Abstract

The turbulence effect models derived with the Rytov theory method cannot be applied in the analysis of moderate-to-strong non-Kolmogorov turbulence. In this work, new expressions of the temporal power spectra of irradiance fluctuations are derived theoretically for optical waves propagating through moderate-to-strong non-Kolmogorov turbulence. They are developed under Andrews’ assumption that small-scale irradiance fluctuations are modulated by large-scale irradiance fluctuations of the optical wave. A wide range of turbulence strength is considered instead of a limited range for weak non-Kolmogorov turbulence. These expressions have general spectral power law values in the range 3 to 4 instead of the standard power law value of 11/3 for Kolmogorov turbulence. Calculations are performed to analyze turbulence strength and turbulence spectral power law’s variations on the final expressions.

© 2013 Optical Society of America

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References

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  1. S. F. Clifford, “Temporal-frequency spectra for a spherical wave propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 61, 1285–1292 (1971).
    [CrossRef]
  2. V. A. Banakh and V. L. Mironov, “Spectra of temporal intensity fluctuations of laser radiation traveling in a turbulent atmosphere,” Sov. J. Quantum Electron. 5, 1178–1182 (1975).
    [CrossRef]
  3. O. P. Lay, “The temporal power spectrum of atmospheric fluctuations due to water vapor,” Astron. Astrophys. Suppl. Ser. 122, 535–545 (1997).
    [CrossRef]
  4. R. Rao, S. Wang, X. Liu, and Z. Gong, “Turbulence spectrum effect on wave temporal-frequency spectra for light propagating through the atmosphere,” J. Opt. Soc. Am. A 16, 2755–2762 (1999).
    [CrossRef]
  5. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
  6. 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]
  7. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
  8. 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]
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    [CrossRef]
  10. 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]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  18. L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE, 1998).

2012 (1)

2010 (1)

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

2007 (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (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]

1999 (2)

1997 (2)

O. P. Lay, “The temporal power spectrum of atmospheric fluctuations due to water vapor,” Astron. Astrophys. Suppl. Ser. 122, 535–545 (1997).
[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]

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]

1975 (1)

V. A. Banakh and V. L. Mironov, “Spectra of temporal intensity fluctuations of laser radiation traveling in a turbulent atmosphere,” Sov. J. Quantum Electron. 5, 1178–1182 (1975).
[CrossRef]

1971 (1)

S. F. Clifford, “Temporal-frequency spectra for a spherical wave propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 61, 1285–1292 (1971).
[CrossRef]

Al-Habash, M. A.

Andrews, L. C.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (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).

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

Bai, X.

Banakh, V. A.

V. A. Banakh and V. L. Mironov, “Spectra of temporal intensity fluctuations of laser radiation traveling in a turbulent atmosphere,” Sov. J. Quantum Electron. 5, 1178–1182 (1975).
[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]

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.

Clifford, S. F.

S. F. Clifford, “Temporal-frequency spectra for a spherical wave propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 61, 1285–1292 (1971).
[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.

Du, W.

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (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 trubulence,” Atmos. Res. 88, 66–77 (2008).
[CrossRef]

Gong, Z.

Gurvich, A. S.

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.

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 trubulence,” 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 trubulence,” 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]

Lay, O. P.

O. P. Lay, “The temporal power spectrum of atmospheric fluctuations due to water vapor,” Astron. Astrophys. Suppl. Ser. 122, 535–545 (1997).
[CrossRef]

Liu, X.

Ma, J.

Mironov, V. L.

V. A. Banakh and V. L. Mironov, “Spectra of temporal intensity fluctuations of laser radiation traveling in a turbulent atmosphere,” Sov. J. Quantum Electron. 5, 1178–1182 (1975).
[CrossRef]

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (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 and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 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]

Rao, R.

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 trubulence,” 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]

Tan, L.

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

Toselli, I.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (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 trubulence,” Atmos. Res. 88, 66–77 (2008).
[CrossRef]

Wang, S.

