Abstract

Based on the Rytov approximation we have developed for weak scintillation conditions a general expression for the temporal averaged variance of irradiance. The present analysis provides, for what we believe is the first time, a firm theoretical basis for the often-observed reduction of irradiance fluctuations of an optical beam due to atmospheric turbulence. Accurate elementary analytic approximations are presented here for plane, spherical and beam waves for predicting the averaging times required to obtain an arbitrary value of the ratio of the standard deviation to the mean of an optical beam propagating through an arbitrary path in the atmosphere. In particular, a novel application of differential absorption measurement for the purpose of measuring column-integrated concentrations of various so-called greenhouse gas (GHG) atmospheric components is considered where the results of our analysis indicates that relatively short averaging times, on the order of a few seconds, are required to reduce the irradiance fluctuations to a value precise enough for GHG measurements of value to climate related studies.

© 2015 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Aperture averaging of optical scintillations in the turbulent atmosphere

James H. Churnside
Appl. Opt. 30(15) 1982-1994 (1991)

Effects of time averaging on optical scintillation in a ground-to-satellite atmospheric propagation

Morio Toyoshima and Kenichi Araki
Appl. Opt. 39(12) 1911-1919 (2000)

References

  • View by:
  • |
  • |
  • |

  1. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, Ed. 2 (SPIE Press Monograph, 2005) Vol. PM152, Chaps. 11 & 12 and references therein.
  2. D. K. Killinger and N. Menyuk, “Effect of turbulence-induced correlation on laser remote sensing errors,” Appl. Phys. Lett. 38(12), 968–970 (1981).
    [Crossref]
  3. W. B. Grant, A. M. Brothers, and J. R. Bogan, “Differential absorption lidar signal averaging,” Appl. Opt. 27(10), 1934–1938 (1988).
    [Crossref] [PubMed]
  4. P. L. Smith and S. M. Beck, “ System and apparatus for monitoring concentration of greenhouse gas,” U.S. Patent US 8614794 B2.
  5. B. Buffett and D. Archer, “Global inventory of methane clathrate: sensitivity to changes in the deep ocean,” Earth Planet. Sci. Lett. 227(3-4), 185–199 (2004).
    [Crossref]
  6. A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Energy and environment. Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
    [PubMed]
  7. M. Toyoshima and K. Araki, “Effects of time averaging on optical scintillation in a ground-to-satellite atmospheric propagation,” Appl. Opt. 39(12), 1911–1919 (2000).
    [Crossref] [PubMed]
  8. M. Toyoda, M. Toyoshima, T. Fukazawa, T. Takahashi, M. Shikatani, and K. Araki, “Measurement of laser-link-scintillation between ETS-VI and a ground station,” Proc. SPIE 2990, 287–295 (1997).
  9. H. Shen, L. Yu, and C. Fan, “Temporal spectrum of atmospheric scintillation and the effects of aperture averaging and time averaging,” Opt. Commun. 330, 160–164 (2014).
    [Crossref]
  10. A. D. Wheelon, Electromagnetic Scintillation: II. Weak Scattering (Cambridge University Press, 2003).
  11. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (Mc-Graw Hill, 1961), trans. by R. A. Silverman.
  12. S. Wolfram, Mathematica (Cambridge University Press, 2012), Version 9.
  13. R. L. Mitchell, “Permanance of the log-normal distribution,” J. Opt. Soc. Am. 58(9), 1267–1272 (1968).
    [Crossref]
  14. D. L. Fried, “Propagation of a spherical wave in a turbulent medium,” J. Opt. Soc. Am. 57(2), 175–180 (1967).
    [Crossref]
  15. R. T. Aiken, “Propagation from a point source in a randomly refractive medium,” Bell Syst. Tech. J. 48(5), 1129–1165 (1969).
    [Crossref]
  16. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, 1978), Vol. 2.
  17. J. D. Shelton, “Turbulence-induced scintillation on gaussian-beam waves: theoretical prediction and observations from a laser-illuminated satellite,” J. Opt. Soc. Am. A 12(10), 2172–2181 (1995).
    [Crossref]
  18. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence, Ed. 2 (SPIE, 2000).
  19. J. L. Bufton, “Comparison of vertical profile turbulence structure with stellar observations,” Appl. Opt. 12(8), 1785–1793 (1973).
    [Crossref] [PubMed]
  20. H. T. Yura and W. G. McKinley, “Aperture averaging of scintillation for a space-to-ground optical communication system,” Appl. Opt. 22(11), 1608–1609 (1983).
    [Crossref] [PubMed]

