Analysis of angle of arrival fluctuations for optical waves’ propagation through weak anisotropic non-Kolmogorov turbulence

Linyan Cui

doi:10.1364/OE.23.006313

Analysis of angle of arrival fluctuations for optical waves’ propagation through weak anisotropic non-Kolmogorov turbulence

Linyan Cui

Optics Express
Vol. 23,
Issue 5,
pp. 6313-6325
(2015)
•https://doi.org/10.1364/OE.23.006313

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Linyan Cui, "Analysis of angle of arrival fluctuations for optical waves’ propagation through weak anisotropic non-Kolmogorov turbulence," Opt. Express 23, 6313-6325 (2015)

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Related Topics
Table of Contents Category
- Atmospheric and Oceanic Optics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Atmospheric optics (010.1290)
- Atmospheric propagation (010.1300)
- Atmospheric turbulence (010.1330)

About this Article
History
- Original Manuscript: October 29, 2014
- Revised Manuscript: February 6, 2015
- Manuscript Accepted: February 12, 2015
- Published: March 2, 2015

Accessible

Open Access

Abstract

Analytical expressions for the variance of angle of arrival (AOA) fluctuations based on the Rytov approximation theory are derived for plane and spherical waves’ propagation through weak anisotropic non-Kolmogorov turbulence atmosphere. The anisotropic spectrum model based on the assumption of circular symmetry in the orthogonal plane throughout the path is adopted and it includes the same degree of anisotropy along the direction of propagation for all the turbulence cells size in the inertial sub-range. The derived expressions consider a single anisotropic coefficient describing the turbulence anisotropic property and a general spectral power law value in the range 3 to 4. They reduce correctly to the previously published analytic expressions for the cases of plane and spherical waves’ propagation through weak isotropic non-Kolmogorov turbulence for the special case of anisotropic factor equaling one. To reduce the complexity of the analytical results, the asymptotic-fit expressions are also derived and they fit well with the close-form ones. These results are useful for understanding the potential impact of deviations from the standard isotropic non-Kolmogorov turbulence atmosphere.

