Abstract

Nowadays it has been accepted that the Kolmogorov model is not the only possible turbulent one in the atmosphere, which has been confirmed by the increasing experimental evidences and some results of theoretical investigation. This has prompted the scientist community to study optical propagation in non-Kolmogorov atmospheric turbulence. In this paper, using a non-Kolmogorov power spectrum which has a more general power law instead of standard Kolmogorov power law value 11/3 and a more general amplitude factor instead of constant value 0.033, the temporal power spectra of the presentative amplitude and phase effects, irradiance and angle of arrival fluctuations, have been derived for horizontal link in weak turbulence. And then the influence of spectral power-law variations on the temporal power spectrum has been analyzed. It is anticipated that this work is helpful to the investigations of atmospheric turbulence and optical wave propagation in the atmospheric turbulence.

© 2010 Optical Society of America

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  1. J. C. Ricklin, S. M. Hammel, F. D. Eaton, and S. L. Lachinova, "Atmospheric channel effects on free-space laser communication," J. Opt. Fiber. Commun. Rep. 3, 111-158 (2006).
    [CrossRef]
  2. K. Kazaura, K. Omae, T. Suzuki, and M. Matsumoto, "Enhancing performance of next generation FSO communication systems using soft computing-based predictions," Opt. Express 14, 4958-4968 (2006).
    [CrossRef] [PubMed]
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    [CrossRef]
  5. D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, "Measurement of optical turbulence in the upper troposphere and lower stratosphere," Proc. SPIE 2110, 43-55 (1994).
    [CrossRef]
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    [CrossRef]
  7. A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, "Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence," Atmospheric Research 88, 66-77 (2008).
    [CrossRef]
  8. 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-1-12 (2006).
    [CrossRef]
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    [CrossRef]
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  14. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, "Optical propagation in non-Kolmogorov atmospheric turbulence," Proc. SPIE 2471, 181-196 (1995).
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    [CrossRef]
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    [CrossRef]
  22. D. M. Winker, "Effect of a finite outer scale on the Zernike decomposition of atmospheric optical turbulence," J. Opt. Soc. Am. A 11, 1568-1574 (1991).
    [CrossRef]
  23. P. Hickson, "Wave-front curvature sensing from a single defocused image," J. Opt. Soc. Am. 11, 1667-1673 (1994).
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  24. A. Erd’elyi, Tables of Integral Transforms, (McGraw-Hill, New York, 1959).
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    [CrossRef]
  30. Gao Chong, Ma Jing, and Tan Liying, "Angle-of-arrival fluctuation of light beam propagation in strong turbulence regime," High Power Laser and Particle Beams 18, 891-894 (2006).
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    [CrossRef]

2009 (1)

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

2008 (1)

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

2006 (3)

Gao Chong, Ma Jing, and Tan Liying, "Angle-of-arrival fluctuation of light beam propagation in strong turbulence regime," High Power Laser and Particle Beams 18, 891-894 (2006).

J. C. Ricklin, S. M. Hammel, F. D. Eaton, and S. L. Lachinova, "Atmospheric channel effects on free-space laser communication," J. Opt. Fiber. Commun. Rep. 3, 111-158 (2006).
[CrossRef]

K. Kazaura, K. Omae, T. Suzuki, and M. Matsumoto, "Enhancing performance of next generation FSO communication systems using soft computing-based predictions," Opt. Express 14, 4958-4968 (2006).
[CrossRef] [PubMed]

2004 (1)

2000 (2)

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

R. Conan, J. Borgnino, A. Ziad, and F. Martin, "Analytical solution for the covariance and for the decorrelation time of the angle of arrival of a wave front corrugated by atmospheric turbulence," J. Opt. Soc. Am. A 17, 1807-1818 (2000).
[CrossRef]

1997 (1)

M. S. Be1en’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, "Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion," Proc. SPIE 3126, 113-123 (1997).
[CrossRef]

1996 (3)

S. S. Moiseev and O. G. Chkhetiani, "Helical scaling in turbulence," JETP 83, 192-198 (1996).

T. Elperin, N. Kleeorin, and I. Rogachevskii, "Isotropic and anisotropic spectra of passive scalar fluctuations in turbulent fluid flow," Phys. Rev. E 53, 3431-3441 (1996).
[CrossRef]

