Abstract

The effects of moderate-to-strong non-Kolmogorov turbulence on the angle of arrival (AOA) fluctuations for plane and spherical waves are investigated in detail both analytically and numerically. New analytical expressions for the variance of AOA fluctuations are derived for moderate-to-strong non-Kolmogorov turbulence. The new expressions cover a wider range of non-Kolmogorov turbulence strength and reduce correctly to previously published analytic expressions for the cases of plane and spherical wave propagation through both weak non-Kolmogorov turbulence and moderate-to-strong Kolmogorov turbulence cases. The final results indicate that, as turbulence strength becomes greater, the expressions developed with the Rytov theory deviate from those given in this work. This deviation becomes greater with stronger turbulence, up to moderate-to-strong turbulence strengths. Furthermore, general spectral power law has significant influence on the variance of AOA fluctuations in non-Kolmogorov turbulence. These results are useful for understanding the potential impact of deviations from the standard Kolmogorv spectrum.

© 2013 Optical Society of America

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2012

2011

2010

2009

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

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

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

Y. Cheon and A. Muschinski, “Closed-form approximations for the angle-of-arrival variance of plane and spherical waves propagating through homogeneous and isotropic turbulence,” J. Opt. Soc. Am. A 24, 415–422 (2007).
[CrossRef]

2006

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

2005

H. T. Eyyuboglu and Y. Baykal, “Analysis of laser multimode content on the angle-of-arrival fluctuations in free-space optics access systems,” Opt. Eng. 44, 056002 (2005).
[CrossRef]

2004

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

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

2002

M. J. Vilcheck, A. E. Reed, H. R. Burris, W. J. Scharpf, C. I. Moore, and M. R. Suite, “Multiple methods for measuring atmospheric turbulence,” Proc. SPIE 4821, 300–309 (2002).
[CrossRef]

2000

1999

E. Masciadri, J. Vernin, and P. Bougeault, “3D mapping of optical turbulence using an atmospheric numerical model. I. A useful tool for the ground-based astronomy,” Astron. Astrophys Suppl. Ser. 137, 185–202 (1999).
[CrossRef]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
[CrossRef]

1997

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

1994

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]

1992

R. J. Hill, “Review of optical scintillation methods of measuring the refractive-index spectrum, inner scale and surface fluxes,” Waves Random Media 2, 179–201 (1992).
[CrossRef]

Al-Habash, M. A.

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, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
[CrossRef]

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

Bai, X.

Baykal, Y.

H. T. Eyyuboglu and Y. Baykal, “Analysis of laser multimode content on the angle-of-arrival fluctuations in free-space optics access systems,” Opt. Eng. 44, 056002 (2005).
[CrossRef]

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]

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

Bin-dang, X.

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]

Bognino, J.

Bougeault, P.

E. Masciadri, J. Vernin, and P. Bougeault, “3D mapping of optical turbulence using an atmospheric numerical model. I. A useful tool for the ground-based astronomy,” Astron. Astrophys Suppl. Ser. 137, 185–202 (1999).
[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]

Burris, H. R.

M. J. Vilcheck, A. E. Reed, H. R. Burris, W. J. Scharpf, C. I. Moore, and M. R. Suite, “Multiple methods for measuring atmospheric turbulence,” Proc. SPIE 4821, 300–309 (2002).
[CrossRef]

Cao, X.

Cheon, Y.

Chong, G.

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

Conan, R.

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.

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]

Eyyuboglu, H. T.

H. T. Eyyuboglu and Y. Baykal, “Analysis of laser multimode content on the angle-of-arrival fluctuations in free-space optics access systems,” Opt. Eng. 44, 056002 (2005).
[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.

Hill, R. J.

R. J. Hill, “Review of optical scintillation methods of measuring the refractive-index spectrum, inner scale and surface fluxes,” Waves Random Media 2, 179–201 (1992).
[CrossRef]

Hopen, C. Y.

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]

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]

Jian-kang, D.

Jie-ning, W.

Jing, M.

M. Jing, G. Chong, and T. Li-Ying, “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]

Lin-yan, C.

Li-Ying, T.

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

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.

Masciadri, E.

