Abstract

Speckle patterns of arbitrary resolution are obtained by applying the sampling theorem to measurements of two orthogonal components of the microwave field transmitted through multiply scattering samples. Core structures of phase singularities, phase critical points, and polarization singularities are explored. We find that equiphase lines connect phase singularities with opposite topological signs except for the bifurcation lines, which run through a phase saddle point, in agreement with predictions by Freund [Phys. Rev. E 25, 2348 (1995) ]. We observe hyperbolic equiphase lines near phase saddle points and elliptical equiphase lines around phase extrema. Polarization singularities of the vector field with the three morphologies predicted are observed.

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  1. J. W. Goodman, Statistical Optics (Wiley, 1985).
  2. J. W. Goodman, Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer, 1984).
  3. J. F. Nye, Nature Focusing and Fine Structure of Light (IOP, 1999).
  4. J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
    [CrossRef]
  5. M. V. Berry, "Disruption of wavefronts: statistics of dislocations in incoherent Gaussian random waves," J. Phys. A 11, 27-37 (1978).
    [CrossRef]
  6. B. I. Halperin, "Statistical mechanics of topological defects," in Physics of Defects, R.Balian, M.Kleman, and J.P.Poirier, eds. (North-Holland, 1981), pp. 813-857.
  7. M. V. Berry, "Singularities in waves and rays," in Physics of Defects, R.Balian, M.Kleman, and J.P.Poirier, eds. (North-Holland, 1981), pp. 453-543.
  8. N. B. Baranova, B. Ya. Zel'dovich, A. V. Mamaev, N. Pilipetskii, and V. V. Shkunov, "Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment)," JETP Lett. 33, 195-199 (1981).
  9. I. Freund, N. Shvartsman, and V. Freilikher, "Optical dislocation networks in highly random media," Opt. Commun. 101, 247-264 (1993).
    [CrossRef]
  10. I. Freund and N. Shvartsman, "Wave-field phase singularities: the sign principle," Phys. Rev. A 50, 5164-5172 (1994).
    [CrossRef]
  11. N. Shvartsman and I. Freund, "Vortices in random wave fields: nearest neighbor anticorrelations," Phys. Rev. Lett. 72, 1008-1011 (1994).
    [CrossRef]
  12. I. Freund, "Saddles, singularities, and extrema in random phase fields," Phys. Rev. E 52, 2348-2359 (1995).
    [CrossRef]
  13. I. Freund, " '1001' correlations in random wave fields," Waves Random Media 8, 119-158 (1998).
    [CrossRef]
  14. M. V. Berry and M. R. Dennis, "Phase singularities in isotropic random waves," Proc. R. Soc. London, Ser. A 456, 2059-2079 (2000).
    [CrossRef]
  15. M. R. Dennis, "Local properties and statistics of phase singularities in generic wavefields," Proc. SPIE 4403, 13-23 (2001).
    [CrossRef]
  16. W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, "Experimental investigation of local properties and statistics of optical vortices in random wave fields," Phys. Rev. Lett. 94, 103902 (2005).
    [CrossRef] [PubMed]
  17. M. R. Dennis, "Correlations and screening of topological charges in Gaussian random fields," J. Phys. A 36, 6611-6628 (2003).
    [CrossRef]
  18. G. Foltin, "The distribution of extremal points of Gaussian scalar fields," J. Phys. A 36, 4561-4580 (2003).
    [CrossRef]
  19. I. Freund and M. Wilkinson, "Critical-point screening in random wave fields," J. Opt. Soc. Am. 15, 2892-2902 (1998).
    [CrossRef]
  20. M. Dennis, "Phase critical point densities in planar isotropic random waves," J. Phys. A 34, L297-L303 (2001).
    [CrossRef]
  21. J. F. Nye and J. V. Hajnal, "The wave structure of monochromatic electromagnetic radiation," Proc. R. Soc. London, Ser. A 409, 21-36 (1987).
    [CrossRef]
  22. M. V. Berry and M. R. Dennis, "Polarization singularities in isotropic random vector waves," Proc. R. Soc. London, Ser. A 457, 141-155 (2001).
    [CrossRef]
  23. M. R. Dennis, "Polarization singularities in paraxial vector fields: morphology and statistics," Opt. Commun. 213, 201-221 (2002).
    [CrossRef]
  24. M. V. Berry, "Index formulae for singular lines of polarization," J. Opt. A, Pure Appl. Opt. 6, 675-678 (2004).
    [CrossRef]
  25. J. V. Hajnal, "Observation of singularities in electric and magnetic fields of freely propagating microwaves," Proc. R. Soc. London, Ser. A 430, 413-421 (1990).
    [CrossRef]
  26. O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, "Interferometric methods in diagnostics of polarization singularities," Phys. Rev. E 65, 036602 (2002).
    [CrossRef]
  27. I. Freund, M. S. Soskin, and A. I. Mokhun, "Elliptic critical points in paraxial optical fields," Opt. Commun. 208, 223-253 (2002).
    [CrossRef]
  28. M. S. Soskin, V. Denisenko, and I. Freund, "Optical polarization singularities and elliptic stationary points," Opt. Lett. 28, 1475-1477 (2003).
    [CrossRef] [PubMed]
  29. M. S. Soskin, V. Denisenko, and R. Egorov, "Topological networks of paraxial ellipse speckle-fields," J. Opt. A, Pure Appl. Opt. 6, S281-S287 (2004).
    [CrossRef]
  30. E. T. Whittaker, "On the functions which are represented by the expansions of the interpolation theory," Proc. R. Soc. Edinburgh, Sect. A: Math. Phys. Sci. 35, 181-194 (1915).
  31. C. E. Shannon, "Communication in the presence of noise," Proc. IRE 37, 10-21 (1949).
    [CrossRef]
  32. B. A. van Tiggelen, D. Anache, and A. Ghysels, "Role of mean free path in spatial phase correlation and nodal screening," Europhys. Lett. 74, 999-1005 (2006).
    [CrossRef]
  33. R. W. Schoonover and T. D. Visser, "Polarization singularities of focused, radially polarized fields," Opt. Express 14, 5733-5745 (2006).
    [CrossRef] [PubMed]

