Scattering by a rough dielectric interface: a modifiedWiener-Hermite expansion approach
Cornel Eftimiu
McDonnell Douglas Research Laboratories, St. Louis, Missouri 63166
Received August 7, 1989; accepted December 19, 1989
An approach to the problem of scattering by the rough surface of an arbitrary, uniform, lossy dielectric medium,
based on a modified Wiener-Hermite functional expansion of the electric and magnetic currents, is presented. The
results obtained for the reflection and backscattering coefficients, as well as for the bistatic cross section, arecompared with the corresponding predictions based on perturbation theory and the Kirchhoff approximation. Theapproach is believed to yield satisfactory answers in a domain that includes the rectangle defined by the inequalities
ka S 2 and 1 • kR 5 10, where a2 is the variance of the surface profile, R is the correlation radius, and k is the wave
number.
1. INTRODUCTION
A modified approach based on the Wiener-Hermite func-tional expansion was recently' employed to investigate elec-tromagnetic scattering by rough conducting surfaces. Thepurpose of the present research is twofold. First, we extendthe research of Ref. 1 to rough surfaces separating free spacefrom an arbitrary, uniform, lossy dielectric medium. Thevariety of cases that can be considered is thereby much
larger, while the special case of perfectly conducting roughsurfaces can be recovered as the limit for the infinite imagi-nary part of the dielectric constant. Second, we encoun-tered in Ref. 1 certain technical difficulties, which wererelated to the presence of integral terms containing noninte-grable singularities. To treat these singularities, we per-formed in Ref. 1 an ad hoc regularization, based on the
explicit assumption that the illuminated area of the surfaceis finite. In this paper we show that an alternative proce-dure, which is less expeditious but theoretically more attrac-tive, is to abandon the idealized concept of a perfectly con-ducting surface and to consider a conductor as a lossy dielec-
tric with a finite, albeit arbitrarily large, conductivity. Aswill be demonstrated, no nonintegrable singularities arepresent if we follow this procedure, and if the conductivity issufficiently large, no dependence on its actual value can benumerically detected.
The geometry of the problem is represented in Fig. 1. Therough interface z = (x, y) will be viewed as a realization of a
stochastic process with an ensemble average ((x, y)) = 0, aconstant variance
a.2 = ( 2(x, y)), (1)
and a given correlation function
c(x,y) = ` 2 ((x,y)(0,0)), (2)
with a correlation radius R defined by the relation
7rR2= J dxdyc(x, y). (3)
In numerical calculations the correlation function was mod-eled by a Gaussian:
C(X, y) = 0- 2 explj-(X2 + y 2)/R 2]. (4)
The medium filling the half space z < t(x, y) is assumed tohave the magnetic permeability of free space but an arbi-trarily given dielectric constant E', which generally is a com-plex number with Im e' > 0. The incident field is a planewave
Ei = e' exp[ik(y sinO - z cos 0)], (5)
where ei = eH - ix for horizontal polarization and e' = evi iYsin 0 + i cos 0 for vertical polarization.
2. SCATTERED AND TRANSMITTED FIELDS
At a point above the surface, the scattered field can becalculated from its general representation in terms of theelectric J and magnetic M surface currents:
Es(r) = 13 J 3& eirx dx'dy'
X e-iK.r IC X J(x', y') + M(x', Y) (6)
where z' = (x', y'), no = [Ao/eo] 1/2, and k is assigned a positive,
arbitrarily small, imaginary part. Note that expression (6)is mathematically valid at any point, above, below, or on thesurface. Of physical interest is the field that is representedin expression (6) and that is not only above the surface butalso far from it, in the so-called far zone, i.e., at points r = r,with fi = (sin OS cos Os, sin OS sin OS, cos OS > 0) and r
indefinitely large. For such points,
Es(r) ik eikr
r 4v r
where 6 is the amplitude of the scattered far field:
60(, 4s) = f X J dx'dy' exp(ikk - r')(71ok X J + M).
(7)
(8)
The transmitted field below the surface likewise can be rep-
in which ' = [o/e'] 1/2 and k' = ker1 /2, with e, = e'/eo and Im k'> 0. Expression (9) is also valid at any point, an observationthat is important because it permits us to formulate integralequations for the determination of the surface currents byusing the extinction theorem.
