Abstract

Phase correlations are studied for neighboring critical points of the intensity in an isotropic Gaussian random wave field. Significant correlations and anticorrelations are found that extend out to at least the fifth nearest neighbors. A theoretical interpretation of the empirical data is attempted within the framework of the phase autocorrelation and the probability-density functions of extended two-dimensional random phase fields. It is found, however, that adaptations of these theoretical models are unable to account satisfactorily, or even qualitatively, for the extensive phase correlations that are present in these fields.

© 1998 Optical Society of America

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  4. R. H. T. Bates, M. J. McDonnell, Image Restoration and Reconstruction (Clarendon, Oxford, 1989).
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    [CrossRef]
  6. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), pp. 219–226.
  7. C. Roddier, F. Roddier, “Wave-front reconstruction from defocused images and the testing of ground-based optical telescopes,” J. Opt. Soc. Am. A 10, 2277–2287 (1993);F. Roddier, C. Roddier, “Wavefront reconstruction using iterative Fourier transforms,” Appl. Opt. 30, 1325–1327 (1991);F. Roddier, “Wavefront sensing and the irradiance transport equation,” Appl. Opt. 29, 1402–1403 (1990).
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  8. R. P. Millane, “Multidimensional phase problems,” J. Opt. Soc. Am. A 13, 725–734 (1996);“Phase-retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990).
    [CrossRef]
  9. T. Mavroidis, J. C. Dainty, “Imaging after double passage through a random screen,” Opt. Lett. 15, 857–859 (1990);T. Mavroidis, C. J. Solomon, J. C. Dainty, “Imaging a coherently illuminated object after double passage through a random screen,” J. Opt. Soc. Am. A 8, 1003–1013 (1991).
    [CrossRef] [PubMed]
  10. I. Freund, “Looking through walls and around corners,” Physica A 168, 49–65 (1990);“Correlation imaging through multiply scattering media,” Phys. Lett. A 147, 502–506 (1990); “Image reconstruction through multiple scattering media,” Opt. Commun. 86, 216–227 (1991); “Diffractometry through multiple scattering media,” Opt. Commun. 87, 5–8 (1991); “Time-reversal symmetry and image reconstruction through multiple scattering media,” J. Opt. Soc. Am. A 9, 456–463 (1992).
    [CrossRef]
  11. H. Kadono, N. Takai, T. Asakura, “Statistical properties of the speckle phase in the diffraction region,” J. Opt. Soc. Am. A 3, 1080–1089 (1986);H. Kadono, T. Asakura, “Statistical properties of the speckle phase in the optical imaging system,” J. Opt. Soc. Am. A 2, 787–792 (1985);H. Kadono, N. Takai, T. Asakura, “Experimental study of the laser speckle phase in the image field,” Opt. Acta 32, 1223–1234 (1985).
    [CrossRef]
  12. B. B. Gorbatenko, I. S. Klimenko, L. A. Maksimova, V. P. Ryabukho, “Statistical properties of the spatial distribution of the phase of a developed speckle field,” Sov. Tech. Phys. Lett. 18, 35–36 (1992) [Pis’ma Zh. Tekh. Fiz. 18, 26–28 (1992)]; “Some statistical properties of the phase difference in the developed speckle-modulated field,” Opt. Spectrosc. 78, 283–286 (1995) [Opt. Spektrosk. 78, 316–319 (1995)].
  13. I. Freund, “Optical vortices in Gaussian random wave fields: statistical probability densities,” J. Opt. Soc. Am. A 11, 1644–1652 (1994);“Critical point level-crossing geometry in random wave fields, J. Opt. Soc. Am. A 14, 1911–1927 (1997);“‘1001’ correlations in random wave fields,” Waves Random Media 8, 119–158 (1998).
    [CrossRef]
  14. N. Shvartsman, I. Freund, “Wave-field phase singularities: near neighbor correlations and anticorrelations,” J. Opt. Soc. Am. A 11, 2710–2718 (1994);“Vortices in random wave fields: nearest neighbor anticorrelations,” Phys. Rev. Lett. 72, 1008–1011 (1994).
    [CrossRef] [PubMed]
  15. I. Freund, D. Kessler, “Phase autocorrelation of random wave fields,” Opt. Commun. 124, 322–332 (1995).
  16. J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
    [CrossRef]
  17. M. Berry, “Singularities in waves and rays,” in Physics of Defects, R. Balian, M. Kleman, J.-P. Poirier, eds. (North-Holland, Amsterdam, 1981), pp. 453–549.
  18. M. Berry, “Disruption of wave-fronts: statistics of dislocations in incoherent Gaussian random waves,” J. Phys. A 11, 27–37 (1978).
    [CrossRef]
  19. N. B. Baranova, B. Ya Zel’dovich, A. V. Mamaev, N. Pilipetskii, V. V. Shkukov, “Dislocations of the wave-front of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981); N. B. Baranova, B. Ya Zel’dovich, “Dislocations of the wave-front surface and zeros of amplitude,” Sov. Phys. JETP 53, 925–929 (1981); N. B. Baranova, A. V. Mamaev, N. Pilipetskii, 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).
    [CrossRef]
  20. I. Freund, N. Shvartsman, V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993);I. Freund, V. Freilikher, “Parameterization of anisotropic vortices,” J. Opt. Soc. Am. A 14, 1902–1910 (1997).
    [CrossRef]
  21. D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), pp. 194–229.
  22. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989);J. M. Huntley, J. R. Buckland, “Characterization of sources of 2π phase discontinuity in speckle interferograms,” J. Opt. Soc. Am. A 12, 1990–1996 (1995).
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  23. D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
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  24. D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960), pp. 335–368.
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  26. J. W. Goodman, Statistical Optics (Wiley, New York, 1995).
  27. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1985), pp. 9–75.
  28. A. Weinberg, “Percolation threshold of a two-dimensional continuum system,” Phys. Rev. B 26, 1352–1361 (1982).
    [CrossRef]
  29. A. Weinberg, B. I. Halperin, “Distribution of maxima, minima, and saddle points of the intensity of laser speckle patterns,” Phys. Rev. B 26, 1362–1368 (1982).
    [CrossRef]
  30. N. Shvarstman, I. Freund, “Speckle spots ride phase saddles sidesaddle,” Opt. Commun. 117, 228–234 (1994).
    [CrossRef]
  31. I. Freund, “Saddles, singularities, and extrema in random phase fields,” Phys. Rev. E 52, 2348–2360 (1995).
    [CrossRef]

