Abstract

Based on the generalized von Kármán spectrum and the extended Rytov theory, new analytic expressions for the variance of angle of arrival (AOA) fluctuations are derived for optical plane and spherical waves propagating through moderate-to-strong non-Kolmogorov turbulence with horizontal path. They consider finite turbulence outer scale and general spectral power law value, and cover a wide range of non-Kolmogorov turbulence strength. When the turbulence outer scale is set to infinite, the new expressions can reduce correctly to previously published analytic expressions [J. Opt. Soc. Am. A , 302188 (2013]. The final results show that the increased turbulence outer scale value enlarges the variance of AOA fluctuations greatly under moderate-to-strong (or strong) non-Kolmogorov turbulence.

© 2014 Optical Society of America

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  1. D. T. Kyrazis, J. B. Wissler, 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]
  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]
  3. 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]
  4. 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, 66–77 (2008).
    [CrossRef]
  5. 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]
  6. M. S. Belen’kii, “Effect of the stratosphere on star image motion,” Opt. Lett. 20, 1359–1361 (1995).
    [CrossRef]
  7. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
    [CrossRef]
  8. 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]
  9. L. Y. Cui, B. D. Xue, X. G. Cao, J. K. Dong, and J. N. Wang, “Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence,” Opt. Express 18, 21269–21283 (2010).
    [CrossRef]
  10. L. Tan, W. Du, and J. Ma, “Effect of the outer scale on the angle of arrival variance for free-space-laser beam corrugated by non-Kolmogorov turbulence,” J. Russ. Laser Res. 30, 552–559 (2009).
    [CrossRef]
  11. B. Xue, L. Cui, W. Xue, X. Bai, and F. Zhou, “Theoretical expressions of the angle-of-arrival variance for optical waves propagating through non-Kolmogorov turbulence,” Opt. Express 19, 8433–8443 (2011).
    [CrossRef]
  12. L. Cui, B. Xue, W. Xue, X. Bai, X. Cao, and F. Zhou, “Atmospheric spectral model and theoretical expressions of irradiance scintillation index for optical wave propagating through moderate-to-strong non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 29, 1091–1098 (2012).
    [CrossRef]
  13. L. Cui, B. Xue, and F. Zhou, “Analytical expressions for the angle of arrival fluctuations for optical waves’ propagation through moderate-to-strong non-Kolmogorov refractive turbulence,” J. Opt. Soc. Am. A 30, 2188–2195 (2013).
    [CrossRef]
  14. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE Optical Engineering, 2005).
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    [CrossRef]
  16. C. Y. Young, A. J. Masino, and F. Thomas, “Phase fluctuations in moderate-to-strong turbulence,” Proc. SPIE 4976, 141–148 (2003).
    [CrossRef]
  17. C. Y. Young, A. J. Masino, F. E. Thomas, and C. J. Subich, “The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory,” Waves Random Media 14, 75–96 (2004).
    [CrossRef]
  18. L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Optical Engineering, 1998).
  19. M. Jing, G. Chong, and L. Y. Tan, “Angle-of-arrival fluctuations in moderate to strong turbulence,” Chin. Phys. 16, 1327–1333 (2007).
    [CrossRef]

2013 (1)

2012 (2)

2011 (1)

2010 (1)

2009 (1)

L. Tan, W. Du, and J. Ma, “Effect of the outer scale on the angle of arrival variance for free-space-laser beam corrugated by non-Kolmogorov turbulence,” J. Russ. Laser Res. 30, 552–559 (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,” Atmos. Res. 88, 66–77 (2008).
[CrossRef]

2007 (2)

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

M. Jing, G. Chong, and L. Y. Tan, “Angle-of-arrival fluctuations in moderate to strong turbulence,” Chin. Phys. 16, 1327–1333 (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]

2004 (1)

C. Y. Young, A. J. Masino, F. E. Thomas, and C. J. Subich, “The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory,” Waves Random Media 14, 75–96 (2004).
[CrossRef]

2003 (1)

C. Y. Young, A. J. Masino, and F. Thomas, “Phase fluctuations in moderate-to-strong turbulence,” Proc. SPIE 4976, 141–148 (2003).
[CrossRef]

1997 (1)

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]

1995 (3)

1994 (1)

D. T. Kyrazis, J. B. Wissler, 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]

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]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE Optical Engineering, 2005).

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

Bai, X.

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, 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]

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]

M. S. Belen’kii, “Effect of the stratosphere on star image motion,” Opt. Lett. 20, 1359–1361 (1995).
[CrossRef]

Bishop, K. P.

D. T. Kyrazis, J. B. Wissler, 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]

Brown, J. M.

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]

Cao, X.

Cao, X. G.

Chong, G.

M. Jing, G. Chong, and L. Y. Tan, “Angle-of-arrival fluctuations in moderate to strong turbulence,” Chin. Phys. 16, 1327–1333 (2007).
[CrossRef]

Cuellar, E.

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

Cui, L.

Cui, L. Y.

Dong, J. K.

Du, W.