G. D. Boreman and C. Dainty, "Zernike expansions for non-Kolmogorov turbulence," J. Opt. Soc. Am. A 13, 517-522 (1996).
[CrossRef]

1995 (5)

1994 (2)

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, "Measurement of optical turbulence in the upper troposphere and lower stratosphere," Proc. SPIE 2110, 43-55 (1994).
[CrossRef]

P. Hickson, "Wave-front curvature sensing from a single defocused image," J. Opt. Soc. Am. 11, 1667-1673 (1994).
[CrossRef]

1991 (1)

D. M. Winker, "Effect of a finite outer scale on the Zernike decomposition of atmospheric optical turbulence," J. Opt. Soc. Am. A 11, 1568-1574 (1991).
[CrossRef]

1971 (1)

1959 (1)

G. K. Batchelor, "Small-scale variation of convected quantities like temperature in turbulent fluid. Part I. General discussion and the case of small conductivity," J. Fluid Mech. 5, 113-133 (1959).
[CrossRef]

Andrews, L. C.

Batchelor, G. K.

G. K. Batchelor, "Small-scale variation of convected quantities like temperature in turbulent fluid. Part I. General discussion and the case of small conductivity," J. Fluid Mech. 5, 113-133 (1959).
[CrossRef]

Be, M. S.

M. S. Be1en’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, "Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion," Proc. SPIE 3126, 113-123 (1997).
[CrossRef]

Beland, R. R.

R. R. Beland, "Some aspects of propagation through weak isotropic non-Kolmogorov turbulence," Proc. SPIE 2375, 1111-1126 (1995).

Belen’kii, M. S.

Bishop, K. P.

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, "Measurement of optical turbulence in the upper troposphere and lower stratosphere," Proc. SPIE 2110, 43-55 (1994).
[CrossRef]

Boreman, G. D.

Borgnino, J.

Chkhetiani, O. G.

S. S. Moiseev and O. G. Chkhetiani, "Helical scaling in turbulence," JETP 83, 192-198 (1996).

Chong, Gao

Gao Chong, Ma Jing, and Tan Liying, "Angle-of-arrival fluctuation of light beam propagation in strong turbulence regime," High Power Laser and Particle Beams 18, 891-894 (2006).

Clifford, S. F.

Conan, R.

Dainty, C.

Du, W.

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

Eaton, F. D.

J. C. Ricklin, S. M. Hammel, F. D. Eaton, and S. L. Lachinova, "Atmospheric channel effects on free-space laser communication," J. Opt. Fiber. Commun. Rep. 3, 111-158 (2006).
[CrossRef]

Elperin, T.

T. Elperin, N. Kleeorin, and I. Rogachevskii, "Isotropic and anisotropic spectra of passive scalar fluctuations in turbulent fluid flow," Phys. Rev. E 53, 3431-3441 (1996).
[CrossRef]

Golbraikh, E.

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

E. Golbraikh and N. S. Kopeika, "Behavior of structure function of refraction coefficients in different trubulent fields," Appl. Opt. 43, 6151-6156 (2004).
[CrossRef] [PubMed]

Gurvich, A. S.

Hammel, S. M.

J. C. Ricklin, S. M. Hammel, F. D. Eaton, and S. L. Lachinova, "Atmospheric channel effects on free-space laser communication," J. Opt. Fiber. Commun. Rep. 3, 111-158 (2006).
[CrossRef]

Hickson, P.

P. Hickson, "Wave-front curvature sensing from a single defocused image," J. Opt. Soc. Am. 11, 1667-1673 (1994).
[CrossRef]

Jiang, W.

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

Jiang, Y.

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

Jing, Ma

Gao Chong, Ma Jing, and Tan Liying, "Angle-of-arrival fluctuation of light beam propagation in strong turbulence regime," High Power Laser and Particle Beams 18, 891-894 (2006).

Kazaura, K.

Keating, D. B.

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, "Measurement of optical turbulence in the upper troposphere and lower stratosphere," Proc. SPIE 2110, 43-55 (1994).
[CrossRef]

Kleeorin, N.