2014 (2)

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Energy and environment. Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[PubMed]

H. Shen, L. Yu, and C. Fan, “Temporal spectrum of atmospheric scintillation and the effects of aperture averaging and time averaging,” Opt. Commun. 330, 160–164 (2014).
[Crossref]

2004 (1)

B. Buffett and D. Archer, “Global inventory of methane clathrate: sensitivity to changes in the deep ocean,” Earth Planet. Sci. Lett. 227(3-4), 185–199 (2004).
[Crossref]

2000 (1)

1997 (1)

M. Toyoda, M. Toyoshima, T. Fukazawa, T. Takahashi, M. Shikatani, and K. Araki, “Measurement of laser-link-scintillation between ETS-VI and a ground station,” Proc. SPIE 2990, 287–295 (1997).

1995 (1)

1988 (1)

1983 (1)

1981 (1)

D. K. Killinger and N. Menyuk, “Effect of turbulence-induced correlation on laser remote sensing errors,” Appl. Phys. Lett. 38(12), 968–970 (1981).
[Crossref]

1973 (1)

1969 (1)

R. T. Aiken, “Propagation from a point source in a randomly refractive medium,” Bell Syst. Tech. J. 48(5), 1129–1165 (1969).
[Crossref]

1968 (1)

1967 (1)

Aiken, R. T.

R. T. Aiken, “Propagation from a point source in a randomly refractive medium,” Bell Syst. Tech. J. 48(5), 1129–1165 (1969).
[Crossref]

Araki, K.

M. Toyoshima and K. Araki, “Effects of time averaging on optical scintillation in a ground-to-satellite atmospheric propagation,” Appl. Opt. 39(12), 1911–1919 (2000).
[Crossref] [PubMed]

M. Toyoda, M. Toyoshima, T. Fukazawa, T. Takahashi, M. Shikatani, and K. Araki, “Measurement of laser-link-scintillation between ETS-VI and a ground station,” Proc. SPIE 2990, 287–295 (1997).

Archer, D.

B. Buffett and D. Archer, “Global inventory of methane clathrate: sensitivity to changes in the deep ocean,” Earth Planet. Sci. Lett. 227(3-4), 185–199 (2004).
[Crossref]

Arent, D.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Energy and environment. Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[PubMed]

Bogan, J. R.

Bradley, R.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Energy and environment. Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[PubMed]

Brandt, A. R.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Energy and environment. Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[PubMed]

Brothers, A. M.

Brown, N. J.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Energy and environment. Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[PubMed]

Buffett, B.

B. Buffett and D. Archer, “Global inventory of methane clathrate: sensitivity to changes in the deep ocean,” Earth Planet. Sci. Lett. 227(3-4), 185–199 (2004).
[Crossref]

Bufton, J. L.

Eardley, D.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Energy and environment. Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[PubMed]

Fan, C.

H. Shen, L. Yu, and C. Fan, “Temporal spectrum of atmospheric scintillation and the effects of aperture averaging and time averaging,” Opt. Commun. 330, 160–164 (2014).
[Crossref]

Fried, D. L.

Fukazawa, T.

M. Toyoda, M. Toyoshima, T. Fukazawa, T. Takahashi, M. Shikatani, and K. Araki, “Measurement of laser-link-scintillation between ETS-VI and a ground station,” Proc. SPIE 2990, 287–295 (1997).

Gopstein, A. M.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Energy and environment. Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[PubMed]

Grant, W. B.

Harriss, R.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Energy and environment. Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[PubMed]

Heath, G. A.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Energy and environment. Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[PubMed]

Jordaan, S. M.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Energy and environment. Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[PubMed]

Killinger, D. K.

D. K. Killinger and N. Menyuk, “Effect of turbulence-induced correlation on laser remote sensing errors,” Appl. Phys. Lett. 38(12), 968–970 (1981).
[Crossref]

Kort, E. A.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Energy and environment. Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[PubMed]

McKinley, W. G.

Menyuk, N.

D. K. Killinger and N. Menyuk, “Effect of turbulence-induced correlation on laser remote sensing errors,” Appl. Phys. Lett. 38(12), 968–970 (1981).
[Crossref]

Mitchell, R. L.

O’Sullivan, F.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Energy and environment. Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[PubMed]

Pétron, G.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Energy and environment. Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[PubMed]

Sasiela, R. J.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence, Ed. 2 (SPIE, 2000).