Specifically, these equations can be written as
Ei(r) + Es(r) = 0 for z < t(x, y) (10)
and
Et(r) = 0
However, Eqs. (10) and (11) do not completely determinethe surface currents and, for this reason, must be supple-mented by explicit requirements of tangentiality, i.e.,
n-J=0 (12)
and
Substituting Eq. (15) into Eqs. (10) and (11) and thentaking the ensemble average, we find that
77oeH(eH' * FO) + oeV'(evi * F) - eH(eV * Go)
+ ev'(eHi - Go) = 2e'cos (16)
and
- ?leH (eH * Fo') + n'ev'(ev' - Go') - e (ev Got)
+ ev'(eH'* Go') = 0, (17)
respectively, in which eH' = i and eV = iy cos ' + i sin 0',where the (generally complex) angle 0' is defined by therelation k' sin 0' = k sin 0. Also in Eq. (17),
GFO') GO) iv 2(A)'
with v = k' cos 0' + k cos 0, and
(9) (A) f I dad3(R(a, 3) c(a, 3)(9) ~B/ (27r' (,
where the tilde indicates the two-dimensional Fourier trans-form.
Similarly, substitution of Eq. (15) into the conditions (12)and (13), and ensemble averaging, yields
(F0, _ ior2 f d rF.(a,) 1 + Fy(a, 11Goz) (27r)2 J ,LGX(a, 0) LGy(a, )J
(18)
If, after the substitution of Eq. (15) in Eqs. (10) and (11), oneinstead multiplies by t* (a, A) before ensemble averaging,one obtains a second set of equations:
k K X {K X [(Kx, KY - k sin 0) - iu(Fo - iu0r2 A)]I
+ K X [G(K, KY - k sin 0) - iu(Go - iua2B)] = 0, (19)
(13) in which K, = -(k2-KX 2-KY
2)1/
2 and u = K, + k sin 0,
where n is the normal at a point of the surface:
i~Z x Xa~iY Y }+I a) + a. 2(14)( Oax ay )[ ( x y
3. MODIFIED WIENER-HERMITEFUNCTIONAL EXPANSION
We shall now seek the surface currents in the form of themodified, truncated-to-first-order Wiener-Hermite expan-sion,
M(x, y) = exprikLy sin 0 - {(x, y)cos j]IG )
+ dx' dy[F(x - x', y - y') D(x/ y')+J x' Y'[G (x - x', y (15)
which has been partly (for the electric current only) used inRef. 1. We refer to this reference for its background andrelation to previous research.
t K X {K' X [(Kx', KY' - k sin 0) - iu'(Fo - iu'A)]j
+ K' X [G(Kx', KY' - k sin 0) - iu'(Go - iu'c2 B)] = 0, (20)
where K = (k' 2 - KX2 - K 2)1/2 and u' = K,' + k cos 0.Similarly, Eqs. (12) and (13) lead to the expression
(21)EG (e 3) = L aG° )FG + 0 oyJPerhaps we should stress again that the vector Eqs. (16),
(17), (19), and (20) are not sufficient for the complete deter-mination of the unknown vectors F, Go, F, and G. Thiscircumstance is due to the fact that each vector equationyields only two effective scalar equations (say, from thevanishing of their x and y components), while the third isredundant, i.e., a linear combination of the other two. Thisis precisely the reason for which conditions (12) and (13) hadto be explicitly enforced, yielding the supplementary Eqs.(18) and (21).
n- M =0,
Cornel Eftimiu
for z > tx, y).
(15)
Vol. 7, No. 5/May 1990/J. Opt. Soc. Am. A 877
Before attempting to solve the obtained equations for thecurrent coefficients, one should note that the unknowns F'and G also appear under a sign of integration, e.g., in thequantities A and B. However, this complication can bereadily addressed.
First, one should recognize that A, = B, = 0, as a conse-
quence of Eq. (21) and because a(a, Al) is an even function ineach of its variables. Second, one must solve the x and ycomponents of the vector Eqs. (19) and (20), also by usingEq. (21), with respect to Fx, Fa, 0&x, and Oy in terms of Ax, Ay,Bx, BY) F0, and Go. The results are given in Appendix A.