T. Elperin, N. Kleeorin, and I. Rogachevskii, "Isotropic and anisotropic spectra of passive scalar fluctuations in turbulent fluid flow," Phys. Rev. E 53, 3431-3441 (1996).
[CrossRef]

Kopeika, N. S.

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

E. Golbraikh and N. S. Kopeika, "Behavior of structure function of refraction coefficients in different trubulent fields," Appl. Opt. 43, 6151-6156 (2004).
[CrossRef] [PubMed]

Kupershmidt, I.

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

Kyrazis, D. T.

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, "Measurement of optical turbulence in the upper troposphere and lower stratosphere," Proc. SPIE 2110, 43-55 (1994).
[CrossRef]

Lachinova, S. L.

J. C. Ricklin, S. M. Hammel, F. D. Eaton, and S. L. Lachinova, "Atmospheric channel effects on free-space laser communication," J. Opt. Fiber. Commun. Rep. 3, 111-158 (2006).
[CrossRef]

Ling, N.

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

Liying, Tan

Gao Chong, Ma Jing, and Tan Liying, "Angle-of-arrival fluctuation of light beam propagation in strong turbulence regime," High Power Laser and Particle Beams 18, 891-894 (2006).

Ma, J.

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

Martin, F.

Matsumoto, M.

Moiseev, S. S.

S. S. Moiseev and O. G. Chkhetiani, "Helical scaling in turbulence," JETP 83, 192-198 (1996).

Omae, K.

Phillips, R. L.

Preble, A. J.

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, "Measurement of optical turbulence in the upper troposphere and lower stratosphere," Proc. SPIE 2110, 43-55 (1994).
[CrossRef]

Rao, C.

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

Ricklin, J. C.

J. C. Ricklin, S. M. Hammel, F. D. Eaton, and S. L. Lachinova, "Atmospheric channel effects on free-space laser communication," J. Opt. Fiber. Commun. Rep. 3, 111-158 (2006).
[CrossRef]

Rogachevskii, I.

T. Elperin, N. Kleeorin, and I. Rogachevskii, "Isotropic and anisotropic spectra of passive scalar fluctuations in turbulent fluid flow," Phys. Rev. E 53, 3431-3441 (1996).
[CrossRef]

Roggemann, M. C.

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

Shtemler, Y.

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

Suzuki, T.

Tan, L.

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

Virtser, A.

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

Welsh, B. M.

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

Winker, D. M.

D. M. Winker, "Effect of a finite outer scale on the Zernike decomposition of atmospheric optical turbulence," J. Opt. Soc. Am. A 11, 1568-1574 (1991).
[CrossRef]

Wissler, J.

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, "Measurement of optical turbulence in the upper troposphere and lower stratosphere," Proc. SPIE 2110, 43-55 (1994).
[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-Kolmogorov turbulence," Opt. Commun. 282, 705-708 (2009).
[CrossRef]

Yu, P. T.

Yu, S.

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

Ziad, A.

Zilberman, A.

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

Appl. Opt. (2)

Atmospheric Research (1)

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

High Power Laser and Particle Beams (1)

Gao Chong, Ma Jing, and Tan Liying, "Angle-of-arrival fluctuation of light beam propagation in strong turbulence regime," High Power Laser and Particle Beams 18, 891-894 (2006).

J. Fluid Mech. (1)

G. K. Batchelor, "Small-scale variation of convected quantities like temperature in turbulent fluid. Part I. General discussion and the case of small conductivity," J. Fluid Mech. 5, 113-133 (1959).
[CrossRef]

J. Mod. Opt. (1)

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

J. Opt. Fiber. Commun. Rep. (1)

J. C. Ricklin, S. M. Hammel, F. D. Eaton, and S. L. Lachinova, "Atmospheric channel effects on free-space laser communication," J. Opt. Fiber. Commun. Rep. 3, 111-158 (2006).
[CrossRef]

J. Opt. Soc. Am. (2)

P. Hickson, "Wave-front curvature sensing from a single defocused image," J. Opt. Soc. Am. 11, 1667-1673 (1994).
[CrossRef]