1996 (1)

1995 (2)

I. Freund, D. Kessler, “Phase autocorrelation of random wave fields,” Opt. Commun. 124, 322–332 (1995).

I. Freund, “Saddles, singularities, and extrema in random phase fields,” Phys. Rev. E 52, 2348–2360 (1995).
[CrossRef]

1994 (3)

1993 (2)

1992 (1)

B. B. Gorbatenko, I. S. Klimenko, L. A. Maksimova, V. P. Ryabukho, “Statistical properties of the spatial distribution of the phase of a developed speckle field,” Sov. Tech. Phys. Lett. 18, 35–36 (1992) [Pis’ma Zh. Tekh. Fiz. 18, 26–28 (1992)]; “Some statistical properties of the phase difference in the developed speckle-modulated field,” Opt. Spectrosc. 78, 283–286 (1995) [Opt. Spektrosk. 78, 316–319 (1995)].

1991 (1)

1990 (2)

T. Mavroidis, J. C. Dainty, “Imaging after double passage through a random screen,” Opt. Lett. 15, 857–859 (1990);T. Mavroidis, C. J. Solomon, J. C. Dainty, “Imaging a coherently illuminated object after double passage through a random screen,” J. Opt. Soc. Am. A 8, 1003–1013 (1991).
[CrossRef] [PubMed]

I. Freund, “Looking through walls and around corners,” Physica A 168, 49–65 (1990);“Correlation imaging through multiply scattering media,” Phys. Lett. A 147, 502–506 (1990); “Image reconstruction through multiple scattering media,” Opt. Commun. 86, 216–227 (1991); “Diffractometry through multiple scattering media,” Opt. Commun. 87, 5–8 (1991); “Time-reversal symmetry and image reconstruction through multiple scattering media,” J. Opt. Soc. Am. A 9, 456–463 (1992).
[CrossRef]

1989 (1)

1986 (1)

1983 (1)

1982 (2)

A. Weinberg, “Percolation threshold of a two-dimensional continuum system,” Phys. Rev. B 26, 1352–1361 (1982).
[CrossRef]

A. Weinberg, B. I. Halperin, “Distribution of maxima, minima, and saddle points of the intensity of laser speckle patterns,” Phys. Rev. B 26, 1362–1368 (1982).
[CrossRef]

1981 (1)

N. B. Baranova, B. Ya Zel’dovich, A. V. Mamaev, N. Pilipetskii, V. V. Shkukov, “Dislocations of the wave-front of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981); N. B. Baranova, B. Ya Zel’dovich, “Dislocations of the wave-front surface and zeros of amplitude,” Sov. Phys. JETP 53, 925–929 (1981); N. B. Baranova, A. V. Mamaev, N. Pilipetskii, 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).
[CrossRef]

1978 (2)

1974 (1)

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

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Asakura, T.