Third, multiplying each of the expressions thus obtained bya function c of appropriate arguments and integrating withrespect to these arguments, one obtains a system of linearalgebraic equations to be solved for Ax, Ay, Bx, and By in
terms of F0 and Go. Finally, we obtain a linear system ofequations for the unknowns F0, Go, F, and G, which, once
solved, yields the surface currents represented in Eq. (15).Before following the procedure that was just described, onewould be well advised to treat the two incident-polarizationcases separately. By considering first the horizontal-polar-ization case, one can easily see from the expression repro-duced in Appendix A that Ay and Bx depend only on Foy, Foe,and Gox. It follows then from Eqs. (16) and (17) that thecomponents Foy, Foz, and Go. satisfy a homogeneous systemof equations with a nonvanishing determinant. Conse-quently, in this case Foy = Fo, = Gox = 0, and hence Ay = Bx =0. The compatibility of this conclusion with the integralexpressions for Ay, Bx, and F0, follows from the observationthat, as can be seen from the explicit expressions reproducedin Appendix A, for this polarization Px(a, A), Gy(a, A), andGa(a, f3) are odd functions of a. As a result only 6 equations,of a total of 12 operating equations, are to be solved for thequantities Fox, Goy, Got, Fx, Gy, and ,z, and these equationsare the only ones needed for determining the surface cur-rents completely. The vertical-polarization case is clearlycomplementary.
4. REFLECTION COEFFICIENTS
In anticipation of the conclusion that reflection is specular,we shall consider only the scattering direction given by theunit vector j = iy sin 0 + i, cos 0, the polarization vectors ofthe reflected field being either eje = ix or evs = - iy cos 0 + i,sin 0.
It follows then from Eq. (8) that
1 2 0RpQ = - 2~ cexp(-2s cos2 )e8.- n0F0 -2is cos 0-A2os0s[ k
- k X (Go - 2is cos 0 k B)]X (24)
where s = (kU)2.In view of the comments made at the end of Section 3, it
can be seen that RpQ = bpQRQ, where, explicitly,
RH - 2 1 2s COS2 0)71FOx + cos OGOY-sin OGO,RH = -2cos 0 exp(-2 c I2 0 no o G~-sn0 0
- 2is cos 0 (Ax + cos 0 By)] (25)
and
Rv = 2 cos 01 exp(-2s cos2 )[Gox cos OnFoy + sin 0nqFo,
+ 2is cos 0 - (nqAy cos 0-BY)k Iy
(26)
are the coefficients for (both incident and reflected) horizon-tal and vertical polarization, respectively.
Before proceeding to a closer examination of Eqs. (25) and(26), it is instructive to consider two limiting special cases.The first case is the limit for large correlation radii, i.e., kR >>1, that are so large that we may substitute in our analyticexpressions for the quantities involved in Eqs. (25) and (26)
a(a, ) (27r)26(a)6(3).
We find then that, for horizontal polarization,
sin 20 cos 0'
sin(0 + 0')
and
2 cos 0 sin 0'
GY sin(0 + 0')
while Go, A - By - 0. For vertical polarization in thislimit,
2 sin 20
oy sin 20 + sin 20'
2 cos 0 sin 20'
sin 20 + sin 20'
and F0z Ay B. 0. Consequently,
eS = - dx'dy' exp(-ikP )e' [oJ-r X M]
has the averaged value
(e' * A) = -27ri cos 05(0)6(0)RpQ,
(22)(27)RH - in(O -E ) exp(- 2s cos0')
kR--w sin(0 + 0')ex(2co 0
and
(23)
in which the two a functions would enforce specularity had itnot been already assumed. The quantity RPQ is by defini-tion the reflection coefficient for the case in which the inci-dent polarization is P (which may be either H or V) and thereflected polarization is Q (also either H or V).
In terms of quantities previously defined and now as-sumed known,
(28)R- tan ( + 0') exp(-2s cos2 0),
which are the well-known expressions resulting from the useof the Kirchhoff approximation (KA).
The second interesting limit is that of small s. If weexpand the quantities F0, Go, A, and B in powers of s andretain terms at most linear in s, a rather lengthy calculationshows that
RH sin( - ) 1- 2s cos 0 (sin 0 cos 0 ) + IH]'sin( (0) I - 0n ' (29)
Cornel Eftimiu
878 J. Opt. Soc. Am. A/Vol. 7, No. 5/May 1990
where'H = d-dj(k2K K ) (a, 03 K sin 0)IH = (2r) 2k | dk d(,B2- KK,) k 2KZ K2 K),
with K = -(k 2- a2 - p2)1/2 and Kx/ = (k'2 - a2 _ 32)1/2
Similarly, in the case of vertical polarization,
- tan(0-') 1 2s cos0 /sin cos 0' +Is'-o tan(0 + 0') L cos2 0-sin 2 0' sin 0' ) 1
(30)
where
Iv= k(2 )2k f dad#a2(cos2 0 - sin2 0') - 2 Sin2 0
KZKz's COS 2 0' + (o sin 2 0 COS 0'1 a(a, #3 - K sin 0)
Both limiting expressions described above coincide with theresults obtained by using the perturbation theory (PT) tothe first order.
