66 IEEE TRANSACTIONS ON A P I ~ N N A S AND PROPAGATION, VOL. AP-15, NO. 1, JANUARY 1967

Propagation of a Spherically Symmetric Mutual a

Coherence Function Through a Random Medium MARK J. BERAN

Abstract-In this paper, an approximate solution obtained by itera- tion is given for the development of a spherically symmetric mutual co- herence function, representing radiation propagating through a statisti- cally homogeneous and isotropic random medium. Only small angle scat- tering about the radial propagation direction is considered. The results are compared to the problem where propagation is principally in a single direction.

H I . INTRODUCTION

N A PREVIOUS paper by Beran [2 ] , a solution was ob- tained by iteration for the development of the mutual coherence function representing radiation propagating

in a half space (z>O) filled with a statistically homogeneous and isotropic random medium. The principal propagation direction was along the positive z axis, and the mutual co- herence function r(xl, x2, 7) depended on only, ( X ~ - X ~ ) ~ f (y1-y~)~~ z1 and z1-22. In this paper, we use a similar iteration procedure to find r(xl, x2, T ) in a statistically homo- geneous and isotropic medium for radiation that initially has a mutual coherence function that is spherically sym- metric. By spherically symmetric we mean here a mutual coherence function that depends on the difference of the angular coordinates only, for fixed radial distance. An ex- ample of such a radiation field would be radiation from a point source. The results of this calculation will be compared to the problem mentioned above.

We will consider only small angle scattering, and, thus, scalar theory will be adequate. The principal propagation direction will be in the positive r direction. The analytic signal V(xl, t) associated with the real field vT(xl, I) satisfies the wave equation in each member of an ensemble represent- ing the random media. That is, we have

.'(x) d*V(X, 1) VV(X, t ) = -

C2 at2

where nz(x) is a function of x in each member of the ensemble. The mutual coherence function is defined as

r(xl, x2, 7) = (v(xl, t + 7 > v * ( ~ 2 , t ) ) (2)

where the brackets indicate a time average. We shall be con- cerned with finding the mutual coherence function after it is averaged over all members of the ensemble. Denoting this ensemble averaged function by { r(xl, x2, 7) 1, it may be shown (see Beran and Parrent [1], ch. 6 ) that { r(xl, x2, T))

Manuscript received April 15, 1966; revised August 10, 1966. The author is with the Tome School, University of Pennsylvania,

Philadelphia, Pa.; he is also Consultant to Technical Operations Re- search, Burlington, Mass.

satisfies the following equation

where

."x) = { n"x) ] + n2'(x)

1202 2 {n"x>f. (4)

(Note that if n(x)=rl+n'(x), then n2'(x)= 2&.) In this paper, we shall assume that the fluctuations in n'(x) about no2 are very small.

Equation (3) is indeterminate as formulated, but it may be utilized if we assume the mean square value of n2'(x) is small and an iteration procedure is employed. Suppose that { r(xl, x2, 7) ] is given initially over some surface r = ro. We seek to find { r(xl, x2,7>) on some surface r = r , (r,>ro). To accomplish this we imagine the space between r = ro and I' = r, to be subdivided by n- 1 spherical surfaces at positions ~ ' ~ = r ~ + A r , r2=ro+2Ar, . . . , rn-l=~,,+(n-l)(Ar). The dis- tance Ar must be taken large compared to the largest scale of inhomogeneities associated with the random medium h1, but small enough to meet the condition that { r(xl, x2, 7 ) } is not changed appreciably in the interval Ar, as a result of the scattering by the random medium. In addition, we shall only be interested in those cases where rn/Ar>>l.

In the interval r j l rSrj+l we suppose that the term on the right-hand side of (3) may be approximated in the following manner

where { rrj(xl, x2, T)) is the solution of (3) in this interval in the absence of scattering. The assumption here is that {r(xl, xp, T ) ] changes little in the interval Ar, as a result of scattering. Thus, in the term on the right-hand side, which represents an effective source for the scattered radiation, the coherence function may be represented by its unscat- tered value. Essentially, this approximation neglects multiple scattering in this interval and results from our assumption of small fluctuations in n2(x).

