November 1966

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA VOLUME 56, NUMBER 11 NOVEMBER 1966

Propagation of the Mutual Coherence Function Through Random Media*

MARK J. BERANTowne School, University of Pennsylvania, Philadelphia, Pennsylvania 19104

and Stanford Electronics Laboratories, Stanford, California(Received 11 March 1966)

In this paper an approximate solution is given for the development of the ensemble-averaged mutual-coherence function {r (X1,X2,T)) as it propagates through statistically homogeneous and isotropic randommedia. Only small-angle scattering about the principal propagation direction z is considered and it is assumedthat {r(xi,x 2,T)} is a function of [(Xi-X2)

2+(yi-ys,)f2]

1 2, zI-Z 2 , and al. Under these conditions, it is

possible to solve the governing equations using an iteration procedure. The solution is valid for long pathlengths. The results are compared to the results given in Chernov, and Tatarski under those conditionswhere it is appropriate to do so.

INDEX HEADINGS: Coherence; Scattering; Refractive index; Atmospheric optics.

A. INTRODUCTION

A CONSIDERABLE amount of analysis'-3 hasbeen performed on the propagation of scalar

waves in statistically homogeneous and isotropic media

* Work supported by AF 04(695)-536 and Army ResearchOffice Contract DA-30-31-124(D)-340.

' L. Chernov, Wave Propagation in a Random Medium (McGraw-Hill Book Co., Inc., New York, 1960).

2V. Tatarski, Wave Propagation in a Turbzlenit Medium (Mc-Graw-Hill Book Co., Inc., New York, 1961).

3V. Tatarski, in International Symposium on the Pine-ScaleStructure of the Atmosphere, Moscow, 1965.

by considering the wave of the form

it(x,)=A (x) expEiS(x)-iwifl.

The correlation functions

BA(XIX2)= {X(X1)X(X2)},

where X(x) = logA (x) and

Bsaxlx2 )= {S(XI)S(X2)l,

are the quantities usually studied. The { } bracketsindicate an ensemble average. The purpose of this paper

1475

46 MARK J. BERAN

is to study instead the propagation of the ensemble-averaged mutual-coherence function {r(PIXP2 ,rM, de-fined as

{ r (xi,x 2,r)) = {(V(xi, + T) V* (X2,/))), (1)

where the brackets ( ) indicate a time average in eachmember of the ensemble. V(x,/) is the analytic signalassociated with the real field Vr(x,t). XVe solve for(r(xjx 2,r)) using an iterative procedure and comparethe results to BA(xlx 2) and Bs(xl,X2 ) under thoseconditions where it is possible to do so. The solutionobtained is valid for large path lengths.

The function Vr(x,s) satisfies the wave equation ineach member of the ensemble. Thus

V2 Vr(x,t) = [n2 (x)/c2][02 Vr(xO/otl2, (2)

where Z2(x) is a function of x in each member of theensemble. We term this function the index of refraction.From Eq. (2) and the definition given in Eq. (1) itmay be shown4 that the following equation governs{fr(Xx 2 ,r))

[V 12 -?(nj/c 2

) (02

/8r2)]

x: [V 2 - (nO/C2

) (02

/0r2

)Jfr (Xi,X 2,))}

- {n 212(xi)n2 '(x0 )(1/c 2) (04/0r4)P (X:,X2 ,r)), (3)

where we have written

n2

(X) = n102

J nl (X)

{ n2 (x)} = n 2.