Expressions (25) and (26) were extensively investigatednumerically over a wide range of statistical parameters kaand kR and for a variety of choices for the relative dielectricconstant er that started, for purposes of validation and crosschecking, with the case of a dielectric constant with a largeimaginary part. We shall present here only a small sampleof results pertaining to the selection er = 20(1 + i).
In Fig. 2(a) we show IRHI2 as a function of k for anincident angle close to the normal ( = 5). Up to approxi-mately ka = 0.8, PT, KA, and our modified Wiener-Hermiteapproach (WH) provide essentially identical results.Where ka is greater than this value, PT breaks down andincreases in a totally unphysical manner, while the KA andthe WH continue to overlap. The latter exhibits only animperceptible dependence on kR.
(a)
0 1.0 2.0
(b)
A -00
Fig. 2. Reflection coefficient for horizontal polarization as a func-tion of k, where e, = 20(1 + i). The numbers associated with PTand WH represent values of kR.
-10
-- 20
0 (dcg)
Fig. 3. Reflection coefficient for horizontal polarization as a func-tion of angle of incidence, where e = 20(1 + i). The numbersassociated with PT and WH represent values of kR.
As the incidence angle increases, the PT dependence on kogradually improves and, as seen in Fig. 2(b), where the num-bers associated with the symbols PT and WH representvalues of kR, a dependence on kR becomes apparent, al-though the vertical scale must be enlarged to make it clearlyvisible. Remarkably, as kR increases, the WH curves slowlymove downward, and the PT curves move upward; this be-havior suggests eventual overlapping at large kR.
The angular dependence on the incidence angle of thereflection coefficient, at fixed values for k, can be betterviewed in Fig. 3. For the considered values of ka, the generalagreement between the KA and the WH is to be expected,but even at such large values of k, PT still shows goodbehavior as the angle approaches grazing values.
Perhaps the most worthwhile conclusion that can bedrawn from these numerical investigations is a simple cau-tionary statement: The mere fact that at small ka onerecovers analytically the PT results, while at large kR onerecovers analytically the KA results, is not sufficient to char-acterize completely the domains of validity of these approxi-mations. An important role is played by the hidden param-eter, which is the angle of incidence. At 50, as indicated inFig. 2(a), the KA and the WH practically coincide even if thecondition, which requires that kR be large and under whichthe KA is believed to be reliable, is not necessarily satisfied.Likewise, even if k a is given values large enough to make PTunlikely to be valid, as done in Fig. 3, PT still practicallycoincides with the KA and the WH at large angles of inci-dence.
We are aware that these observations imply that coinci-dence with the WH presumes a guarantee of validity. Of
of-
a L:
\\\ 2 -
\ \ P5\ r \\
\ \ I
-0 - 'II~ \\4 1~~ \ \ ' . 2\
Cornel Eftimiu
Vol. 7, No. 5/May 1990/J. Opt. Soc. Am. A 879
course, there is at present no such guarantee, and one couldbe obtained only by a direct comparison of the WH withmeasured data. The best we can do now is to express thebelief that, because the WH correctly reproduces the KAand PT in their generally accepted domains of validity, theWH stands a good chance of bridging the gap between theKA and PT in a manner that permits us to refer to the WH asa unified theoretical approach.
