Abstract

Under certain conditions, light-wave propagation through turbulent media causes a specific type of phase distortion: so-called phase dislocations. A salient feature of phase dislocations is an appearance of zones where the phase turns out to be a multivalued function of coordinates. The problem of turbulence-induced phase dislocations is considered. Both a theoretical treatment and simulations based on the numerical solution of a parabolic equation are used for estimation of the dislocation density. Various turbulence conditions, ranging from weak to very strong ones, are considered as well as the dependences on wavelength, and the inner scales of turbulence are presented. An empirical formula for the dislocation density suitable for a wide range of turbulent and propagation conditions is derived. The results obtained can be useful for both atmospheric and adaptive optics.

© 1998 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).
    [CrossRef]
  2. M. V. Berry, “Disruption of wavefront: statistics of dislocations in incoherent Gaussian random waves,” J. Phys. A 11, 27–37 (1978).
    [CrossRef]
  3. 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).
  4. N. B. Baranova, B. Ya. Zel’dovich, “Dislocations of the wave-front surface and zeros of the amplitude,” Zh. Eksp. Teor. Fiz. 80, 1789–1797 (1981).
  5. N. B. Baranova, A. V. Mamaev, N. Pilipetskii, V. V. Shkukov, B. Ya. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. 73, 525–528 (1983).
    [CrossRef]
  6. J. F. Nye, J. V. Hajnal, J. H. Hannay, “Phase saddles and dislocations in two-dimensional waves such as tides,” Proc. R. Soc. London Ser. A 417, 7–20 (1988).
    [CrossRef]
  7. 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).
    [CrossRef] [PubMed]
  8. I. Freund, N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
    [CrossRef] [PubMed]
  9. N. Shvartsman, I. Freund, “Vortices in random wave fields: nearest neighbor anticorrelations,” Phys. Rev. Lett. 72, 1008–1011 (1994).
    [CrossRef] [PubMed]
  10. I. Freund, “Optical vortices in Gaussian random wave fields: statistical probability densities,” J. Opt. Soc. Am. A 11, 1644–1652 (1994).
    [CrossRef]
  11. I. Freund, N. Shvartsman, “Structural correlations in Gaussian random wave fields,” Phys. Rev. E 51, 3770–3773 (1995).
    [CrossRef]
  12. I. Freund, “Saddles, singularities, and extrema in random fields,” Phys. Rev. E 52, 2348–2360 (1995).
    [CrossRef]
  13. I. Freund, “Amplitude topological singularities in random electromagnetic wavefields,” Phys. Lett. A 198, 139–144 (1995).
    [CrossRef]
  14. N. R. Heckenberg, M. Vaupel, J. T. Malos, C. O. Weiss, “Optical-vortex pair creation and annichilation and helical astigmatism of a nonplanar ring resonator,” Phys. Rev. A 54, 2369–2378 (1996).
    [CrossRef] [PubMed]
  15. E. Abramochkin, V. Volostnikov, “Spiral-type beams: optical and quantum aspects,” Opt. Commun. 125, 302–323 (1996).
    [CrossRef]
  16. I. Freund, “Vortex derivatives,” Opt. Commun. 137, 118–126 (1997).
    [CrossRef]
  17. D. L. Fried, J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).
    [CrossRef] [PubMed]
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  19. V. A. Tartakovski, N. N. Mayer, “Phase dislocations and minimal phase representation of the wave function,” Atmos. Oceanic Opt. 8, 231–235 (1995).
  20. B. V. Fortes, V. Lukin, “The effects of wavefront dislocations on the atmospheric adaptive optical systems performance,” in Optics for Science and New Technology. Part II. Lasers and Laser Spectroscopy, J. Chang, J. Lee, C. Nan, eds., Proc. SPIE2778, 1002–1003 (1996).
  21. V. Aksenov, V. Banakh, O. Tikhomirova, “Potential and vortex features of optical speckle fields,” Atmos. Oceanic Opt. 9, 1450–1456 (1996).
  22. V. A. Tartakovski, N. N. Mayer, “Focal spot in the presence of phase dislocations,” Atmos. Oceanic Opt. 9, 1457–1460 (1996).
  23. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2, Chap. 20.
  24. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984), Chap. 11.
  25. V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation, NSF Report TT-68-50464 (U.S. National Science Foundation, Washington, D.C., 1968).
  26. J. M. Martin, S. M. Flatte, “Intensity images and statistics from numerical simulation of wave propagation through 3-D random media,” Appl. Opt. 27, 2111–2126 (1988).
    [CrossRef] [PubMed]
  27. R. L. Fante, “Propagation of electromagnetic waves through turbulent plasma using transport theory,” IEEE Trans. Antennas Propag. AP-21, 750–755 (1975).
  28. G. Parry, P. N. Pusey, “K distribution in atmospheric propagation of laser light,” J. Opt. Soc. Am. 69, 796–798 (1979).
    [CrossRef]
  29. A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon & Breach, New York, 1990), Vol. 3, p. 40.

