Abstract

We present a detailed investigation, qualitative and quantitative, on how the atmospheric turbulence with a non-Kolmogorov power spectrum affects the major statistics of stochastic electromagnetic beams, such as the spectral composition and the states of coherence and polarization. We suggest a detailed survey on how these properties evolve on propagation of beams generated by electromagnetic Gaussian Schell-model sources, depending on the fractal constant α of the atmospheric power spectrum.

© 2010 Optical Society of America

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  9. F. Daludier and C. Sidi, “Direct evidence of sheets in the atmospheric temperature field,” J. Atmos. Sci. 51, 237-248 (1994).
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  10. H. Luce, F. Daludier, M. Crochet, and C. Sidi, “Direct comparison between in situ and VHF oblique radar measurements of refractive index spectra: a new successful attempt,” Radio Sci. 31, 1487-1500 (1996).
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  12. A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66-77 (2008).
    [CrossRef]
  13. M. S. Belenkii, S. J. Karis, J. M. BrownII, and R. Q. Fugate, “Experimental study of the effect of non Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
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  16. B. Joseph, A. Mahalov, B. Nicolaenko, and K. L. Tse, “Variability of turbulence and its outer scales in a model tropopause jet,” J. Atm. Sci. 61, 621-643 (2004).
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  17. A. Mahalov, B. Nicolaenko, K.L. Tse, and B. Joseph, “Eddy mixing in jet-stream turbulence under stronger stratification,” Geophys. Res. Lett. 31, L23111 (2004).
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  18. C. Rao, W. Jiang, and N. Ling, “Adaptive-Optics Compensation by Distributed Beacons for Non-Kolmogorov Turbulence,” Appl. Opt. 40, 3441–3449 (2001).
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  19. O. Korotkova, N. Farwell, and A. Mahalov, “The effect of the jet-stream on the intensity of laser beams propagating along slanted paths in the upper layers of the turbulent atmosphere,” Waves in Random Media 19, 692–702 (2009).
    [CrossRef]
  20. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Engineering 47, 026003 (February 2008).
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    [CrossRef]
  23. W. Du, S. Yu, L. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctiations for wave propagation trough non-Kolmogorov turbulence,” Opt. Commun. 282, 705–708 (2009).
    [CrossRef]
  24. L. Tan,W. Du, J. Ma, S. Yu, and Q. Han, “Log-amplitude variance for a Gaussian-beam wave propagating trough non-Kolmogorov turbulence,” Opt. Express 18, 451–461 (2010).
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  26. A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47, 6385–6391 (2008).
    [CrossRef] [PubMed]
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    [CrossRef]
  28. O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of partially coherent beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225–230 (2004).
    [CrossRef]
  29. M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent EM beams propagating through turbulent atmosphere,” Waves in Random Media 14, 513–523 (2004).
    [CrossRef]
  30. O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere,” Waves Random Complex Media 15, 353–364 (2005).
    [CrossRef]
  31. X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15, 16909–16915 (2007).
    [CrossRef] [PubMed]
  32. W. Gao, “Changes of polarization of light beams on propagation trough tissue,” Opt. Commun. 260, 749–754 (2006).
    [CrossRef]
  33. W. Gao and O. Korotkova, “Changes in the state of polarization of a random electro-magnetic beam propagating through tissue,” Opt. Commun. 270, 474–478 (2007).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  38. H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005).
    [CrossRef]
  39. W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271, 1–8 (2007).
    [CrossRef]
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  41. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342-2348 (2008).
  42. O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94, 681-690 (2009).
    [CrossRef]

2010 (2)

2009 (4)

O. Korotkova, N. Farwell, and A. Mahalov, “The effect of the jet-stream on the intensity of laser beams propagating along slanted paths in the upper layers of the turbulent atmosphere,” Waves in Random Media 19, 692–702 (2009).
[CrossRef]

W. Du, S. Yu, L. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctiations for wave propagation trough non-Kolmogorov turbulence,” Opt. Commun. 282, 705–708 (2009).
[CrossRef]

J. Pu and O. Korotkova, “Propagation of the degree of cross-polarization in the turbulent atmosphere” Opt. Commun. 282, 1691–1698 (2009).
[CrossRef]

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94, 681-690 (2009).
[CrossRef]

2008 (4)

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342-2348 (2008).