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References

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D. T. Kyrazis, J. B. Wissler, 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]
M. S. Belen’kii, S. J. Karis, J. M. Brown, 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]
M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U (2006).
[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(1), 66–77 (2008).
[Crossref]
A. S. Gurvich and M. S. Belen’kii, “Influence of stratospheric turbulence on infrared imaging,” J. Opt. Soc. Am. A 12, 2517–2522 (1995).
M. S. Belen’kii, “Effect of the stratosphere on star image motion,” Opt. Lett. 20(12), 1359–1361 (1995).
[Crossref] [PubMed]
B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in Non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[Crossref]
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 (2007).
[Crossref]
B. D. Xue, L. Y. Cui, W. F. Xue, X. Z. Bai, and F. G. Zhou, “Theoretical expressions of the angle-of-arrival variance for optical waves propagating through non-Kolmogorov turbulence,” Opt. Express 19(9), 8433–8443 (2011).
[Crossref] [PubMed]
L. Y. Tan, W. H. Du, and J. Ma, “Effect of the outer scale on the angle of arrival variance for free-space-laser beam corrugated by non-Kolmogorov turbulence,” J. Russ. Laser Res. 30(6), 552–559 (2009).
[Crossref]
L. Y. Cui, B. D. Xue, S. L. Zheng, W. F. Xue, X. Z. Bai, X. G. Cao, and F. G. Zhou, “Atmospheric spectral model and theoretical expressions of irradiance scintillation index for optical wave propagating through moderate-to-strong non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 29(6), 1091–1098 (2012).
[Crossref] [PubMed]
X. Yi, Z. J. Liu, and P. Yue, “Inner- and outer-scale effects on the scintillation index of an optical wave propagating through moderate-to-strong non-Kolmogorov turbulence,” Opt. Express 20(4), 4232–4247 (2012).
[Crossref] [PubMed]
L. Y. Cui, B. D. Xue, and F. G. Zhou, “Analytical expressions for the angle of arrival fluctuations for optical waves’ propagation through moderate-to-strong non-Kolmogorov refractive turbulence,” J. Opt. Soc. Am. A 30(11), 2188–2195 (2013).
[Crossref] [PubMed]
L. Y. Cui, B. D. Xue, X. G. Cao, and F. G. Zhou, “Angle of arrival fluctuations considering turbulence outer scale for optical waves’ propagation through moderate-to-strong non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 31(4), 829–835 (2014).
[Crossref] [PubMed]
R. M. Manning, “An anisotropic turbulence model for wave propagation near the surface of the Earth,” IEEE Trans. Antenn. Propag. 34(2), 258–261 (1986).
[Crossref]
F. D. Eaton and G. D. Nastrom, “Preliminary estimates of the vertical profiles of inner and outer scales from White Sands Missile Range, New Mexico, VHF radar observations,” Radio Sci. 33(4), 895–903 (1998).
[Crossref]
G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]
L. Biferale and I. Procaccia, “Anisotropic contribution to the statistics of the atmospheric boundary layer,” Phys. Rep. 414, 43–164 (2005).
[Crossref]
M. S. Belen’kii, 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. Belen’kii, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999).
[Crossref]
C. Robert, J. M. Conan, V. Michau, J. B. Renard, C. Robert, and F. Dalaudier, “Retrieving parameters of the anisotropic refractive index fluctuations spectrum in the stratosphere from balloon-borne observations of stellar scintillation,” J. Opt. Soc. Am. A 25(2), 379–393 (2008).
[Crossref] [PubMed]
L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinova, V. P. Lukin, A. P. Rostov, B. V. Fortes, and A. P. Yankov, “Investigation of turbulence spectrum anisotropy in the ground atmospheric layer; preliminary results,” Atmos. Oceanic Opt. 8, 993–996 (1995).
M. S. Belen’kii, 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]
A. D. Wheelon, Electromagnetic Scintillation I. Geometric Optics (Cambridge University, 2001).
I. Toselli, B. Agrawal, and S. Restaino, “Light propagation through anisotropic turbulence,” J. Opt. Soc. Am. A 28(3), 483–488 (2011).
[Crossref] [PubMed]
V. S. Gudimetla, R. B. Holmes, and J. F. Riker, “Analytical expressions for the log-amplitude correlation function for plane wave propagation in anisotropic non-Kolmogorov refractive turbulence,” J. Opt. Soc. Am. A 29(12), 2622–2627 (2012).
[Crossref] [PubMed]
L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013).
[Crossref]
V. S. Gudimetla, R. B. Holmes, and J. F. Riker, “Analytical expressions for the log-amplitude correlation function for spherical wave propagation through anisotropic non-Kolmogorov atmosphere,” J. Opt. Soc. Am. A 31(1), 148–154 (2014).
[Crossref] [PubMed]
L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]
I. Toselli, “Introducing the concept of anisotropy at different scales for modeling optical turbulence,” J. Opt. Soc. Am. A 31(8), 1868–1875 (2014).
[Crossref] [PubMed]
Y. Cheon and A. Muschinski, “Closed-form approximations for the angle-of-arrival variance of plane and spherical waves propagating through homogeneous and isotropic turbulence,” J. Opt. Soc. Am. A 24(2), 415–422 (2007).
[Crossref] [PubMed]
L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Optical Engineering, 1998).
L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering, 2005).
W. H. Du, S. Yu, L. Y. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctuations for wave propagation through non-Kolmogorov turbulence,” Opt. Commun. 282(5), 705–708 (2009).
[Crossref]

2014 (4)

L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]

V. S. Gudimetla, R. B. Holmes, and J. F. Riker, “Analytical expressions for the log-amplitude correlation function for spherical wave propagation through anisotropic non-Kolmogorov atmosphere,” J. Opt. Soc. Am. A 31(1), 148–154 (2014).
[Crossref] [PubMed]

L. Y. Cui, B. D. Xue, X. G. Cao, and F. G. Zhou, “Angle of arrival fluctuations considering turbulence outer scale for optical waves’ propagation through moderate-to-strong non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 31(4), 829–835 (2014).
[Crossref] [PubMed]

