146 OPTICS LETTERS / Vol. 12, No. 3 / March 1987

Geometrical theory of diffraction for high-frequency coherencefunctions in a weakly random medium

R. Mazar and L. B. Felsen

Department of Electrical Engineering and Computer Science, Polytechnic University, Farmingdale, New York 11735

Received June 26, 1986; accepted December 12, 1986

We propose a theory of propagation and diffraction of the statistical moments of high-frequency fields in thepresence of interfaces, boundaries, or general scattering objects embedded within a medium with weak large-scalerandom fluctuations. The formulation utilizes the deterministic ray paths and initial conditions of the geometricaltheory of diffraction when stochastic effects are omitted to construct initial conditions for ray-centered transportequations governing appropriate statistical measures of the field when stochastic effects are included. The basicprinciples of the problem strategy are discussed and demonstrated on canonical examples.

The localization of high-frequency wave propagationaround ray trajectories, and the reflection, refraction,and (or) diffraction of these local plane-wave fields byboundaries, inhomogeneities, and (or) scattering cen-ters, has been combined through the geometrical the-ory of diffraction (GTD) into one of the most effectivemeans for analyzing high-frequency wave phenomenain complex deterministic environments.' It is pro-posed to incorporate these constructs into a stochasticpropagation and diffraction theory for statistical mo-ments of the high-frequency field in a randomly fluc-tuating medium, provided that the correlation lengthI, of the fluctuations is large compared with the localwavelength X = 27r/k = 27rc/w in the fluctuation-freebackground, with k being the local wave number, c thelocal wave speed, and co the radian frequency. Thecanonical problems of deterministic GTD, which yieldthe (uniform, if necessary) local reflection, refraction,and diffraction coefficients descriptive of variouspropagation and scattering events, can still be utilizedbecause the restriction X << 1, carves out sufficientlylarge deterministic blobs, within which the necessarylocal conditions can be established that relate incom-ing to outgoing wave fields. Thus one gains access to amuch larger class of high-frequency problems in a ran-dom medium than at present.

The major analytical building blocks that we haveemployed so far in pursuit of this objective are (a) localcoordinates centered on the GTD ray trajectories inthe deterministic background environment; (b) multi-scale expansions in these coordinates to chart thepropagation properties of statistical measures (someof which are new; see below) of the parabolically for-mulated ray fields; (c) Kirchhoff or physical-optics(PO) approximations, generated by GTD fields, toestablish initial conditions for fields reflected fromextended smooth surfaces; and (d) point-scatterer so-lutions to establish GTD initial conditions for smallscatterers and edges. If the background is refracting,the extracted phase in the parabolic approximation isalong the curved ray paths, thereby extending previ-

ous analyses based on straight parallel backgroundrays.2' 3 We have already shown 3 that we can solve forthe second- and higher even-order coherence func-tions under these generalized forward-propagationconditions, which also accommodate uniformly the be-havior near caustics. This solution strategy must nowbe explored in the more general context of reflectionand diffraction.

To account for correlation of incident and backwardreflected or diffracted fields that traverse the samepropagation volume, we introduce as appropriate sta-tistical objects two-point random functions (TPRF's)and corresponding higher-order functions instead ofthe usual statistical moments. Under certain condi-tions, by pairing combinations of random and auxilia-ry deterministic fields, designated as two-point fieldfunctions (TPFF's) and corresponding functions ofhigher order, we have also solved for the random fielditself. In what follows, we construct the solutions forthe coherence functions, TPRF's, TPFF's, etc. in arandom medium with homogeneous background, lean-ing heavily on previous results, which merely requiresimple extension. We then consider simple canonicalproblems, in the sense of the GTD, to demonstrateimplementation of our approach.

With U(r, a) denoting the full wave field, we consid-er the parabolically approximated wave function u(r,a) = U(r, o)exp(-ike) transported along a straight raypath with range coordinate a- and paraxial transversevector coordinate r. In the paraxial region near theray, using the scaled coordinates i = r/1, and a = a/ln,u(r, a') - (Ii, a) satisfies the parabolic equation4'5

&a- 2kl, R u + ikInh(f, a)U, (1)

with the initial condition i(i) = -S(i). Here, superim-posed upon a unit background is a weakly fluctuatingrefractive index fi characterized by the correlationlength I, and satisfying the conditions Iil << 1, (h) = 0,k 2 (n2) 1n 2 << 1, where ( ' denotes the ensemble average.The refractive-index fluctuations are assumed to be

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March 1987 / Vol. 12, No. 3 / OPTICS LETTERS 147

described by an effective correlation function of thetype Bn(r1 - r2 , li - u2) = AJI(ri - r2)/l]a6(61 - ff2)[Eq. (2.9) of Ref. 4].

