Abstract

In this paper, the concept of anisotropy at different atmospheric turbulence scales is introduced. A power spectrum and its associated structure function with inner and outer scale effects and anisotropy are also shown. The power spectrum includes an effective anisotropic parameter ζeff to describe anisotropy, which is useful for modeling optical turbulence when a non-Kolmogorov power law and anisotropy along the direction of propagation are present.

© 2014 Optical Society of America

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  5. 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]
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  10. R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375, 6–16 (1995).
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    [CrossRef]
  12. D. T. Kyrazis, F. D. Eaton, D. G. Black, W. T. Black, and R. A. Black,. “The balloon ring: a high-performance, low-cost instrumentation platform for measuring atmospheric turbulence profiles,” Proc. SPIE 7463, 746308 (2009).
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  15. I. Toselli, B. Agrawal, and S. Restaino, “Light propagation through anisotropic turbulence,” J. Opt. Soc. Am. A 28, 483–488 (2011).
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    [CrossRef]
  17. A. S. Gurvich and A. I. Kon, “Aspect sensitivity of radar returns from anisotropic turbulent irregularities,” J. Electromagn. Waves Appl. 7, 1343–1353 (1993).
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    [CrossRef]
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    [CrossRef]
  21. 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, 895–903 (1998).
    [CrossRef]
  22. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).
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    [CrossRef]
  26. A. S. Gurvich and V. Kan, “Radio wave fluctuations in satellite–atmosphere–satellite links: estimates from stellar scintillation observations and their comparison with experimental data,” Atmos. Oceanic Phys. 33, 284–292 (1997).
  27. 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. Eng. 47, 026003 (2008).
    [CrossRef]
  28. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov turbulence,” IEEE Trans. Antennas Propag. 57, 1783–1788 (2009).
    [CrossRef]
  29. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation in non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
    [CrossRef]
  30. V. S. Rao 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, 148–154 (2014).
    [CrossRef]

2014 (1)

2013 (2)

S. Gladysz, K. Stein, E. Sucher, and D. Sprung, “Measuring non-Kolmogorov turbulence,” Proc. SPIE 8890, 889013 (2013).
[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]

2012 (1)

2011 (1)

2009 (2)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov turbulence,” IEEE Trans. Antennas Propag. 57, 1783–1788 (2009).
[CrossRef]

D. T. Kyrazis, F. D. Eaton, D. G. Black, W. T. Black, and R. A. Black,. “The balloon ring: a high-performance, low-cost instrumentation platform for measuring atmospheric turbulence profiles,” Proc. SPIE 7463, 746308 (2009).
[CrossRef]

2008 (3)

2007 (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation in 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 (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, and C. L. Osmon, “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, 895–903 (1998).
[CrossRef]

1997 (1)

A. S. Gurvich and V. Kan, “Radio wave fluctuations in satellite–atmosphere–satellite links: estimates from stellar scintillation observations and their comparison with experimental data,” Atmos. Oceanic Phys. 33, 284–292 (1997).

1995 (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]

R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375, 6–16 (1995).

1994 (3)

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

A. I. Kon, “Qualitative theory of amplitude and phase fluctuations in a medium with anisotropic turbulent irregularities,” Waves Random Complex Media 4, 297–306 (1994).
[CrossRef]

1993 (1)

A. S. Gurvich and A. I. Kon, “Aspect sensitivity of radar returns from anisotropic turbulent irregularities,” J. Electromagn. Waves Appl. 7, 1343–1353 (1993).

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, 169–175 (1992).
[CrossRef]

1985 (1)

G. D. Nastrom and K. S. Gage, “A climatology of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft,” J. Atmos. Sci. 42, 950–960 (1985).
[CrossRef]

1970 (1)

1962 (1)

B. Bolgiano, “Structure of turbulence in stratified media,” J. Geophys. Res. 67, 3015–3023 (1962).
[CrossRef]

1960 (1)

L. R. Tsvang, “Measurements of the spectrum of temperature fluctuations in the free atmosphere,” Izvestiya Akademii Nauk SSSR, Geofizicheskaya 1, 1117–1120 (1960).

Agrawal, B.

Andrews, L. C.

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, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov turbulence,” IEEE Trans. Antennas Propag. 57, 1783–1788 (2009).
[CrossRef]

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. Eng. 47, 026003 (2008).
[CrossRef]

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

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

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]

Beland, R. R.

R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375, 6–16 (1995).

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

Black, D. G.

