Abstract

In the past decades, both the increasing experimental evidence and some results of theoretical investigation on non-Kolmogorov turbulence have been reported. This has prompted the study of optical propagation in non-Kolmogorov atmospheric turbulence. In this paper, based on the thin phase screen model and a non-Kolmogorov power spectrum which owns a generalized power law instead of standard Kolmogorov power law value 11/3 and a generalized amplitude factor instead of constant value 0.033, the temporal power spectrum of irradiance fluctuations for a Gaussian-beam wave is derived in the weak fluctuation regime for a horizontal path. The analytic expressions are obtained and then used to analyze the influence of spectral power law variations on the temporal power spectrum of irradiance fluctuations.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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  24. Z. Zeng, X. Luo, A. Xia, Y. Zhang, and C. Sun, “Rytov variance equivalence through extended atmospheric turbulence and an arbitrary thickness phase screen in non-Kolmogorov turbulence,” Optik (Stuttg.) 125(15), 4092–4097 (2014).
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    [Crossref] [PubMed]
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2015 (1)

2014 (2)

M. Yao, I. Toselli, and O. Korotkova, “Propagation of electromagnetic stochastic beams in anisotropic turbulence,” Opt. Express 22(26), 31608–31619 (2014).
[Crossref] [PubMed]

Z. Zeng, X. Luo, A. Xia, Y. Zhang, and C. Sun, “Rytov variance equivalence through extended atmospheric turbulence and an arbitrary thickness phase screen in non-Kolmogorov turbulence,” Optik (Stuttg.) 125(15), 4092–4097 (2014).
[Crossref]

2012 (1)

2011 (2)

2010 (1)

2008 (2)

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47(34), 6385–6391 (2008).
[Crossref] [PubMed]

2007 (2)

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, 026003 (2007).
[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]

2006 (1)

G. Wang, “A new random-phase-screen time series simulation algorithm for dynamically atmospheric turbulence wave-front generator,” Proc. SPIE 6027, 602716 (2006).
[Crossref]

2005 (1)

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Lidar studies of aerosols and non-Kolmogorov turbulence in the Mediterranean troposphere,” Proc. SPIE 5987, 598702 (2005).
[Crossref]

2004 (1)

1999 (1)

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]

1997 (2)

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

L. C. Andrews, R. L. Phillips, and A. R. Weeks, “Propagation of a Gaussian-beam wave through a random phase screen,” Waves Random Media 7(2), 229–244 (1997).
[Crossref]

1996 (3)

G. D. Boreman and C. Dainty, “Zernike expansions for non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 13(3), 517–522 (1996).
[Crossref]

S. S. Moiseev and O. G. Chkhetiani, “Helical scaling in turbulence,” Sov. Phys. JETP 83(1), 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 Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 53(4), 3431–3441 (1996).
[Crossref] [PubMed]

1995 (4)

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

J. D. Shelton, “Turbulence-induced scintillation on Gaussian-beam waves: theoretical predictions and observations from a laser-illuminated satellite,” J. Opt. Soc. Am. A 12(10), 2172–2181 (1995).
[Crossref]

L. C. Andrews and W. B. Miller, “Single-pass and double-pass propagation through complex paraxial optical systems,” J. Opt. Soc. Am. A 12(1), 137–150 (1995).
[Crossref]

1989 (1)

Agrawal, B.

Andrews, L. C.

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]

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, 026003 (2007).
[Crossref]

L. C. Andrews, R. L. Phillips, and A. R. Weeks, “Propagation of a Gaussian-beam wave through a random phase screen,” Waves Random Media 7(2), 229–244 (1997).
[Crossref]

L. C. Andrews and W. B. Miller, “Single-pass and double-pass propagation through complex paraxial optical systems,” J. Opt. Soc. Am. A 12(1), 137–150 (1995).
[Crossref]

Baykal, Y.

Beland, R. R.

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

Belen’kii, M. S.

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, 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]

Boreman, G. D.

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, 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]

Chen, C.

Chkhetiani, O. G.

S. S. Moiseev and O. G. Chkhetiani, “Helical scaling in turbulence,” Sov. Phys. JETP 83(1), 192–198 (1996).

Cui, L.

Dainty, C.

Du, W.

Elperin, T.