2006

B. A. van Tiggelen, D. Anache, and A. Ghysels, "Role of mean free path in spatial phase correlation and nodal screening," Europhys. Lett. 74, 999-1005 (2006).
[CrossRef]

R. W. Schoonover and T. D. Visser, "Polarization singularities of focused, radially polarized fields," Opt. Express 14, 5733-5745 (2006).
[CrossRef] [PubMed]

2005

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, "Experimental investigation of local properties and statistics of optical vortices in random wave fields," Phys. Rev. Lett. 94, 103902 (2005).
[CrossRef] [PubMed]

2004

M. V. Berry, "Index formulae for singular lines of polarization," J. Opt. A, Pure Appl. Opt. 6, 675-678 (2004).
[CrossRef]

M. S. Soskin, V. Denisenko, and R. Egorov, "Topological networks of paraxial ellipse speckle-fields," J. Opt. A, Pure Appl. Opt. 6, S281-S287 (2004).
[CrossRef]

2003

M. S. Soskin, V. Denisenko, and I. Freund, "Optical polarization singularities and elliptic stationary points," Opt. Lett. 28, 1475-1477 (2003).
[CrossRef] [PubMed]

M. R. Dennis, "Correlations and screening of topological charges in Gaussian random fields," J. Phys. A 36, 6611-6628 (2003).
[CrossRef]

G. Foltin, "The distribution of extremal points of Gaussian scalar fields," J. Phys. A 36, 4561-4580 (2003).
[CrossRef]

2002

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, "Interferometric methods in diagnostics of polarization singularities," Phys. Rev. E 65, 036602 (2002).
[CrossRef]

