Abstract

Measurements of the level of turbulence Cn2 have been successfully performed with the optical scintillometer. The success of this instrument is based on the observed fact that the variance of aperture averaged scintillation is described by weak scattering theory even for conditions in which strong scintillation is observed for point detectors. However, for sufficiently long propagation paths, the aperture averaged variance is affected by strong scattering. The effects of strong scattering are calculated theoretically and compared to experimental results. The physics of this regime are discussed and the important parameters investigated. The new range of validity of the optical scintillometer is discussed.

© 1990 Optical Society of America

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References

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  1. G. R. Ochs, Ting-i Wang, “Finite Aperture Optical Scintillometer for Profiling Wind and Cn2,” Appl. Opt. 17, 3774–3778 (1978).
    [CrossRef] [PubMed]
  2. G. R. Ochs, W. D. Cartwright, “An Optical Device for Path-Averaged Measurements of Cn2,” Proc. Soc. Photo-Instrum. Eng. 277, 2–5 (1981).
  3. Ting-i Wang, G. R. Ochs, S. F. Clifford, “A Saturation-Resistant Optical Scintillometer to Measure CN2,” J. Opt. Soc. Am. 68, 334–338 (1978).
    [CrossRef]
  4. G. R. Ochs, S. F. Clifford, Ting-i Wang, “Laser Wind Sensing: the Effects of Saturation of Scintillation,” Appl. Opt. 15, 403–408 (1976).
    [CrossRef] [PubMed]
  5. G. R. Ochs, R. J. Hill, “Optical-Scintillation Method of Measuring Turbulence Inner Scale,” Appl. Opt. 24, 2430–2432 (1985).
    [CrossRef] [PubMed]
  6. R. J. Hill, G. R. Ochs, “Fine Calibration of Large-Aperture Optical Scintillometers and an Optical Estimate of the Inner Scale of Turbulence,” Appl. Opt. 17, 3608–3612 (1978).
    [CrossRef] [PubMed]
  7. R. J. Hill, “Saturation Resistance and Inner-Scale Resistance of a Large-Aperture Scintillometer: a Case Study,” Appl. Opt. 20, 3822–3824 (1981).
    [CrossRef] [PubMed]
  8. S. F. Clifford, G. R. Ochs, R. S. Lawrence, “Saturation of Optical Scintillation by Strong Turbulence,” J. Opt. Soc. Am. 64, 148–154 (1974).
    [CrossRef]
  9. R. G. Frehlich, S. M. Wandzura, R. J. Hill, “Log–Amplitude Covariance for Waves Propagating through very Strong Turbulence,” J. Opt. Soc. Am. A 4, 2158–2161 (1987).
    [CrossRef]
  10. J. L. Codona, R. G. Frehlich, “Scintillation from Extended Incoherent Sources,” Radio Sci. 22, 469–480 (1987).
    [CrossRef]
  11. J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Moment Equation and Path Integral Techniques for Wave Propagation in Random Media,” J. Math. Phys. 27, 171–177 (1986).
    [CrossRef]
  12. V. U. Zavorotnyi, “Strong Fluctuations of Electromagnetic Waves in a Random Medium with Finite Longitudinal Correlation of the Inhomogeneities,” Zh. Eksp. Teor. Fiz. 75, 56–65 (1978) [Sov. Physics JETP 48, 27–31 (1978)].
  13. J. H. Churnside, R. J. Hill, “Probability Density of Irradiance for Strong Path-Integrated Refractive Turbulence,” J. Opt. Soc. Am. A 4, 727–733 (1987).
    [CrossRef]
  14. R. J. Hill, J. H. Churnside, D. H. Sliney, “Measured Statistics of Laser Beam Scintillation in Strong Refractive Turbulence Relevant to Eye Safety,” Health Phys. 53, 639–647 (1987).
    [CrossRef] [PubMed]
  15. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Keter, Jerusalem, 1971), p. 65–66.
  16. R. J. Hill, “Models of the Scalar Spectrum for Turbulent Advection,” J. Fluid Mech. 88, 541–562 (1978).
    [CrossRef]
  17. G. R. Ochs, R. J. Hill, A Study of Factors Influencing the Calibration of Optical Cn2 Meters, NOAA Tech. Memo., ERL WPL-106 (1982) (available from the authors or National Technical Information Service).

