Abstract

Expressions for the variance and the power spectral density of turbulence-induced log-amplitude fluctuations are derived for Gaussian-beam waves in the regime of weak scattering. This formulation includes effects that are due to turbulence strength variations along the propagation path, offset of the observation point from the beam axis, and sensitivity to focus and beam diameter. Comparison of theoretical results with observed scintillation during experiments with a laser-illuminated satellite reveals good agreement.

© 1995 Optical Society of America

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References

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  1. V. A. Banakh, V. L. Mironov, Lidar in a Turbulent Atmosphere (Artech House, Boston, 1987).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  5. T. W. Lawrence, D. M. Goodman, E. M. Johansson, J. P. Fitch, “Speckle imaging of satellites at the U.S. Air Force Maui Optical Station,” Appl. Opt. 31, 6307–6321 (1992).
    [CrossRef] [PubMed]
  6. A. Labreyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).
  7. R. J. Sasiela, J. D. Shelton, “Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations,” J. Math. Phys. 34, 2572–2617 (1993).
    [CrossRef]
  8. R. J. Sasiela, J. D. Shelton, “Transverse spectral filtering and Mellin transform techniques applied to the effect of outer scale on tilt anisoplanatism,” J. Opt. Soc. Am. A 10, 646–660 (1993).
    [CrossRef]
  9. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, New York, 1994).
    [CrossRef]
  10. V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).
  11. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.
  12. F. P. Carlson, “Application of optical scintillation measurements to turbulence diagnostics,” J. Opt. Soc. Am. A 59, 1343–1347 (1969).
    [CrossRef]
  13. A. Ishimaru, “The beam wave case in remote sensing,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer, New York, 1978), Chap. 5.
    [CrossRef]
  14. W. B. Miller, J. C. Ricklin, L. C. Andrews, “Log-amplitude and wave structure function: a new perspective for Gaussian beams,” J. Opt. Soc. Am. A 10, 661–672 (1993).
    [CrossRef]
  15. M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillation in a turbulent medium,” Waves Random Media 4, 243–273 (1994).
    [CrossRef]
  16. A. Ishimaru, “Fluctuations of a focused beam wave for atmospheric turbulence probing,” Proc. IEEE 57, 407–414 (1969).
    [CrossRef]
  17. A. Ishimaru, “Temporal frequency spectra of multifrequency waves in Turbulent atmosphere,” IEEE Trans. Antennas Propag. AP-20, 10–19 (1972).
    [CrossRef]
  18. C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortions using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
    [CrossRef]
  19. D. V. Murphy, C. A. Primmerman, B. G. Zollars, H. T. Barclay, “Experimental demonstration of atmospheric compensation using multiple synthetic beacons,” Opt. Lett. 16, 1797–1799 (1991).
    [CrossRef] [PubMed]
  20. D. P. Greenwood, C. A. Primmerman, “Adaptive optics research at Lincoln Laboratory,” Lincoln Lab. J. 5, 3–24 (1992).
  21. D. V. Murphy, “Atmospheric-turbulence compensation experiments using cooperative beacons,” Lincoln Lab. J. 5, 25–44 (1992).
  22. Yu. N. Barabanenkov, Yu. A. Kravtsov, V. D. Ozrin, A. I. Saichev, “Enhanced backscattering in optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1991), Chap. 2.
    [CrossRef]

1994

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillation in a turbulent medium,” Waves Random Media 4, 243–273 (1994).
[CrossRef]

1993

1992

D. P. Greenwood, C. A. Primmerman, “Adaptive optics research at Lincoln Laboratory,” Lincoln Lab. J. 5, 3–24 (1992).

D. V. Murphy, “Atmospheric-turbulence compensation experiments using cooperative beacons,” Lincoln Lab. J. 5, 25–44 (1992).

T. W. Lawrence, D. M. Goodman, E. M. Johansson, J. P. Fitch, “Speckle imaging of satellites at the U.S. Air Force Maui Optical Station,” Appl. Opt. 31, 6307–6321 (1992).
[CrossRef] [PubMed]

1991

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortions using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

D. V. Murphy, C. A. Primmerman, B. G. Zollars, H. T. Barclay, “Experimental demonstration of atmospheric compensation using multiple synthetic beacons,” Opt. Lett. 16, 1797–1799 (1991).
[CrossRef] [PubMed]

1977

1972

A. Ishimaru, “Temporal frequency spectra of multifrequency waves in Turbulent atmosphere,” IEEE Trans. Antennas Propag. AP-20, 10–19 (1972).
[CrossRef]

1970

A. Labreyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

1969

F. P. Carlson, “Application of optical scintillation measurements to turbulence diagnostics,” J. Opt. Soc. Am. A 59, 1343–1347 (1969).
[CrossRef]

A. Ishimaru, “Fluctuations of a focused beam wave for atmospheric turbulence probing,” Proc. IEEE 57, 407–414 (1969).
[CrossRef]

Andrews, L. C.

