Abstract

Theoretical and experimental investigations have shown that the atmospheric turbulence exhibits both anisotropic and non-Kolmogorov properties. In this work, two theoretical atmosphere refractive-index fluctuations spectral models are derived for optical waves propagating through anisotropic non-Kolmogorov atmospheric turbulence. They consider simultaneously the finite turbulence inner and outer scales and the asymmetric property of turbulence eddies in the orthogonal xy-plane throughout the path. Two anisotropy factors which parameterize the asymmetry of turbulence eddies in both horizontal and vertical directions are introduced in the orthogonal xy-plane, so that the circular symmetry assumption of turbulence eddies in the xy-plane is no longer required. Deviations from the classic 11/3 power law behavior in the spectrum model are also allowed by assuming power law value variations between 3 and 4. Based on the derived anisotropic spectral model and the Rytov approximation theory, expressions for the variance of angle of arrival (AOA) fluctuations are derived for optical plane and spherical waves propagating through weak anisotropic non-Kolmogorov turbulence. Calculations are performed to analyze the derived spectral models and the variance of AOA fluctuations.

© 2015 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles

References

  • View by:
  • |
  • |
  • |

  1. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (trans.for NOAA by Israel Program for Scientific Translations, Jerusalem, 1971).
  2. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical Propagation in Non-Kolmogorov Atmospheric Turbulence,” Proc. SPIE 2471, 181–196 (1995).
    [Crossref]
  3. 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]
  4. L. Y. Cui, B. D. Xue, X. G. Cao, J. K. Dong, and J. N. Wang, “Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(20), 21269–21283 (2010).
    [Crossref] [PubMed]
  5. B. Xue, L. Cui, W. Xue, X. Bai, and F. Zhou, “Generalized modified atmospheric spectral model for optical wave propagating through non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 28(5), 912–916 (2011).
    [Crossref] [PubMed]
  6. S. Fu, L. Y. Tan, J. Ma, and Y. P. Zhou, “Effect of non-Kolmogorov turbulence on angle-of-arrival fluctuations of starlight,” J. Russ. Laser Res. 31(4), 332–337 (2010).
    [Crossref]
  7. W. Du, S. Yu, L. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctuations for wave propagation through non-Kolmolgorov turbulence,” Opt. Commun. 282(5), 705–708 (2009).
    [Crossref]
  8. L. Tan, W. 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]
  9. B. Xue, L. Cui, W. Xue, X. Bai, and F. 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]
  10. L. Y. Cui, X. G. Cao, B. D. Xue, and L. Cao, “Analysis of angle-of-arrival fluctuations for optical waves propagating through weak non-Kolmogorov turbulence,” Opt. Commun. 305, 36–41 (2013).
    [Crossref]
  11. 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]
  12. 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]
  13. 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]
  14. L. Biferale and I. Procaccia, “Anisotropy in turbulent flows and in turbulent transport,” Phys. Rep. 414, 43–164 (2005).
    [Crossref]
  15. 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]
  16. 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]
  17. 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]
  18. 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).
  19. A. D. Wheelon, Electromagnetic Scintillation I. Geometric Optics (Cambridge University, 2001).
  20. 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]
  21. I. Toselli, B. Agrawal, and S. Restaino, “Light propagation through anisotropic turbulence,” J. Opt. Soc. Am. A 28(3), 483–488 (2011).
    [Crossref] [PubMed]
  22. 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]
  23. 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]
  24. 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]
  25. L. Cui, “Analysis of angle of arrival fluctuations for optical waves’ propagation through weak anisotropic non-Kolmogorov turbulence,” Opt. Express 23(5), 6313–6325 (2015).
    [Crossref] [PubMed]
  26. 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]
  27. I. Toselli and O. Korotkova, “General scale-dependent anisotropic turbulence and its impact on free space optical communication system performance,” J. Opt. Soc. Am. A 32(6), 1017–1025 (2015).
    [Crossref] [PubMed]
  28. 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]
  29. J. Borgnino, J. Martin, and A. F. Ziad, “Effects of a finite spatial-coherence outer scale on the covariances of angle-of-arrival fluctuations,” Opt. Commun. 91(3–4), 267–279 (1992).
    [Crossref]
  30. 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]
  31. L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Optical Engineering, 1998).
  32. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media. (SPIE Optical Engineering, 2005).