T. Elperin, N. Kleeorin, and I. Rogachevskii, “Isotropic and anisotropic spectra of passive scalar fluctuations in turbulent fluid flow,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 53(4), 3431–3441 (1996).
[Crossref] [PubMed]

Ferrero, V.

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, 026003 (2007).
[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]

Fugate, R. Q.

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, 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]

Gerçekcioglu, H.

Golbraikh, E.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47(34), 6385–6391 (2008).
[Crossref] [PubMed]

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Lidar studies of aerosols and non-Kolmogorov turbulence in the Mediterranean troposphere,” Proc. SPIE 5987, 598702 (2005).
[Crossref]

E. Golbraikh and N. S. Kopeika, “Behavior of structure function of refraction coefficients in different turbulent fields,” Appl. Opt. 43(33), 6151–6156 (2004).
[Crossref] [PubMed]

Hanson, S. G.

Jiang, Y.

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, 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]

Kleeorin, N.

T. Elperin, N. Kleeorin, and I. Rogachevskii, “Isotropic and anisotropic spectra of passive scalar fluctuations in turbulent fluid flow,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 53(4), 3431–3441 (1996).
[Crossref] [PubMed]

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,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47(34), 6385–6391 (2008).
[Crossref] [PubMed]

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Lidar studies of aerosols and non-Kolmogorov turbulence in the Mediterranean troposphere,” Proc. SPIE 5987, 598702 (2005).
[Crossref]

E. Golbraikh and N. S. Kopeika, “Behavior of structure function of refraction coefficients in different turbulent fields,” Appl. Opt. 43(33), 6151–6156 (2004).
[Crossref] [PubMed]

Korotkova, O.

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,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

Liu, R.

Lou, Y.

Luo, X.

Z. Zeng, X. Luo, A. Xia, Y. Zhang, and C. Sun, “Rytov variance equivalence through extended atmospheric turbulence and an arbitrary thickness phase screen in non-Kolmogorov turbulence,” Optik (Stuttg.) 125(15), 4092–4097 (2014).
[Crossref]

Ma, J.

Miller, W. B.

Moiseev, S. S.

S. S. Moiseev and O. G. Chkhetiani, “Helical scaling in turbulence,” Sov. Phys. JETP 83(1), 192–198 (1996).

Osmon, C. L.

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.

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, 026003 (2007).
[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 A. R. Weeks, “Propagation of a Gaussian-beam wave through a random phase screen,” Waves Random Media 7(2), 229–244 (1997).
[Crossref]

Restaino, S.

Rogachevskii, I.

T. Elperin, N. Kleeorin, and I. Rogachevskii, “Isotropic and anisotropic spectra of passive scalar fluctuations in turbulent fluid flow,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 53(4), 3431–3441 (1996).
[Crossref] [PubMed]

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]

Shelton, J. D.

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,” Atmos. Res. 88(1), 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]

Sun, C.

Z. Zeng, X. Luo, A. Xia, Y. Zhang, and C. Sun, “Rytov variance equivalence through extended atmospheric turbulence and an arbitrary thickness phase screen in non-Kolmogorov turbulence,” Optik (Stuttg.) 125(15), 4092–4097 (2014).
[Crossref]

Tan, L.

Tong, S.

Toselli, I.

M. Yao, I. Toselli, and O. Korotkova, “Propagation of electromagnetic stochastic beams in anisotropic turbulence,” Opt. Express 22(26), 31608–31619 (2014).
[Crossref] [PubMed]

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

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, 026003 (2007).
[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]

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,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

Wang, G.

G. Wang, “A new random-phase-screen time series simulation algorithm for dynamically atmospheric turbulence wave-front generator,” Proc. SPIE 6027, 602716 (2006).
[Crossref]

Weeks, A. R.

L. C. Andrews, R. L. Phillips, and A. R. Weeks, “Propagation of a Gaussian-beam wave through a random phase screen,” Waves Random Media 7(2), 229–244 (1997).
[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]

Xia, A.

Z. Zeng, X. Luo, A. Xia, Y. Zhang, and C. Sun, “Rytov variance equivalence through extended atmospheric turbulence and an arbitrary thickness phase screen in non-Kolmogorov turbulence,” Optik (Stuttg.) 125(15), 4092–4097 (2014).
[Crossref]

Yang, H.