5. DIFFUSE FIELD INTENSITY
For surfaces that are significantly rough, the coherent re-sponse to the incident field, which is strictly specular andquantitatively described by the reflection coefficient, be-comes largely unimportant. In such cases, the quantity ofprimary interest is the intensity of the diffuse, incoherentcomponent of the scattered field, which is usually represent-ed quantitatively by the bistatic cross section
Using these partial results and introducing the notationin which & is as defined in Eq. (8) and A stands for theilluminated area. The latter may remain largely unspeci-fied because it drops out of the final expressions; one mustonly assume it to be finite and sufficiently large to prevent asizable effect of its edges on the far field. In the followingdiscussion we shall also consider only the radiation in theplane of incidence ( = r/2), both because this plane iswhere most of the response is present and because in thisway a certain simplification of the algebra occurs. We shallthus write
with p = cos Os + cos 0.We must now, as required in Eq. (31), calculate the ensem-
ble average of the squared absolute value of Eq. (32). Tothis end, it is convenient to calculate separately the followingaverages:
([T0(xy) - (To)I[To*(XY) - (To*)I)
= lexp[sp'c(x - x', y - y')] - 1jexp(-sp2 ), (35)
and also
M2(a, 1) = M,(a, 0) + MO,
MW(a, 13; a', 1') = M(a, 1) + M(a', 13')
- M,(a + a'; 1 +1'),
we find for the cross section the general expression
o-PQ(0 ) = 4- e sP{IQo2Mo + 2kpa 2
(43)
(44)
X ImLQO (2 )2 d d13 Q*(a, 1)M,(a, M)(a, O]
+ Y2J da d13Q(a, 312M 2(a, 13)c(a, 1)
- S p| dad1da'd13'Q(a, 13)Q*(a', 13')
(45)
Before proceeding any further with our examination ofexpression (45), let us consider for the diffuse radiation as
(38)
(39)
(40)
(41)
(42)
Cornel Eftimiu
X Wal 0; a" OX a(al VW, 01) -
880 J. Opt. Soc. Am. A/Vol. 7, No. 5/May 1990
well the special limits taken in Section 4 for the reflectioncoefficients.
If the correlation radius is sufficiently large to permit thereplacement of the Fourier transform of the correlationfunction by a Dirac function, then, because Q(0, 0) = 0.
k2 2
apQ - - esP IQo2MO. (46)k - 4
This expression vanishes if P d Q, i.e., if the incident andscattered polarizations are different. If P = Q = H, we findthat
kH 2 - 1 sin cos' - sin ' cos 0s 2
kRH -co S sin(0 + 0')
(47)
and if P = Q = V it follows that
VV '' - cos2 e IP tan(0 - 0')kR-co 7r tan(0 + 0')
cos S-cos 0 2sin 0 COS 0 + sin 0' COS 0'tM (48)
A comparison of the above expressions with the KA re-sults shows coincidence in the specular direction (Os = 0) buta difference consisting of a factor of cos-4 0 in the backscat-ter direction. The limiting expressions (47) and (48) thusindicate a behavior that is different from that of the KA onlyif cos 0 is significantly different from unity, i.e., away fromthe normal direction. We are not inclined to view this dis-crepancy as a serious flaw of our approach because the KA iswell known to be valid only for incidence angles that are notsignificantly (less than -20°) different from the normal di-rection.
For slightly rough surfaces, i.e., to the first order in s =(ka-)2, expression (45) reduces to
a-PQ - k sa(0, kq) PQO(O) + k Q(°)( kq) (49)
where the superscript to Qo and Q indicates that these quan-tities are obtained by setting s = 0 in their expressions. Thecross-polarized cross sections again vanish in this limit aswell. For P = Q = H we find that
a-HH ' - k2 M(0, kq)cos 2 0 cos2 aska-0 Vl
X I(Er-1)[cos 0 + (r - sin2 0)1/211]l
X [cos OS + (er - sin2 Os)1/21112, (50)
while for P = Q = V
a-VV - k2a(0, kq)cos 2
0 cos2 Aska--0 r
x (e -1)[(er - sin2 0)1/ 2 (Er _ sin2 O)1/2 - er sin 0 sin OS] 2
ler cos 0 + (r - sin2 0)1/2] [er cos 0s + (er - sin2Os)1/2]
(51)
These expressions coincide with the first-order PT results.Returning now to the evaluation of Eq. (45), we clearly
must substitute for QO and Q their explicit expressions andperform the indicated integrations. This calculation is, byfar, easier said than done. Moreover, the complexity of theformulas and the volume of operations involved are suchthat we will be able to present samples of the numericalresults only in graphic form. Nevertheless, a few observa-tions regarding the computational aspect are in order.
First, to evaluate the quantities MO, Ml, M2, and M 3 , whichare defined in Eqs. (40) and (41) and Eqs. (43) and (44), wemust explicitly evaluate the quantity AY, which is defined inEq. (42). [Note also that MO = W(0, 0).] If the exponentialcontaining the correlation function is expanded in a powerseries and the Gaussian model in Eq. (3) is adopted, then
Second, one should closely examine the terms involved inthe quantity IQ(a, 3)12 because this expression is the sourceof the difficulties encountered in Ref. 1 and mentioned inSection 1. Indeed, these difficulties can be readily traced tothe presence in Q(a, ) of terms typically containing a factorsuch as
k2 K2 '- k' 2 K, k2 (k'2 - a 2- 2)1/2 + k' 2(k2
- a 2- 2)1/2
In the limit of a perfectly conducting surface, i.e., when Im k'- , the above expression reduces to (k2 - a2 - 2)-1/2. As
exhibited, the singularity is benign (it was also present in theexpression of the reflection coefficient) because it is integra-ble. However, in Eq. (45) such terms appear in squaredabsolute values and hence lead to a nonintegrable singular-ity. Yet, one can see that as long as k' (real or complex) isdifferent from k, the above term is completely innocuous.Therefore, to recover the conducting surface case withoutencountering any calculational difficulties, all one has to dois to let Im k' be large (we varied it from 103 to 106 withoutnoticing appreciable differences) but finite. Evidently, thesurface then will not be perfectly conducting, but it will bejust as conducting or more conducting than any surfaceunder practical consideration.