Banakh, V. A.

V. A. Banakh, V. L. Mironov, Lidar in a Turbulent Atmosphere (Artech House, Boston, 1987).

Barabanenkov, Yu. N.

Yu. N. Barabanenkov, Yu. A. Kravtsov, V. D. Ozrin, A. I. Saichev, “Enhanced backscattering in optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1991), Chap. 2.
[CrossRef]

Barclay, H. T.

D. V. Murphy, C. A. Primmerman, B. G. Zollars, H. T. Barclay, “Experimental demonstration of atmospheric compensation using multiple synthetic beacons,” Opt. Lett. 16, 1797–1799 (1991).
[CrossRef] [PubMed]

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortions using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

Bufton, J.

Carlson, F. P.

F. P. Carlson, “Application of optical scintillation measurements to turbulence diagnostics,” J. Opt. Soc. Am. A 59, 1343–1347 (1969).
[CrossRef]

Charnotskii, M. I.

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillation in a turbulent medium,” Waves Random Media 4, 243–273 (1994).
[CrossRef]

Fitch, J. P.

Goodman, D. M.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Greenwood, D. P.

D. P. Greenwood, C. A. Primmerman, “Adaptive optics research at Lincoln Laboratory,” Lincoln Lab. J. 5, 3–24 (1992).

Ishimaru, A.

A. Ishimaru, “Temporal frequency spectra of multifrequency waves in Turbulent atmosphere,” IEEE Trans. Antennas Propag. AP-20, 10–19 (1972).
[CrossRef]

A. Ishimaru, “Fluctuations of a focused beam wave for atmospheric turbulence probing,” Proc. IEEE 57, 407–414 (1969).
[CrossRef]

A. Ishimaru, “The beam wave case in remote sensing,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer, New York, 1978), Chap. 5.
[CrossRef]

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

Iyer, R.

Johansson, E. M.

Kravtsov, Yu. A.

Yu. N. Barabanenkov, Yu. A. Kravtsov, V. D. Ozrin, A. I. Saichev, “Enhanced backscattering in optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1991), Chap. 2.
[CrossRef]

Labreyrie, A.

A. Labreyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Lawrence, T. W.

Miller, W. B.

Mironov, V. L.

V. A. Banakh, V. L. Mironov, Lidar in a Turbulent Atmosphere (Artech House, Boston, 1987).

Murphy, D. V.

D. V. Murphy, “Atmospheric-turbulence compensation experiments using cooperative beacons,” Lincoln Lab. J. 5, 25–44 (1992).

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortions using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

D. V. Murphy, C. A. Primmerman, B. G. Zollars, H. T. Barclay, “Experimental demonstration of atmospheric compensation using multiple synthetic beacons,” Opt. Lett. 16, 1797–1799 (1991).
[CrossRef] [PubMed]

Ozrin, V. D.

Yu. N. Barabanenkov, Yu. A. Kravtsov, V. D. Ozrin, A. I. Saichev, “Enhanced backscattering in optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1991), Chap. 2.
[CrossRef]

Page, D. A.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortions using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

Primmerman, C. A.

D. P. Greenwood, C. A. Primmerman, “Adaptive optics research at Lincoln Laboratory,” Lincoln Lab. J. 5, 3–24 (1992).

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortions using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

D. V. Murphy, C. A. Primmerman, B. G. Zollars, H. T. Barclay, “Experimental demonstration of atmospheric compensation using multiple synthetic beacons,” Opt. Lett. 16, 1797–1799 (1991).
[CrossRef] [PubMed]

Ricklin, J. C.

Saichev, A. I.

Yu. N. Barabanenkov, Yu. A. Kravtsov, V. D. Ozrin, A. I. Saichev, “Enhanced backscattering in optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1991), Chap. 2.
[CrossRef]

Sasiela, R. J.

R. J. Sasiela, J. D. Shelton, “Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations,” J. Math. Phys. 34, 2572–2617 (1993).
[CrossRef]

R. J. Sasiela, J. D. Shelton, “Transverse spectral filtering and Mellin transform techniques applied to the effect of outer scale on tilt anisoplanatism,” J. Opt. Soc. Am. A 10, 646–660 (1993).
[CrossRef]

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, New York, 1994).
[CrossRef]

Shelton, J. D.