With the asterisk denoting the complex conjugate,we now define the TPRF P(ri, Y2, a) = u(i1 , a9u*(r2 , a)and the TPFF rf(tl, r2, a) = i(i', ojif*(i2, a), whereU satisfies Eq. (1) and Uf is a solution of Eq. (1) in theabsence of fluctuations. In the coordinates a, D = (i1+ i2)/2, s = klt(il - r2), the propagation equation forr can be written as

ar = iv- * V7r + ikl,2

(F{~-j 2 D]~ (2)

with the initial condition r,(, i) = t4(1)tis* (i2). Forrf, the second f term in Eq. (2) is omitted and theinitial condition is given by rfs(fi, s) = Us(il)Ufs*(?2).

Then, following Refs. 6-8 in the choice of the expan-sion parameter e = (kln)-1 and introducing the notionof a propagator into the two-scale expansion formal-ism, we obtain for the solution of Eq. (2)

r(p, g, a) = J... J d2 vd2 qI'(v, J)G(-p, s, klv, iq, 0)

(3a)

=2r)I J*. J d2Pd 2 qd2~Ps(V, P)

X G(D, so, a, IV, iq, 0)exp(ip- * q), (3b)

rP(v, p) = jj d2qP6vY, q)exp(-i ), (3c)

where G(Pi2, 2 , 2IP1, s1, WI) G(211) is the two-scaleexpansion TPRF propagator,

G(211)- 1 exp_____(27r) c2 c 14 exp[i _ _2

2l jX J... J d2 -1 d2 s 2E(tal, a 2 )

X expQE-1 AJ dfl([R+ + y('Y, 2)a li

- &[R + Y(a2, al), (4)

~(t a6) = Jf d2 'tzujt')Gu(itIi', 0), (5a)

Gu(rX, -2 if, ) = ( 1)]3i [ exp[ 021]

X ft.. . d 2 -&ld2o&2 E(- 1 , a&2)

X exp ic-l Ad>h[fKl

- rK2+ Y(t1, Z2), t v (5b)

These results can now be used for construction of allstatistical moments of the field. We have verified sofar that the 2mth-order coherence functions obtainedby averaging products of r in Eq. (3) agree with thosededuced in Ref. 3 by the two-scale solution of theaveraged moment equations, when the background ishomogeneous. We have also verified that the second-order coherence function constructed from pairing thefields in Eqs. (5), and subsequent averaging, agreeswith that in Refs. 4 and 5. Furthermore, we haveconstructed the expression for the fourth-order statis-tical moment of a spherical wave using the propagatorGo in Eq. (5).

Canonical problems furnish the GTD initial condi-tions after an encounter in terms of the incoming fieldbefore the encounter. Knowing the new initial condi-tions, one may then employ the solutions above tocalculate the reflected, diffracted, etc. coherence func-tions. We shall illustrate the procedure on two simpleexamples.

First, from a given source distribution, we considerbackreflection along the reference ray incident nor-mally onto a plane boundary at -j = wr with local reflec-tion coefficient specified by the function f(). Here, itis necessary to allow for correlation between the for-ward- and backward-propagating fields. We employGTD to establish the PO initial conditions for thereflected field Ur and the TPRF 1r. Then the reflect-ed field Ur and the TPRF at the observation plane -ocan be obtained by using the formal solutions of Refs.9-13 and the explicit propagators G11 and G of Eqs. (5)and (4):

Ur(r Yo = ..= f. d2 Fld 2t2 Us(t)fQ(2)G( 2, jrIfl, )

(6)

where R4 = (W2 + Es2/2)K1 -(Di ± es1/2)K2, Ki = -

W)Af2 - a1), i = 1, 2, (db &2)= Bt6I2 -al -al(l+ 072)/(2l62 - d1 J) + ca2/2, and E(a 1 , a2) = exp ti(a2

-

zil 2)/[2e('U2-j .]}. The solutions in Eqs. (3) and (4)also apply to rf, provided that the second h term in theexponent of Eq. (4) is omitted and P8 (v, iq) is replacedby rft(v, i). Since the auxiliary field Ut in rf is struc-tured so that uf(r, a) - 1 when tfij = 1 (this is truebecause of our plane-wave choice of Of), the solutionfor the random field can be recovered by imposing thisinitial condition:

rW' i, S04 = I ... J d2 pld21 d2 hP2d

2§2rP(pW, S1)

(+ 2 ) ?( 2 - ES2)

X G(V2 , S2, d31D1, Sg 0)

X G(D2, _g2, 02I1P S. CYo)- (7)

The integrations fi, Di, 9s (i = 1, 2) extend over thesource and reflector planes, respectively. The average

X G(Y2, - If, - ),cr co

148 OPTICS LETTERS / Vol. 12, No. 3 / March 1987

reflected field and intensity distribution follow by ap-plying averaging to Eqs. (6) and (7).