Shelton, J. D.

Shen, H.

H. Shen, L. Yu, and C. Fan, “Temporal spectrum of atmospheric scintillation and the effects of aperture averaging and time averaging,” Opt. Commun. 330, 160–164 (2014).
[Crossref]

Shikatani, M.

M. Toyoda, M. Toyoshima, T. Fukazawa, T. Takahashi, M. Shikatani, and K. Araki, “Measurement of laser-link-scintillation between ETS-VI and a ground station,” Proc. SPIE 2990, 287–295 (1997).

Stucky, G. D.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Energy and environment. Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[PubMed]

Takahashi, T.

M. Toyoda, M. Toyoshima, T. Fukazawa, T. Takahashi, M. Shikatani, and K. Araki, “Measurement of laser-link-scintillation between ETS-VI and a ground station,” Proc. SPIE 2990, 287–295 (1997).

Tans, P.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Energy and environment. Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[PubMed]

Toyoda, M.

M. Toyoda, M. Toyoshima, T. Fukazawa, T. Takahashi, M. Shikatani, and K. Araki, “Measurement of laser-link-scintillation between ETS-VI and a ground station,” Proc. SPIE 2990, 287–295 (1997).

Toyoshima, M.

M. Toyoshima and K. Araki, “Effects of time averaging on optical scintillation in a ground-to-satellite atmospheric propagation,” Appl. Opt. 39(12), 1911–1919 (2000).
[Crossref] [PubMed]

M. Toyoda, M. Toyoshima, T. Fukazawa, T. Takahashi, M. Shikatani, and K. Araki, “Measurement of laser-link-scintillation between ETS-VI and a ground station,” Proc. SPIE 2990, 287–295 (1997).

Wilcox, J.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Energy and environment. Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[PubMed]

Wofsy, S.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Energy and environment. Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[PubMed]

Yu, L.

H. Shen, L. Yu, and C. Fan, “Temporal spectrum of atmospheric scintillation and the effects of aperture averaging and time averaging,” Opt. Commun. 330, 160–164 (2014).
[Crossref]

Yura, H. T.

Appl. Opt. (4)

Appl. Phys. Lett. (1)

D. K. Killinger and N. Menyuk, “Effect of turbulence-induced correlation on laser remote sensing errors,” Appl. Phys. Lett. 38(12), 968–970 (1981).
[Crossref]

Bell Syst. Tech. J. (1)

R. T. Aiken, “Propagation from a point source in a randomly refractive medium,” Bell Syst. Tech. J. 48(5), 1129–1165 (1969).
[Crossref]

Earth Planet. Sci. Lett. (1)

B. Buffett and D. Archer, “Global inventory of methane clathrate: sensitivity to changes in the deep ocean,” Earth Planet. Sci. Lett. 227(3-4), 185–199 (2004).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

H. Shen, L. Yu, and C. Fan, “Temporal spectrum of atmospheric scintillation and the effects of aperture averaging and time averaging,” Opt. Commun. 330, 160–164 (2014).
[Crossref]

Proc. SPIE (1)

M. Toyoda, M. Toyoshima, T. Fukazawa, T. Takahashi, M. Shikatani, and K. Araki, “Measurement of laser-link-scintillation between ETS-VI and a ground station,” Proc. SPIE 2990, 287–295 (1997).

Science (1)

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Energy and environment. Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[PubMed]

Other (7)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, Ed. 2 (SPIE Press Monograph, 2005) Vol. PM152, Chaps. 11 & 12 and references therein.

P. L. Smith and S. M. Beck, “ System and apparatus for monitoring concentration of greenhouse gas,” U.S. Patent US 8614794 B2.

A. D. Wheelon, Electromagnetic Scintillation: II. Weak Scattering (Cambridge University Press, 2003).

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (Mc-Graw Hill, 1961), trans. by R. A. Silverman.

