Abstract

We have developed approximate expressions for the aperture-averaging factor of optical scintillation in the turbulent atmosphere. For large apertures and weak path-integrated turbulence with a small inner scale, the variance of signal fluctuations is proportional to the −7/3 power of the ratio of the aperture diameter to the Fresnel zone size. If the inner scale is large, the variance is proportional to the −7/3 power of the ratio of the aperture diameter to the inner scale. In strong path-integrated turbulence, two scales develop. That portion of the variance associated with the smaller scale is proportional to the −2 power of the ratio of the aperture diameter to the phase coherence length. That portion of the variance associated with the larger scale is proportional to the −7/3 power of the ratio of the aperture diameter to the scattering disk. These simple approximations are within a factor of 2 of the measurements.

© 1991 Optical Society of America

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References

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  1. D. L. Fried, “Aperture Averaging of Scintillation,” J. Opt. Soc. Am. 57, 169–175 (1967).
    [CrossRef]
  2. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Chap. 13.
  3. M. E. Gracheva, A. S. Gurvich, “Averaging Effect of the Receiving Aperture on Fluctuations in Light Intensity,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 12, 253–255 (1969).
  4. R. F. Lutomirski, R. E. Huschke, W. C. Meecham, H. T. Yura, “Degradation of Laser Systems by Atmospheric Turbulence,” Report R-1171-ARPA/RC, Rand Corp., Santa Monica, CA (June1973).
  5. A. G. Kjelaas, P. E. Nordal, “Scintillation Noise Reduction by Aperture Averaging in a Long-Path Laser Absorption Spectrometer,” Appl. Opt. 21, 2481–2488 (1982).
    [CrossRef] [PubMed]
  6. D. L. Fried, “Theoretical Analysis of Aperture Averaging,” Report DR-015, Optical Science Consultants, Yorba Linda, CA (Oct.1973).
  7. A. I. Kon, “Averaging of Spherical-Wave Fluctuations over a Receiving Aperture,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 12, 149–152 (1969).
  8. R. F. Lutomirski, H. T. Yura, “Aperture-Averaging Factor of a Fluctuating Light Signal,” J. Opt. Soc. Am. 59, 1247–1248 (1969).
    [CrossRef]
  9. H. T. Yura, W. G. McKinley, “Aperture Averaging of Scintillation for Space-to-Ground Optical Communication Applications,” Appl. Opt. 22, 1608–1609 (1983).
    [CrossRef] [PubMed]
  10. S. J. Wang, Y. Baykal, M. A. Plonus, “Receiver-Aperture Averaging Effects for the Intensity Fluctuation of a Beam Wave in the Turbulent Atmosphere,” J. Opt. Soc. Am. 73, 831–837 (1983).
    [CrossRef]
  11. D. H. Hohn, “Effects of Atmospheric Turbulence on the Transmission of a Laser Beam at 6328 Å. I: Distribution of Intensity,” Appl. Opt. 5, 1427–1431 (1966).
    [CrossRef] [PubMed]
  12. D. L. Fried, G. E. Meyers, M. P. Keister, “Measurements of Laser-Beam Scintillation in the Atmosphere,” J. Opt. Soc. Am. 57, 787–797 (1967).
    [CrossRef]
  13. G. E. Homstad, J. W. Strohbehn, R. H. Berger, J. M. Heneghan, “Aperture-Averaging Effects for Weak Scintillations,” J. Opt. Soc. Am. 64, 162–165 (1974).
    [CrossRef]
  14. R. S. Iyer, J. L. Bufton, “Aperture Averaging Effects in Stellar Scintillation,” Opt. Commun. 22, 377–381 (1977).
    [CrossRef]
  15. S. H. Reiger, “Starlight Scintillation and Atmospheric Turbulence,” Astron. J. 68, 395–406 (1963).
    [CrossRef]
  16. A. T. Young, “Photometric Error Analysis. VI. Confirmation of Reiger’s Theory of Scintillation,” Astron. J. 72, 747–753 (1967).
    [CrossRef]
  17. A. T. Young, “Aperture Filtering and Saturation of Scintillation,” J. Opt. Soc. Am. 60, 248–250 (1970).
    [CrossRef]
  18. J. R. Kerr, “Experiments on Turbulence Characteristics and Multiwavelength Scintillation Phenomena,” J. Opt. Soc. Am. 62, 1040–1049 (1972).
    [CrossRef]
  19. K. S. Gochelashvily, V. I. Shishov, “Multiple Scattering of Light in a Turbulent Medium,” Opt. Acta 18, 767–777 (1971).
    [CrossRef]
  20. K. S. Gochelashvily, V. G. Pevgov, V. I. Shishov, “Saturation of Fluctuations of the Intensity of Laser Radiation at Large Distances in a Turbulent Atmosphere (Fraunhofer Zone of Transmitter),” Sov. J. Quantum Electron. 4, 632–637 (1974).
    [CrossRef]
  21. A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser Irradiance Propagation in Turbulent Media,” Proc. IEEE 63, 790–810 (1975).
    [CrossRef]
  22. R. L. Fante, “Inner-Scale Size Effect on the Scintillations of Light in the Turbulent Atmosphere,” J. Opt. Soc. Am. 73, 277–281 (1983).
    [CrossRef]
  23. R. G. Frehlich, “Intensity Covariance of a Point Source in a Random Medium with a Kolmogorov Spectrum and an Inner Scale of Turbulence,” J. Opt. Soc. Am. A 4, 360–366 (1987).
    [CrossRef]
  24. W. A. Coles, R. G. Frehlich, “Simultaneous Measurements of Angular Scattering and Intensity Scintillation in the Atmosphere,” J. Opt. Soc. Am. 72, 1042–1048 (1982).
    [CrossRef]
  25. J. H. Churnside, S. F. Clifford, “Lognormal Rician Probability-Density Function of Optical Scintillations in the Turbulent Atmosphere,” J. Opt. Soc. Am. A 4, 1923–1930 (1987).
    [CrossRef]
  26. R. S. Lawrence, J. W. Strohbehn, “A Survey of Clear-Air Propagation Effects Relevant to Optical Communications,” Proc. IEEE 58, 1523–1545 (1970).
    [CrossRef]
  27. R. J. Hill, “Models of the Scalar Spectrum for Turbulent Advection,” J. Fluid Mech. 88, 541–562 (1978).
    [CrossRef]
  28. R. J. Hill, S. F. Clifford, “Modified Spectrum of Atmospheric Temperature Fluctuations and its Application to Optical Propagation,” J. Opt. Soc. Am. 68, 892–899 (1978).
    [CrossRef]
  29. J. H. Churnside, “A Spectrum of Refractive Turbulence in the Turbulent Atmosphere,” J. Mod. Opt. 37, 13–16 (1990).
    [CrossRef]
  30. S. F. Clifford, H. T. Yura, “Equivalence of Two Theories of Strong Optical Scintillation,” J. Opt. Soc. Am. 64, 1641–1644 (1974).
    [CrossRef]
  31. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1970), Chaps. 6 and 13.
  32. G. R. Ochs, W. D. Cartwright, D. D. Russell, “Optical Cn2 Instrument Model II,” NOAA Tech. Memo. ERL WPL-51 (National Technical Information Service, 5285 Port Royal Rd., Springfield, VA, 22161, 1979), WPL51:PB 80-209000.
  33. G. R. Ochs, R. J. Hill, “Optical-Scintillation Method of Measuring Turbulence Inner Scale,” Appl. Opt. 24, 2430–2432 (1985).
    [CrossRef] [PubMed]