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47, 6385–6391 (2008).
[CrossRef] [PubMed]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Engineering 47, 026003 (February 2008).
[CrossRef]

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66-77 (2008).
[CrossRef]

2007 (3)

X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15, 16909–16915 (2007).
[CrossRef] [PubMed]

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271, 1–8 (2007).
[CrossRef]

W. Gao and O. Korotkova, “Changes in the state of polarization of a random electro-magnetic beam propagating through tissue,” Opt. Commun. 270, 474–478 (2007).
[CrossRef]

2006 (1)

W. Gao, “Changes of polarization of light beams on propagation trough tissue,” Opt. Commun. 260, 749–754 (2006).
[CrossRef]

2005 (2)

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005).
[CrossRef]

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere,” Waves Random Complex Media 15, 353–364 (2005).
[CrossRef]

2004 (4)

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of partially coherent beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225–230 (2004).
[CrossRef]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent EM beams propagating through turbulent atmosphere,” Waves in Random Media 14, 513–523 (2004).
[CrossRef]

B. Joseph, A. Mahalov, B. Nicolaenko, and K. L. Tse, “Variability of turbulence and its outer scales in a model tropopause jet,” J. Atm. Sci. 61, 621-643 (2004).
[CrossRef]

A. Mahalov, B. Nicolaenko, K.L. Tse, and B. Joseph, “Eddy mixing in jet-stream turbulence under stronger stratification,” Geophys. Res. Lett. 31, L23111 (2004).
[CrossRef]

2001 (2)

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schellmodel beams,” J. Opt. A: Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

C. Rao, W. Jiang, and N. Ling, “Adaptive-Optics Compensation by Distributed Beacons for Non-Kolmogorov Turbulence,” Appl. Opt. 40, 3441–3449 (2001).
[CrossRef]

2000 (1)

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[CrossRef]

1999 (1)

M. S. Belenkii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396-406 (1999).
[CrossRef]

1997 (1)

M. S. Belenkii, S. J. Karis, J. M. BrownII, and R. Q. Fugate, “Experimental study of the effect of non Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[CrossRef]

1996 (1)

H. Luce, F. Daludier, M. Crochet, and C. Sidi, “Direct comparison between in situ and VHF oblique radar measurements of refractive index spectra: a new successful attempt,” Radio Sci. 31, 1487-1500 (1996).
[CrossRef]

1995 (3)

1994 (3)

D. F. V. James, “Change of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11, 1641-1649 (1994).
[CrossRef]

D. T. Kyrazis, J. Wissler, D. D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[CrossRef]

F. Daludier and C. Sidi, “Direct evidence of sheets in the atmospheric temperature field,” J. Atmos. Sci. 51, 237-248 (1994).
[CrossRef]

1989 (1)

F. Dalaudier, M. Crochet, and C. Sidi, “Direct comparison between in situ and radar measurements of temperature fluctuation spectra: a puzzling results,” Radio Sci. 24, 311-324 (1989).
[CrossRef]

1972 (1)

1970 (1)

1941 (2)

A. N. Kolmogorov, “The local structure of turbulence in an incompressible viscous fluid for very large Reynolds numbers,” C. R. Acad. Sci. URSS 30, 301–305 (1941).

A. N. Kolmogorov, “Dissipation of energy in the locally isotropic turbulence,” C. R. Acad. Sci. URSS 32, 16–18 (1941).

Andrews, L. C.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Engineering 47, 026003 (February 2008).
[CrossRef]

Barchers, J. D.

M. S. Belenkii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396-406 (1999).
[CrossRef]

Belenkii, M. S.

M. S. Belenkii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396-406 (1999).
[CrossRef]

M. S. Belenkii, S. J. Karis, J. M. BrownII, and R. Q. Fugate, “Experimental study of the effect of non Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[CrossRef]

A. S. Gurvich and M. S. Belenkii, “Influence of stratospheric turbulence on infrared imaging,” J. Opt. Soc. Am. A 12, 2517-2522 (1995).
[CrossRef]

M. S. Belenkii, “Influence of stratospheric turbulence on infrared imaging,” J. Opt. Soc. Am. A 12, 2517-2522 (1995).
[CrossRef]

Bishop, K. P.

D. T. Kyrazis, J. Wissler, D. D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[CrossRef]

Borghi, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schellmodel beams,” J. Opt. A: Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

Brown, J. M.

