Coherence Function for Waves in Random Media* DAVID M. CHASE

TRG Incorporated, Melville, New York† 11749 (Received 25 March 1965)

IN a recent paper Hufnagel and Stanley1 seemed to prove a remarkable result concerning the lateral spatial correlation

(coherence function) of the field associated with a wave propa­gated through a medium having random fluctuations in refractive index. The incident wave was assumed plane, and the field within the medium statistically stationary with respect to lateral dis­placement in planes normal to the direction of incidence. According to the stated result, the lateral correlation of the field, i.e., the statistical expectation of the product of the field at two laterally separated points at a fixed time,2 is given correctly by computing it as though ray optics applied to each realization, even when the path length is so long that the condition for validity of ray optics

November 1965 L E T T E R S T O T H E E D I T O R 1559

is certainly not satisfied. As a corollary, it was concluded that the effects of diffraction and of scintillation exactly canceled.

In fact, however, the quoted result of this otherwise useful paper is incorrectly inferred. We proceed, using the notation and definitions of HS. From Eqs. (4.2) and (4.3) of HS, which followed with certain approximations from the wave equation,3 we obtain the following differential equation involving the field-modification factor A (pi,z) and the index fluctuation N(pi,z) at two points pi in the plane at z

where

and a prime denotes differentiation with respect to the coordinate in the direction of incidence z. Equation (4.4) of HS is obtained by taking expectations in Eq. (1). Solving (1) as a first-order linear differential equation for ƒ(z), we find4

This equation, of course, does not constitute an explicit solution for A*(p2,z)A(p1,z)( = ƒ), since the function h(z') in the right member also involves the unknown field A. Taking expectations in Eq. (3) and normalizing by assuming ƒ(0) = l, we obtain a formal expression for the desired field correlation in the plane at z

where

and

In the approximation of ray optics the second of the two terms in (4) would be the only one contributing. Now, as pointed out in HS, under the assumption that A (p,z) is stationary for changes in p, it is true that (h(z)) = 0, where h was defined in (2c) and occurs in the first term of (4). This point, however, evidently does not imply the vanishing of the correlation occurring in the first term of (4). Hence, no proof of the claimed result has been supplied.5

The possible nonvanishing of the correlation in the first term in Eq. (4) is clear from the following. The field at any point A (p1,z') and the phase integral φ1 (z',z) over the path beyond are statistically dependent, on account of the dependence of A (p1,z') on the index fluctuation at prior points z"<z', the dependence of φi(z',z) on the fluctuation at points z">z', and the correlation (over a limited range of separations) of the index fluctuation at z" <z' with that at z">z'.

Nevertheless, as noted below, ray optics does have statistical validity in at least some regime outside that of its applicability to individual realizations. Indeed, no counterexample to its validity seems available within the scope of present approximate solutions, and the result may prove approximately valid in general.6 I t remains, however, to establish conclusively, in terms of the perti­nent parameters, the extent of that regime and the corresponding accuracy of approximation.

Ray optics has statistical validity, in particular, within the range of applicability of first-order perturbation theory,7 as can be shown immediately from the work of Tatarski.6,8 His work and the inference of statistical validity apply, in fact, where the wave equation is treated by the Rytov method, which aspires to validity

also outside the perturbation regime.9 In view of the cogent criticism of HS on this latter point, however, it has not been demonstrated that statistical ray optics has this broader applicability.

Ray optics has statistical validity in the following further instance.10 Suppose the wave at z = 0 has already undergone propagation through a random medium, so that ƒ(0)[=A*(p2,0) ×A (p1,0)] depends on p1, and p2 [but with A (p,z) laterally sta­tistically stationary], and suppose the index fluctuations vanish for all s > 0 ; then taking expectations in Eq. (3) yields

i.e., the coherence function is preserved in vacuum propagation. † A subsidiary of Control Data Corporation. * This research was partially sponsored by the Electronic Technology

Division, Air Force Avionics Laboratory, Research and Technology Divi­sion, Air Force Systems Command, Wright-Patterson Air Force Base, Ohio, under contract AF33(615)-1973. 1 R. Iî. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964); hereafter referred to as HS. 2 In the actual turbulent atmosphere, the statistical average may be replaced by a time average over the fluctuations of the index field. 3 These approximations include most notably a sagittal approximation for whose validity it suffices that

where z is the maximum distance to an inhomogeneity along a ray from the field point, l0 is the microscale, and λ( =2π/k) the wavelength. It is assumed from the start that λ≪lo. V. I. Tatarski, Wave Propagation in a Turbulent Medium, translated by R. A. Silverman (McGraw-Hill Book Company, Inc., New York, 1961), p. 127.

4 The suggestion to derive Eq. (4) in this fashion is due to Dr. Herbert Steinberg. 5 Setting pi=p2, we see that Eq. (4) does imply (|A(z)|2) = l, i.e., the expectation of the field intensity is unaffected by the random medium. This result represents the energy conservation theorem for the process. If we permit complex, as well as real, refractive indices, corresponding to ab­sorption, Eq. (4) has φ2 replaced by φ2*, and the conservation theorem then involves the imaginary part of φ.

6 1 am indebted to Dr. Hufnagel for correcting my impression on this point, with particular reference to the perturbation approximation men­tioned below. Dr. Hufnagel is understood to be working on evaluation of the first term of Eq. (4) under more general conditions. 7 For a discussion of the perturbation approach, see, for example, HS, p. 54. Equations (3.6) and (3.8) of this reference contain an excess factor —i in the second terms. 8 The pertinent basic result of Tatarski (see Ref. 3), in his notation, is that

FA (K.O) +Fs(K.O) =2πk2LФn(K); this result is given by his Eqs. (7.50), (7.51). [These require that L≫p, where L is path length in the random medium and p ( = | p1 – P2 |) the separation argument in the coherence func­tion, and are trivially generalizable to the case of a medium that is only locally homogeneous.]

9 Dr. Hufnagel (unpublished paper). 10 This implication of Eq. (3) was called to my attention by Dr. Hufnagel.

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