993).
[CrossRef]

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]

Feinberg, J.

Feit, M. D.

Fetter, A. L.

A. L. Fetter, “Vortices in an imperfect Bose gas,” Phys. Rev. 138, A429–A431 (1965).
[CrossRef]

Firth, W. J.

W. J. Firth and A. Lord, “Two-dimensional solitons in a Kerr cavity,” J. Mod. Opt. 43, 1071–1077 (1996).
[CrossRef]

G. S. 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]

Fleck, Jr., J. A.

Freilikher, V.

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

Freund, I.

I. Freund, “Optical vortices in Gaussian random wave fields: statistical probability densities,” J. Opt. Soc. Am. A 11, 1644–1652 (1994).
[CrossRef]

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

Gahagan, K. T.

Gahl, A.

G. L. Lippi, T. Ackemann, L. M. Hoffer, A. Gahl, and W. Lange, “Interplay of linear and nonlinear effects in the formation of optical vortices in a nonlinear resonator,” Phys. Rev. A 48, R4043–R4046 (1993).
[CrossRef]

Garrison, J. C.

E. M. Wright, R. Y. Chiao, and J. C. Garrison, “Optical anyons: atoms trapped on electromagnetic vortices,” Chaos Solitons Fractals 4, 1797–1803 (1994).
[CrossRef]

Gil, L.

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

Ginzburg, V. L.

V. L. Ginzburg and L. P. Pitaevskii, “On the theory of superfluidity,” Zh. Eksp. Teor. Fiz. 34, 1240–1245 (1958) [ Sov. Phys. JETP 7, 858–861 (1958)].

Goubau, G.

G. Goubau and F. Schwering, “On the guided 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]

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]

Haus, H. A.

W. G. Wagner, H. A. Haus, and J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
[CrossRef]

He, H.

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high-efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

Heckenberg, N. R.

N. R. Heckenberg, M. Vaupel, J. T. Malos, and C. O. Weiss, “Optical-vortex pair creation and annihilation and helical astigmatism of a nonplanar ring resonator,” Phys. Rev. A 54, 1–10 (1996).

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[CrossRef] [PubMed]

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high-efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

D. Y. Tang, N. R. Heckenberg, and C. O. Weiss, “Phase-dependent helical pattern formation in a laser,” Opt. Commun. 114, 95–100 (1995).
[CrossRef]

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[CrossRef]

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]

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinsztein-Dunlop, R. McDuff, C. O. Weiss, and Chr. Tamm, “Interferometric measurements of phase singularities in the output of a visible laser,” J. Mod. Opt. 38, 2531–2541 (1991).
[CrossRef]

Hoffer, L. M.

G. L. Lippi, T. Ackemann, L. M. Hoffer, A. Gahl, and W. Lange, “Interplay of linear and nonlinear effects in the formation of optical vortices in a nonlinear resonator,” Phys. Rev. A 48, R4043–R4046 (1993).
[CrossRef]

Indebetouw, G.

G. Indebetouw and D. R. Korwan, “Model of vortices nucleation in a photorefractive phase-conjugate resonator,” J. Mod. Opt. 41, 941–950 (1994).
[CrossRef]

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

Jarutis, V.

K. Staliunas, A. Berzanskis, and V. Jarutis, “Vortex statistics in optical speckle fields,” Opt. Commun. 120, 23–28 (1995).
[CrossRef]

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]

Khonina, S. N.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

Kivshar, Yu. S.

Korwan, D. R.

G. Indebetouw and D. R. Korwan, “Model of vortices nucleation in a photorefractive phase-conjugate resonator,” J. Mod. Opt. 41, 941–950 (1994).
[CrossRef]

Kotlyar, V. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

Kriege, E.

T. Ackemann, E. Kriege, and W. Lange, “Phase singularities via nonlinear beam propagation in sodium vapor,” Opt. Commun. 115, 339–346 (1995).
[CrossRef]

Kristensen, M.

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

Lange, W.