M. S. Belenkii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396-406 (1999).
[CrossRef]

M. S. Belenkii, S. J. Karis, J. M. BrownII, and R. Q. Fugate, “Experimental study of the effect of non Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[CrossRef]

Cai, Y.

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94, 681-690 (2009).
[CrossRef]

Clifford, S. F.

Crochet, M.

H. Luce, F. Daludier, M. Crochet, and C. Sidi, “Direct comparison between in situ and VHF oblique radar measurements of refractive index spectra: a new successful attempt,” Radio Sci. 31, 1487-1500 (1996).
[CrossRef]

F. Dalaudier, M. Crochet, and C. Sidi, “Direct comparison between in situ and radar measurements of temperature fluctuation spectra: a puzzling results,” Radio Sci. 24, 311-324 (1989).
[CrossRef]

Dalaudier, F.

F. Dalaudier, M. Crochet, and C. Sidi, “Direct comparison between in situ and radar measurements of temperature fluctuation spectra: a puzzling results,” Radio Sci. 24, 311-324 (1989).
[CrossRef]

Daludier, F.

H. Luce, F. Daludier, M. Crochet, and C. Sidi, “Direct comparison between in situ and VHF oblique radar measurements of refractive index spectra: a new successful attempt,” Radio Sci. 31, 1487-1500 (1996).
[CrossRef]

F. Daludier and C. Sidi, “Direct evidence of sheets in the atmospheric temperature field,” J. Atmos. Sci. 51, 237-248 (1994).
[CrossRef]

Dogariu, A.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere,” Waves Random Complex Media 15, 353–364 (2005).
[CrossRef]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent EM beams propagating through turbulent atmosphere,” Waves in Random Media 14, 513–523 (2004).
[CrossRef]

Du, W.

L. Tan,W. Du, J. Ma, S. Yu, and Q. Han, “Log-amplitude variance for a Gaussian-beam wave propagating trough non-Kolmogorov turbulence,” Opt. Express 18, 451–461 (2010).
[CrossRef] [PubMed]

W. Du, S. Yu, L. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctiations for wave propagation trough non-Kolmogorov turbulence,” Opt. Commun. 282, 705–708 (2009).
[CrossRef]

Du, X.

Farwell, N.

O. Korotkova, N. Farwell, and A. Mahalov, “The effect of the jet-stream on the intensity of laser beams propagating along slanted paths in the upper layers of the turbulent atmosphere,” Waves in Random Media 19, 692–702 (2009).
[CrossRef]

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Engineering 47, 026003 (February 2008).
[CrossRef]

Fugate, R. Q.

M. S. Belenkii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396-406 (1999).
[CrossRef]

M. S. Belenkii, S. J. Karis, J. M. BrownII, and R. Q. Fugate, “Experimental study of the effect of non Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[CrossRef]

Gao, W.

W. Gao and O. Korotkova, “Changes in the state of polarization of a random electro-magnetic beam propagating through tissue,” Opt. Commun. 270, 474–478 (2007).
[CrossRef]

W. Gao, “Changes of polarization of light beams on propagation trough tissue,” Opt. Commun. 260, 749–754 (2006).
[CrossRef]

Golbraikh, E.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47, 6385–6391 (2008).
[CrossRef] [PubMed]

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66-77 (2008).
[CrossRef]

Gori, F.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schellmodel beams,” J. Opt. A: Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

Guo, H.

Gurvich, A. S.

Han, Q.

James, D. F. V.

Jiang, W.

C. Rao, W. Jiang, and N. Ling, “Adaptive-Optics Compensation by Distributed Beacons for Non-Kolmogorov Turbulence,” Appl. Opt. 40, 3441–3449 (2001).
[CrossRef]

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[CrossRef]

Jiang, Y.

W. Du, S. Yu, L. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctiations for wave propagation trough non-Kolmogorov turbulence,” Opt. Commun. 282, 705–708 (2009).
[CrossRef]

Joseph, B.

B. Joseph, A. Mahalov, B. Nicolaenko, and K. L. Tse, “Variability of turbulence and its outer scales in a model tropopause jet,” J. Atm. Sci. 61, 621-643 (2004).
[CrossRef]

A. Mahalov, B. Nicolaenko, K.L. Tse, and B. Joseph, “Eddy mixing in jet-stream turbulence under stronger stratification,” Geophys. Res. Lett. 31, L23111 (2004).
[CrossRef]

Karis, S. J.

