Abstract

The effect of anisotropic Kolmogorov turbulence on the log-amplitude correlation function for plane-wave fields is investigated using analysis, numerical integration, and simulation. A new analytical expression for the log-amplitude correlation function is derived for anisotropic Kolmogorov turbulence. The analytic results, based on the Rytov approximation, agree well with a more general wave-optics simulation based on the Fresnel approximation as well as with numerical evaluations, for low and moderate strengths of turbulence. The new expression reduces correctly to previously published analytic expressions for isotropic turbulence. The final results indicate that, as asymmetry becomes greater, the Rytov variance deviates from that given by the standard formula. This deviation becomes greater with stronger turbulence, up to moderate turbulence strengths. The anisotropic effects on the log-amplitude correlation function are dominant when the separation of the points is within the Fresnel length. In the direction of stronger turbulence, there is an enhanced dip in the correlation function at a separation close to the Fresnel length. The dip is diminished in the weak-turbulence axis, suggesting that energy redistribution via focusing and defocusing is dominated by the strong-turbulence axis. The new analytical expression is useful when anisotropy is observed in relevant experiments.

© 2012 Optical Society of America

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2011 (1)

2008 (1)

2006 (1)

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

2004 (1)

V. E. Ostashev, D. K. Wilson, and G. H. Goedecke, “Spherical wave propagation through inhomogeneous, anisotropic turbulence: log-amplitude and phase correlations,” J. Acoust. Soc. Am. 115, 120–130 (2004).
[CrossRef]

2001 (1)

V. E. Ostashev and D. K. Wilson, “Log-amplitude and phase fluctuations of a plane wave propagating through anisotropic, inhomogeneous turbulence,” Acta Acust. Acust. (Acta Acustica united with Acustica) 87, 685–694 (2001).

1999 (1)

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]

1995 (3)

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).

V. P. Lukin, “Investigation of the anisotropy of the atmospheric turbulence spectrum in the low-frequency range,” Proc. SPIE 2471, 347–354 (1995).
[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).
[CrossRef]

1994 (1)

V. P. Lukin, “Investigation of some peculiarities in the structure of large scale atmospheric turbulence,” Proc. SPIE 2200, 384–395 (1994).
[CrossRef]

1992 (1)

B. V. Fortes and V. P. Lukin, “Modeling of the image observed through the turbulent atmosphere,” Proc. SPIE 1668, 477–488 (1992).
[CrossRef]

1990 (2)

A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, M. K. H. Manarov, A. I. Pokhomov, Yu. V. Romanenko, S. A. Savchenko, S. I. Serova, and Y. G. Titov, “Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars,” Izv. Atmos. Ocean. Phys. 26, 5–16(1990).

J. M. Martin and S. M. Flatté, “Simulation of point-source scintillation through three-dimensional random media,” J. Opt. Soc. Am. A 7, 838–847 (1990).
[CrossRef]

1988 (1)

1986 (1)

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

1970 (1)

1966 (1)

Agrawal, B.

Aleksandrov, A. P.

A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, M. K. H. Manarov, A. I. Pokhomov, Yu. V. Romanenko, S. A. Savchenko, S. I. Serova, and Y. G. Titov, “Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars,” Izv. Atmos. Ocean. Phys. 26, 5–16(1990).

Andrews, Larry C.

Larry C. Andrews and R. L. Phillips, Laser Beam Propagation in Random Media (SPIE, 2005).

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).

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 A. V. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U (2006).
[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]

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

Beresnev, L. A.

M. Vorontsov, G. W. Carhart, V. S. R. Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149 km propagation path using multi-wavelength laser beacons,” in Proceedings of the Advanced Maui Optical and Space Surveillance Technologies Conference, P. Kervan, ed. (Maui Economic Development Board, 2010), pp. 184–195.

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, 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]

Carhart, G. W.

M. Vorontsov, G. W. Carhart, V. S. R. Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149 km propagation path using multi-wavelength laser beacons,” in Proceedings of the Advanced Maui Optical and Space Surveillance Technologies Conference, P. Kervan, ed. (Maui Economic Development Board, 2010), pp. 184–195.

