Abstract

The influence of aperture averaging on the two-wavelength intensity covariance function was experimentally determined for visible (0.63 μm) and infrared (1.06 μm) collinear, approximately spherical beams which propagated through the earth’s turbulent atmosphere. Range varied from 1300 to 3250 m, and due to the prevailing atmospheric conditions, most measurements were made in the strong turbulence regimes. Results show that (1) the covariance function monotonically decreases as the receiver aperture size increases; (2) the correlation coefficient attains high values (≧0.7) even for a relatively small aperture size of 5 mm; (3) while the single wavelength probability distribution of the intensity is approxiamtely lognormal, the experimental two-wavelength conditional probabilities are higher than those predicted by the lognormal model.

© 1985 Optical Society of America

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References

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  1. V. I. Tatarski, “The Effect of the Turbulent Atmosphere on Wave Propagation,” Israel Program for Scientific Translations, Jerusalem (1971).
  2. R. Fante, “Electromagnetic Beam Propagation in Turbulent Media,” Proc IEEE 63, 1669 (1975).
    [CrossRef]
  3. R. Fante, “Electromagnetic Beam Propagation in Turbulent Media: An Update,” Proc IEEE 68, 1424 (1980).
    [CrossRef]
  4. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.
  5. G. E. Homstad, J. W. Strohbehn, R. H. Berger, J. M. Heneghan, “Aperture-Averaging Effects for Weak Scintillations,” J. Opt. Soc. Am. 64, 162 (1974).
    [CrossRef]
  6. W. P. Brown, “Fourth Moment of a Wave Propagating in a Random Medium,” J. Opt. Soc. Am. 62, 966 (1972).
    [CrossRef]
  7. J. R. Dunphy, J. R. Kerr, “Scintillation Measurments for Large Integrated-Path Turbulence,” J. Opt. Soc. Am. 63, 981 (1973).
    [CrossRef]
  8. D. L. Fried, G. E. Mevers, M. P. Keister, “Measurements of Laser-Beam Scintillation in the Atmosphere,” J. Opt. Soc. Am. 57, 787 (1967).
    [CrossRef]
  9. R. E. Hufnagel, “Restoration of Atmospherically Degraded Images” (National Academy of Sciences, Washington, D.C., 1966), Vol. 3, Appendix 2, p. 11.
  10. D. L. Fried, “Probability of Getting a Lucky Short-Exposure Image Through Turbulence,” J. Opt. Soc. Am. 68, 1651 (1978).
    [CrossRef]
  11. A. Ishimaru, “Temporal Frequency Spectra of Multifrequency Waves in Turbulent Atmosphere,” IEEE Trans. Antennas Propag. AP-20, 10 (1972).
    [CrossRef]
  12. I. M. Fuks, “Correlation of the Fluctuations of Frequency Spaced Signals in a Randomly Inhomogeneous Medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 17, 1665 (1974).
  13. Y. Baykal, M. A. Plonus, “Two-Source, Two-Frequency Spherical Wave Structure Functions in Atmospheric Turbulence,” J. Opt. Soc. Am. 70, 1278 (1980).
    [CrossRef]
  14. M. Tamir, E. Azoulay, S. Tsur, U. Halavee, “Aperture-Averaged Spectral Correlations of Beams in a Turbulent Atmosphere,” Appl. Opt. 23, 2359 (1984).
    [CrossRef] [PubMed]
  15. A. S. Gurvich, V. Kan, V. Pokasov, “Two-Frequency Fluctuations of Light Intensity in a Turbulent Medium,” Opt. Acta 26, 555 (1979).
    [CrossRef]
  16. R. J. Hill, S. F. Clifford, “Modified Spectrum of Atmospheric Temperature Fluctuations and Its Application to Optical Propagation,” J. Opt. Soc. Am. 68, 892 (1978).
    [CrossRef]
  17. R. L. Phillips, L. C. Andrews, “Measured Statistics of Laser-Light Scattering in Atmospheric Turbulence,” J. Opt. Soc. Am. 71, 1440 (1981).
    [CrossRef]
  18. G. Parry, “Measurement of Atmospheric Turbulence Induced Intensity Fluctuations in a Laser Beam,” Opt. Acta 28, 715 (1981).
    [CrossRef]
  19. M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, V. V. Pokasov, “Similarity Relations for Strong Fluctuations of the Intensity of Light Propagating in a Turbulent Medium,” Sov. Phys. JETP 40, 1011 (1975).
  20. R. J. Hill, “Theory of Saturation of Optical Scintillation by Strong Turbulence: Plane-Wave Variance and Covariance and Spherical-Wave Covariance,” J. Opt. Soc. Am. 72, 212 (1982).
    [CrossRef]
  21. G. Parry, P. N. Pusey, “K Distributions in Atmospheric Propagation of Laser Light,” J. Opt. Soc. Am. 69, 796 (1979).
    [CrossRef]

