Abstract

The Markov approximation for waves in random media specifies that, under strong scintillation conditions, the optical field of unbounded waves has a normal probability distribution with zero mean. Using the coherence function provided by the Markov approximation, we calculate statistics of the phase of the optical field that accounts for the presence of multiple phase dislocations. We also develop and test a Monte Carlo model that generates the phase samples obeying these statistics. In contrast to numerous phase models described in the literature, this model generates discontinuous phase samples that contain optical vortices.

© 2014 Optical Society of America

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References

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  1. M. I. Charnotskii, “Common omission and misconceptions of wave propagation in turbulence: discussion,” J. Opt. Soc. Am. A 29, 711–720 (2012).
    [CrossRef]
  2. M. E. Gracheva and A. S. Gurvich, “Strong fluctuations in the intensity of light propagated through the atmosphere close to the earth,” Radiophys. Quantum Electron. 8, 511–515 (1965).
  3. G. R. Ochs, R. R. Bergman, and J. R. Snyder, “Laser-beam scintillation over horizontal paths from 5.5 to 145 kilometers,” J. Opt. Soc. Am. 59, 231–234 (1969).
    [CrossRef]
  4. N. Perlot, D. Giggenbach, H. Henniger, J. Horwath, M. Knapek, and K. Zettl, “Measurements of the beam-wave fluctuations over a 142  km atmospheric path,” Proc. SPIE 6304, 63041O (2006).
    [CrossRef]
  5. M. A. Vorontsov, G. W. Carhart, V. S. Rao Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149  km propagation path using multi-wavelength laser beacons,” in Proceedings of the 2010 Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS Conference) (Curran, 2010), paper E18.
  6. A. S. Gurvich, M. E. Gorbunov, O. V. Fedorova, G. Kirchengast, V. Proschek, G. González Abad, and K. A. Tereszchuk, “Spatiotemporal structure of a laser beam over 144  km in a Canary Islands experiment,” Appl. Opt. 51, 7374–7383 (2012).
    [CrossRef]
  7. M. Charnotskii, “Weak and strong off-axis beam scintillations and beam wander for propagation in turbulence,” Proc. SPIE 7685, 768502 (2010).
    [CrossRef]
  8. V. I. Tatarskii and V. U. Zavorotnyi, “Strong fluctuations in light propagation in a randomly inhomogeneous medium,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1980), pp. 207–256.
  9. V. I. Tatarskii, “Light propagation in a medium with random refractive index inhomogeneities in the Markov random process approximation,” Sov. Phys. JETP 29, 1133–1139 (1969).
  10. V. U. Zavorotnyi, V. I. Klyatskin, and V. I. Tatarskii, “Strong fluctuations of electromagnetic waves in randomly inhomogeneous media,” Sov. Phys. JETP 46, 252–259 (1977).
  11. V. I. Klyatskin and V. I. Tatarskii, “Statistical theory of light propagation in a turbulent medium. (Review),” Radiophys. Quantum Electron. 15, 1095–1112 (1972).
    [CrossRef]
  12. S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation Through Random Media (Springer, 1989).
  13. M. Charnotskii, “Asymptotic analysis of finite-beam scintillation in a turbulent medium,” Waves Random Media 4, 243–273 (1994).
    [CrossRef]
  14. M. Rosenblatt, Gaussian and Non-Gaussian Linear Time Series and Random Fields (Springer, 2000).
  15. M. Charnotskii, “Sparse Spectrum model for a turbulent phase,” J. Opt. Soc. Am. A 30, 479–488 (2013).
    [CrossRef]
  16. M. Charnotskii, “Statistics of the Sparse Spectrum turbulent phase,” J. Opt. Soc. Am. A 30, 2455–2465 (2013).
    [CrossRef]
  17. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
  18. J. W. Goodman, Statistical Optics (Wiley, 2000).
  19. R. F. Lutomirski and H. T. Yura, “Wave structure function and mutual coherence function of an optical wave in a turbulent atmosphere,” J. Opt. Soc. Am. 61, 482–487 (1971).
    [CrossRef]
  20. M. Charnotskii, “Statistical modeling of the point spread function for imaging in turbulence,” Opt. Eng. 51, 101706 (2012).
    [CrossRef]
  21. D. W. Oesch, D. J. Sanchez, and C. L. Matson, “The aggregate behavior of branch points—measuring the number and velocity of atmospheric turbulence layers,” Opt. Express 18, 22377–22392 (2010).
    [CrossRef]
  22. I. Freund, “Saddles, singularities, and extrema in random phase fields,” Phys. Rev. E 52, 2348–2360 (1995).
    [CrossRef]
  23. M. V. Berry, “Disruption of wavefronts: statistics of dislocations in incoherent Gaussian random waves,” J. Phys. A 11, 27–37 (1978).
    [CrossRef]
  24. N. B. Baranova and B. Ya. Zel’dovich, “Dislocations of the wave-front surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–929 (1981).
  25. V. V. Voitsekhovich, D. Kouznetsov, and D. Kh. Morzov, “Density of turbulence-induced phase dislocations,” Appl. Opt. 37, 4525–4535 (1998).
    [CrossRef]
  26. M. Charnotskii, “Intensity fluctuations of flat-topped beam in non-Kolmogorov weak turbulence: comment,” J. Opt. Soc. Am. A 29, 1838–1840 (2012).
    [CrossRef]