Clifford, S. F.

S. F. Clifford, “The classical theory of wave propagation in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, 1978), pp. 9–43.

Cloud, J. D.

Conan, J.-M.

Consortini, A.

Cuellar, E.

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

Dalaudier, F.

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).

Flatté, S. M.

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).

B. V. Fortes and V. P. Lukin, “Modeling of the image observed through the turbulent atmosphere,” Proc. SPIE 1668, 477–488 (1992).
[CrossRef]

Fried, D. L.

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]

Goedecke, G. H.

V. E. Ostashev, D. K. Wilson, and G. H. Goedecke, “Spherical wave propagation through inhomogeneous, anisotropic turbulence: log-amplitude and phase correlations,” J. Acoust. Soc. Am. 115, 120–130 (2004).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1965), p. 973.

Grechko, G. M.

A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, M. K. H. Manarov, A. I. Pokhomov, Yu. V. Romanenko, S. A. Savchenko, S. I. Serova, and Y. G. Titov, “Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars,” Izv. Atmos. Ocean. Phys. 26, 5–16(1990).

Gudimetla, V. S. R.

M. Vorontsov, G. W. Carhart, V. S. R. Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149 km propagation path using multi-wavelength laser beacons,” in Proceedings of the Advanced Maui Optical and Space Surveillance Technologies Conference, P. Kervan, ed. (Maui Economic Development Board, 2010), pp. 184–195.

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).
[CrossRef]

A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, M. K. H. Manarov, A. I. Pokhomov, Yu. V. Romanenko, S. A. Savchenko, S. I. Serova, and Y. G. Titov, “Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars,” Izv. Atmos. Ocean. Phys. 26, 5–16(1990).

Hughes, K. A.

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

Kan, V.

A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, M. K. H. Manarov, A. I. Pokhomov, Yu. V. Romanenko, S. A. Savchenko, S. I. Serova, and Y. G. Titov, “Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars,” Izv. Atmos. Ocean. Phys. 26, 5–16(1990).

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]

Lachinova, S. L.

M. Vorontsov, G. W. Carhart, V. S. R. Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149 km propagation path using multi-wavelength laser beacons,” in Proceedings of the Advanced Maui Optical and Space Surveillance Technologies Conference, P. Kervan, ed. (Maui Economic Development Board, 2010), pp. 184–195.

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).

Liu, J.

M. Vorontsov, G. W. Carhart, V. S. R. Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149 km propagation path using multi-wavelength laser beacons,” in Proceedings of the Advanced Maui Optical and Space Surveillance Technologies Conference, P. Kervan, ed. (Maui Economic Development Board, 2010), pp. 184–195.

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).

V. P. Lukin, “Investigation of the anisotropy of the atmospheric turbulence spectrum in the low-frequency range,” Proc. SPIE 2471, 347–354 (1995).
[CrossRef]

V. P. Lukin, “Investigation of some peculiarities in the structure of large scale atmospheric turbulence,” Proc. SPIE 2200, 384–395 (1994).
[CrossRef]

B. V. Fortes and V. P. Lukin, “Modeling of the image observed through the turbulent atmosphere,” Proc. SPIE 1668, 477–488 (1992).
[CrossRef]

V. P. Lukin, “Investigation of the anisotropy of the atmospheric turbulence spectrum in the low frequency range,” in Proceedings of International Geoscience and Remote Sensing Symposium, 1996. Remote Sensing for a Sustainable Future (IEEE, 1996), Vol. 1, pp. 22–24.

Manarov, M. K. H.

A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, M. K. H. Manarov, A. I. Pokhomov, Yu. V. Romanenko, S. A. Savchenko, S. I. Serova, and Y. G. Titov, “Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars,” Izv. Atmos. Ocean. Phys. 26, 5–16(1990).

Manning, R. M.

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

Martin, J. M.

Michau, V.

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]

Ostashev, V. E.

