Abstract

Current theoretical temporal power spectra models of an optical wave have been developed for terrestrial environments. The interactions between humidity and temperature fluctuations in the marine atmospheric environments make the marine atmospheric turbulence particularly challenging, and the optical waves’ propagation through marine turbulence exhibits a different behavior with respect to terrestrial propagation. In this paper, the temporal power spectra of irradiance scintillation under weak marine atmospheric turbulence, which is one of the key temporal statistics to describe the correlation of irradiance fluctuations at different time instances, is investigated in detail both analytically and numerically. Closed-form expressions for the temporal power spectra of irradiance scintillation are derived for infrared plane and spherical waves under weak marine atmospheric turbulence, and they consider physically the influences of finite turbulence inner and outer scales. The final results indicate that the marine atmospheric turbulence brings more effects on the irradiance scintillation than the terrestrial atmospheric turbulence.

© 2014 Optical Society of America

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References

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  1. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, 1971).
  2. 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]
  3. 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]
  4. 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]
  5. 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]
  6. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
  7. W. H. Du, L. Y. Tan, J. Ma, and Y. J. Jiang, “Temporal-frequency spectra for optical wave propagating through non-Kolmogorov turbulence,” Opt. Express 18, 5763–5775 (2010).
    [CrossRef]
  8. L. Cui, B. Xue, and X. Cao, “Analysis of optical waves propagating through moderate-to-strong non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 30, 1738–1745 (2013).
    [CrossRef]
  9. C. A. Friehe, J. C. La Rue, F. H. Champagne, C. H. Gibson, and G. F. Dreyer, “Effects of temperature and humidity fluctuations on the optical refractive index in the marine boundary layer,” J. Opt. Soc. Am. 65, 1502–1511 (1975).
    [CrossRef]
  10. R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges (atmospheric effects on radio propagation),” Radio Sci. 13, 953–961 (1978).
    [CrossRef]
  11. K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 1, 173–184 (2008).
    [CrossRef]
  12. F. Strömqvist Vetelino, K. Grayshan, and C. Y. Young, “Inferring path average Cn2 values in the marine environment,” J. Opt. Soc. Am. A 24, 3198–3206 (2007).
    [CrossRef]
  13. I. Toselli, B. Agrawal, and S. Restaino, “Gaussian beam propagation in maritime atmospheric turbulence: long term beam spread and beam wander analysis,” Proc. SPIE 7814, 78140R (2010).
    [CrossRef]
  14. L. Cui, B. Xue, and F. Zhou, “Atmospheric turbulence MTF for infrared optical waves’ propagation through marine atmospheric turbulence,” Infrared Phys. Technol. 65, 24–29 (2014).
    [CrossRef]
  15. L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE, 1998).

2014 (1)

L. Cui, B. Xue, and F. Zhou, “Atmospheric turbulence MTF for infrared optical waves’ propagation through marine atmospheric turbulence,” Infrared Phys. Technol. 65, 24–29 (2014).
[CrossRef]

2013 (1)

2010 (2)

W. H. Du, L. Y. Tan, J. Ma, and Y. J. Jiang, “Temporal-frequency spectra for optical wave propagating through non-Kolmogorov turbulence,” Opt. Express 18, 5763–5775 (2010).
[CrossRef]

I. Toselli, B. Agrawal, and S. Restaino, “Gaussian beam propagation in maritime atmospheric turbulence: long term beam spread and beam wander analysis,” Proc. SPIE 7814, 78140R (2010).
[CrossRef]

2008 (2)

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 1, 173–184 (2008).
[CrossRef]

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 (1)

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]

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]

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]

1978 (1)

R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges (atmospheric effects on radio propagation),” Radio Sci. 13, 953–961 (1978).
[CrossRef]

1975 (1)

Agrawal, B.

I. Toselli, B. Agrawal, and S. Restaino, “Gaussian beam propagation in maritime atmospheric turbulence: long term beam spread and beam wander analysis,” Proc. SPIE 7814, 78140R (2010).
[CrossRef]

Andrews, L. C.

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).

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]

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.

Champagne, F. H.

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.

L. Cui, B. Xue, and F. Zhou, “Atmospheric turbulence MTF for infrared optical waves’ propagation through marine atmospheric turbulence,” Infrared Phys. Technol. 65, 24–29 (2014).
[CrossRef]

L. Cui, B. Xue, and X. Cao, “Analysis of optical waves propagating through moderate-to-strong non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 30, 1738–1745 (2013).
[CrossRef]

Dreyer, G. F.