I. Toselli, “Introducing the concept of anisotropy at different scales for modeling optical turbulence,” J. Opt. Soc. Am. A 31(8), 1868–1875 (2014).
[Crossref] [PubMed]

2013 (2)

L. Y. Cui, B. D. Xue, and F. G. Zhou, “Analytical expressions for the angle of arrival fluctuations for optical waves’ propagation through moderate-to-strong non-Kolmogorov refractive turbulence,” J. Opt. Soc. Am. A 30(11), 2188–2195 (2013).
[Crossref] [PubMed]

L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013).
[Crossref]

2012 (3)

X. Yi, Z. J. Liu, and P. Yue, “Inner- and outer-scale effects on the scintillation index of an optical wave propagating through moderate-to-strong non-Kolmogorov turbulence,” Opt. Express 20(4), 4232–4247 (2012).
[Crossref] [PubMed]

L. Y. Cui, B. D. Xue, S. L. Zheng, W. F. Xue, X. Z. Bai, X. G. Cao, and F. G. Zhou, “Atmospheric spectral model and theoretical expressions of irradiance scintillation index for optical wave propagating through moderate-to-strong non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 29(6), 1091–1098 (2012).
[Crossref] [PubMed]

V. S. Gudimetla, R. B. Holmes, and J. F. Riker, “Analytical expressions for the log-amplitude correlation function for plane wave propagation in anisotropic non-Kolmogorov refractive turbulence,” J. Opt. Soc. Am. A 29(12), 2622–2627 (2012).
[Crossref] [PubMed]

2011 (2)

I. Toselli, B. Agrawal, and S. Restaino, “Light propagation through anisotropic turbulence,” J. Opt. Soc. Am. A 28(3), 483–488 (2011).
[Crossref] [PubMed]

B. D. Xue, L. Y. Cui, W. F. Xue, X. Z. Bai, and F. G. Zhou, “Theoretical expressions of the angle-of-arrival variance for optical waves propagating through non-Kolmogorov turbulence,” Opt. Express 19(9), 8433–8443 (2011).
[Crossref] [PubMed]

2009 (2)

W. H. Du, S. Yu, L. Y. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctuations for wave propagation through non-Kolmogorov turbulence,” Opt. Commun. 282(5), 705–708 (2009).
[Crossref]

L. Y. Tan, W. H. Du, and J. Ma, “Effect of the outer scale on the angle of arrival variance for free-space-laser beam corrugated by non-Kolmogorov turbulence,” J. Russ. Laser Res. 30(6), 552–559 (2009).
[Crossref]

2008 (2)

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(1), 66–77 (2008).
[Crossref]

C. Robert, J. M. Conan, V. Michau, J. B. Renard, C. Robert, and F. Dalaudier, “Retrieving parameters of the anisotropic refractive index fluctuations spectrum in the stratosphere from balloon-borne observations of stellar scintillation,” J. Opt. Soc. Am. A 25(2), 379–393 (2008).
[Crossref] [PubMed]

2007 (2)

Y. Cheon and A. Muschinski, “Closed-form approximations for the angle-of-arrival variance of plane and spherical waves propagating through homogeneous and isotropic turbulence,” J. Opt. Soc. Am. A 24(2), 415–422 (2007).
[Crossref] [PubMed]

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 (2007).
[Crossref]

2006 (1)

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U (2006).
[Crossref]

2005 (1)

L. Biferale and I. Procaccia, “Anisotropic contribution to the statistics of the atmospheric boundary layer,” Phys. Rep. 414, 43–164 (2005).
[Crossref]

1999 (3)

M. S. Belen’kii, 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. Belen’kii, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999).
[Crossref]

1998 (1)

F. D. Eaton and G. D. Nastrom, “Preliminary estimates of the vertical profiles of inner and outer scales from White Sands Missile Range, New Mexico, VHF radar observations,” Radio Sci. 33(4), 895–903 (1998).
[Crossref]

1997 (1)

M. S. Belen’kii, S. J. Karis, J. M. Brown, 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]

1995 (4)

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

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinova, V. P. Lukin, A. P. Rostov, B. V. Fortes, and A. P. Yankov, “Investigation of turbulence spectrum anisotropy in the ground atmospheric layer; preliminary results,” Atmos. Oceanic Opt. 8, 993–996 (1995).