2015 (2)

2014 (3)

2013 (2)

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]

L. Y. Cui, X. G. Cao, B. D. Xue, and L. Cao, “Analysis of angle-of-arrival fluctuations for optical waves propagating through weak non-Kolmogorov turbulence,” Opt. Commun. 305, 36–41 (2013).
[Crossref]

2012 (1)

2011 (3)

2010 (2)

L. Y. Cui, B. D. Xue, X. G. Cao, J. K. Dong, and J. N. Wang, “Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(20), 21269–21283 (2010).
[Crossref] [PubMed]

S. Fu, L. Y. Tan, J. Ma, and Y. P. Zhou, “Effect of non-Kolmogorov turbulence on angle-of-arrival fluctuations of starlight,” J. Russ. Laser Res. 31(4), 332–337 (2010).
[Crossref]

2009 (2)

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

L. Tan, W. 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 (1)

2007 (2)

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]

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]

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, “Anisotropy in turbulent flows and in turbulent transport,” Phys. Rep. 414, 43–164 (2005).
[Crossref]

1999 (2)

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]

1995 (2)

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).

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

1992 (2)

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]

J. Borgnino, J. Martin, and A. F. Ziad, “Effects of a finite spatial-coherence outer scale on the covariances of angle-of-arrival fluctuations,” Opt. Commun. 91(3–4), 267–279 (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.

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]

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]

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]

Antoshkin, L. V.

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).

Bai, X.

Barchers, J. D.

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]

Belen’kii, M. S.

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]

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]

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]

Biferale, L.

L. Biferale and I. Procaccia, “Anisotropy in turbulent flows and in turbulent transport,” Phys. Rep. 414, 43–164 (2005).
[Crossref]

Borgnino, J.

J. Borgnino, J. Martin, and A. F. Ziad, “Effects of a finite spatial-coherence outer scale on the covariances of angle-of-arrival fluctuations,” Opt. Commun. 91(3–4), 267–279 (1992).
[Crossref]

Botygina, N. N.

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).

Brown, J. M.

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]

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]

Cao, L.

L. Y. Cui, X. G. Cao, B. D. Xue, and L. Cao, “Analysis of angle-of-arrival fluctuations for optical waves propagating through weak non-Kolmogorov turbulence,” Opt. Commun. 305, 36–41 (2013).
[Crossref]

Cao, X. G.

L. Y. Cui, X. G. Cao, B. D. Xue, and L. Cao, “Analysis of angle-of-arrival fluctuations for optical waves propagating through weak non-Kolmogorov turbulence,” Opt. Commun. 305, 36–41 (2013).
[Crossref]

L. Y. Cui, B. D. Xue, X. G. Cao, J. K. Dong, and J. N. Wang, “Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(20), 21269–21283 (2010).
[Crossref] [PubMed]

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]

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]

Cuellar, E.

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]

Cui, L.

Cui, L. Y.

L. Y. Cui, X. G. Cao, B. D. Xue, and L. Cao, “Analysis of angle-of-arrival fluctuations for optical waves propagating through weak non-Kolmogorov turbulence,” Opt. Commun. 305, 36–41 (2013).
[Crossref]

L. Y. Cui, B. D. Xue, X. G. Cao, J. K. Dong, and J. N. Wang, “Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(20), 21269–21283 (2010).
[Crossref] [PubMed]

Dalaudier, F.

Dong, J. K.

Du, W.

L. Tan, W. 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]

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

Eaton, F. D.

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]

Emaleev, O. N.

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).

Ferrero, V.

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]

Fortes, B. V.

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).

Fu, S.

S. Fu, L. Y. Tan, J. Ma, and Y. P. Zhou, “Effect of non-Kolmogorov turbulence on angle-of-arrival fluctuations of starlight,” J. Russ. Laser Res. 31(4), 332–337 (2010).
[Crossref]

Fugate, R. Q.

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]

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.

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.

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]

Jiang, Y.

W. Du, S. Yu, L. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctuations for wave propagation through non-Kolmolgorov 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.

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]

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]

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]

Korotkova, O.

Lavrinova, L. N.

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).

Leclerc, T.

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]

Lukin, V. P.

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).

Ma, J.

S. Fu, L. Y. Tan, J. Ma, and Y. P. Zhou, “Effect of non-Kolmogorov turbulence on angle-of-arrival fluctuations of starlight,” J. Russ. Laser Res. 31(4), 332–337 (2010).
[Crossref]

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

L. Tan, W. 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]

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]

Martin, J.