The governing equation in the interval rj< r<rj+l is thus

BERAN: PROPAGATION OF MUTUAL COHERENCE FUNCTION 67

where

u(x1, x2) = { n 2 ' ( X l ) n 2 ' ( X 2 ) } .

To find { r(x1, x2, T ) ) I we use (6) to find { r(xl, x2, T I ] I p T l in terms of {r(xl, x P , ~ ) } I + = T o ; { r(xl, x2, T I } I wr2 in terms of { r(xl, x2, T ) } I and so on until we find { I'(xl, xz, T)} I ,=T,, in terms of { r(xl, x 2 , T ) ) I p r n - l .

We shall consider only spherically symmetric solutions so that { r ( x l , X P , T ) } = { r(rl, 81, 4 1 ; r2, 02, h ; T ) } (where r, 8, 4 are spherical coordinates) will only be a function of rla = r2 -r4 T I , and the difference in angular coordinates. In this paper we shall assume that any coherence lengths resulting from scattering of the radiation will be very small compared to Ar and, hence, the arc length As at constant r may be approximated by

\

As12 = r2(& - 8 1 ) ~ + rz sin2 81(& - q51)2.

Thus we assume

{ r(xl, x2, T ) ] = { r h , rl, AS^^, 1. To be consistent with the above assumptions we must as-

sume that the statistics of the n"x) are homogeneous and isotropic. Thus we have

XI, ~ 2 ) = U ( R 1 z )

where

R12 = x z - XI; R12 = I R l z I . I The assumption of small-angle scattering demands that

(XJlm) <<1 where X, is a characteristic wavelength and I, is the minimum scale associated with the random inhomogene- ities. In order to obtain a solution, we will also require that ( X C l M / h 2 ) <<I.

11. SOLUTION OF EQUATIONS To solve (6) it is conveneint to consider the Fourier trans-

form of { r(x1, x2, T ) } defined as m

I ?(x1, x2, 1 = J r(xl, x?, 1 e 2 a w 7 . (7) -m

{ f ( x l , xz, v)] satisfies the equation

(v12 + k n 2 ) (vZ2 + L a ) { f(xl, ~ 2 , V) } = k40(x1, X J { f r j ( x l , x?, v)) (interval rj < r < r j + ~ ) ;

k,2 = . (8) n02m

To find { r(x1, x2, T) 1 from { f(x1, x2, v) } we need only invert (7). For the spherically symmetric case, { f ( x l , x z , v)} = { f(r1, r12, AsE, v) 1. For convenience, we will let n o = 1 in the remainder of the paper.

{ f'(r1, rE, AsE, v)) changes as the radiation propagates

from rl=rj to rl=ri+l for two reasons: the radiation is diverging, and the radiation is being scattered. The diver- gence effect occurs in the absence of a random media and results simply from the condition of spherical symmetry. It is absent in the case of propagation that is principally along the z direction. We shall calculate the two effects separately. To include the diverging effect we must choose the proper homogeneous solution to (8). The scattering is represented in the particular solution.

A . Homogeneous Solution We seek a spherically symmetric solution of the homo-

geneous equation

(VI2 + k2)(V22 + k') { f a ( x 1 , X ? , v ) } = 0. (9)

(V22 + k2) { f H ( X 1 , X') v) } = 0. (10)

To solve this equation we first seek a solution of the equation

A formal solution of this equation in the interval rj < r < rJ+l is (see Stratton [3])'

.cos m(42 - 42s) { f (XSj , , xs j2 , v) 1 (11)

where xsj i is a point on the spherical surface r=r , with angular coordinates dis and 8;s. P,," is the associated Le- gendre polynomial and em is the Neumann factor for which en= 1, en=2 (n>O). A,(*) is the Hankel function of the fist kind.

