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

V. I. Tatarski, see Ref. 1, p. 209.

A. Kolmogoroff, in Turbulence, Classic Papers on Statistical Theory, S. K. Friedlander and L. Topper, Eds. (Interscience Publishers, John Wiley & Sons, Inc., New York, 1961), p. 151.

V. I. Tatarski, see Ref. 1, Ch. 3.

We use α0 as a measure of the size of the laser beam. Precisely defined it is the standard deviation of the amplitude distribution across the laser beam. 2−12α0 is the standard deviation of the intensity.

V. I. Tatarski. see Ref. 1, Ch. 9.

V. I. Tatarski, see Ref. 1, Ch. 13.

The data for Fig. 5 are atypical only in the sense that they represent the noise observed with the smallest collector aperture, and therefore with the largest amplification factor in the analog computer. This makes the noise distribution appear wider than typical. As a consequence, the statement that the distribution is gaussian is easier to check.

If the scintillation is described by a probability density function ps(i), and the noise is characterized by a probability density function pn(j), where i and j are continuous variables, then ps+n(i), the probability density function characteristic of the measured scintillation plus noise is given by the convolution integralps+n(i)=∫−∞+∞ps(j)pn(i−j)dj.We replaced the continuous probability density functions and this convolution integral by the quantized probabilities Ps(i), Pn(j), and Ps+n(i), and the equivalent formulaPs+n(i)=∑f=−∞+∞Ps(i)Pn(i−j).Strictly speaking this represents an infinite number of simultaneous equations and an infinite number of unknowns, the unknowns being the Ps(j) (j= 0, 1, 2, …). These unknowns represent the true scintillation statistics, uncontaminated by equipment noise. Because Pn is fairly sharply peaked, and Ps+n goes to zero outside some finite range, we could truncate the set of equations and unknowns so that we could deal with only 99 simultaneous equations and 99 unknowns. The solution of these equations was achieved by a rather slow, (e.g., 12h on a Recomp III) iterative computer program.Using the fact that pn is gaussian, we might have obtained a solution more quickly by inverting the convolution integral, i.e., Fourier transform the convolution, divide by the transform of the gaussian, which is also gaussian, and then Fourier transform back. Then, when converted to a quantized form, the matrix inversion of the matrix Pn(i− j) (which is a matrix in i and j) would be obtained and the solution of the original simultaneous equations could be accomplished with little effort. Unfortunately realization was too late in the project to be of use. R. A. Jones,“An Automated Edge Gradient Technique,” May 1966 Annual Conf. SPSE, suggested the double Fourier transform concept.

Although the log-normal distribution appears to fit the data exceptionally well, it should be pointed out that if the intensity-signal voltage was converted to a log-intensity voltage using the analog computer, and if this was entered into the pulse-height analyzer, then on the pulse-height analyzer’s CRT display of probability density a very minor, but quite obvious secondary peak appeared consistently on all the laser-scintillation data runs. The peak was always very weak and occurred at a higher intensity than the mode of the distribution. The peak was sufficiently far from the mode that the probability density was less than one-fifth of the peak probability density. When intensity (not log-intensity) was fed into the pulse-height analyzer the peak was never prominent enough to be clearly identified. We have been unable to find any explanation for this secondary peak. Because the log-normal probability-distribution type of plot appears to“suppress” this minor anomaly, we have ignored it in our data processing.

If we consider the results of Ref. 14, assume the applicability of infinite-plane-wave statistics, and recall that (4Z/k)12≃5.7 cm, we see that the reduction of the variance of the intensity with a 10-cm aperture would be equivalent to the summation of approximately 15 regions of independent scintillation. (If we assume spherical-wave-propagation statistics to be applicable, we may estimate that the number of independent regions would be between five and ten.) The summation of that many independent variations should, it would seem, remove the distribution from the log-normal case and probably bring it close to a normal distribution.

The least-squares fit was carried out over the data region 5≤i≤60 to avoid the logarithmic magnification of quantization errors for very small values of i−j¯, and to avoid the low-probability region for large i, which contained very few data. (Although 104 samples were taken per second by the pulse-height analyzer, the scintillation bandwidth was between 200–1000 Hz, so probably no more than 1000 independent data points were taken per second. With our 60-sec measurement of scintillation, only a few independent samples contributed to the scintillation data beyond the 99.9 percentile.) Because the density of data points vs log(i−j¯) increases as i without any corresponding increase of the density of data, we used a 1/i weighting in the region 5≤i≤60, for the curve fitting.

For this experiment kαs2/z=3.66×10−3 which is clearly much less than unity. Hence Eq. (1.1) is applicable.

As part of a different project we made measurements over a 300-m path about 1.5 m above an asphalt road during the middle of a hot summer day and observed values of CN2 nearly as large as the largest in Table II.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Co., New York1961), Chs. 7 and 8.

Figures 14, 15, and 16, but excluding sunrise or sunset data.

Sunrise and sunset data from Figs. 14, 15, and 16.

The use of temporal dispersion, i.e., time-delay redundancy, does not appear to be a reasonable alternative to spatial diversity. The frequency of fading is sufficiently low that the memory required for the necessary time-delay of even a 10-kHz information bandwidth would be prohibitively large. This leaves only spatial dispersion, i.e., the use of large optics—or less attractive, the use of several links. (The use of chromatic dispersion, i.e., the use of many or a wide band of wavelengths, is not a reasonable alternative since the“chromatic correlation of scintillation” extends over about an octave.)

In the absence of atmospheric turbulence such transmitter and receiver dimensions would result in most of the beam being collected by the receiver aperture. In such a situation it is easy to see how small perturbations which redistributed energy inside the beam and perhaps“spilled” some of the energy outside the beam boundary, but still inside the collector, would produce only negligible scintillation. The dimensional arguments which lead us to the suggested transmitter and receiver dimensions are, however, considerably more general than that. We expect substantial reduction of scintillation even if the atmosphere so spreads the beam that only a small fraction of it is captured by the collection aperture.

A. L. Buck, in Proceedings of the Conference on Atmospheric Limitations to Optical Propagation, Boulder, Colorado (1965).

The usefulness of a large-diameter transmitter in reducing scintillation has been discussed in Ref. 4, in conjunction with the use of a point detector and transmission of a beam focused on the detector. Here we have recommended making the transmitter diameter about equal to (4Z/k)12 (with a comparable receiver diameter). In Ref. 4, significant reductions were not obtained (with the point detector) until the transmitter diameter was made significantly larger than (4Z/k)12. It may be that these two results represent a smooth continuation of each other for the case of a finite-size collector.