J. Borgnino, J. Martin, and A. F. Ziad, “Effects of a finite spatial-coherence outer scale on the covariances of angle-of-arrival fluctuations,” Opt. Commun. 91(3–4), 267–279 (1992).
[Crossref]

Michau, V.

Muschinski, A.

Nastrom, G. D.

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]

Osmon, C. L.

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]

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]

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]

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]

Procaccia, I.

L. Biferale and I. Procaccia, “Anisotropy in turbulent flows and in turbulent transport,” Phys. Rep. 414, 43–164 (2005).
[Crossref]

Renard, J. B.

Restaino, S.

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.

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).

Rye, V. A.

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]

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]

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.

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

L. Tan, W. 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]

Tan, L. Y.

S. Fu, L. Y. Tan, J. Ma, and Y. P. Zhou, “Effect of non-Kolmogorov turbulence on angle-of-arrival fluctuations of starlight,” J. Russ. Laser Res. 31(4), 332–337 (2010).
[Crossref]

Toselli, I.

Wang, J. N.

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]

Xie, W.

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

Xue, B.

Xue, B. D.

L. Y. Cui, X. G. Cao, B. D. Xue, and L. Cao, “Analysis of angle-of-arrival fluctuations for optical waves propagating through weak non-Kolmogorov turbulence,” Opt. Commun. 305, 36–41 (2013).
[Crossref]

L. Y. Cui, B. D. Xue, X. G. Cao, J. K. Dong, and J. N. Wang, “Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(20), 21269–21283 (2010).
[Crossref] [PubMed]

Xue, W.

Yankov, A. P.

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).

Yu, S.

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

Zhou, F.

Zhou, Y. P.

S. Fu, L. Y. Tan, J. Ma, and Y. P. Zhou, “Effect of non-Kolmogorov turbulence on angle-of-arrival fluctuations of starlight,” J. Russ. Laser Res. 31(4), 332–337 (2010).
[Crossref]

Ziad, A. F.

J. Borgnino, J. Martin, and A. F. Ziad, “Effects of a finite spatial-coherence outer scale on the covariances of angle-of-arrival fluctuations,” Opt. Commun. 91(3–4), 267–279 (1992).
[Crossref]

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)

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).

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 (8)

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]

B. Xue, L. Cui, W. Xue, X. Bai, and F. Zhou, “Generalized modified atmospheric spectral model for optical wave propagating through non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 28(5), 912–916 (2011).
[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]

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]

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]

I. Toselli and O. Korotkova, “General scale-dependent anisotropic turbulence and its impact on free space optical communication system performance,” J. Opt. Soc. Am. A 32(6), 1017–1025 (2015).
[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]

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]

J. Russ. Laser Res. (2)

S. Fu, L. Y. Tan, J. Ma, and Y. P. Zhou, “Effect of non-Kolmogorov turbulence on angle-of-arrival fluctuations of starlight,” J. Russ. Laser Res. 31(4), 332–337 (2010).
[Crossref]

L. Tan, W. 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]

Opt. Commun. (3)

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

L. Y. Cui, X. G. Cao, B. D. Xue, and L. Cao, “Analysis of angle-of-arrival fluctuations for optical waves propagating through weak non-Kolmogorov turbulence,” Opt. Commun. 305, 36–41 (2013).
[Crossref]

J. Borgnino, J. Martin, and A. F. Ziad, “Effects of a finite spatial-coherence outer scale on the covariances of angle-of-arrival fluctuations,” Opt. Commun. 91(3–4), 267–279 (1992).
[Crossref]

Opt. Express (3)

Phys. Rep. (1)

L. Biferale and I. Procaccia, “Anisotropy in turbulent flows and in turbulent transport,” Phys. Rep. 414, 43–164 (2005).
[Crossref]

Proc. SPIE (7)

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]

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]

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]

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]

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]

Radio Sci. (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]

Other (4)

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (trans.for NOAA by Israel Program for Scientific Translations, Jerusalem, 1971).

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).

A. D. Wheelon, Electromagnetic Scintillation I. Geometric Optics (Cambridge University, 2001).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1 The ratios between the anisotropic non-Kolmogorov turbulence spectra and the isotropic non-Kolmogorov turbulence spectra ( l 0 =1mm , L 0 =10m ). (a): α=10/3 ; (b): α=11/3 ; (c): α=3.9 .
Fig. 2
Fig. 2 Variance of AOA fluctuations as a function of μ y with different α and μ x values for both plane and spherical waves. (a): α=10/3 ; (b): α=11/3 ; (c): α=3.9 .
Fig. 3
Fig. 3 Variance of AOA fluctuations as a function of L 0 with different μ x and μ y values for both plane and spherical waves ( α=10/3 ). (a): μ x =1, μ y =2,5,10 ; (b): μ x =2, μ y =2,5,10 ; (c): μ x = μ y =1,2,10 .

