Y. L. Feinberg, Propagation of Radiowaves along the Earth’s Surface (Akad. Nauk., Moscow, USSR, 1961).

Reference 1, p. 211.

Reference 1, Eq. (9.43). The variance of the log amplitude, 〈χ2〉, is one fourth of σ2, the variance of the log irradiance.

The plane divergence of the cw laser was ~0.77 mrad and ~1.2 mrad for the pulsed laser.

G. E. Mevers, private communication. In these experiments, an argon laser beam was propagated to a diffuse target board. The scattered return light was monitored temporally by a receiver near the transmitter.

A system with a divergent beam has a complex equivalence to a diffraction-limited transmitter with an appropriately adjusted aperture and propagation path.

Reference 18, Fig. 4, p. 760.

The primary difficulty encountered in the spatial measurement arises from the variation in low-spatial-frequency content of the laser cross sections due both to profile effects and, in some cases, to beam breakup. Power-spectrum computations of irradiance scans show that the principal power is located at low frequencies. Statistical variation in low frequencies due to beam effects can contribute substantially to the spread in the variance plots, since the variance is given by the integral of the log-irradiance power spectrum. Variations in the saturation plot are also imposed by the reliability of the Cn measurement. Although the experiments were performed over level, open terrain, specially prepared for such experiments, there were still statistical variations of Cn along the optical path. In addition, experiment limitations permit only about a factor-or-two reliability under very weak-turbulence conditions. In spite of these constraints, we believe the phenomenon and general trend of the saturation effect to be accurately portrayed.

F. P. Carlson, private communication. Unpublished results indicate that beam-wave theory may extend the region of applicability of the Rytov procedure.

Reference 1, Fig. 11, p. 140.

Reference 1, Eqs. (7.87) and (7.88), p. 151.

In the limit of a large inner scale, the turbulence spectrum and the optical filter function would have little common area, and the irradiance fluctuations would approach zero (Ref. 1, Fig. 10, p. 139). If such variations exist in the turbulence spectrum, performance predictions of atmospheric-optical systems based on the single parameter Cn may prove to be inadequate.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw–Hill Book Co., New York, 1960), Ch. 9.

A more complicated form of the saturation equation was fitted by Tatarski to the data of Gracheva and Gurvich. The shape of the modified curve differs from the general saturation equation in that it follows the plane-wave equation, Eq. (1), up to a value of about unity before falling off to saturate at a maximum value near two. deWolf incorrectly evaluated the general formulation [Ref. 15, Eq. (10), p. 463] by using the plane-wave equation to compute σ12.

Reference 1, p. 55 ff.

P. H. Deitz, in Modern Optics, J. Fox, Ed. (Polytechnic Press, New York, 1967).

The XM–23 E2 Range Finder, supplied by Frankford Arsenal, Philadelphia, Pa.

Cn= CT(−79×10−6p/T2), where p is pressure in millibars and T is absolute temperature. The apparatus used for the Cn measurements is manufactured by Flow Corporation and consists of two high-speed temperature probes, an anemometer bridge, model HWB-3, a secondary voltage supply, model HWI-3, and a random signal voltmeter, model 12A-1.

Reference 15. deWolf has argued that I follows a Rice distribution, which, while tending to a log-normal distribution for weak fluctuations, tends to a Rayleigh distribution in the limit of strong fluctuations and/or long paths.

Where P(I) = 1−exp(−I).

Reference 1, Eq. (12.1).

Equation (1) is the Rytov-approximated estimate of the logarithmic ratio of variance to square mean.

Gracheva and Gurvich eliminated similar statistics from their graphs (Ref. 9).