Third, one should recognize that by far the most taxingnumerical work is required in the evaluation of the fourfoldintegral in Eq. (45). Double integrals do not pose much of aproblem, particularly if one realizes that, when polar coordi-nates are introduced, the angular integration can be per-formed analytically (typically, in terms of Bessel functions.)A close examination of M3 reveals that all of its terms permitthe fourfold integral to separate into products of two doubleintegrals, with one exception, namely, the term containing9(a + a', 13 + '). To treat this particular contribution, wechose to examine the possibility of its being likewise ex-pressed in products of double integrals; we found that, in-deed, this reduction can be done but at a price. When ourmodel from Eq. (3) is adopted, we get
with ,.mm'(a, 1) = am(# -kq/2)m'.In this manner, we give the expression 9(a + a', 3 + ') the
form of a (triple!) sum, each of the terms being the product
between a function of a and 13 only and a function of a' and 1'only. As a result, the fourfold integral in Eq. (45) is com-
pletely reduced to a number of products of double integrals.The size of this number clearly depends on the input valuesfor k- and kR-the larger these values, the larger the num-ber of terms necessary to produce an acceptable accuracy.Even if the individual double integrals can be evaluated
without much effort, if their number becomes too large oneeventually reaches a feasibility limit. All told, our computer
(VAX 8650) began to protest when we entered values largerthan ka = 2 and/or larger than kR = 10. Quite possibly, a
bigger machine and/or a programming virtuosity that is be-yond our capability might break these limits.
One of our principal concerns in a numerical investigation
of the cross section in Eq. (45) was to validate the resultsobtained in Ref. 1 for perfectly conducting surfaces. Asmentioned in Section 1, in Ref. 1 we first regularized thethird integral in Eq. (45), which was written in the limit Im k'- c, and then estimated that both the regularized integraland the fourfold integral contribute negligibly in the consid-ered ranges of parameters (ha- < 1, 1 < kR < 7). We thus
began the numerical evaluation of Eq. (45) by assigning alarge value to Im k'. Recalculating the results that werepresented in Ref. 1, we were able to confirm in toto our
estimates of the neglected terms. Some of these calcula-tions are shown in Fig. 4, in which we give the bistatic cross
section for er = 1 + i2000 and 0 = 450, for a selection of values
of ka and kR. As can be seen for k- = 0.33, no difference canbe detected between WH and WH-, which are the crosssections calculated with and without the last two terms inEq. (45), respectively, while the differences evident at ka =0.66 are for the most part less than 1 dB. Sizable differences
become manifest, as expected, at larger values of ka. Thevalues for ha and kR that are considered in Fig. 4 were also
chosen to correspond to some cases that are considered in
Ref. 2, in which the domain of the validity of the KA, and tosome extent of PT as well, is discussed by comparison with
so-called exact results obtained by Monte Carlo simulation.Although only the one-dimensional, rough-conducting-sur-faces case was considered in Ref. 2, we have no reason toexpect qualitative differences when two-dimensional roughsurfaces are considered. The parameters k- and kR in Fig.4(a) correspond to a case for which neither the KA nor PT
Fig. 4. Bistatic cross section as a function of scattering angle at 0 = 45, where e, = 1 + i2000.