1990

J. H. Churnside, “A Spectrum of Refractive Turbulence in the Turbulent Atmosphere,” J. Mod. Opt. 37, 13–16 (1990).
[CrossRef]

1987

1985

1983

1982

1978

1977

R. S. Iyer, J. L. Bufton, “Aperture Averaging Effects in Stellar Scintillation,” Opt. Commun. 22, 377–381 (1977).
[CrossRef]

1975

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser Irradiance Propagation in Turbulent Media,” Proc. IEEE 63, 790–810 (1975).
[CrossRef]

1974

K. S. Gochelashvily, V. G. Pevgov, V. I. Shishov, “Saturation of Fluctuations of the Intensity of Laser Radiation at Large Distances in a Turbulent Atmosphere (Fraunhofer Zone of Transmitter),” Sov. J. Quantum Electron. 4, 632–637 (1974).
[CrossRef]

G. E. Homstad, J. W. Strohbehn, R. H. Berger, J. M. Heneghan, “Aperture-Averaging Effects for Weak Scintillations,” J. Opt. Soc. Am. 64, 162–165 (1974).
[CrossRef]

S. F. Clifford, H. T. Yura, “Equivalence of Two Theories of Strong Optical Scintillation,” J. Opt. Soc. Am. 64, 1641–1644 (1974).
[CrossRef]

1972

1971

K. S. Gochelashvily, V. I. Shishov, “Multiple Scattering of Light in a Turbulent Medium,” Opt. Acta 18, 767–777 (1971).
[CrossRef]

1970

R. S. Lawrence, J. W. Strohbehn, “A Survey of Clear-Air Propagation Effects Relevant to Optical Communications,” Proc. IEEE 58, 1523–1545 (1970).
[CrossRef]

A. T. Young, “Aperture Filtering and Saturation of Scintillation,” J. Opt. Soc. Am. 60, 248–250 (1970).
[CrossRef]

1969

R. F. Lutomirski, H. T. Yura, “Aperture-Averaging Factor of a Fluctuating Light Signal,” J. Opt. Soc. Am. 59, 1247–1248 (1969).
[CrossRef]

M. E. Gracheva, A. S. Gurvich, “Averaging Effect of the Receiving Aperture on Fluctuations in Light Intensity,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 12, 253–255 (1969).

A. I. Kon, “Averaging of Spherical-Wave Fluctuations over a Receiving Aperture,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 12, 149–152 (1969).

1967

1966

1963

S. H. Reiger, “Starlight Scintillation and Atmospheric Turbulence,” Astron. J. 68, 395–406 (1963).
[CrossRef]

Baykal, Y.

Berger, R. H.

Bufton, J. L.

R. S. Iyer, J. L. Bufton, “Aperture Averaging Effects in Stellar Scintillation,” Opt. Commun. 22, 377–381 (1977).
[CrossRef]

Bunkin, F. V.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser Irradiance Propagation in Turbulent Media,” Proc. IEEE 63, 790–810 (1975).
[CrossRef]

Cartwright, W. D.

G. R. Ochs, W. D. Cartwright, D. D. Russell, “Optical Cn2 Instrument Model II,” NOAA Tech. Memo. ERL WPL-51 (National Technical Information Service, 5285 Port Royal Rd., Springfield, VA, 22161, 1979), WPL51:PB 80-209000.