Yao, M.

Yura, H. T.

Zeng, Z.

Z. Zeng, X. Luo, A. Xia, Y. Zhang, and C. Sun, “Rytov variance equivalence through extended atmospheric turbulence and an arbitrary thickness phase screen in non-Kolmogorov turbulence,” Optik (Stuttg.) 125(15), 4092–4097 (2014).
[Crossref]

Zhang, Y.

Z. Zeng, X. Luo, A. Xia, Y. Zhang, and C. Sun, “Rytov variance equivalence through extended atmospheric turbulence and an arbitrary thickness phase screen in non-Kolmogorov turbulence,” Optik (Stuttg.) 125(15), 4092–4097 (2014).
[Crossref]

Zilberman, A.

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47(34), 6385–6391 (2008).
[Crossref] [PubMed]

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Lidar studies of aerosols and non-Kolmogorov turbulence in the Mediterranean troposphere,” Proc. SPIE 5987, 598702 (2005).
[Crossref]

Appl. Opt. (2)

Atmos. Res. (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,” Atmos. Res. 88(1), 66–77 (2008).
[Crossref]

J. Opt. Soc. Am. A (5)

Opt. Express (4)

Opt. Lett. (1)

Optik (Stuttg.) (1)

Z. Zeng, X. Luo, A. Xia, Y. Zhang, and C. Sun, “Rytov variance equivalence through extended atmospheric turbulence and an arbitrary thickness phase screen in non-Kolmogorov turbulence,” Optik (Stuttg.) 125(15), 4092–4097 (2014).
[Crossref]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

T. Elperin, N. Kleeorin, and I. Rogachevskii, “Isotropic and anisotropic spectra of passive scalar fluctuations in turbulent fluid flow,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 53(4), 3431–3441 (1996).
[Crossref] [PubMed]

Proc. SPIE (8)

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

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Lidar studies of aerosols and non-Kolmogorov turbulence in the Mediterranean troposphere,” Proc. SPIE 5987, 598702 (2005).
[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, 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]

G. Wang, “A new random-phase-screen time series simulation algorithm for dynamically atmospheric turbulence wave-front generator,” Proc. SPIE 6027, 602716 (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, 026003 (2007).
[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]

Sov. Phys. JETP (1)

S. S. Moiseev and O. G. Chkhetiani, “Helical scaling in turbulence,” Sov. Phys. JETP 83(1), 192–198 (1996).

Waves Random Media (1)

L. C. Andrews, R. L. Phillips, and A. R. Weeks, “Propagation of a Gaussian-beam wave through a random phase screen,” Waves Random Media 7(2), 229–244 (1997).
[Crossref]

Other (2)

L. C. Andrews and R. L. Phillips, Laser beam propagation through random media, 2nd ed. (SPIE Optical Engineering, 2005).

L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Optical Engineering, 1998).

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Figures (7)

Fig. 1
Fig. 1 Propagation geometry for a random phase screen.
Fig. 2
Fig. 2 Normalized temporal power spectrum of irradiance fluctuations as a function of ω/ω0 for various values of Θ0.
Fig. 3
Fig. 3 Normalized temporal power spectrum of irradiance fluctuations as a function of Λ0 for various values of α.
Fig. 4
Fig. 4 Normalized temporal power spectrum of irradiance fluctuations as a function of ω/ω0 for various values of Λ0, where Λ0 < Λm.
Fig. 5
Fig. 5 Normalized temporal power spectrum of irradiance fluctuations as a function of ω/ω0 for various values of Λ0, where Λ0 > Λm.
Fig. 6
Fig. 6 Normalized temporal power spectrum of irradiance fluctuations as a function of ω/ω0 for various values of α.
Fig. 7
Fig. 7 Normalized temporal power spectrum of irradiance fluctuations as a function of ω/ω0 for various values of r/W.