Baranova, N. B.

N. B. Baranova, B. Ya Zel’dovich, A. V. Mamaev, N. Pilipetskii, V. V. Shkukov, “Dislocations of the wave-front of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981); N. B. Baranova, B. Ya Zel’dovich, “Dislocations of the wave-front surface and zeros of amplitude,” Sov. Phys. JETP 53, 925–929 (1981); N. B. Baranova, A. V. Mamaev, N. Pilipetskii, 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).
[CrossRef]

Bates, R. H. T.

R. H. T. Bates, W. R. Fright, “Composite two-dimensional phase restoration procedure,” J. Opt. Soc. Am. 73, 358–365 (1983).
[CrossRef]

R. H. T. Bates, M. J. McDonnell, Image Restoration and Reconstruction (Clarendon, Oxford, 1989).

Berry, M.

M. Berry, “Disruption of wave-fronts: statistics of dislocations in incoherent Gaussian random waves,” J. Phys. A 11, 27–37 (1978).
[CrossRef]

M. Berry, “Singularities in waves and rays,” in Physics of Defects, R. Balian, M. Kleman, J.-P. Poirier, eds. (North-Holland, Amsterdam, 1981), pp. 453–549.

Berry, M. V.

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

Bone, D. J.

Dainty, J. C.

Fienup, J. R.

Freilikher, V.

I. Freund, N. Shvartsman, V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993);I. Freund, V. Freilikher, “Parameterization of anisotropic vortices,” J. Opt. Soc. Am. A 14, 1902–1910 (1997).
[CrossRef]

Freund, I.

I. Freund, “Saddles, singularities, and extrema in random phase fields,” Phys. Rev. E 52, 2348–2360 (1995).
[CrossRef]

I. Freund, D. Kessler, “Phase autocorrelation of random wave fields,” Opt. Commun. 124, 322–332 (1995).

I. Freund, “Optical vortices in Gaussian random wave fields: statistical probability densities,” J. Opt. Soc. Am. A 11, 1644–1652 (1994);“Critical point level-crossing geometry in random wave fields, J. Opt. Soc. Am. A 14, 1911–1927 (1997);“‘1001’ correlations in random wave fields,” Waves Random Media 8, 119–158 (1998).
[CrossRef]

N. Shvartsman, I. Freund, “Wave-field phase singularities: near neighbor correlations and anticorrelations,” J. Opt. Soc. Am. A 11, 2710–2718 (1994);“Vortices in random wave fields: nearest neighbor anticorrelations,” Phys. Rev. Lett. 72, 1008–1011 (1994).
[CrossRef] [PubMed]

N. Shvarstman, I. Freund, “Speckle spots ride phase saddles sidesaddle,” Opt. Commun. 117, 228–234 (1994).
[CrossRef]

I. Freund, N. Shvartsman, V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993);I. Freund, V. Freilikher, “Parameterization of anisotropic vortices,” J. Opt. Soc. Am. A 14, 1902–1910 (1997).
[CrossRef]

I. Freund, “Looking through walls and around corners,” Physica A 168, 49–65 (1990);“Correlation imaging through multiply scattering media,” Phys. Lett. A 147, 502–506 (1990); “Image reconstruction through multiple scattering media,” Opt. Commun. 86, 216–227 (1991); “Diffractometry through multiple scattering media,” Opt. Commun. 87, 5–8 (1991); “Time-reversal symmetry and image reconstruction through multiple scattering media,” J. Opt. Soc. Am. A 9, 456–463 (1992).
[CrossRef]

Fright, W. R.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1995).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1985), pp. 9–75.

Gorbatenko, B. B.