Churnside, J. H.

Clifford, S. F.

Coles, W. A.

Fante, R. L.

Frehlich, R. G.

Fried, D. L.

Gochelashvily, K. S.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser Irradiance Propagation in Turbulent Media,” Proc. IEEE 63, 790–810 (1975).
[CrossRef]

K. S. Gochelashvily, V. G. Pevgov, V. I. Shishov, “Saturation of Fluctuations of the Intensity of Laser Radiation at Large Distances in a Turbulent Atmosphere (Fraunhofer Zone of Transmitter),” Sov. J. Quantum Electron. 4, 632–637 (1974).
[CrossRef]

K. S. Gochelashvily, V. I. Shishov, “Multiple Scattering of Light in a Turbulent Medium,” Opt. Acta 18, 767–777 (1971).
[CrossRef]

Gracheva, M. E.

M. E. Gracheva, A. S. Gurvich, “Averaging Effect of the Receiving Aperture on Fluctuations in Light Intensity,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 12, 253–255 (1969).

Gurvich, A. S.

M. E. Gracheva, A. S. Gurvich, “Averaging Effect of the Receiving Aperture on Fluctuations in Light Intensity,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 12, 253–255 (1969).

Heneghan, J. M.

Hill, R. J.

Hohn, D. H.

Homstad, G. E.

Huschke, R. E.

R. F. Lutomirski, R. E. Huschke, W. C. Meecham, H. T. Yura, “Degradation of Laser Systems by Atmospheric Turbulence,” Report R-1171-ARPA/RC, Rand Corp., Santa Monica, CA (June1973).

Iyer, R. S.

R. S. Iyer, J. L. Bufton, “Aperture Averaging Effects in Stellar Scintillation,” Opt. Commun. 22, 377–381 (1977).
[CrossRef]

Keister, M. P.

Kerr, J. R.

Kjelaas, A. G.

Kon, A. I.

A. I. Kon, “Averaging of Spherical-Wave Fluctuations over a Receiving Aperture,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 12, 149–152 (1969).

Lawrence, R. S.

R. S. Lawrence, J. W. Strohbehn, “A Survey of Clear-Air Propagation Effects Relevant to Optical Communications,” Proc. IEEE 58, 1523–1545 (1970).
[CrossRef]

Lutomirski, R. F.

R. F. Lutomirski, H. T. Yura, “Aperture-Averaging Factor of a Fluctuating Light Signal,” J. Opt. Soc. Am. 59, 1247–1248 (1969).
[CrossRef]

R. F. Lutomirski, R. E. Huschke, W. C. Meecham, H. T. Yura, “Degradation of Laser Systems by Atmospheric Turbulence,” Report R-1171-ARPA/RC, Rand Corp., Santa Monica, CA (June1973).

McKinley, W. G.

Meecham, W. C.

R. F. Lutomirski, R. E. Huschke, W. C. Meecham, H. T. Yura, “Degradation of Laser Systems by Atmospheric Turbulence,” Report R-1171-ARPA/RC, Rand Corp., Santa Monica, CA (June1973).

Meyers, G. E.

Nordal, P. E.

Ochs, G. R.

G. R. Ochs, R. J. Hill, “Optical-Scintillation Method of Measuring Turbulence Inner Scale,” Appl. Opt. 24, 2430–2432 (1985).
[CrossRef] [PubMed]

G. R. Ochs, W. D. Cartwright, D. D. Russell, “Optical Cn2 Instrument Model II,” NOAA Tech. Memo. ERL WPL-51 (National Technical Information Service, 5285 Port Royal Rd., Springfield, VA, 22161, 1979), WPL51:PB 80-209000.

Pevgov, V. G.

K. S. Gochelashvily, V. G. Pevgov, V. I. Shishov, “Saturation of Fluctuations of the Intensity of Laser Radiation at Large Distances in a Turbulent Atmosphere (Fraunhofer Zone of Transmitter),” Sov. J. Quantum Electron. 4, 632–637 (1974).
[CrossRef]

Plonus, M. A.

Prokhorov, A. M.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser Irradiance Propagation in Turbulent Media,” Proc. IEEE 63, 790–810 (1975).
[CrossRef]

Reiger, S. H.

S. H. Reiger, “Starlight Scintillation and Atmospheric Turbulence,” Astron. J. 68, 395–406 (1963).
[CrossRef]

Russell, D. D.

G. R. Ochs, W. D. Cartwright, D. D. Russell, “Optical Cn2 Instrument Model II,” NOAA Tech. Memo. ERL WPL-51 (National Technical Information Service, 5285 Port Royal Rd., Springfield, VA, 22161, 1979), WPL51:PB 80-209000.

Shishov, V. I.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser Irradiance Propagation in Turbulent Media,” Proc. IEEE 63, 790–810 (1975).
[CrossRef]

K. S. Gochelashvily, V. G. Pevgov, V. I. Shishov, “Saturation of Fluctuations of the Intensity of Laser Radiation at Large Distances in a Turbulent Atmosphere (Fraunhofer Zone of Transmitter),” Sov. J. Quantum Electron. 4, 632–637 (1974).
[CrossRef]

K. S. Gochelashvily, V. I. Shishov, “Multiple Scattering of Light in a Turbulent Medium,” Opt. Acta 18, 767–777 (1971).
[CrossRef]

Strohbehn, J. W.