A. S. Gurvich and M. S. Belen’kii, “Influence of stratospheric turbulence on infrared imaging,” J. Opt. Soc. Am. A 12, 2517–2522 (1995).

M. S. Belen’kii, “Effect of the stratosphere on star image motion,” Opt. Lett. 20(12), 1359–1361 (1995).
[Crossref] [PubMed]

1994 (1)

D. T. Kyrazis, J. B. Wissler, 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]

1992 (1)

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

1986 (1)

R. M. Manning, “An anisotropic turbulence model for wave propagation near the surface of the Earth,” IEEE Trans. Antenn. Propag. 34(2), 258–261 (1986).
[Crossref]

Agrawal, B.

I. Toselli, B. Agrawal, and S. Restaino, “Light propagation through anisotropic turbulence,” J. Opt. Soc. Am. A 28(3), 483–488 (2011).
[Crossref] [PubMed]

Andrews, L. C.

L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]

Antoshkin, L. V.

Bai, X. Z.

Barchers, J. D.

Belen’kii, M. S.

A. S. Gurvich and M. S. Belen’kii, “Influence of stratospheric turbulence on infrared imaging,” J. Opt. Soc. Am. A 12, 2517–2522 (1995).

M. S. Belen’kii, “Effect of the stratosphere on star image motion,” Opt. Lett. 20(12), 1359–1361 (1995).
[Crossref] [PubMed]

Biferale, L.

L. Biferale and I. Procaccia, “Anisotropic contribution to the statistics of the atmospheric boundary layer,” Phys. Rep. 414, 43–164 (2005).
[Crossref]

Bishop, K. P.

Botygina, N. N.

Brown, J. M.

Cao, X. G.

Cheon, Y.

Conan, J. M.

Crabbs, R.

L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]

Cuellar, E.

Cui, L. Y.

Dalaudier, F.

Du, W. H.

W. H. Du, S. Yu, L. Y. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctuations for wave propagation through non-Kolmogorov turbulence,” Opt. Commun. 282(5), 705–708 (2009).
[Crossref]

Eaton, F. D.

Emaleev, O. N.

Ferrero, V.

Fortes, B. V.

Fugate, R. Q.

Golbraikh, E.

Grechko, G. M.

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

Gudimetla, V. S.

Gurvich, A. S.

A. S. Gurvich and M. S. Belen’kii, “Influence of stratospheric turbulence on infrared imaging,” J. Opt. Soc. Am. A 12, 2517–2522 (1995).

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

Holmes, R. B.

Hughes, K. A.

Jiang, Y.

W. H. Du, S. Yu, L. Y. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctuations for wave propagation through non-Kolmogorov turbulence,” Opt. Commun. 282(5), 705–708 (2009).
[Crossref]

Kan, V.

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

Karis, S. J.

Keating, D. B.

Kireev, S. V.

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

Kopeika, N. S.

Kupershmidt, I.

Kyrazis, D. T.

Lavrinova, L. N.

Leclerc, T.

Liu, Z. J.

Lukin, V. P.

Ma, J.

W. H. Du, S. Yu, L. Y. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctuations for wave propagation through non-Kolmogorov turbulence,” Opt. Commun. 282(5), 705–708 (2009).
[Crossref]

Manning, R. M.

R. M. Manning, “An anisotropic turbulence model for wave propagation near the surface of the Earth,” IEEE Trans. Antenn. Propag. 34(2), 258–261 (1986).
[Crossref]

Michau, V.

Muschinski, A.

Nastrom, G. D.

Osmon, C. L.

Phillips, R. L.

L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]

Preble, A. J.

Procaccia, I.

L. Biferale and I. Procaccia, “Anisotropic contribution to the statistics of the atmospheric boundary layer,” Phys. Rep. 414, 43–164 (2005).
[Crossref]

Renard, J. B.

Restaino, S.

I. Toselli, B. Agrawal, and S. Restaino, “Light propagation through anisotropic turbulence,” J. Opt. Soc. Am. A 28(3), 483–488 (2011).
[Crossref] [PubMed]

Riker, J. F.