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, W.

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

Astron. Astrophys. Suppl. Ser. (1)

O. P. Lay, “The temporal power spectrum of atmospheric fluctuations due to water vapor,” Astron. Astrophys. Suppl. Ser. 122, 535–545 (1997).
[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 trubulence,” Atmos. Res. 88, 66–77 (2008).
[CrossRef]

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

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (5)

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, “Scintillation index of optical plane wave propagating through non Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (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]

Sov. J. Quantum Electron. (1)

V. A. Banakh and V. L. Mironov, “Spectra of temporal intensity fluctuations of laser radiation traveling in a turbulent atmosphere,” Sov. J. Quantum Electron. 5, 1178–1182 (1975).
[CrossRef]

Other (3)

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

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

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

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Figures (5)

Fig. 1.
Fig. 1.

Normalized spatial covariance functions of irradiance fluctuations as the function of kρ2/L for plane and spherical waves with α=10/3. (a) Plane wave and (b) spherical wave.

Fig. 2.
Fig. 2.

Normalized spatial covariance functions of irradiance fluctuations as the function of kρ2/L for plane and spherical waves with α=11/3. (a) Plane wave and (b) spherical wave.

Fig. 3.
Fig. 3.

Normalized spatial covariance functions of irradiance fluctuations as the function of kρ2/L for plane and spherical waves with α=3.9. (a) Plane wave and (b) spherical wave.

Fig. 4.
Fig. 4.

Temporal power spectra of irradiance fluctuations ωtSI(pl)(ω,α) and ωSI(pl)(ω,α)/2πσI2 as the function of ω/ωt for a plane wave with α=10/3. (a) ωtSI(pl)(ω,α) and (b) ωSI(pl)(ω,α)/2πσI2.

Fig. 5.
Fig. 5.

Temporal power spectra of irradiance fluctuations ωtSI(pl)(ω,α) and ωSI(pl)(ω,α)/2πσI2 as the function of ω/ωt for plane wave with α=11/3. (a) ωtSI(pl)(ω,α) and (b) ωSI(pl)(ω,α)/2πσI2.

Equations (65)