T. Ackemann, E. Kriege, and W. Lange, “Phase singularities via nonlinear beam propagation in sodium vapor,” Opt. Commun. 115, 339–346 (1995).
[CrossRef]

G. L. Lippi, T. Ackemann, L. M. Hoffer, A. Gahl, and W. Lange, “Interplay of linear and nonlinear effects in the formation of optical vortices in a nonlinear resonator,” Phys. Rev. A 48, R4043–R4046 (1993).
[CrossRef]

Law, C. T.

C. T. Law and G. A. Swartzlander, Jr., “Polarized optical vortex solitons: instabilities and dynamics in Kerr nonlinear media,” Chaos Solitons Fractals 4, 1759–1766 (1994).
[CrossRef]

G. A. Swartzlander, Jr., and C. T. Law, “The optical vortex soliton,” Opt. Photonics News 4 (12), 10 (1993).
[CrossRef]

C. T. Law and G. A. Swartzlander, Jr., “Optical vortex solitons and the stability of dark soliton stripes,” Opt. Lett. 18, 586–588 (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]

Lippi, G. L.

G. L. Lippi, T. Ackemann, L. M. Hoffer, A. Gahl, and W. Lange, “Interplay of linear and nonlinear effects in the formation of optical vortices in a nonlinear resonator,” Phys. Rev. A 48, R4043–R4046 (1993).
[CrossRef]

Lord, A.

W. J. Firth and A. Lord, “Two-dimensional solitons in a Kerr cavity,” J. Mod. Opt. 43, 1071–1077 (1996).
[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]

Malos, J. T.

N. R. Heckenberg, M. Vaupel, J. T. Malos, and C. O. Weiss, “Optical-vortex pair creation and annihilation and helical astigmatism of a nonplanar ring resonator,” Phys. Rev. A 54, 1–10 (1996).

Mamaev, A. V.

A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetry breaking to spatial turbulence,” Phys. Rev. A 54, 870–879 (1996).
[CrossRef] [PubMed]

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]

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Break-up of two-dimensional bright spatial solitons due to transverse modulation instability,” Europhys. Lett. 35, 25–30 (1996).
[CrossRef]

N. B. Baranova, A. V. Mamaev, N. F. Pilipetskii, V. V. Shkukov, and B. Ya. Zel’dovich, “Wavefront dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. 73, 525–528 (1983).
[CrossRef]

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetskii, and V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” Pis’ma Zh. Eksp. Teor. Fiz. 33, 206–210 (1981) [ JETP Lett. 33, 195–199 (1981)].

Marburger, J. H.

W. G. Wagner, H. A. Haus, and J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
[CrossRef]

McDonald, G. S.

G. S. 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]

McDuff, R.

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[CrossRef]

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]

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinsztein-Dunlop, R. McDuff, C. O. Weiss, and Chr. Tamm, “Interferometric measurements of phase singularities in the output of a visible laser,” J. Mod. Opt. 38, 2531–2541 (1991).
[CrossRef]

Mitchell, D. J.

Morin, M.

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

M. Morin, G. Duree, and M. Segev, “Waveguides formed by quasi-steady-state photorefractive spatial solitons,” Opt. Lett. 20, 2066–2068 (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]

Nepomnyashchy, A. A.

L. M. Pismen and A. A. Nepomnyashchy, “On interaction of spiral waves,” Physica D 54, 183–193 (1992).
[CrossRef]

Neshev, D.

I. Velchev, A. Dreischuh, D. Neshev, and S. Dinev, “Interactions of optical vortex solitons superimposed on different background beams,” Opt. Commun. 130, 385–392 (1996).
[CrossRef]

Nore, C.

C. Nore, M. E. Brachet, and S. Fauve, “Numerical study of hydrodynamics using the nonlinear Schrödinger (NLS) equation,” Physica D 65, 154–162 (1993).
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J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
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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).
[CrossRef] [PubMed]

Pilipetskii, N. F.

N. B. Baranova, A. V. Mamaev, N. F. Pilipetskii, V. V. Shkukov, and B. Ya. Zel’dovich, “Wavefront dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. 73, 525–528 (1983).
[CrossRef]

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetskii, and V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” Pis’ma Zh. Eksp. Teor. Fiz. 33, 206–210 (1981) [ JETP Lett. 33, 195–199 (1981)].