M. S. Belenkii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396-406 (1999).
[CrossRef]

M. S. Belenkii, S. J. Karis, J. M. BrownII, and R. Q. Fugate, “Experimental study of the effect of non Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[CrossRef]

Keating, D. D. B.

D. T. Kyrazis, J. Wissler, D. D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[CrossRef]

Kerr, J. R.

Kolmogorov, A. N.

A. N. Kolmogorov, “The local structure of turbulence in an incompressible viscous fluid for very large Reynolds numbers,” C. R. Acad. Sci. URSS 30, 301–305 (1941).

A. N. Kolmogorov, “Dissipation of energy in the locally isotropic turbulence,” C. R. Acad. Sci. URSS 32, 16–18 (1941).

Kopeika, N. S.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47, 6385–6391 (2008).
[CrossRef] [PubMed]

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66-77 (2008).
[CrossRef]

Korotkova, O.

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94, 681-690 (2009).
[CrossRef]

J. Pu and O. Korotkova, “Propagation of the degree of cross-polarization in the turbulent atmosphere” Opt. Commun. 282, 1691–1698 (2009).
[CrossRef]

O. Korotkova, N. Farwell, and A. Mahalov, “The effect of the jet-stream on the intensity of laser beams propagating along slanted paths in the upper layers of the turbulent atmosphere,” Waves in Random Media 19, 692–702 (2009).
[CrossRef]

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342-2348 (2008).

X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15, 16909–16915 (2007).
[CrossRef] [PubMed]

W. Gao and O. Korotkova, “Changes in the state of polarization of a random electro-magnetic beam propagating through tissue,” Opt. Commun. 270, 474–478 (2007).
[CrossRef]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005).
[CrossRef]

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere,” Waves Random Complex Media 15, 353–364 (2005).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of partially coherent beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225–230 (2004).
[CrossRef]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent EM beams propagating through turbulent atmosphere,” Waves in Random Media 14, 513–523 (2004).
[CrossRef]

Kupershmidt, I.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66-77 (2008).
[CrossRef]

Kyrazis, D. T.

D. T. Kyrazis, J. Wissler, D. D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[CrossRef]

Lawrence, R. S.

Ling, N.

C. Rao, W. Jiang, and N. Ling, “Adaptive-Optics Compensation by Distributed Beacons for Non-Kolmogorov Turbulence,” Appl. Opt. 40, 3441–3449 (2001).
[CrossRef]

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[CrossRef]

Liu, L.

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271, 1–8 (2007).
[CrossRef]

Lu, W.

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271, 1–8 (2007).
[CrossRef]

Luce, H.

H. Luce, F. Daludier, M. Crochet, and C. Sidi, “Direct comparison between in situ and VHF oblique radar measurements of refractive index spectra: a new successful attempt,” Radio Sci. 31, 1487-1500 (1996).
[CrossRef]

Luo, B.

Ma, J.

L. Tan,W. Du, J. Ma, S. Yu, and Q. Han, “Log-amplitude variance for a Gaussian-beam wave propagating trough non-Kolmogorov turbulence,” Opt. Express 18, 451–461 (2010).
[CrossRef] [PubMed]

W. Du, S. Yu, L. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctiations for wave propagation trough non-Kolmogorov turbulence,” Opt. Commun. 282, 705–708 (2009).
[CrossRef]

Mahalov, A.

O. Korotkova, N. Farwell, and A. Mahalov, “The effect of the jet-stream on the intensity of laser beams propagating along slanted paths in the upper layers of the turbulent atmosphere,” Waves in Random Media 19, 692–702 (2009).
[CrossRef]

B. Joseph, A. Mahalov, B. Nicolaenko, and K. L. Tse, “Variability of turbulence and its outer scales in a model tropopause jet,” J. Atm. Sci. 61, 621-643 (2004).
[CrossRef]

A. Mahalov, B. Nicolaenko, K.L. Tse, and B. Joseph, “Eddy mixing in jet-stream turbulence under stronger stratification,” Geophys. Res. Lett. 31, L23111 (2004).
[CrossRef]

Mondello, A.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schellmodel beams,” J. Opt. A: Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

Nicolaenko, B.