R. J. Sasiela, J. D. Shelton, “Transverse spectral filtering and Mellin transform techniques applied to the effect of outer scale on tilt anisoplanatism,” J. Opt. Soc. Am. A 10, 646–660 (1993).
[CrossRef]

R. J. Sasiela, J. D. Shelton, “Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations,” J. Math. Phys. 34, 2572–2617 (1993).
[CrossRef]

Tatarski, V. I.

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

Taylor, L.

Zollars, B. G.

D. V. Murphy, C. A. Primmerman, B. G. Zollars, H. T. Barclay, “Experimental demonstration of atmospheric compensation using multiple synthetic beacons,” Opt. Lett. 16, 1797–1799 (1991).
[CrossRef] [PubMed]

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortions using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

Appl. Opt.

Astron. Astrophys.

A. Labreyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

IEEE Trans. Antennas Propag.

A. Ishimaru, “Temporal frequency spectra of multifrequency waves in Turbulent atmosphere,” IEEE Trans. Antennas Propag. AP-20, 10–19 (1972).
[CrossRef]

J. Math. Phys.

R. J. Sasiela, J. D. Shelton, “Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations,” J. Math. Phys. 34, 2572–2617 (1993).
[CrossRef]

J. Opt. Soc. Am. A

Lincoln Lab. J.

D. P. Greenwood, C. A. Primmerman, “Adaptive optics research at Lincoln Laboratory,” Lincoln Lab. J. 5, 3–24 (1992).

D. V. Murphy, “Atmospheric-turbulence compensation experiments using cooperative beacons,” Lincoln Lab. J. 5, 25–44 (1992).

Nature (London)

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortions using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

Opt. Lett.

Proc. IEEE

A. Ishimaru, “Fluctuations of a focused beam wave for atmospheric turbulence probing,” Proc. IEEE 57, 407–414 (1969).
[CrossRef]

Waves Random Media

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillation in a turbulent medium,” Waves Random Media 4, 243–273 (1994).
[CrossRef]

Other

V. A. Banakh, V. L. Mironov, Lidar in a Turbulent Atmosphere (Artech House, Boston, 1987).

Yu. N. Barabanenkov, Yu. A. Kravtsov, V. D. Ozrin, A. I. Saichev, “Enhanced backscattering in optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1991), Chap. 2.
[CrossRef]

A. Ishimaru, “The beam wave case in remote sensing,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer, New York, 1978), Chap. 5.
[CrossRef]

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, New York, 1994).
[CrossRef]

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

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

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Figures (12)

Fig. 1
Fig. 1

Turbulence strength weighting function as a function of transmit diameter for an initially collimated beam with the observations on the beam axis, a range of 500 km, and λ = 0.5 μm.

Fig. 2
Fig. 2

Turbulence strength weighting function as a function of offset from beam axis for an initially collimated 3-cm beam at a range of 500 km with λ = 0.5 μm (FN = 0.003).

Fig. 3
Fig. 3

Turbulence strength weighting function as a function of beam focus for a 3-cm beam with observations on the beam axis at a range of 500 km with λ = 0.5 μm (FN = 0.003).

Fig. 4
Fig. 4

Log-amplitude variance as a function of beam diameter and range for a collimated beam, observations on the beam axis, a zenith angle of 0, and a HV turbulence model.

Fig. 5
Fig. 5

Log-amplitude variance as a function of beam diameter and displacement of the receiver from the beam axis for a collimated beam, a range of 500 km, a zenith angle of 0, and a HV turbulence model. Radial displacement is normalized by the beam’s diffraction-limited half-width, and log-amplitude variance is normalized by the variance of a point source.

Fig. 6
Fig. 6

Log-amplitude variance as a function of focus for various transmit beam diameters with the receiver on the beam axis, a range of 500 km, a zenith angle of 0, and a HV turbulence model.

Fig. 7
Fig. 7

Scintillation PSD dependence upon transmit beam diameter. The assumed parameters are λ = 0.5 μm, L = 500 km, R0 = ∞, |r| = 0, a zenith angle of 0, v(z) as described in the text, and a HV turbulence strength model.

Fig. 8
Fig. 8

Scintillation PSD dependence upon radial displacement of the measurement point. The assumed parameters are λ = 0.5 μm, L = 500 km, R0 = ∞, D0 = 3 cm, a zenith angle of 0, v(z) as described in the text, and a HV turbulence strength model.