S. F. Clifford, "Temporal-frequency spectra for a spherica wave propagating through atmospheirc turbulence," J. Opt. Soc. Am. 61, 1285-1292 (1971).
[CrossRef]

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

JETP (1)

S. S. Moiseev and O. G. Chkhetiani, "Helical scaling in turbulence," JETP 83, 192-198 (1996).

Opt. Commun. (1)

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

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. E (1)

T. Elperin, N. Kleeorin, and I. Rogachevskii, "Isotropic and anisotropic spectra of passive scalar fluctuations in turbulent fluid flow," Phys. Rev. E 53, 3431-3441 (1996).
[CrossRef]

Proc. SPIE (4)

R. R. Beland, "Some aspects of propagation through weak isotropic non-Kolmogorov turbulence," Proc. SPIE 2375, 1111-1126 (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]

M. S. Be1en’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, "Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion," Proc. SPIE 3126, 113-123 (1997).
[CrossRef]

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, "Measurement of optical turbulence in the upper troposphere and lower stratosphere," Proc. SPIE 2110, 43-55 (1994).
[CrossRef]

Other (10)

G. Wang, "A new random-phase-screen time series simulation algorithm for dynamically atmospheric turbulence wave-front generator," Proc. SPIE 6027, 602716-1-12 (2006).
[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-1-12 (2006).
[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," Proc. SPIE 6457, 64570T-1-11 (2007).
[CrossRef]

L. C. Andrews and R.L. Phillips, Laser Propagation through Random Media (SPIE Optical Engineering Press, Bellingham, 1998).

V. I. Tatarski, Wave Propagating in a Turbulent Medium, (McGraw-Hill, New York, 1961).

A. Erd’elyi, Tables of Integral Transforms, (McGraw-Hill, New York, 1959).

C. Ho and A. Wheelon, Power Spectrum of Atmospheric Scintillation for the Deep Space Network Goldstone Ka-band Downlink, (Jet Propulsion Laboratory, California, 2004).
[PubMed]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1965).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, New York, 1978).

W. L. Wolf and G. J. Zissis, The Infrared Handbook (Office of Naval Research, Washington, 1978).

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Figures (7)

Fig. 1.
Fig. 1.

Schematic layout of the baseline ρ⃗, the transverse wind velocity ν⃗, the observation axis x of the AOA fluctuations, and the angle β between ρ⃗ and x axis.

Fig. 2.
Fig. 2.

The temporal power spectrum of AOA fluctuations scaled by the corresponding variance as a function of the frequency ratio ω/ω0 for several values of power law α, with (a) for a plane wave, (b) for a spherical wave.

Fig. 3.
Fig. 3.

The temporal power spectrum of AOA fluctuations scaled by the corresponding variance as a function of the frequency ratio ω/ω0 for several values of the receiver aperture D, with (a) for a plane wave, (b) for a spherical wave.

Fig. 4.
Fig. 4.

The temporal power spectrum of AOA fluctuations scaled by the corresponding variance as a function of the frequency ratio ω/ω0 for several values of β, with (a) for a plane wave, (b) for a spherical wave.

Fig. 5.
Fig. 5.

The temporal power spectrum of AOA fluctuations scaled by the corresponding variance as a function of the frequency ratio ω/ω0 for power law α = 11/3. The solid curve represents a spherical wave. The circle denotes a plane wave.

Fig. 6.
Fig. 6.

The temporal power spectrum of irradiance fluctuations scaled by the corresponding variance as a function of the frequency ratio ω/ω0 for several values of power law α, with (a) for a plane wave, (b) for a spherical wave.

Fig. 7.
Fig. 7.

The temporal power spectrum of irradiance fluctuations scaled by the corresponding variance as a function of the frequency ratio ω/ω0 for power law α = 11/3. The dashed curve represents a plane wave, the dotted curve represents a spherical wave.