(Note that if n(x) = fi+n'(x) then n2t(x) = Man'.)Equation (3) is indeterminate as it is given and some

assumptions are necessary in order to allow us to findr(PxIx 2,r)} from Eq. (3).To state the type of problem we consider, we suppose

the random medium fills the upper half-space (z>0) ineach member of the ensemble. As a boundary condition(r(xl,x 2,r)7 is specified over the plane z=0 and theproblem is to determine (r(xix 2,r)) when xi and x2lie in the upper half-space. We consider here only thecase when (r(xi,yi,zi; x2,y2,z2; r)} J z=z2=o is of the form(I r(r 12 ,,0,rT)J, where X12 X2 -X 1, Y12= Y2yi, andr12

2 =x122+y12 2. Since we have stated that we consider

only homogeneous and isotropic media, the generalsolution is of the form

(r~r12,Z12,Z1,7')j,

where Z1 2 =Z2-zl. The special case of a plane mono-chromatic wave impinging on a random medium isgiven by the boundary condition

{(ra2 ,0,Or)= e-2ivr

'Mark Beran and G. B. Parrent, Jr., Theory of Partial Co-herence (Prentice-Hall, Inc., Englewood Cliffs, NT. J., 1964),Ch. 6.

We further restrict ouTselves to the case where thesmallest scale of the inhomogeneities I is large comparedto a characteristic wavelength X, and the index-of-re-fraction deviations are small. That is, X/«<<1 and{ (nI2 )2}/(,,o 2)2<<k . Further, we are only interested indistances z that are very large compared to 1,1., thelargest scale associated with the inhomogeneities, where6, satisfies the inequality XIAI/l 2K<l.

The assumption X/K1<l is usually made when con-sidering the propagation of light through the atmos-phere. It follows from this assumption that the scatteredlight is confined to a small solid angle about the z axisand considerable mathematical simplification followsfrom this condition. In the propagation of sound, it ispossible to use scalar theory even if the conditionX/l<<1 is not met. For light propagation, however, it isinappropriate to use scalar theory unless the scatteringis confined to small angles.

B. ITERATIVE PROCEDURE

We solve Eq. (3) by the following iterative procedure:Divide the region 0 to L by a series of M-1 infiniteplanes located at the z coordinates Az, 2Az, **(M-1)Az, where Az=L/M. The number M is chosenlarge enough so that in the interval jAz to (j+1)Az,(r(x1,x 2 ,r)} is governed by the approximate equation

[V 12

-(noi/c2

) (2/0r2)]

X [V22 - (nO2/c 2) (a2/a02)]{Ir (xx,x2,r))

=-{X (X1)X2'(X2)) (1/C 2) (0 4 /4')(P{A. 2 (lxbX 2 ,r)}, (4)

whererJA2(xlx 2,r)

is the solution of Eq. (4) in the interval jAz <z< (j+1)Az, when there is no scattering in this interval.

The minimum condition on M, which we determinefrom the solution, is that for each spectral component

Azk2/4 f u(O,O,Zl2)dzl 2 = Azk

2j(0,0)<<1,

where

Az= (L/M)

of (X12) {n2'(x)n 2' (X2)}, (X12 = X2 - X1)

and k is the wavenumber. We also require I4/AzK<l.To find {r(xbx2,r), given {ro(X1,X2,r)}, we first

solve Eq. (4) for {rP2 (xix2,7)}. Using (T'AZ(xl,x2,r)) asa boundary condition, we then use Eq. (4) to find{ r2An (X1,X 2,r) }. Continuing this procedure until z = MAz,we find (r(xi,x 2,r)). As the reader will see, the require-ment of spatial homogeneity yields solutions in eachinterval jAz to (j+1)Az that are identical in form.This allows us to eliminate easily the intermediatestages of the solution and find {r(xs,x2 ,r)} in terms of{ro(X1,x2 ,r))-

1476 Vol. 56

November1966 PROPAGATION OF COHERENCE THROUGH RANDOM MEDIA 1477

C. SOLUTION

It is easier to work with the Fourier transform of{r(xi,x2 ,r)}, called the mutual power spectrum anddenoted by {f'(xI,x2 ,v)}. It is defined by the relation

{P (x1,x 2,)) = Jr (Xl,X 2 , T)3 exp k27riv r)dT. (5)

From Eq. (4) we find that the function {t(x 1 ,x2,v)}satisfies the equation [for the interval jAz to (j+1)Az]

(VW2 +kn2) (V 2 2+kn2){P(x 1,x2 ,V)}

= k4 a-(Xlx 2 )(rFAz(xlx 2 ,v)}, (6)where

kn2= (no2.v/c)2; k2= (27p/C)2

.