S. Wolfram, Mathematica (Cambridge University Press, 2012), Version 9.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, 1978), Vol. 2.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence, Ed. 2 (SPIE, 2000).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1 The finite averaging time spatial wave number weighting function FT(K;s) defined by (2.8) as a function of its argument.
Fig. 2
Fig. 2 The temporal averaging factor for plane, spherical and beam waves for constant turbulence conditions. Plane and spherical waves correspond to the limit F N , and 0, respectively, while the blue and red curves pertain to the exact and approximate results of Sec. 3.
Fig. 3
Fig. 3 The ratio of the Rytov beam wave irradiance variance to that of a spherical wave as a function of the transmitter Fresnel number.
Fig. 4
Fig. 4 A comparison between the numerical and analytic fit for the temporal scale T0 as a function of Fresnel number.
Fig. 5
Fig. 5 A comparison of the exact and approximate path weighting function given by Eq. (38) and Eq. (39), respectively.
Fig. 6
Fig. 6 The standard deviation of irradiance normalized to the mean as a function of averaging time. The horizontal black line is at the 1% level, and the red curve indicates the 1 / T1/2 scaling dependence.

Equations (43)

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

I(T)= 1 T 0 T I(t)dt
I(T) = 1 T 0 T I(t) dt= 1 T 0 T I 0 dt= I 0 ,
σ I 2 (T)= ( 1 T 0 T I( t 1 )d t 1 I(T) ) ( 1 T 0 T I( t 2 )d t 2 I(T) ) = 1 T 2 0 T d t 1 0 T d t 2 C I (τ)
C I (τ)= [I( t 1 ) I 0 ][I( t 2 ) I 0 ]
C I (τ)=16 π 2 k 2 0 L ds 0 dKK Φ n (K;s) F χ (K;s) J 0 (KV(s)τ)
F χ (K;s)= sin 2 [ K 2 η(s) 2k ]
σ I 2 (T)=16 π 2 k 2 0 L ds 0 dKK Φ n (K;s) F χ (K;s) F T (K;s)
F T (K;s)= F 2 1 ( 1 2 , 3 2 ;2, [KTV(s)] 2 /4 )
σ I 2 (T)= 1 2π W(ω) sinc 2 (ωT/2) dω,
σ 1 2 ( T )= σ I 2 ( 0 )A( T ),
A( T )= σ I 2 ( T ) σ I 2 ( 0 ) ,
A( T )= 0 L ds 0 dK K Φ n ( K;s ) F χ ( K;s ) F T ( K;s ) 0 L ds 0 dK K Φ n ( K;s ) F χ ( K;s ) .
A( T )= W( ω ) sin c 2 ( ωT/2 )dω W( ω ) dω .
A(T)= 0 L ds 0 dK sin 2 [ K 2 s 2k ] F 2 1 ( 1 2 , 3 2 ;2, (KTV) 2 /4 )/ K 8/3 0 L ds 0 dK sin 2 [ K 2 s 2k ]/ K 8/3 .
A(T)= 1 1728( 3 1) [ 162Γ( 5 6 ) T N 5/3 +1728( 3 1)π F 3 4 ( 11 12 , 5 12 , 1 4 ; 1 2 ,1, 5 4 , 3 2 ; T N 4 256 ) +66( 3 +1)π T N 2 F 3 4 ( 5 12 , 1 12 , 3 4 ; 3 2 , 3 2 , 7 4 ,2; T N 4 256 ) ]
A( T ) 1 1+ ( T N /1.95 ) 2