G. E. Homstad, J. W. Strohbehn, R. H. Berger, J. M. Heneghan, “Aperture-Averaging Effects for Weak Scintillations,” J. Opt. Soc. Am. 64, 162–165 (1974).
[CrossRef]

R. S. Lawrence, J. W. Strohbehn, “A Survey of Clear-Air Propagation Effects Relevant to Optical Communications,” Proc. IEEE 58, 1523–1545 (1970).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Chap. 13.

Wang, S. J.

Young, A. T.

A. T. Young, “Aperture Filtering and Saturation of Scintillation,” J. Opt. Soc. Am. 60, 248–250 (1970).
[CrossRef]

A. T. Young, “Photometric Error Analysis. VI. Confirmation of Reiger’s Theory of Scintillation,” Astron. J. 72, 747–753 (1967).
[CrossRef]

Yura, H. T.

Appl. Opt.

Astron. J.

S. H. Reiger, “Starlight Scintillation and Atmospheric Turbulence,” Astron. J. 68, 395–406 (1963).
[CrossRef]

A. T. Young, “Photometric Error Analysis. VI. Confirmation of Reiger’s Theory of Scintillation,” Astron. J. 72, 747–753 (1967).
[CrossRef]

Izv. Vyssh. Uchebn. Zaved. Radiofiz.

M. E. Gracheva, A. S. Gurvich, “Averaging Effect of the Receiving Aperture on Fluctuations in Light Intensity,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 12, 253–255 (1969).

A. I. Kon, “Averaging of Spherical-Wave Fluctuations over a Receiving Aperture,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 12, 149–152 (1969).

J. Fluid Mech.

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

J. Mod. Opt.

J. H. Churnside, “A Spectrum of Refractive Turbulence in the Turbulent Atmosphere,” J. Mod. Opt. 37, 13–16 (1990).
[CrossRef]

J. Opt. Soc. Am.

A. T. Young, “Aperture Filtering and Saturation of Scintillation,” J. Opt. Soc. Am. 60, 248–250 (1970).
[CrossRef]

J. R. Kerr, “Experiments on Turbulence Characteristics and Multiwavelength Scintillation Phenomena,” J. Opt. Soc. Am. 62, 1040–1049 (1972).
[CrossRef]

G. E. Homstad, J. W. Strohbehn, R. H. Berger, J. M. Heneghan, “Aperture-Averaging Effects for Weak Scintillations,” J. Opt. Soc. Am. 64, 162–165 (1974).
[CrossRef]

S. F. Clifford, H. T. Yura, “Equivalence of Two Theories of Strong Optical Scintillation,” J. Opt. Soc. Am. 64, 1641–1644 (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, 892–899 (1978).
[CrossRef]

W. A. Coles, R. G. Frehlich, “Simultaneous Measurements of Angular Scattering and Intensity Scintillation in the Atmosphere,” J. Opt. Soc. Am. 72, 1042–1048 (1982).
[CrossRef]

R. L. Fante, “Inner-Scale Size Effect on the Scintillations of Light in the Turbulent Atmosphere,” J. Opt. Soc. Am. 73, 277–281 (1983).
[CrossRef]

S. J. Wang, Y. Baykal, M. A. Plonus, “Receiver-Aperture Averaging Effects for the Intensity Fluctuation of a Beam Wave in the Turbulent Atmosphere,” J. Opt. Soc. Am. 73, 831–837 (1983).
[CrossRef]

D. L. Fried, G. E. Meyers, M. P. Keister, “Measurements of Laser-Beam Scintillation in the Atmosphere,” J. Opt. Soc. Am. 57, 787–797 (1967).
[CrossRef]

D. L. Fried, “Aperture Averaging of Scintillation,” J. Opt. Soc. Am. 57, 169–175 (1967).
[CrossRef]

R. F. Lutomirski, H. T. Yura, “Aperture-Averaging Factor of a Fluctuating Light Signal,” J. Opt. Soc. Am. 59, 1247–1248 (1969).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Acta

K. S. Gochelashvily, V. I. Shishov, “Multiple Scattering of Light in a Turbulent Medium,” Opt. Acta 18, 767–777 (1971).
[CrossRef]

Opt. Commun.

R. S. Iyer, J. L. Bufton, “Aperture Averaging Effects in Stellar Scintillation,” Opt. Commun. 22, 377–381 (1977).
[CrossRef]

Proc. IEEE

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser Irradiance Propagation in Turbulent Media,” Proc. IEEE 63, 790–810 (1975).
[CrossRef]

R. S. Lawrence, J. W. Strohbehn, “A Survey of Clear-Air Propagation Effects Relevant to Optical Communications,” Proc. IEEE 58, 1523–1545 (1970).
[CrossRef]

Sov. J. Quantum Electron.

K. S. Gochelashvily, V. G. Pevgov, V. I. Shishov, “Saturation of Fluctuations of the Intensity of Laser Radiation at Large Distances in a Turbulent Atmosphere (Fraunhofer Zone of Transmitter),” Sov. J. Quantum Electron. 4, 632–637 (1974).
[CrossRef]

Other

R. F. Lutomirski, R. E. Huschke, W. C. Meecham, H. T. Yura, “Degradation of Laser Systems by Atmospheric Turbulence,” Report R-1171-ARPA/RC, Rand Corp., Santa Monica, CA (June1973).