Equations (49)

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Φ n κ α = A ( α ) C ˜ n 2 κ α , 2 π / L 0 κ 2 π / l 0 , 3 < α < 4 ,
A ( α ) = 1 4 π 2 Γ ( α 1 ) cos ( απ 2 ) ,
Φ n ( κ ) = 0.033 C n 2 κ 11 / 3 ,
W θ ω β = 4 0 C θ t β cos ( ωt ) dt .
C θ ρ β = π k 2 0 κ 3 W ϕ ( κ ) G D ( κ ) [ J 0 ( ρκ ) cos ( 2 β ) J 2 ( ρκ ) ] ,
C θ t β = π k 2 0 κ 3 W ϕ ( κ ) G D ( κ ) [ J 0 ( νtκ ) cos ( 2 β ) J 2 ( νtκ ) ] .
W ϕ ( pl ) ( κ ) = 2 π k 2 0 L Φ n ( κ ) cos 2 ( κ 2 z 2 k ) dz ,
W ϕ ( pl ) κ α = 2 π k 2 0 L Φ n κ α cos 2 ( κ 2 z 2 k ) dz .
C θ ( pl ) α t β = 2 π 2 0 κ 3 Φ n κ α G D ( κ ) [ J 0 ( νt κ ) cos ( 2 β ) J 2 ( νt κ ) ]
× 0 L cos 2 ( κ 2 z 2 k ) dzdκ .
W θ ( pl ) α ω β = 8 π 2 0 κ 3 Φ n κ α G D ( κ ) [ J 0 ( νtκ ) cos ( 2 β ) J 2 ( νtκ ) ]
× 0 L cos 2 ( κ 2 z 2 k ) 0 cos ( ωt ) dκdzdt .
0 J 0 ( ax ) cos ( bx ) dx = { ( a 2 b 2 ) 1 / 2 , 0 < b < a , 0 , b > a ,
0 J 2 n ( ax ) cos ( bx ) dx = { ( 1 ) n ( a 2 b 2 ) 1 / 2 T 2 n ( b / a ) , 0 < b < a , 0 , b > a ,
T m ( z ) = cos ( m cos 1 z ) ,
W θ ( pl ) α ω β = 8 π 2 0 dz ω / ν κ 3 Φ n κ α G D ( κ ) cos 2 ( κ 2 z 2 k )
× { [ ( νκ ) 2 ω 2 ] 1 / 2 + cos ( 2 β ) [ ( νκ ) 2 ω 2 ] 1 / 2
× [ 2 ( ω νκ ) 2 1 ] } .
G D ( κ ) exp ( c 2 D 2 κ 2 4 ) ,
W θ ( pl ) α ω β = 8 π 2 A ( α ) C ˜ n 2 0 L dz ω / ν κ 3 α exp ( c 2 D 2 κ 2 4 )
× cos 2 ( κ 2 z 2 k ) { [ ( νκ ) 2 ω 2 ] 1 / 2 + cos ( 2 β )
× [ ( νκ ) 2 ω 2 ] 1 / 2 [ 2 ( ω νκ ) 2 1 ] } .
0 ( t + a ) 2 μ 1 ( t b ) 2 ν 1 exp ( pt ) dt
= { 0 , 0 < t < b , Γ ( 2 ν ) ( a + b ) μ + ν 1 p μ ν e p ( a b ) / 2 W μ ν , μ + ν 1 / 2 ( bp + ap ) , t > b ,
W θ ( pl ) α ω β = 7.09 π 2 A ( α ) C ˜ n 2 L 1 + α 4 k 3 α 4 ω 0 ( ω ω 0 ) 1 α 2 ( c 2 D 2 4 ) α 5 4
× exp [ k c 2 D 2 8 L ( ω ω 0 ) 2 ] { [ 1 cos ( 2 β ) ] W 3 α 4 , 3 α 4 [ k c 2 D 2 4 L ( ω ω 0 ) 2 ]
+ 2 cos ( 2 β ) ω ω 0 ( k c 2 D 2 4 L ) 1 2 W 1 α 4 , 1 α 4 [ k c 2 D 2 4 L ( ω ω 0 ) 2 ] } ,
