Abstract

An experiment has been performed to confirm the proportionality between log-amplitude variance and the 7/6 power of wavenumber predicted by Tatarski for horizontal propagation from a spherical-wave transmitter to a point detector. The validity of this proportionality was tested for two wavelengths: 0.632 and 10.6 μ. Beams from a helium-neon and a CO2 laser were simultaneously transmitted over a folded 1.2-km horizontal path and were detected with a photomultiplier and a gold-doped germanium detector. The primary scintillation statistic, log-amplitude variance, was evaluated for each wavelength with a digital computer and the ratio of variances at 0.632 and 10.6 μ was found to be in close agreement with predictions. Power spectral density, autocorrelation, and cumulative probability density were also evaluated for each wavelength. Scintillation statistics at 10.6 μ were found to be log normal, as in the visible.

© 1969 Optical Society of America

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References

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  1. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw–Hill Book Co., New York, 1961).
  2. D. L. Fried, J. Opt. Soc. Am. 57, 175 (1967).
    [Crossref]
  3. D. H. Hohn, Appl. Opt. 5, 1427 (1966).
    [Crossref]
  4. T. S. Chu, Appl. Opt. 6, 163 (1967).
    [Crossref]
  5. M. E. Gracheva and A. S. Gurvich, Radiofizika 8, 717 (1965).
  6. D. L. Fried, G. E. Mevers, and M. P. Keister, J. Opt. Soc. Am. 57, 787 (1967); J. Opt. Soc. Am. 58, 1164E (1968).
    [Crossref]
  7. X¯ and S2 are maximum-likelihood estimators for μx and σx2. The amount of bias in S2 becomes insignificant for large samples.
  8. J. S. Bendat and A. G. Piersol, Measurement and Analysis of Random Data (John Wiley & Sons, Inc., New York, 1966).
  9. Reference 8, Ch. 5.
  10. In a strict sense, the equivalent bandwidth referred to in Eqs. (8) and (9) should be the bandwidth of the log-irradiance signal and not the bandwidth of the irradiance fluctuations. However, our data-reduction system did not permit an evaluation of the spectrum of log irradiance, so the equivalent bandwidth of the irradiance signal was used. It can be shown that under weak scintillation conditions (i.e., when the irradiance fluctuations are less than the mean irradiance), logarithmic amplification does not significantly change the shape of the power spectral density.

1967 (3)

1966 (1)

1965 (1)

M. E. Gracheva and A. S. Gurvich, Radiofizika 8, 717 (1965).

Bendat, J. S.

J. S. Bendat and A. G. Piersol, Measurement and Analysis of Random Data (John Wiley & Sons, Inc., New York, 1966).

Chu, T. S.

Fried, D. L.

Gracheva, M. E.

M. E. Gracheva and A. S. Gurvich, Radiofizika 8, 717 (1965).

Gurvich, A. S.

M. E. Gracheva and A. S. Gurvich, Radiofizika 8, 717 (1965).

Hohn, D. H.

Keister, M. P.

Mevers, G. E.

Piersol, A. G.

J. S. Bendat and A. G. Piersol, Measurement and Analysis of Random Data (John Wiley & Sons, Inc., New York, 1966).

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw–Hill Book Co., New York, 1961).

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

Radiofizika (1)

M. E. Gracheva and A. S. Gurvich, Radiofizika 8, 717 (1965).

Other (5)

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw–Hill Book Co., New York, 1961).

X¯ and S2 are maximum-likelihood estimators for μx and σx2. The amount of bias in S2 becomes insignificant for large samples.

J. S. Bendat and A. G. Piersol, Measurement and Analysis of Random Data (John Wiley & Sons, Inc., New York, 1966).

Reference 8, Ch. 5.

In a strict sense, the equivalent bandwidth referred to in Eqs. (8) and (9) should be the bandwidth of the log-irradiance signal and not the bandwidth of the irradiance fluctuations. However, our data-reduction system did not permit an evaluation of the spectrum of log irradiance, so the equivalent bandwidth of the irradiance signal was used. It can be shown that under weak scintillation conditions (i.e., when the irradiance fluctuations are less than the mean irradiance), logarithmic amplification does not significantly change the shape of the power spectral density.

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Figures (5)

Fig. 1
Fig. 1

Ratio of log-amplitude variances for He–Ne laser and CO2 laser wavelengths.

Fig. 2
Fig. 2

Dual-wavelength normalized power spectral density. Resolution bandwidth = 5 Hz, each curve normalized to the same area. Log-amplitude variances for these data are σl2 · 632 = 0.200, σl210.6 = 0.00693.

Fig. 3
Fig. 3

Normalized autocorrelation function for dual-wavelength scintillation. Fitting the data with the exponential eατ gives α(CO2) = 71.4, α(He–Ne) = 312.

Fig. 4
Fig. 4

Typical analog scintillation signal. He–Ne laser beam, above; CO2 laser beam, below. Log-amplitude variances are: 0.200 (He–Ne), 0.00668 (CO2).

Fig. 5
Fig. 5

Cumulative probability of log amplitude at 10.6 μ. Data plotted on a gaussian abscissa scale.

Tables (1)

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Table I Experimental parameters.

Equations (10)

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l ( t ) = 1 2 log e [ I ( t ) / I 0 ] ,
σ l 2 k 7 / 6 z 11 / 6 C N 2 ,
X ¯ 1 / n i = 1 n x i
S 2 1 / n i = 1 n ( x i - X ¯ ) 2 .
E [ X ¯ ] = μ x
E [ S 2 ] = [ ( n - 1 ) / n ] σ x 2
E [ S 2 - E ( S 2 ) ] 2 = variance S 2 = m 4 - m 2 2 n - 2 ( m 4 - 2 m 2 2 ) n 2 + m 4 - 3 m 2 2 n 3 ,
n = 2 B e q T ,
B e q = R x ( 0 ) / [ 2 - R x ( τ ) d τ ] ,
F ( l ) = - l f ( u ) d u ,