1984

1982

1981

R. L. Phillips, L. C. Andrews, “Measured Statistics of Laser-Light Scattering in Atmospheric Turbulence,” J. Opt. Soc. Am. 71, 1440 (1981).
[CrossRef]

G. Parry, “Measurement of Atmospheric Turbulence Induced Intensity Fluctuations in a Laser Beam,” Opt. Acta 28, 715 (1981).
[CrossRef]

1980

1979

G. Parry, P. N. Pusey, “K Distributions in Atmospheric Propagation of Laser Light,” J. Opt. Soc. Am. 69, 796 (1979).
[CrossRef]

A. S. Gurvich, V. Kan, V. Pokasov, “Two-Frequency Fluctuations of Light Intensity in a Turbulent Medium,” Opt. Acta 26, 555 (1979).
[CrossRef]

1978

1975

R. Fante, “Electromagnetic Beam Propagation in Turbulent Media,” Proc IEEE 63, 1669 (1975).
[CrossRef]

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, V. V. Pokasov, “Similarity Relations for Strong Fluctuations of the Intensity of Light Propagating in a Turbulent Medium,” Sov. Phys. JETP 40, 1011 (1975).

1974

I. M. Fuks, “Correlation of the Fluctuations of Frequency Spaced Signals in a Randomly Inhomogeneous Medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 17, 1665 (1974).

G. E. Homstad, J. W. Strohbehn, R. H. Berger, J. M. Heneghan, “Aperture-Averaging Effects for Weak Scintillations,” J. Opt. Soc. Am. 64, 162 (1974).
[CrossRef]

1973

1972

A. Ishimaru, “Temporal Frequency Spectra of Multifrequency Waves in Turbulent Atmosphere,” IEEE Trans. Antennas Propag. AP-20, 10 (1972).
[CrossRef]

W. P. Brown, “Fourth Moment of a Wave Propagating in a Random Medium,” J. Opt. Soc. Am. 62, 966 (1972).
[CrossRef]

1967

Andrews, L. C.

Azoulay, E.

Baykal, Y.

Berger, R. H.

Brown, W. P.

Clifford, S. F.

Dunphy, J. R.

Fante, R.

R. Fante, “Electromagnetic Beam Propagation in Turbulent Media: An Update,” Proc IEEE 68, 1424 (1980).
[CrossRef]

R. Fante, “Electromagnetic Beam Propagation in Turbulent Media,” Proc IEEE 63, 1669 (1975).
[CrossRef]

Fried, D. L.

Fuks, I. M.

I. M. Fuks, “Correlation of the Fluctuations of Frequency Spaced Signals in a Randomly Inhomogeneous Medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 17, 1665 (1974).

Gracheva, M. E.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, V. V. Pokasov, “Similarity Relations for Strong Fluctuations of the Intensity of Light Propagating in a Turbulent Medium,” Sov. Phys. JETP 40, 1011 (1975).

Gurvich, A. S.

A. S. Gurvich, V. Kan, V. Pokasov, “Two-Frequency Fluctuations of Light Intensity in a Turbulent Medium,” Opt. Acta 26, 555 (1979).
[CrossRef]

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, V. V. Pokasov, “Similarity Relations for Strong Fluctuations of the Intensity of Light Propagating in a Turbulent Medium,” Sov. Phys. JETP 40, 1011 (1975).

Halavee, U.

Heneghan, J. M.

Hill, R. J.

Homstad, G. E.

Hufnagel, R. E.

R. E. Hufnagel, “Restoration of Atmospherically Degraded Images” (National Academy of Sciences, Washington, D.C., 1966), Vol. 3, Appendix 2, p. 11.

Ishimaru, A.

A. Ishimaru, “Temporal Frequency Spectra of Multifrequency Waves in Turbulent Atmosphere,” IEEE Trans. Antennas Propag. AP-20, 10 (1972).
[CrossRef]

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

Kan, V.

A. S. Gurvich, V. Kan, V. Pokasov, “Two-Frequency Fluctuations of Light Intensity in a Turbulent Medium,” Opt. Acta 26, 555 (1979).
[CrossRef]

Kashkarov, S. S.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, V. V. Pokasov, “Similarity Relations for Strong Fluctuations of the Intensity of Light Propagating in a Turbulent Medium,” Sov. Phys. JETP 40, 1011 (1975).

Keister, M. P.

Kerr, J. R.

Mevers, G. E.

Parry, G.

