Abstract

Simple, closed-form analytical expressions are given for the statistical probability densities of the six parameters that define an optical vortex (phase singularity) in a Gaussian random wave field. Good agreement is found between calculation and a computer simulation that generates these vortices.

© 1994 Optical Society of America

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  1. J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
  2. M. V. Berry, “Disruption of wavefronts: statistics of dislocations in incoherent Gaussian random waves,” J. Phys. A 11, 27–37 (1978).
  3. F. J. Wright, “Wavefront dislocations and their analysis using catastrophe theory,” in Structural Stability in Physics, W. Guttinger, H. Eikemeier, eds. (Springer-Verlag, Berlin, 1979), pp. 141–156.
  4. M. Berry, “Singularities in waves and rays,” in Physics of Defects, R. Balian, M. Kleman, J.-P. Poirier, eds. (North-Holland, Amsterdam, 1981), pp. 453–543.
  5. J. F. Nye, “The motion and structure of dislocations in wave-fronts,” Proc. R. Soc. London Ser. A 378, 219–239 (1981).
  6. F. J. Wright, J. F. Nye, “Dislocations in diffraction patterns: continuous waves and pulses,” Phil. Trans. R. Soc. London Ser. A 305, 339–382 (1982).
  7. Polarization singularities have been discussed by J. F. Nye, “Dislocations and disclinations in transverse electromagnetic waves,” in Physics of Defects, R. Balian, M. Kleman, J.-P. Poirier, eds. (North-Holland, Amsterdam, 1981), pp. 545–549;“Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. R. Soc. London Ser. A 387, 105–132 (1983);“Lines of circular polarization in electromagnetic fields,” Proc. R. Soc. London Ser. A 389, 279–290 (1983);J. F. Nye, J. V. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. London Ser. A 409, 21–36 (1987);J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. I. Theory,” Proc. R. Soc. London Ser. A 414, 433–446 (1987);“Singularities in the transverse fields of electromagnetic waves. II. Observations on the electric field,” Proc. R. Soc. London Ser. A 414, 447–468 (1987).
  8. P. Coullet, L. Gil, J. Lega, “Defect-mediated turbulence,” Phys. Rev. Lett. 62, 1619–1622 (1987).
  9. P. Coullet, L. Gil, F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
  10. G. Goren, I. Procaccia, S. Rasenat, V. Steinberg, “Interactions and dynamics of topological defects: theory and experiments near the onset of weak turbulence,” Phys. Rev. Lett. 63, 1237–1240 (1989).
  11. F. T. Arecchi, G. Giacomelli, P. L. Ramazza, S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
  12. L. Gil, J. Lega, J. L. Meunier, “Statistical properties of defect-mediated turbulence,” Phys. Rev. A 41, 1138–1141 (1990).
  13. M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991);“Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43, 5114–5120 (1991).
  14. F. T. Arecchi, G. Giacomelli, P. L. Ramazza, S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749–3752 (1991).
  15. P. L. Ramazza, S. Residori, G. Giacomelli, F. T. Arecchi, “Statistics of topological defects in linear and nonlinear optics,” Europhys. Lett. 19, 475–480 (1992).
  16. G. Indebetouw, S. R. Liu, “Defect-mediated spatial complexity and chaos in a phase-conjugate resonator,” Opt. Commun. 91, 321–330 (1992).
  17. G. S. McDonald, K. S. Syed, W. J. Firth, “Optical vortices in beam propagation through a self-focussing medium,” Opt. Commun. 94, 469–476 (1992).
  18. G. A. Swartzlander, C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503–2506 (1992).
  19. R. Neubecker, M. Kreuzer, T. Tschudi, “Phase defects in a nonlinear Fabry–Perot resonator,” Opt. Commun. 96, 117–122 (1993).
  20. L. Gil, “Vector order parameter for an unpolarized laser and its vectorial topological defects,” Phys. Rev. Lett. 70, 162–165 (1993).
  21. J. M. Kosterlitz, D. J. Thouless, “Two-dimensional physics,” in Progress in Low Temperature Physics, D. F. Brewer, ed. (North-Holland, Amsterdam, 1978), Vol. VII B, pp. 371–433.
  22. N. D. Mermin, “The topological theory of defects in ordered media,” Rev. Mod. Phys. 51, 591–648 (1979).
  23. N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. Pilipetskii, V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981).
  24. N. B. Baranova, A. V. Mamaev, N. Pilipetsky, V. V. Shkunov, B. Ya. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. 73, 525–528 (1983).
  25. I. Freund, N. Shvartsman, V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
  26. Since we assume a linearly polarized wave, formally, a longitudinal component is required in order for ∇ · Ψ= 0 also to be satisfied (see Ref. 7, first entry). As a practical matter this longitudinal component is unmeasurable, and normally it may be neglected. Taking u=x̂, for our case the full formal solution of Maxwell’s equations is ψ(x,y,z)=[x̂F(x,y)−ẑ(i/k)∂F(x,y)/∂x]exp(−ikz), where x̂(ẑ) is a unit vector oriented along the x axis (z axis).
  27. E. Ochoa, J. W. Goodman, “Statistical properties of ray directions in a monochromatic speckle pattern,” J. Opt. Soc. Am. 73, 943–949 (1983).
  28. Handbook of Mathematical Functions, Natl. Bur. Stand. Appl. Math. Ser. No. 55, M. Abramowitz, I. Stegun, eds. (U.S. Government Printing Office, Washington, D.C., 1964), Chap. 9.
  29. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 2.