2013 (2)

2012 (4)

2010 (2)

D. W. Oesch, D. J. Sanchez, and C. L. Matson, “The aggregate behavior of branch points—measuring the number and velocity of atmospheric turbulence layers,” Opt. Express 18, 22377–22392 (2010).
[CrossRef]

M. Charnotskii, “Weak and strong off-axis beam scintillations and beam wander for propagation in turbulence,” Proc. SPIE 7685, 768502 (2010).
[CrossRef]

2006 (1)

N. Perlot, D. Giggenbach, H. Henniger, J. Horwath, M. Knapek, and K. Zettl, “Measurements of the beam-wave fluctuations over a 142  km atmospheric path,” Proc. SPIE 6304, 63041O (2006).
[CrossRef]

1998 (1)

1995 (1)

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

1994 (1)

M. Charnotskii, “Asymptotic analysis of finite-beam scintillation in a turbulent medium,” Waves Random Media 4, 243–273 (1994).
[CrossRef]

1981 (1)

N. B. Baranova and B. Ya. Zel’dovich, “Dislocations of the wave-front surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–929 (1981).

1978 (1)

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

1977 (1)

V. U. Zavorotnyi, V. I. Klyatskin, and V. I. Tatarskii, “Strong fluctuations of electromagnetic waves in randomly inhomogeneous media,” Sov. Phys. JETP 46, 252–259 (1977).

1972 (1)

V. I. Klyatskin and V. I. Tatarskii, “Statistical theory of light propagation in a turbulent medium. (Review),” Radiophys. Quantum Electron. 15, 1095–1112 (1972).
[CrossRef]

1971 (1)

1969 (2)

G. R. Ochs, R. R. Bergman, and J. R. Snyder, “Laser-beam scintillation over horizontal paths from 5.5 to 145 kilometers,” J. Opt. Soc. Am. 59, 231–234 (1969).
[CrossRef]

V. I. Tatarskii, “Light propagation in a medium with random refractive index inhomogeneities in the Markov random process approximation,” Sov. Phys. JETP 29, 1133–1139 (1969).

1965 (1)

M. E. Gracheva and A. S. Gurvich, “Strong fluctuations in the intensity of light propagated through the atmosphere close to the earth,” Radiophys. Quantum Electron. 8, 511–515 (1965).

Baranova, N. B.

