R. E. Hufnagel and N. R. Stanley, Ref. 1, Fig. 6.

Normally the incomplete gamma function as defined in (3.3) is denoted by a lower case gamma. The capital gamma is used to denote the integral from b to ∞. To avoid confusion with the function defined in (2.7) we have violated this usage.

By “conservation of average irradiance” we refer to the fact that the average irradiance is independent of the strength of the turbulence in the propagation path, and is the same even if there is no atmospheric turbulence at all. It depends only on the source and the distance to the source.

We use the term “resolution” of an optical system in a manner equivalent to the electrical engineers’ use of the term “bandwidth” of a filter (or other linear electronic circuit). Quantitatively, we consider it to be the integral over the frequency domain of the MTF, though somewhat variant quantitative forms are equally suitable here, just as they are in electrical engineering. Since the image-frequency domain is two-dimensional, resolution is measured in cycles squared per unit area (or per unit solid angle), while the electronic signal-frequency domain being one-dimensional, bandwidth is measured in cycles per unit time. Just as risetime can be computed from the reciprocal bandwidth, a resolvable area or length (or a resolvable solid angle or angle) can be computed from the reciprocal resolution.

Although the “wave-structure function,” defined by Eq. (2.11), is expressible in terms of the more conventional phase and log-amplitude-structure function, the much greater simplicity of evaluation of the “wave-structure function” and its central place in the calculation of many atmospheric-optical effects make it appear as a physically significant quantity in its own right, rather than simply the combination of two physically significant quantities. The “wave-structure function” can be equated with the mean-square magnitude of the difference of the two phasors (containing phase and log-amplitude variations), at x and x′, correcting the optical electric-field vector for fluctuations associated with propagation in a randomly inhomogeneous medium.

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

R. A. Schmeltzer, Quart. Appl. Math. (to be published).

The dependence of the covariance on r is the dominant aspect of the function. The covariance vanishes for values of r much greater than some “correlation distance.” The dependence on s is very weak, (in particular cases there may be no dependence on s), and contains only the fact that the statistics which determine the covariance may change slowly along the path of propagation. There is never any significant change in the covariance when s changes by an amount of the order of the “correlation distance.”

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

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

Tatarski, Ref. 5, p. 269, note d of Chap. 6.