Equations (47)

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

Φ n_von ( κ,α )= A( α ) C ^ n 2 ( κ 2 + κ 0 2 ) α/2 exp( κ 2 κ l 2 ), ( 0κ<, 3<α<4 ) .
Φ n_exp ( κ,α )=A( α ) C ^ n 2 κ α [ 1exp( κ 2 κ 0 2 ) ]exp( κ 2 κ l 2 ) , ( 0κ<, 3<α<4 ) .
Φ n ( κ,α )= 1 4 π 2 κ 2 0 sin( κR ) κR R [ R 2 D n ( R,α ) R ]dR.
Φ n_aniso ( κ,α, μ x , μ y )= μ x μ y 1 4 π 2 ( κ' ) 2 0 sin( κ ' R ' ) κ ' R ' R ' [ ( R' ) 2 D n ( R ' ,α ) R ' ]d R ' .
Φ n ( κ ' ,α )= 1 4 π 2 ( κ' ) 2 0 sin( κ ' R ' ) κ ' R ' R ' [ ( R' ) 2 D n ( R ' ,α ) R ' ]d R ' .
Φ n_aniso_exp ( κ,α, μ x , μ y )= μ x μ y Φ n_exp ( κ ' ,α ),
Φ n_aniso_von ( κ,α, μ x , μ y )= μ x μ y Φ n_von ( κ ' ,α ).
D n ( R ' ,α )=8π 0 ( κ ' ) 2 Φ n_exp ( κ ' ,α )( 1 sin κ ' R ' κ ' R ' )d κ ' .
A ^ ( α )= Γ( α1 ) 4 π 2 sin[ ( α3 ) π 2 ].
c ' ( α )= [ π A ^ ( α )Γ( α 2 + 3 2 )( 3α 3 ) ] 1 α5 .
Φ n_aniso_von ( κ,α, μ x , μ y )= A ^ ( α ) C ^ n 2 μ x μ y ( μ x 2 κ x 2 + μ y 2 κ y 2 + κ z 2 + κ 0 '2 ) α/2 exp( μ x 2 κ x 2 + μ y 2 κ y 2 + κ z 2 κ l '2 ),
Φ n_aniso_exp ( κ,α, μ x , μ y )= μ x μ y A ^ ( α ) C ^ n 2 ( μ x 2 κ x 2 + μ y 2 κ y 2 + κ z 2 ) α/2 [ 1exp( μ x 2 κ x 2 + μ y 2 κ y 2 + κ z 2 κ 0 '2 ) ]exp( μ x 2 κ x 2 + μ y 2 κ y 2 + κ z 2 κ l '2 ).
C θ ( ρ,β )=π k 2 0 κ 3 W ϕ ( κ ) G D ( κ ) [ J 0 ( ρκ )cos( 2β ) J 2 ( ρκ ) ]dκ,
G D ( κ )exp( b 2 D 2 κ 2 4 ),b=0.52.
W ϕ( pl ) ( κ )=2π k 2 0 L Φ n_aniso_exp ( κ,α, u x , u y ) cos 2 ( κ 2 z 2k )dz ,
W ϕ( sp ) ( κ )=2π k 2 0 L Φ n_aniso_exp ( κ,α, u x , u y ) ( z L ) 2 cos 2 [ κ 2 z( Lz ) 2kL ]dz .
σ 2 = C θ ( ρ=0,β=0 )=π k 2 0 κ 3 W φ ( κ ) G D ( κ )dκ .
σ ( pl ) 2 ( α, μ x , μ y )= πL 2 0 0 κ 2 Φ n_aniso_exp ( κ,α, μ x , μ y ) [ 1+ k L κ 2 sin( L κ 2 k ) ]exp[ b 2 D 2 κ 2 4 ]d κ x d κ y ,
σ ( sp ) 2 ( α, μ x , μ y )= πL 2 0 0 0 1 dξ κ 2 Φ n_aniso_exp ( κ,α, μ x , μ y ) [ 1+cos( κ 2 ξ( 1ξ )L k ) ] ξ 2 exp[ b 2 D 2 κ 2 ξ 2 4 ]d κ x d κ y .