Cornel Eftimiu
882 J. Opt. Soc. Am. A/Vol. 7, No. 5/May 1990
i -so-2(
-4(
40 80 0 40 so0 (dcg) 0 (dc)
Fig. 5. Backscattering cross section for HH polarization as a func-tion of the angle of incidence, where e = 20(1 + i).
should be assumed to be valid approximations. Our ap-proach produces a result visibly different from the PT pre-diction up to 200 and that coincides with the KA predictiononly for a small angular portion, from 200 to approximately500. It should also be noted that the maximum of the crosssection is clearly displaced from the specular value, which isan effect that is frequently observed. It is important quali-tatively that the PT curve at the beginning of the angularinterval be situated below the WH curve, while the KA curveis above WH and outside the range where they coincide.These features are in agreement with the behavior of the so-
C
5
1
-10
(a)
ka- *2.0 ,,,---- -_
W -,>- WH
I, ,' Pr_
,, _ __ _H "
. -
k.: 1.0,- 'I
10 kR 7 A* "
XYW FT
460 -20 20 600, (g)
called exact results as shown in Figs. 7 and 8(b) of Ref. 2. Ifha is doubled in value, as shown in Fig. 4(b), then the relativebehavior of the WH and the KA remains about the same, butPT becomes definitely inapplicable. This case again com-pares the WH favorably with the so-called exact resultsgiven in Figs. 6 and 8(a) of Ref. 2. A further increase of ha tothe value of 1.33 [Fig. 4(c)] leads to a completely aberrant PTprediction with a WH/KA deviation markedly apparentonly at the ends of the angular interval, as one would expectfrom Fig. 4 of Ref. 2 (note, in particular, the abrupt drop ofthe so-called exact result at the upper end of the angularinterval). The peculiar shape of the WH curve at the begin-ning of the angular interval is further accented if the valuefor ha is increased, as seen in Fig. 4(d). Whether this unusu-al dependence represents real behavior or is just a numericalartifact is, we believe, still an open question.
Addressing now the general case of a rough interface be-tween free space and a lossy, dielectric medium, we selectedfor illustration purposes what might be considered a typicalvalue for er, namely, 20(1 + i). In Figs. 5 and 6 we show thebackscattering and bistatic scattering (for 0 = r/6) crosssections, respectively, for four sets of values for (ka, kR),corresponding to the four corners of the domain consideredin Ref. 1. The lower-left corner of this rectangle is generallybelieved to be occupied by the domain of validity of PT,while the upper-right corner belongs to the domain of theKA. No distinct curves for WH and WH- are shown; thedifferences between them at any angle are, in this region, lessthan 0.1 dB.
The agreement between the KA and the WH at ha = 1 andkR = 7 appears good in both figures. In backscattering wefind complete overlap from 0 to 400, while in bistatic scat-tering the agreement is within 1 dB or so over the entireangular interval. For the same value of kR but for a muchsmaller ka = 0.1, the WH/KA agreement extends only up toapproximately 200 or so, with increasing (note the scale)deviations beyond this angle. For ka = 0.1 and kR = 1, theagreement between the WH and PT is good (note the scale)although not perfect, particularly at large angles. For k- =1 and kR = 1, neither the KA nor PT can be considered valid,and the WH behaves quite differently from either of them.
(C)
200, (dg)
Fig. 6. Bistatic scattering cross section at 0 = 300 as a function of the scattering angle, where e, = 20(1 + i).
(d)
kR7
Pub 11 WH \
(b) (d)
Cornel Eftimiu
ff
ao
I
I
aS
Vol. 7, No. 5/May 1990/J. Opt. Soc. Am. A 883
As shown, there is an isotropic tendency over a wide angularinterval, which should not be unexpected for the representedphysical situation.
Perhaps it is worth stressing that we have found no reli-able means for validating the WH except for cases like those
specifically indicated above. However, the recent trend 2 -4
to perform Monte Carlo simulations of so-called exact re-sults makes us hopeful that statistically well-characterizeddata for two-dimensionally rough surfaces will become avail-
able for validation of approximate schemes, such as the
scheme presented in this paper. We would be glad to coop-erate in any such efforts.
6. CONCLUDING REMARKS
We have presented in this paper an approach to the scatter-ing problem for the rough surface of an arbitrary, uniform,lossy medium, based on a modified Wiener-Hermite expan-sion to first order of the electric and magnetic surface cur-rents. At this time, we believe that this approach providessatisfactory results, at least in a domain defined by theinequalities ha 2 and 1 5 kR 5 10. The numerical calcula-tions for the diffuse response were restricted to the HHpolarization case, although consideration of other polariza-tion cases involves only additional computer time. Given
the sheer volume of required calculations, the extension ofthis approach by considering higher-order terms, which is inprinciple a straightforward procedure, might not be practi-cally feasible. An avenue for improved prediction capabili-ty, which is probably less demanding and possibly morefruitful, lies in the numerical aspect of the present develop-ment.