1987 (4)

J. L. Codona, R. G. Frehlich, “Scintillation from Extended Incoherent Sources,” Radio Sci. 22, 469–480 (1987).
[CrossRef]

R. J. Hill, J. H. Churnside, D. H. Sliney, “Measured Statistics of Laser Beam Scintillation in Strong Refractive Turbulence Relevant to Eye Safety,” Health Phys. 53, 639–647 (1987).
[CrossRef] [PubMed]

J. H. Churnside, R. J. Hill, “Probability Density of Irradiance for Strong Path-Integrated Refractive Turbulence,” J. Opt. Soc. Am. A 4, 727–733 (1987).
[CrossRef]

R. G. Frehlich, S. M. Wandzura, R. J. Hill, “Log–Amplitude Covariance for Waves Propagating through very Strong Turbulence,” J. Opt. Soc. Am. A 4, 2158–2161 (1987).
[CrossRef]

1986 (1)

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Moment Equation and Path Integral Techniques for Wave Propagation in Random Media,” J. Math. Phys. 27, 171–177 (1986).
[CrossRef]

1985 (1)

1982 (1)

G. R. Ochs, R. J. Hill, A Study of Factors Influencing the Calibration of Optical Cn2 Meters, NOAA Tech. Memo., ERL WPL-106 (1982) (available from the authors or National Technical Information Service).

1981 (2)

G. R. Ochs, W. D. Cartwright, “An Optical Device for Path-Averaged Measurements of Cn2,” Proc. Soc. Photo-Instrum. Eng. 277, 2–5 (1981).

R. J. Hill, “Saturation Resistance and Inner-Scale Resistance of a Large-Aperture Scintillometer: a Case Study,” Appl. Opt. 20, 3822–3824 (1981).
[CrossRef] [PubMed]

1978 (5)

1976 (1)

1974 (1)

Cartwright, W. D.

G. R. Ochs, W. D. Cartwright, “An Optical Device for Path-Averaged Measurements of Cn2,” Proc. Soc. Photo-Instrum. Eng. 277, 2–5 (1981).

Churnside, J. H.

J. H. Churnside, R. J. Hill, “Probability Density of Irradiance for Strong Path-Integrated Refractive Turbulence,” J. Opt. Soc. Am. A 4, 727–733 (1987).
[CrossRef]

R. J. Hill, J. H. Churnside, D. H. Sliney, “Measured Statistics of Laser Beam Scintillation in Strong Refractive Turbulence Relevant to Eye Safety,” Health Phys. 53, 639–647 (1987).
[CrossRef] [PubMed]

Clifford, S. F.

Codona, J. L.

J. L. Codona, R. G. Frehlich, “Scintillation from Extended Incoherent Sources,” Radio Sci. 22, 469–480 (1987).
[CrossRef]

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Moment Equation and Path Integral Techniques for Wave Propagation in Random Media,” J. Math. Phys. 27, 171–177 (1986).
[CrossRef]

Creamer, D. B.

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Moment Equation and Path Integral Techniques for Wave Propagation in Random Media,” J. Math. Phys. 27, 171–177 (1986).
[CrossRef]

Flatté, S. M.

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Moment Equation and Path Integral Techniques for Wave Propagation in Random Media,” J. Math. Phys. 27, 171–177 (1986).
[CrossRef]

Frehlich, R. G.

R. G. Frehlich, S. M. Wandzura, R. J. Hill, “Log–Amplitude Covariance for Waves Propagating through very Strong Turbulence,” J. Opt. Soc. Am. A 4, 2158–2161 (1987).
[CrossRef]

J. L. Codona, R. G. Frehlich, “Scintillation from Extended Incoherent Sources,” Radio Sci. 22, 469–480 (1987).
[CrossRef]

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Moment Equation and Path Integral Techniques for Wave Propagation in Random Media,” J. Math. Phys. 27, 171–177 (1986).
[CrossRef]

Henyey, F. S.

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Moment Equation and Path Integral Techniques for Wave Propagation in Random Media,” J. Math. Phys. 27, 171–177 (1986).
[CrossRef]

Hill, R. J.

Lawrence, R. S.

Ochs, G. R.

Sliney, D. H.

R. J. Hill, J. H. Churnside, D. H. Sliney, “Measured Statistics of Laser Beam Scintillation in Strong Refractive Turbulence Relevant to Eye Safety,” Health Phys. 53, 639–647 (1987).
[CrossRef] [PubMed]

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Keter, Jerusalem, 1971), p. 65–66.

Wandzura, S. M.

Wang, Ting-i

Zavorotnyi, V. U.

V. U. Zavorotnyi, “Strong Fluctuations of Electromagnetic Waves in a Random Medium with Finite Longitudinal Correlation of the Inhomogeneities,” Zh. Eksp. Teor. Fiz. 75, 56–65 (1978) [Sov. Physics JETP 48, 27–31 (1978)].