N. B. Baranova and B. Ya. Zel’dovich, “Dislocations of the wave-front surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–929 (1981).

Beresnev, L. A.

M. A. Vorontsov, G. W. Carhart, V. S. Rao Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149  km propagation path using multi-wavelength laser beacons,” in Proceedings of the 2010 Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS Conference) (Curran, 2010), paper E18.

Bergman, R. R.

Berry, M. V.

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

Carhart, G. W.

M. A. Vorontsov, G. W. Carhart, V. S. Rao Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149  km propagation path using multi-wavelength laser beacons,” in Proceedings of the 2010 Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS Conference) (Curran, 2010), paper E18.

Charnotskii, M.

M. Charnotskii, “Statistics of the Sparse Spectrum turbulent phase,” J. Opt. Soc. Am. A 30, 2455–2465 (2013).
[CrossRef]

M. Charnotskii, “Sparse Spectrum model for a turbulent phase,” J. Opt. Soc. Am. A 30, 479–488 (2013).
[CrossRef]

M. Charnotskii, “Intensity fluctuations of flat-topped beam in non-Kolmogorov weak turbulence: comment,” J. Opt. Soc. Am. A 29, 1838–1840 (2012).
[CrossRef]

M. Charnotskii, “Statistical modeling of the point spread function for imaging in turbulence,” Opt. Eng. 51, 101706 (2012).
[CrossRef]

M. Charnotskii, “Weak and strong off-axis beam scintillations and beam wander for propagation in turbulence,” Proc. SPIE 7685, 768502 (2010).
[CrossRef]

M. Charnotskii, “Asymptotic analysis of finite-beam scintillation in a turbulent medium,” Waves Random Media 4, 243–273 (1994).
[CrossRef]

Charnotskii, M. I.

Fedorova, O. V.

Freund, I.

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

Giggenbach, D.

N. Perlot, D. Giggenbach, H. Henniger, J. Horwath, M. Knapek, and K. Zettl, “Measurements of the beam-wave fluctuations over a 142  km atmospheric path,” Proc. SPIE 6304, 63041O (2006).
[CrossRef]

González Abad, G.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 2000).

Gorbunov, M. E.

Gracheva, M. E.

M. E. Gracheva and A. S. Gurvich, “Strong fluctuations in the intensity of light propagated through the atmosphere close to the earth,” Radiophys. Quantum Electron. 8, 511–515 (1965).

Gurvich, A. S.

A. S. Gurvich, M. E. Gorbunov, O. V. Fedorova, G. Kirchengast, V. Proschek, G. González Abad, and K. A. Tereszchuk, “Spatiotemporal structure of a laser beam over 144  km in a Canary Islands experiment,” Appl. Opt. 51, 7374–7383 (2012).
[CrossRef]

M. E. Gracheva and A. S. Gurvich, “Strong fluctuations in the intensity of light propagated through the atmosphere close to the earth,” Radiophys. Quantum Electron. 8, 511–515 (1965).

Henniger, H.

N. Perlot, D. Giggenbach, H. Henniger, J. Horwath, M. Knapek, and K. Zettl, “Measurements of the beam-wave fluctuations over a 142  km atmospheric path,” Proc. SPIE 6304, 63041O (2006).
[CrossRef]

Horwath, J.

N. Perlot, D. Giggenbach, H. Henniger, J. Horwath, M. Knapek, and K. Zettl, “Measurements of the beam-wave fluctuations over a 142  km atmospheric path,” Proc. SPIE 6304, 63041O (2006).
[CrossRef]

Kirchengast, G.

Klyatskin, V. I.

V. U. Zavorotnyi, V. I. Klyatskin, and V. I. Tatarskii, “Strong fluctuations of electromagnetic waves in randomly inhomogeneous media,” Sov. Phys. JETP 46, 252–259 (1977).

V. I. Klyatskin and V. I. Tatarskii, “Statistical theory of light propagation in a turbulent medium. (Review),” Radiophys. Quantum Electron. 15, 1095–1112 (1972).
[CrossRef]

Knapek, M.