1993 (3)

R. Neubecker, M. Kreuzer, T. Tschudi, “Phase defects in a nonlinear Fabry–Perot resonator,” Opt. Commun. 96, 117–122 (1993).

L. Gil, “Vector order parameter for an unpolarized laser and its vectorial topological defects,” Phys. Rev. Lett. 70, 162–165 (1993).

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

1992 (4)

P. L. Ramazza, S. Residori, G. Giacomelli, F. T. Arecchi, “Statistics of topological defects in linear and nonlinear optics,” Europhys. Lett. 19, 475–480 (1992).

G. Indebetouw, S. R. Liu, “Defect-mediated spatial complexity and chaos in a phase-conjugate resonator,” Opt. Commun. 91, 321–330 (1992).

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

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

1991 (2)

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991);“Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43, 5114–5120 (1991).

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749–3752 (1991).

1990 (2)

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).

L. Gil, J. Lega, J. L. Meunier, “Statistical properties of defect-mediated turbulence,” Phys. Rev. A 41, 1138–1141 (1990).

1989 (2)

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

G. Goren, I. Procaccia, S. Rasenat, V. Steinberg, “Interactions and dynamics of topological defects: theory and experiments near the onset of weak turbulence,” Phys. Rev. Lett. 63, 1237–1240 (1989).

1987 (1)

P. Coullet, L. Gil, J. Lega, “Defect-mediated turbulence,” Phys. Rev. Lett. 62, 1619–1622 (1987).

1983 (2)

1982 (1)

F. J. Wright, J. F. Nye, “Dislocations in diffraction patterns: continuous waves and pulses,” Phil. Trans. R. Soc. London Ser. A 305, 339–382 (1982).

1981 (2)

J. F. Nye, “The motion and structure of dislocations in wave-fronts,” Proc. R. Soc. London Ser. A 378, 219–239 (1981).

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. Pilipetskii, V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981).

1979 (1)

N. D. Mermin, “The topological theory of defects in ordered media,” Rev. Mod. Phys. 51, 591–648 (1979).

1978 (1)

M. V. Berry, “Disruption of wavefronts: statistics of dislocations in incoherent Gaussian random waves,” J. Phys. A 11, 27–37 (1978).

1974 (1)

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

Arecchi, F. T.

P. L. Ramazza, S. Residori, G. Giacomelli, F. T. Arecchi, “Statistics of topological defects in linear and nonlinear optics,” Europhys. Lett. 19, 475–480 (1992).

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749–3752 (1991).

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).

Baranova, N. B.

N. B. Baranova, A. V. Mamaev, N. Pilipetsky, V. V. Shkunov, B. Ya. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. 73, 525–528 (1983).

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. Pilipetskii, V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981).

Battipede, F.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991);“Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43, 5114–5120 (1991).

Berry, M.

M. Berry, “Singularities in waves and rays,” in Physics of Defects, R. Balian, M. Kleman, J.-P. Poirier, eds. (North-Holland, Amsterdam, 1981), pp. 453–543.

Berry, M. V.

