6
JOURNAL OF TIHE OPTICAL SOCIETY OF AMERICA Propagation of Mutual Coherence in Turbulent Media Using the Local-Independence Approximation* LEONARD S. TAYLORAND STEPHEN MELLAIANt Electrical Engineering Department, University of Maryland, College Park, Maryland 20742 (Received 18 April 1970) We treat the scalar wave equation in a medium with stochastic permittivity using the local-independence approximation. The equation is solved by a new method to yield an exact solution of the differential equa- tion. It is shown that this is identical to the formula for the mutual coherence that is obtained from the com- plete Born series for the wave field assuming local independence. The relation of the local-independence solution to the complete solution is discussed. INDEx HEADINGS: Inhomogeneous media; Atmospheric optics; Coherence. It is now generally recognized that the problem of the propagation and scattering of electromagnetic waves in random media is mathematically related to problems that appear in many branches of physics.'-' Among the mathematical characteristics that are shared by these problems is that they lead formally to an infinite hierarchy of equations for the statistical moments. The problem is to reduce suitably the infinite set by some truncation procedure. Historically, these trun- cation procedures have appeared, often in quite dis- guised form, under the names of various approxi- mations, methods, hypotheses, and techniques. It has often been difficult to identify the truncation procedure that has been employed and its range of validity. One of the most important of these truncation pro- cedures has been based on the assumption that the local fluctuations of the field, ,6(r), are statistically independent of the local variations of index of refraction or dielectric permittivity be(r). That is, it is assumed that (5E (r)&E (r')\6 (r) ,* (r') ) = (be (r)5,E (' (() r (r') )- This assumption truncates the hierarchy of equations, leaving a fourth-order differential equation for the propagation of mutual coherence. Our work is devoted to a study of the solution of this differential equation. The method employed in this paper depends upon a formal result of single-scatter theory that was derived by Bremmer, 4 which he called the "spectacular result" of Born theory. Because the method we use is closely related to the single-scatter formulation, we are able to obtain a result that can be identified with the com- plete Born series for the mutual coherence, using the local-independence approximation. Our final formula agrees with the formula first derived by Beran 5 ' 6 using a different method of solution of the differential equation but avoids the convergence difficulties and the assumptions 6 that were previously required. More- over, by considering the relation of the local-inde- pendence differential equation to an approximate equation for the propagator of the mutual-coherence field, 7 we are able to answer the question of why the present approach yields the same formula as the re- normalization technique used by Brown. 8 THEORY In this paper, we consider fields propagating in a medium with a randomly varying dielectric constant. In such a medium, the wave equation may be written as V-4'(r,t) = [e (r)/c2] (a 2 /at 2 )[E/i(r,t)]-s (r,t), (1) where the permittivity e(r) is a random function of position and s(r,t) represents the sources. Assume that the incident radiation consists of monochromatic plane waves with wavenumber ko in free space and that the permittivity has the form e(r) = 0 +5E(r)], (2) where e is the mean permittivity of the medium. The wave equation can be written in the form (V 2 +k 2 ) ,6(r) =-k 28e(r),6(r)-s(r), where k 2 = ko 2 . (3) (4) If we now use the operator notation L = V 2 +k 2 and rewrite Eq. (3), at the point r', take the complex conjugate, and combine, we have the equation for the propagation of the mutual coherence in the random dielectric, (5) In writing the above equation, we have assumed that no external sources are present. Note that the Green's function for the operator on the left-hand side of Eq. (3) is Go(r,r') =expik | r-r' /41r I r-r'[. (6) Thus Eq. (3) can now be written in the form of an 236 VOLUMIE 61, NUMBER 2 FEBRUARY 1971 LL'*(V, (r),P* (r')) = kl6,E (r)3,E (r')VI (r)q/* (r')).

Propagation of Mutual Coherence in Turbulent Media Using the Local-Independence Approximation

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JOURNAL OF TIHE OPTICAL SOCIETY OF AMERICA

Propagation of Mutual Coherence in Turbulent Media Using theLocal-Independence Approximation*

LEONARD S. TAYLOR AND STEPHEN MELLAIANt

Electrical Engineering Department, University of Maryland, College Park, Maryland 20742(Received 18 April 1970)

We treat the scalar wave equation in a medium with stochastic permittivity using the local-independenceapproximation. The equation is solved by a new method to yield an exact solution of the differential equa-tion. It is shown that this is identical to the formula for the mutual coherence that is obtained from the com-plete Born series for the wave field assuming local independence. The relation of the local-independencesolution to the complete solution is discussed.INDEx HEADINGS: Inhomogeneous media; Atmospheric optics; Coherence.