N. Perlot, D. Giggenbach, H. Henniger, J. Horwath, M. Knapek, and K. Zettl, “Measurements of the beam-wave fluctuations over a 142  km atmospheric path,” Proc. SPIE 6304, 63041O (2006).
[CrossRef]

Kouznetsov, D.

Kravtsov, Yu. A.

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation Through Random Media (Springer, 1989).

Lachinova, S. L.

M. A. Vorontsov, G. W. Carhart, V. S. Rao Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149  km propagation path using multi-wavelength laser beacons,” in Proceedings of the 2010 Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS Conference) (Curran, 2010), paper E18.

Liu, J.

M. A. Vorontsov, G. W. Carhart, V. S. Rao Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149  km propagation path using multi-wavelength laser beacons,” in Proceedings of the 2010 Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS Conference) (Curran, 2010), paper E18.

Lutomirski, R. F.

Matson, C. L.

Morzov, D. Kh.

Ochs, G. R.

Oesch, D. W.

Perlot, N.

N. Perlot, D. Giggenbach, H. Henniger, J. Horwath, M. Knapek, and K. Zettl, “Measurements of the beam-wave fluctuations over a 142  km atmospheric path,” Proc. SPIE 6304, 63041O (2006).
[CrossRef]

Proschek, V.

Rao Gudimetla, V. S.

M. A. Vorontsov, G. W. Carhart, V. S. Rao Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149  km propagation path using multi-wavelength laser beacons,” in Proceedings of the 2010 Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS Conference) (Curran, 2010), paper E18.

Rehder, K.

M. A. Vorontsov, G. W. Carhart, V. S. Rao Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149  km propagation path using multi-wavelength laser beacons,” in Proceedings of the 2010 Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS Conference) (Curran, 2010), paper E18.

Riker, J. F.

M. A. Vorontsov, G. W. Carhart, V. S. Rao Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149  km propagation path using multi-wavelength laser beacons,” in Proceedings of the 2010 Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS Conference) (Curran, 2010), paper E18.

Rosenblatt, M.

M. Rosenblatt, Gaussian and Non-Gaussian Linear Time Series and Random Fields (Springer, 2000).

Rytov, S. M.

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation Through Random Media (Springer, 1989).

Sanchez, D. J.

Snyder, J. R.

Stevenson, E.

M. A. Vorontsov, G. W. Carhart, V. S. Rao Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149  km propagation path using multi-wavelength laser beacons,” in Proceedings of the 2010 Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS Conference) (Curran, 2010), paper E18.

Tatarskii, V. I.

V. U. Zavorotnyi, V. I. Klyatskin, and V. I. Tatarskii, “Strong fluctuations of electromagnetic waves in randomly inhomogeneous media,” Sov. Phys. JETP 46, 252–259 (1977).

V. I. Klyatskin and V. I. Tatarskii, “Statistical theory of light propagation in a turbulent medium. (Review),” Radiophys. Quantum Electron. 15, 1095–1112 (1972).
[CrossRef]

V. I. Tatarskii, “Light propagation in a medium with random refractive index inhomogeneities in the Markov random process approximation,” Sov. Phys. JETP 29, 1133–1139 (1969).

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation Through Random Media (Springer, 1989).

V. I. Tatarskii and V. U. Zavorotnyi, “Strong fluctuations in light propagation in a randomly inhomogeneous medium,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1980), pp. 207–256.

Tereszchuk, K. A.

Voitsekhovich, V. V.

Vorontsov, M. A.

M. A. Vorontsov, G. W. Carhart, V. S. Rao Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149  km propagation path using multi-wavelength laser beacons,” in Proceedings of the 2010 Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS Conference) (Curran, 2010), paper E18.

Weyrauch, T.