For convenience, we take no= 1 in the remainder of thepaper.

We perform the iteration procedure described aboveusing {t(xl,x2 ,v)} and formally invert Eq. (5) at the con-clusion of the calculation. We first calculate the scat-tered portion of (f(xbx2 ,V)}, denoted by (rF(xbx2 ,v)),in the interval jAz to (j±l)Az. Using the free-spaceGreen's functions, we find {f'(xl,x2 ,v)} from Eq. (6).

We have

k4 r exp{-ik[R(x2,xv,)-R(xl,xv,)]}

=- I(4 J V)2 J iv, R(x1 ,xv')R(x 2 1xv8 )

Xo(xvxv',X) { PjA(xv' ,Xv , v)}dxv'dxv', (7)

in the interval jAz to (j+1)Az. Here the integrationover V' and V" is over the space contained between theplanes z= jAz and z= (j+± 1)zz. R(x,xv) is the distancefrom the observation point x to a point xv in thescattering volume.

We consider here only the case of small-angle scatter-ing and Eq. (7) simplifies a great deal. We expand

R(x,xv)= [(Z-ZV)2 + (x-xv) 2 + (y-yv) 2 ]1

and find

R(x,xv) (zzv)+2[(xx v)2+ (y-yV)2 ]/ (Z-Zv), (8)

since I z-zvl>>[(x-xV)2+ (y-yV) 2]k. The neglectedterms in kR are of order AzX 3/14. Hence Az<<14/X3 mustbe satisfied.

Substituting Eq. (8) into Eq. (7) yields

k4___r exp~ik(zv--zv')]{rSX1,X2 V))}==-exp[ik#Z1-Z2)1J

(4,r)2 I vt (ZI-ZV')(Z2-ZV")

ik f[(xl-xv)'+ (yi- yv,)2] :(x2 - xv")2 + (y2-yV")2]

Xe xp- 2- (xv ,xv",) ({ jiz(xv',xv, v) }ldxvdxv,,. (9)

It is possible to find the solution of Eq. (9) by comparing it to known solutions.2 We proceed directly, however,since the solution is easily obtained.

We rewrite Eq. (9) in terms of the variables

(10)prXvo.

In terms of these variables, we have

=k4 epk(z-~)f exp (iks,,)

J(X1,X2,4)} =- exp[i-(zS-Z2)]zz)(47r)2 (Z1- Pz) (Z2-Sz- pz)

ikr (xl-p.)2+ (y1- po)2 (X2- SZ-p )2+ (Y2-Sy- py)2

X<e xpa- -2 - (Z1-Adz,) (Z2-Sz-pz) -

, (s){I jA3 (s,p,v))dsdp, (11)

where {f'(xix 2 ,v)j depends upon X12, Y12, Z12 and z; (x12=x 2-xi). The Z12 dependence in small-angle scatteringover a characteristic distance IM is simply exp(-ikzi 2 ). Thus, for purposes of evaluating Eq. (11) we need onlyknow {(fiA(s,,s,,0,p 2,v)}. Similarly we are interested in calculating {t'8 (x 1,x2 ,v)) only when Z1=Z2. We also notethat in the small-angle approximation the exponential term

(X2-S.-p.) 2 + (y2S,_py) 2 ]

2 L (Z 2 -Sz-pZ)

may be replaced by

ik (X2-S.-p.)2 + (y2S-8pY)2

2 L (Z2-pz)

MARK J. BERAN

since the neglected term is of order k(x 2 -s.-p )2

S2 /z2 The term (X2-Sl-zP) is of order (X/sz)z2 and thus

k X-=-pe)252/IZ2= (X/5z,) (Sls<<

In addition, (Z2 -s 2 -p 3 ) may be replaced by (z2 -pz) in the denominator of the integrand.Equation (11) thus becomes

k4 P expx(iksa)