It is now generally recognized that the problem of thepropagation and scattering of electromagnetic wavesin random media is mathematically related to problemsthat appear in many branches of physics.'-' Amongthe mathematical characteristics that are shared bythese problems is that they lead formally to an infinitehierarchy of equations for the statistical moments.The problem is to reduce suitably the infinite set bysome truncation procedure. Historically, these trun-cation procedures have appeared, often in quite dis-guised form, under the names of various approxi-mations, methods, hypotheses, and techniques. It hasoften been difficult to identify the truncation procedurethat has been employed and its range of validity.

One of the most important of these truncation pro-cedures has been based on the assumption that thelocal fluctuations of the field, ,6(r), are statisticallyindependent of the local variations of index of refractionor dielectric permittivity be(r). That is, it is assumedthat

(5E (r)&E (r')\6 (r) ,* (r') ) = (be (r)5,E (' (() r (r') )-

This assumption truncates the hierarchy of equations,leaving a fourth-order differential equation for thepropagation of mutual coherence. Our work is devotedto a study of the solution of this differential equation.

The method employed in this paper depends upon aformal result of single-scatter theory that was derivedby Bremmer,4 which he called the "spectacular result"of Born theory. Because the method we use is closelyrelated to the single-scatter formulation, we are ableto obtain a result that can be identified with the com-plete Born series for the mutual coherence, using thelocal-independence approximation. Our final formulaagrees with the formula first derived by Beran5' 6 usinga different method of solution of the differentialequation but avoids the convergence difficulties andthe assumptions6 that were previously required. More-over, by considering the relation of the local-inde-pendence differential equation to an approximate

equation for the propagator of the mutual-coherencefield,7 we are able to answer the question of why thepresent approach yields the same formula as the re-normalization technique used by Brown.8

THEORY

In this paper, we consider fields propagating in amedium with a randomly varying dielectric constant.In such a medium, the wave equation may be written as

V-4'(r,t) = [e (r)/c2] (a2 /at2)[E/i(r,t)]-s (r,t), (1)

where the permittivity e(r) is a random function ofposition and s(r,t) represents the sources. Assume thatthe incident radiation consists of monochromatic planewaves with wavenumber ko in free space and that thepermittivity has the form

e(r) = 0 +5E(r)], (2)

where e is the mean permittivity of the medium. Thewave equation can be written in the form

(V2+k2 ) ,6(r) =-k 28e(r),6(r)-s(r),where

k2 = ko2 .

(3)

(4)

If we now use the operator notation L = V2 +k2 andrewrite Eq. (3), at the point r', take the complexconjugate, and combine, we have the equation for thepropagation of the mutual coherence in the randomdielectric,

(5)

In writing the above equation, we have assumedthat no external sources are present.

Note that the Green's function for the operator onthe left-hand side of Eq. (3) is

Go(r,r') =expik | r-r' /41r I r-r'[. (6)

Thus Eq. (3) can now be written in the form of an

236

VOLUMIE 61, NUMBER 2 FEBRUARY 1971

LL'*(V, (r),P* (r')) = kl�6,E (r)3,E (r')VI (r)q/* (r')).

PROPAGATION OF MUTUAL COHERENCE

integral equation as4

V,6(r) -k2j Go(r,r')5E(r')V/'(r')dv(r')= j G(rr')s(r')dv(r')+ Go(r,r')-V (r')-V(r')-Go(r,r') dor(r'). (7)

We assume that we know certain statistical properties of the random dielectric. The ensemble average of bEis taken to be zero everywhere in the dielectric so that

(5e(r)) =0.

We also note that the surface integral in Eq. (7) approaches zero as the surface becomes infinite. 'using Eq. (7), we write an integral equation for the mutual coherence,

(,6~ (r)i,6*(r')) = k2j j Go(r,ri)Go* (r',r 2) (be (r1)I,6 (rl)s* (r2))dv (rl)dv (r 2)

+k2 f Go* (r',r 2)Go(r,rl) (6E(r 2)/* (r 2)s (r,))dv (rl)dv (r2)

+k4f f Go (r,ri)Go* (r',r 2 ) (6E (rl)aE (r2) ,6(ri)Vl* (r 2))dv (rl)dv (r 2)

f f Go(r,r1 )Go* (r',r 2) (s (rl)s (r 2))dv (rl)dv(r2)