I. Freund, M. S. Soskin, and A. I. Mokhun, "Elliptic critical points in paraxial optical fields," Opt. Commun. 208, 223-253 (2002).
[CrossRef]

M. R. Dennis, "Polarization singularities in paraxial vector fields: morphology and statistics," Opt. Commun. 213, 201-221 (2002).
[CrossRef]

2001

M. V. Berry and M. R. Dennis, "Polarization singularities in isotropic random vector waves," Proc. R. Soc. London, Ser. A 457, 141-155 (2001).
[CrossRef]

M. R. Dennis, "Local properties and statistics of phase singularities in generic wavefields," Proc. SPIE 4403, 13-23 (2001).
[CrossRef]

M. Dennis, "Phase critical point densities in planar isotropic random waves," J. Phys. A 34, L297-L303 (2001).
[CrossRef]

2000

M. V. Berry and M. R. Dennis, "Phase singularities in isotropic random waves," Proc. R. Soc. London, Ser. A 456, 2059-2079 (2000).
[CrossRef]

1998

I. Freund and M. Wilkinson, "Critical-point screening in random wave fields," J. Opt. Soc. Am. 15, 2892-2902 (1998).
[CrossRef]

I. Freund, " '1001' correlations in random wave fields," Waves Random Media 8, 119-158 (1998).
[CrossRef]

1995

I. Freund, "Saddles, singularities, and extrema in random phase fields," Phys. Rev. E 52, 2348-2359 (1995).
[CrossRef]

1994

I. Freund and N. Shvartsman, "Wave-field phase singularities: the sign principle," Phys. Rev. A 50, 5164-5172 (1994).
[CrossRef]

N. Shvartsman and I. Freund, "Vortices in random wave fields: nearest neighbor anticorrelations," Phys. Rev. Lett. 72, 1008-1011 (1994).
[CrossRef]

1993

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

1990

J. V. Hajnal, "Observation of singularities in electric and magnetic fields of freely propagating microwaves," Proc. R. Soc. London, Ser. A 430, 413-421 (1990).
[CrossRef]

1987

J. F. Nye and J. V. Hajnal, "The wave structure of monochromatic electromagnetic radiation," Proc. R. Soc. London, Ser. A 409, 21-36 (1987).
[CrossRef]

1981

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

1978

M. V. Berry, "Disruption of wavefronts: statistics of dislocations in incoherent Gaussian random waves," J. Phys. A 11, 27-37 (1978).
[CrossRef]

1974

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

1949

C. E. Shannon, "Communication in the presence of noise," Proc. IRE 37, 10-21 (1949).
[CrossRef]

1915

E. T. Whittaker, "On the functions which are represented by the expansions of the interpolation theory," Proc. R. Soc. Edinburgh, Sect. A: Math. Phys. Sci. 35, 181-194 (1915).

Anache, D.

B. A. van Tiggelen, D. Anache, and A. Ghysels, "Role of mean free path in spatial phase correlation and nodal screening," Europhys. Lett. 74, 999-1005 (2006).
[CrossRef]

Angelsky, O. V.

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, "Interferometric methods in diagnostics of polarization singularities," Phys. Rev. E 65, 036602 (2002).
[CrossRef]

Baranova, N. B.

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

Berry, M. V.

M. V. Berry, "Index formulae for singular lines of polarization," J. Opt. A, Pure Appl. Opt. 6, 675-678 (2004).
[CrossRef]

M. V. Berry and M. R. Dennis, "Polarization singularities in isotropic random vector waves," Proc. R. Soc. London, Ser. A 457, 141-155 (2001).
[CrossRef]

M. V. Berry and M. R. Dennis, "Phase singularities in isotropic random waves," Proc. R. Soc. London, Ser. A 456, 2059-2079 (2000).
[CrossRef]

M. V. Berry, "Disruption of wavefronts: statistics of dislocations in incoherent Gaussian random waves," J. Phys. A 11, 27-37 (1978).
[CrossRef]

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

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

Denisenko, V.