E. Masciadri, J. Vernin, and P. Bougeault, “3D mapping of optical turbulence using an atmospheric numerical model. I. A useful tool for the ground-based astronomy,” Astron. Astrophys Suppl. Ser. 137, 185–202 (1999).
[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]

Moore, C. I.

M. J. Vilcheck, A. E. Reed, H. R. Burris, W. J. Scharpf, C. I. Moore, and M. R. Suite, “Multiple methods for measuring atmospheric turbulence,” Proc. SPIE 4821, 300–309 (2002).
[CrossRef]

Muschinski, A.

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, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16, 1417–1429 (1999).
[CrossRef]

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]

Reed, A. E.

M. J. Vilcheck, A. E. Reed, H. R. Burris, W. J. Scharpf, C. I. Moore, and M. R. Suite, “Multiple methods for measuring atmospheric turbulence,” Proc. SPIE 4821, 300–309 (2002).
[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]

Scharpf, W. J.

M. J. Vilcheck, A. E. Reed, H. R. Burris, W. J. Scharpf, C. I. Moore, and M. R. Suite, “Multiple methods for measuring atmospheric turbulence,” Proc. SPIE 4821, 300–309 (2002).
[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]

Suite, M. R.

M. J. Vilcheck, A. E. Reed, H. R. Burris, W. J. Scharpf, C. I. Moore, and M. R. Suite, “Multiple methods for measuring atmospheric turbulence,” Proc. SPIE 4821, 300–309 (2002).
[CrossRef]

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]

Tatarski, V. I.

V. I. Tatarski, Propagation of Waves in a Turbulent Medium (Nauka, 1967).

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, trans. for NOAA (Israel Program for Scientific Translations, 1971).

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]

Vernin, J.

E. Masciadri, J. Vernin, and P. Bougeault, “3D mapping of optical turbulence using an atmospheric numerical model. I. A useful tool for the ground-based astronomy,” Astron. Astrophys Suppl. Ser. 137, 185–202 (1999).
[CrossRef]

Vilcheck, M. J.

M. J. Vilcheck, A. E. Reed, H. R. Burris, W. J. Scharpf, C. I. Moore, and M. R. Suite, “Multiple methods for measuring atmospheric turbulence,” Proc. SPIE 4821, 300–309 (2002).
[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]

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]

Xiao-guang, C.

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]

Xue, B.

Xue, W.

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]

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]

Zhou, F.

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

Astron. Astrophys Suppl. Ser.

E. Masciadri, J. Vernin, and P. Bougeault, “3D mapping of optical turbulence using an atmospheric numerical model. I. A useful tool for the ground-based astronomy,” Astron. Astrophys Suppl. Ser. 137, 185–202 (1999).
[CrossRef]

Atmos. Res.

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.

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

J. Opt. Soc. Am. A

Opt. Commun.

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. Eng.

H. T. Eyyuboglu and Y. Baykal, “Analysis of laser multimode content on the angle-of-arrival fluctuations in free-space optics access systems,” Opt. Eng. 44, 056002 (2005).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

M. J. Vilcheck, A. E. Reed, H. R. Burris, W. J. Scharpf, C. I. Moore, and M. R. Suite, “Multiple methods for measuring atmospheric turbulence,” Proc. SPIE 4821, 300–309 (2002).
[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]

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Figures (3)

Fig. 1.
Fig. 1.

Variance of AOA fluctuations derived in this study and variance of AOA fluctuations derided from the Rytov theory for plane and spherical waves with different α values. (a) α=10/3, (b) α=11/3, and (c) α=3.9.

Fig. 2.
Fig. 2.

Variance of AOA fluctuations and large-scale variance of AOA fluctuations for plane and spherical waves with different α values: (a) α=10/3, (b) α=11/3, (c) α=3.9.

Fig. 3.
Fig. 3.

Comparison of the variance of AOA fluctuations derived in this work with those in [11] for Kolmogorov strong turbulence: (a) plane wave and (b) spherical wave.