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).
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L. M. Pismen and A. A. Nepomnyashchy, “On interaction of spiral waves,” Physica D 54, 183–193 (1992).
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L. P. Pitaevskii, “Vortex lines in an imperfect Bose gas,” Zh. Eksp. Teor. Fiz. 40, 646–651 (1961) [ Sov. Phys. JETP 13, 451–454 (1961)].

V. L. Ginzburg and L. P. Pitaevskii, “On the theory of superfluidity,” Zh. Eksp. Teor. Fiz. 34, 1240–1245 (1958) [ Sov. Phys. JETP 7, 858–861 (1958)].

Poladian, L.

Powles, R.

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]

Rocca, F.

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

Roux, F. S.

Rubinsztein-Dunlop, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
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H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high-efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[CrossRef]

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinsztein-Dunlop, R. McDuff, C. O. Weiss, and Chr. Tamm, “Interferometric measurements of phase singularities in the output of a visible laser,” J. Mod. Opt. 38, 2531–2541 (1991).
[CrossRef]

Saffman, M.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Break-up of two-dimensional bright spatial solitons due to transverse modulation instability,” Europhys. Lett. 35, 25–30 (1996).
[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]

A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetry breaking to spatial turbulence,” Phys. Rev. A 54, 870–879 (1996).
[CrossRef] [PubMed]

Salamo, G.

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

Schwering, F.

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

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M. Morin, G. Duree, and M. Segev, “Waveguides formed by quasi-steady-state photorefractive spatial solitons,” Opt. Lett. 20, 2066–2068 (1995).
[CrossRef] [PubMed]

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

Shinkaryev, M. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

Shkukov, V. V.

N. B. Baranova, A. V. Mamaev, N. F. Pilipetskii, V. V. Shkukov, and B. Ya. Zel’dovich, “Wavefront dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. 73, 525–528 (1983).
[CrossRef]

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetskii, and V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” Pis’ma Zh. Eksp. Teor. Fiz. 33, 206–210 (1981) [ JETP Lett. 33, 195–199 (1981)].

Shvartsman, N.

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

Slekys, G.

G. Slekys, K. Staliunas, and C. O. Weiss, “Motion of phase singularities in a class-B laser,” Opt. Commun. 119, 433–446 (1995).
[CrossRef]

Smith, C. P.

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[CrossRef]

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]

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinsztein-Dunlop, R. McDuff, C. O. Weiss, and Chr. Tamm, “Interferometric measurements of phase singularities in the output of a visible laser,” J. Mod. Opt. 38, 2531–2541 (1991).
[CrossRef]

Snyder, A. W.

Soifer, V. A.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

Soskin, M. S.

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

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

V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis’ma Zh. Eksp. Teor. Fiz. 52, 1037–1039 (1990) [ JETP Lett. 52, 429–431 (1990)].

Staliunas, K.

K. Staliunas and C. O. Weiss, “Nonstationary vortex lattices in large-aperture class B lasers,” J. Opt. Soc. Am. B 12, 1142–1149 (1995).
[CrossRef]

K. Staliunas, A. Berzanskis, and V. Jarutis, “Vortex statistics in optical speckle fields,” Opt. Commun. 120, 23–28 (1995).
[CrossRef]

G. Slekys, K. Staliunas, and C. O. Weiss, “Motion of phase singularities in a class-B laser,” Opt. Commun. 119, 433–446 (1995).
[CrossRef]

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

C. O. Weiss, H. R. Telle, and K. Staliunas, “Restless optical vortex,” Phys. Rev. A 47, R1616–R1619 (1993).
[CrossRef] [PubMed]

K. Staliunas, “Optical vortices during three-way nonlinear coupling,” Opt. Commun. 91, 82–86 (1992).
[CrossRef]

K. Staliunas, “Dynamics of optical vortices in a laser beam,” Opt. Commun. 90, 123–127 (1992).
[CrossRef]

Swartzlander, Jr., G. A.