M. S. Soskin, V. Denisenko, and R. Egorov, "Topological networks of paraxial ellipse speckle-fields," J. Opt. A, Pure Appl. Opt. 6, S281-S287 (2004).
[CrossRef]

M. S. Soskin, V. Denisenko, and I. Freund, "Optical polarization singularities and elliptic stationary points," Opt. Lett. 28, 1475-1477 (2003).
[CrossRef] [PubMed]

Dennis, M.

M. Dennis, "Phase critical point densities in planar isotropic random waves," J. Phys. A 34, L297-L303 (2001).
[CrossRef]

Dennis, M. R.

M. R. Dennis, "Correlations and screening of topological charges in Gaussian random fields," J. Phys. A 36, 6611-6628 (2003).
[CrossRef]

M. R. Dennis, "Polarization singularities in paraxial vector fields: morphology and statistics," Opt. Commun. 213, 201-221 (2002).
[CrossRef]

M. R. Dennis, "Local properties and statistics of phase singularities in generic wavefields," Proc. SPIE 4403, 13-23 (2001).
[CrossRef]

M. V. Berry and M. R. Dennis, "Polarization singularities in isotropic random vector waves," Proc. R. Soc. London, Ser. A 457, 141-155 (2001).
[CrossRef]

M. V. Berry and M. R. Dennis, "Phase singularities in isotropic random waves," Proc. R. Soc. London, Ser. A 456, 2059-2079 (2000).
[CrossRef]

Egorov, R.

M. S. Soskin, V. Denisenko, and R. Egorov, "Topological networks of paraxial ellipse speckle-fields," J. Opt. A, Pure Appl. Opt. 6, S281-S287 (2004).
[CrossRef]

Foltin, G.

G. Foltin, "The distribution of extremal points of Gaussian scalar fields," J. Phys. A 36, 4561-4580 (2003).
[CrossRef]

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.

M. S. Soskin, V. Denisenko, and I. Freund, "Optical polarization singularities and elliptic stationary points," Opt. Lett. 28, 1475-1477 (2003).
[CrossRef] [PubMed]

I. Freund, M. S. Soskin, and A. I. Mokhun, "Elliptic critical points in paraxial optical fields," Opt. Commun. 208, 223-253 (2002).
[CrossRef]

I. Freund, " '1001' correlations in random wave fields," Waves Random Media 8, 119-158 (1998).
[CrossRef]

I. Freund and M. Wilkinson, "Critical-point screening in random wave fields," J. Opt. Soc. Am. 15, 2892-2902 (1998).
[CrossRef]

I. Freund, "Saddles, singularities, and extrema in random phase fields," Phys. Rev. E 52, 2348-2359 (1995).
[CrossRef]

I. Freund and N. Shvartsman, "Wave-field phase singularities: the sign principle," Phys. Rev. A 50, 5164-5172 (1994).
[CrossRef]

N. Shvartsman and I. Freund, "Vortices in random wave fields: nearest neighbor anticorrelations," Phys. Rev. Lett. 72, 1008-1011 (1994).
[CrossRef]

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

Ghysels, A.

B. A. van Tiggelen, D. Anache, and A. Ghysels, "Role of mean free path in spatial phase correlation and nodal screening," Europhys. Lett. 74, 999-1005 (2006).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer, 1984).

J. W. Goodman, Statistical Optics (Wiley, 1985).

Hajnal, J. V.

J. V. Hajnal, "Observation of singularities in electric and magnetic fields of freely propagating microwaves," Proc. R. Soc. London, Ser. A 430, 413-421 (1990).
[CrossRef]

J. F. Nye and J. V. Hajnal, "The wave structure of monochromatic electromagnetic radiation," Proc. R. Soc. London, Ser. A 409, 21-36 (1987).
[CrossRef]

Halperin, B. I.

