For the concept of MTF to be meaningful, a reasonable-size isoplanatism patch must exist, and we so assume. Though the MTF is conventionally defined in terms of the amplitude of the image of a unit-amplitude sine-wave test pattern in the patch, we work with the MTF defined in terms of the Fourier spectrum of the image response to a unit impulse in the isoplanatism patch. So long as the isoplanatism patch is large enough, there is, in effect, no difference in the definitions. For the unit impulse we use what is nominally an infinite plane wave, though we could equally well use a spherical wave with an adjustment of the image plane, or any other wavefront that should produce a point image.

D. L. Fried, J. Opt. Soc. Am. 55, 1427 (1965); 56, 410E (1966).

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon Press Ltd., Oxford, 1964), p. 385, Equation (38).

We consider the combination of turbulent atmosphere in the propagation path and the lens itself as the image-forming optical system.

V. I. Tatarski, Wave Propagation it a Turbulent Medium (McGraw-Hill Book Co., New York, 1961), p. 209. This discusses only the distribution of *l*, but the same argument can be easily modified to apply to ø.

Because atmospheric turbulence, or any refractive inhomogeneity can only redistribute radiant energy, not absorb it, for an infinite plane wave (or spherical wave), the average irradiance reaching any point must be independent of the strength of the turbulence. Otherwise, the average of the irradiance over a large collecting surface would not be a constant; but it must be, since all of the energy has to reach this surface. From Note 5, we see that the irradiance fluctuations are distributed in a log-normal manner. This and conservation of energy are compatible only if the center of the distribution, determined by *l*, is related to the variance of the distribution *C*_{1}(0). (Of course, when there is no turbulence, *l* and *C*_{i}(0) are both zero.)

D. M. Chase, J. Opt. Soc. Am. 56, 33 (1966).

The derivation of (3.13) with slightly different words would be sufficient to prove that exp[-½D(*r*)] is exactly equal to the mutual coherence function as used by Hufnagel and Stanley (cf. Ref. 10).

The general approach of Sec. 3 to compute the long-exposure MTF has been used by E. A. Trabka, J. Opt. Soc. Am. 56, 128 (1966), but by assuming that *l*=0, he had to allow the normalization term, corresponding to *B* in the paper, to depend on the strength of the turbulence, which it should not.

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

Dr. G. R. Heidbreder, of Aerospace Corp., has informed me that he has been able to show that, at least for a one-dimensional aperture, assumption II must be viewed as an approximation, and has found a weak correlation between ø(*u*) - *au* and *a*. As he has shown, however, the inaccuracy in assumption II drops out in the v-integration when assumption II is used to obtain a result like (4.8), providing that ½([ø(v) - a•v]-[ø(v-λ*R*f)-a•(v-λ*R*f)])^{2} is small enough that the approximation *e*^{x}≈1+*x* can be made. Whenever the short-exposure MTF is not too severely degraded by atmospheric turbulence, this smallness condition is satisfied.

D. L. Fried and J. D. Cloud, "Optical Propagation in the Atmosphere: Theoretical Evaluation and Experimental Determination of the Phase Structure Function," presented at the Conference on Atmospheric Limitations to Optical Propagation at the U. S. National Bureau of Standards, CRPL, 18–19 March 1965.

V. I. Tatarski, Ref. 5, Eq. (8.20).

Though previously published results prove the validity of (5.1) for only an infinite plane wave, it can be shown that (5.1) is equally applicable to the propagation of a spherical wave. The coefficient *A* is different. For horizontal propagation, it can be shown that the spherical-wave coefficient is exactly 3/8 of the coefficient for an infinite plane wave propagating over the same path.

The Rytov approximation was introduced and so attributed by Tatarski, Ref. 5, p. 269, though we have been unable to find such an approximation discussed in Rytov's rather lengthy "source" paper.

D. L. Fried, Ref. 2. Appendix C.

An example of this for the infinite plane-wave propagation case is provided by Tatarski, Ref. 5, Eqs. (8.20), (8.21), and (8.22).

Dr. R. E. Hufnagel has informed me that he has arrived at a result equivalent to (5.9a), through an analysis differing in substantial features from that presented here.

E. L. O'Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co.. Reading, Massachusetts, 1963), p. 106.

D. M. Chase, J. Opt. Soc. Am. 55, 1559 (1965).

E. Djurle and A. Bäck, J. Opt. Soc. Am. 51, 1029 (1961).

C. E. Coulman, J. Opt. Soc. Am. 55, 806 (1965).

C. B. Rogers, J. Opt. Soc. Am. 55, 1151 (1965).