(8)

[hus,

We now make the approximation of local inde-pendence.7 That is, we assume that the field and thefluctuations of the dielectric constant are statisticallyindependent at any point r. Thus we introduce theapproximate solution /IL(r) that satisfies the statisticalrelations

(i/L (r)L*(r')se(r)be(r')) = (5e5 (r)&e (r'))X (4L (r)k* (r')) (10)

(&E(r)iL(r)) = (&e(r))(VL(r))

and from Eq. (5)

(11)

LLI*(/IL (r)L*(r -k4(8e (r)re (r'))K/L (r)/L* (r')). (12)

This is the e ation that shall be solved in this paper.Assuming that no independent sources are present, weobtain from Eqs. (9), (10), and (11) the formal, generalsolution of Eq. (12),

+k4j j Go(r,ri)Go* (r',r 2) (65E(r1)3e(r2))

(h6(r1)PL*(r2))dV(rj)dv(r2), (13)

where 0bo(r)Qbo*(r') is the homogeneous solution.We note that, at present, there are no known mathe-

matical criteria that determine the range of usefulnessof the statistical-independence approximation. Inmedia that are stratified, or in which there is a sharpgradient of the permittivity, specular reflection occurs.In these cases, the field and the permittivity are

(9)

certainly not statistically independent. However, ifthe statistics of the medium are spatially stationaryover regions large compared to the wavelength in thedielectric medium, statistical independence is rea-sonably assumed. A criterion derived by Kieburtz9

used as an analogy is a spheroidal lens of refractive index[n2 (r)Ii, whose principal axes have lengths equal tothe correlation distances of the refractive-index fluc-tuations along each cartesian axis. For a medium witha single correlation scale, this criterion reduces to

(6,-:2) Y Z, (14)

a very weak condition. It has been shown7 that Eq.(12) does not contain self-interaction processes and,therefore, does not yield a complete solution for themutual coherence. We will discuss this point later.

We now return to find a solution of Eq. (12). As-suming homogeneous and isotropic turbulence, wemay write the average permittivity in terms of acorrelation function that depends upon the distancebetween points and the scale of turbulence. In thispaper, we will be concerned with single-scale turbulencemodels only. Thus

r ((6e)1)Cl r-r'l /1), (15)

where C( r-r'1/1) is the normalized autocorrelationfunction of the fluctuations of permittivity. Thus, fromEq. (15),

LL'*(/I(r)QL* (r')) =k 4((1e)2)C( |r-r' l/l)X(i/L(r)i/,L*(r')). (16)

237February 1971

L. S. TAYLOR AND S. A MELLMAN V 6

We now employ a perturbation technique. Considerfirst the solution ,Po(r) of the homogeneous equation

ObviouslyLPo (r) = 0.

LL'*(4Po (r)4Po* (r')) = 0.

(17)

(18)

Next, consider the solution XIB of the equation

LVB6(r) = -k2Re(r)q/o(r). (19)

The function +,B(r) is formally equivalent to the firstBorn solution. From Eq. (19), we obtain

LL'*(PB (r)'PB*(r')) = k4((66)2)C( |r-r' /1)XPo(r)4Po*(r'). (20)

We emphasize here that our usage of SOB does not inany way imply restrictions due to the limited range inwhich PUB is a valid solution of the wave equation. Weemploy APB in a formal manner only, as a solution ofEq. (19). Subtract Eq. (20) from Eq. (16), and addand subtract the quantity (41B(r)V1B*(r')) on the right,and obtain

[LL'* - 4((3C)2)C( I r-r'{ /l)]D(r,r')=k4~((E2))C( I r-r' I /1)[(IPB (r)is*(r'))

-^60r)\0*(r)],(21)where, by definition

D (r,r')-=_(AL (r)46L* (r') )- (X (r)XP* (r') ). (22)

We observe that the right-hand side of Eq. (21)contains only known functions, whereas on the left-hand side we have the operator

LL'*-k 4 ( (6e)2)C -( r' I /1) = V2V'2+k2(V2+V' 2)+ k 4[- ((3e)2)C( r-r' /I)]. (23)

We assume that ((6E) 2)<<« so that we may institute aperturbation procedure in which the last term in thebrackets on the right of Eq. (23) is neglected in thefirst iteration step, and we solve

LL'*Do(rr') - k4((6e)2 )C( lr-r' |/)

We now proceed to iterate the equation again, setting

D = Do+Dl'. (25)

Subtracting Eq. (24) from Eq. (21) and substitutingEq. (25), we find

LL'*D,'(r,r') =k4((6,E)2)C( |r-r' |l/) (Do+D,'). (26)

We again assume that ((a3e)2)<<1, whence the approxi-mate solution of Eq. (26) is

LL'*Di(r,r') = k4 ((8e)2)C( jr-r' | /1)Do(r,r'). (27)

Successive iterations lead to the result

LL'*D, (r,r') = k4( (aE)2 )C( j r-r' /I)Dnl.(r,r') (28)

and

D=, D,.