B. I. Halperin, "Statistical mechanics of topological defects," in Physics of Defects, R.Balian, M.Kleman, and J.P.Poirier, eds. (North-Holland, 1981), pp. 813-857.

Hanson, S. G.

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, "Experimental investigation of local properties and statistics of optical vortices in random wave fields," Phys. Rev. Lett. 94, 103902 (2005).
[CrossRef] [PubMed]

Mamaev, A. V.

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

Miyamoto, Y.

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, "Experimental investigation of local properties and statistics of optical vortices in random wave fields," Phys. Rev. Lett. 94, 103902 (2005).
[CrossRef] [PubMed]

Mokhun, A. I.

I. Freund, M. S. Soskin, and A. I. Mokhun, "Elliptic critical points in paraxial optical fields," Opt. Commun. 208, 223-253 (2002).
[CrossRef]

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, "Interferometric methods in diagnostics of polarization singularities," Phys. Rev. E 65, 036602 (2002).
[CrossRef]

Mokhun, I. I.

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, "Interferometric methods in diagnostics of polarization singularities," Phys. Rev. E 65, 036602 (2002).
[CrossRef]

Nye, J. F.

J. F. Nye and J. V. Hajnal, "The wave structure of monochromatic electromagnetic radiation," Proc. R. Soc. London, Ser. A 409, 21-36 (1987).
[CrossRef]

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

J. F. Nye, Nature Focusing and Fine Structure of Light (IOP, 1999).

Pilipetskii, N.

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

Schoonover, R. W.

Shannon, C. E.

C. E. Shannon, "Communication in the presence of noise," Proc. IRE 37, 10-21 (1949).
[CrossRef]

Shkunov, V. V.

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

Shvartsman, N.

I. Freund and N. Shvartsman, "Wave-field phase singularities: the sign principle," Phys. Rev. A 50, 5164-5172 (1994).
[CrossRef]

N. Shvartsman and I. Freund, "Vortices in random wave fields: nearest neighbor anticorrelations," Phys. Rev. Lett. 72, 1008-1011 (1994).
[CrossRef]

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

Soskin, M. S.

M. S. Soskin, V. Denisenko, and R. Egorov, "Topological networks of paraxial ellipse speckle-fields," J. Opt. A, Pure Appl. Opt. 6, S281-S287 (2004).
[CrossRef]

M. S. Soskin, V. Denisenko, and I. Freund, "Optical polarization singularities and elliptic stationary points," Opt. Lett. 28, 1475-1477 (2003).
[CrossRef] [PubMed]

I. Freund, M. S. Soskin, and A. I. Mokhun, "Elliptic critical points in paraxial optical fields," Opt. Commun. 208, 223-253 (2002).
[CrossRef]

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, "Interferometric methods in diagnostics of polarization singularities," Phys. Rev. E 65, 036602 (2002).
[CrossRef]

Takeda, M.

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, "Experimental investigation of local properties and statistics of optical vortices in random wave fields," Phys. Rev. Lett. 94, 103902 (2005).
[CrossRef] [PubMed]

van Tiggelen, B. A.

B. A. van Tiggelen, D. Anache, and A. Ghysels, "Role of mean free path in spatial phase correlation and nodal screening," Europhys. Lett. 74, 999-1005 (2006).
[CrossRef]

Visser, T. D.

Wang, W.

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, "Experimental investigation of local properties and statistics of optical vortices in random wave fields," Phys. Rev. Lett. 94, 103902 (2005).
[CrossRef] [PubMed]

Whittaker, E. T.

E. T. Whittaker, "On the functions which are represented by the expansions of the interpolation theory," Proc. R. Soc. Edinburgh, Sect. A: Math. Phys. Sci. 35, 181-194 (1915).

Wilkinson, M.

I. Freund and M. Wilkinson, "Critical-point screening in random wave fields," J. Opt. Soc. Am. 15, 2892-2902 (1998).
[CrossRef]

Zel'dovich, B. Ya.