Equations (51)

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Φ n ( κ,α )=A( α ) C ˜ n 2 κ α , 2π/ L 0 κ 2π/ l 0 , 3<α<4,
A( α )= 1 4 π 2 Γ( α1 )cos( απ 2 ),
Φ n ( κ )=0.033 C n 2 κ 11/3 ,
U 0 ( r,0 )= a 0 exp( r 2 W 0 2 i k r 2 2 F 0 ),
U( r,L )= U 0 ( r,L )exp[ ψ( r,L ) ]= U 0 ( r,L )exp[ ψ 1 ( r,L )+ ψ 2 ( r,L )+ ],
U 0 ( r,L )= a 0 Θ 0 +i Λ 0 exp( ikL r 2 W 2 i k r 2 2F )
Θ 0 =1 L F 0 , Λ 0 = 2L k W 0 2 .
Θ=1+ L F = Θ 0 Θ 0 2 + Λ 0 2 , Θ ¯ =1Θ, Λ= 2L k W 2 = Λ 0 Θ 0 2 + Λ 0 2 .
E 1 ( 0,0 )= ψ 2 ( r,L ) + 1 2 ψ 1 2 ( r,L ) =2 π 2 k 2 0 L 0 κ Φ n ( κ,z ) dκdz,
E 2 ( r 1 , r 2 )= ψ 1 ( r 1 ,L ) ψ 1 ( r 2 ,L ) =4 π 2 k 2 0 L 0 κ Φ n ( κ,z ) exp[ ΛL κ 2 ( 1z/L ) 2 k ] × J 0 { κ| [ 1 Θ ¯ ( 1z/L ) ]p2iΛ( 1z/L )r | }dκdz,
E 3 ( r 1 , r 2 )= ψ 1 ( r 1 ,L ) ψ 1 ( r 2 ,L ) =4 π 2 k 2 0 L 0 κ Φ n ( κ,z )exp{ iL κ 2 k ( 1z/L )[ 1 Θ ¯ ( 1z/L ) ] } ×exp[ ΛL κ 2 ( 1z/L ) 2 k ] J 0 { κρ[ 1( Θ ¯ +iΛ )( 1z/L ) ] }dκdz ,
1 z L = d 3 ( 1+ d 2 η ), 0η1 ,
E 1 ( 0,0 )=2 π 2 k 2 L d 2 d 3 0 1 0 κ Φ n ( κ,η ) dκdη,
E 2 ( r 1 , r 2 )=4 π 2 k 2 L d 2 d 3 0 1 0 κ Φ n ( κ,η ) exp[ ΛL κ 2 d 3 2 ( 1+ d 2 η ) 2 k ] × J 0 { κ| [ 1 Θ ¯ d 3 ( 1+ d 2 η ) ]p2iΛ d 3 ( 1+ d 2 η )r | }dκdη,
E 3 ( r 1 , r 2 )=4 π 2 k 2 L d 2 d 3 0 1 0 κ Φ n ( κ,η )exp{ iL κ 2 k d 3 ( 1+ d 2 η )[ 1 Θ ¯ d 3 ( 1+ d 2 η ) ] } ×exp[ ΛL κ 2 d 3 2 ( 1+ d 2 η ) 2 k ] J 0 { κρ[ 1( Θ ¯ +iΛ ) d 3 ( 1+ d 2 η ) ] }dκdη .
E 1 ( 0,0 )=2 π 2 k 2 L d 2 d 3 0 κ Φ n ( κ )dκ ,
E 2 ( r 1 , r 2 )=4 π 2 k 2 L d 2 d 3 0 κ Φ n ( κ ) exp( ΛL d 3 2 κ 2 k ) × J 0 [ κ| ( 1 Θ ¯ d 3 )p2iΛ d 3 r | ]dκ,
E 3 ( r 1 , r 2 )=4 π 2 k 2 L d 2 d 3 0 κ Φ n ( κ )exp[ iL κ 2 k d 3 ( 1 Θ ¯ d 3 ) ] ×exp( ΛL d 3 2 κ 2 k ) J 0 [ κρ( 1 Θ ¯ d 3 iΛ d 3 ) ]dκ .
σ R 2 =8 π 2 k 2 L 0 1 0 κ Φ n ( κ )[ 1cos( L κ 2 ξ k ) ] dκdξ.