V. E. Ostashev, D. K. Wilson, and G. H. Goedecke, “Spherical wave propagation through inhomogeneous, anisotropic turbulence: log-amplitude and phase correlations,” J. Acoust. Soc. Am. 115, 120–130 (2004).
[CrossRef]

V. E. Ostashev and D. K. Wilson, “Log-amplitude and phase fluctuations of a plane wave propagating through anisotropic, inhomogeneous turbulence,” Acta Acust. Acust. (Acta Acustica united with Acustica) 87, 685–694 (2001).

Phillips, R. L.

Larry C. Andrews and R. L. Phillips, Laser Beam Propagation in Random Media (SPIE, 2005).

Pokhomov, A. I.

A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, M. K. H. Manarov, A. I. Pokhomov, Yu. V. Romanenko, S. A. Savchenko, S. I. Serova, and Y. G. Titov, “Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars,” Izv. Atmos. Ocean. Phys. 26, 5–16(1990).

Rehder, K.

M. Vorontsov, G. W. Carhart, V. S. R. Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149 km propagation path using multi-wavelength laser beacons,” in Proceedings of the Advanced Maui Optical and Space Surveillance Technologies Conference, P. Kervan, ed. (Maui Economic Development Board, 2010), pp. 184–195.

Renard, J.-B.

Restaino, S.

Riker, J. F.

M. Vorontsov, G. W. Carhart, V. S. R. Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149 km propagation path using multi-wavelength laser beacons,” in Proceedings of the Advanced Maui Optical and Space Surveillance Technologies Conference, P. Kervan, ed. (Maui Economic Development Board, 2010), pp. 184–195.

Robert, C.

Romanenko, Yu. V.

A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, M. K. H. Manarov, A. I. Pokhomov, Yu. V. Romanenko, S. A. Savchenko, S. I. Serova, and Y. G. Titov, “Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars,” Izv. Atmos. Ocean. Phys. 26, 5–16(1990).

Ronchi, L.

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, A. V.

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

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1965), p. 973.

Sasiela, Richard J.

Richard J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, 1994).

Savchenko, S. A.

A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, M. K. H. Manarov, A. I. Pokhomov, Yu. V. Romanenko, S. A. Savchenko, S. I. Serova, and Y. G. Titov, “Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars,” Izv. Atmos. Ocean. Phys. 26, 5–16(1990).

Schmidt, J. D.

J. D. Schmidt, Numerical Simulation Of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).

Serova, S. I.

A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, M. K. H. Manarov, A. I. Pokhomov, Yu. V. Romanenko, S. A. Savchenko, S. I. Serova, and Y. G. Titov, “Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars,” Izv. Atmos. Ocean. Phys. 26, 5–16(1990).

Stefanutti, L.

Stevenson, E.

M. Vorontsov, G. W. Carhart, V. S. R. Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149 km propagation path using multi-wavelength laser beacons,” in Proceedings of the Advanced Maui Optical and Space Surveillance Technologies Conference, P. Kervan, ed. (Maui Economic Development Board, 2010), pp. 184–195.

Titov, Y. G.

A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, M. K. H. Manarov, A. I. Pokhomov, Yu. V. Romanenko, S. A. Savchenko, S. I. Serova, and Y. G. Titov, “Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars,” Izv. Atmos. Ocean. Phys. 26, 5–16(1990).

Toselli, I.

Vorontsov, M.

M. Vorontsov, G. W. Carhart, V. S. R. Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149 km propagation path using multi-wavelength laser beacons,” in Proceedings of the Advanced Maui Optical and Space Surveillance Technologies Conference, P. Kervan, ed. (Maui Economic Development Board, 2010), pp. 184–195.

Weyrauch, T.

M. Vorontsov, G. W. Carhart, V. S. R. Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149 km propagation path using multi-wavelength laser beacons,” in Proceedings of the Advanced Maui Optical and Space Surveillance Technologies Conference, P. Kervan, ed. (Maui Economic Development Board, 2010), pp. 184–195.

Wheelon, A. D.

A. D. Wheelon, Electromagnetic Scintillation (Cambridge University, 2001), Vol. 1, Chaps. 2 and 4.

Wheelon, Albert D.