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

Europhys. Lett.

B. A. van Tiggelen, D. Anache, and A. Ghysels, "Role of mean free path in spatial phase correlation and nodal screening," Europhys. Lett. 74, 999-1005 (2006).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

M. S. Soskin, V. Denisenko, and R. Egorov, "Topological networks of paraxial ellipse speckle-fields," J. Opt. A, Pure Appl. Opt. 6, S281-S287 (2004).
[CrossRef]

M. V. Berry, "Index formulae for singular lines of polarization," J. Opt. A, Pure Appl. Opt. 6, 675-678 (2004).
[CrossRef]

J. Opt. Soc. Am.

I. Freund and M. Wilkinson, "Critical-point screening in random wave fields," J. Opt. Soc. Am. 15, 2892-2902 (1998).
[CrossRef]

J. Phys. A

M. Dennis, "Phase critical point densities in planar isotropic random waves," J. Phys. A 34, L297-L303 (2001).
[CrossRef]

M. V. Berry, "Disruption of wavefronts: statistics of dislocations in incoherent Gaussian random waves," J. Phys. A 11, 27-37 (1978).
[CrossRef]

M. R. Dennis, "Correlations and screening of topological charges in Gaussian random fields," J. Phys. A 36, 6611-6628 (2003).
[CrossRef]

G. Foltin, "The distribution of extremal points of Gaussian scalar fields," J. Phys. A 36, 4561-4580 (2003).
[CrossRef]

JETP Lett.

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

Opt. Commun.

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

M. R. Dennis, "Polarization singularities in paraxial vector fields: morphology and statistics," Opt. Commun. 213, 201-221 (2002).
[CrossRef]

I. Freund, M. S. Soskin, and A. I. Mokhun, "Elliptic critical points in paraxial optical fields," Opt. Commun. 208, 223-253 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

I. Freund and N. Shvartsman, "Wave-field phase singularities: the sign principle," Phys. Rev. A 50, 5164-5172 (1994).
[CrossRef]

Phys. Rev. E

I. Freund, "Saddles, singularities, and extrema in random phase fields," Phys. Rev. E 52, 2348-2359 (1995).
[CrossRef]

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, "Interferometric methods in diagnostics of polarization singularities," Phys. Rev. E 65, 036602 (2002).
[CrossRef]

Phys. Rev. Lett.

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, "Experimental investigation of local properties and statistics of optical vortices in random wave fields," Phys. Rev. Lett. 94, 103902 (2005).
[CrossRef] [PubMed]

N. Shvartsman and I. Freund, "Vortices in random wave fields: nearest neighbor anticorrelations," Phys. Rev. Lett. 72, 1008-1011 (1994).
[CrossRef]

Proc. IRE

C. E. Shannon, "Communication in the presence of noise," Proc. IRE 37, 10-21 (1949).
[CrossRef]

Proc. R. Soc. Edinburgh, Sect. A: Math. Phys. Sci.

E. T. Whittaker, "On the functions which are represented by the expansions of the interpolation theory," Proc. R. Soc. Edinburgh, Sect. A: Math. Phys. Sci. 35, 181-194 (1915).

Proc. R. Soc. London, Ser. A

J. V. Hajnal, "Observation of singularities in electric and magnetic fields of freely propagating microwaves," Proc. R. Soc. London, Ser. A 430, 413-421 (1990).
[CrossRef]

J. F. Nye and J. V. Hajnal, "The wave structure of monochromatic electromagnetic radiation," Proc. R. Soc. London, Ser. A 409, 21-36 (1987).
[CrossRef]

M. V. Berry and M. R. Dennis, "Polarization singularities in isotropic random vector waves," Proc. R. Soc. London, Ser. A 457, 141-155 (2001).
[CrossRef]