D. L. Fried, “Theoretical Analysis of Aperture Averaging,” Report DR-015, Optical Science Consultants, Yorba Linda, CA (Oct.1973).

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1970), Chaps. 6 and 13.

G. R. Ochs, W. D. Cartwright, D. D. Russell, “Optical Cn2 Instrument Model II,” NOAA Tech. Memo. ERL WPL-51 (National Technical Information Service, 5285 Port Royal Rd., Springfield, VA, 22161, 1979), WPL51:PB 80-209000.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Chap. 13.

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Figures (15)

Fig. 1
Fig. 1

Aperture-averaging factor A vs ratio of aperture radius D/2 to the Fresnel zone size (L/k)1/2 for plane-wave propagation through weak turbulence with the small inner scale. The solid line is the exact formula, and the dashed line is the recommended approximation.

Fig. 2
Fig. 2

Aperture-averaging factor A vs ratio of aperture diameter D to the inner scale l0 for plane-wave propagation through weak turbulence with a large inner scale. The solid line is the exact formula using the Tatarskii spectrum, the points are exact values using the Hill spectrum, and the dashed line is the recommended approximation.

Fig. 3
Fig. 3

Aperture-averaging factor A vs ratio of inner scale l0 to the Fresnel zone size (L/K)1/2 for plane-wave propagation with the aperture size as labeled. The solid lines are the exact formula using the Hill spectrum, and the dashed lines are the approximate values.

Fig. 4
Fig. 4

Aperture-averaging factor A vs ratio of the aperture radius D/2 to the Fresnel zone size (L/K)1/2 for spherical-wave propagation through weak turbulence with the small inner scale. The solid line is the exact formula, and the dashed line is the recommended approximation.

Fig. 5
Fig. 5

Aperture-averaging factor A vs ratio of aperture diameter D to the inner scale l0 for spherical-wave propagation through weak turbulence with a large inner scale. The solid line is the exact formula, and the dashed line is the recommended approximation.

Fig. 6
Fig. 6

Aperture-averaging factor A vs ratio of the aperture radius D/2 to the phase coherence length ρ0 for plane-wave propagation through a strong turbulence with a small inner scale and three different values of irradiance variance σ I 2. The solid lines are the exact results, and the dashed lines represent the recommended approximation.

Fig. 7
Fig. 7

Aperture-averaging factor A vs ratio of the aperture radius D/2 to the phase coherence length ρ0 for plane-wave propagation through strong turbulence with l0 = 10ρ0 and three different values of irradiance variance σ I 2. The solid lines are the exact results, and the dashed lines represent the recommended approximation.

Fig. 8
Fig. 8

Aperture-averaging factor A vs ratio of the aperture radius D/2 to the phase coherence length ρ0 for spherical-wave propagation through strong turbulence and three different values of irradiance variance σ I 2. The solid lines are the exact results, and the dashed lines represent the recommended approximation.

Fig. 9
Fig. 9

Aperture-averaging factor A vs ratio of the aperture radius D/2 to the phase coherence length ρ0 for spherical-wave propagation through strong turbulence with l0 = 10ρ0 and three different values of irradiance variance σ I 2. The solid lines are the exact results, and the dashed lines represent the recommended approximation.

Fig. 10
Fig. 10

Aperture-averaging factor A vs ratio of the aperture diameter D to the inner scale l0. The points represent data taken at 100 m, and the curve is the approximate formula for a spherical wave, weak turbulence, and large l0.

Fig. 11
Fig. 11

Aperture-averaging factor A vs ratio of aperture diameter D to inner scale l0. The points represent data taken at 250 m, and the curve is the approximate formula for a spherical wave, weak turbulence, and large l0.

Fig. 12
Fig. 12

Aperture-averaging factor A vs ratio of the aperture radius D/2 to Fresnel zone size (L/k)1/2. The points represent data taken at 250 m, and the curve is the approximate formula for a spherical wave, weak turbulence, and small l0.

Fig. 13
Fig. 13

Aperture-averaging factor A vs ratio of the aperture radius D/2 to the Fresnel zone size (L/k)1/2. The points represent data taken at 500 m, and the curve is the approximate formula for a spherical wave, weak turbulence, and small l0.

Fig. 14
Fig. 14

Aperture-averaging factor A vs ratio of the aperture radius D/2 to the coherence length ρ0. The points represent data taken at 1000 m, the solid curve is the strong-turbulence approximation, and the dashed curve is the weak-turbulence approximation.

Fig. 15
Fig. 15

Aperture-averaging factor A vs ratio of the aperture radius D/2 to the coherence length ρ0. The points represent data taken at 1000 m, the solid curve is the strong-turbulence approximation, and the dashed curve is the weak-turbulence approximation.