Equations (55)

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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.
βa2=Dω(D)(kD)2,
Dωp(ρ)=8π2k20Ldz0[1J0(κρ)]Φn1(κ)κdκ,
Dωs(ρ)=8π2k2L01dξ0[1J0(κρξ)]Φn1(κ)κdκ.
Dωp(ρ,α)=8π2k20Ldz0[1J0(κρ)]Φn1(κ,α)κdκ.
Dωp(ρ,α)=Dωp_l(ρ,α)+Dωp_s(ρ,α),
Dωp_l(ρ,α)=8π2k20Ldz0[1J0(κρ)]×Φn(κ,α)GX(κ,α)κdκ,
Dωp_s(ρ,α)=8π2k20Ldz0[1J0(κρ)]×Φn(κ,α)GY(κ,α)κdκ.
0Jv(ax)exp(p2x2)xμ1dx=avΓ(μ+ν2)2ν+1pμ+νΓ(ν+1)F11(μ+ν2;ν+1;a24p2),(μ+ν>0,Re(p2)>0),
0xν+1Jν(ax)(x2+b2)μ+1dx=aμbνμ2μΓ(μ+1)Kμν(ab),(a,b>0,1<Re(ν)<Re(2μ+32)),
Dωp_l(ρ,α)=4σR(pl)2ηX(pl)1α/2β1(α)Γ(1α2)×[1F11(1α2;1;kρ2ηX(pl)4L)],
Dωp_s(ρ,α)=8σR(pl)2ηY(pl)1α/2β1(α)[1α21Γ(α/2)×(kρ2ηY(pl)4L)α/21Kα/21(kρ2ηY(pl)L)].
Dωp(ρ,α)=8σR(pl)2β1(α){12ηX(pl)1α/2Γ(1α2)×[1F11(1α2;1;kρ2ηX(pl)4L)]+ηY(pl)1α/2×[1α21Γ(α/2)(kρ2ηY(pl)4L)α/21×Kα/21(kρ2ηY(pl)L)]}.
βa2pl=8σR(pl)2β1(α)(kD)2{12ηX(pl)1α/2Γ(1α2)×[1F11(1α2;1;kD2ηX(pl)4L)]+ηY(pl)1α/2×[1α21Γ(α/2)(kD2ηY(pl)4L)α/21×Kα/21(kD2ηY(pl)L)]}.
βa2pl=4σR(pl)2β1(α)(kD)2ηX(pl)1α/2Γ(1α2)×[1F11(1α2;1;kD2ηX(pl)4L)].
1F11(1α2;1;x)(1α2)x{1+[2(α2)Γ(α/2)]2α4x}α/22.
βa2pl=Γ(2α2)σR(pl)2β1(α)kLηX(pl)2α/2×{1+[2(α2)Γ(α/2)]2α4kD2ηX(pl)4L}α/22.
βa2pl=π2A(α)LC^n2Γ(2α/2)(D2/4)2α/2[2(α2)Γ(α/2)].
βa2pl=1.81C^n2k1/6L5/6ηX(pl)1/6(1+0.058kD2ηX(pl)L)1/6.
Dωs(ρ,α)=8π2k2L01dξ0[1J0(κρξ)]Φn1(κ,α)κdκ.
Dωs(ρ,α)=Dωs_l(ρ,α)+Dωs_s(ρ,α).
Dωs_l(ρ,α)=8π2k20Ldz0[1J0(κρξ)]×Φn(κ,α)GX(κ,α)κdκ,
Dωs_s(ρ,α)=8π2k20Ldz0[1J0(κρξ)]×Φn(κ,α)GY(κ,α)κdκ.
Dωs_l(ρ,α)=4σR(sp)2ηX(sp)1α/2β2(α)01Γ(1α2)×[1F11(1α2;1;kρ2ξ2ηX(sp)4L)]dξ,
Dωs_s(ρ,α)=8σR(sp)2ηY(sp)1α/2β2(α)01[1α21Γ(α/2)×(kρ2ηY(sp)ξ24L)α412Kα/21(kρ2ηY(sp)ξ2L)]dξ.
F11(a;b;z)=n=0(a)nzn(b)nn!,
Kp(z)=π2Ip(z)Ip(z)sinpπ,Ip(z)=n=0(z/2)2n+pn!Γ(n+p+1),
F22(a,b;c,d;z)=n=0(a)n(c)nzn(b)n(d)nn!,F21(a;b,c;z)=n=0(a)nzn(b)n(c)nn!,