Fig. 9
Fig. 9

Scintillation PSD dependence upon focus. The assumed parameters are λ = 0.5 μm, L = 500 km, R0 = ∞, D0 = 6 cm, r = 0 (observation is on the beam axis), v(z) as described in the text, and a HV turbulence strength model. The observation is assumed to occur at a zenith angle of 0.

Fig. 10
Fig. 10

Timeline of range and zenith angle from June 3, 1990, satellite pass.

Fig. 11
Fig. 11

Timeline of predicted and observed log-intensity variance from a satellite pass on June 3, 1990.

Fig. 12
Fig. 12

Predicted and observed log-intensity PSD from a satellite pass on June 3, 1990.

Tables (2)

Tables Icon

Table 1 Range Dependence of Scintillation Altitude Weighting Function Assuming an Initially Collimated Beam

Tables Icon

Table 2 SWAT Illuminator/Receiver and LACE Retroreflector Characteristics

Equations (28)

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u 0 ( ρ ) = exp ( - 4 ρ 2 D 0 2 - i k 0 ρ 2 2 R 0 ) ,
σ χ 2 ( r ) = 0.2073 π k 0 2 ( sec ζ ) 0 ( cos ζ ) L d z C n 2 ( z ) × 0 d κ κ - 8 / 3 exp ( - κ / κ i 2 ) × [ J 0 ( 2 i γ i κ r ) exp { γ i κ 2 [ L - ( sec ζ ) z ] k 0 } - Re ( exp { i γ κ 2 [ L - ( sec ζ ) z ] k 0 } ) ] ,
γ r = F N 2 ( 1 - L / R 0 ) [ 1 - ( sec ζ ) z / R 0 ] + ( sec ζ ) z / L F N 2 ( 1 - L / R 0 ) 2 + 1 ,
γ i = F N { ( 1 - L / R 0 ) ( sec ζ ) z / L - [ 1 - ( sec ζ ) z / R 0 ] } F N 2 ( 1 - L / R 0 ) 2 + 1 .
γ r = F N 2 [ ( 1 - L / R 0 ) 2 + ( sec ζ ) z / R 0 ( 1 - L / R 0 ) ] + [ 1 - ( sec ζ ) z / L ] F N 2 ( 1 - L / R 0 ) 2 + 1 ,
γ i = - F N ( sec ζ ) z / L F N 2 ( 1 - L / R 0 ) 2 + 1 .
σ χ 2 ( r ) = 2.175 k 0 2 0 ( cos ζ ) L d z C n 2 ( z ) × [ Re ( B 5 / 3 ) - A 5 / 3 F 1 1 ( - 5 6 ; 1 ; γ i 2 r 2 A 2 ) ] ,
S ( ω ) = 1.303 k 0 2 ω 8 / 3 0 ( cos ζ ) L d z C n 2 ( z ) v 5 / 3 ( z ) ( Q 1 - Q 2 ) ,
Q 1 = 0 d c c U ( c - 1 ) c 2 - 1 c - 5 / 3 exp ( - A c 2 ) J 0 ( C c ) ,
Q 2 = Re [ 0 d c c U ( c - 1 ) c 2 - 1 c - 5 / 3 exp ( - B c 2 ) ] ,
A = { - γ i [ L - ( sec ζ ) z ] k 0 + 1 κ i 2 } [ ω v ( z ) ] 2 ,
B = { i γ [ L - ( sec ζ ) z ] k 0 + 1 κ i 2 } [ ω v ( z ) ] 2 ,
C = i 2 γ i r ω v ( z ) .
Q 1 = π 2 m = 0 ( - 1 ) m m ! Γ ( m + 1 ) ( C 2 4 A ) m 1 2 π i × δ - i δ + i d s Γ [ s + 4 3             s + m s + 11 6 ] A - s ,
Γ [ a b c d ] = Γ ( a ) Γ ( b ) Γ ( c ) Γ ( d ) .