1997

I. Freund, “Vortex derivatives,” Opt. Commun. 137, 118–126 (1997).
[CrossRef]

1996

N. R. Heckenberg, M. Vaupel, J. T. Malos, C. O. Weiss, “Optical-vortex pair creation and annichilation and helical astigmatism of a nonplanar ring resonator,” Phys. Rev. A 54, 2369–2378 (1996).
[CrossRef] [PubMed]

E. Abramochkin, V. Volostnikov, “Spiral-type beams: optical and quantum aspects,” Opt. Commun. 125, 302–323 (1996).
[CrossRef]

V. Aksenov, V. Banakh, O. Tikhomirova, “Potential and vortex features of optical speckle fields,” Atmos. Oceanic Opt. 9, 1450–1456 (1996).

V. A. Tartakovski, N. N. Mayer, “Focal spot in the presence of phase dislocations,” Atmos. Oceanic Opt. 9, 1457–1460 (1996).

1995

I. Freund, N. Shvartsman, “Structural correlations in Gaussian random wave fields,” Phys. Rev. E 51, 3770–3773 (1995).
[CrossRef]

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

I. Freund, “Amplitude topological singularities in random electromagnetic wavefields,” Phys. Lett. A 198, 139–144 (1995).
[CrossRef]

V. Lukin, B. Fortes, “The influence of wave front dislocations on phase conjugation instability at thermal blooming compensation,” Atmos. Oceanic Opt. 8, 223–230 (1995).

V. A. Tartakovski, N. N. Mayer, “Phase dislocations and minimal phase representation of the wave function,” Atmos. Oceanic Opt. 8, 231–235 (1995).

1994

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

N. Shvartsman, I. Freund, “Vortices in random wave fields: nearest neighbor anticorrelations,” Phys. Rev. Lett. 72, 1008–1011 (1994).
[CrossRef] [PubMed]

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

1992

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).
[CrossRef] [PubMed]

1988

J. M. Martin, S. M. Flatte, “Intensity images and statistics from numerical simulation of wave propagation through 3-D random media,” Appl. Opt. 27, 2111–2126 (1988).
[CrossRef] [PubMed]

J. F. Nye, J. V. Hajnal, J. H. Hannay, “Phase saddles and dislocations in two-dimensional waves such as tides,” Proc. R. Soc. London Ser. A 417, 7–20 (1988).
[CrossRef]

1983

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).

N. B. Baranova, B. Ya. Zel’dovich, “Dislocations of the wave-front surface and zeros of the amplitude,” Zh. Eksp. Teor. Fiz. 80, 1789–1797 (1981).

1979

1978

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

1975

R. L. Fante, “Propagation of electromagnetic waves through turbulent plasma using transport theory,” IEEE Trans. Antennas Propag. AP-21, 750–755 (1975).

1974

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

Abramochkin, E.

E. Abramochkin, V. Volostnikov, “Spiral-type beams: optical and quantum aspects,” Opt. Commun. 125, 302–323 (1996).
[CrossRef]

Aksenov, V.