κ x = q x μ x = qcosθ μ x , κ y = q y μ y = qsinθ μ y ,q= q x 2 + q y 2 . d κ x d κ y = d q x d q y μ x μ y = qdqdθ μ x μ y , Φ n_aniso_exp ( κ,α, μ x , μ y )= μ x μ y A ^ ( α ) C ^ n 2 q α [ 1exp( q 2 κ 0 '2 ) ]exp( q 2 κ l '2 ).
σ ( pl ) 2 ( α, μ x , μ y )= πL 2 A ^ ( α ) C ^ n 2 0 2π 0 q 3α ( cos 2 θ μ x 2 + sin 2 θ μ y 2 ) [ 1exp( q 2 κ 0 '2 ) ]exp( q 2 κ l '2 ) [ 1+ k L q 2 ( cos 2 θ μ x 2 + sin 2 θ μ y 2 ) 1 sin( L q 2 k ( cos 2 θ μ x 2 + sin 2 θ μ y 2 ) ) ] exp[ b 2 D 2 q 2 4 ( cos 2 θ μ x 2 + sin 2 θ μ y 2 ) ]dqdθ,
σ ( sp ) 2 ( α, μ x , μ y )= πL 2 A ^ ( α ) C ^ n 2 0 2π 0 0 1 q 3α ( cos 2 θ μ x 2 + sin 2 θ μ y 2 )[ 1exp( q 2 κ 0 '2 ) ] exp( q 2 κ l '2 )[ 1+cos( q 2 ξ( 1ξ )L k ( cos 2 θ μ x 2 + sin 2 θ μ y 2 ) ) ] ξ 2 exp[ b 2 D 2 q 2 ξ 2 4 ( cos 2 θ μ x 2 + sin 2 θ μ y 2 ) ]dqdθdξ.
σ ( pl ) 2 ( α, μ x , μ y )= πL 2 A ^ ( α ) C ^ n 2 0 2π 0 q 1 3α ( cos 2 θ μ x 2 + sin 2 θ μ y 2 ) 2α 2 [ 1exp( q 1 2 κ 0 '2 ( cos 2 θ μ x 2 + sin 2 θ μ y 2 ) 1 ) ]exp[ q 1 2 κ l '2 ( cos 2 θ μ x 2 + sin 2 θ μ y 2 ) 1 ] [ 1+ k L q 1 2 sin( L q 1 2 k ) ]exp[ b 2 D 2 q 1 2 4 ]d q 1 dθ,
σ ( sp ) 2 ( α, μ x , μ y )= πL 2 A ^ ( α ) C ^ n 2 0 2π 0 0 1 q 1 3α ( cos 2 θ μ x 2 + cos 2 θ μ y 2 ) 2α 2 [ 1exp( q 1 2 κ 0 '2 ( cos 2 θ μ x 2 + cos 2 θ μ y 2 ) 1 ) ]exp[ q 1 2 κ l '2 ( cos 2 θ μ x 2 + cos 2 θ μ y 2 ) 1 ] [ 1+cos( q 1 2 ξ( 1ξ )L k ) ] ξ 2 exp[ b 2 D 2 q 1 2 ξ 2 4 ]d q 1 dθdξ.
F 2 1 ( a,b;c;z )= Γ( c ) Γ( b )Γ( cb ) 0 1 t b1 ( 1t ) cb1 ( 1tz ) a dt,
σ ( pl ) 2 ( α, μ x , μ y )= π 2 A ^ ( α ) C ^ n 2 LΓ( 2 α 2 ) 0 2π dθ ( cos 2 θ μ x 2 + sin 2 θ μ y 2 ) α2 2 { [ b 2 D 2 4 + 1 κ l '2 ( cos 2 θ μ x 2 + sin 2 θ μ y 2 ) 1 ] 2+ α 2 [ b 2 D 2 4 +( 1 κ 0 '2 + 1 κ l '2 ) ( cos 2 θ μ x 2 + sin 2 θ μ y 2 ) 1 ] 2+ α 2 },
σ sp 2 ( α, μ x , μ y )= π 2 A ^ ( α ) C ^ n 2 L 1 3 Γ( 2 α 2 ) 0 2π dθ( cos 2 θ μ x 2 + sin 2 θ μ y 2 ) { ( 1 κ l '2 ) 2+ α 2 F 2 1 ( 2 α 2 , 3 2 ; 5 2 ; b 2 D 2 κ l 2 4 ( cos 2 θ μ x 2 + sin 2 θ μ y 2 ) ) ( 1 κ l '2 + 1 κ 0 '2 ) 2+ α 2 F 2 1 ( 2 α 2 , 3 2 ; 5 2 ; b 2 D 2 4 ( 1 κ l '2 + 1 κ 0 '2 ) 1 ( cos 2 θ μ x 2 + sin 2 θ μ y 2 ) ) }.