G. Parry, “Measurement of Atmospheric Turbulence Induced Intensity Fluctuations in a Laser Beam,” Opt. Acta 28, 715 (1981).
[CrossRef]

G. Parry, P. N. Pusey, “K Distributions in Atmospheric Propagation of Laser Light,” J. Opt. Soc. Am. 69, 796 (1979).
[CrossRef]

Phillips, R. L.

Plonus, M. A.

Pokasov, V.

A. S. Gurvich, V. Kan, V. Pokasov, “Two-Frequency Fluctuations of Light Intensity in a Turbulent Medium,” Opt. Acta 26, 555 (1979).
[CrossRef]

Pokasov, V. V.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, V. V. Pokasov, “Similarity Relations for Strong Fluctuations of the Intensity of Light Propagating in a Turbulent Medium,” Sov. Phys. JETP 40, 1011 (1975).

Pusey, P. N.

Strohbehn, J. W.

Tamir, M.

Tatarski, V. I.

V. I. Tatarski, “The Effect of the Turbulent Atmosphere on Wave Propagation,” Israel Program for Scientific Translations, Jerusalem (1971).

Tsur, S.

Appl. Opt.

IEEE Trans. Antennas Propag.

A. Ishimaru, “Temporal Frequency Spectra of Multifrequency Waves in Turbulent Atmosphere,” IEEE Trans. Antennas Propag. AP-20, 10 (1972).
[CrossRef]

Izv. Vyssh. Uchebn. Zaved. Radiofiz.

I. M. Fuks, “Correlation of the Fluctuations of Frequency Spaced Signals in a Randomly Inhomogeneous Medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 17, 1665 (1974).

J. Opt. Soc. Am.

W. P. Brown, “Fourth Moment of a Wave Propagating in a Random Medium,” J. Opt. Soc. Am. 62, 966 (1972).
[CrossRef]

J. R. Dunphy, J. R. Kerr, “Scintillation Measurments for Large Integrated-Path Turbulence,” J. Opt. Soc. Am. 63, 981 (1973).
[CrossRef]

G. E. Homstad, J. W. Strohbehn, R. H. Berger, J. M. Heneghan, “Aperture-Averaging Effects for Weak Scintillations,” J. Opt. Soc. Am. 64, 162 (1974).
[CrossRef]

R. J. Hill, S. F. Clifford, “Modified Spectrum of Atmospheric Temperature Fluctuations and Its Application to Optical Propagation,” J. Opt. Soc. Am. 68, 892 (1978).
[CrossRef]

D. L. Fried, “Probability of Getting a Lucky Short-Exposure Image Through Turbulence,” J. Opt. Soc. Am. 68, 1651 (1978).
[CrossRef]

R. L. Phillips, L. C. Andrews, “Measured Statistics of Laser-Light Scattering in Atmospheric Turbulence,” J. Opt. Soc. Am. 71, 1440 (1981).
[CrossRef]

R. J. Hill, “Theory of Saturation of Optical Scintillation by Strong Turbulence: Plane-Wave Variance and Covariance and Spherical-Wave Covariance,” J. Opt. Soc. Am. 72, 212 (1982).
[CrossRef]

D. L. Fried, G. E. Mevers, M. P. Keister, “Measurements of Laser-Beam Scintillation in the Atmosphere,” J. Opt. Soc. Am. 57, 787 (1967).
[CrossRef]

G. Parry, P. N. Pusey, “K Distributions in Atmospheric Propagation of Laser Light,” J. Opt. Soc. Am. 69, 796 (1979).
[CrossRef]

Y. Baykal, M. A. Plonus, “Two-Source, Two-Frequency Spherical Wave Structure Functions in Atmospheric Turbulence,” J. Opt. Soc. Am. 70, 1278 (1980).
[CrossRef]

Opt. Acta

A. S. Gurvich, V. Kan, V. Pokasov, “Two-Frequency Fluctuations of Light Intensity in a Turbulent Medium,” Opt. Acta 26, 555 (1979).
[CrossRef]

G. Parry, “Measurement of Atmospheric Turbulence Induced Intensity Fluctuations in a Laser Beam,” Opt. Acta 28, 715 (1981).
[CrossRef]

Proc IEEE

R. Fante, “Electromagnetic Beam Propagation in Turbulent Media,” Proc IEEE 63, 1669 (1975).
[CrossRef]

R. Fante, “Electromagnetic Beam Propagation in Turbulent Media: An Update,” Proc IEEE 68, 1424 (1980).
[CrossRef]

Sov. Phys. JETP

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, V. V. Pokasov, “Similarity Relations for Strong Fluctuations of the Intensity of Light Propagating in a Turbulent Medium,” Sov. Phys. JETP 40, 1011 (1975).