L. Tan, W. Du, and J. Ma, “Effect of the outer scale on the angle of arrival variance for free-space-laser beam corrugated by non-Kolmogorov turbulence,” J. Russ. Laser Res. 30, 552–559 (2009).
[CrossRef]

Ferrero, V.

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

Fugate, R. Q.

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]

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

Gurvich, A. S.

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]

Jing, M.

M. Jing, G. Chong, and L. Y. Tan, “Angle-of-arrival fluctuations in moderate to strong turbulence,” Chin. Phys. 16, 1327–1333 (2007).
[CrossRef]

Karis, S. J.

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]

Keating, D. B.

D. T. Kyrazis, J. B. Wissler, 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]

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, 66–77 (2008).
[CrossRef]

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, 66–77 (2008).
[CrossRef]

Kyrazis, D. T.

D. T. Kyrazis, J. B. Wissler, 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]

Liu, Z.

Ma, J.

L. Tan, W. Du, and J. Ma, “Effect of the outer scale on the angle of arrival variance for free-space-laser beam corrugated by non-Kolmogorov turbulence,” J. Russ. Laser Res. 30, 552–559 (2009).
[CrossRef]

Masino, A. J.

C. Y. Young, A. J. Masino, F. E. Thomas, and C. J. Subich, “The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory,” Waves Random Media 14, 75–96 (2004).
[CrossRef]

C. Y. Young, A. J. Masino, and F. Thomas, “Phase fluctuations in moderate-to-strong turbulence,” Proc. SPIE 4976, 141–148 (2003).
[CrossRef]

Phillips, R. L.

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 and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE Optical Engineering, 2005).

Preble, A. J.

D. T. Kyrazis, J. B. Wissler, 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]

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]

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]

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, 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]

Subich, C. J.

C. Y. Young, A. J. Masino, F. E. Thomas, and C. J. Subich, “The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory,” Waves Random Media 14, 75–96 (2004).
[CrossRef]

Tan, L.

L. Tan, W. Du, and J. Ma, “Effect of the outer scale on the angle of arrival variance for free-space-laser beam corrugated by non-Kolmogorov turbulence,” J. Russ. Laser Res. 30, 552–559 (2009).
[CrossRef]

Tan, L. Y.

M. Jing, G. Chong, and L. Y. Tan, “Angle-of-arrival fluctuations in moderate to strong turbulence,” Chin. Phys. 16, 1327–1333 (2007).
[CrossRef]

Thomas, F.

C. Y. Young, A. J. Masino, and F. Thomas, “Phase fluctuations in moderate-to-strong turbulence,” Proc. SPIE 4976, 141–148 (2003).
[CrossRef]

Thomas, F. E.

C. Y. Young, A. J. Masino, F. E. Thomas, and C. J. Subich, “The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory,” Waves Random Media 14, 75–96 (2004).
[CrossRef]

Toselli, I.

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, 66–77 (2008).
[CrossRef]

Wang, J. N.

Welsh, B. M.

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

Wissler, J. B.

D. T. Kyrazis, J. B. Wissler, 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]

Xue, B.

Xue, B. D.

Xue, W.

Yi, X.

Young, C. Y.

C. Y. Young, A. J. Masino, F. E. Thomas, and C. J. Subich, “The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory,” Waves Random Media 14, 75–96 (2004).
[CrossRef]

C. Y. Young, A. J. Masino, and F. Thomas, “Phase fluctuations in moderate-to-strong turbulence,” Proc. SPIE 4976, 141–148 (2003).
[CrossRef]

Yue, P.

Zhou, F.

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

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, 66–77 (2008).
[CrossRef]

Chin. Phys. (1)

M. Jing, G. Chong, and L. Y. Tan, “Angle-of-arrival fluctuations in moderate to strong turbulence,” Chin. Phys. 16, 1327–1333 (2007).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Russ. Laser Res. (1)

L. Tan, W. Du, and J. Ma, “Effect of the outer scale on the angle of arrival variance for free-space-laser beam corrugated by non-Kolmogorov turbulence,” J. Russ. Laser Res. 30, 552–559 (2009).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Proc. SPIE (6)

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

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

D. T. Kyrazis, J. B. Wissler, 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]

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]

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]

C. Y. Young, A. J. Masino, and F. Thomas, “Phase fluctuations in moderate-to-strong turbulence,” Proc. SPIE 4976, 141–148 (2003).
[CrossRef]

Waves Random Media (1)

C. Y. Young, A. J. Masino, F. E. Thomas, and C. J. Subich, “The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory,” Waves Random Media 14, 75–96 (2004).
[CrossRef]

Other (2)

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

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE Optical Engineering, 2005).

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Figures (5)

Fig. 1.
Fig. 1.

Variance of AOA fluctuations derived in this study and those obtained with the Rytov theory for plane and spherical waves (α=10/3). (a) plane wave; (b) spherical wave.

Fig. 2.
Fig. 2.

Variance of AOA fluctuations derived in this study and those obtained with the Rytov theory for plane and spherical waves (α=11/3). (a) Plane wave, (b) spherical wave.

Fig. 3.
Fig. 3.