A Study of Factors Influencing the Calibration of Optical Cn2 Meters (1)

G. R. Ochs, R. J. Hill, A Study of Factors Influencing the Calibration of Optical Cn2 Meters, NOAA Tech. Memo., ERL WPL-106 (1982) (available from the authors or National Technical Information Service).

Appl. Opt. (5)

Health Phys. (1)

R. J. Hill, J. H. Churnside, D. H. Sliney, “Measured Statistics of Laser Beam Scintillation in Strong Refractive Turbulence Relevant to Eye Safety,” Health Phys. 53, 639–647 (1987).
[CrossRef] [PubMed]

J. Fluid Mech. (1)

R. J. Hill, “Models of the Scalar Spectrum for Turbulent Advection,” J. Fluid Mech. 88, 541–562 (1978).
[CrossRef]

J. Math. Phys. (1)

J. L. Codona, D. B. Creamer, S. M. Flatté, R. G. Frehlich, F. S. Henyey, “Moment Equation and Path Integral Techniques for Wave Propagation in Random Media,” J. Math. Phys. 27, 171–177 (1986).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Proc. Soc. Photo-Instrum. Eng. (1)

G. R. Ochs, W. D. Cartwright, “An Optical Device for Path-Averaged Measurements of Cn2,” Proc. Soc. Photo-Instrum. Eng. 277, 2–5 (1981).

Radio Sci. (1)

J. L. Codona, R. G. Frehlich, “Scintillation from Extended Incoherent Sources,” Radio Sci. 22, 469–480 (1987).
[CrossRef]

Zh. Eksp. Teor. Fiz. (1)

V. U. Zavorotnyi, “Strong Fluctuations of Electromagnetic Waves in a Random Medium with Finite Longitudinal Correlation of the Inhomogeneities,” Zh. Eksp. Teor. Fiz. 75, 56–65 (1978) [Sov. Physics JETP 48, 27–31 (1978)].

Other (1)

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Keter, Jerusalem, 1971), p. 65–66.

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Figures (7)

Fig. 1
Fig. 1

Normalized scaling functions (a) g(0.15,0.1,δ) and (b) g(0.15,0.2,δ) vs δ for the Tatarskii spectrum (solid line) and Hill spectrum (dashed line).

Fig. 2
Fig. 2

Normalized scaling functions g(0.15,0.1,δ) (solid line) and g(0.4,0.1,δ) (dashed line) vs δ for the Tatarskii spectrum (a) and Hill spectrum (b).

Fig. 3
Fig. 3

Ratio of the scaling function G calculated with the Gaussian limit of the function H to the exact calculation of G vs δ for the β = 0.15 and γ = 0.1 (a) and γ = 0.05 (b). The Tatarskii spectrum (solid line) and the Hill spectrum (dashed line) were used in the calculations.

Fig. 4
Fig. 4

Cumulative spectral weighting function Y(κ) vs κ for various values of the strength of scattering δ and the typical case of β = 0.15 and γ = 0.1 for the Tatarskii spectrum.

Fig. 5
Fig. 5

Normalized path weighting functions W(x) vs the normalized propagation distance x for various values of the strength of scattering δ and the typical case of β = 0.15 and γ = 0.1 and the Tatarskii spectrum.

Fig. 6
Fig. 6

Level of turbulence C n 2 vs the scintillometer variance σ2 for a propagation distance of L = 1000 m and the Hill spectrum for inner scale of 5 mm (solid line) and 10 mm (dashed line) and for (a) R = 7.5 cm, (b) R = 10 cm, and (c) R = 15 cm.

Fig. 7
Fig. 7

Estimates of C n 2 from 5-cm radii scintillometer vs estimates from a 10-cm radii scintillometer (○) and theoretical predictions for λ0 = (——), λ0 = 5 mm and Tatarskii spectrum(. . . .), λ0 = 10 mm and Tatarskii spectrum (– – – –), λ0 = 5 mm and Hill spectrum, (— — —), λ0 = 10 mm and Hill spectrum (– · – ·), and ideal operation (—— —— ——).