W ϕ ( sp ) ( κ ) = 2 π k 2 0 L Φ n ( κ ) ( z L ) 2 cos 2 [ κ 2 z ( L z ) 2 kL ] dz .
W ϕ ( sp ) κ α = 2 π k 2 0 L Φ n κ α ( z L ) 2 cos 2 [ κ 2 z ( L z ) 2 kL ] dz .
C θ ( sp ) α t β = 2 π 2 0 κ 3 Φ n κ α G D ( κ ) [ J 0 ( νtκ ) cos ( 2 β ) J 2 ( νtκ ) ]
× 0 L cos 2 [ κ 2 z ( L z ) 2 kL ] ( z L ) 2 dzdκ .
W θ ( sp ) α ω β = 2.36 π 2 A ( α ) C ˜ n 2 L 1 + α 4 k 3 α 4 ω 0 ( ω ω 0 ) 1 α 2 ( c 2 D 2 4 ) α 5 4
× exp [ k c 2 D 2 8 L ( ω ω 0 ) 2 ] { [ 1 cos ( 2 β ) ] W 3 α 4 , 3 α 4 [ k c 2 D 2 4 L ( ω ω 0 ) 2 ]
+ 2 cos ( 2 β ) ω ω 0 ( k c 2 D 2 4 L ) 1 2 W 1 α 4 , 1 α 4 [ k c 2 D 2 4 L ( ω ω 0 ) 2 ] } .
W I ω L = 4 0 C I t L cos ( ωt ) dt .
C I ( pl ) t L = 8 π 2 k 2 0 L 0 κ Φ n ( κ ) J 0 ( νtκ ) [ 1 cos ( κ 2 ( L z ) k ) ] dκdz .
C I ( pl ) ( α , t , L ) = 8 π 2 k 2 0 L 0 κ Φ n κ α J 0 ( νtκ ) [ 1 cos ( κ 2 ( L z ) k ) ] dκdz .
W I ( pl ) ( α , ω , L ) = 32 π 2 k 2 0 L dz 0 0 κ Φ n κ α J 0 ( νtκ )
× [ 1 cos ( κ 2 ( L z ) k ) ] cos ( ωt ) dt .
W I ( pl ) α ω L = 16 π 2 k 6 α 2 A ( α ) C ˜ n 2 L α 2 ω 0 ( ω ω 0 ) 1 α Γ ( 1 2 ) Γ ( α 1 2 ) Γ ( α 2 ) Re { 1
2 F 2 ( 1 , 2 α 2 ; 3 2 , 3 α 2 ; i 1 2 ( ω ω 0 ) 2 ) Γ ( α 2 ) Γ ( 1 α 2 ) Γ ( 1 + α 2 ) Γ ( α 1 2 ) Γ ( 3 + α 2 )
× ( i 1 2 ( ω ω 0 ) 2 ) α 1 2 1 F 1 ( 1 2 ; α + 3 2 ; i 1 2 ( ω ω 0 ) 2 ) } .
C I ( sp ) t L = 8 π 2 k 2 0 L 0 κ Φ n ( κ ) J 0 ( νtκ ) [ 1 cos ( κ 2 z ( L z ) Lk ) ] dκdz .
C I ( sp ) ( α , t , L ) = 8 π 2 k 2 0 L 0 κ Φ n κ α J 0 ( νtκ ) [ 1 cos ( κ 2 z ( L z ) Lk ) ] dκdz .
W I ( sp ) ( α , ω , L ) = 32 π 2 k 2 0 L dz 0 0 κ Φ n κ α J 0 ( νtκ )
× [ 1 cos ( κ 2 z ( L z ) Lk ) ] cos ( ωt ) dt .
W I ( sp ) α ω L = 16 π 2 k 6 α 2 A ( α ) C ˜ n 2 L α 2 ω 0 ( ω ω 0 ) 1 α Γ ( 1 2 ) Γ ( α 1 2 ) Γ ( α 2 ) Re { 1
2 F 2 ( 1 , 2 α 2 ; 3 2 , 3 α 2 ; i 1 4 ( ω ω 0 ) 2 ) 1 2 Γ ( α 2 ) Γ ( 1 α 2 ) Γ ( 1 + α 2 ) Γ ( α 1 2 ) Γ ( 2 + α 2 )
× ( i 1 4 ( ω ω 0 ) 2 ) α 1 2 1 F 1 ( 1 2 ; α + 2 α ; i 1 4 ( ω ω 0 ) 2 ) } .

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