A. Mahalov, B. Nicolaenko, K.L. Tse, and B. Joseph, “Eddy mixing in jet-stream turbulence under stronger stratification,” Geophys. Res. Lett. 31, L23111 (2004).
[CrossRef]

B. Joseph, A. Mahalov, B. Nicolaenko, and K. L. Tse, “Variability of turbulence and its outer scales in a model tropopause jet,” J. Atm. Sci. 61, 621-643 (2004).
[CrossRef]

Ochs, G. R.

Osmon, C. L.

M. S. Belenkii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396-406 (1999).
[CrossRef]

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Engineering 47, 026003 (February 2008).
[CrossRef]

Piquero, G.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schellmodel beams,” J. Opt. A: Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

Preble, A. J.

D. T. Kyrazis, J. Wissler, D. D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[CrossRef]

Pu, J.

J. Pu and O. Korotkova, “Propagation of the degree of cross-polarization in the turbulent atmosphere” Opt. Commun. 282, 1691–1698 (2009).
[CrossRef]

Rao, C.

C. Rao, W. Jiang, and N. Ling, “Adaptive-Optics Compensation by Distributed Beacons for Non-Kolmogorov Turbulence,” Appl. Opt. 40, 3441–3449 (2001).
[CrossRef]

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[CrossRef]

Roggemann, M. C.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[CrossRef]

Roychowdhury, H.

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005).
[CrossRef]

Salem, M.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere,” Waves Random Complex Media 15, 353–364 (2005).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of partially coherent beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225–230 (2004).
[CrossRef]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent EM beams propagating through turbulent atmosphere,” Waves in Random Media 14, 513–523 (2004).
[CrossRef]

Santarsiero, M.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schellmodel beams,” J. Opt. A: Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

Shtemler, Y.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66-77 (2008).
[CrossRef]

Sidi, C.

H. Luce, F. Daludier, M. Crochet, and C. Sidi, “Direct comparison between in situ and VHF oblique radar measurements of refractive index spectra: a new successful attempt,” Radio Sci. 31, 1487-1500 (1996).
[CrossRef]

F. Daludier and C. Sidi, “Direct evidence of sheets in the atmospheric temperature field,” J. Atmos. Sci. 51, 237-248 (1994).
[CrossRef]

F. Dalaudier, M. Crochet, and C. Sidi, “Direct comparison between in situ and radar measurements of temperature fluctuation spectra: a puzzling results,” Radio Sci. 24, 311-324 (1989).
[CrossRef]

Simon, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schellmodel beams,” J. Opt. A: Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

Stribling, B. E.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[CrossRef]

Sun, J.

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271, 1–8 (2007).
[CrossRef]

Tan, L.

L. Tan,W. Du, J. Ma, S. Yu, and Q. Han, “Log-amplitude variance for a Gaussian-beam wave propagating trough non-Kolmogorov turbulence,” Opt. Express 18, 451–461 (2010).
[CrossRef] [PubMed]

W. Du, S. Yu, L. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctiations for wave propagation trough non-Kolmogorov turbulence,” Opt. Commun. 282, 705–708 (2009).
[CrossRef]

Toselli, I.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Engineering 47, 026003 (February 2008).
[CrossRef]

Tse, K. L.

B. Joseph, A. Mahalov, B. Nicolaenko, and K. L. Tse, “Variability of turbulence and its outer scales in a model tropopause jet,” J. Atm. Sci. 61, 621-643 (2004).
[CrossRef]

Tse, K.L.

A. Mahalov, B. Nicolaenko, K.L. Tse, and B. Joseph, “Eddy mixing in jet-stream turbulence under stronger stratification,” Geophys. Res. Lett. 31, L23111 (2004).
[CrossRef]

Virtser, A.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66-77 (2008).
[CrossRef]

Watson, E.

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94, 681-690 (2009).
[CrossRef]

Welsh, B. M.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[CrossRef]

Wissler, J.

D. T. Kyrazis, J. Wissler, D. D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[CrossRef]

Wolf, E.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere,” Waves Random Complex Media 15, 353–364 (2005).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of partially coherent beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225–230 (2004).
[CrossRef]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent EM beams propagating through turbulent atmosphere,” Waves in Random Media 14, 513–523 (2004).
[CrossRef]

Wu, G.