B. B. Gorbatenko, I. S. Klimenko, L. A. Maksimova, V. P. Ryabukho, “Statistical properties of the spatial distribution of the phase of a developed speckle field,” Sov. Tech. Phys. Lett. 18, 35–36 (1992) [Pis’ma Zh. Tekh. Fiz. 18, 26–28 (1992)]; “Some statistical properties of the phase difference in the developed speckle-modulated field,” Opt. Spectrosc. 78, 283–286 (1995) [Opt. Spektrosk. 78, 316–319 (1995)].

Halperin, B. I.

A. Weinberg, B. I. Halperin, “Distribution of maxima, minima, and saddle points of the intensity of laser speckle patterns,” Phys. Rev. B 26, 1362–1368 (1982).
[CrossRef]

Huntley, J. M.

Kadono, H.

Kessler, D.

I. Freund, D. Kessler, “Phase autocorrelation of random wave fields,” Opt. Commun. 124, 322–332 (1995).

Klimenko, I. S.

B. B. Gorbatenko, I. S. Klimenko, L. A. Maksimova, V. P. Ryabukho, “Statistical properties of the spatial distribution of the phase of a developed speckle field,” Sov. Tech. Phys. Lett. 18, 35–36 (1992) [Pis’ma Zh. Tekh. Fiz. 18, 26–28 (1992)]; “Some statistical properties of the phase difference in the developed speckle-modulated field,” Opt. Spectrosc. 78, 283–286 (1995) [Opt. Spektrosk. 78, 316–319 (1995)].

Maksimova, L. A.

B. B. Gorbatenko, I. S. Klimenko, L. A. Maksimova, V. P. Ryabukho, “Statistical properties of the spatial distribution of the phase of a developed speckle field,” Sov. Tech. Phys. Lett. 18, 35–36 (1992) [Pis’ma Zh. Tekh. Fiz. 18, 26–28 (1992)]; “Some statistical properties of the phase difference in the developed speckle-modulated field,” Opt. Spectrosc. 78, 283–286 (1995) [Opt. Spektrosk. 78, 316–319 (1995)].

Mamaev, A. V.

N. B. Baranova, B. Ya Zel’dovich, A. V. Mamaev, N. Pilipetskii, V. V. Shkukov, “Dislocations of the wave-front of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981); N. B. Baranova, B. Ya Zel’dovich, “Dislocations of the wave-front surface and zeros of amplitude,” Sov. Phys. JETP 53, 925–929 (1981); N. B. Baranova, A. V. Mamaev, N. Pilipetskii, 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).
[CrossRef]

Mavroidis, T.

McDonnell, M. J.

R. H. T. Bates, M. J. McDonnell, Image Restoration and Reconstruction (Clarendon, Oxford, 1989).

Middleton, D.

D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960), pp. 335–368.

Millane, R. P.

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), pp. 219–226.

Nye, J. F.

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

Pilipetskii, N.

N. B. Baranova, B. Ya Zel’dovich, A. V. Mamaev, N. Pilipetskii, V. V. Shkukov, “Dislocations of the wave-front of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981); N. B. Baranova, B. Ya Zel’dovich, “Dislocations of the wave-front surface and zeros of amplitude,” Sov. Phys. JETP 53, 925–929 (1981); N. B. Baranova, A. V. Mamaev, N. Pilipetskii, 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).
[CrossRef]

Robinson, D. W.

D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), pp. 194–229.

Roddier, C.

Roddier, F.

Ryabukho, V. P.

B. B. Gorbatenko, I. S. Klimenko, L. A. Maksimova, V. P. Ryabukho, “Statistical properties of the spatial distribution of the phase of a developed speckle field,” Sov. Tech. Phys. Lett. 18, 35–36 (1992) [Pis’ma Zh. Tekh. Fiz. 18, 26–28 (1992)]; “Some statistical properties of the phase difference in the developed speckle-modulated field,” Opt. Spectrosc. 78, 283–286 (1995) [Opt. Spektrosk. 78, 316–319 (1995)].

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Shkukov, V. V.

N. B. Baranova, B. Ya Zel’dovich, A. V. Mamaev, N. Pilipetskii, V. V. Shkukov, “Dislocations of the wave-front of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981); N. B. Baranova, B. Ya Zel’dovich, “Dislocations of the wave-front surface and zeros of amplitude,” Sov. Phys. JETP 53, 925–929 (1981); N. B. Baranova, A. V. Mamaev, N. Pilipetskii, 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).
[CrossRef]

Shvarstman, N.

N. Shvarstman, I. Freund, “Speckle spots ride phase saddles sidesaddle,” Opt. Commun. 117, 228–234 (1994).
[CrossRef]

Shvartsman, N.