K. T. Gahagan and G. A. Swartzlander, Jr., “Optical vortex trapping of particles,” Opt. Lett. 21, 827–829 (1996).
[CrossRef] [PubMed]

C. T. Law and G. A. Swartzlander, Jr., “Polarized optical vortex solitons: instabilities and dynamics in Kerr nonlinear media,” Chaos Solitons Fractals 4, 1759–1766 (1994).
[CrossRef]

C. T. Law and G. A. Swartzlander, Jr., “Optical vortex solitons and the stability of dark soliton stripes,” Opt. Lett. 18, 586–588 (1993).
[CrossRef]

G. A. Swartzlander, Jr., and C. T. Law, “The optical vortex soliton,” Opt. Photonics News 4 (12), 10 (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]

Syed, K. S.

G. S. 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.

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinsztein-Dunlop, R. McDuff, C. O. Weiss, and Chr. Tamm, “Interferometric measurements of phase singularities in the output of a visible laser,” J. Mod. Opt. 38, 2531–2541 (1991).
[CrossRef]

Tang, D. Y.

D. Y. Tang, N. R. Heckenberg, and C. O. Weiss, “Phase-dependent helical pattern formation in a laser,” Opt. Commun. 114, 95–100 (1995).
[CrossRef]

Telle, H. R.

C. O. Weiss, H. R. Telle, and K. Staliunas, “Restless optical vortex,” Phys. Rev. A 47, R1616–R1619 (1993).
[CrossRef] [PubMed]

Tikhonenko, V.

Tredicce, J. 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]

Uspleniev, G. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

van der Veen, H. E. L. O.

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, 123–132 (1993).
[CrossRef]

Vasnetsov, M. V.

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

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

V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis’ma Zh. Eksp. Teor. Fiz. 52, 1037–1039 (1990) [ JETP Lett. 52, 429–431 (1990)].

Vaughan, J. M.

Vaupel, M.

N. R. Heckenberg, M. Vaupel, J. T. Malos, and C. O. Weiss, “Optical-vortex pair creation and annihilation and helical astigmatism of a nonplanar ring resonator,” Phys. Rev. A 54, 1–10 (1996).

Velchev, I.

I. Velchev, A. Dreischuh, D. Neshev, and S. Dinev, “Interactions of optical vortex solitons superimposed on different background beams,” Opt. Commun. 130, 385–392 (1996).
[CrossRef]

Volostnikov, V.

E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
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W. G. Wagner, H. A. Haus, and J. H. Marburger, “Large-scale self-trapping of optical beams in the paraxial ray approximation,” Phys. Rev. 175, 256–266 (1968).
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Wegener, M. J.

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[CrossRef]

Weiss, C. O

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]

Weiss, C. O.

N. R. Heckenberg, M. Vaupel, J. T. Malos, and C. O. Weiss, “Optical-vortex pair creation and annihilation and helical astigmatism of a nonplanar ring resonator,” Phys. Rev. A 54, 1–10 (1996).

K. Staliunas and C. O. Weiss, “Nonstationary vortex lattices in large-aperture class B lasers,” J. Opt. Soc. Am. B 12, 1142–1149 (1995).
[CrossRef]

D. Y. Tang, N. R. Heckenberg, and C. O. Weiss, “Phase-dependent helical pattern formation in a laser,” Opt. Commun. 114, 95–100 (1995).
[CrossRef]

G. Slekys, K. Staliunas, and C. O. Weiss, “Motion of phase singularities in a class-B laser,” Opt. Commun. 119, 433–446 (1995).
[CrossRef]

C. O. Weiss, H. R. Telle, and K. Staliunas, “Restless optical vortex,” Phys. Rev. A 47, R1616–R1619 (1993).
[CrossRef] [PubMed]

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinsztein-Dunlop, R. McDuff, C. O. Weiss, and Chr. Tamm, “Interferometric measurements of phase singularities in the output of a visible laser,” J. Mod. Opt. 38, 2531–2541 (1991).
[CrossRef]

White, A. G.

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]

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinsztein-Dunlop, R. McDuff, C. O. Weiss, and Chr. Tamm, “Interferometric measurements of phase singularities in the output of a visible laser,” J. Mod. Opt. 38, 2531–2541 (1991).
[CrossRef]

Willetts, D. V.

Woerdman, J. P.

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

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, 123–132 (1993).
[CrossRef]

Wright, E. M.

E. M. Wright, R. Y. Chiao, and J. C. Garrison, “Optical anyons: atoms trapped on electromagnetic vortices,” Chaos Solitons Fractals 4, 1797–1803 (1994).
[CrossRef]

Yariv, A.