Equations (37)

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F = - I ( y ) P ( y ) d y ,
σ 2 = [ F - F ] 2 F 2 = - - C ( y 1 - y 2 ) P ( y 1 ) P ( y 2 ) d y 1 d y 2 [ - P ( y ) d y ] 2 - 1 ,
C ( x ) = I ( y ) I ( y + x ) I 2
σ 2 = 8 π k 2 0 L - V [ q ( 1 - z 1 / L ) ] 2 T ( q z 1 / L ) 2 Φ n ( q , q z = 0 , z ) × sin 2 [ q 2 z 1 2 k ( 1 - z 1 / L ) ] H ( q , z 1 ) d q d z 1 ,
H ( q , z 1 ) = exp { - 0 L d [ q k h ( z , z 1 ) , z ] d z } ,
V ( q ) = - S ( x ) exp ( - i q · x ) d x - S ( x ) d x ,
T ( q ) = - P ( y ) exp ( - i q · y ) d y - P ( y ) d y ,
h ( z , z 1 ) = z ( z 1 L - 1 )             z < z 1 = z 1 ( z L - 1 ) ,             z > z 1 .
d ( s , z ) = 4 π k 2 - [ 1 - cos ( s · q ) ] Φ n ( q , q z = 0 , z ) d q
σ ln I 2 = ln ( 1 + σ 2 ) .
Φ n ( q ) = A ( α ) C n 2 q - α - 2 f ( q λ 0 ) ,
A ( α ) = Γ ( α + 1 ) 4 π 2 sin [ ( α - 1 ) π / 2 ]
Φ n ( q ) = A ( α ) C n 2 q - α - 2 exp ( - l 0 2 q 2 / 4 ) ,
E ( α ) = [ 2 Γ ( 3 / 2 ) ( α - 1 ) Γ ( 1 + α / 2 ) ] 1 / ( 3 - α )
d ( s , z ) = 8 π 2 A ( α ) C n 2 ( z ) k 2 λ 0 a g ( x / λ 0 ) ,
g ( x ) = 0 q - α - 1 f ( q ) [ 1 - J 0 ( q x ) ] d q ,
g ( x ) = x 2 4 0 q 1 - α f ( q ) d q C ( α ) x 2             x λ 0 ,
g ( x ) = Γ ( 1 - α / 2 ) 2 α α Γ ( 1 + α / 2 ) x α             x λ 0 .
D ( s ) = 0 L d [ s z / L , z ] d z .
D ( s ) = 8 π 2 A ( α ) C n 2 k 2 λ 0 2 L 0 1 g ( s z / λ 0 ) d z .
D ( s ) = 8 3 π 2 A ( α ) C ( α ) C n 2 k 2 λ 0 α - 2 L s 2             s λ 0 ,
D ( s ) = B ( α ) C n 2 k 2 L s α             s λ 0 ,
β ( α ) = 8 π 2 A ( α ) Γ ( 1 - α / 2 ) 2 α α ( α + 1 ) Γ ( 1 + α / 2 ) .
ρ s 2 = 3 8 π 2 A ( α ) C ( α ) λ 0 2 - α C n 2 k 2 L = J ( α ) λ 0 2 - α C n 2 k 2 L             ρ 0 L 0 ,
ρ z = [ B ( α ) C n 2 k 2 L ] - 1 / α             ρ 0 L 0 .
R S = R f 2 ρ 0 ,
H ( q , x ) = exp { - L 0 1 d [ q L k x ( 1 - x ) z ] d z } ,
σ 2 = L 3 R α - 4 C n 2 G ( β , γ , δ ) ,
G ( β , γ , δ ) = 256 π 2 A ( α ) β - 4 0 1 0 { J 1 [ κ ( 1 - x ) ] J 1 ( κ x ) κ 2 x ( 1 - x ) } 2 × κ - α - 1 sin 2 [ κ 2 β 2 2 x ( 1 - x ) ] f ( γ κ ) H ( κ , x , β , γ , δ ) d κ d x ,
H ( κ , x , β , γ , δ ) = exp { - 3 δ 2 γ 2 C ( α ) β 4 0 1 g [ β 2 κ γ x ( 1 - x ) z ] d z } .
H ( κ , x , β , γ , δ ) = exp { - [ δ κ x ( 1 - x ) ] 2 }             ρ 0 λ 0 ,
H ( κ , x , β , γ , δ ) = exp { - [ δ 2 κ x ( 1 - x ) ] α }             ρ 0 λ 0 ,
G ( β , γ , δ ) = 0 U ( κ ) d κ .
Y ( κ ) = 0 κ U ( x ) d x 0 U ( x ) d x .
G ( β , γ , δ ) = 0 1 Z ( x ) d x
W ( x ) = Z ( x ) / G ( β , γ , δ ) .
C ^ n 2 = 5.6373 σ ln I 2 L - 3 R 7 / 3 .

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