Tables (2)

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Table I Aperture Averaging Factor for Plane Waves

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Table II Aperture Averaging Factor for Spherical Waves

Equations (91)

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A = 16 π D 2 0 D C I ( ρ ) C I ( 0 ) [ cos - 1 ( ρ D ) - ρ D ( 1 - ρ 2 D 2 ) 1 / 2 ] ρ d ρ ,
A = 16 π 0 1 C I ( D y ) C I ( 0 ) [ cos - 1 y - y ( 1 - y 2 ) 1 / 2 ] y d y .
C I ( ρ ) = 16 π 2 k 2 0 d K K Φ n ( K ) 0 L d z J 0 ( K ρ s ) sin 2 [ K 2 ( L - z ) s 2 k ] ,
Φ n ( K ) = 0.033 C n 2 K - 11 / 3 ,
C I ( ρ ) = ( 0.033 ) 16 π 2 k 2 C n 2 0 d K K - 8 / 3 0 L d z J 0 ( K ρ ) × sin 2 [ K 2 ( L - z ) 2 k ] ,
C I ( 0 ) = 1.23 k 7 / 6 L 11 / 6 C n 2 .
A = 21.6 k 5 / 6 L - 11 / 6 0 d K K - 8 / 3 0 L d z sin 2 [ K 2 ( L - z ) 2 k ] × 0 1 d y y J 0 ( K D y ) [ cos - 1 y - y ( 1 - y 2 ) 1 / 2 ] .
A = 8.47 ( k D 2 4 L ) 5 / 6 0 d u u - 14 / 3 J 1 2 ( u ) [ 1 - k D 2 4 L u 2 sin ( 4 L u 2 k D 2 ) ] ,
A = 4.24 4 L k D 2 0 d v v - 17 / 6 ( 1 - sin v v ) J 1 2 [ ( k D 2 v 4 L ) 1 / 2 ] .
A 1.41 ( k D 2 4 L ) - 7 / 6 0 d u u - 2 / 3 J 1 2 ( u ) = 0.932 ( k D 2 4 L ) - 7 / 6 .
A = [ 1 + 1.07 ( k D 2 4 L ) 7 / 6 ] - 1 .
C I 2 ( 0 ) = 1.23 k 7 / 6 L 11 / 6 11 6 0 1 d x ( 1 - x ) 5 / 6 C n 2 ( x L ) .
A = [ 1 + 1.07 C ( k D 2 4 L ) 7 / 6 ] - 1 ,
C = 11 0 1 d x ( 1 - x ) 5 / 6 C n 2 ( x L ) 18 0 1 d x ( 1 - x ) 2 C n 2 ( x L ) ,
Φ n ( K ) = 0.033 C n 2 K - 11 / 3 exp ( - 0.0285 K 2 l 0 2 ) .
C I ( ρ ) = ( 0.033 ) 16 π 2 k 2 C n 2 0 d K K - 8 / 3 exp ( - 0.0285 K 2 l 0 2 ) × 0 L d z J 0 ( K ρ ) sin 2 [ K 2 ( L - z ) 2 k ] .
C I ( ρ ) = ( 0.033 ) 4 3 π 2 L 3 C n 2 0 d K K 4 / 3 J 0 ( K ρ ) exp ( - 0.0285 K 2 l 0 2 ) .
C I ( 0 ) = 12.8 L 3 C n 2 l 0 - 7 / 3 .
A = 0.686 ( D l 0 ) - 7 / 3 0 d u u - 2 / 3 J 1 2 ( u ) exp ( - 0.1141 l 0 2 u 2 D 2 ) .
A = 0.453 ( D l 0 ) - 7 / 3 .
A = [ 1 + 2.21 ( D l 0 ) 7 / 3 ] - 1 .
Φ n ( K ) = 0.033 C n 2 K - 11 / 3 { exp ( - 1.29 K 2 l 0 2 ) + 1.45 exp [ - 0.97 ( ln K l 0 - 0.452 ) 2 ] } .
C I ( 0 ) = 38.3 L 3 0 1 d x ( 1 - x ) 2 C n 2 ( x L ) l 0 - 7 / 3 ( x L ) ,
A = [ 1 + 2.21 C D 7 / 3 ] - 1 ,
C = 0 1 d x ( 1 - x ) 2 C n 2 ( x L ) l 0 - 7 / 3 ( x L ) 0 1 d x ( 1 - x ) 2 C n 2 ( x L ) .
C I ( ρ ) = ( 0.033 ) 16 π 2 k 2 C n 2 0 d K K - 8 / 3 0 L d z J 0 ( K ρ z L ) × sin 2 [ K 2 z ( L - z ) 2 k L ] .
C I ( 0 ) = 0.497 k 7 / 6 L 11 / 6 C n 2 .
A = 53.4 k 5 / 6 L - 11 / 6 0 d K K - 8 / 3 0 L d z sin 2 [ K 2 z ( L - z ) 2 k L ] × 0 1 d y y J 0 ( K D z y L ) [ cos - 1 y - y ( 1 - y 2 ) 1 / 2 ] .