Q 1 = π 2 m = 0 n = 0 ( - 1 ) m ( - 1 ) n m ! n ! ( C 2 4 A ) m × ( Γ [ m - n - 4 3 m + 1             - n - 1 2 ] A n + 4 / 3 + Γ [ - m - n + 4 3 m + 1             - m - n + 11 6 ] A m + n ) .
Q 1 = π 3 m = 0 1 m ! Γ ( m + 1 ) ( C 2 4 A ) m { Γ [ 1 2 - m + 7 3 ] A 4 / 3 × F 1 1 ( 1 2 ; - m + 7 3 ; - A ) + ( - 1 ) m 2 Γ [ m - 5 6 m - 1 3 ] A m × F 1 1 ( - m - 5 6 ; m - 1 3 ; - A ) } .
Q 2 = π 2 Re ( 1 2 π i ɛ - i ɛ - i d s Γ [ s + 4 3             s s + 11 6 ] B - s ) .
Q 2 = π 2 Re { n = 0 ( - 1 ) n n ! ( Γ [ - m + 4 3 - m + 11 6 ] + B 4 / 3 Γ [ - m - 4 3 - m + 1 2 ] ) B n } = π 3 Re { 1 2 Γ [ - 5 6 - 1 3 ] F 1 1 ( - 5 6 ; - 1 3 ; - B ) + Γ [ 1 2 7 3 ] B 4 / 3 F 1 1 ( 1 2 ; - 7 3 ; - B ) } .
S ( ω ) = 0.668 k 0 2 ω 8 / 3 0 ( cos ζ ) L d z C n 2 ( z ) v 5 / 3 ( z ) × ( m = 0 1 ( m ! ) 2 ( C 2 4 A ) m { Γ [ 1 2 - m + 7 3 ] A 4 / 3 × F 1 1 ( 1 2 ; - m + 7 3 ; - A ) + ( - 1 ) m 2 Γ [ m - 5 6 m - 1 3 ] A m × F 1 1 ( m - 5 6 ; m - 1 3 ; - A ) } - Re { 1 2 Γ [ - 5 6 - 1 3 ] F 1 1 ( - 5 6 ; - 1 3 ; - B ) + Γ [ 1 2 7 3 ] B 4 / 3 F 1 1 ( 1 2 ; 7 3 ; - B ) } ) .
S ( ω ) ~ 1.1548 k 0 2 ω 8 / 3 0 ( cos ζ ) L d z C n 2 ( z ) v 5 / 3 ( z ) × { J 0 ( C ) exp ( - A ) / A - Re [ exp ( - B ) / B ] }
S ( ω ) ~ 1.1548 k 0 5 / 2 ω 11 / 3 0 ( cos ζ ) L d z C n 2 ( z ) v 8 / 3 ( z ) × [ I 0 [ 2 γ i r ω v ( z ) ] exp { γ i ( L - z ) - k 0 / κ i 2 k 0 [ ω v ( z ) ] 2 } [ - γ i ( L - z ) + k 0 / κ i 2 ] 1 / 2 - Re ( exp { - i γ ( L - z ) - k 0 / κ i 2 k 0 [ ω v ( z ) ] 2 } [ i γ ( L - z ) + k 0 / κ i 2 ] 1 / 2 ) ] ,
σ χ , A 2 = 1.303 k 0 2 0 L d z C n 2 ( z ) 0 d κ κ - 8 / 3 × sin 2 [ z ( L - z ) 2 k 0 L κ 2 ] [ 2 J 1 ( κ D S z / 2 L ) κ D S z / 2 L ] 2 ,
σ χ , A 2 = - 0.3656 k 0 7 / 6 0 L d z C n 2 ( z ) z 5 / 6 ( 1 - z L ) 5 / 6 × { 0.5324 [ k 0 z D S 2 2 L ( L - z ) ] 5 / 6 + m = 0 ( - 1 ) n n ! [ k 0 z D S 2 2 L ( L - z ) ] n × Γ [ n 2 - 5 11             n + 3 2 - n 2 + 11 12             n + 2             n + 3 ] 2 } .
σ χ , A 2 0.5631 k 0 7 / 6 0 L d z C n 2 ( z ) z 5 / 6 ( 1 - z L ) 5 / 6 × { 1.0 - 0.3457 [ k 0 z D S 2 2 L ( L - z ) ] 5 / 6 + 0.3888 [ k 0 z D S 2 2 L ( L - z ) ] + 0.0009 [ k 0 z D S 2 2 L ( L - z ) ] 2 } ,
σ χ , A 2 = 1.303 k 0 2 0 L d z C n 2 ( z ) 0 L d κ κ - 8 / 3 × sin 2 ( z ( L - z ) 2 k 0 L κ 2 ) { 2 J 1 [ κ D R ( L - z ) / 2 L ] κ D R ( L - z ) / 2 L } 2 .
σ χ , A 2 σ χ 2 ( D C D R ) 7 / 3 ,
D C = 0.957 ( μ 2 μ 5 / 6 ) 3 / 7 λ

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