D. T. Kyrazis, F. D. Eaton, D. G. Black, W. T. Black, and R. A. Black,. “The balloon ring: a high-performance, low-cost instrumentation platform for measuring atmospheric turbulence profiles,” Proc. SPIE 7463, 746308 (2009).
[CrossRef]

Black, R. A.

D. T. Kyrazis, F. D. Eaton, D. G. Black, W. T. Black, and R. A. Black,. “The balloon ring: a high-performance, low-cost instrumentation platform for measuring atmospheric turbulence profiles,” Proc. SPIE 7463, 746308 (2009).
[CrossRef]

Black, W. T.

D. T. Kyrazis, F. D. Eaton, D. G. Black, W. T. Black, and R. A. Black,. “The balloon ring: a high-performance, low-cost instrumentation platform for measuring atmospheric turbulence profiles,” Proc. SPIE 7463, 746308 (2009).
[CrossRef]

Bolgiano, B.

B. Bolgiano, “Structure of turbulence in stratified media,” J. Geophys. Res. 67, 3015–3023 (1962).
[CrossRef]

Brown, J. M.

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]

Charnotskii, M.

Conan, J. M.

Consortini, A. A.

Crabbs, R.

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]

Dalaudier, F.

Eaton, F. D.

D. T. Kyrazis, F. D. Eaton, D. G. Black, W. T. Black, and R. A. Black,. “The balloon ring: a high-performance, low-cost instrumentation platform for measuring atmospheric turbulence profiles,” Proc. SPIE 7463, 746308 (2009).
[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, 895–903 (1998).
[CrossRef]

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov turbulence,” IEEE Trans. Antennas Propag. 57, 1783–1788 (2009).
[CrossRef]

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. Eng. 47, 026003 (2008).
[CrossRef]

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

Gage, K. S.

G. D. Nastrom and K. S. Gage, “A climatology of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft,” J. Atmos. Sci. 42, 950–960 (1985).
[CrossRef]

Gladysz, S.

S. Gladysz, K. Stein, E. Sucher, and D. Sprung, “Measuring non-Kolmogorov turbulence,” Proc. SPIE 8890, 889013 (2013).
[CrossRef]

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, 169–175 (1992).
[CrossRef]

Gurvich, A. S.

A. S. Gurvich and V. Kan, “Radio wave fluctuations in satellite–atmosphere–satellite links: estimates from stellar scintillation observations and their comparison with experimental data,” Atmos. Oceanic Phys. 33, 284–292 (1997).

A. S. Gurvich and A. I. Kon, “Aspect sensitivity of radar returns from anisotropic turbulent irregularities,” J. Electromagn. Waves Appl. 7, 1343–1353 (1993).

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, 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]

Kan, V.

A. S. Gurvich and V. Kan, “Radio wave fluctuations in satellite–atmosphere–satellite links: estimates from stellar scintillation observations and their comparison with experimental data,” Atmos. Oceanic Phys. 33, 284–292 (1997).

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, 169–175 (1992).
[CrossRef]

Karis, S. J.

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, and C. L. Osmon, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999).
[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]

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, 169–175 (1992).
[CrossRef]

Kon, A. I.

A. I. Kon, “Qualitative theory of amplitude and phase fluctuations in a medium with anisotropic turbulent irregularities,” Waves Random Complex Media 4, 297–306 (1994).
[CrossRef]

A. S. Gurvich and A. I. Kon, “Aspect sensitivity of radar returns from anisotropic turbulent irregularities,” J. Electromagn. Waves Appl. 7, 1343–1353 (1993).

Kopeika, N. S.

Kravtsov, Yu. A.

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation Through Random Media (Springer, 1988).

Kyrazis, D. T.

D. T. Kyrazis, F. D. Eaton, D. G. Black, W. T. Black, and R. A. Black,. “The balloon ring: a high-performance, low-cost instrumentation platform for measuring atmospheric turbulence profiles,” Proc. SPIE 7463, 746308 (2009).
[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]

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]

Michau, V.

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, 895–903 (1998).
[CrossRef]

G. D. Nastrom and K. S. Gage, “A climatology of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft,” J. Atmos. Sci. 42, 950–960 (1985).
[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, and C. L. Osmon, “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, 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, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov turbulence,” IEEE Trans. Antennas Propag. 57, 1783–1788 (2009).
[CrossRef]

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. Eng. 47, 026003 (2008).
[CrossRef]

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

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

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]

Procaccia, I.

