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

S. Gardner, 1964 IEEE International Convention Record, Part 6, p. 337.

J. O. Hinze, Turbulence (McGraw-Hill Book Company, Inc., New York, 1959).

D. L. Fried, J. Opt. Soc. Am. 55, 1427 (1965). This work was called to my attention following completion of that reported here.

The degree of approach to perfect tracking as a function of frequency response of the control system is remarked in Sec. 7.

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

The basic equations were given also in Ref. 2.

E.g., see L. Landau and E. Lifshitz, The Classical Theory of Fields (Addison-Wesley Publishing Company, Inc., Reading, Mass., 1951).

This model becomes more realistic, but pertinent only for aperture sizes *a࣠l*_{0}, when *l*_{e} refers to the minimum eddy size (microscale *l*_{0}) and δø_{0} is replaced by (3.44)^{½}*kC*_{n}L^{½}*l*_{0}^{⅚}, in which *C*_{n} is the coefficient referred to at Eq. (37) below [see Ref. 1, Eq. (6.64)].

Some examination of the validity of this supposition could presumably be effected on the basis of the approximation for Φ_{1}–Φ_{2} finally employed below [see Eq. (31)].

The similar parameter *r*_{0} of Ref. 4 is related to *a*_{e} by *r*_{0} = 2.1*a*_{e}.

More generally, *C*_{n}^{2} may not be constant in space, and *C*_{n}^{2}*L* must then be replaced by the integral along the propagation path, ∫_{0}^{L}dsC_{n}^{2}(*s*).

The stated rough upper limit on *a* for validity of (46) is established by requiring 〈(ø_{1}-ø_{2})^{2}〉 ≲ 1 for |ρ_{1}-ρ_{2}| = 2*a*, and the limit for (47) by increasing the former limit in accord with the reduction of the area average of 〈(Φ_{1}-Φ_{2})^{2}〉 relative to 〈(ø_{1}-ø_{2})^{2}〉. The error associated with (47) is smaller than that with (46) for equal *a* but roughly the same for equal power reduction.

The sagittal condition^{1}*L«l*_{0}^{4}/λ^{3} and the condition λ«l_{0} are still imposed.

Dr. R. E. Hufnagel has noted this statistical validity of ray optics in the perturbation and Rytov approximations on the basis of Tatarski's work^{1} and pointed out the likelihood of more general validity (unpublished). General validity is not correctly proved in Ref. 6.^{16}

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

For the fixed-axis case, the asymptotic signal-to-noise ratio is equal to the ratio ih the absence of propagation disturbance when the aperture radius in the latter instance is equal to (1.084)^{1/2}a_{e}. The effective radius so defined is thus nearly equal to a_{e}, justifying the appellation for a_{e}.

The value 3.4 agrees with that obtained for the mathematically identical quantity in Ref. 4. In Ref. 4, as here, no computation is made that would yield the power-reduction factors outside the moderate-aperture approximation of Eq. (46) and (47).

In the absence of propagation effects, for example, L(k_{1},k_{2}) = (2π)^{2}a^{2}(k_{1}k_{2})^{-1}J_{1}(k_{1}a)J_{1}(k_{2}a).

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953), p. 1324.

Bateman Manuscript Project, Tables of Integral Transforms (McGraw-Hill Book Company, Inc., New York, 1954), Vol. 2, p. 24, No. (7).

Reference 21, Vol. 1, p. 331, No. (33).

To obtain Eq. (A34), multiply Eq. (3) of Ref. 16 by exp[*ik* × (ζ_{1}-ζ_{2})], take expectations, and perform the double integration over the aperture. The left member, by the second equation following (A20), is identified as the left member of Eq. (A34), and the right, since Φ=ø+*k*ζ, as the right.