Abstract

The variance of the log-amplitude of a laser beam is evaluated for a horizontal propagation path through the atmosphere. The treatment is based upon results obtained by Schmeltzer. It is found that the logamplitude variance can be separated into two factors, one of which is simply the log-amplitude variance of a spherical wave, as derived by Tatarski. The second factor, which contains the dependence upon the size α<sub>0</sub> of the transmitted beam, can be written as a function of kα<sub>0</sub><sup>2</sup>/z, where k is the wave number and z is the path length. This second factor shows a significant oscillation around kα<sub>0</sub><sup>2</sup>/z > 1 when the transmitted beam is collimated, and starts to roll off strongly for kα<sub>0</sub><sup>2</sup>/z> 1 when the transmitted beam is focused at a range z.

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  1. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Company, Inc., New York, 1961).
  2. L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill Book Company, New York, 1960).
  3. R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).
  4. D. L. Fried, J. Opt. Soc. Am. 57, 175 (1967).
  5. R. A. Schmeltzer, Quart. Appl. Math. 24, 339 (1967).
  6. Equation (2.2) is obtained from Schmeltzer's Eq. (7.8) with τ set equal to zero to convert a covariance to a variance.
  7. A. Kolmogorov, in Turbulence, Classic Papers on Statistical Theory, Ed. by S. K. Friedlander and L. Topper (Interscience Publishers, Inc., New York, 1961), p. 151.
  8. D. L. Fried, J. Opt. Soc. Am. 56, 1380 (1966), Eq. (2.21).
  9. I. Goldstein, et al., Proc. IEEE 53, 1172 (1965).
  10. J. I. Davis, Appl. Opt. 5, 139 (1966).
  11. A. Erdelyi et al., Table of Integral Transforms (McGraw-Hill Book Company, New York, 1954), Vol. I, Eq. 4.3(1).
  12. A. Erdelyi et al., Table of Integral Transforms (McGraw-Hill Book Company, New York, 1954), Vol. I, p. 373.
  13. A. Erdelyi et al., Higher Transcendental Functions (McGraw-Hill Book Co., New York, 1953), Vol. I, Chaps. 2 and 5.
  14. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Company, New York, 1961), Eq. (9.43).
  15. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Company, New York, 1961), Eq. (7.94).
  16. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U. S. Government Printing Office, Washington, D. C., 1964) (Dover Publications, Inc., New York, 1965), Eq. (15.3.9).
  17. D. L. Fried and J. D. Cloud, J. Opt. Soc. Am. 56, 1667 (1966).
  18. Although the Rytov approximation in the explicit form used by Tatarski is not apparent in Schmeltzer's work, a careful reading of the latter work, especially Secs. 3 and 5, makes it apparent that the results Schmeltzer has obtained are equivalent to those he would have obtained with a formal invocation of the Rytov approximation.

Kolmogorov, A.

A. Kolmogorov, in Turbulence, Classic Papers on Statistical Theory, Ed. by S. K. Friedlander and L. Topper (Interscience Publishers, Inc., New York, 1961), p. 151.

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U. S. Government Printing Office, Washington, D. C., 1964) (Dover Publications, Inc., New York, 1965), Eq. (15.3.9).

Chernov, L. A.

L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill Book Company, New York, 1960).

Cloud, J. D.

D. L. Fried and J. D. Cloud, J. Opt. Soc. Am. 56, 1667 (1966).

Davis, J. I.

J. I. Davis, Appl. Opt. 5, 139 (1966).

Erdelyi, A.

A. Erdelyi et al., Table of Integral Transforms (McGraw-Hill Book Company, New York, 1954), Vol. I, p. 373.

A. Erdelyi et al., Higher Transcendental Functions (McGraw-Hill Book Co., New York, 1953), Vol. I, Chaps. 2 and 5.

A. Erdelyi et al., Table of Integral Transforms (McGraw-Hill Book Company, New York, 1954), Vol. I, Eq. 4.3(1).

Fried, D. L.

D. L. Fried and J. D. Cloud, J. Opt. Soc. Am. 56, 1667 (1966).

D. L. Fried, J. Opt. Soc. Am. 56, 1380 (1966), Eq. (2.21).

D. L. Fried, J. Opt. Soc. Am. 57, 175 (1967).

Goldstein, I.

I. Goldstein, et al., Proc. IEEE 53, 1172 (1965).

Hufnagel, R. E.

R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).

Schmeltzer, R. A.

R. A. Schmeltzer, Quart. Appl. Math. 24, 339 (1967).

Stanley, N. R.

R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U. S. Government Printing Office, Washington, D. C., 1964) (Dover Publications, Inc., New York, 1965), Eq. (15.3.9).

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Company, Inc., New York, 1961).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Company, New York, 1961), Eq. (9.43).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Company, New York, 1961), Eq. (7.94).

Other (18)

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Company, Inc., New York, 1961).

L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill Book Company, New York, 1960).

R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).

D. L. Fried, J. Opt. Soc. Am. 57, 175 (1967).

R. A. Schmeltzer, Quart. Appl. Math. 24, 339 (1967).

Equation (2.2) is obtained from Schmeltzer's Eq. (7.8) with τ set equal to zero to convert a covariance to a variance.

A. Kolmogorov, in Turbulence, Classic Papers on Statistical Theory, Ed. by S. K. Friedlander and L. Topper (Interscience Publishers, Inc., New York, 1961), p. 151.

D. L. Fried, J. Opt. Soc. Am. 56, 1380 (1966), Eq. (2.21).

I. Goldstein, et al., Proc. IEEE 53, 1172 (1965).

J. I. Davis, Appl. Opt. 5, 139 (1966).

A. Erdelyi et al., Table of Integral Transforms (McGraw-Hill Book Company, New York, 1954), Vol. I, Eq. 4.3(1).

A. Erdelyi et al., Table of Integral Transforms (McGraw-Hill Book Company, New York, 1954), Vol. I, p. 373.

A. Erdelyi et al., Higher Transcendental Functions (McGraw-Hill Book Co., New York, 1953), Vol. I, Chaps. 2 and 5.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Company, New York, 1961), Eq. (9.43).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Company, New York, 1961), Eq. (7.94).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U. S. Government Printing Office, Washington, D. C., 1964) (Dover Publications, Inc., New York, 1965), Eq. (15.3.9).

D. L. Fried and J. D. Cloud, J. Opt. Soc. Am. 56, 1667 (1966).

Although the Rytov approximation in the explicit form used by Tatarski is not apparent in Schmeltzer's work, a careful reading of the latter work, especially Secs. 3 and 5, makes it apparent that the results Schmeltzer has obtained are equivalent to those he would have obtained with a formal invocation of the Rytov approximation.

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