Other

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

R. E. Hufnagel, “Restoration of Atmospherically Degraded Images” (National Academy of Sciences, Washington, D.C., 1966), Vol. 3, Appendix 2, p. 11.

V. I. Tatarski, “The Effect of the Turbulent Atmosphere on Wave Propagation,” Israel Program for Scientific Translations, Jerusalem (1971).

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Figures (6)

Fig. 1
Fig. 1

Experimental setup: L1,L2, lasers; SL, beam forming lens; M1, plane mirror; S1,S2, dichroic beam splitters; D, variable circular aperture; F1,F2, narrowband interference filters; T1,T2, focusing lenses; P1,P2, photodiodes; A1,A2, amplifiers; ADC1,ADC2, analog-to-digital converters; Rockwell AIM 65, microcomputer.

Fig. 2
Fig. 2

Measured normalized variance of intensity is plotted as a function of the parameter β 0 = [ 0 . 5 C n 2 k 7 / 6 L 11 / 6 ] 1 / 2 for two aperture sizes.

Fig. 3
Fig. 3

Measured normalized variance and the two-wavelength covariance for (a) L = 1300 m and (b) L = 3250 m as a function of the aperture size. The data points shown represent averages over all sets of measurements, with a normalized standard deviation of ~15%.

Fig. 4
Fig. 4

Experimental correlation coefficient for (a) L = 1300 m and (b) L = 3250 m as a function of the aperture size. The data points represent averages over all sets of measurements, and the resulting standard deviations are indicated by the bars.

Fig. 5
Fig. 5

Sample time records of the two signals: (a) λ = 1.064 μm, (b) λ = 0.6328 μm. The range was 1300 m and the aperture size was 20 mm.

Fig. 6
Fig. 6

Probability, prob{I0.63 ≧ 〈I0.63〉[1 + ασ I 0.63)]}. (×), and the conditional probability of I1.06 to exceed 〈I1.06〉[1 + ασ I 1.06)] given that I0.63 exceeds 〈I0.63〉[1 + ασ I 0.63)] (with the same α) (solid circles) as a function of α for L = 1300 m. The solid and dashed lines represent the lognormal models of Eqs. (8) and (12), respectively. Each record length was 10,000 points. Also note that the received intensity is always positive, (a) D = 10 mm; (b) D = 20 mm; (c) D = 30 mm.

Equations (12)

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σ I 2 = I 2 I 2 I 2 ,
cov ( λ 1 , λ 2 ) = β 2 = [ I ( λ 1 ) I ( λ 1 ) ] [ I ( λ 2 ) I ( λ 2 ) ] I ( λ 1 ) I ( λ 2 )
ρ ( λ 1 , λ 2 ) = cov ( λ 1 , λ 2 ) σ I ( λ 1 ) σ I ( λ 2 )
σ I 2 ( λ i ) = I 2 ( λ i ) I ( λ i ) 2 I ( λ i ) 2 , i = 1 , 2
C n 2 = ln ( 1 + σ I 2 ) ( 0 . 5 k 7 / 6 L 11 / 6 ) ,
P ( I ¯ 1 . 06 | I ¯ 0 . 63 , ρ ) = prob [ I 1 . 06 I ¯ 1 . 06 ; given that I 0 . 63 I ¯ 0 . 63 ] .
α = I ( λ i ) I ( λ i ) σ I ( λ i ) .
p ( I ) = [ 2 π σ ln I I ] 1 exp [ ( ln I ln I ) 2 / ( 2 σ ln I 2 ) ] ,
p ( I 1 . 06 , I 0 . 63 , ρ ) = [ 2 π σ ln I 1 . 06 σ ln I 0 . 63 I 1 . 06 I 0 . 63 ] 1 × ( 1 ρ 2 ) 1 / 2 exp [ F 2 2 ( 1 ρ 2 ) ] ,
F 2 = [ ln I 1 . 06 ln I 1 . 06 σ ln I 1 . 06 ] 2 + [ ln I 0 . 63 ln I 0 . 63 σ ln I 0 . 63 ] 2 2 ρ [ ln I 1 . 06 ln I 1 . 06 σ ln I 1 . 06 ] [ ln I 0 . 63 ln I 0 . 63 σ ln I 0 . 63 ] .
P ( I ¯ 1 . 06 , I ¯ 0 . 63 , ρ ) = I ¯ 1 . 06 d I 1 . 06 I ¯ 0 . 63 d I 0 . 63 p ( I 1 . 06 , I 0 . 63 , ρ )
P ( I ¯ 1 . 06 | I ¯ 0 . 63 , ρ ) = P ( I ¯ 1 . 06 , I ¯ 0 . 63 , ρ ) / P ( I ¯ 0 . 63 ) .

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