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

Rao Gudimetla, V. S.

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]

Ronchi, L.

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]

Rytov, S. M.

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation Through Random Media (Springer, 1988).

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, 169–175 (1992).
[CrossRef]

Sidi, C.

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

Sprung, D.

S. Gladysz, K. Stein, E. Sucher, and D. Sprung, “Measuring non-Kolmogorov turbulence,” Proc. SPIE 8890, 889013 (2013).
[CrossRef]

Stefanutti, L.

Stein, K.

S. Gladysz, K. Stein, E. Sucher, and D. Sprung, “Measuring non-Kolmogorov turbulence,” Proc. SPIE 8890, 889013 (2013).
[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]

Sucher, E.

S. Gladysz, K. Stein, E. Sucher, and D. Sprung, “Measuring non-Kolmogorov turbulence,” Proc. SPIE 8890, 889013 (2013).
[CrossRef]

Tatarskii, V. I.

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation Through Random Media (Springer, 1988).

Toselli, I.

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

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov turbulence,” IEEE Trans. Antennas Propag. 57, 1783–1788 (2009).
[CrossRef]

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. Eng. 47, 026003 (2008).
[CrossRef]

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

Tsvang, L. R.

L. R. Tsvang, “Measurements of the spectrum of temperature fluctuations in the free atmosphere,” Izvestiya Akademii Nauk SSSR, Geofizicheskaya 1, 1117–1120 (1960).

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]

Ziberman, 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, 169–175 (1992).
[CrossRef]

Appl. Opt. (2)

Atmos. Oceanic Phys. (1)

A. S. Gurvich and V. Kan, “Radio wave fluctuations in satellite–atmosphere–satellite links: estimates from stellar scintillation observations and their comparison with experimental data,” Atmos. Oceanic Phys. 33, 284–292 (1997).

IEEE Trans. Antennas Propag. (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov turbulence,” IEEE Trans. Antennas Propag. 57, 1783–1788 (2009).
[CrossRef]

Izvestiya Akademii Nauk SSSR, Geofizicheskaya (1)

L. R. Tsvang, “Measurements of the spectrum of temperature fluctuations in the free atmosphere,” Izvestiya Akademii Nauk SSSR, Geofizicheskaya 1, 1117–1120 (1960).

J. Atmos. Sci. (2)

G. D. Nastrom and K. S. Gage, “A climatology of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft,” J. Atmos. Sci. 42, 950–960 (1985).
[CrossRef]

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

J. Electromagn. Waves Appl. (1)

A. S. Gurvich and A. I. Kon, “Aspect sensitivity of radar returns from anisotropic turbulent irregularities,” J. Electromagn. Waves Appl. 7, 1343–1353 (1993).

J. Geophys. Res. (1)

B. Bolgiano, “Structure of turbulence in stratified media,” J. Geophys. Res. 67, 3015–3023 (1962).
[CrossRef]

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

Opt. Eng. (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. Eng. 47, 026003 (2008).
[CrossRef]

Phys. Rep. (1)

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

Proc. SPIE (10)

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, and C. L. Osmon, “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]

R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375, 6–16 (1995).

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]

D. T. Kyrazis, F. D. Eaton, D. G. Black, W. T. Black, and R. A. Black,. “The balloon ring: a high-performance, low-cost instrumentation platform for measuring atmospheric turbulence profiles,” Proc. SPIE 7463, 746308 (2009).
[CrossRef]

S. Gladysz, K. Stein, E. Sucher, and D. Sprung, “Measuring non-Kolmogorov turbulence,” Proc. SPIE 8890, 889013 (2013).
[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).
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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]

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

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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, 895–903 (1998).
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A. I. Kon, “Qualitative theory of amplitude and phase fluctuations in a medium with anisotropic turbulent irregularities,” Waves Random Complex Media 4, 297–306 (1994).
[CrossRef]

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L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation Through Random Media (Springer, 1988).

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Figures (7)

Fig. 1.
Fig. 1.

(a) Anisotropy along the direction of propagation, z. (b) Anisotropy at different scales of turbulence cells.

Fig. 2.
Fig. 2.

Plot of Eq. (5) as a function of the scale size Rxy with L0=100m, ζmax=ζ(L0)=30, and Riso=0.1m.

Fig. 3.
Fig. 3.

Scaled power spectrum as a function of the wavenumber for several power law values α with ζeff=1 and κz=0.

Fig. 4.
Fig. 4.