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

Zel’dovich, B. Ya.

N. B. Baranova, A. V. Mamaev, N. F. Pilipetskii, V. V. Shkukov, and B. Ya. Zel’dovich, “Wavefront dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. 73, 525–528 (1983).
[CrossRef]

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetskii, and V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” Pis’ma Zh. Eksp. Teor. Fiz. 33, 206–210 (1981) [ JETP Lett. 33, 195–199 (1981)].

Zozulya, A. A.

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]

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Break-up of two-dimensional bright spatial solitons due to transverse modulation instability,” Europhys. Lett. 35, 25–30 (1996).
[CrossRef]

A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetry breaking to spatial turbulence,” Phys. Rev. A 54, 870–879 (1996).
[CrossRef] [PubMed]

Appl. Opt. (1)

Biophys. J. (1)

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in array optic regime,” Biophys. J. 61, 569–582 (1992).
[CrossRef] [PubMed]

Chaos Solitons Fractals (4)

E. M. Wright, R. Y. Chiao, and J. C. Garrison, “Optical anyons: atoms trapped on electromagnetic vortices,” Chaos Solitons Fractals 4, 1797–1803 (1994).
[CrossRef]

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]

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

C. T. Law and G. A. Swartzlander, Jr., “Polarized optical vortex solitons: instabilities and dynamics in Kerr nonlinear media,” Chaos Solitons Fractals 4, 1759–1766 (1994).
[CrossRef]

Europhys. Lett. (1)

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Break-up of two-dimensional bright spatial solitons due to transverse modulation instability,” Europhys. Lett. 35, 25–30 (1996).
[CrossRef]

IRE Trans. Antennas Propag. (1)

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

J. Mod. Opt. (7)

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinsztein-Dunlop, R. McDuff, C. O. Weiss, and Chr. Tamm, “Interferometric measurements of phase singularities in the output of a visible laser,” J. Mod. Opt. 38, 2531–2541 (1991).
[CrossRef]

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

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

G. Indebetouw and D. R. Korwan, “Model of vortices nucleation in a photorefractive phase-conjugate resonator,” J. Mod. Opt. 41, 941–950 (1994).
[CrossRef]

W. J. Firth and A. Lord, “Two-dimensional solitons in a Kerr cavity,” J. Mod. Opt. 43, 1071–1077 (1996).
[CrossRef]

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high-efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

J. Opt. Soc. Am. (3)

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

J. Opt. Soc. Am. B (3)

JETP Lett. (2)

V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis’ma Zh. Eksp. Teor. Fiz. 52, 1037–1039 (1990) [ JETP Lett. 52, 429–431 (1990)].

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetskii, and V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” Pis’ma Zh. Eksp. Teor. Fiz. 33, 206–210 (1981) [ JETP Lett. 33, 195–199 (1981)].

Opt. Commun. (15)

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

G. Slekys, K. Staliunas, and C. O. Weiss, “Motion of phase singularities in a class-B laser,” Opt. Commun. 119, 433–446 (1995).
[CrossRef]

E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[CrossRef]

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

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

I. Velchev, A. Dreischuh, D. Neshev, and S. Dinev, “Interactions of optical vortex solitons superimposed on different background beams,” Opt. Commun. 130, 385–392 (1996).
[CrossRef]

T. Ackemann, E. Kriege, and W. Lange, “Phase singularities via nonlinear beam propagation in sodium vapor,” Opt. Commun. 115, 339–346 (1995).
[CrossRef]

K. Staliunas, “Dynamics of optical vortices in a laser beam,” Opt. Commun. 90, 123–127 (1992).
[CrossRef]

K. Staliunas, “Optical vortices during three-way nonlinear coupling,” Opt. Commun. 91, 82–86 (1992).
[CrossRef]

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

K. Staliunas, A. Berzanskis, and V. Jarutis, “Vortex statistics in optical speckle fields,” Opt. Commun. 120, 23–28 (1995).
[CrossRef]

D. Y. Tang, N. R. Heckenberg, and C. O. Weiss, “Phase-dependent helical pattern formation in a laser,” Opt. Commun. 114, 95–100 (1995).
[CrossRef]

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

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, 123–132 (1993).
[CrossRef]

Opt. Lett. (8)

Opt. Photonics News (1)

G. A. Swartzlander, Jr., and C. T. Law, “The optical vortex soliton,” Opt. Photonics News 4 (12), 10 (1993).
[CrossRef]

Opt. Quantum Electron. (1)

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[CrossRef]

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[CrossRef]

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[CrossRef]

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Other (9)

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L. G. Gouy, “Sur la propagation anomale des ondes,” Ann. Chim. Phys. Ser. 6 24, 145–213 (1891).