Robert, C.

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]

Rostov, A. P.

Rye, V. A.

Savchenko, S. A.

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

Shtemler, Y.

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]

Tan, L. Y.

W. H. Du, S. Yu, L. Y. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctuations for wave propagation through non-Kolmogorov turbulence,” Opt. Commun. 282(5), 705–708 (2009).
[Crossref]

Toselli, I.

I. Toselli, “Introducing the concept of anisotropy at different scales for modeling optical turbulence,” J. Opt. Soc. Am. A 31(8), 1868–1875 (2014).
[Crossref] [PubMed]

I. Toselli, B. Agrawal, and S. Restaino, “Light propagation through anisotropic turbulence,” J. Opt. Soc. Am. A 28(3), 483–488 (2011).
[Crossref] [PubMed]

Virtser, A.

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. B.

Xie, W.

W. H. Du, S. Yu, L. Y. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctuations for wave propagation through non-Kolmogorov turbulence,” Opt. Commun. 282(5), 705–708 (2009).
[Crossref]

Xue, B. D.

Xue, W. F.

Yankov, A. P.

Yi, X.

Yu, S.

W. H. Du, S. Yu, L. Y. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctuations for wave propagation through non-Kolmogorov turbulence,” Opt. Commun. 282(5), 705–708 (2009).
[Crossref]

Yue, P.

Zheng, S. L.

Zhou, F. G.

Zilberman, A.

Adv. Space Res. (1)

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

Atmos. Oceanic Opt. (1)

Atmos. Res. (1)

IEEE Trans. Antenn. Propag. (1)

R. M. Manning, “An anisotropic turbulence model for wave propagation near the surface of the Earth,” IEEE Trans. Antenn. Propag. 34(2), 258–261 (1986).
[Crossref]

J. Opt. Soc. Am. A (10)

I. Toselli, B. Agrawal, and S. Restaino, “Light propagation through anisotropic turbulence,” J. Opt. Soc. Am. A 28(3), 483–488 (2011).
[Crossref] [PubMed]

I. Toselli, “Introducing the concept of anisotropy at different scales for modeling optical turbulence,” J. Opt. Soc. Am. A 31(8), 1868–1875 (2014).
[Crossref] [PubMed]

A. S. Gurvich and M. S. Belen’kii, “Influence of stratospheric turbulence on infrared imaging,” J. Opt. Soc. Am. A 12, 2517–2522 (1995).

J. Russ. Laser Res. (1)

Opt. Commun. (1)

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Fig. 1 Aperture-averaged AOA variance, normalized by

{\hat{C}}_{n}^{2} L D^{α - 4}

, versus

q

for plane and spherical waves with different anisotropic factor

ς

values. (a): Plane wave; (b): Spherical wave.

Download Full Size | PPT Slide | PDF

Fig. 2 Aperture-averaged AOA variance, normalized by

{\hat{C}}_{n}^{2} L D^{α - 4}

, versus

q

for plane and spherical waves with different anisotropic factor

ς

and general spectral power law values. (a): Plane wave; (b): Spherical wave.

Download Full Size | PPT Slide | PDF

Fig. 3 Difference (%) between the close-form expression with the asymptotic fit results with different anisotropic factor

ς

and

α

values. (a): Differences as a function of

ς

; (b): Differences as a function of

α

Download Full Size | PPT Slide | PDF

Fig. 4 The influence of anisotropic factor on the variance of AOA fluctuations with different spectral power law values. (a): Plane wave; (b): Spherical wave.

Download Full Size | PPT Slide | PDF

Fig. 5 The influence of general spectral power law values on the variance of AOA fluctuations with different anisotropic factor values. (a): Plane wave; (b): Spherical wave.

Download Full Size | PPT Slide | PDF

Equations (40)

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Φ_{n_i s o t r o p i c} (κ, α) = A (α) \cdot {\hat{C}}_{n}^{2} \cdot \begin{matrix} κ^{- α}, & (\begin{matrix} 0 \leq κ < \infty, & 3 < α < 4 \end{matrix}) . \end{matrix}

A (α) = \frac{1}{4 π^{2}} Γ (α - 1) \cos [\frac{α π}{2}] .