M. V. Berry, “Disruption of wavefronts: statistics of dislocations in incoherent Gaussian random waves,” J. Phys. A 11, 27–37 (1978).

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

Brambilla, M.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991);“Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43, 5114–5120 (1991).

Coullet, P.

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

P. Coullet, L. Gil, J. Lega, “Defect-mediated turbulence,” Phys. Rev. Lett. 62, 1619–1622 (1987).

Firth, W. J.

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

Freilikher, V.

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

Freund, I.

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

Giacomelli, G.

P. L. Ramazza, S. Residori, G. Giacomelli, F. T. Arecchi, “Statistics of topological defects in linear and nonlinear optics,” Europhys. Lett. 19, 475–480 (1992).

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749–3752 (1991).

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).

Gil, L.

L. Gil, “Vector order parameter for an unpolarized laser and its vectorial topological defects,” Phys. Rev. Lett. 70, 162–165 (1993).

L. Gil, J. Lega, J. L. Meunier, “Statistical properties of defect-mediated turbulence,” Phys. Rev. A 41, 1138–1141 (1990).

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

P. Coullet, L. Gil, J. Lega, “Defect-mediated turbulence,” Phys. Rev. Lett. 62, 1619–1622 (1987).

Goodman, J. W.

Goren, G.

G. Goren, I. Procaccia, S. Rasenat, V. Steinberg, “Interactions and dynamics of topological defects: theory and experiments near the onset of weak turbulence,” Phys. Rev. Lett. 63, 1237–1240 (1989).

Indebetouw, G.

G. Indebetouw, S. R. Liu, “Defect-mediated spatial complexity and chaos in a phase-conjugate resonator,” Opt. Commun. 91, 321–330 (1992).

Kosterlitz, J. M.

J. M. Kosterlitz, D. J. Thouless, “Two-dimensional physics,” in Progress in Low Temperature Physics, D. F. Brewer, ed. (North-Holland, Amsterdam, 1978), Vol. VII B, pp. 371–433.

Kreuzer, M.

R. Neubecker, M. Kreuzer, T. Tschudi, “Phase defects in a nonlinear Fabry–Perot resonator,” Opt. Commun. 96, 117–122 (1993).

Law, C. T.

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

Lega, J.

L. Gil, J. Lega, J. L. Meunier, “Statistical properties of defect-mediated turbulence,” Phys. Rev. A 41, 1138–1141 (1990).

P. Coullet, L. Gil, J. Lega, “Defect-mediated turbulence,” Phys. Rev. Lett. 62, 1619–1622 (1987).

Liu, S. R.

G. Indebetouw, S. R. Liu, “Defect-mediated spatial complexity and chaos in a phase-conjugate resonator,” Opt. Commun. 91, 321–330 (1992).

Lugiato, L. A.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991);“Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43, 5114–5120 (1991).

Mamaev, A. V.

N. B. Baranova, A. V. Mamaev, N. Pilipetsky, V. V. Shkunov, B. Ya. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. 73, 525–528 (1983).

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. Pilipetskii, V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981).

McDonald, G. S.

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

Mermin, N. D.

N. D. Mermin, “The topological theory of defects in ordered media,” Rev. Mod. Phys. 51, 591–648 (1979).

Meunier, J. L.

L. Gil, J. Lega, J. L. Meunier, “Statistical properties of defect-mediated turbulence,” Phys. Rev. A 41, 1138–1141 (1990).

Neubecker, R.

R. Neubecker, M. Kreuzer, T. Tschudi, “Phase defects in a nonlinear Fabry–Perot resonator,” Opt. Commun. 96, 117–122 (1993).

Nye, J. F.

F. J. Wright, J. F. Nye, “Dislocations in diffraction patterns: continuous waves and pulses,” Phil. Trans. R. Soc. London Ser. A 305, 339–382 (1982).

J. F. Nye, “The motion and structure of dislocations in wave-fronts,” Proc. R. Soc. London Ser. A 378, 219–239 (1981).

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

Polarization singularities have been discussed by J. F. Nye, “Dislocations and disclinations in transverse electromagnetic waves,” in Physics of Defects, R. Balian, M. Kleman, J.-P. Poirier, eds. (North-Holland, Amsterdam, 1981), pp. 545–549;“Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. R. Soc. London Ser. A 387, 105–132 (1983);“Lines of circular polarization in electromagnetic fields,” Proc. R. Soc. London Ser. A 389, 279–290 (1983);J. F. Nye, J. V. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. London Ser. A 409, 21–36 (1987);J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. I. Theory,” Proc. R. Soc. London Ser. A 414, 433–446 (1987);“Singularities in the transverse fields of electromagnetic waves. II. Observations on the electric field,” Proc. R. Soc. London Ser. A 414, 447–468 (1987).