A( T )= 0 L ds 0 dK sin 2 [ K 2 s( 1s/L ) sk ] F 2 1 ( 1 2 , 3 2 ;2, ( KTV ) 2 /4 )/ K 8/3 0 L ds 0 dK sin 2 [ K 2 s( 1s/L ) sk ]/ K 8/3 .
A(T)= 0 1 du u 5/6 (1u) 5/6 Q(T;u) ,
Q(T;u)= 1 3168( 3 1)π [ 162Γ( 5 6 ) ( T N 2 u( 1u ) ) 5/6 +3168( 3 1 )π F 3 4 ( 5 12 , 1 12 , 1 4 ; 1 2 ,1, 5 4 , 3 2 ; T N 4 256 u 2 ( 1u ) 2 ) +55( 3 +1)π T N 2 F 3 4 ( 1 12 , 7 12 , 3 4 ; 3 2 , 3 2 , 7 4 ,2; T N 4 256 u 2 ( 1u ) 2 ) ]
A(T) 1 1+ ( T N /1.09) 2
u 0 ( r )=exp[ 4 r 2 D 2 ik r 2 2 R 0 ]
σ B (0)=8 π 2 k 2 0 L ds 0 dKK Φ n ( K,s )( exp[ γ i K 2 ( Ls ) k ]Re{ exp[ iγ K 2 ( Ls ) k ] } ) =4 π 2 0 L ds γ r 2 ( Ls ) 2 0 dK K 5 Φ n ( K,s ) sinc 2 [ γ r K 2 ( Ls ) 2k ]exp[ γ i K 2 ( Ls ) k ] ,
γ= γ r +i γ i
γ r = F N 2 ( 1L/ R 0 )( 1s/ R 0 )+s/L 1+ F N 2 ( 1L/ R 0 ) , γ i = F N [ ( 1L/ R 0 )s/L( 1S/ R 0 ) ] 1+ F N 2 ( 1L/ R 0 ) ,
σ B ( 0 )=2.606 k 7/6 C n 2 L 11/6 Re{ 1 16 Γ( 5 6 ) [ 3 ( F N F N 2 +1 ) 5/6 + 3 2 i ( i F N F N +1 ) 5/6 ( ( F N +i ) 2 F 1 ( 1, 5 3 ; 11 6 ; i F N +1 ) F N +i ) ] }
W(ω)=1.303 k 2 ω 8/3 0 L C n 2 (s) V 5/3 (s)( Q 1 Re[ Q 2 ] ) ds,
Q 1 = 0 1 dx exp[A x 2 ] x 8/3 x 2 1 , Q 2 = 0 1 dx exp[B x 2 ] x 8/3 x 2 1 ,
A=[ γ i (Ls) k ][ω/V(s)] 2 ,B=[ i γ(Ls) k ][ω/V(s)] 2 .
A B ( T )= dω sin c 2 ( ωT/2 ) 0 L ds C n 2 ( s ) V 5/3 ( s )( Q 1 Re[ Q 2 ] )/ ω 8/3 dω 0 L ds C n 2 ( s ) V 5/3 ( s )( Q 1 Re[ Q 2 ] )/ ω 8/3 ,
A B ( T )= dω sin c 2 ( ωT/2 ) 0 L ds( Q 1 Re[ Q 2 ] )/ ω 8/3 dω 0 L ds( Q 1 Re[ Q 2 ] )/ ω 8/3 ,
Q 1 = 1 2 ( A 4/3 Γ( 4 3 ) F 1 1 ( 1 2 ; 7 3 ;A )+ π Γ( 4 3 ) F 1 1 ( 5 6 ; 1 3 ;A ) Γ( 11 6 ) )
A B ( T ) 1 1+ ( T N / T 0 ) 2 ,
T 0 ( F N )=1.09exp[ 0.665 F N 1/2 ]+1.95( 1exp[ 0.555 F N 1/2 ] ),
V 0 ( h )=5+37exp[ ( h12 ) 2 25 ]( m/s ),
V= V 0 sinE [ 1+ cot 2 E sin 2 ( θ θ 0 ) ] 1/2 .
σ I 2 ( 0 )=2.25 k 7/6 0 L ds C n 2 ( h[ s ] ) s 5/6 ( 1s/L ) 5/6
A( T )= 0 L ds C n 2 [ h( s ) ] s 5/6 ( 1s/L ) 5/6 R( T;s ) 0 L ds C n 2 [ h( s ) ] s 5/6 ( 1s/L ) 5/6 ,
R( T;s )= 9( 1+ 3 ) r 5/3 Γ( 5 6 ) 352π + F 3 4 ( 5 12 , 1 12 , 1 4 ; 1 2 ,1, 5 4 , 3 2 ; r 4 256 ) + 5 288 ( 2+ 3 ) r 2 F 3 4 ( 1 12 , 7 12 , 3 4 ; 3 2 ,1, 7 4 ,2; r 4 256 )
R( T;s )= 1 1+0.154 r 2
A( T ) 1 1+ ( T 3.45 λcscE ) 2
1 T 2 0 T d t 1 0 T d t 2 J 0 ( KV( s )( t 1 t 2 ) ) = 0 1 d μ 1 0 1 d μ 2 J 0 ( KTV( s )( μ 1 μ 2 ) )
J 0 ( KTV( s )( μ 1 μ 2 ) )= 1 2π 0 2π dϕcos[ KTV( s )( μ 1 μ 2 )cosϕ ]
1 T 2 0 T d t 1 0 T d t 2 J 0 ( γ( t 1 t 2 ) )= 1 2π 0 2π dϕ4 sec 2 ϕ sin 2 ( γ 2 cosϕ ) / γ 2 = F 2 1 ( 1 2 , 3 2 ;2; γ 2 4 ),

Metrics