(P., (X,2 ) (47r) J~2 (Z- p.)2 0's{j-SP })

Xdsdpzj f[exp {- [(x-p0)2 - (X2-Pc -S)2+ (yI pu) 2 Y2-Py-s )23 } ]dpdp. 12)

In the integral over px, the terms in p1/ and p,2 cancel out when the exponential is expanded. The result of

integration over px and p, is thus

(27r)a/k2i o(Z sp)26 (SaXd2)6 (S-Yl2) -

rntegrating over s. and s, then yields

{t,(XIx2 ,Pv)} I z2=2 z= J l| (Xl2,yl2,s3)exp (iks3 ) { Paz (xI2,y12,s.,pz, v)} dsdpz.

Integration over pz yields (since tja, is independent of p,)

(P., xx2,v) Jf = z= - j a (X12,y12,S 3 ) eXp (iksz)) (PjA(Xl2,yI2,s 2,z,P)}dsz, (jAzz<z< (j+ 1)Az),4M

t13)

where z'=z-jAz.Finally, we let

{PjAz(X12,y12,SzZv)PI = exp(-ikes){(ta 2z(xI2,yl 2,0,z,v) .

This form is permissible since in Eq. (14) only valuesof szl'M contribute to the integral. In this distance,significant diffraction does not occur. This form gives5

(P8 (Xu2)y12,O,zv)j =k 2z'( PjAz(x12,Ys2,O z, }a )}(x2yYi2),

(jAz<z< (j+l)Az), (15)where

1 (XI2,yI2) = d- dsza(x 2 ,yi 2 ,Sz)-

To find the entire function P(x2,Y12 ,0,z,v) in thisinterval we note that the intensity of the scatteredradiation ("at frequency v") iso

{tf(O,O,Oz,v)} =k2z(FrjA.(o,o,o,z,v}.@r(OO). (16)

Since the scattering is confined to small angles, con-servation of energy demands that

(t(O,O,O,z,v)} = {Pj z(O,O,Oz,v)I. (17)

Hence the intensity of the unscattered radiation,

'Since we assume a(aly2,Y1) =a(r,2) and {I'O(X,2,Y12,0,0,O)}= Pro(rl2,OOv) I then f rPa,(xl2,y2,O,O,V) } = {rjj,-(r12,0,0,v) }. Wewill for the most part continue to display X12 and yi2 separately,to exhibit the full three-dimensional nature of the problem.

S since we are using analytic signals this is actually twice theintensity for real fields.

(14)

denoted by (t,(0,0,0,z,v)}, is at position z,

(P{Az(O,O,O,z,v)) [1-k2z'6(0,0)].

This result has been given by a number of writers.In the statistically homogeneous problem {(PA1 (x12 ,

y12,0,z,v)} may be derived by considering the superposi-tion of an angular spectrum of incoherent plane waves.For small-angle scattering, the power in each plane waveis reduced by the same amount and hence we have

[Pu(XI2jY12)0)ZO)P)

= {rja3 (xi2,yi2,0,zv)})[ l-k 2z'5(O,O)J. (18)

Next we note that

(P) =

since, when Az/LvP>1, the scattered and unscatteredwaves are uncorrelated except for energy conservation.The only correlation that could be introduced wouldbe in the neighborhood of the scattering volume. WhenAz/la1>>, however, the scattered waves result fromthe sum of a great number of uncorrelated scatteringvolumes and hence are uncorrelated to the initial un-scattered waves. This may be proven directly by notingthat the homogeneous solution of Eq. (4) is of the form

since (rjAPg(x1,X 2,r)) depends on x1 -x2 . The completesolution to Eq. (4) is obtained by adding this term tofTI, (x1,x2,r)}, which is the particular solution. From the

1478 Vol. 56

November1966 PROPAGATION OF COHERENCE THROUGH RANDOM MEDIA 1479

energy-conservation argument just given we find that

F(z)= 1-k 2 z'&(0,O);