V. Aksenov, V. Banakh, O. Tikhomirova, “Potential and vortex features of optical speckle fields,” Atmos. Oceanic Opt. 9, 1450–1456 (1996).

Arecchi, F. T.

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).
[CrossRef] [PubMed]

Banakh, V.

V. Aksenov, V. Banakh, O. Tikhomirova, “Potential and vortex features of optical speckle fields,” Atmos. Oceanic Opt. 9, 1450–1456 (1996).

Baranova, N. B.

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

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. Pilipetskii, V. V. Shkukov, “Dislocations of the wavefront 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 the amplitude,” Zh. Eksp. Teor. Fiz. 80, 1789–1797 (1981).

Berry, M. V.

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

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

Brychkov, Yu. A.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon & Breach, New York, 1990), Vol. 3, p. 40.

Fante, R. L.

R. L. Fante, “Propagation of electromagnetic waves through turbulent plasma using transport theory,” IEEE Trans. Antennas Propag. AP-21, 750–755 (1975).

Flatte, S. M.

Fortes, B.

V. Lukin, B. Fortes, “The influence of wave front dislocations on phase conjugation instability at thermal blooming compensation,” Atmos. Oceanic Opt. 8, 223–230 (1995).

Fortes, B. V.

B. V. Fortes, V. Lukin, “The effects of wavefront dislocations on the atmospheric adaptive optical systems performance,” in Optics for Science and New Technology. Part II. Lasers and Laser Spectroscopy, J. Chang, J. Lee, C. Nan, eds., Proc. SPIE2778, 1002–1003 (1996).

Freund, I.

I. Freund, “Vortex derivatives,” Opt. Commun. 137, 118–126 (1997).
[CrossRef]

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

I. Freund, “Amplitude topological singularities in random electromagnetic wavefields,” Phys. Lett. A 198, 139–144 (1995).
[CrossRef]

I. Freund, N. Shvartsman, “Structural correlations in Gaussian random wave fields,” Phys. Rev. E 51, 3770–3773 (1995).
[CrossRef]

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

N. Shvartsman, I. Freund, “Vortices in random wave fields: nearest neighbor anticorrelations,” Phys. Rev. Lett. 72, 1008–1011 (1994).
[CrossRef] [PubMed]

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

Fried, D. L.

Giacomelli, G.

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).
[CrossRef] [PubMed]

Hajnal, J. V.

J. F. Nye, J. V. Hajnal, J. H. Hannay, “Phase saddles and dislocations in two-dimensional waves such as tides,” Proc. R. Soc. London Ser. A 417, 7–20 (1988).
[CrossRef]

Hannay, J. H.

J. F. Nye, J. V. Hajnal, J. H. Hannay, “Phase saddles and dislocations in two-dimensional waves such as tides,” Proc. R. Soc. London Ser. A 417, 7–20 (1988).
[CrossRef]

Heckenberg, N. R.

N. R. Heckenberg, M. Vaupel, J. T. Malos, C. O. Weiss, “Optical-vortex pair creation and annichilation and helical astigmatism of a nonplanar ring resonator,” Phys. Rev. A 54, 2369–2378 (1996).
[CrossRef] [PubMed]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2, Chap. 20.

Lukin, V.

V. Lukin, B. Fortes, “The influence of wave front dislocations on phase conjugation instability at thermal blooming compensation,” Atmos. Oceanic Opt. 8, 223–230 (1995).

B. V. Fortes, V. Lukin, “The effects of wavefront dislocations on the atmospheric adaptive optical systems performance,” in Optics for Science and New Technology. Part II. Lasers and Laser Spectroscopy, J. Chang, J. Lee, C. Nan, eds., Proc. SPIE2778, 1002–1003 (1996).

Malos, J. T.

N. R. Heckenberg, M. Vaupel, J. T. Malos, C. O. Weiss, “Optical-vortex pair creation and annichilation and helical astigmatism of a nonplanar ring resonator,” Phys. Rev. A 54, 2369–2378 (1996).
[CrossRef] [PubMed]

Mamaev, A. V.