Du, W. H.

Friehe, C. A.

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]

Gibson, C. H.

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]

Grayshan, K.

Grayshan, K. J.

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 1, 173–184 (2008).
[CrossRef]

Hill, R. J.

R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges (atmospheric effects on radio propagation),” Radio Sci. 13, 953–961 (1978).
[CrossRef]

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. J.

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]

La Rue, J. C.

Ma, J.

Phillips, R. L.

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]

Restaino, S.

I. Toselli, B. Agrawal, and S. Restaino, “Gaussian beam propagation in maritime atmospheric turbulence: long term beam spread and beam wander analysis,” Proc. SPIE 7814, 78140R (2010).
[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]

Strömqvist Vetelino, F.

Tan, L. Y.

Tatarskii, V. I.

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

Toselli, I.

I. Toselli, B. Agrawal, and S. Restaino, “Gaussian beam propagation in maritime atmospheric turbulence: long term beam spread and beam wander analysis,” Proc. SPIE 7814, 78140R (2010).
[CrossRef]

Vetelino, F. S.

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 1, 173–184 (2008).
[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]

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.

L. Cui, B. Xue, and F. Zhou, “Atmospheric turbulence MTF for infrared optical waves’ propagation through marine atmospheric turbulence,” Infrared Phys. Technol. 65, 24–29 (2014).
[CrossRef]

L. Cui, B. Xue, and X. Cao, “Analysis of optical waves propagating through moderate-to-strong non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 30, 1738–1745 (2013).
[CrossRef]

Young, C. Y.

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 1, 173–184 (2008).
[CrossRef]

F. Strömqvist Vetelino, K. Grayshan, and C. Y. Young, “Inferring path average Cn2 values in the marine environment,” J. Opt. Soc. Am. A 24, 3198–3206 (2007).
[CrossRef]

Zhou, F.

L. Cui, B. Xue, and F. Zhou, “Atmospheric turbulence MTF for infrared optical waves’ propagation through marine atmospheric turbulence,” Infrared Phys. Technol. 65, 24–29 (2014).
[CrossRef]

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]

Infrared Phys. Technol. (1)

L. Cui, B. Xue, and F. Zhou, “Atmospheric turbulence MTF for infrared optical waves’ propagation through marine atmospheric turbulence,” Infrared Phys. Technol. 65, 24–29 (2014).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Express (1)

Proc. SPIE (4)

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]

I. Toselli, B. Agrawal, and S. Restaino, “Gaussian beam propagation in maritime atmospheric turbulence: long term beam spread and beam wander analysis,” Proc. SPIE 7814, 78140R (2010).
[CrossRef]

Radio Sci. (1)

R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges (atmospheric effects on radio propagation),” Radio Sci. 13, 953–961 (1978).
[CrossRef]

Waves Random Complex Media (1)

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 1, 173–184 (2008).
[CrossRef]

Other (3)

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

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 (6)

Fig. 1.
Fig. 1.

f(κ,κl,κ0) for both marine and terrestrial turbulence cases (L0=50m, l0=1mm).

Fig. 2.
Fig. 2.

ω0WI(ω,l0,L0)/σR2 as a function of ω/ω0 with different turbulence inner scale values under marine atmospheric turbulence (λ=1.55μm, L=2000m, L0=50m); (a) plane wave and (b) spherical wave.

Fig. 3.
Fig. 3.

ωWI(ω,l0,L0)/2πσR2 as a function of ω/ω0 with different turbulence inner scale values (λ=1.55μm, L=2000m, L0=50m); (a) plane wave and (b) spherical wave.

Fig. 4.
Fig. 4.

ω0WI(ω,l0,L0)/σR2 as a function of ω/ω0 with different turbulence inner values under terrestrial atmospheric turbulence (λ=1.55μm, L=2000m, L0=50m); (a) plane wave and (b) spherical wave.

Fig. 5.
Fig. 5.

Comparisons between the results derived for both marine and terrestrial turbulences with different turbulence inner scale values (λ=1.55μm, L=2000m, L0=50m); (a) l0=1mm, (b) l0=2mm, (c) l0=3mm, and (d) l0=4mm.

Fig. 6.
Fig. 6.

ω0WI(ω,l0,L0)/σR2 as a function of ω/ω0 with different wavelengths under marine atmospheric turbulence (l0=1mm, L=2000m, L0=50m); (a) plane wave and (b) spherical wave.