(t(x2iy 12,0,zv)} is thus of the form

(19){ N12vY12A0ZN)OI= {1fA.(X12,Y12,0,ZP)}

X[En+k2Z'e(X12,Y12) -k2s'0 (0,0)],

(jAZl<z< (j+1)Az)and

(P91) Az(X12,y12 ,0 + 1) Az, v)

= (jAZ&xz2yY2,0,JAZP) } [l+Aza(xi2sys2)], (20)

a (x12,y12) = k2[a(xs2,y12 )- (0,0)]

From Eq. (20) we find

= ( JO (X12,Y12,0,0,v )} [1 + (L/IM)a (Xy12)], (21)

which for large M yields approximately

(X (x2iy12 ,0,LIv)} = (t(xs 2 ,yi2 A0,O)v)}

X exp{k2L[6f(X12,Y12) -(0,0)]} .7 (22)

From Eq. (22), (r(x12,yi2,0,L,,r)} is calculated fromthe relation

({ rtX12,y2, 0 ,Lr)} = f (F (X12,Y12,O,L,v))}e 2ri,-rdr. (23)

This procedure is a good approximation only if(F'xX2,y2,0,z,r)} does not change appreciably in anydistance As. We see from Eq. (18) that this conditionis assured if k2AZj(0,0)<<1. We remember also thatlM,/Az<<l, so that strictly speaking the exponentialform can never be exact. In distances of the order of 1x,however, the coherence in the xy planes is remains essen-tially constant.

D. COMPARISON WITH TATARSKI (1961)

When k2L5(0,0)<<l, Eq. (22) yields the same resultsfor the scattered radiation as those given by Tatarski.This corresponds to the small-perturbation solution and

{t(xi,x 2 ,v)} = I[BA (xi,x2)+Bs(X1,X2)], (24)

where I is the intensity. This may be seen by notingthat the sum of Eqs. (7.62) and (7.63) in Tatarski'swork yields

BA (XlX 2 ) +Bs Xl,X2) = 4r2kLf Jo (,r12)At4&)dy, (25)

'This expression -was obtained previously by Hufnagel andStanley' without adequate proof. See Eqs. (3.5), (4.6), (5.3), and(5.10) of their paper. See Chase [D. M. Chase, J. Opt. Soc. Am.55, 1559 (1965)] for comments on the Hufnagel and Stanleyderivation.. Nte added in proof: See G. Keller [Astron. J. 58,113 (1953)] for the derivation of a similar expression using geo-metrical optics and independence arguments.

where cI,. (p) is the spectrum of the index-of-refractionfield. [In Tatarski's work

r(x12,y12,0,OO)=I;

that is, he considers a plane wave impinging on therandom medium.] It may be shown that

oXs2,yY2) =(r2) = 4rf Jo(Ar12)t'(p)pdM (26)

from the relation

&( = fjt (ri2,s,)ds2

=X dsJ db(,)exp(t.-s)dt]. (27)

We cannot compare Eq. (22) with any of the formulasin Tatarski2 since despite the claims made for theRytov method in this book, it is good for only smallperturbations. This was indicated by Hufnagel andStanley8 and emphasized by Tatarski.

In Chernov,1 Eq. (22) is derived for an incident planewave based upon the assumption of joint gaussianstatistics for log A and S. Since the Rytov approxima-tion can be shown to hold for only small fluctuations,there is no a priori justification for the assumption ofgaussian statistics for long path lengths. The derivationgiven above shows that Eq. (22) is correct whether ornot the joint statistics of log A and S are gaussian.