F 2 1 ( a,b;c;z )= Γ( c )Γ( ba ) Γ( b )Γ( ca ) ( z ) a ( 1+ a( 1c+a ) 1b+a 1 z ) + Γ( c )Γ( ab ) Γ( a )Γ( cb ) ( z ) b ( 1+ b( 1c+b ) 1a+b 1 z ),| z |1
F 2 1 ( a,b;c;z )=1+ ab c z,| z |1
σ ( pl ) 2 ( α, μ x , μ y )= π 2 A ^ ( α ) C ^ n 2 LΓ( 2 α 2 ) 0 2π dθ ( cos 2 θ μ x 2 + sin 2 θ μ y 2 ) α 2 1 { ( b 2 D 2 4 ) 2+ α 2 [ b 2 D 2 4 + 1 κ 0 '2 ( cos 2 θ μ x 2 + sin 2 θ μ y 2 ) 1 ] 2+ α 2 },
σ sp 2 ( α, μ x , μ y )= π 2 A ^ ( α ) C ^ n 2 L 1 3 Γ( 2 α 2 ) 0 2π dθ ( cos 2 θ μ x 2 + sin 2 θ μ y 2 ) α 2 1 { 3 α1 ( b 2 D 2 4 ) α 2 2 [ κ 0 '2 ( cos 2 θ μ x 2 + sin 2 θ μ y 2 ) ] 2 α 2 [ 1 3 5 ( 2 α 2 ) b 2 D 2 κ 0 '2 4 ( cos 2 θ μ x 2 + sin 2 θ μ y 2 ) ] }.
rati o von = Φ n_aniso_von ( κ,α, μ x , μ y ) Φ n_von ( κ,α ) ,
rati o exp = Φ n_aniso_exp ( κ,α, μ x , μ y ) Φ n_exp ( κ,α ) .
1 sin κ ' R ' κ ' R ' = n=1 ( 1 ) n1 ( 2n+1 )! ( κ ' ) 2n ( R ' ) 2n ,
Γ( x )= 0 κ x1 e κ dκ ( κ>0,x>0 ) ,
F 1 1 ( a;b;z )= n=0 ( a ) n z n ( b ) n n! ,
( a ) n = Γ( a+n ) Γ( a ) =a( a+1 )( a+n1 ),
D n ( R ' ,α )=4π A ^ ( α ) C ^ n 2 ( κ l ' ) 3α Γ( α 2 + 3 2 )[ 1 F 1 1 ( α 2 + 3 2 ; 3 2 ; ( R ' ) 2 ( κ l ' ) 2 4 ) ].
D n ( R,α )={ C ^ n 2 l 0 α5 R 2 0R l 0 C ^ n 2 R α3 l 0 R L 0 .
D n ( R ' ,α )={ C ^ n 2 l 0 α5 ( R ' ) 2 0 R ' l 0 C ^ n 2 ( R ' ) α3 l 0 R ' L 0 .
F 1 1 ( a;b;x ) Γ( b ) Γ( ba ) x a ( x1 )
D n ( R ' ,α )4π A ^ ( α ) C ^ n 2 Γ( α 2 + 3 2 ) Γ( 3/2 ) Γ( α/2 ) ( 1 2 ) α3 ( R ' ) α3 ( l 0 R ' L 0 )
Γ( α+1 )=αΓ( α ),Γ( 1α )Γ( α )= π sin( πα ) , Γ( α )Γ( α+ 1 2 )= 2 12α π Γ( 2α ).
A ^ ( α )= Γ( α1 ) 4 π 2 sin[ ( α3 ) π 2 ].
F 1 1 ( a;b;x ) n=0 1 ( a ) n z n ( b ) n n! = 1+ a b x ( x1 )
D n ( R ' ,α )π A ^ ( α ) C ^ n 2 ( κ l ' ) 5α ( R ' ) 2 [ Γ( α 2 + 3 2 )( 3α 3 ) ] ( 0 R ' l 0 ) .
c ' ( α )= [ π A ^ ( α )Γ( α 2 + 3 2 )( 3α 3 ) ] 1 α5 .

Metrics