When krj3>1, both h,(l)(krz) and h,(l)(krj) may be ap- proximated by an exponential form if kri>n. We have

1 hn(l)(kr) - (-i)n+IeLkr.

kr

If terms beyond n = krj are neghgible, (12) may be substi- tuted in (1 1) and we find

(12)

{ ~ B ( X S ; ~ , XP, v)) = - eGt(n-n){ f(xsjl, XS,.;, v) 1 (13)

where xs j : is defined so that xq and xsjpl have the same angular coordinates. This means that in the approxima- tion krj>>l the coherence between a point on the surface xsjl and an exterior point x2 is the same as (rj/r2)eck(fl-rj) times the coherence between the two surface points xsjl and xsjpI. kri>>l is a geometric optics-type approximation and the result is not surprising.

Equation (12) may be used in (1 1) for hncl)(krz) and kc1)(krj), provided that ff(xsjl, xSj2, v)} does not have

ri r2

1 Note that rz lies between ri and r,+l. Strictly speaking, we should use a different symbol to distinguish this coordinate from the coordinate denoting the spherical surface rz. We felt, however, that it was unneces- sary to do this, since it is clear from the context what is meant.

68 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION JANUARY

characteristic correlation lengths of the order of a wave- length. For correlation lengths that are large compared to a wavelength, the terms in the (1 1 ) summation, for which n> krj, are negligible. This may be seen by expanding { f'xsjl, x s j n , Y) 1 in spherical harmonics over the surface r = r j . In this expansion only terms with n values less than rj/Is will be significant, Is being the minimum characteristic correlation length associated with { f(xsj,, xsj2, Y)), Since the spherical harmonics are orthogonal there will be neg- ligible contribution to ( 1 1 ) when n- krj, since Isk>>l.

We will assume here that lsk>>l and this will be borne out by the consistency of the solution obtained. Thus (13) will be considered a satisfactory solution to (10). To fmd the complete solution to (9) we may now write it in the form

( 0 2 ' + k2)(V12 + k2) { PH(XI, X Z , Y)} = 0 (14)

and seek a solution of the equation

(V12 + k2) { FH(X1, x2, v) ] = 0. (15)

Instead, however, of using {fII(.sj,, xsj2, v)) as a bound- ary condition to solve (15), we now use {f(xSjl, x2, v) }, which is given by (13). Proceeding exactly as before we find that

rj2

w - 2 { f ~ ( x 1 , XZ, Y)} = - e'"*?){ f(xsj;, xsj,I, Y) 1. (16)

The solution to (15) may be made somewhat more gen- eral than that given in (16). A solution of the form

{ MX1, xa, 1 - ~ ( ~ ~ 1 2.- c i k ( r 2 - r , )

.2 - I f(XS,.,I, xsj2/, .) 1 (17)

r is possible. In the absence of scattering resulting from a random medium, enera conservation demands (in this geometric optics-type approximation)

XI, v)} = { f(x1, XI, v)} = - {f(~sj;, VI) ; (18) rj2

that is, the intensity is reduced because the area of the spherical surface at rl is greater than at rj . In this case $'(xz)= 1. When, however, there is scattering, F(xz) must have a different value. This is necessary if we are to account for the radiation energy lost to the primary radiation pass- ing through the surface r = r j as it travels from r = r j to r= rj+1. F(x) will be explicitly calculated in Section 11-C.

B. Particular Solution The particular solution of (8) may be written (see Beran

and Parrent [ l ] )

1 f,h, x21 1 k4

- -- L,, L , -

(4n)Z exp { -ik [R(x? , xp-.) - I Z ( X ~ , xp)] 1

R(x1, xv,)R(xz, XP.) c(xPj xF*.>

' { P,(XV, X V , v) ] d x p dxv. (19)

V represents the volume between the spherical surfaces rj and rj+l.

take into account the spherical nature of the solution in the interval jAr< r<( f+ 1)Ar. When, however, we use the condition (r,/Ar)>>l in addition to the small scattering assumption, then in most of the intervals we may approxi- mate the shell jAr<r<(j+ 1)Ar by two parallel planes for purposes of integration in (19). This integral has been evaluated in Beran [2], and previously in Tatarski [4]. The result is, in ( I , 0, 4) coordinates,

Solving the integral given in (19) is very difficult if we =

{ FP(X~, XZ, Y)) = k21.'Z(Asjlz){ ~ ( x s , ~ ' , X S ~ ~ I , Y)} (20)

where r'=r-jAr and

+ (Asj12)2)dr12.