Albert D. Wheelon, Electromagnetic Scintillation (Cambridge University, 2001), Vol. 2, Chap. 3.

Wilson, D. K.

V. E. Ostashev, D. K. Wilson, and G. H. Goedecke, “Spherical wave propagation through inhomogeneous, anisotropic turbulence: log-amplitude and phase correlations,” J. Acoust. Soc. Am. 115, 120–130 (2004).
[CrossRef]

V. E. Ostashev and D. K. Wilson, “Log-amplitude and phase fluctuations of a plane wave propagating through anisotropic, inhomogeneous turbulence,” Acta Acust. Acust. (Acta Acustica united with Acustica) 87, 685–694 (2001).

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).

Acta Acust. Acust. (1)

V. E. Ostashev and D. K. Wilson, “Log-amplitude and phase fluctuations of a plane wave propagating through anisotropic, inhomogeneous turbulence,” Acta Acust. Acust. (Acta Acustica united with Acustica) 87, 685–694 (2001).

Appl. Opt. (2)

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. Antennas Propag. (1)

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

Izv. Atmos. Ocean. Phys. (1)

A. P. Aleksandrov, G. M. Grechko, A. S. Gurvich, V. Kan, M. K. H. Manarov, A. I. Pokhomov, Yu. V. Romanenko, S. A. Savchenko, S. I. Serova, and Y. G. Titov, “Spectra of temperature variations in the stratosphere as indicated by satellite-borne observation of the twinkling of stars,” Izv. Atmos. Ocean. Phys. 26, 5–16(1990).

J. Acoust. Soc. Am. (1)

V. E. Ostashev, D. K. Wilson, and G. H. Goedecke, “Spherical wave propagation through inhomogeneous, anisotropic turbulence: log-amplitude and phase correlations,” J. Acoust. Soc. Am. 115, 120–130 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Proc. SPIE (5)

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[CrossRef]

B. V. Fortes and V. P. Lukin, “Modeling of the image observed through the turbulent atmosphere,” Proc. SPIE 1668, 477–488 (1992).
[CrossRef]

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[CrossRef]

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

Other (9)

M. Vorontsov, G. W. Carhart, V. S. R. Gudimetla, T. Weyrauch, E. Stevenson, S. L. Lachinova, L. A. Beresnev, J. Liu, K. Rehder, and J. F. Riker, “Characterization of atmospheric turbulence effects over 149 km propagation path using multi-wavelength laser beacons,” in Proceedings of the Advanced Maui Optical and Space Surveillance Technologies Conference, P. Kervan, ed. (Maui Economic Development Board, 2010), pp. 184–195.

A. D. Wheelon, Electromagnetic Scintillation (Cambridge University, 2001), Vol. 1, Chaps. 2 and 4.

Albert D. Wheelon, Electromagnetic Scintillation (Cambridge University, 2001), Vol. 2, Chap. 3.

V. P. Lukin, “Investigation of the anisotropy of the atmospheric turbulence spectrum in the low frequency range,” in Proceedings of International Geoscience and Remote Sensing Symposium, 1996. Remote Sensing for a Sustainable Future (IEEE, 1996), Vol. 1, pp. 22–24.

S. F. Clifford, “The classical theory of wave propagation in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, 1978), pp. 9–43.

Larry C. Andrews and R. L. Phillips, Laser Beam Propagation in Random Media (SPIE, 2005).

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1965), p. 973.

Richard J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, 1994).

J. D. Schmidt, Numerical Simulation Of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).

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Figures (7)

Fig. 1.
Fig. 1.

Comparison of numerical evaluation of Eq. (36) using quasi-Monte-Carlo approach (crosses) with corresponding values from the analytical expressions (solid line) for at three different values of m at several values of separation of points, normalized to Fresnel length.

Fig. 2.
Fig. 2.

Comparison of the log-amplitude correlation values from the analytical expressions herein and the Fried–Cloud expression for the special case of isotropic turbulence.

Fig. 3.
Fig. 3.

Comparison of the wave-optics simulation and analytical log-amplitude correlation data [(left) X-axis data, (right) Y-axis data]. Range=30km, wavelength=0.737μm, Cn2=1017m2/3, anisotropy ε=0.5.