Ochoa, E.

Penna, V.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991);“Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43, 5114–5120 (1991).

Pilipetskii, N.

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. Pilipetskii, V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981).

Pilipetsky, N.

Prati, F.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991);“Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43, 5114–5120 (1991).

Procaccia, I.

G. Goren, I. Procaccia, S. Rasenat, V. Steinberg, “Interactions and dynamics of topological defects: theory and experiments near the onset of weak turbulence,” Phys. Rev. Lett. 63, 1237–1240 (1989).

Ramazza, P. L.

P. L. Ramazza, S. Residori, G. Giacomelli, F. T. Arecchi, “Statistics of topological defects in linear and nonlinear optics,” Europhys. Lett. 19, 475–480 (1992).

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749–3752 (1991).

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).

Rasenat, S.

G. Goren, I. Procaccia, S. Rasenat, V. Steinberg, “Interactions and dynamics of topological defects: theory and experiments near the onset of weak turbulence,” Phys. Rev. Lett. 63, 1237–1240 (1989).

Residori, S.

P. L. Ramazza, S. Residori, G. Giacomelli, F. T. Arecchi, “Statistics of topological defects in linear and nonlinear optics,” Europhys. Lett. 19, 475–480 (1992).

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749–3752 (1991).

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).

Rocca, F.

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

Shkukov, V. V.

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. Pilipetskii, V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981).

Shkunov, V. V.

Shvartsman, N.

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

Steinberg, V.

G. Goren, I. Procaccia, S. Rasenat, V. Steinberg, “Interactions and dynamics of topological defects: theory and experiments near the onset of weak turbulence,” Phys. Rev. Lett. 63, 1237–1240 (1989).

Swartzlander, G. A.

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

Syed, K. S.

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

Tamm, C.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991);“Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43, 5114–5120 (1991).

Thouless, D. J.

J. M. Kosterlitz, D. J. Thouless, “Two-dimensional physics,” in Progress in Low Temperature Physics, D. F. Brewer, ed. (North-Holland, Amsterdam, 1978), Vol. VII B, pp. 371–433.

Tschudi, T.

R. Neubecker, M. Kreuzer, T. Tschudi, “Phase defects in a nonlinear Fabry–Perot resonator,” Opt. Commun. 96, 117–122 (1993).

Weiss, C. O.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991);“Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43, 5114–5120 (1991).

Wright, F. J.

F. J. Wright, J. F. Nye, “Dislocations in diffraction patterns: continuous waves and pulses,” Phil. Trans. R. Soc. London Ser. A 305, 339–382 (1982).

F. J. Wright, “Wavefront dislocations and their analysis using catastrophe theory,” in Structural Stability in Physics, W. Guttinger, H. Eikemeier, eds. (Springer-Verlag, Berlin, 1979), pp. 141–156.

Zel’dovich, B. Ya.

N. B. Baranova, A. V. Mamaev, N. Pilipetsky, V. V. Shkunov, B. Ya. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. 73, 525–528 (1983).

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. Pilipetskii, V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981).

Europhys. Lett. (1)

P. L. Ramazza, S. Residori, G. Giacomelli, F. T. Arecchi, “Statistics of topological defects in linear and nonlinear optics,” Europhys. Lett. 19, 475–480 (1992).

J. Opt. Soc. Am. (2)

J. Phys. A (1)

M. V. Berry, “Disruption of wavefronts: statistics of dislocations in incoherent Gaussian random waves,” J. Phys. A 11, 27–37 (1978).

JETP Lett. (1)

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. Pilipetskii, V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981).

Opt. Commun. (5)

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

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

G. Indebetouw, S. R. Liu, “Defect-mediated spatial complexity and chaos in a phase-conjugate resonator,” Opt. Commun. 91, 321–330 (1992).

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

R. Neubecker, M. Kreuzer, T. Tschudi, “Phase defects in a nonlinear Fabry–Perot resonator,” Opt. Commun. 96, 117–122 (1993).