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

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

Marichev, O. I.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon & Breach, New York, 1990), Vol. 3, p. 40.

Martin, J. M.

Mayer, N. N.

V. A. Tartakovski, N. N. Mayer, “Focal spot in the presence of phase dislocations,” Atmos. Oceanic Opt. 9, 1457–1460 (1996).

V. A. Tartakovski, N. N. Mayer, “Phase dislocations and minimal phase representation of the wave function,” Atmos. Oceanic Opt. 8, 231–235 (1995).

Nye, J. F.

J. F. Nye, J. V. Hajnal, J. H. Hannay, “Phase saddles and dislocations in two-dimensional waves such as tides,” Proc. R. Soc. London Ser. A 417, 7–20 (1988).
[CrossRef]

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

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984), Chap. 11.

Parry, G.

Pilipetskii, N.

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

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

Prudnikov, A. P.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon & Breach, New York, 1990), Vol. 3, p. 40.

Pusey, P. N.

Ramazza, P. L.

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).
[CrossRef] [PubMed]

Residori, S.

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).
[CrossRef] [PubMed]

Shkukov, V. V.

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

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

Shvartsman, N.

I. Freund, N. Shvartsman, “Structural correlations in Gaussian random wave fields,” Phys. Rev. E 51, 3770–3773 (1995).
[CrossRef]

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

N. Shvartsman, I. Freund, “Vortices in random wave fields: nearest neighbor anticorrelations,” Phys. Rev. Lett. 72, 1008–1011 (1994).
[CrossRef] [PubMed]

Tartakovski, V. A.

V. A. Tartakovski, N. N. Mayer, “Focal spot in the presence of phase dislocations,” Atmos. Oceanic Opt. 9, 1457–1460 (1996).

V. A. Tartakovski, N. N. Mayer, “Phase dislocations and minimal phase representation of the wave function,” Atmos. Oceanic Opt. 8, 231–235 (1995).

Tatarski, V. I.

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation, NSF Report TT-68-50464 (U.S. National Science Foundation, Washington, D.C., 1968).

Tikhomirova, O.

V. Aksenov, V. Banakh, O. Tikhomirova, “Potential and vortex features of optical speckle fields,” Atmos. Oceanic Opt. 9, 1450–1456 (1996).

Vaughn, J. L.

Vaupel, M.

N. R. Heckenberg, M. Vaupel, J. T. Malos, C. O. Weiss, “Optical-vortex pair creation and annichilation and helical astigmatism of a nonplanar ring resonator,” Phys. Rev. A 54, 2369–2378 (1996).
[CrossRef] [PubMed]

Volostnikov, V.

E. Abramochkin, V. Volostnikov, “Spiral-type beams: optical and quantum aspects,” Opt. Commun. 125, 302–323 (1996).
[CrossRef]

Weiss, C. O.

N. R. Heckenberg, M. Vaupel, J. T. Malos, C. O. Weiss, “Optical-vortex pair creation and annichilation and helical astigmatism of a nonplanar ring resonator,” Phys. Rev. A 54, 2369–2378 (1996).
[CrossRef] [PubMed]

Zel’dovich, B. Ya.

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

N. B. Baranova, B. Ya. Zel’dovich, “Dislocations of the wave-front surface and zeros of the amplitude,” Zh. Eksp. Teor. Fiz. 80, 1789–1797 (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).

Appl. Opt.

Atmos. Oceanic Opt.

V. Aksenov, V. Banakh, O. Tikhomirova, “Potential and vortex features of optical speckle fields,” Atmos. Oceanic Opt. 9, 1450–1456 (1996).

V. A. Tartakovski, N. N. Mayer, “Focal spot in the presence of phase dislocations,” Atmos. Oceanic Opt. 9, 1457–1460 (1996).

V. Lukin, B. Fortes, “The influence of wave front dislocations on phase conjugation instability at thermal blooming compensation,” Atmos. Oceanic Opt. 8, 223–230 (1995).