D_{n} (R, α, ς) = A (α) {\hat{C}}_{n}^{2} {(\frac{x^{2} + y^{2}}{ς^{2}} + z^{2})}^{α / 2}, l_{0} ≪ R ≪ L_{0}, 3 < α < 4.

Φ_{n} (κ, α, ς) = A (α) \cdot {\hat{C}}_{n}^{2} \cdot ς^{2} \cdot {(κ^{'})}^{- α} = A (α) \cdot {\hat{C}}_{n}^{2} \cdot ς^{2} \cdot {[κ_{z}^{2} + ς^{2} (κ_{x}^{2} + κ_{y}^{2})]}^{- \frac{α}{2}} .

Φ_{n} (κ, α, ς) = A (α) \cdot {\hat{C}}_{n}^{2} \cdot ς^{2 - α} \cdot κ^{- α}, κ = \sqrt{κ_{x}^{2} + κ_{y}^{2}} .

C_{θ} (ρ, β) = π k^{- 2} \int_{0}^{\infty} κ^{3} W_{ϕ} (κ) G_{D} (κ) [J_{0} (ρ κ) - \cos (2 β) J_{2} (ρ κ)] d κ

G_{D} (κ) \approx \exp (- \frac{b^{2} D^{2} κ^{2}}{4}), b = 0.52.

σ_{}^{2} = C_{θ} (ρ = 0, β = 0) = π k^{- 2} \int_{0}^{\infty} κ^{3} W_{φ} (κ) G_{D} (κ) d κ,

W_{ϕ (p l)} (κ) = 2 π k^{2} \int_{0}^{L} Φ_{n} (κ, α, ς) \cos^{2} (\frac{κ^{2} z}{2 k}) d z,

σ_{(p l)}^{2} = 2 π^{2} \int_{0}^{\infty} κ^{3} Φ_{n} (κ, α, ς) G_{D} (κ) d κ \int_{0}^{L} \cos^{2} (\frac{κ^{2} z}{2 k}) d z .

σ_{(p l)}^{2} = π^{2} \int_{0}^{\infty} d κ \int_{0}^{1} d z κ^{3} Φ_{n} (κ, α, ς) [1 + \frac{k}{L κ^{2}} \sin (\frac{L κ^{2}}{k})] \exp [- \frac{b^{2} D^{2} κ^{2}}{4}],

Γ (x) = \int_{0}^{\infty} κ^{x - 1} \cdot e^{- κ} d κ,

\int_{0}^{\infty} κ^{μ - 1} \exp (- a κ) \sin (b κ) d κ = \frac{Γ (μ)}{{(a^{2} + b^{2})}^{μ / 2}} \sin [μ \tan^{- 1} (\frac{b}{a})], μ > - 1, a > 0

\begin{matrix} σ_{(p l)}^{2} (α, ζ) = \frac{π^{2} A (α) {\hat{C}}_{n}^{2} L ς^{2 - α}}{2} \cdot {Γ (2 - \frac{α}{2}) {(\frac{b^{2} D^{2}}{4})}^{\frac{α}{2} - 2} + \\ \frac{k}{L} \frac{Γ (1 - \frac{α}{2})}{{(\frac{b^{4} D^{4}}{16} + \frac{L^{2}}{k^{2}})}^{\frac{2 - α}{4}}} \cdot \sin [\frac{2 - α}{2} \tan^{- 1} (\frac{4 L}{k b^{2} D^{2}})]} . \end{matrix}

σ_{(p l)}^{2} (α, ζ) = r_{p} (q, α, ζ) {\hat{C}}_{n}^{2} L D^{α - 4},

\begin{matrix} r_{p} (q, α, ζ) = \frac{π^{2} A (α) ς^{2 - α}}{2} Γ (2 - \frac{α}{2}) {(\frac{b}{2})}^{α - 4} \\ \times {1 + \frac{2}{α - 2} {(\frac{π}{2})}^{2 - α / 2} b^{4 - α} q^{4 - α} {(1 + \frac{b^{4} q^{4} π^{2}}{4})}^{\frac{α - 2}{4}} \cdot \sin [\frac{α - 2}{2} \tan^{- 1} (\frac{2}{b^{2} q^{2} π})]} . \end{matrix}