APPENDIX A
In terms of the notation introduced in Section 3, it followsfrom Eqs. (19)-(21) that
PX(KX, Ky-k sin 0)
F '2 u2 K 2-2 u 2 K'+ 2 (K,' + KZ)(U' U )1
l h/2 KZ - h2 KZ' Y h2 K2 - k2K J
Xa2A~-KI~ (K2,' + K2) (U'2 - u2)a-A
KXy S2 K k/ KZ - k 2K A
+ i~k (U' + U) h 2 K' - h 2K2' a-2 Bx
U + U /2k' 2K ' h2 K,- KK +KZK. ia-Bk k h'2K2 - h2a,' / Y
+ ih cos OFo0 - iKxk sin 0 K2 Foy
- iKx(1 + h cos0 h 2k- k2 ,y.k - K2
- K2K2 - kcos i co-
2K 2'- k2K+ i k 2 Gox + i k Ky(Ky k sin a)
-KZKZ -K, 2k Z Z -iKcos OGO,
FY(Kx, Ky- k sin 0) = -KxKy
rh'2u2Kz -kU2 +L + K.
Lh'2K - kz
(KZ' + K,)(U'2 -U 2) a-2Ak KZ- k 2K X
2 K, ' + K2 )(U ' 2 - u2) AX- h2K K ' -ak2K 5
+ U + U (K2 h2K2' -h2K2 2+ kh Ky h'2K - k2K2' + K2K 'B
Kx~y kh'2Kk- 2 KZ 2-(u' + U - h2K 2
7h~o h' K 2 h2
+ik COSOK~ sin h'K 2K')o
-iKy( +kcos0 'o2 -h2k2Z- h2
K2' O
i k 2k2(K0' - K2) -K K2 (k 2 - k2 )(K2' + K2)
+-k k'2K - k2K ' x
+-Kx(sinO - Ky *o, + X K cos 0 G02,
X(KX Ky - h sin 0) = 1K0Xy k 2 k2 a2Ax
+ f h 2K K 2 ) -_ _ _ _ _ _ _ _- _ _
X7k (K x jV Z ) K -k 2 K' A k 2 K2 -k2K /
X [(K X2
- KZKZ')(h2 COS2
0- KZKZ' -U 2 - U'2 )
K2 (k2 cos2 0 - KZKZ')] a-2B X KX K2Z - K2
X (2k2 cos 2 -2KzK' -U2 -u' 2 )a- 2 By
[7k (K2' - K)hK~ sin 01-inh~l + 2k 2 k2 KZ'
-ink 2 COS 0 K2 K2 K Fox
+ Kkcos ' + K2K')(k2 - hGoxcos 0 h'2K2 - h2
K2 '
X iK2 K Go - iK O,Y h,2K2 - h 2K2' x
Cornel Eftimiu
884 J. Opt. Soc. Am. A/Vol. 7, No. 5/May 1990
0y(K, K - k sin 0) = -(K2 K5K5') U - U , AXu'2 u2 k i2K 2 K
+ kKK U -U2 2 A + K KZ -Kk t-kZ k ' X Y' h/2K5 -k2K 1
X (2k 2 cos2 0 - 2KZKZ U _ U 2 )a 2BX + K5 - K2
ht2 K 2- 2K2'
X [(K 2 - KK')(k 2 COS2 0 - KKZ - U2- U 2 ) - KX (k2 cos 0
- KZKZ')] a-2 B + ikFox + ink2 sin 0 hK I' hK FoyK 2 'K 2 Z -k'2 -K2
+ ink KX cos 0 h'2 - k2K Fo0 - iKXKY h 2 K2 - 2 K 2 Ox
+ i(k cos 0 + K + K' + X h'2 _ k2K - iK Go,.
Because (KX, K - k sin 0) is an even function of Kx, multi-plication of PX(Kx, Ky - k sin 0) and GY(KX, Ky - k sin 0) by (Kx,Ky- k sin 0) and integration with respect to Kx and Ky yields
two liner equations for Ax and By in terms of Fox, GO, andGoz. Similarly, multiplication of Fy(Kx, Ky - k sin 0) andGx(Kx, Ky - k sin 0) by C(Kx, Ky - k sin 0) integration leads totwo equations for A, and Bx in terms of Foy, Foz, and Gox.
ACKNOWLEDGMENT
This research was conducted under the McDonnell DouglasIndependent Research and Development Program.
REFERENCES
1. C. Eftimiu, "Modified Wiener-Hermite expansion in rough-sur-face scattering," J. Opt. Soc. Am. A 6, 1584-1594 (1989).
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