Xie, W.

W. Du, S. Yu, L. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctiations for wave propagation trough non-Kolmogorov turbulence,” Opt. Commun. 282, 705–708 (2009).
[CrossRef]

Yang, Q.

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271, 1–8 (2007).
[CrossRef]

Yu, S.

Zhao, D.

Zhu, Y.

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271, 1–8 (2007).
[CrossRef]

Zilberman, A.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47, 6385–6391 (2008).
[CrossRef] [PubMed]

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66-77 (2008).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (1)

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94, 681-690 (2009).
[CrossRef]

Atmos. Res. (1)

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66-77 (2008).
[CrossRef]

C. R. Acad. Sci. URSS (2)

A. N. Kolmogorov, “The local structure of turbulence in an incompressible viscous fluid for very large Reynolds numbers,” C. R. Acad. Sci. URSS 30, 301–305 (1941).

A. N. Kolmogorov, “Dissipation of energy in the locally isotropic turbulence,” C. R. Acad. Sci. URSS 32, 16–18 (1941).

Geophys. Res. Lett. (1)

A. Mahalov, B. Nicolaenko, K.L. Tse, and B. Joseph, “Eddy mixing in jet-stream turbulence under stronger stratification,” Geophys. Res. Lett. 31, L23111 (2004).
[CrossRef]

J. Atm. Sci. (1)

B. Joseph, A. Mahalov, B. Nicolaenko, and K. L. Tse, “Variability of turbulence and its outer scales in a model tropopause jet,” J. Atm. Sci. 61, 621-643 (2004).
[CrossRef]

J. Atmos. Sci. (1)

F. Daludier and C. Sidi, “Direct evidence of sheets in the atmospheric temperature field,” J. Atmos. Sci. 51, 237-248 (1994).
[CrossRef]

J. Mod. Opt. (1)

C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schellmodel beams,” J. Opt. A: Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Opt. Commun. (8)

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005).
[CrossRef]

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271, 1–8 (2007).
[CrossRef]

W. Gao, “Changes of polarization of light beams on propagation trough tissue,” Opt. Commun. 260, 749–754 (2006).
[CrossRef]

W. Gao and O. Korotkova, “Changes in the state of polarization of a random electro-magnetic beam propagating through tissue,” Opt. Commun. 270, 474–478 (2007).
[CrossRef]

W. Du, S. Yu, L. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctiations for wave propagation trough non-Kolmogorov turbulence,” Opt. Commun. 282, 705–708 (2009).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of partially coherent beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225–230 (2004).
[CrossRef]

J. Pu and O. Korotkova, “Propagation of the degree of cross-polarization in the turbulent atmosphere” Opt. Commun. 282, 1691–1698 (2009).
[CrossRef]

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342-2348 (2008).

Opt. Engineering (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Engineering 47, 026003 (February 2008).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Proc. SPIE (4)

M. S. Belenkii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396-406 (1999).
[CrossRef]

M. S. Belenkii, S. J. Karis, J. M. BrownII, and R. Q. Fugate, “Experimental study of the effect of non Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[CrossRef]

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[CrossRef]

D. T. Kyrazis, J. Wissler, D. D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[CrossRef]

Radio Sci. (2)

F. Dalaudier, M. Crochet, and C. Sidi, “Direct comparison between in situ and radar measurements of temperature fluctuation spectra: a puzzling results,” Radio Sci. 24, 311-324 (1989).
[CrossRef]

H. Luce, F. Daludier, M. Crochet, and C. Sidi, “Direct comparison between in situ and VHF oblique radar measurements of refractive index spectra: a new successful attempt,” Radio Sci. 31, 1487-1500 (1996).
[CrossRef]

Waves Random Complex Media (1)

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere,” Waves Random Complex Media 15, 353–364 (2005).
[CrossRef]

Waves in Random Media (2)

O. Korotkova, N. Farwell, and A. Mahalov, “The effect of the jet-stream on the intensity of laser beams propagating along slanted paths in the upper layers of the turbulent atmosphere,” Waves in Random Media 19, 692–702 (2009).
[CrossRef]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent EM beams propagating through turbulent atmosphere,” Waves in Random Media 14, 513–523 (2004).
[CrossRef]

Other (5)

V. I. Tatarski, Wave Propagation in a Turbulent Medium (Nauka, Moscow, 1967).

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E–1–12 (2007).
[CrossRef]

E. Wolf, Intoduction to the Theories of Coherence and Polarization of Light (Cambridge University Press, Cambridge, 2007).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).