Equations (41)

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Φn,Ma(κ)=0.033Cn2κ11/3f(κ,κl,κ0),(0κ<),
f(κ,κl,κ0)=[10.061κκl+2.836(κκl)7/6]×[1exp(κ2κ02)]exp(κ2κl2),
WI(ω)=40CI(t)cos(ωt)dt.
CI(pl)(ρ)=8π2k20L0κΦn,Ma(κ)J0(ρκ)[1cos(κ2(Lz)k)]dκdz,
CI(sp)(ρ)=8π2k20L0κΦn,Ma(κ)J0(ρκ)[1cos(κ2z(Lz)Lk)]dκdz,
CI(pl)(t)=8π2k20L0κΦn,Ma(κ)J0(νtκ)[1cos(κ2(Lz)k)]dκdz,
CI(sp)(t)=8π2k20L0κΦn,Ma(κ)J0(νtκ)[1cos(κ2z(Lz)Lk)]dκdz.
WI(pl)(ω,l0,L0)=32π2k20L00κΦn,Ma(κ)J0(νtκ)×[1cos(κ2(Lz)k)]cos(ωt)dκdzdt.
0J0(ax)cos(bx)dx={(a2b2)1/20<b<a0b>a,
WI(pl)(ω,l0,L0)=32π2k2Lω/νκΦn,Ma(κ)×[1sin(κ2L/k)κ2L/k](ν2κ2ω2)1/2dκ.
U(a;c;z)=1Γ(a)0eztta1(1+t)ca1dt,a>0,Re(z)>0,
WI(pl)(ω,l0,L0)=7.52σR(pl)2ω0·(ω2ω02)4/3[gpl(1)(ω,l0,L0)gpl(2)(ω,l0,L0)],
gpl(i)(ω,l0,L0)=exp(Ai)U(12;13;Ai)(ω2ω02)1Im[exp(Bi)U(12;43;Bi)]0.061Ai1/2{exp(Ai)U(12;16;Ai)(ω2ω02)1Im[exp(Bi)U(12;56;Bi)]}+2.836Ai7/12{exp(Ai)U(12;14;Ai)(ω2ω02)1Im[exp(Bi)U(12;34;Bi)]},
σR(pl)2=1.23Cn2k7/6L11/6,A1=ω2v2(1κl2),A2=ω2v2(1κl2+1κ02),B1=ω2v2(1κl2iLk),B2=ω2v2(1κl2+1κ02i).
WI(pl)(ω,l0,L0)=7.52σR(pl)2ω0·(ω2ω02)4/3[gpl(1)(ω,l0,L0)].
gpl(1)(ω,l0,L0)=Γ(4/3)Γ(11/6)(ω2ω02)1Im[exp(iω2ω02)U(12;43;iω2ω02)].
WI(pl)(ω,l0,L0)=7.52σR(pl)2ω0(ω2ω02)4/3×[Γ(4/3)Γ(11/6)(ω2ω02)1Im[exp(iω2ω02)U(12;43;iω2ω02)]].
WI(sp)(ω,l0,L0)=32π2k20L00κΦn,Ma(κ)J0(νtκ)[1cos(κ2z(Lz)Lk)]cos(ωt)dκdz.
WI(sp)(ω,l0,L0)=32π2k20Lω/νκΦn,Ma(κ)×[1cos(κ2z(Lz)Lk)](v2κ2ω2)1/2dκdz.
WI(sp)(ω,l0,L0)=1.87Cn2π2k2Lv·(ωv)8/3×01[gsp(1)(ω,l0,L0)gsp(2)(ω,l0,L0)]dξ,
gsp(i)(ω,l0,L0)=exp(Ai)U(12;13;Ai)Re[exp(Bi)U(12;13;Bi)]0.061κl(ωv){exp(Ai)U(12;16;Ai)Re[exp(Bi)U(12;16;Bi)]}+2.836κl7/6(ωv)7/6{exp(Ai)U(12;14;Ai)Re[exp(Bi)U(12;14;Bi)]};
A1=ω2v2(1κl2),B1=ω2v2(1κl2iLξ(1ξ)k),A2=ω2v2(1κl2+1κ02),B2=ω2v2(1κl2+1κ02iLξ(1ξ)k).
WI(sp)(ω,l0,L0)=18.6σR(sp)2ω0(ω2ω02)4/3[gsp(1)(ω,l0,L0)gsp(2)(ω,l0,L0)],