σ R 2 ( α )=8 π 2 k 2 L 0 1 0 κ Φ n ( κ,α )[ 1cos( L κ 2 ξ k ) ] dκdξ.
Φ n ( κ,α )=A( α ) C ˜ n 2 exp( κ 2 / κ m 2 ) ( κ 2 + κ 0 2 ) α/2 , 0κ< , 3<α<4,
U( a;c;z )= 1 Γ( a ) 0 e zt t a1 ( 1+t ) ca1 dt, a>0 , Re( z )>0 ,
U( a;c;z ) Γ( 1c ) Γ( 1+ac ) + Γ( c1 ) Γ( a ) z 1c , | z |1 ,
σ R 2 ( α )Re[ 8 α A( α ) π 2 Γ( 1 α 2 ) ( i ) α/21 ] C ˜ n 2 k 3α/2 L α/2 ,
Φ n ( κ,α )=A( α ) c ˜ n 2 exp( κ 2 / κ m 2 ) ( κ 2 + κ 0 2 ) α/2 ,
σ ^ R 2 ( α )=8 π 2 k 2 L d 2 d 3 0 κ Φ n ( κ,α )[ 1cos( L κ 2 d 3 k ) ] dκ Re[ 4 π 2 A( α )Γ( 1 α 2 ) ( i ) α/21 ] c ˜ n 2 k 3α/2 L α/2 d 2 d 3 α/2 .
C ˜ n 2 = α 2 d 2 d 3 α/2 c ˜ n 2 = α 2 d 3 α/21 L 2 L c ˜ n 2 .
W I ( ω )=4 0 C I ( t )cos( ωt )dt.
C I ( ρ )=2Re[ E 2 ( r 1 , r 2 )+ E 3 ( r 1 , r 2 ) ] = 8 π 2 k 2 L d 2 d 3 0 κ Φ n ( κ )exp( ΛL d 3 2 κ 2 k ) Re{ J 0 [ κ| ( 1 Θ ¯ d 3 )p2iΛ d 3 r | ] exp[ iL κ 2 d 3 k ( 1 Θ ¯ d 3 ) ] J 0 [ κρ( 1 Θ ¯ d 3 iΛ d 3 ) ] }dκ.
d 3 =0.670.17Θ.
Re[ J 0 ( | xiy | ) ]= J 0 ( x ) I 0 ( y )+2 n=1 ( 1 ) n J 2n ( x ) I 2n ( y ) cos( 2nφ ),
Re{ J 0 [ κ| ( 1 Θ ¯ d 3 )p2iΛ d 3 r | ] }=Re{ J 0 [ κ| γp/2+ γ p/2+γr γ r | ] } = J 0 ( κ v t ) I 0 ( 2Λκ d 3 r ),
J 0 [ κρ( 1 Θ ¯ d 3 iΛ d 3 ) ]= J 0 ( γκρ )= J 0 ( κ v t ),
C I ( t )=8 π 2 k 2 L d 2 d 3 0 κ Φ n ( κ ) exp( ΛL d 3 2 κ 2 k ) J 0 ( κ v t ) ×{ I 0 ( 2Λκ d 3 r )cos[ L κ 2 k d 3 ( 1 Θ ¯ d 3 ) ] }dκ .
C I ( t,α )=8 π 2 k 2 L d 2 d 3 0 κ Φ n ( κ,α ) exp( ΛL d 3 2 κ 2 k ) J 0 ( κ v t ) ×{ I 0 ( 2Λκ d 3 r )cos[ L κ 2 k d 3 ( 1 Θ ¯ d 3 ) ] }dκ .
W I ( ω,r,α )=32 π 2 k 2 L d 2 d 3 0 0 κ Φ n ( κ,α ) exp( ΛL d 3 2 κ 2 k ) J 0 ( κ v t )cos( ωt ) ×{ I 0 ( 2Λκ d 3 r )cos[ L κ 2 k d 3 ( 1 Θ ¯ d 3 ) ] }dκ dt.
W I ( ω,r,α )= W I,l ( ω,α )+ W I,r ( ω,r,α ),
W I,l ( ω,α )=32 π 2 k 2 L d 2 d 3 0 0 κ Φ n ( κ,α ) exp( ΛL d 3 2 κ 2 k ) J 0 ( κ v t )cos( ωt ) ×{ 1cos[ L κ 2 k d 3 ( 1 Θ ¯ d 3 ) ] }dκ dt,