V. A. Tartakovski, N. N. Mayer, “Phase dislocations and minimal phase representation of the wave function,” Atmos. Oceanic Opt. 8, 231–235 (1995).

IEEE Trans. Antennas Propag.

R. L. Fante, “Propagation of electromagnetic waves through turbulent plasma using transport theory,” IEEE Trans. Antennas Propag. AP-21, 750–755 (1975).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. A

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

JETP Lett.

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.

E. Abramochkin, V. Volostnikov, “Spiral-type beams: optical and quantum aspects,” Opt. Commun. 125, 302–323 (1996).
[CrossRef]

I. Freund, “Vortex derivatives,” Opt. Commun. 137, 118–126 (1997).
[CrossRef]

Phys. Lett. A

I. Freund, “Amplitude topological singularities in random electromagnetic wavefields,” Phys. Lett. A 198, 139–144 (1995).
[CrossRef]

Phys. Rev. A

N. R. Heckenberg, M. Vaupel, J. T. Malos, C. O. Weiss, “Optical-vortex pair creation and annichilation and helical astigmatism of a nonplanar ring resonator,” Phys. Rev. A 54, 2369–2378 (1996).
[CrossRef] [PubMed]

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

Phys. Rev. E

I. Freund, N. Shvartsman, “Structural correlations in Gaussian random wave fields,” Phys. Rev. E 51, 3770–3773 (1995).
[CrossRef]

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

Phys. Rev. Lett.

N. Shvartsman, I. Freund, “Vortices in random wave fields: nearest neighbor anticorrelations,” Phys. Rev. Lett. 72, 1008–1011 (1994).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. A

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

J. F. Nye, J. V. Hajnal, J. H. Hannay, “Phase saddles and dislocations in two-dimensional waves such as tides,” Proc. R. Soc. London Ser. A 417, 7–20 (1988).
[CrossRef]

Zh. Eksp. Teor. Fiz.

N. B. Baranova, B. Ya. Zel’dovich, “Dislocations of the wave-front surface and zeros of the amplitude,” Zh. Eksp. Teor. Fiz. 80, 1789–1797 (1981).

Other

B. V. Fortes, V. Lukin, “The effects of wavefront dislocations on the atmospheric adaptive optical systems performance,” in Optics for Science and New Technology. Part II. Lasers and Laser Spectroscopy, J. Chang, J. Lee, C. Nan, eds., Proc. SPIE2778, 1002–1003 (1996).

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon & Breach, New York, 1990), Vol. 3, p. 40.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2, Chap. 20.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984), Chap. 11.

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation, NSF Report TT-68-50464 (U.S. National Science Foundation, Washington, D.C., 1968).

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Figures (9)

Fig. 1
Fig. 1

Examples of (a) single linear vortex and (b) phase dislocation.

Fig. 2
Fig. 2

Notation used for the calculations.

Fig. 3
Fig. 3

Examples of phase dislocations obtained from simulations.

Fig. 4
Fig. 4

Dislocation density (top graph) and scintillation index (bottom graph) versus propagation length. Fifty phase screens were used.

Fig. 5
Fig. 5

Dislocation density (top graph) and variance of log-amplitude derivative (bottom graph) versus propagation length. The region of rapid growth of the density is shown. Seventy-five phase screens were used.

Fig. 6
Fig. 6

Dislocation density (top graph) and scintillation index (bottom graph) versus propagation length. 150 phase screens were used.

Fig. 7
Fig. 7

Dislocation density (top graph) and variance of log-amplitude derivative (bottom graph) versus propagation length. The region of rapid growth of the density is shown. 250 phase screens were used.

Fig. 8
Fig. 8

Dislocation density versus wavelength. Twenty-five phase screens were used.

Fig. 9
Fig. 9

Dislocation density versus inner scale of the turbulence. Twenty-five phase screens were used.