σR(sp)2=0.49Cn2k7/6L11/6.
gsp(i)(ω,l0,L0)=hsp(i)(12,13,z)0.061Ai1/2hsp(i)(12,16,z)+2.836Ai7/12hsp(i)(12,14,z),
hsp(1)(a,c,z)=exp(A1)U(a;c;A1)Re{Γ(1c)Γ(1+ac)F22(ca,1;c,32;i4ω2ω02)+12Γ(c1)Γ(2c)πΓ(a)Γ(2c+1/2)(i4ω2ω02)1cF11(1a;2c+12;i4ω2ω02)},
hsp(2)(a,c,z)=exp(A2)U(a;c;A2)Γ(1c)Γ(1+ac)F11(ca;c;ω2v2κ02)Γ(c1)Γ(a)(ω2v2κ02)1cF11(1a;2c;ω2v2κ02).
WI(sp)(ω,l0,L0)=18.6σR(sp)2ω0(ω2ω02)4/3[gsp(1)(ω,l0,L0)].
gsp(1)(ω,l0,L0)=Γ(4/3)Γ(11/6)Re{1F22(65,1;13,32;i4ω2ω02)12Γ(4/3)Γ(7/3)Γ(11/6)Γ(17/6)Γ(4/3)(i4ω2ω02)4/3F11(12;176;i4ω2ω02)}.
WI(sp)(ω,l0,L0)=18.6σR(sp)2(α)ω0(ω2ω02)4/3Γ(4/3)Γ(11/6)Re{1F22(65,1;13,32;i4ω2ω02)12Γ(4/3)Γ(7/3)Γ(11/6)Γ(17/6)Γ(4/3)(i4ω2ω02)4/3F11(12;176;i4ω2ω02)}.
WI(sp)(ω,l0,L0)=1.87C^n2π2k2Lv·(ωv)8/301[gsp(1)(ω,l0,L0)gsp(2)(ω,l0,L0)]dξ,
gsp(i)(ω,l0,L0)=exp(Ai)U(12;13;Ai)Re[exp(Bi)U(12;13;Bi)]0.061κl(ωv){exp(Ai)U(12;16;Ai)Re[exp(Bi)U(12;16;Bi)]}+2.836κl7/6(ωv)7/6{exp(Ai)U(12;14;Ai)Re[exp(Bi)U(12;14;Bi)]},
U(a;c;z)=Γ(1c)Γ(1+ac)F11(a;c;z)+Γ(c1)Γ(a)z1cF11(1+ac;2c;z),
ezF11(a;c;z)=F11(ca;c;z),
exp(Bi)U(a;c;Bi)=Γ(1c)Γ(1+ac)F11(ca;c;Bi)+Γ(c1)Γ(a)Bi1c·F11(1a;2c;Bi).
F11(a;c;z)=n=0(a)nzn(c)nn!,
01exp(Bi)U(a;c;Bi)dξ=01Re[Γ(1c)Γ(1+ac)F11(ca;c;Bi)+Γ(c1)Γ(a)Bi1cF11(1a;2c;Bi)]dξ=Γ(1c)Γ(1+ac)n=0(ca)n(1)n(c)nn!01Re[Bin]dξ+Γ(c1)Γ(a)n=0(1a)n(1)n(2c)nn!01Re[Bi1c+n]dξ.
F12(a,b;c;z)=Γ(c)Γ(b)·Γ(cb)01tb1·(1t)cb1·(1tz)adt,
F12(a,b;c;1)=Γ(c)Γ(cab)Γ(ca)·Γ(cb).
01Re[exp(B1)U(a;c;B1)]dξ=Re{Γ(1c)Γ(1+ac)n=0(ca)n(c)nn!(iω2L4v2k)n(1)n(3/2)n+Γ(c1)Γ(a)Γ(2c)π2Γ(2c+1/2)n=0(1a)n(2c+1/2)nn!(iω2L4v2k)1c+n}=Re{Γ(1c)Γ(1+ac)F22(ca,1;c,3/2;iω2L4v2k)+Γ(c1)Γ(a)Γ(2c)π2Γ(2c+1/2)(iω2L4v2k)1cF11(1a;2c+1/2;iω2L4v2k)},
01exp(B2)U(a;c;B2)dξ=Γ(1c)Γ(1+ac)F11(ca;c;ω2v2κ02)+Γ(c1)Γ(a)(ω2v2κ02)1cF11(1a;2c;ω2v2κ02).

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