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A new method of evaluating the mutual-coherence function for propagation in a randomly inhomogeneous medium like the atmosphere is presented. The new method, which is highly physical, as distinct from a mathematical approach, does not involve the treatment of any differential equations. Instead, the treatment is based on decomposition of a randomly distorted wavefront into a set of plane waves with random amplitudes. These plane waves constitute orthogonal modes. Propagation in a random medium is treated as the physical process of diffusion of amplitude (or energy) between the modes, and a short-path-propagator function for this diffusion is developed. From the short-path-propagator function, a long-path-propagator function is easily obtained, and from this the mutual-coherence function is computed. Starting from the known short-path mutual-coherence function, which is known to be accurate, the mutual-coherence function for long paths is obtained. The results are in agreement with previous results, all of whose derivations have recently been subject to criticism. Because this derivation is not a mathematical exercise, it should not be subject to any of these or similar criticisms, which were primarily questions of mathematical rigor.

Measurements have been made of the scintillation of a laser beam after propagating over an 8-km path near the ground. The measurements were made with collection apertures ranging from 1 mm to 1 m in diameter. The probability distribution of the scintillation was found to be log-normal for all collector diameters. The log-normal variance decreased smoothly for diameters from 1 mm to about 10 cm, but showed no decrease between 10 cm and 1 m. A hypothesis is offered which explains those results which are anomalous in terms of present theories. Measurements of the log-normal variance were made over extensive periods on three days, during winter, spring, and summer. From these the atmosphere’s refractive-index structure constant, CN^2, was computed. Values of CN^2 were found to be fairly constant over the full range of measurements, falling between 3 and 6×10^−15m^−23, except near sunrise and sunset, when values of about 1×10^−15m^−23 were more typical.

The results concerning the log-amplitude covariance for propagation of an infinite plane wave through a turbulent medium as derived by Tatarski, are developed further in analytic form. The numerical-integration data obtained by Tatarski for horizontal propagation are replaced by analytic results. Propagation of starlight down through the atmosphere is studied using a function fitted to Hufnagel’s data for the vertical distribution of turbulence. Here, also, analytic results are obtained, representing the first available data for the covariance of log-amplitude fluctuations of starlight. Results for the phase-structure functions for both types of propagation path are presented.

The turbulence of the atmosphere places an upper limit on the quality of an image of ground objects obtained by long-exposure photography from high altitudes in the atmosphere or in space. (By making the imaging optics good enough, the film resolution fine enough, and the platform stable enough, this limit could be approached but not exceeded.) A useful quantity for indicating the magnitude of this limit is the integral of the MTF associated with the turbulence. Treating the integral as a two-dimensional bandwidth, one-half the inverse of its square root can be associated with a resolution length, or angle, in the same manner that an electrical engineer associates a rise time with one-half the inverse of the bandwidth of an R–C filter. Based on published data for typical strength of atmospheric turbulence, the integral of the MTF was calculated as a function of altitude and the corresponding resolution computed. This resolution is shown to correspond to a length of about 4.6 cm on the ground. It is shown that as an observer goes deeper into space, this limiting ground resolution remains constant, but the diameter of the optics needed to approach the limit goes up. Graphs of achievable ground resolution at any altitude and of the diameter of the optics needed to approach this limit are presented.

A theoretical foundation is developed for relating the statistics of wave distortion to optical resolution. The average resolution of very-long- and very-short-exposure images is studied in terms of the phase- and log-amplitude-structure functions, whose sum we call the “wave-structure function.” Those results which are comparable are in agreement with the findings of Hufnagel and Stanley who studied the average modulation transfer function of long-exposure images. It is found that the average short-exposure resolution can be significantly better than the average long-exposure resolution, but only if the wave distortion does not include substantial intensity variation.

A precise statistical definition is established for the geometric “shape” of a randomly distorted wavefront. Relationships between the phase-structure function and the statistics governing the shape are derived. The most significant portion of wavefront distortion caused by atmospheric turbulence is a random tilting of the plane-wave front. A procedure is outlined for calculating the influence of wavefront distortion on optical systems. Estimates are formed of the effect of wave-front distortion on photographic resolution and optical heterodyne efficiency.

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 log-amplitude 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_0 of the transmitted beam, can be written as a function of kα_0^2/z, where k is the wave number and z is the path length. This second factor shows a significant oscillation around kα_0^2/z = 1 when the transmitted beam is collimated, and starts to roll off strongly for kα_0^2/z>1 when the transmitted beam is focused at a range z.

The relationship between the statistics of log-amplitude fluctuations and irradiance fluctuations due to atmospheric turbulence is derived. This is used to evaluate the effect of use of a large aperture diameter in reducing the variance of a fluctuating signal. Curves for the reduction factor are presented. From these, an irradiance-fluctuation correlation distance is evaluated. This distance, unlike the correlation distance for log-amplitude fluctuations, is found to be a function of the log-amplitude variance. A particular example of the application of these results to a space-to-ground communications systems performance is worked out.

The log-amplitude covariance and the phase-structure function, associated with the statistics of propagation of a spherical wave in a turbulent medium, are derived for a propagation path over which the statistics of turbulence are constant. Analytic and graphical representation of the results are presented. The theoretical prediction of the zero crossing of the log-amplitude covariance for a spherical wave is found to be in much better agreement with Tatarski’s experimental results than were the results of the infinite-plane wave analysis—as might have been expected, in retrospect, from consideration of the details of Tatarski’s experiment.

The problem of scintillation of a laser beam because of atmospheric turbulence, which has previously-been studied for a horizontal propagation path, is studied in this paper for a path from a ground-based transmitter to a measurement point in space. The results are presented in terms of the log-amplitude variance Cl(0), from which the intensity variance can be computed. The log-amplitude variance is found to be separable into a strength factor Cls(0) associated with the use of a zero-diameter source, and an aperture factor which measures the reduction of Cl(0) caused by use of a large-aperture transmitter. Values of the aperture factor are calculated.