Variance of AOA fluctuations derived in this study and those obtained with the Rytov theory for plane and spherical waves (α=3.9). (a) Plane wave, (b) spherical wave.

Fig. 4.
Fig. 4.

Variance of AOA fluctuations as a function of a general spectral power law with different turbulence outer scale values under moderate-to-strong turbulence. (a) Plane wave, (b) spherical wave.

Fig. 5.
Fig. 5.

Variance of AOA fluctuations derived in this study and those obtained in [19] for plane and spherical waves with different turbulence outer scale values. (a) Plane wave, (b) spherical wave.

Equations (28)

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

Φn1(κ,α)=Φn(κ,α)G(κ,α),(2π/L0κ2π/l0,3<α<4),
Φn(κ,α)=A(α)·C^n2·κα,(2π/L0κ2π/l0,3<α<4),
A(α)=Γ(α1)4π2sin[(α3)π2],
G(κ,α)=GX(κ,α)+GY(κ,α),
GX(κ,α)=exp[κ2κX2(α)],GY(κ,α)=κα[κ2+κY2(α)]α/2.
GX(κ,l0,L0,α)=f(κ,l0,α)g(κ,L0)exp(κ2κX2),GY(κ,α)=κα[κ2+κY2(α)]α/2.
GX(κ,l0,L0,α)=exp(κ2κm2)[exp(κ2κX2)exp(κ2κX02)].
Φn1(κ,L0,α)A(α)·C^n2·κα[exp(κ2κX2)exp(κ2κX02)].
βa2=Dω(D)(kD)2.
Dωp(ρ)=8π2k20Ldz0[1J0(κρ)]Φn1(κ)κdκ,
Dωs(ρ)=8π2k2L01dξ0[1J0(κρξ)]Φn1(κ)κdκ.
Dωp(ρ,L0,α)=8π2k20Ldz0[1J0(κρ)]Φn1(κ,L0,α)κdκ.
J0(x)=n=0(1)nn!·Γ(n+1)·(x2)2n,
Dωp(ρ,L0,α)=8π2k2A(α)C^n2×0L{n=1(1)n1n!·(1)n·(ρ2)2n·0κ2nα+1·GX(κ,L0,α)dκ}dz.
Γ(x)=0κx1·eκdκ(κ>0,x>0),
F11(a;b;z)=n=0(a)n·zn(b)n·n!,
Dωp(ρ,L0,α)=4σR(pl)2β1(α)Γ(1α2)×{ηX(pl)1α/2[1F11(1α2;1;kρ2ηX(pl)4L)]ηX0(pl)1α/2[1F11(1α2;1;kρ2ηX0(pl)4L)]},
σR(pl)2(α)=β1A(α)C^n2π2k3α/2Lα/2,β1(α)=4Γ(α2)sin(πα4).
1F11(1α2;1;x)(1α2)x{1+[2(α2)Γ(α/2)]2α4x}α/22,
Dωp(ρ,L0,α)=σR(pl)2β1(α)Γ(2α2)kρ2L×{ηX(pl)2α/2{1+[2(α2)Γ(α/2)]2α4kρ2ηX(pl)4L}α/22ηX0(pl)2α/2{1+[2(α2)Γ(α/2)]2α4kρ2ηX0(pl)4L}α/22}.
βa2pl=σR(pl)2kL·β1(α)Γ(2α2)×{ηX(pl)2α/2{1+[2(α2)Γ(α/2)]2α4kD2ηX(pl)4L}α/22ηX0(pl)2α/2{1+[2(α2)Γ(α/2)]2α4kD2ηX0(pl)4L}α/22}.
Dωs(ρ,L0,α)=8π2k20Ldξ0[1J0(κρz/L)]Φn1(κ,L0,α)κdκ.
F22(a,b;c,d;z)=n=0(a)n(b)n·zn(c)n(d)n·n!,
Dωs_l(ρ,L0,α)=4σR(sp)2β2(α)Γ(1α2){ηX(sp)1α/2[1F22(1α2,12;1,32;kρ2ηX(sp)4L)]ηX0(sp)1α/2[1F22(1α2,12;1,32;kρ2ηX0(sp)4L)]}.
σR(sp)2=β2A(α)C^n2π2k3α/2Lα/2,β2(α)=4Γ(1α2)sin(πα4)Γ2(α/2)Γ(α).
1F22(1α2,12;1,32;x)(2α6)x{1+[6(α1)(α2)Γ(α/2)]2α4x}α/22.
Dωs(ρ,L0,α)=σR(sp)23β2(α)Γ(2α2)kρ2L×{ηX(sp)2α/2{1+[6(α1)(α2)Γ(α/2)]2α4kρ2ηX(sp)4L}α/22ηX0(sp)2α/2{1+[6(α1)(α2)Γ(α/2)]2α4kρ2ηX0(sp)4L}α/22}.
βa2sp=σR(sp)23kL·β2(α)Γ(2α2){ηX(sp)2α/2{1+[6(α1)(α2)Γ(α/2)]2α4kD2ηX(sp)4L}α/22ηX0(sp)2α/2{1+[6(α1)(α2)Γ(α/2)]2α4kD2ηX0(sp)4L}α/22}.

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