Next, we consider a small scatterer whose dimensiona is much smaller than the wavelength and which has adiffraction pattern that is cylindrically (two-dimen-sional) or spherically (three-dimensional) isotropic.Small scatterers are useful also for describing edge ortip diffraction since high-frequency edge- or tip-dif-fracted fields away from these scattering centers (andoutside ray-transition regions near reflection andshadow boundaries) exhibit cylindrical or sphericalspreading, respectively, modified, however, by a dif-fraction pattern. Within the framework of GTD,these directional features are expressed by incident(0e) and observation (0d) angle and dependent edge- ortip-diffraction coefficients D(Gd, Qi, k).1 To accommo-date both isolated small scatterers and edge or tipdiffraction, the relevant canonical problem is that of apostulated small scatterer with nonisotropic scatter-ing characteristics. In the weak-scattering regime,the diffracted random field and the TPRF can beobtained from Eqs. (6) and (7) on assigning to thefunction f(i) the form f(fr) = l1,,D(Gd, Oil, h)ci), wherethe diffraction coefficient is assumed to be knownfrom the solution of the appropriate canonical prob-lem.1 The distance coordinate o0 is now measuredalong the ray with angular displacement ad connectingthe scatterer and the observer. In the strong-scatter-ing regime, owing to multiple ray interference, it maybe preferable to deal with the spectra generated by thesource distribution. For the TPRF, the spectral ap-proach can be pursued by assuming an effective sourcewith angular intensity distribution given by

1rrs(V, P) = inc h2 6(tV)ID0 (p, O; 1)!2 ,

Do(-, Oi; k) = D(Od + 0, oi, k)M (8)

The slope parameter p = tan 0 (see Ref. 3) measuresangular deviation 0 from the central ray tagged by thediffraction angle Od, and Ii,, is the intensity of theincident field. If the incident wave is created by apoint or a line source, then Ilc = AG(0, 0, frf0, 0, 0),where G is the three- or two-dimensional form of thepropagator in Eq. (4) and -r is the distance measuredalong the ray connecting the source with the scatterer.Here, A = Ik 2/l1 2 for the point source and A = Ik/llfor the line source, with I, denoting the sourcestrength. Substituting Eq. (8) with these source con-ditions into Eq. (3b), we obtain in the three-dimen-sional case for the average diffracted intensity Id('o,,T) = (rr(DO = ro, s0 = 0, e)) in the perpendicularobservation plane e,, emanating from the point -do onthe reference diffracted ray identified by Gd:

Id(-ro, do) = (2ir)2 j4 j d2.d 2.. D , k)1

X exp(iT - i) (G(0, 0, ff0, 0, 0)

X G(rT, 0, fe,10, q, 0)). (9)

This expression, which contains formally the correctpropagator G, becomes explicit (though cumbersome)

when the approximate two-scale expansion form inEq. (4) is applied. In the two-dimensional case, theconstant term before the integral is Ik 2/(21D,2) alldouble integrals are replaced by single integrals, andthe diffraction coefficient Do and the three-dimen-sional propagator G are replaced by their two-dimen-sional analogs.

The preceding formulations apply generally for re-flection and diffraction processes that correlate ingo-ing and outgoing events. In the diffraction problem, ifthe incident ray (er) and the diffracted ray to theobserver (JO) traverse substantially different trajec-tories, then, because of decorrelation, the average ofthe product (GG) in Eq. (9) or in the averaged term ofEqs. (6) and (7) can be split into the product of aver-ages. For backscattering, where ir and -co overlap,(GG) must generally be averaged as a product. If thepoint scatterer is isotropic so that D0(p, As; h) = 1, thenEq. (9) reduces in the perturbation and saturationlimits to results obtained elsewhere.14' 15 In the ab-sence of refractive-index fluctuations, Eq. (9) becomesId(ro, -a.) = Ii.,k211Do(il/%y, ib; k)I2I(2irlnt)2, in agree-ment with the results of deterministic GTD. Thesecanonical building blocks have been used for the moresubstantial problem of wedge diffraction.1 6

This research has been sponsored in part by theRome Air Development Center, Hanscom Air ForceBase, through a grant administered by the Joint Ser-vices Electronics Program under contract no. F-49620-85-0-0078 and in part by the U.S. Office of NavalResearch, Underwater Acoustics, under contract no.N-00014-79-0013.

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