Generalized coherence functions for propagation in a random medium

Abstract

We have derived the two-source, two-frequency mutual coherence functions for a spherical wave propagating in a random medium for the case when the receivers do not necessarily lie in the same transverse plane.

Full text

1326 J. Opt. Soc. Am. A/Vol. 3, No. 8/August 1986
Generalized coherence functions for propagation in a
random medium
Ronald L. Fante
Avco Systems/Textron, Wilmington, Massachusetts 01887
Received October 26, 1985; accepted April 9, 1986
We have derived the two-source, two-frequency mutual coherence functions for a spherical wave propagating in a
random medium for the case when the receivers do not necessarily lie in the same transverse plane.
It is now well known",2that the electric field produced by a
planar source located in a weakly fluctuating random medi-
um can be calculated by employing the extended Huygens-
Fresnel principle. That is, for a medium in which the fluc-
tuations in the index of refraction are small in comparison
with the mean index and in which the spatial size of all
random inhomogeneities is large in comparison with a wave-
length, the electric field at a point (xo, yo, z) due to a source
field E(xl, y, 0) is given by (see Fig. 1)
E(po, zo; k) = Pi E(pl, 0)
X exp [i
2z (Po -p) 2+ .(po, Z0, P10; k)
(1)
where it is assumed that zo2
>> (p -p) 2.Also, po -- (xo, yo),
P1 (xi, Yl), k = 2r/X is the wavelength, and A
-- X(Po,
zo, p1,
0;
k) + iS(po, zo, p, 0;
k) is the additional complex phase (due to
the random medium) of a spherical wave propagating from
(p1,
0) to (po, z). It can be shown that 3
I dz' II d2p' nl(x', y',
27r do .z (z -Z')
r oik
( -p,)2(p
-p) 2(Po
-p) 211
X exp. ~
I+ / (2)
2 [z - z' Z/ J
where n(x', y', z') is the fluctuation in the index of refrac-
tion. It is found that Eq. (2) is valid, provided that (x 2) << 1,
where ( ) denotes an ensemble average. In this same do-
main of validity, x and S are Gaussian random variables.
The generalized mutual coherence function
(E(po z, k)E*(p/o, z'O, k'))
and other moments of the field can be computed by using
Eq. (1) along
with the fact that x and S are Gaussian random
variables. Consequently, all moments can be expressed in
terms of (xx'), (SS'), (xS'), and (x'S), where (xx') (x(po,
zo,
p, 0; k)x(p'o, z'o, p'l, 0, k')), etc. and the points (po,
z),
(p'o, z'o), etc. are shown in Fig. 1.
The purpose of this paper is to present the results for the
generalized two-source, two-frequency spherical-wave co-
herence functions (xx'), (SS'), etc. By employing Eq. (2)
and paralleling previous4analyses, it is readily shown (the
details are very lengthy and will be omitted) that, provided
that (x 2) << 1, kzo >>
1, k'zo >> 1, klo >>
1, and k'0>>
1, where
lo is the smallest size (often called the inner scale) of the
random inhomogeneities, the desired coherence functions
are
(XX') ' 22kk' Idv KdK4(I, )J0(Ka)
(5'{j L L(
X K2V 1/ 1/ V\
yOS
-[-1 ---- u
2
k' ZI k~ zO
FCSK20
[
V 1_V)+( V )]
{(SX')} 2 JZ°
{(SW) =7 Zrkk' fo
dv fo KdK'P(, 0)JO(Ka)
X in- -1- -11-- i
{ 2 k'( Z'oJ k ( Zo!]
-- snK2
V
[ 1-V ) 1_V )
+ sin
where it has been assumed that z'o 2 z. Also,
a Z0
--)+Pi P
(3)
(4)
(5)
J(.. .) is the zero-order Bessel function, and (K, 0) = (K,
Ky,
0), where (Kx,, Ky, K) is the wave-number spectrum of the
index-of-refraction fluctuations. The structure functions,
DXX
= ((X -x')2), etc., can be obtained directly from Eqs.
(3) and (4).
For Kolmogorov turbulence
VW ) 0.033Cn 2(v)exp(-K 2/Km2)
[, 2+ L -2]11/6
where Lo is the outer scale size of the random inhomogenei-
ties, C2is the index-of-refraction structure constant, and Km
is related to the inner scale lo as Km = 5 9 1/1o. When c is
0740-3232/86/081326-02$02.00 © 1986 Optical Society of America
Ronald L. Fante
WI