Dωs_l(ρ,α)=4σR(sp)2β2(α)ηX(sp)1α/2Γ(1α2)×[1F22(1α2,12;1,32;kρ2ηX(sp)4L)],
Dωs_s(ρ,α)=8σR(sp)2β2(α){1α2ηY(sp)1α/2ηY(sp)1α/2α2×F21(12;32,2α2;kρ2ηY(sp)4L)+Γ(2α/2)(α2)(α1)Γ(α/2)(kρ24L)α/21×F21(α12;α2,α+12;kρ2ηY(sp)4L)}.
Dωs(ρ,α)=8σR(sp)2β2(α){12ηX(sp)1α/2Γ(1α2)×[1F22(1α2,12;1,32;kρ2ηX(sp)4L)]+ηY(sp)1α/2α2ηY(sp)1α/2α2F21(12;32,2α2;kρ2ηY(sp)4L)+Γ(2α/2)(α2)(α1)Γ(α/2)(kρ24L)α/21×F21(α12;α2,α+12;kρ2ηY(sp)4L)}.
βa2sp=8σR(sp)2β2(α)(kD)2{12ηX(sp)1α/2Γ(1α2)×[1F22(1α2,12;1,32;kD2ηX(sp)4L)]+ηY(sp)1α/2α2ηY(sp)1α/2α2F21(12;32,2α2;kD2ηY(sp)4L)+Γ(2α/2)(α2)(α1)Γ(α/2)(kD24L)α/21×F21(α12;α2,α+12;kD2ηY(sp)4L)}.
βa2sp=4σR(sp)2β2(α)(kD)2ηX(sp)1α/2Γ(1α2)×[1F22(1α2,12;1,32;kD2ηX(sp)4L)].
1F22(1α2,12;1,32;x)(2α6)x{1+[6(α1)(α2)Γ(α/2)]2α4x}α/22.
βa2sp=13Γ(2α2)σR(sp)2β2(α)kLηX(sp)2α/2×{1+[6(α1)(α2)Γ(α2)]2α4kD2ηX(sp)4L}α/22.
βa2sp=1(α1)π2A(α)LC^n2Γ(2α/2)(D2/4)2α/2[2(α2)Γ(α/2)].
βa2sp=0.603C^n2k1/6L5/6ηX(sp)1/6{1+0.0285kD2ηX(sp)L}1/6.
κX(pl)2(α)=kLηX(pl)(α),κY(pl)2(α)=kLηY(pl)(α),
κX(sp)2(α)=kLηX(sp)(α),κY(sp)2(α)=kLηY(sp)(α).
ηX(pl)(α)=[3β1(α)×0.492Γ(3α/2)]26α[1+fX(pl)(α)σR(pl)4α2(α)]1,
ηY(pl)(α)=[0.51(α2)β1(α)8]22α[1+fY(pl)(α)σR(pl)4α2(α)],
ηX(sp)(α)=[15β2(α)×0.49Γ(3α/2)]26α[1+fX(sp)(α)σR(sp)4α2(α)]1,
ηY(sp)(α)=[0.51(α2)β2(α)8]22α[1+fY(sp)(α)σR(sp)4α2(α)],
fX(pl)(α)=[r1(α)I1(α)0.98]2α6,fY(pl)(α)=(ln20.51)22α.
fX(sp)(α)=[r2(α)I2(α)0.98]2α6,fY(sp)(α)=(ln20.51)22α.
σR(pl)2(α)=β1A(α)Cn2π2k3α/2Lα/2,β1(α)=4Γ(α2)sin(πα4).
σR(sp)2=β2A(α)Cn2π2k3α/2Lα/2,β2(α)=4Γ(1α2)sin(πα4)Γ2(α/2)Γ(α).
r1(α)=2(3α)(α10)α2α2[Γ(1α/2)Γ(α/2)]α6α2Γ(6αα2)[β1(α)]82αα2,
r2(α)=2(3α)(α10)α2α2[Γ(1α/2)Γ(α/2)]α6α2Γ(6αα2)[β2(α)]82αα2,
I1(α)=F12(6αα2,α3;α2;α2α1)α3,
I2(α)=(α1)6αα2Γ2(α3)Γ(2α6).

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