M. A. Vorontsov, G. W. Carhart, V. S. Rao Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149  km propagation path using multi-wavelength laser beacons,” in Proceedings of the 2010 Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS Conference) (Curran, 2010), paper E18.

Yura, H. T.

Zavorotnyi, V. U.

V. U. Zavorotnyi, V. I. Klyatskin, and V. I. Tatarskii, “Strong fluctuations of electromagnetic waves in randomly inhomogeneous media,” Sov. Phys. JETP 46, 252–259 (1977).

V. I. Tatarskii and V. U. Zavorotnyi, “Strong fluctuations in light propagation in a randomly inhomogeneous medium,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1980), pp. 207–256.

Zel’dovich, B. Ya.

N. B. Baranova and B. Ya. Zel’dovich, “Dislocations of the wave-front surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–929 (1981).

Zettl, K.

N. Perlot, D. Giggenbach, H. Henniger, J. Horwath, M. Knapek, and K. Zettl, “Measurements of the beam-wave fluctuations over a 142  km atmospheric path,” Proc. SPIE 6304, 63041O (2006).
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

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).
[CrossRef]

Opt. Eng. (1)

M. Charnotskii, “Statistical modeling of the point spread function for imaging in turbulence,” Opt. Eng. 51, 101706 (2012).
[CrossRef]

Opt. Express (1)

Phys. Rev. E (1)

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

Proc. SPIE (2)

N. Perlot, D. Giggenbach, H. Henniger, J. Horwath, M. Knapek, and K. Zettl, “Measurements of the beam-wave fluctuations over a 142  km atmospheric path,” Proc. SPIE 6304, 63041O (2006).
[CrossRef]

M. Charnotskii, “Weak and strong off-axis beam scintillations and beam wander for propagation in turbulence,” Proc. SPIE 7685, 768502 (2010).
[CrossRef]

Radiophys. Quantum Electron. (2)

V. I. Klyatskin and V. I. Tatarskii, “Statistical theory of light propagation in a turbulent medium. (Review),” Radiophys. Quantum Electron. 15, 1095–1112 (1972).
[CrossRef]

M. E. Gracheva and A. S. Gurvich, “Strong fluctuations in the intensity of light propagated through the atmosphere close to the earth,” Radiophys. Quantum Electron. 8, 511–515 (1965).

Sov. Phys. JETP (3)

V. I. Tatarskii, “Light propagation in a medium with random refractive index inhomogeneities in the Markov random process approximation,” Sov. Phys. JETP 29, 1133–1139 (1969).

V. U. Zavorotnyi, V. I. Klyatskin, and V. I. Tatarskii, “Strong fluctuations of electromagnetic waves in randomly inhomogeneous media,” Sov. Phys. JETP 46, 252–259 (1977).

N. B. Baranova and B. Ya. Zel’dovich, “Dislocations of the wave-front surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–929 (1981).

Waves Random Media (1)

M. Charnotskii, “Asymptotic analysis of finite-beam scintillation in a turbulent medium,” Waves Random Media 4, 243–273 (1994).
[CrossRef]

Other (6)

M. Rosenblatt, Gaussian and Non-Gaussian Linear Time Series and Random Fields (Springer, 2000).

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation Through Random Media (Springer, 1989).

V. I. Tatarskii and V. U. Zavorotnyi, “Strong fluctuations in light propagation in a randomly inhomogeneous medium,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1980), pp. 207–256.

M. A. Vorontsov, G. W. Carhart, V. S. Rao Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149  km propagation path using multi-wavelength laser beacons,” in Proceedings of the 2010 Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS Conference) (Curran, 2010), paper E18.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

J. W. Goodman, Statistical Optics (Wiley, 2000).

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Figures (7)

Fig. 1.
Fig. 1.

PDFs of phase differences for different values of coherence γ.

Fig. 2.
Fig. 2.

Solid curve: phase structure function [Eq. (15)]. Dashed lines: asymptotes [Eq. (16)].

Fig. 3.
Fig. 3.