Note h j x = { I xsjl'-xsj2'l 1 Remember that xsji' is deiined SO that xi and xsji' have the same angular coordi- nates. I

TO insure conservation of energy for small angle scatter- ing we must have, for x1=xZ=x,

1 Tj2 t h , v) } = - r2 { f(xsJ} (22)

and thus using (21)

rj2 rj2 - r2 = F(r) - r2 + k%-'z(o).

Therefore,

F(r ) = 1 - kZr'Z(0) - ' rj2

(23)

where the angular coordinates of xl and xz are the same at both r= jAr and r= (j+ 1)Ar. Asllz has the same angular difference coordinates as xl-x2. 1

1967 BERAN: PROPAGATION OF MUTUAL COHERENCE FUNCTION 69

To make the angular and radial dependencies more explicit we will now write

where cr)12=sp12/rp. Thus, in this notation we have

* { f(rj, WZ, ] . (25)

The relation between { f(rn, w ~ ~ , Y)} and { f(ro, w12, v)) is obtained from (25). We have

{ f (rn, ~ 1 2 , Y) 1 = + k2Ar[5(r,-lw12) - i(O)]]

This product may be evaluated approximately when (Ar/v) <<1 by letting Ar4. In this case (25) becomes

,im [I f(ri+1, ~ 1 2 , 1 - I f(rj, ~ 1 2 , ] Ar-0 Ar 1

or

The solution of (28) is

Oexp (k‘ [ o r [5(r’wlz) - O(O)]d#) . (29)

f r(r, WU, T ) } may be obtained from { f ( r , oU, v)} by Fourier inversion.

111. CO~ARISON OF SOLUTION TO PREVIOUS RESULTS

In Beran [2] the problem of radiation propagating prin- cipally along the z axis was treated. Here { f(xl, x2, v)] was taken to depend on { f(zlz, z, p ~ , Y) 1 where pU= (x1-xz)2 +(y~--y~)~. The result for ZE=O was

{ f ( z , plz, Y) = { f(0, p12, v) ] es2z(Q(p~2)-~(o)) (30)

which is very similar to (29).

If we set

g12 = 0

Pi2 = x12 = rw12 z = r

then we see the term

k2 lor [ 0 ( x12 ;) - ;(O)] dr‘

has a smaller negative value than the term

k2(r - ro) [ ~ ( x d - ~(0) J for fixed value of xE. Thus, the coherence loss because of propagation through the random medium is less in the spherical problem. In the spherical problem there is, moreover, a gain in coherence resulting from the spherical expansion of the wave.

For example, suppose

where A = { [ ~ P ( X ~ ) ] ~ 1. Then

where B = 2aA. Using (31) in (30) and (29), we find (when r>>ro)

(propagation principally in the z direction, Y = z).

(spherically-symmetric problem). (33)

The ratio of the exponential terms is

a factor which is always less than one. Hence, the expres- sion in (33) decays more slowly than the expression in (32).

The application of this type of formalism to atmospheric propagation was discussed in Beran [2]. The reader is referred to this paper, since (29) has essentially the same range of applicability as (30).

REFERENCES [l] M. Beran and G. Parrent, Jr., Theory of Partial Coherence. Engle-

wood Cliffs, N. J. : Prentice-Hall, 1964. [2] M. Beran, “Propagation of the mutual coherence function through

random media,” to be published in J. Opt. Soc. Am., 1966. [3] J. Stratton, Electrompzefic Theory. New York: McGraw-Hill,

1941. [4] V. Tatarski, Waoe Propugution in u Twbdent Medium. New York:

McGraw-Hill, 1961.