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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.
BI(ρ)=exp[BlnX(ρ)+BlnY(ρ)]1,
bI(ρ)=BI(ρ)BI(0)BI(ρ)σI2.
BI(pl)(ρ,α)=8π2k2L010κΦn1(κ,α)J0(κρ)[1cos(Lκ2ξk)]dκdξ,
BI(pl)(ρ,α)=exp[BlnX(pl)(ρ,α)+BlnY(pl)(ρ,α)]1,
BlnX(pl)(ρ,α)=8π2k2L010κΦn(κ,α)·GX(κ,α)·J0(κρ)[1cos(Lκ2ξk)]dκdξ,
BlnY(pl)(ρ,α)=8π2k2L010κΦn(κ,α)·GY(κ,α)·J0(κρ)[1cos(Lκ2ξk)]dκdξ.
BlnX(pl)(ρ,α)=23π2k3α2Lα2·A(α)C^n2·0(η)2α2·exp[ηηX(pl)(α)]·J0(kηLρ)dη,
BlnY(pl)(ρ,α)=4π2k3α2Lα2·A(α)C^n2·01(η+ηY(pl))α/2·J0(kηLρ)dη.
0Jv(ax)exp(p2x2)xμ1dx=av·Γ(μ+ν2)2ν+1pμ+νΓ(ν+1)·F11(μ+ν2;ν+1;a24p2),(μ+ν>0,Re(p2)>0),
0xν+1·Jν(ax)(x2+b2)μ+1dx=aμ·bνμ2μ·Γ(μ+1)·Kμν(ab),(a,b>0,1<ν<2μ+32).
BlnX(pl)(ρ,α)=σlnX(pl)2(α)·F11(3α2;1;kρ2ηX(pl)(α)4L),
BlnY(pl)(ρ,α)=α2Γ(α/2)·σlnY(pl)2(α)·(kρ2ηY(pl)(α)4L)α412·Kα/21(kρ2ηY(pl)(α)L),
BI(pl)(ρ,α)=exp[σlnX(pl)2(α)·F11(3α2;1;kρ2ηX(pl)4L)+α2Γ(α/2)·σlnY(pl)2(α)·(kρ2ηY(pl)4L)α412·Kα/21(kρ2ηY(pl)L)]1.
bI(pl)(ρ,α)=BI(pl)(ρ,α)BI(pl)(0,α)BI(pl)(ρ,α)σI(pl)2(α),
Bsp(ρ,α)=8π2k2L010κΦn1(κ,α)J0(κξρ){1cos[Lκ2ξ(1ξ)k]}dκdξ,
BI(sp)(ρ,α)=exp[BlnX(sp)(ρ,α)+BlnY(sp)(ρ,α)]1,
BlnX(sp)(ρ,α)=8π2k2L010κΦn(κ,α)·GX(κ,α)·J0(κξρ){1cos[Lκ2ξ(1ξ)k]}dκdξ,
BlnY(sp)(ρ,α)=8π2k2L010κΦn(κ,α)·GY(κ,α)·J0(κξρ){1cos[Lκ2ξ(1ξ)k]}dκdξ.
BlnX(sp)(ρ,α)=2π2k3α/2Lα/2·A(α)C^n2·010η2α/2×exp[ηηX(sp)]·J0(kηLξρ)ξ2(1ξ)2dηdξ,
BlnY(sp)(ρ,α)=4π2k3α/2Lα/2A(α)C^n2·0101(η+ηY(sp))α/2·J0(kηLξρ)·dηdξ.
J0(x)=n=0(1)nn!·Γ(n+1)·(x2)2n,
0eatxt1=Γ(t)at,(t>0,a>0),
F12(A,B;C;Z)=Γ(C)Γ(B)·Γ(CB)01tB1·(1t)CB1·(1tZ)Adt,
F33(a,b,c;d,e,f;z)=n=0(a)n·(b)n·(c)n·zn(d)n·(e)n·(f)n·n!,
(a)n=Γ(a+n)Γ(a)=a(a+1)(a+n1).
BlnX(sp)(ρ,α)=σlnX(sp)2(α)·F33(3α2,32,2;72,3,1;kρ2ηX(sp)4L),
0xν+1·Jν(ax)(x2+b2)μ+1dx=aμ·bνμ2μ·Γ(μ+1)·Kμν(ab),(a,b>0,1<ν<2μ+32),
BlnY(sp)(ρ,α)=8Γ(α/2)π2k3α/2Lα/2A(α)C^n2·(k4Lρ)α21·ηY(sp)12α4·01ξα21·Kα/21(kηY(sp)Lξρ)dξ.
Kp(z)=π2·Ip(z)Ip(z)sinpπ,Ip(z)=n=0(z/2)2n+pn!·Γ(n+p+1),
F21(a;b,c;z)=n=0(a)n·zn(b)n·(c)n·n!,