σ_{(p l)}^{2} (q ≪ 1, α, ζ) = r_{p} (q ≪ 1, α, ζ) {\hat{C}}_{n}^{2} L D^{α - 4},

r_{p} (q ≪ 1, α, ζ) = \frac{π^{2} A (α) ς^{2 - α}}{2} Γ (2 - \frac{α}{2}) {(\frac{b}{2})}^{α - 4} .

σ_{(p l)}^{2} (q ≫ 1, α, ζ) = r_{p} (q ≫ 1, α, ζ) {\hat{C}}_{n}^{2} L D^{α - 4},

r_{p} (q ≫ 1, α, ζ) = π^{2} A (α) ς^{2 - α} Γ (2 - \frac{α}{2}) {(\frac{b}{2})}^{α - 4} .

σ_{(p l)}^{2} (q < 1, α, ζ) = r_{p} (q < 1, α, ζ) {\hat{C}}_{n}^{2} L D^{α - 4},

\begin{matrix} r_{p} (q < 1, α, ζ) = \frac{π^{2} A (α) ς^{2 - α}}{2} Γ (2 - \frac{α}{2}) {(\frac{b}{2})}^{α - 4} \\ \times {1 + \frac{2}{α - 2} {(\frac{π}{2})}^{2 - α / 2} b^{4 - α} q^{4 - α} \cdot \sin [\frac{π (α - 2)}{4}]} . \end{matrix}

W_{ϕ (s p)} (κ) = 2 π k^{2} \int_{0}^{L} Φ_{n} (κ, α, ς) {(\frac{z}{L})}^{2} \cos^{2} [\frac{κ^{2} z (L - z)}{2 k L}] d z .

σ_{(s p)}^{2} = π^{2} L \int_{0}^{\infty} d κ \int_{0}^{1} d ξ κ^{3} Φ_{n} (κ, α, ς) [1 + \cos (\frac{κ^{2} ξ (1 - ξ) L}{k})] ξ^{2} \exp [- \frac{b^{2} D^{2} κ^{2} ξ^{2}}{4}],

{}_{2}F_{1} (A, B; C; Z) = \frac{Γ (C)}{Γ (B) \cdot Γ (C - B)} \int_{0}^{1} t^{B - 1} \cdot {(1 - t)}^{C - B - 1} \cdot {(1 - t Z)}^{- A} d t .

\begin{matrix} σ_{s p}^{2} (α, ζ) = \frac{π^{2} A (α) {\hat{C}}_{n}^{2} L ς^{2 - α}}{2} \cdot Γ (2 - \frac{α}{2}) {\frac{1}{α - 1} {(\frac{b^{2} D^{2}}{4})}^{α / 2 - 2} + \\ Re [{(i \frac{L}{k})}^{- \frac{4 - α}{2}} \cdot \frac{2}{2 + α} \cdot {}_{2}F_{1} (\frac{4 - α}{2}, \frac{2 + α}{2}; \frac{4 + α}{2}; 1 + i \frac{b^{2} D^{2} k}{4 L})]} . \end{matrix}

σ_{s p}^{2} (α, ζ) = r_{s} (q, α, ς) {\hat{C}}_{n}^{2} L D^{α - 4},

\begin{matrix} r_{s} (q, α, ζ) = \frac{π^{2} A (α) ς^{2 - α}}{2 (α - 1)} \cdot Γ (2 - \frac{α}{2}) {(\frac{b^{2}}{4})}^{α / 2 - 2} {1 + \frac{2 (α - 1)}{2 + α} \times \\ Re [{(\frac{2 i}{π b^{2} q^{2}})}^{- \frac{4 - α}{2}} \cdot {}_{2}F_{1} (\frac{4 - α}{2}, \frac{2 + α}{2}; \frac{4 + α}{2}; 1 + i \frac{π b^{2} q^{2}}{2})]} . \end{matrix}