Fig. 4.
Fig. 4.

Comparison of the wave-optics simulation and analytical log-amplitude correlation data [(left) X-axis data, (right) Y-axis data]. Range=60km, wavelength=0.737μm, Cn2=1017m2/3, anisotropy ε=1.0.

Fig. 5.
Fig. 5.

Dependence of Y-axis log-amplitude autocorrelation function on the anisotropy factor. Range=60km, wavelength=0.737μm, Cn2=0.5×1017m2/3. Wave-optics simulation results: solid, anisotropy ε=0; dotted, ε=1; dashed, ε=2; dotted–dashed, ε=4. Analytic results: circles, ε=0; squares, ε=1; triangles, ε=2; crosses, ε=4.

Fig. 6.
Fig. 6.

Comparison of the wave-optics simulation and analytical log-amplitude correlation data for a long-range laser communication link [(left) X-axis data, (right) Y-axis data]. Range=200km, wavelength=1.55μm, and Cn2=0.1×1017m2/3, anisotropy ε=1.0.

Fig. 7.
Fig. 7.

(Left) X- and Y-log-amplitude correlations, range=100km, wavelength=0.737μm wavelength, Cn2=0.275×1017m2/3, anisotropy ε=2. Solid, X simulation; dotted, Y simulation; crosses, X analytic; circles, Y analytic. (Right) Variability for three different wave-optics simulation runs.

Equations (38)

