Z. I. Feizulin and Y. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radiophys. Quantum Electron. 10, 33–35 (1967).

A. I. Kon and V. I. Tatarskii, "On the theory of the propagation of partially coherent light beams in a turbulent atmosphere," Radiophys. Quantum Electron. 15, 1187–1192 (1972).

A. I. Kon and Z. I. Feizulin, "Fluctuations in the parameters of spherical waves propagating in a turbulent atmosphere," Radiophys. Quantum Electron. 13, 51–53 (1970).

For a demonstration of the accuracy of this approximation, let us consider a coherent plane wave in a random medium. It is well known (Ref. 4, p. 445) that the plane-wave coherence length derived from the five-thirds law is ρ_{c} = (8/3)^{−3/5}ρ_{0} (ρ_{0} being the spherical wave coherence length). The result in section *V*, using the quadratic approximation, shows that ρ_{c} = ρ_{0}/√3, which is within a 4% agreement. One should notice, however, that although this approximation is reasonable for the second moment, it results in a significant difference for the higher-order moments.

In Young's experiment, this corresponds to calculating the visibility from the interference pattern on the optical axis.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice Hall, Englewood Cliff, N.J., 1964).

We note that, for a statistically homogeneous source distribution,the relation of the form Γ^{s}_{2}(r_{1}, r_{2}) ≡ 〈*u*_{s}(r_{1})*u**_{s}(r_{2})〉_{s}= *F*(r_{1} − r_{2}) should hold for all position vectors r_{1} and r_{2} in the source plane *z* = 0, implying a source of infinite dimension. Obviously, this requirement is never satisfied in practice. In this paper we assume "statistical homogeneity" for a finite source in the approximate sense that the above relation holds whenever the point r_{1} and r_{2} both lie within the source aperture.

A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 2 (Academic, New York, 1978), p. 422.

In the literature the term "mutual coherence function" is sometimes used synonymously with "mutual intensity function." We note that the latter is a special case of the former, i.e., τ = 0 in Eq.(3).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, VA, 1971).

A. C. Shell, "The multiple plate antenna," Doctoral Dissertation, Massachusetts Institute of Technology, 1961 (unpublished).

M. Born and E. Wolf, Principles of Optics, 5th Ed. (Pergamon, Oxford, 1975).

Special note should be taken in interpreting the results of Lee *et al*., about the dominant scale size for the focused case in very weak turbulence. The "coherent" TEM_{00} source with radius α_{T} is focused on the diffuse target on which the spot size reduces to the order α_{F} = λ*F*/πα_{T} (assuming negligible turbulence effect). The scattered beam, having suffered a random phase delay from point to point over the target, then acts as an incoherent planar source propagating to the receiver. Since the receiver falls definitely into the far field of the new (secondary) source as in Eq. (21), the coherence length is ~2*L*/*k*α_{F} = α_{T}, as predicted in Ref. 5.