N. Shvartsman, I. Freund, “Wave-field phase singularities: near neighbor correlations and anticorrelations,” J. Opt. Soc. Am. A 11, 2710–2718 (1994);“Vortices in random wave fields: nearest neighbor anticorrelations,” Phys. Rev. Lett. 72, 1008–1011 (1994).
[CrossRef] [PubMed]

I. Freund, N. Shvartsman, V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993);I. Freund, V. Freilikher, “Parameterization of anisotropic vortices,” J. Opt. Soc. Am. A 14, 1902–1910 (1997).
[CrossRef]

Takai, N.

Weinberg, A.

A. Weinberg, “Percolation threshold of a two-dimensional continuum system,” Phys. Rev. B 26, 1352–1361 (1982).
[CrossRef]

A. Weinberg, B. I. Halperin, “Distribution of maxima, minima, and saddle points of the intensity of laser speckle patterns,” Phys. Rev. B 26, 1362–1368 (1982).
[CrossRef]

Zel’dovich, B. Ya

N. B. Baranova, B. Ya Zel’dovich, A. V. Mamaev, N. Pilipetskii, V. V. Shkukov, “Dislocations of the wave-front of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981); N. B. Baranova, B. Ya Zel’dovich, “Dislocations of the wave-front surface and zeros of amplitude,” Sov. Phys. JETP 53, 925–929 (1981); N. B. Baranova, A. V. Mamaev, N. Pilipetskii, 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).
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

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

H. Kadono, N. Takai, T. Asakura, “Statistical properties of the speckle phase in the diffraction region,” J. Opt. Soc. Am. A 3, 1080–1089 (1986);H. Kadono, T. Asakura, “Statistical properties of the speckle phase in the optical imaging system,” J. Opt. Soc. Am. A 2, 787–792 (1985);H. Kadono, N. Takai, T. Asakura, “Experimental study of the laser speckle phase in the image field,” Opt. Acta 32, 1223–1234 (1985).
[CrossRef]

I. Freund, “Optical vortices in Gaussian random wave fields: statistical probability densities,” J. Opt. Soc. Am. A 11, 1644–1652 (1994);“Critical point level-crossing geometry in random wave fields, J. Opt. Soc. Am. A 14, 1911–1927 (1997);“‘1001’ correlations in random wave fields,” Waves Random Media 8, 119–158 (1998).
[CrossRef]

N. Shvartsman, I. Freund, “Wave-field phase singularities: near neighbor correlations and anticorrelations,” J. Opt. Soc. Am. A 11, 2710–2718 (1994);“Vortices in random wave fields: nearest neighbor anticorrelations,” Phys. Rev. Lett. 72, 1008–1011 (1994).
[CrossRef] [PubMed]

C. Roddier, F. Roddier, “Wave-front reconstruction from defocused images and the testing of ground-based optical telescopes,” J. Opt. Soc. Am. A 10, 2277–2287 (1993);F. Roddier, C. Roddier, “Wavefront reconstruction using iterative Fourier transforms,” Appl. Opt. 30, 1325–1327 (1991);F. Roddier, “Wavefront sensing and the irradiance transport equation,” Appl. Opt. 29, 1402–1403 (1990).
[CrossRef] [PubMed]

R. P. Millane, “Multidimensional phase problems,” J. Opt. Soc. Am. A 13, 725–734 (1996);“Phase-retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990).
[CrossRef]

J. Phys. A (1)

M. Berry, “Disruption of wave-fronts: statistics of dislocations in incoherent Gaussian random waves,” J. Phys. A 11, 27–37 (1978).
[CrossRef]

JETP Lett. (1)

N. B. Baranova, B. Ya Zel’dovich, A. V. Mamaev, N. Pilipetskii, V. V. Shkukov, “Dislocations of the wave-front of a speckle-inhomogeneous field (theory and experiment),” JETP Lett. 33, 195–199 (1981); N. B. Baranova, B. Ya Zel’dovich, “Dislocations of the wave-front surface and zeros of amplitude,” Sov. Phys. JETP 53, 925–929 (1981); N. B. Baranova, A. V. Mamaev, N. Pilipetskii, 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).
[CrossRef]

Opt. Commun. (3)

I. Freund, N. Shvartsman, V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993);I. Freund, V. Freilikher, “Parameterization of anisotropic vortices,” J. Opt. Soc. Am. A 14, 1902–1910 (1997).
[CrossRef]