BlnY(sp)(ρ,α)=σlnY(sp)2(α)·F21(12;2α2,32;ρ2kηY(sp)4L)σlnY(sp)2(α)·Γ(2α/2)Γ(α/2)·(α1)·(kρ2ηY(sp)4L)α/21·F21(α12;α2,α+12;ρ2kηY(sp)4L),
BI(sp)(ρ,α)=exp[σlnX(sp)2(α)·F33(3α2,32,2;72,3,1;kρ2ηX(sp)4L)+σlnY(sp)2(α)·F21(12;2α2,32;kρ2ηY(sp)4L)Γ(2α/2)Γ(α/2)·(α1)σlnY(sp)2(α)(kρ2ηY(sp)4L)α/21·F21(α12;α2,α+12;kρ2ηY(sp)4L)]1.
bI(sp)(ρ,α)=BI(sp)(ρ,α)BI(sp)(0,α)BI(sp)(ρ,α)σI(sp)2(α),
SI(ω)=2BI(τ,L)eiωτdτ=40BI(τ,L)cos(ωτ)dτ.
BI(pl)(τ,α)=exp[σlnX(pl)2(α)·F11(3α2;1;kV2τ2ηX(pl)4L)+α2Γ(α/2)·σlnY(pl)2(α)·(kV2τ2ηY(pl)4L)α/41/2·Kα/21(kV2τ2ηY(pl)L)]1.
SI(pl)(ω,α)=40BI(pl)(τ,α)cosωτdτ.
SI(pl)(ω,α)=4ωt0BI(pl)(s/ωt,α)cos(ωsωt)ds.
Bsp(τ,α)=8π2k2L010κΦn(κ,α)J0(κVτ){1cos[Lκ2ξ(1ξ)k]}dκdξ.
BI(sp)(τ,α)=exp[σlnX(sp)2(α)·F11(3α2;1;kV2τ2ηX(sp)4L)+α2Γ(α/2)·σlnY(sp)2(α)·(kV2τ2ηY(sp)4L)α/41/2·Kα/21(kV2τ2ηY(sp)L)]1.
SI(sp)(ω,α)=4ωt0BI(sp)(s/ωt,α)cos(ωsωt)ds.
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[1+fX(pl)(α)·σR(pl)4α2(α)],
ηY(pl)(α)=[0.51×(α2)·β1(α)8]22α·[1+fY(pl)(α)·σR(pl)4α2(α)],
ηX(sp)(α)=[30β2(α)×0.492Γ(3α/2)]26α·11+fX(sp)(α)·[σR(sp)2(α)]2α2,
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)2×0.49]2α6,fY(pl)(α)=(ln20.51)22α,
fX(sp)(α)=[r2(α)·I2(α)2×0.49]2α6,fY(sp)(α)=(ln20.51)22α,
σR(pl)2(α)=β1·A(α)Cn2π2k3α/2Lα/2,β1(α)=4·Γ(α2)·sin(πα4),
σR(sp)2=β2·A(α)Cn2π2k3α/2Lα/2,β2(α)=4·Γ(1α2)·sin(πα4)·Γ2(α/2)Γ(α),
r1(α)=1α2·[2](3α)(α10)α2·[Γ(1α/2)Γ(α/2)]α6α2·Γ(6αα2)·[β1(α)]82αα2,
r2(α)=1α2·[2](3α)(α10)α2·[Γ(1α/2)Γ(α/2)]α6α2·Γ(6αα2)·[β2(α)]82αα2,
I1(α)=1α3F12(6αα2,α3;α2;α2α1),
I2(α)=(α1)6αα2·Γ2(α3)Γ(2α6).
σlnX(pl)2(α)=2Γ(3α/2)3β1(α)·ηX(pl)3α2(α)·σR(pl)2(α),
σlnY(pl)2(α)=8(α2)β1(α)·ηY(pl)1α2(α)·σR(pl)2(α),
σlnX(sp)2(α)=2Γ(3α/2)30β2(α)·ηX(sp)3α2(α)·σR(sp)2(α),
σlnY(sp)2(α)=8(α2)β2(α)·ηY(sp)1α2(α)·σR(sp)2(α).
σI(pl)2(α)=exp[0.49σR(pl)2(α)[1+fX(pl)(α)·σR(pl)4α2(α)]3α2+0.51σR(pl)2(α)[1+fY(pl)(α)·σR(pl)4α2(α)]α21]1,
σI(sp)2(α)=exp[0.49σR(sp)2(α)[1+fX(sp)(α)·σR(sp)4α2(α)]3α2+0.51σR(sp)2(α)[1+fY(sp)(α)·σR(sp)4α2(α)]α21]1.

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