We consider a plane wave propagating in the zdirection. That is, we take the boundary condition ofour differential equation to correspond to an incidentplane wave at z=0, propagating into the right half-space. The solution to the homogeneous wave equationis

,Po(r)'Po*(r') = eik(z'). (30)

The formal Born solution may be written in the form(6A, 80 real)

VIB (r) = eik{l+aA (r)+iao(r)]. (31)

Equation (31) defines 3A(r) and 60p(r), but we notethat for small fluctuations they can be, respectively,identified as the amplitude and phase fluctuation.Noting that (3A(r))=(&k(r))=0, we have

('PB(r)'PB*(r'))=ei(z-z')[l+(SA (r)5A (r'))+(&p(r)&k(r'))], (32)

where the term expik(z-z') is the plane-wave homo-geneous solution. Combining Eqs. (24), (30), and (31),we find, upon inverting,

Do(r,r') =k4 ((5E)2 )

X fG(rri)G*(r',r 2)C( r-r2 /) expEik(Z1 -Z2)]

X[(5A (r1)3i (r 2 ))+ (&(ri)&k(r 2)>]drldr 2 . (33)

We note that we may apply the following relation tothe formal Born solution ERef. 5, Eqs. (1), (3), (5), and(15)] when z>>l,

(34)

where by definition

C (x/l,y/l)= dwC([(x//)2+(y/l)2 +w2 ]}. (35)

Thus, we obtain explicitly

k 6((5e)2) 21 lexpik r-rli exp-ik|r'-r 2 |

32J2 J I r-ril I r'-r 2 l

XexpEik(z 1-Z 2 )]ZlC( [rj-r2 I /1)

XC[(x1-x2 )/l, (y1-y2 )/fldridr2. (36)

We assume that the slab thickness d<<K4/X3, so that wemay use the sagittal approximation,

Ir-ri! =z-zl+2E(X-XI)2+ (y-y1)2]/(z-z1) (37)

r-r2 I =z'-Z 2 +2[(X'-X2 )2 + (y'-y2) 2 ]/ (Z'-Z2 ). (38)

238 Vol. 61

(M (r)bA='k 2IZOE(x-411, (y-y')/IJ((6E)2)'

PROPAGATION OF MUTUAL COHERENCE

Thus,

_k_ ((_)_)_Irf e (2 Z)Z, fl[(x-x,)+±(y-yl)2 (X'- X2)2+(Y'-Y2)2±Do(~ ) exp-3 27r2 Ji (z-z1) (z'-Z2) 2 z-Z z' -Z2

XWe w tr-r2r /e)-a(Xs-e2)/d, (Yi-Y2)/s]ymrSdr2-

We now transform to the center-of-mass and relative coordinate systems. Set

u=X1-X 2 V=yl-y2 U= (Xl+X2)/2

Then, Eq. (39) is written as

D rl) e-((z2-eir

2 o)Z

3 27r' Ii,

V= (yl+y2)/2 .

rI r r r ik(X-U-"12)2+(y- V-vl2)2V V V Vexp-

JZ -J zZ1) (Z' -Z2) 2 - z-j

(x'- U+u/2)2+ (y' - V+v/2)21Z -Z 2 v

(41)

where we have ignored backscatter contributions. We can integrate over the variables U and V using an identitygiven by Chernov.10 This integration yields

, k5 ((6E)2 )2leik (z-z'

Do (rr )=1--I 67ri

Irr rr Z1

JO JOJJZ-Z-Zl+Z2

ikr (U+X'-X) 2+ (V+y - Y)IXexp- (CE) Vi, v(Zi-Z 2)/l]C(u/l,v/l)dudvdz1dz2. (42)2 (Z-Z'-Z 1+Z2)

Assuming thatkl2>>z, z', (43)

we may use the method of stationary phase to integrateover the variables u and v in Eq. (42). This yields

k4 ( (e6E)2)2leik (-zZ')

Do (r,r') =8

xj j ziC[(x-x')/l, (y-y')/l, (z-z')/l]

XC[(x-x')/l, (y-y')/l]dzdz2. (44)Equation (43) is a restrictive assumption that isnecessary to evaluate (42). However, the scale, 1, thatappears in Eq. (42) is the integral scale and, if thepresent results are to be used in connection with aKolmogorov model," it should not be taken as theinner scale. The integral scale for the Kolmogorovmodel' 2 is (7/10) louter so that Eq. (43) will be validfor ranges of several hundred meters in atmosphericturbulence.