E. SOME REMARKS ABOUT THE SOLUTION

1. Plane-Wave Spectrum

From Eq. (24) we see that the function (t(xix 2,v)}does not give separate information about phase andamplitude. However, the concept of phase is ratherambiguous for large fluctuations and it is much moreuseful to use ((Xrx 2 ,v)) to determine the spectrum ofplane waves in any plane z= constant. This is accom-plished by taking the spatial transform of {t(x 1,x2 ,V)}.We have

{P (k.,ky,z,v)} = f exp(ikx2+ikyyl2)

X (P (x'17y12,z,v)}dxu2dyn2, (28)

where we take =12=0 and for convenience suppress theZ12 argument. For this transform relation to representa plane-wave spectrum we must have k 2=4+k+kY.By isotropy {P(k.,kyz,v)) is a function of onlykX2+kY 2= k2. The angle of arrival, 0, (0<cl) of planewaves with wave number k, is 6=k7 /k. Thus a plot of(t(kr,z,v))} vs kr/k yields the plane-wave spectrum at z.

8 R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52(1964).

MARK J. BERAN

Since {t(x12 ,yn2,z,v)} depends on only r12, we have,after some manipulation,

(29a)(f (krzv) } = 27/' Jo (krrl2) { I'rlZv) r12dr12

and

{ (kr,z,v)} = 2rf Jo (k~rl2) (P (r12,0,v)}

X exp(k2z[6(r12)-jk0)]}r12dr12. (29b)

2. Initial Plane-Wave Solution

From Eq. (22) we note again that

{r((,z,v)} I= r(0,0,v))

(again setting z1?=0 and suppressing it). Further we.see that as r12 ->

{t(o,z,v)) = {f( ,0,v)} expE-k 2 zi(0)].

This gives a nonzero solution only if (t(co,o,O)} wasnonzero, that is, if there was finite energy in a planewave traveling in some direction. For an initial planewave in the z direction, k2j(0) is the attenuationcoefficient.

For (P(r12,0,v)} =(v), the plane-wave case, we have

{t(r 12 ,Z,V)})=(v) expfk 2 z[a(r 1 2) -(0)]M .

The factorexp k 2z[J(r12) - (M)

gives the loss of coherence as a result of the passage ofthe plane wave through the random media. Sincej(r 12 )- (0) is negative, the larger the distance z theless the average coherence interval. If a(x) is known,this factor may be easily calculated.

3. Relation to Atmospheric Propagation Problem

It is appropriate to ask whether Eq. (22) is usefulfor studying propagation of visible light through theatmosphere. For convenience, let us consider horizontalpropagation to avoid having to treat the atmosphericvariations in the vertical direction. Near the earth'ssurface we may take 5 r 100 m, 1= 10 cm, n"'2 10-16.For k= 105 we find then

k2[f(0)= 102 Cm'.

The simultaneous conditions

k2

f(0)zK<<I

and

lMl/LXZ<K1

thus cannot be met. This means that if large eddies inthe atmosphere are responsible for an important partof the scattering of light waves, the analysis given inthis paper is not appropriate for studying atmosphericpropagation problems.

If we study Eq. (22), however, we see that the variouseddy sizes contribute to the coherence function throughthe exponential term

exp{k2 z[6(rl 2 ) - j (0)]

When r12 is small we see the effects of the small eddvsizes and when r12 is o forder 1M we see the effects of thelarge eddies. If jk2z[&F(r12 )-(0)]I>>1 for r12/l1,C<lthen the small eddies have a dominant effects on thecoherence function. This is true since the termk2z[j(r12)-e(0)] is negative and generally increases inmagnitude as r12 increases. In this limit the effects ofthe large eddies are suppressed by the exponentialdecay as Y12 increases.

When the large eddies are unimportant, we mayarbitrarily assume that ¢P(a) is equal to zero below somevalue pi= (I/ls). Is is a characteristic size associatedwith the small eddies. If Is is taken to be 100 cm then

k2

(0) = 104 cm1

and the conditionsk2a(0)zAz<<1

Is/Az<<l

may be met for z= 103 cm.As a last remark we should note that if the above

theory is to be used for beam-propagation studies, thenthe width of the beam must be much larger than Is.The above value of 1sz 100 cm was not meant to becharacteristic of all cases and indeed a value ls; 10 cmmay be more appropriate if z is very large. Even in thislatter case, however, we would require minimun beamdiameters of the order of 30-50 cm for the theory to beused without modification.

1480 Vol. 56