N. Shvarstman, I. Freund, “Speckle spots ride phase saddles sidesaddle,” Opt. Commun. 117, 228–234 (1994).
[CrossRef]

I. Freund, D. Kessler, “Phase autocorrelation of random wave fields,” Opt. Commun. 124, 322–332 (1995).

Opt. Lett. (2)

Optik (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Phys. Rev. B (2)

A. Weinberg, “Percolation threshold of a two-dimensional continuum system,” Phys. Rev. B 26, 1352–1361 (1982).
[CrossRef]

A. Weinberg, B. I. Halperin, “Distribution of maxima, minima, and saddle points of the intensity of laser speckle patterns,” Phys. Rev. B 26, 1362–1368 (1982).
[CrossRef]

Phys. Rev. E (1)

I. Freund, “Saddles, singularities, and extrema in random phase fields,” Phys. Rev. E 52, 2348–2360 (1995).
[CrossRef]

Physica A (1)

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Figures (5)

Fig. 1
Fig. 1

(a) Phase autocorrelation function C ϕϕ(r/ L coh) between two points separated by r, where L coh is the transverse coherence length of the field. (b) PDF P for the phase difference ϕ2 - ϕ1 between points 1 and 2. P2- ϕ1) depends on the separation r between the points, and shown are graphs for r/ L coh = 0.5, 1.0, 1.340. These separations correspond, respectively, to the initial decay of C ϕϕ, its first zero, and its first (negative) minimum.

Fig. 2
Fig. 2

PDF for the separation r between near-neighbor intensity maxima for (a) the first NN’s (1-NN) and (b) the fifth NN’s (5-NN). The histograms are the measured data, the solid curves represent the cubic-spline fits to the histograms, and the crosses indicate the probabilities calculated for a random distribution of points with the same number density as the intensity maxima.

Fig. 3
Fig. 3

PDF’s P for the phase differences ϕ2 - ϕ1 between near-neighbor intensity critical points. P2 - ϕ1) depends on the critical-point type chosen for the central point (type 1) and the type chosen for its NN (type 2). In this figure both critical-point types are chosen to be the same. Phase differences between (a) maxima and their first NN maxima, (b) maxima and their fifth NN maxima, (c) saddle points and their first NN saddle points, (d) saddle points and their fifth NN saddle points, (e) minima and their first NN minima, and (f) minima and their fifth NN minima.

Fig. 4
Fig. 4

As in Fig. 3, but here critical-point types 1 and 2 are chosen to be different. Phase differences between (a) maxima and their first NN saddle points, (b) maxima and their fifth NN saddle points, (c) minima and their first NN maxima, (d) minima and their fifth NN maxima, (e) minima and their first NN saddle points, and (f) minima and their fifth NN saddle points.

Fig. 5
Fig. 5

Enlarged, small portion of a contour map of the phase field with superimposed intensity critical points. The centers of the vortices (phase singularities) are the points from which contours of constant phase radiate outward to form a phase star (pinwheel). The phase (labels not shown) circulates from 0–2π counterclockwise (clockwise) for positive (negative) vortices, with all phase contours including branch cuts [heavy curves (see Section 2)] beginning on one vortex and ending on another of opposite sign: (a) NN intensity maxima (circles) and (b) intensity minima (circles) and their NN intensity saddle points (triangles). In both (a) and (b) the straight lines connect NN’s.

Tables (1)

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Table 1 NN Separations 〈r〉/Lcoh of Intensity Critical Pointsa

Equations (6)

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C ϕ ϕ = ³/₂ 2 γ + 4 γ 2 - Γ , γ = 1 2 π sin - 1 μ ,     Γ = 1 2 π 2 n = 1 μ 2 n n 2 ,
μ r = 2 J 1 κ R s r / κ R s r .
P ϕ 2 - ϕ 1 = 1 - | μ | 2 4 π 2 1 - α 2 - 3 / 2 a   sin - 1   a + π a / 2 + 1 - a 2 , a = μ   cos ϕ 2 - ϕ 1 .
P rand n r = 2 π η r π η rand r 2 n - 1 n - 1 ! exp - π η rand r 2 ,
r rand n = 2 n - 1 ! ! 2 n n - 1 !   η rand - 1 / 2 .
P calc n ϕ 2 - ϕ 1 = 0   P spline n r P ϕ 2 - ϕ 1 d r ,

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