We make the coordinate transformation

B=zi,B = Zl-Z2. (45)

The integration in Eq. (44) now becomes an integrationover the trapezoid with vertices (0,0), (z,z'), (z, z-z'),

and (0,z') in the B-13 plane. It is now convenient toassume that z-z'<<l<<z (we shall later take z=z'), sothat we may replace the trapezoid by the parallelogram(0,0), (z,z'), (z,0), (0, -z'). Consequently

k4 ( (6 e)2)2leik (z-z')

Do(r~r)= 8 C[(x-x')/l, (y-y')/l]

xJJ BC[(x-x')/l, (y-y')1l,011]dBd#

z' o

+J BCE(x-x')Il, (y-y')1l,,B/IjdBd0 . (46)

The upper limit of integration for the inner portion ofthe second integral may also be replaced by z' inaccordance with the assumption of this paragraph andwe find, upon integrating over B and combining,

k4 ( (6e) 2)2 e k (z-z' )12Do(rr') = C[(x-x')/l, (y-y')/l]

8

X {lz"2C[(x-x')/l, (y-y')/l]

+z'f BC[(x-x')/l, (y-y')/l,3/l~d,3. (47)

(39)

(40)

February 1971 239

L. S. TAYLOR AND S. MELLMIAN

The term (y-y') in Eq. (47) is of order i/z' comparedto the first and may be dropped. The final result forDo(r,r') is

Do= 2!(2)2 C{(x-x')/l, (y-y')/l]eik*(z-'). (48)

We can now perform the second iteration step toobtain D1. We use Eq. (27) to obtain

k8(()2)312 f fexp[ik I r-r l ] exp--ik r'-r l ]2!(2)2.16,r21 I lr-r1 1 lr'-r

XZi2C2 [(XI1-X2 )/1, (Y1-y2)/1/

XC(Irl-rr 2 //) exp[ik @' -z2)Jdrldr 2. (49)

The integration proceeds exactly as before, yielding

k6 ( (bC)2)3I3Z3Di= ~ C3[(x-x')/l, (y-y')/fl] (50)

3! -(2)3

The nth term is given by

k2(n+2) ((3e) 2 ) n+2Zn+21n+2

(n +2) !2 n+2

XC,,+2[(X-Xt) //| (y-yt)/lgeik(z-d). (51)

Finally, we set z=z' to obtain the local-independencesolution for the transverse coherence,

<AIL (r)#IL* (r'))

.o k2m ( (5,E)2) mslmlt

- - C"'(x-x')/1, (y-y')/1Jrn=O mn!200

k2=exp -((6,E)2)1zC[(XX')1J, (y-y')1l]. (52)

CONCLUSION

Equation (52) is a solution of the differential equa-tion (12), including a homogeneous solution whichrepresents a plane wave. Before any normalization ofthis result is considered, it is important to note thatEq. (12) corresponds to an arbitrarily truncated series7for the bilocal propagator, (LL*%)-l. In this truncation,the "self-interaction" terms7 have been dropped.Maxwell's equations are energy conserving, but arbi-trary truncations of the type under discussion neednot conserve energy. Using the homogeneous solutionof Eq. (12), for example, we may indeed insert energyconservation in the zeroth step, adjusting the multi-plying constant so that (J| 1B(r) 12) = 1. However, wefind it impossible to impose energy conservation in thesucceeding iteration steps, in which the particularsolutions are functions of z. The problem may be re-solved using a suggestion of Bourret.7 The effect of theself-interaction terms is reintroduced by the simplebut inexact device of multiplying the above solutionby a function F(x-x', y-y', z). Because we haverestricted our analysis to small scattering angles, thisfunction can be a function of only z, not of the trans-verse coordinates. Now, using (I i(r) I2) = 1, we find

(Qb(r)4b(r')) =exp( 2hklz{C[(x-x')/l, (y-y')/l]-C(0,0))). (53)

This is the result which has been derived by severalinvestigators.5 8'13 4 In the present formulation, itbecomes evident why this result can appear both fromthe local-independence equation and from the renor-malization procedure.8 The local-independence equationdoes not account for all the terms that are included inthe renormalization process and depends upon amendingthe solution of the original differential equation in orderto conform to energy conservation. This feature appearsexplicitly in our direct procedure. This fundamentalweakness of the local-independence method is importantin considering the validity of the method in relation toother techniques, and in the use of this method in thecalculation of the fourth-order coherence.