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ϕ(K)=0.033Cn2abc(a2Kx2+b2Ky2+c2Kz2+K02)11/6,
ϕn3d(K)=η2N(h)2Cw(h)K5[1+(lwK)2],
ϕn(K)=0.033Cn2(a2Kx2+b2Ky2+Kz2)11/6
ϕ(K)=0.033Cn2(ab)11/6(a2Kx2+b2Ky2)11/6.
ϕ(K)=0.033Cn2K11/3(1+ε2)11/12(1+ε2sin2(θK))11/6
Bχ(ρ)=χ(ρ1)χ(ρ1+ρ)=k2eiK·(ρ1ρ)+iK·ρ10zdz0zdzsin[K2(zz)2k]sin[K2(zz)2k]dν(K,z)dν*(K,z),
dν(K,z)dν*(K,z)=δ(KK)Fn(K,zz)d2Kd2K.
Bχ(ρ,z)=k2d2KeiK·ρ0zdz0zdzsin[K2(zz)2k]sin[K2(zz)2k]Fn(K,zz).
Fn(K,zz)=+dKzϕn(K,Kz)cos(Kz(zz)),
Bχ(ρ)=k2d2KeiK·ρ0LdξLξ/2L+ξ/2dη{cos[K2ξ2k]cos[K2η2k]}Fn(K,ξ).
Bχ(ρ)=k2d2KeiK·ρLξ/2L+ξ/2dη{1cos[K2η2k]}0LdξFn(K,ξ).
Bχ(ρ)=k2Ld2KeiK·ρ{1kK2Lsin[K2Lk]}0LdξFn(K,ξ).
0LdξFn(K,ξ)=πϕn(K,0),
Bχ(ρ)=k2d2KeiK·ρ{1kK2Lsin[K2Lk]}πϕ(K).
Bχ(ρ,θρ)=πk2L0KdK02πdθKeiKρcos(θKθρ){1kK2Lsin[K2Lk]}ϕ(K)=2π2k2L0KdKJ0(Kρ){1kK2Lsin[K2Lk]}ϕ(K).
Bχ(ρ,θρ)=πk2L0.033Cn2(1+ε2)11/12d2KeiK·ρ{1kK2Lsin[K2Lk]}K11/3(1+ε2sin2(θK))11/6=0.033πk2LCn2(1+ε2)11/120dKK8/3{1kK2Lsin[K2Lk]}02πdθKeiKρcos(θKθρ)1(1+ε2sin2(θK))11/6.
exp(iKρcos(θρθK))=J0(Kρ)+2m=1imJm(Kρ)cos(m(θρθK)).
02πdθK1(1+ε2sin2(θK))11/6[J0(Kρ)+2m=1imJm(Kρ)cos(m(θρθK))]=J0(Kρ)π+πdθK1(1+ε2sin2(θK))11/6+2m=1imJm(Kρ)cos(mθρ)π+πdθKcos(mθK)(1+ε2sin2(θK))11/6.
02πdθK1(1+ε2sin2(θK))11/6[J0(Kρ)+2m=1imJm(Kρ)cos(m(θρθK))]=J0(Kρ)π+πdθK1(1+ε2sin2(θK))11/6+2m=1(1)mJ2m(Kρ)cos(2mθρ)π+πdθKcos(2mθK)(1+ε2sin2(θK))11/6.
Bχ(ρ,θρ)=0.033πk2LCn2(1+ε2)11/12F(0,ε2)0dKK8/3{1kK2Lsin[K2Lk]}J0(Kρ)F(0,ε2)+2(0.033πk2LCn2)(1+ε2)11/12m=1(1)mcos(2mθρ)F(m,ε2)0dKK8/3{1kK2Lsin[K2Lk]}J2m(Kρ),
F(m,ε2)=π+πdθKcos(2mθK)(1+ε2sin2(θK))11/6.
Bχ(ρ,θρ)=120.033πk7/6L11/6Cn2(1+ε2)11/12F(0,ε2)0dqq11/6{1sin(q)q}J0((kLρ2)1/2q1/2)+(0.033πk7/6L11/6Cn2)(1+ε2)11/12m=1(1)mcos(2mθρ)F(m,ε2)0dqq11/6{1sin(q)q}J2m((kLρ2)1/2q1/2).
Qm=0dqq11/6{1sin(q)q}J2m((kLρ2)1/2q1/2).
Qm1=0dqq11/6J2m((kLρ2)1/2q1/2),
Qm2=0dqq17/6sin(q)J2m((kLρ2)1/2q1/2).
Qm1=2(kLρ2)5/60dxx8/3J2m(x)=2(kLρ2)5/628/3Γ[5/6+mm+11/6].
Qm1=(k8Lρ2)5/6217/6πΓ[5/12+m/2m/2+1/12m/2+11/12m/2+17/123/2].
Qm2=π2πids(kLρ2)s2s17/6Γ[s/25/12s+ms/2+23/12s+1+m].
Qm2=217/6π2πids(k8Lρ2)2sΓ[s5/12s+m/2s+m/2+1/2s+23/12s+m/2+1/2s+m/2+1].
s=m/2+n1withn1=0,1,2s=m/2+1/2+n2withn2=0,1,2s=5/12n3withn3=0.
Qm2=Qm2a+Qm2b,
Qm2a=π×217/6n=0(1)nn!(k8Lρ2)2n+mΓ[n+1/25/12+n+m/2n+m+1/2n+m+123/12nm/2].
Qm2a=217/6π(k8Lρ2)mΓ[m/25/121/2m+1/2m+123/12m/2]{F23(m/25/12,m/211/12;1/2,m+1/2,m+1;(k8Lρ2)2)}.
Qm2b=π×217/6n=0(1)nn!(k8Lρ2)2n+m+1Γ[n1/25/12+n+m/2+1/2n+m+1n+m+3/223/12nm/21/2].
Qm2b=217/6π(k8Lρ2)m+1Γ[m/2+1/121/2m+1m+3/217/12m/2]{F22(m/2+1/12,m/25/12;3/2,m+1,m+3/2;(k8Lρ2)2)}.
0dqq11/6{1sin(q)q}J2m((kLρ2)1/2q1/2)=Qm1Qm2aQm2b.
Bχ(ρ,θρ)=0.0165πk7/6L11/6Cn2(1+ε2)11/12F(0,ε2)[Qm1+Qm2a+Qm2b]m=0+(0.033πk7/6L11/6Cn2)(1+ε2)11/12m=1(1)mcos(2mθρ)F(m,ε2)[Qm1Qm2aQm2b].
σR2=0.3072Cn2k7/6L11/6.

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