Spectra of the optical field, Eqs. (23) and (26), for several values of the spectral exponent α.

Fig. 4.
Fig. 4.

Phase sample with discontinuities and dislocations. White curves represent zero lines of the real or imaginary parts of the field.

Fig. 5.
Fig. 5.

Phase structure functions for different values of the spectral exponent α. Solid curves: statistics of model-generated samples. Dashed curves: theoretical results [Eq. (15)].

Fig. 6.
Fig. 6.

Number of phase dislocations per rC×rC area, as a function of dimensionless cut-off frequency for α=5/3. Diamonds: statistics drawn from phase samples. Solid line: best-fit power-law dependence.

Fig. 7.
Fig. 7.

Number of phase dislocations per rC×rC area as a function of spectral exponent α. Dimensionless cut-off frequency KmrC=300 for α=5/3. Triangles: statistics drawn from phase samples. Solid line: best-fit power-law dependence.

Equations (34)

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

In=n![1+n(n1)β2+],
u(r)=0,u(r1)u(r2)=0
γ(r1r2)=u(r1)u*(r2)=exp[12D(r1r2)],
D(r)=4πk20Ldzd2KΦn(K,κz=0,z)×{1cos[w(z)K·r]}.
D(r)=2d2KFD(K)[1cos(κ·r)],
p(v,w)=1πexp(v2w2).
p(A,φ)=1πAexp(A2),p(A)=2Aexp(A2),p(φ)=12π,φ[π,π].
φ=0,φ2=π23,A=π2,In=n!.
v(r1)v(r2)=w(r1)w(r2)=12exp[12D(r1r2)],u(r1)w(r2)=0.
p(v1,w1,v2,w2)=1π2(1γ2)×exp[v12+w12+v22+w222γ(v1v2+w1w2)(1γ2)],
u(r1)=A(r1)exp[iψ(r1,r2)+i2θ(r1,r2)],u(r2)=A(r2)exp[iψ(r1,r2)i2θ(r1,r2)].
p(A1,A2,ψ,θ)=A1A2π2(1γ2)×exp[A12+A122γA1A2cosθ(1γ2)].
[2πθ0,θφθ][0θ2π,θφθ]
p(θ)=(1γ2)(2π|θ|)4π2(1γ2cos2θ)×[1+2γcosθ1γ2cos2θarctan1+γcosθ1γcosθ].
Dφ(d)=(1ed)π0πθ(2π|θ|)dθ(1edcos2θ)×[1+2ed/2cosθ1edcos2θarctan1+ed/2cosθ1ed/2cosθ].
Dφ(d)={2πd,d12π2/3,d1.
u(r)=Ren=1Nanexp(iKn·r)
an=0,anam=0,anam*=snδmn,
γ(r)=d2KFγ(K)exp(iK·r),
n=1Nsnpn(K)=Fγ(K).
pn(K)d2K=pn(K,α)kdkdα=pn(K)dkdα2π,
n=1Nsnpn(K)=2πKFγ(K).
Fγ(K)=12πrdrexp[12D(r)]J0(Kr),
D(r)=(rrC)α,1<α<2,
FD(K)=C(α)Kα2rCα,C(α)=Γ(1+α2)2α2απΓ(1α2).
Fγ(K)C(α)Kα2rCα.
sn=2πKn1KnKdKFγ(K).
sn=0rdrexp[12(rrC)α][KnJ1(Knr)Kn1J1(Kn1r)]
sn=Γ(1+α2)2α1Γ(1α2)rCα[Kn1αKnα].
pn(K)=1KnKn1,
NXY=12πγxx(r)[γyy(r)[Imγy(r)]2][Reγxy(r)]2r=0.
NXY=116π[D(r)2(D(r))2]D(r)r|r=0,
FD(K)={C(α)(KrC)α2,K<Km0,K>Km,
D(r)=O(r2rCαKmα2)

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