Equations (33)

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E z ,   ρ = A z ,   ρ   exp iS z ,   ρ , A z ,   ρ = E 1 2 z ,   ρ + E 2 2 z ,   ρ 1 / 2 , S z ,   ρ = const .   +   atan 2 E 2 z ,   ρ ,   E 1 z ,   ρ ,
S ρ = atan 2 E 20 + E 2 x x + E 2 y y ,   E 10 + E 1 x x + E 1 y y , ρ = x ,   y ,
S x ,   y = atan 2 y ,   x .
S x ,   y = atan 2 y ,   x - x 0 + atan 2 y ,   x + x 0 .
λ 0 = lim r 0 P 0 G r π r 2 .
E 1 x ,   y = E 10 + E 1 x x + E 1 y y , E 2 x ,   y = E 20 + E 2 x x + E 2 y y ,
E 10 + E 1 x x 0 + E 1 y y 0 = 0 , E 20 + E 2 x x 0 + E 2 y y 0 = 0 .
x 0 = - E 10 E 2 y - E 20 E 1 y E 1 x E 2 y - E 1 y E 2 x ,     y 0 = E 10 E 2 x - E 20 E 1 x E 1 x E 2 y - E 1 y E 2 x .
E 1 = A   cos   S ,     E 2 = A   sin   S ,     χ = ln A / A 0 ,
x 0 = - S y χ x S y - χ y S x ,     y 0 = S x χ x S y - χ y S x ,
tan   φ = - S x / S y .
x 0 = - 1 / χ x ,     y 0 = 0 .
P 0 G r = 2   1 / r d χ x W χ x χ x ,
λ d = lim r 0 1 π r 2 1 / r d χ x W χ x χ x .
β R 2 = I 2 / I 2 - 1 = 8.7 C n 2 k 2 L κ m - 5 / 3 - 1 + 6 11   D 5 / 6 1 + 1 D 2 11 / 12 × sin 11 6 arctan D , D = L κ m 2 k ,     κ m = 5.92 l 0 ,
β R 2 = 1.23 C n 2 k 7 / 6 L 11 / 6 .
W χ x χ x = 0 d A     d A x W 2 A ,   A x δ χ x - A x A = 0 d AAW 2 A ,   A χ x ,
W 2 A ,   A x = 1 2 π I σ A x A   exp - A 2 2 I - A x 2 2 σ A x 2 ,
W χ x χ x = σ A x 2 I χ x 2 + σ A x 2 I - 3 / 2 .
λ d = 1 2 π σ A x 2 I .
W χ x χ x = 1 2 π σ χ x exp - χ x 2 2 σ χ x 2 ,
λ d = lim r 0 1 2 π r 2 1 - erf 1 2 σ χ x r ,
W 2 A ,   A x = 4 I - α + 1 / 2 α α + 1 / 2 σ A x 2 π Γ α × A α K α - 1 2 α I A exp - A x 2 2 σ A x 2 ,
α = 2 β 2 - 1 ,     β 2 > 1 .
W χ x χ x = 2 α - 2 / 2 I - α / 2 σ A x α α α / 2 Γ α + 1 / 2 Γ α   | χ x | - α - 1 × exp α σ A x 2 χ x 2 I W - α + 1 / 2 , α - 1 / 2 2 α σ A x 2 χ x 2 I ,
λ d = σ A x 2 π 3 - β 2 I ,     1 < β 2 < 3 .
λ d = η 1 - erf 1 2 σ χ x 2 r c ,
b χ ρ 1 - 12.3 ρ 2 λ L - 5 / 6 l 0 - 1 / 3 = 1 - 1.47 ρ 2 D 1 / 6 k / L ,
r c = 0.82 aD - 1 / 12 L / k ,
η = b σ χ x 2 D α ,
λ d = D - 1 / 12 σ χ x 2 2 π 2 1 - erf π D 1 / 12 4 σ χ x L / k .
2 ik   E z ,   ρ z + Δ E z ,   ρ + 2 k 2 ñ z ,   ρ E z ,   ρ = 0 ,
Φ n κ = 0.033 C n 2 κ - 11 / 3 exp - κ 2 / κ m 2 , κ m = 5.92 / l 0 ,

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