A = 41.9 ( k D 2 4 L ) 5 / 6 0 d u u - 14 / 3 J 1 2 ( u ) 0 1 d x x 5 / 3 × sin 2 [ 1 2 4 L k D 2 ( 1 x - 1 ) u 2 ] .
A = [ 1 + 0.214 ( k D 2 4 L ) 7 / 6 ] - 1 ,
C I 2 ( 0 ) = 0.497 k 7 / 6 L 11 / 6 Γ ( 11 3 ) Γ 2 ( 11 6 ) 0 1 d x x 5 / 6 ( 1 - x ) 5 / 6 C n 2 ( x L ) ,
A = [ 1 + 0.214 C ( k D 2 4 L ) 7 / 6 ] - 1 ,
C = 2 Γ ( 2 3 ) 0 1 d x x 5 / 6 ( 1 - x ) 5 / 6 C n 2 ( x L ) Γ 2 ( 11 6 ) 0 1 d x x - 1 / 3 ( 1 - x ) 2 C n 2 ( x L ) .
C I ( ρ ) = ( 0.033 ) 16 π 2 k 2 C n 2 0 d K K - 8 / 3 exp ( - 0.0285 K 2 l 0 2 ) × 0 L d z J 0 ( K ρ z L ) sin 2 [ K 2 z ( L - z ) 2 k L ] .
C I ( ρ ) = ( 0.033 ) 4 π 2 C n 2 L - 2 0 d K K 4 / 3 × exp ( - 0.0285 K 2 l 0 2 ) 0 L d z z 2 ( L - z ) 2 × J 0 ( K ρ z L ) .
C I ( 0 ) = 1.28 L 3 C n 2 l 0 - 7 / 3 .
A = 20.6 ( D l 0 ) - 7.3 0 d u u - 2 / 3 J 1 2 ( u ) 0 1 d x x - 1 / 3 ( 1 - x ) 2 × exp ( - 0.114 l 0 2 u 2 D 2 x 2 ) ,
A = [ 1 + 0.109 ( D l 0 ) 7 / 3 ] - 1 ,
C I ( 0 ) = 38.3 L 3 0 1 d x x 2 ( 1 - x ) 2 C n 2 ( x L ) l 0 - 7 / 3 ( x L ) .
A = [ 1 + 0.109 C D 7 / 3 ] - 1 ,
C = 0 1 d x x 2 ( 1 - x ) 2 C n 2 ( x L ) l 0 - 7 / 3 ( x L ) 0 1 d x x 2 ( 1 - x ) 2 C n 2 ( x L ) .
C I ( ρ ) = exp [ - ( ρ ρ 0 ) 5 / 3 ] + 1 2 N 3 ( k ρ 0 2 L ) 1 / 3 [ b 1 ( ρ ) + b 2 ( ρ ) ] ,
ρ 0 = ( 1.46 k 2 L C n 2 ) - 3 / 5 .
N 3 = 9 ( 2 ) 5 / 3 5 π sin ( 5 π 6 ) Γ 2 ( 11 6 ) Γ ( 7 5 ) F 2 1 ( 7 5 , 2 3 ; 5 3 ; 5 8 ) = 1.22 ,
b 1 ( ρ ) = 7 3 0 1 d x x 4 / 3 J 0 ( k ρ ρ 0 x / L ) .
b 2 = exp [ - ( ρ ρ 0 ) 5 / 3 ] .
σ I 2 = C I ( 0 ) = 1 + 1.22 ( k ρ 0 2 L ) 1 / 3 .
A 1 = 16 π σ I 2 + 1 2 σ I 2 0 1 d y y [ cos - 1 y - y ( 1 - y 2 ) 1 / 2 ] exp [ - ( D y ρ 0 ) 5 / 3 ] ,
A 2 = 56 L 2 3 k 2 D 2 ρ 0 2 σ I 2 - 1 2 σ I 2 0 1 d x x - 2 / 3 J 1 2 ( k D ρ 0 x 2 L ) .
A 1 = σ I 2 + 1 2 σ I 2 .
A 1 = σ I 2 + 1 2 σ I 2 [ 1.10 ( 2 ρ 0 D ) 2 ] .
A 1 = σ I 2 + 1 2 σ I 2 [ 1 + 0.908 ( D 2 ρ 0 ) 2 ] - 1 .
A 2 = σ I 2 - 1 2 σ I 2 .
A 2 = σ I 2 - 1 2 σ I 2 [ 6.16 ( 2 L k ρ 0 D ) 7 / 3 ] ,
A 2 = σ I 2 - 1 2 σ I 2 [ 1 + 0.162 ( k ρ 0 D 2 L ) 7 / 3 ] - 1 .
A = σ I 2 + 1 2 σ I 2 [ 1 + 0.908 ( D 2 ρ 0 ) 2 ] - 1 + σ I 2 - 1 2 σ I 2 [ 1 + 0.162 ( k ρ 0 D 2 L ) 7 / 3 ] - 1 ,
ρ 0 = [ 1.46 k 2 0 L d z C n 2 ( z ) ] - 3 / 5 .
C I ( ρ ) = exp [ - ( ρ ρ 0 ) 2 ] + 1 2 N 3 ( k ρ 0 l 0 L ) 1 / 3 [ b 1 ( ρ ) + b 2 ( ρ ) ] ,
ρ 0 = ( 1.20 k 2 C n 2 L l 0 - 1 / 3 ) - 1 / 2 ,
b 1 ( ρ ) = 0.897 ( L k ρ 0 ) 7 / 3 0 1 d τ τ 2 0 d K K 4 / 3 × exp [ - L 2 k 2 ρ 0 2 K 2 τ 2 ( 1 - 2 3 τ ) ] J 0 ( K ρ ) .
b 2 ( ρ ) = exp [ - ( ρ ρ 0 ) 2 ] .