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See, for example, M. S. El Naschie, ed., “Special issue on nonlinear optical structures, patterns, chaos,” Chaos Solitons Fractals 4(8/9) (1994).

P. G. de Gennes, Superconductivity of Metals and Alloys (Addison-Wesley, Reading, Mass., 1989); R. J. Donnelly, Quantized Vortices in Helium II (Cambridge U. Press, New York, 1991).

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G. A. Swartzlander, Jr., Z. Sacks, X. Zhang, D. Rozas, and C. T. Law, “Formation and propagation of optical vortices,” in Digest of the International Quantum Electronics Conference, 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), p. 31; D. Rozas, Z. S. Sacks, and G. A. Swartzlander, Jr., “Experimental observations of fluid-like motion of optical vortices,” Phys. Rev. Lett. (to be published).

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Figures (14)

Fig. 1
Fig. 1

Intensity profiles of (a) an r vortex and (b) a tanh vortex upon a Gaussian background beam of size w0, showing the respective large and small vortex core sizes. The images are shown with a logarithmic gray-scale pallet, whereas a linear scale is used for the line plots (this applies to the other figures, unless noted).

Fig. 2
Fig. 2

Three-dimensional sketch of the position of the vortex in the transverse xy plane, propagating in the z direction with wave vector kkzzˆ. The wave vector at the center of vortex 2 has a transverse component kkz.

Fig. 3
Fig. 3

Transverse coordinates of the jth vortex, located at point (Rj, Θj) with respect to the origin (assumed to be at the center of the background beam). The amplitude and the phase of the vortex are expressed in terms of coordinates (rj, θj).

Fig. 4
Fig. 4

Two tanh vortices of size wv and separation d placed symmetrically about the center of a Gaussian background beam of size w0.

Fig. 5
Fig. 5

Two r vortices of separation d placed symmetrically about the center of a Gaussian background beam of size w0.

Fig. 6
Fig. 6

Phase profile of two vortices of charge m=-1 in the initial transverse plane, z=0, separated by a distance d. A linear gray-scale pallet was used in this case, where black (white) corresponds to a phase of 0 (2π).

Fig. 7
Fig. 7

Near-field intensity profiles at a propagation distance z/Z0=0.05 showing (a) negligible ringing when wv=57 µm and (b) significant vortex radiation when wv=0. The size of the initial background Gaussian field is w0=488 µm.

Fig. 8
Fig. 8

Off-center r vortex of charge m=+1 in a Gaussian beam (w0=488 µm) initially placed at point (x0, y0) =(r0, 0), where r0=244 µm. The beam is shown at a distance z=Z0. The projection of the vortex trajectory in the transverse plane, shown by the white open squares, is a straight line. The vortex advances at a uniform rate given by r0/Z0.

Fig. 9
Fig. 9

Off-center tanh vortex of charge m=+1 in a Gaussian beam (w0=488 µm) initially placed at point (x0, y0) =(r0, 0), where r0=244 µm. The beam is shown at a distance z=Z0. The projection of the vortex trajectory in the transverse plane, shown by the white open circles (whose radii depict the size of the advancing vortex), is a straight line. As for an r vortex, the tanh vortex advances at a uniform rate given by r0/Z0.

Fig. 10
Fig. 10

Two r vortices of topological charge m=+1 and separation d=2r0=w0=488 µm initially placed at points (±r0, 0). The beam, shown at a distance z=Z0, shows the vortex rotation effect as described by the dynamic Gouy phase. The white open squares mark the positions of the vortices at propagation distances z=0, Z0, 2Z0, 3Z0. The trajectories are straight lines, as in Fig. 8.