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

M. V. Berry and M. R. Dennis, "Phase singularities in isotropic random waves," Proc. R. Soc. London, Ser. A 456, 2059-2079 (2000).
[CrossRef]

Proc. SPIE

M. R. Dennis, "Local properties and statistics of phase singularities in generic wavefields," Proc. SPIE 4403, 13-23 (2001).
[CrossRef]

Waves Random Media

I. Freund, " '1001' correlations in random wave fields," Waves Random Media 8, 119-158 (1998).
[CrossRef]

Other

B. I. Halperin, "Statistical mechanics of topological defects," in Physics of Defects, R.Balian, M.Kleman, and J.P.Poirier, eds. (North-Holland, 1981), pp. 813-857.

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

J. W. Goodman, Statistical Optics (Wiley, 1985).

J. W. Goodman, Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer, 1984).

J. F. Nye, Nature Focusing and Fine Structure of Light (IOP, 1999).

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Figures (7)

Fig. 1
Fig. 1

Illustration of experimental setup. Alumina spheres within Styrofoam shells are contained within a copper tube. A 2.5 cm extension at the end of the tube ensures that the antenna sensitivity is not influenced by the varying distance to elements of the sample as the wire antenna is translated over the cross-section of the tube.

Fig. 2
Fig. 2

(a) an example of a speckle pattern is shown with equiphase lines with values of phase φ = n π 4 rad with n = 0 7 shown as different colors (online) indicated in the color bar. Phase singularities with different signs are shown with red and green (online) dots. (b) Magnification of the region near the singularity within the dashed rectangle in (a). Elliptical contours of intensity (white) and circular contours of current magnitude (green online) are found.

Fig. 3
Fig. 3

Sheets of the zeros of the in-phase (red) and out-of-phase (blue online) parts of the field in a space with two spatial dimensions and one frequency dimension. The green (online) curves at the intersections of these sheets are lines of singularities.

Fig. 4
Fig. 4

Probability distribution of distance to the nearest neighbors with different signs.

Fig. 5
Fig. 5

(a) and (b) are closer views of the regions marked by the dashed elliptical frame and the letter “A,” respectively, in Fig. 2a. Equiphase lines from 0 to 2 π with interval π 16 are shown with different colors online. (a) Phase saddle point (white dot) is located between a pair of phase singularities with the same sign. The bifurcation curves passing through the phase saddle point are marked with white dashed white curves. (b) An extreme phase point is marked with a white dot. (c) and (d) are magnification of the region around the phase saddle point and the phase minimum shown in (a) and (b), respectively. The phase saddle point and minimum are again marked with a white dot, around which equiphase curves are shown. The equiphase lines passing through the phase saddle point, i.e., the bifurcation lines, are indicated with thicker lines.

Fig. 6
Fig. 6

Measurement of state of polarization and observation of polarization singularities. Dots are C points and thick black curves are L lines. Thin black curves and red (online) curves are zero contours of T 1 and T, respectively.

Fig. 7
Fig. 7

Three types of C points: (a) lemon; (b) monstar; (c) star, which are magnified pictures of the C points a, b, and c marked in Fig. 6.

Equations (7)

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E ( x , y ) = E x ( x , y ) x + E y ( x , y ) y ,
E ( x , y ) = A ( x , y ) exp [ i φ ( x , y ) ] = Re E ( x , y ) + i Im E ( x , y ) ,
E ( r ) r E ( 0 ) .
I ( r ) = E ( r ) 2 [ r Re E ( 0 ) ] 2 + [ r Im E ( 0 ) ] 2 ,
J ( r ) = Im E * ( r ) E ( r ) Ω ( 0 ) × r ,
φ ( x , y ) = φ 0 + 1 2 [ x 2 2 φ ( 0 , 0 ) x 2 + 2 x y 2 φ ( 0 , 0 ) x y + y 2 2 φ ( 0 , 0 ) y 2 ] .
Re E exp ( i χ ) = p cos χ + q sin χ ,

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