W I,r ( ω,r,α )=32 π 2 k 2 L d 2 d 3 0 0 κ Φ n ( κ,α ) exp( ΛL d 3 2 κ 2 k ) J 0 ( κ v t )cos( ωt ) ×[ I 0 ( 2Λκ d 3 r )1 ]dκdt.
0 J 0 ( ax )cos( bx )dx={ ( a 2 b 2 ) 1/2 , 0<b<a, 0, b>a,
W I,l ( ω,α )=32 π 2 k 2 L d 2 d 3 ω/ v κ Φ n ( κ,α ) exp( ΛL d 3 2 κ 2 k ) ( κ 2 v 2 ω 2 ) 1/2 ×{ 1cos[ L κ 2 k d 3 ( 1 Θ ¯ d 3 ) ] }dκ ,
W I,r ( ω,r,α )=32 π 2 k 2 L d 2 d 3 ω/ v κ Φ n ( κ,α ) exp( ΛL d 3 2 κ 2 k ) ( κ 2 v 2 ω 2 ) 1/2 ×[ I 0 ( 2Λκ d 3 r )1 ]dκ.
Φ n ( κ,α )=A( α ) c ˜ n 2 κ α ,
I p ( x )= n=0 ( x/2 ) 2n+p n!Γ( n+p+1 ) , | x |< ,
W I,l ( ω,α )=16A( α ) c ˜ n 2 π 2 k 2 L d 2 d 3 ω 1α v 2α Γ( 1 2 )Re [ exp( ΛL d 3 2 ω 2 k v 2 )U( 1 2 ; 3 2 α 2 ; ΛL d 3 2 ω 2 k v 2 ) exp( a 1 L ω 2 k v 2 )U( 1 2 ; 3 2 α 2 ; a 1 L ω 2 k v 2 ) ],
W I,r ( ω,r,α )=16A( α ) c ˜ n 2 π 2 k 2 L d 2 d 3 ω 1α v 2α Γ( 1 2 ) × n=1 ( Λ d 3 rω ) 2n ( n! ) 2 v 2n exp( ΛL d 3 2 ω 2 k v 2 )U( 1 2 ;n+ 3 2 α 2 ; ΛL d 3 2 ω 2 k v 2 ),
U( a;c;z )= Γ( 1c ) Γ( 1+ac ) F 1 1 ( a;c;z )+ Γ( c1 ) Γ( a ) z 1c F 1 1 ( 1+ac;2c;z ),
e z F 1 1 ( a;c;z )= F 1 1 ( ca;c;z ),
W I ( ω,r,α )= W I,l ( ω,α )+ W I,r ( ω,r,α ),
W I,l ( ω,α )= 32A( α ) C ˜ n 2 π 2 k 3-α/2 L α/2 α ω 0 d 3 α/21 ×Re{ Γ( 1 2 )Γ( α 2 1 2 ) Γ( α 2 ) ( ω ω 0 ) 1α [ F 1 1 ( 1 α 2 ; 3 2 α 2 ; Λ d 3 2 ω 2 ω 0 2 ) F 1 1 ( 1 α 2 ; 3 2 α 2 ; a 1 ω 2 ω 0 2 ) ] +Γ( 1 2 α 2 )[ ( Λ d 3 2 ) α 2 1 2 F 1 1 ( 1 2 ; α 2 + 1 2 ; Λ d 3 2 ω 2 ω 0 2 ) a 1 α 2 1 2 F 1 1 ( 1 2 ; α 2 + 1 2 ; a 1 ω 2 ω 0 2 ) ] },
W I,r ( ω,r,α )= 32A( α ) C ˜ n 2 π 2 k 3-α/2 L α/2 α ω 0 d 3 α/21 ( ω ω 0 ) 1α × n=1 ( 2Λ d 3 2 ) n ( n! ) 2 ( r W ) 2n ( ω ω 0 ) 2n [ Γ( 1 2 )Γ( α 2 n 1 2 ) Γ( α 2 n ) F 1 1 ( n+1 α 2 ;n+ 3 2 α 2 ; Λ d 3 2 ω 2 ω 0 2 ) +Γ( n+ 1 2 α 2 ) ( Λ d 3 2 ω 2 ω 0 2 ) α 2 n 1 2 F 1 1 ( 1 2 ; α 2 + 1 2 n; Λ d 3 2 ω 2 ω 0 2 ) ],

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