Fig. 11
Fig. 11

(a) Two tanh vortices of topological charge m=+1 and separation d=2r0=137 µm initially placed at points (±r0, 0) upon a Gaussian background beam of size w0=488 µm. The beam, shown at a distance z=0.10×Z0, shows the vortex rotation effect. The projection of the vortex trajectories onto the transverse plane is depicted in (b), where filled diamonds mark propagation intervals of Δz=Z0/10.

Fig. 12
Fig. 12

Angle of rotation ϕ versus propagation distance z in linear and nonlinear media for a pair of tanh vortices of size wv =57 µm and separation d upon a Gaussian background beam of size w0: (a) d=137 µm, w0=1464 µm (Z0=8.4 m); (b) same as (a) except that d=390 µm; (c) same as (a) except that w0=488 µm (Z0=0.94 m).

Fig. 13
Fig. 13

Initial rate of tanh vortex pair rotation Ω(z=0) plotted for different separation distances d and background beam sizes w0. The dashed curves are theoretical curves from Eq. (25) with no adjustable parameters. Note that Fig. 12 indicates that Ω(z=0) is the same for both linear and nonlinear propagation.

Fig. 14
Fig. 14

Propagation of OVS’s of charge m=+1, with initial conditions (a) d=137 µm, wv=57 µm, w0=1464 µm. A large clockwise rotation is evident over a short distance in (b), with a rotation angle greater than 180° demonstrated in (c). The projections of vortex trajectories onto the transverse plane are shown in (d), where filled diamonds mark propagation intervals of Δz=Z0/50.

Equations (27)

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E(r, θ, z)=E0Gbg(r, z)exp[iΦ(r, z)]×A(r, z)exp(imθ)exp(-ikz)=E0u(r, θ, z)exp(-ikz),
A(r, z=0)=tanh(r/wv),
A(r, z=0)=(r/wr)|m|,
ΦG(z)=arctan(z/Z0)
Gbg(r, z)Am(r, z)=π1/22rw0Z0z1/2Z0R(z)3/2×[Iν(-)(γ)-Iν(+)(γ)]exp-12r2w2(z),
Φ(r, z)=(π/2)(3/2-m)+(Z0/z)(r/w0)2×[1-(1/2)w02/w2(z)]-(3/2)arctan(z/Z0),
-2ik(u/z)+2u+2k2(n2E02/n0)|u|2u=0,
-s/z+k·k=2ρ1/2/ρ1/2-P/ρ,
(1/2)ρ/z+·(ρk)=0,
k=-s=-θˆr-1s/θ-rˆs/r.
E(r, θ, z=0)=E0 exp(imθ)[1+(r/L)cos θ]r/wr,
r(δz)=sgn(m)(δz/k2)k×(Gbg-1Gbg)r=0,
E(r, θ, z)=exp[i(m1θ1+m2θ2)-ikz],
ϕ(z)Θj(z)-Θj(z=0).
k=θˆ1/r1+θˆ2/r2,
tan ψ2=k(r2=0)/k=λ/(2πd),
ϕ=2l2/d=(2/d)z tan ψ2.
Ωddϕ/dz=λ/(πd2)=1/ZV.
E(r, θ, 0)=E0 exp(-r2/w02)exp(-ikz)j=1MAj(rj, 0)×exp(imjθj),
Rj(z)=Rj(0)(1+z2/Z02)1/2,
Θj(z)=Θj(0)-sgn(m)ΦG(z),
yj(z)=xj(z)tan[Θj(0)-sgn(m)ΦG(z)],
Ωr=dϕ/dz=-sgn(m)Z0-1[w0/w(z)]2=-sgn(m)Z0-1[1+(z/Z0)2]-1.
|A(r, z=0)|r=d/2=2A(r, z=0)×(r/w02+1/{wv sinh[(r+d/2)/wv]})|r=d/2.
Ωtanh(z=0)=Ωd+Ω(z=0)=(λ/π){d-2+w0-2-2/[dwv sinh(2d/wv)]}.
wv=1.270k-1(n0/ΔnNL)-1/2,
k(NL)=-ΦNL=+(kzn2/n0)|G|2

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