March 1968 L E T T E R S T O T H E E D I T O R 4 3 1

Propagation of the Coherence Function through Random Media

M. BERAN* Department of Electronics, The Weizmann Institute of Science, Rehovoth, Israel

(Received 5 October 1967) INDEX HEADINGS : Coherence; Inhomogeneous media; Atmospheric optics.

IN my recent paper1 I derived the relation

where Г(r12, z+∆z) is the coherence function at position z+Az and Г(r12,z) is the coherence function at position z. Here z is some arbitrary distance from the plane in which the radiation impinges on the random medium; ∆z is chosen so that Δzα(r12)≪l, where

in which σ(x) is the index-of-refraction correlation function. From this relation I concluded that, as ∆z = (z/M) → 0, the solu­tion of Eq. (1), in terms of the boundary condition, Г(r12,0), is1

In this note I want to give a detailed proof of Eq. (2), since its validity has been questioned.2 Equation (1) may be considered to be the first terms of a power-series solution in ∆z. Thus, we may write

where the Cj(r12,0) are functions of ri2 and z=0. At position z = 2∆z, we have

Using Eq. (3), we obtain from this

For position z=MAz, we find

We shall make the assumption here that the Cj(r12,n∆z) are bounded for all n∆z; in particular as n∆z∙→ ∞. The nature of this assumption will be discussed at the conclusion of this note. Subject to this assumption, we find that as M → ∞ (for fixed z, this corresponds to ∆z∙→ 0 since z=M∆z) the solution of Eq. (5) is

To see this, consider first the second-order contributions to Г(r12,M∆z) from both the term Г(r12,0)[1 + (z/M)α(r12)]M and the terms containing the Cj's. As M → ∞, we have

Thus as, M → ∞, the number of terms of order [∆zα (r12) ] 2 is proportional to M2.

On the other hand, we have

Thus, in this case, the number of terms of order [∆zα(r12)]2

is proportional to M as M → ∞. We see, therefore, that as M → ∞ the second-order contribution is given by the term Г(r12,0)[1+(z/M)α(r12)M.

Similar reasoning shows that in the case of terms of order [∆zα(r12)]k, k≪M, the number of terms contributed by the Г(r120) term is proportional to Mk while the number of terms contributed by the Cj terms is proportional to Mk-l.

If M is chosen large enough, we can, for fixed z, find the solution to any order of approximation by choosing only the terms where k « M . Therefore, Eq. (6) may be considered to be the solution of Eq. (1) for all z, to any order we choose. I t is noted that this is exactly how we would proceed to utilize Eq. (1) if we sought a numerical solution.

As I mentioned in my previous paper, the only difficulty with this procedure is that ∆z must be much greater than the maximum correlation scale associated with the random medium. This

432 L E T T E R S TO T H E E D I T O R Vol.58

method is thus suitable only as an approximation to the physical problem and its applicability depends on the magnitude of the relevant physical parameters.

Lastly, we must consider the nature of the assumption that all the Cj(r12,n∆z) remain bounded for all nAz. If, for example, Cq(r12,n∆z) grew at a rate proportional to n∆z the above proof would not hold.

We have used the relation

If a class of functions Cq(r12,n∆z) does not remain bounded as n∆z→ ∞ then for any small finite values, δ(r12) = Δzα(r12), the solution for Г [ r 1 2 , ( n + 1 ) Δ Z ] will be given by

in the limit n∆z → ∞. [We note that T(r12,n∆z) remains bounded since Г(0,n∆z) is proportional to the intensity.]

This possibility cannot be excluded, but it is pathological. I t means that in the range z = n∆z to Z = ( n + 1 ) Δ Z , (n→ ∞) the incident radiation is of such form that a small nonzero value of the perturbation parameter, ∆zα(r12), is not sufficient to insure the validity of the usual perturbation solution. Since this same radia­tion may impinge on a new random medium of similar structure, it means that even the initial perturbation solution is cast in doubt for certain classes of radiation. In fact, the commonly used plane-wave perturbation solution itself is open to question, since now it is not clear which classes of radiation will produce such an effect.

In sum then, we note that the proof given in this paper is based essentially on the assumption that below some small finite value of ∆zα(r12), the perturbation solution

provides the dominant correction for all radiation fields. The writer would like to thank the reviewer for bringing to his

attention the necessity for bounding Cj(r12,n∆z). * Fulbright fellow on leave from the Towne School, University of

Pennsylvania, Philadelphia 19104. 1 M. Beran, J. Opt. Soc. Am. 56, 1475 (1966). The exponent M was inadvertently omitted from Eq. (21), which should read as does Eq. (2) of this note.

2 L. S. Taylor, J. Opt. Soc. Am. 57, 304 (1967).