E. Wolf, “The influence of Young’s interferences experiment on the development of statistical optics,” in Progress in Optics, E. Wolf, ed. (Elsevier B. V., 2007), 50 pp. 251–273.

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Figures (6)

Fig. 1.
Fig. 1.

The normalized spectral density SN and the spectral degree of polarization P as functions of α for r = 0, z = 1 km, Ax = 1.

Fig. 2.
Fig. 2.

The normalized spectral density SN as a function of distance z for r = 0, Ax = 1, and different α: α = 3.01 (solid curve), α = 3.1 (dashed curve), α = 3.67 (dotted curve) and α = 4.99 (dot-dashed curve); (a) on a semilog scale, (b) on a log scale.

Fig. 3.
Fig. 3.

The spectral degree of polarization P as a function of distance z for r = 0, Ax = 1, and different α: α = 3.01 (solid curve), α = 3.1 (dashed curve), α = 3.67 (dotted curve) and α = 4.99 (dot-dashed curve); (a) on a semilog scale, (b) on a log scale.

Fig. 4.
Fig. 4.

The absolute value of the spectral degree of coherence as a function of distance z for r = 10-3 m, Ax = 1.3 and different α: α = 3.01 (solid curve), α = 3.1 (dashed curve), α = 3.67 (dotted curve) and α = 4.99 (dot-dashed curve); (a) on a basic scale, (b) on a log scale.

Fig. 5.
Fig. 5.

The absolute value of the spectral degree of coherence as a function of α for z = 1 km, Ax = 1.3, r = 5×10-4 m (solid curve), r = 10-3 m (dashed curve), r = 5×10-3 m (dotted curve) and r = 10-2 m (dot-dashed curve).

Fig. 6.
Fig. 6.

The absolute value of the spectral degree of coherence as a function of r for z = 1 km, Ax = 1.3 and different α: α = 3.01 (solid curve), α = 3.1 (dashed curve), α = 3.67 (dotted curve) and α = 4.99 (dot-dashed curve).

Equations (18)

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

W0(r10,r20;ω)=[Ei*(r10;ω)Ej(r20;ω)],(i,j=x,y) ,
Wij(r1,r2;ω)=(k2πz)2∫∫∫∫Wij0(r10,r20;ω)K(r10,r20,r1,r2;ω)d2r10d2r20,
K(r10,r20,r1,r2;ω)=exp[ik(r1r10)2(r2r20)22z]
×exp{π2k2z3[(r1r2)2+(r1r2)(r10r20)+(r10r20)2]0κ3Φn(κ)dκ},
Φn(κ)=A(α)C̃n2exp[κ2/κm2](κ2+κ02)α/2,0κ<,3<α<5,
c(α)=[Γ(5α2)A(α)23π]1/(α5).
A(α)=14π2 Γ (α1) cos (απ2) ,
I=0κ3Φn(κ)dκ=A(α)2(α2)C̃n2[κm2αβexp(κ02κm2)Γ(2α2,κ02κm2)2κ04α],
S(r;ω)=TrW(r,r;ω),
η(r1,r2;ω)=TrW(r1,r2;ω)S(r1;ω)S(r2;ω),
P(r;ω)=14DetW(r,r;ω)[TrW(r,r;ω)]2.
Wij0(r10,r20;ω)=BijIiIjexp[(r10)2+(r20)24σ2]exp[(r10r20)22δij2],
Bxx=Byy=1,Bxy=Byx,φxy=φyx,δxy=δyx,
max{δxx;δyy}δxymin{δxxBxy;δyyBxy}.
Wij(r1,r2;ω)=BijIiIjΔij2(z)exp[(r1+r2)28σ2Δij2(z)]exp[ik(r22r12)2Rij(z)]
×exp{[12Δij2(z)(14σ2+1δij2)+13π2k2zI(1+σ2)π4k2z4I218σ2Δij2(z)](r1r2)2},
Δij2(z)=1+z2k2σ2 (14σ2+1δij2) +2π2z3I3σ2 ,
Rij(z)=σ2Δij2(z)zσ2Δij2(z)+13π2z3Iσ2,

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