σ_{s p}^{2} (q ≪ 1, α, ζ) = r_{p} (q ≪ 1, α, ζ) {\hat{C}}_{n}^{2} L D^{α - 4} .

r_{p} (q ≪ 1, α, ζ) = \frac{π^{2} A (α) ς^{2 - α}}{2 (α - 1)} \cdot Γ (2 - \frac{α}{2}) {(\frac{b^{2}}{4})}^{α / 2 - 2} .

\begin{matrix} {}_{2}F_{1} (a, b; c; z) = \frac{Γ (c) Γ (b - a)}{Γ (b) Γ (c - a)} {(- z)}^{- a} {}_{2}F_{1} (a, 1 - c + a; 1 - b + a; 1 / z) \\ + \frac{Γ (c) Γ (a - b)}{Γ (a) Γ (c - b)} {(- z)}^{- b} {}_{2}F_{1} (b, 1 - c + b; 1 - a + b; 1 / z) \end{matrix}

σ_{s p}^{2} (q ≫ 1, α, ζ) = r_{s} (q ≫ 1, α, ζ) {\hat{C}}_{n}^{2} L D^{α - 4},

r_{s} (q ≫ 1, α, ζ) = \frac{π^{2} A (α) ς^{2 - α}}{(α - 1)} \cdot Γ (2 - \frac{α}{2}) {(\frac{b^{2}}{4})}^{α / 2 - 2} .

{}_{2}F_{1} (a, b; c; 1) = \frac{Γ (c) Γ (c - a - b)}{Γ (c - a) Γ (c - b)} .

σ_{s p}^{2} (q < 1, α, ζ) = r_{s} (q < 1, α, ζ) {\hat{C}}_{n}^{2} L D^{α - 4},

\begin{matrix} r_{s} (q < 1, α, ζ) = \frac{π^{2} A (α) ς^{2 - α}}{2 (α - 1)} \cdot Γ (2 - \frac{α}{2}) {(\frac{b^{2}}{4})}^{α / 2 - 2} \\ \times {1 + \frac{2 (α - 1)}{2 + α} Re [{(\frac{2 i}{π b^{2} q^{2}})}^{- \frac{4 - α}{2}} \cdot \frac{Γ (2 + α / 2) Γ (α / 2 - 1)}{Γ (α)}]} . \end{matrix}

σ_{(p l)}^{2} (q, α, ζ) \approx r_{p} (q, α, ζ) {\hat{C}}_{n}^{2} L D^{α - 4},

σ_{s p}^{2} (q, α, ζ) \approx r_{s} (q, α, ζ) {\hat{C}}_{n}^{2} L D^{α - 4},

r_{p} (q, α, ζ) = {\begin{matrix} \begin{matrix} \frac{π^{2} A (α) ς^{2 - α}}{2} Γ (2 - \frac{α}{2}) {(\frac{b}{2})}^{α - 4} \\ \times {1 + \frac{2}{α - 2} {(\frac{π}{2})}^{2 - α / 2} b^{4 - α} q^{4 - α} \cdot \sin [\frac{π (α - 2)}{4}]}, \end{matrix} & q \leq 1 \\ π^{2} A (α) ς^{2 - α} Γ (2 - \frac{α}{2}) {(\frac{b}{2})}^{α - 4} . & q > 1 \end{matrix}

r_{s} (q, α, ζ) = {\begin{matrix} \begin{matrix} \frac{π^{2} A (α) ς^{2 - α}}{2 (α - 1)} \cdot Γ (2 - \frac{α}{2}) {(\frac{b^{2}}{4})}^{α / 2 - 2} \\ \times {1 + \frac{2 (α - 1)}{2 + α} Re [{(\frac{2 i}{π b^{2} q^{2}})}^{- \frac{4 - α}{2}} \cdot \frac{Γ (2 + α / 2) Γ (α / 2 - 1)}{Γ (α)}]}, \end{matrix} & q \leq 1 \\ \frac{π^{2} A (α) ς^{2 - α}}{(α - 1)} \cdot Γ (2 - \frac{α}{2}) {(\frac{b^{2}}{4})}^{α / 2 - 2} . & q > 1 \end{matrix}