APPENDIX

In order to clarify the relationship of our result to the most general series for the mutual-coherence functionwe begin by writing a purely formal solution for the Born equation,

(/" (r)#n* (r')) =Vo(r)5o* (r') +k4((5e)2)ffGo(r,ri)Go* (r',r 2 )C (ri,r2)wo (ri)Vo* (r2)drldr2 .

Also, write a purely formal solution for Do. From Eq. (24),

Do= k4 ( (6E)2) fGo(r,r)Go* (r',r2)C(r1 ,r2) ( ()t (r1 )'*0 (r 2))- 1o(ri)#o* (r2)}drldr2 -

Similarly, from Eq. (27),

Di (rr') - k4( (8E)2) fJGo (r,ri)Go* (r',r 2 )C(ri,r 2 )Do (ri,r2 )drldr 2.

(Al)

(A2)

(AS)

Vol. 61

PROPAGATION OF MUTUAL COHERENCE

Next substitute Eq. (Al) into Eq. (A2), and obtain

Do (r,r') = k( ((6e)2)2 f fffGo (r,ri)Go* (r',r2)C (r1,r2)Go (ri,r3)Go* (r2,r4)C (r3,r4)V10 (r3 ),6O* (r4)dridr2 dr3 dr4. (A4)

Using Eqs. (A3) and (A4), we find

Di(r,r') =k12((6,E)2)3 ffff

and we see that

D.(r,r') = (k4)n+2((,e)2)n+2f2n+4

/Go (rri)Go* (r',r 2)Go (rir3)Go* (r 2,r4)

XGo (r3,r5)Go* (r 4,r6)C (r1,r2)C (r3 ,r4)C (r5 ,r6)!tk0 (r5)iko* (r 6)dr *... dr6

... fGo(r,ri)Go*(r',r 2) ... Go(r2n+Ir 2n+3)

XGo* (r2n+2,r2n+ 4 )C (ri,r 2) . . C (r2n+3,r2.+ 4)Vt60 (r2n+ 3)Vo* (r 2 n+4 )dr *... dr 2 n+4 .

Combining Eqs. (22), (29), and (Al), we obtain

IGO (r,ri)Go* (r',r 2)C (rir 2)C (r1 ,r2 )t'O (r,)fbo* (r 2)dridr 2

2) n+2j . f Go (r,ri)Go* (r',r2) ... Go (r2n+1r2i,+3)2n+4+n- (k

-=0

XGo* (r2.+2,r2.+4)C (ri,r2) ... C (r2n+3,r2n+4)V0 (r2f+3)V,6o* (r2n+4 )dri ... dr2n+4- (A7)

This expression is a formal series for the local-independence solution. It is identical to the formula for the mutualcoherence that we obtained by use of the complete Born series for the wave field assuming local independence.

REFERENCES

* Research supported by the Atmospheric Sciences Section,National Science Foundation, NSF Grant GA-1481. Paper pre-sented at the Spring 1970 meeting of the Optical Society ofAmerica, Philadelphia, Pennsylvania, J. Opt. Soc. Am. 60, 741A(1970).

t Work submitted in partial fulfillment of the requirements forthe degree of Doctor of Philosophy in Electrical Engineering.

' M. Kac, Probability and Related Topics in Physical Sciences(Wiley-Interscience, New York, 1969).

2 J. R. Klauder and E. C. D. Sudarshan, Fundamentals ofQuantum Optics (Benjamin, New York, 1969).

3 M. J. Beran, Statistical Continuum Theories (Wiley-Inter-science, New York, 1968).

4 H. Bremmer, in Quasi-Optics, edited by J. Fox (Polytechnic,Brooklyn, New York, 1964).

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(Dover, New York, 1960)."1V. I. Tatarski, Wave Propagation in a Turbulent Medium

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(IPL (r)4L* (r')) =Vo(r)Vo*(r') +k4�(&,)2)