Phil. Trans. R. Soc. London Ser. A (1)

F. J. Wright, J. F. Nye, “Dislocations in diffraction patterns: continuous waves and pulses,” Phil. Trans. R. Soc. London Ser. A 305, 339–382 (1982).

Phys. Rev. A (2)

L. Gil, J. Lega, J. L. Meunier, “Statistical properties of defect-mediated turbulence,” Phys. Rev. A 41, 1138–1141 (1990).

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystals,” Phys. Rev. A 43, 5090–5113 (1991);“Transverse laser patterns. II. Variational principle for pattern selection, spatial multistability, and laser hydrodynamics,” Phys. Rev. A 43, 5114–5120 (1991).

Phys. Rev. Lett. (6)

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Proc. R. Soc. London Ser. A (2)

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Other (7)

Since we assume a linearly polarized wave, formally, a longitudinal component is required in order for ∇ · Ψ= 0 also to be satisfied (see Ref. 7, first entry). As a practical matter this longitudinal component is unmeasurable, and normally it may be neglected. Taking u=x̂, for our case the full formal solution of Maxwell’s equations is ψ(x,y,z)=[x̂F(x,y)−ẑ(i/k)∂F(x,y)/∂x]exp(−ikz), where x̂(ẑ) is a unit vector oriented along the x axis (z axis).

Handbook of Mathematical Functions, Natl. Bur. Stand. Appl. Math. Ser. No. 55, M. Abramowitz, I. Stegun, eds. (U.S. Government Printing Office, Washington, D.C., 1964), Chap. 9.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 2.

F. J. Wright, “Wavefront dislocations and their analysis using catastrophe theory,” in Structural Stability in Physics, W. Guttinger, H. Eikemeier, eds. (Springer-Verlag, Berlin, 1979), pp. 141–156.

M. Berry, “Singularities in waves and rays,” in Physics of Defects, R. Balian, M. Kleman, J.-P. Poirier, eds. (North-Holland, Amsterdam, 1981), pp. 453–543.

Polarization singularities have been discussed by J. F. Nye, “Dislocations and disclinations in transverse electromagnetic waves,” in Physics of Defects, R. Balian, M. Kleman, J.-P. Poirier, eds. (North-Holland, Amsterdam, 1981), pp. 545–549;“Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. R. Soc. London Ser. A 387, 105–132 (1983);“Lines of circular polarization in electromagnetic fields,” Proc. R. Soc. London Ser. A 389, 279–290 (1983);J. F. Nye, J. V. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. London Ser. A 409, 21–36 (1987);J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. I. Theory,” Proc. R. Soc. London Ser. A 414, 433–446 (1987);“Singularities in the transverse fields of electromagnetic waves. II. Observations on the electric field,” Proc. R. Soc. London Ser. A 414, 447–468 (1987).

J. M. Kosterlitz, D. J. Thouless, “Two-dimensional physics,” in Progress in Low Temperature Physics, D. F. Brewer, ed. (North-Holland, Amsterdam, 1978), Vol. VII B, pp. 371–433.

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Figures (7)

Fig. 1
Fig. 1

(a) Geometry of the real/imaginary tangent plane; angle ρ measures the orientation of the plane relative to the laboratory x axis, and tan ψ is the slope of the plane. (b) Vortex internal coordinates x′,y′; p measures the rotation of the vortex coordinate system relative to the laboratory x,y frame, Σ measures the angle between y′ and x′, and skewness parameter σ measures the deviation from orthogonality of the internal vortex frame.

Fig. 2
Fig. 2

Phase maps of a single positive vortex illustrating the effects of vortex anisotropy α and skewness σ. The linear equiphases of the phase star are plotted at increments of 22.5°. (a) α = 1, σ = 0°, isotropic vortex; (b) α = 5,σ = 0°, anisotropic vortex; (c) α = 1,σ = 60°, skewed vortex; (d) α = 5,σ = 60°, general distorted vortex, (a′)–(d′) Calculated two-beam interferograms25 corresponding to (a)–(d).

Fig. 3
Fig. 3

Probability density Pρ of vortex rotation angle ρ measured in degrees. The solid line is the theory of Eq. (13), with π = 180°.

Fig. 4
Fig. 4

Probability density Pσ of the vortex skewness parameter σ measured in degrees. The solid line is the theory of Eq. (15), with π = 180°.