σ I 2 = 1 + 1.21 ( k ρ 0 l 0 L ) 1 / 3 .
A 1 = 16 π σ I 2 + 1 2 σ I 2 0 1 d y y [ cos - 1 y - y ( 1 - y 2 ) 1 / 2 ] exp [ - ( D y / ρ 0 ) 2 ] ,
A 2 = 3.59 ( 2 L k ρ 0 D ) 7 / 3 σ I 2 - 1 2 σ I 2 0 1 d τ τ 2 0 d u u - 2 / 3 J 1 2 ( u ) × exp [ - ( 2 L k ρ 0 D ) 2 u 2 τ 2 ( 1 - 2 3 τ ) ] .
A 1 = σ I 2 + 1 2 σ I 2 .
A 1 = σ I 2 + 1 2 σ I 2 ( 2 ρ 0 D ) 2 ,
A 1 = σ I 2 + 1 2 σ I 2 [ 1 + ( D 2 ρ 0 ) 2 ] - 1 .
A 2 = σ I 2 - 1 2 σ I 2 .
A 2 = σ I 2 - 1 2 σ I 2 [ 0.790 ( 2 L k ρ 0 D ) 7 / 3 ] ,
A 2 = σ I 2 - 1 2 σ I 2 [ 1 + 1.27 ( k ρ 0 D 2 L ) 7 / 3 ] - 1 .
A = σ I 2 + 1 2 σ I 2 [ 1 + ( D 2 ρ 0 ) 2 ] - 1 + σ I 2 - 1 2 σ I 2 [ 1 + 1.27 ( k ρ 0 D 2 L ) 7 / 3 ] - 1 .
σ I 2 = 1 + 10.6 ( L k ρ 0 l 0 ) 2 .
A 2 = σ I 2 - 1 2 σ I 2 ( 1.20 ) ( 2 L k ρ 0 D ) 7 / 3 0 d u u - 2 / 3 J 1 2 ( u ) × exp [ - 0.0285 ( 2 l 0 D ) 2 u 2 ]
ρ 0 = [ 1.20 k 2 0 L d z C n 2 ( z ) l 0 - 1 / 3 ( z ) ] - 1 / 2 ,
ρ 0 = ( 0.545 k 2 L C n 2 ) - 3 / 5 ,
N 3 = 3 ( 2 ) 8 / 3 5 π ( 8 3 ) 7 / 5 sin ( 5 π 6 ) Γ 2 ( 11 6 ) Γ ( 7 5 ) × Γ 2 ( 2 3 ) / Γ ( 4 3 ) = 3.86.
b 1 ( ρ ) = 0.915 0 1 d x x - 1 / 3 ( 1 - x ) 2 0 d τ × τ 4 / 3 J 0 ( k ρ ρ 0 τ L ) exp [ - τ 5 / 3 ( 1 - x ) 5 / 3 ] .
b 2 ( ρ ) = exp [ - ( ρ ρ 0 ) 5 / 3 ] .
σ I 2 = 1 + 3.86 ( k ρ 0 2 L ) 1 / 3 .
A 1 = 16 π σ I 2 + 1 2 σ I 2 0 1 d y y [ cos - 1 y - y ( 1 - y 2 ) 1 / 2 ] exp [ - ( D y ρ 0 ) 5 / 3 ] ,
A 2 = 3.66 ( 2 L k D ρ 0 ) 7 / 3 σ I 2 - 1 2 σ I 2 0 d u u - 2 / 3 J 1 2 ( u ) 0 1 d x x 2 ( 1 - x ) - 1 / 3 × exp [ - ( 2 L k D ρ 0 ) 5 / 3 u 5 / 3 x 5 / 3 ] .
A = σ I 2 + 1 2 σ I 2 [ 1 + 0.908 ( D 2 ρ 0 ) 2 ] - 1 + σ I 2 - 1 2 σ I 2 [ 1 + 0.613 ( k D ρ 0 2 L ) 7 / 3 ] - 1 .
ρ 0 = ( 0.545 k 2 C n 2 L l 0 - 1 / 3 ) - 1 / 2 ,
b 1 ( ρ ) = 1.05 ( L k ρ 0 ) 7 / 3 0 1 d x x 2 ( 1 - x ) 2 0 d K K 4 / 3 × exp [ - L 2 k 2 ρ 0 2 K 2 x 2 ( 1 - x ) 2 ] J 0 ( K x ρ ) .
b 2 ( ρ ) = exp [ - ( ρ ρ 0 ) 2 ] .
σ I 2 = 1 + 2.27 ( k ρ 0 l 0 L ) 1 / 3 .
A 1 = 16 π σ I 2 + 1 2 σ I 2 0 1 d y y [ cos - 1 y - y ( 1 - y 2 ) 1 / 2 ] exp [ - ( D y ρ 0 ) 2 ] ,
A 2 = 4.20 ( 2 L k D ρ 0 ) 7 / 3 σ I 2 - 1 2 σ I 2 0 1 d x x 2 ( 1 - x ) - 1 / 3 0 d u u - 2 / 3 J 1 2 ( u ) × exp [ - ( 2 L k D ρ 0 ) 2 x 2 u 2 ] .
A = σ I 2 + 1 2 σ I 2 [ 1 + ( D 2 ρ 0 ) 2 ] - 1 + σ I 2 - 1 2 σ I 2 [ 1 + 0.534 ( k D ρ 0 2 L ) 7 / 3 ] - 1 .
σ I 2 = 1 + 2.34 ( L k ρ 0 l 0 ) 2 .
A 2 = 4.20 ( 2 L k D ρ 0 ) 7 / 3 σ I 2 - 1 2 σ I 2 0 1 d x x 2 ( 1 - x ) - 1 / 3 0 d u u - 2 / 3 J I 2 ( u ) × exp [ - 0.0285 ( 2 l 0 D ) 2 u 2 / x 2 ] .

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