Scaled power spectrum as a function of wavenumber for several power law values α and two cases of anisotropy: ζeff=2 and ζeff=5.

Fig. 5.
Fig. 5.

Scaled power spectrum as a function of wavenumber and power law values α with ζeff=13.89; the rescaling is visible along the α axis.

Fig. 6.
Fig. 6.

Scaled power spectrum as a function of wavenumber and anisotropy ζeff with α=11/3; the rescaling is visible along the ζeff axis.

Fig. 7.
Fig. 7.

Sketch of structure function.

Equations (33)

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ζ={ζ(Rxy),RisoRxyL01,l0RxyRiso,
Dn(R)=Cn2·R23=Cn2·[Rxy2ζ2(Rxy)+z2]13l0RxyL0.
Reff=RisoL0Rxy·ζ(Rxy)·dRxyRisoL0ζ(Rxy)·dRxy,
ζ=ζ(Rxy)=(ζmax1L0Riso)·(RxyRiso)+1,
Reff=L0+Riso2andζeff=1+ζmax2.
ζ=ζ(Rxy)=ζmax1(L0Riso)2(RxyRiso)2+1,
Reff=2(L02+L0Riso+Riso2)3Riso(L0+Riso)3(L0Riso),
Φn(κ,α)=A(α)·C˜n2·ζeff21(ζeff2·κxy2+κz2+κ02)α2·exp(ζeff2·κxy2+κz2κH2),κ>0,3<α<5,
C(α)={π·A(α)·Γ(32α2)·(3α3)}1α5,3<α<5,
A(α)=Γ(α1)4π2cos(απ2),3<α<5,
{x=ζeff·xy=ζeff·yz=zdR=ζeff2dR,
Φn(κ)=14π2κ20sin(κR)κR·R(R2Dn(R)R)·ζeff2dR,
Dn(R)=Cn2·R23,l0RL0.
Φn(κ)=ζeff2·0.033·Cn2·κ113,κ>0,
Φn(κ)=ζeff2·0.033·Cn2·(ζeff2κxy2+κz2)116,κ>0.
Dn(R,α)=8π0κ2·Φn(κ,α)·(1sin(κR)κR)dκ.
Φn(κ,α)=A(α)·C˜n2·ζeff2·1(κ2+κ02)α2·exp(κ2κH2)κ>0,3<α<5.
Dn(R,α)=8π0κ2·Φn(κ,α)·(1sin(κR)κR)1ζeff2dκ=4π·A(α)·C˜n2·κH3α·Γ(3α2)·[1F11(3α2;32;R2κH24)]
R=Rxy2+z2=Rxy2ζeff2+z2.
Dn(R,α)=C˜n2·Rα3=C˜n2·(Rxy2ζeff2+z2)α32,l0RL0,
σI_plane2(DG)=4π2·k2·A(α).(16DG2)1α2·Γ(1α2)·ζeff2α·h0HC˜n2(h){1[16·(hh0)k·DG2]α21·[(k·DG216·(hh0))2+1]12·(α21)·cos[(α21)·arctg(16·(hh0)k·DG2)]}dh,
Dn(R,α)=8π0κ2·Φn(κ,α)·(1sin(κR)κR)dκ.
x=ζeff·xκx=ζeff·κxy=ζeff·yκy=ζeff·κyz=zκz=κz,
Dn(R,α)=8π·A(α)·C˜n2·ζeff2·n=11(2n+1)!R2n·0κ2α+2n·exp(κ2κH2)·1ζeff2·dκ.
Dn(R,α)=C˜n2·Rα3,l0RL0,3<α<5.
Dn(R,α)=C˜n2·(Rxy2ζeff2+z2)α3=C˜n2·Rα3,l0RL0.
F11(a;b;x)Γ(b)Γ(ba)·xa.
Dn(R,α)=4π·A(α)·C˜n2·Γ(32α2)·κHα3·[1Γ(32)Γ(α2)(R·κH2)α3]4π·A(α)·C˜n2·Γ(32α2)·Γ(32)Γ(α2)(R2)α3.
A(α)2α34π·[Γ(32α2)·Γ(32)Γ(α2)]1.
Dn(R,α)=C˜n2·R2·l0α5,0Rl0;3<α<5.
Dn(R,α)=Dn(R,α),{0Rl0ζeff=1.
F11(a;b;x)1ab·x.
Dn(R,α)=4π·A(α)·C˜n2·κH5α·R24·Γ(32α2)·(3α3).

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