Fig. 5
Fig. 5

Probability density Pa/〈a of normalized vortex amplitude a/a〉. The solid curve is the theory of Eq. (18).

Fig. 6
Fig. 6

Probability density Pα of vortex anisotropy α. The solid curve is the theory of Eq. (21).

Fig. 7
Fig. 7

Joint probability density of vortex amplitude a and anisotropy α: (a) scatter plot of a/a〉 versus α, (b) gray-scale coded map of Eq. (23).

Equations (30)

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f re ( x , y ) = r x 0 ( x x 0 ) + r y 0 ( y y 0 ) ,
f im ( x , y ) = i x 0 ( x x 0 ) + i y 0 ( y y 0 ) ,
sgn ( υ ) = sgn ( d φ / d θ ) θ = 0 ,
tan φ ( x , y ) = f im ( x , y ) f re ( x , y ) = i x 0 + i y 0 tan θ r x 0 + r y 0 tan θ ,
sgn ( υ ) = sgn ( | r x 0 r y 0 i x 0 i y 0 | ) .
F ( x , y ) = a { ( x x 0 ) cos ρ + ( y y 0 ) sin ρ + i α [ ( x x 0 ) sin ( ρ + σ ) + ( y y 0 ) cos ( ρ + σ ) ] } .
0 a ,
α ,
π ρ π ,
π / 2 σ π / 2 .
sgn ( υ ) = sgn ( α ) ,
F ( x , y ) = n = 1 N a n exp [ i ( x u n + y υ n + φ n ) ] ,
ρ = arctan ( r y 0 / r x 0 ) ,
σ = arctan ( i x 0 / i y 0 ) ρ ,
a = r x 0 / cos ρ = r y 0 / sin ρ ,
α = i x 0 / [ a sin ( ρ + σ ) ] = i y 0 / [ a cos ( ρ + σ ) ] .
η vortex = 1 / ( 2 A coh ) .
p ( f re , f im , r x , r y , i x , i y ) = 1 8 π 3 σ 2 b 2 exp ( f re 2 + f im 2 2 σ 2 r x 2 + r y 2 + i x 2 + i y 2 2 b ) ,
p ( r x 0 , r y 0 , i x 0 , i y 0 ) = [ d f re d f im δ ( f re 2 π σ 2 ) × δ ( f im 2 π σ 2 ) p ( f re , f im , r x , r y , i x , i y ) ] r x = r x 0 , r y = r y 0 i x = i x 0 , i y = i y 0 = 1 4 π 2 b 2 exp [ ( r x 0 ) 2 + ( r y 0 ) 2 + ( i x 0 ) 2 + ( i y 0 ) 2 2 b ] .
P ρ ( ρ ) = 1 2 π Θ ( π | ρ | ) ,
P σ ( σ ) = K 0 π d ρ im 0 π d ρ re δ ( ρ im ρ re π / 2 | σ | ) ,
P σ ( σ ) = 2 π ( 1 2 | σ | π ) Θ ( π / 2 | σ | ) .
a 2 = ( r x 0 ) 2 + ( r y 0 ) 2 .
P a 2 ( a 2 ) = d i x 0 d i y 0 d r x 0 d r y 0 × δ [ a 2 ( r x 0 ) 2 ( r y 0 ) 2 ] p ( r x 0 , r y 0 , i x 0 , i y 0 ) .
P a / a ( a a ) = π a 2 a exp [ π ( a 2 a ) 2 ] ,
α 2 = ( i x 0 ) 2 + ( i y 0 ) 2 ( r x 0 ) 2 + ( r y 0 ) 2 ,
P α 2 ( α 2 ) = d i x 0 d i y 0 d r x 0 d r y 0 × δ [ α 2 ( i x 0 ) 2 + ( i y 0 ) 2 ( r x 0 ) 2 + ( r y 0 ) 2 ] p ( r x 0 , r y 0 , i x 0 , i y 0 ) .
P α ( α ) = | α | ( 1 + α 2 ) 2 .
P a , α ( a , α ) = a 3 | α | 2 b 2 exp [ a 2 ( 1 + α 2 ) 2 b ] .
P a , α , ρ , σ ( a , α , ρ , σ ) = P a , α ( a , α ) P ρ ( ρ ) P σ ( σ ) .

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