L. Tsang and J. Kong, "Wave theory for microwave remote sensing of a half-space random medium with three dimensional variations," Radio Sci. 14, 359–369 (1979).

K. Furutsu, "Multiple scattering of waves in a medium of randomly distributed particles and derivation of the transport equation," Radio Sci. 10, 29–44 (1975).

R. Fante, "Some physical insights into beam propagation in strong turbulence," Radio Sci. 15, 757–762 (1980).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978); see Chap. 14, where Twersky's theory is summarized.

V. Frisch, "Wave propagation in random media," in Probabilistic Methods in Applied Mathematics, A. Barucha-Reid, ed. (Academic, New York, 1968).

V. Tatarskii, The Effect of the Turbulent Atmosphere on Wave Propagation (National Technical Information Source, Springfield, Virginia, 1971); Secs. 60 and 61.

It is demonstrated in Refs. 1 and 2 that the approximation of Eq. (1) by formula (2) is equivalent to the ladder approximation to the Bethe-Salpeter equation. Consequently, we have given this decomposition the name ladder approximation.

For example, when evaluating 〈*EE**〉 by taking moments of the integral form of the wave equation, we find that we need to evaluate terms of the form 〈∊(**r**′)∊(**r**′)*E**(**r**)〉, where ∊(**r**′) is the relative permittivity fluctuation at **r**′. If ∊ is a Gaussian random variable, the Novikov-Furutsu theorem (see Ref. 1) gives [equation].where δ/δ∊ is a variational derivative. It is readily demonstrated that δ*E*(**r**)/δ∊(**r**″)= -*k*^{2}*E*(**r**″G(**r**, **r**″), where G(**r**, **r**″) is therandom Green's function and satisfies {∇2+ *k*^{2}[1 + ∊(**r**)]}G(**r**,**r**′)= δ(**r** - **r**′). Therefore 〈∊(**r**′)E(**r**′)E*(**r**)〉=-*k*^{2}∋∋∞∞∋d_{3}r″〈∊(**r**′)×(**r**)G(**r**, **r**)E(**r**)E*(**r**)+G*(**r**,**r**)E(**r**′)E*(**r**″)〉so we see that terms of the form of Eq. (1) must be evaluated.

It is demonstrated in Sec. 37 of Ref. 1 that the representation in formula (4) is valid provided that the spatial scale size of the inhomogeneities is large in comparison with the signal wavelength. In particular, it is shown that formula (4) approximately includes multiple small-angle scatterings. Clearly, however, formula (4) cannot be used in the limit when the inhomogeneities are small in comparison with a wavelength, because both formulas (3) and (4) assume that individual backscatterings are negligible. This is equivalent to Twersky's assumption (see Ref. 3, Section 14-1) that there are no propagation paths that go through a scatterer more than once.

It is not necessary to assume that ø has zero mean. When ø has a nonzero mean we can simply lump it in with G_{0}〈…〉 in Eq. (7). That is, 〈G(**r**′, **r**″)〉 = [G_{0}(**r**′, **r**″)exp(i-〈ø〉)]exp(-½〈ø^{2} (**r**′, **r**″)〉), where ø is now the fluctuation of the phase about the mean value 〈ø〉

V. Tatarskii, Wave Propagation in a Turbulent Medium (Dover, New York, 1967), Chap. 6.

The geometric-optics method gives valid results for the real and imaginary parts of the phase over propagation paths less than l_{0}^{2}λ, where l_{0} is the scale size of the smallest inhomogeneities and λ is the wavelength. For propagation paths that are large in comparison with l_{0}^{2}λ, the geometric-optics method gives incorrect results for the imaginary part of ø but gives solutions for the real part of ø that are in agreement with results obtained by both the method of smooth perturbations and the Markov approximation method. Consequently, geometric optics gives useful results for the real part of ø even for large propagation lengths (i.e., paths large compared with *l*_{0}^{2}λ).

It is shown is Sec. 39 [see formula (23)] of Ref. 1 that the ray paths can be approximated by straight lines if 〈*n*^{2}〉≪1.

These conclusions do not hold in the backscatter cone (i.e., in a cone of angle *L/R* « 1 centered about θ + γ = 0 in Fig. 1). However, this is unimportant because (1) it represents only a small small fraction of the total integral over ρ and ρ′, and (2) backscattering usually is insignificant in the limit when l is larger than a wavelength, as we have assumed.

R. Fante, "Mutual coherence function and frequency spectrum of a laser beam propagating through atmospheric turbulence," J. Opt. Soc. Am. 64, 592–598 (1974); seeEq. (24).

For the case when *kl* < 1 we expect that a sufficient condition for 〈*E*_{i}(**r**″)*E*_{j}*(**r**)*G*_{kl}(**r**′, **r**″)〉 〈(E_{i} (**r**″)*E*_{j}*(**r**)〉 〈*G*_{kl}(**r**′, **r**″)〉 is that 〈*n*^{2}〉 ≪ 1, where *E*_{i} is the ith component of the vector **E**, *G*_{ij} is the *ij* component of the Green's tensor, etc. (We did not need to consider this tensor case when kl »≫ 1 because cross polarization is then tinimportant.) Probably for kl sufficiently small in comparison with unity the ladder approximation is valid even for 〈n^{2}〉 > 1, but we have been unable to prove this rigorously.

R. Fante, "Electromagnetic beam propagation in turbulent media: an update," Proc. IEEE 68, 1424–1443 (1980).