Backscatter cross sections for randomly oriented metallicflakes at optical frequencies: full wave approach
Ezekiel Bahar and Mary Ann Fitzwater
The backscatter cross sections for randomly oriented metallic flakes are derived using the full wave ap-proach. The metallic flakes are characterized by their surface height spectral density function. Both spec-ular point and Bragg scattering at optical frequencies are accounted for in a self-consistent manner. It isshown that the average normalized backscatter cross sections (per unit volume) for the randomly orientedmetallic flakes are larger than that of metallic spheres.
1. Introduction
The purpose of this investigation is to determine theaverage normalized backscatter cross sections for ran-domly oriented metallic flakes. The irregular-shapedflake is characterized by its surface height spectraldensity function, and its lateral dimension is assumedto be larger than both the wavelength of the incidentelectromagnetic field and the correlation distance of therandom rough surface. Thus, scattering by the edgesof the metallic flakes is ignored.1
The full wave approach which accounts for bothspecular point scattering and Bragg scattering in aself-consistent manner is used to express the total crosssection of the flake as a weighted sum of two cross sec-tions (see Sec. II). The first is associated with thelarge-scale spectral components of the surface of theflakes, and the second is associated with its small-scalespectral components. The unit vector normal to themean surface of the flake is characterized by the polarangle OF and azimuth angle OF (see Sec. III). Througha suitable choice of the coordinate system, the averagewith respect to the azimuth angle OF is evaluated ana-lytically, while the average over the polar angle F isevaluated numerically (see illustrative examples Sec.IV). It is shown that the average backscatter crosssection/unit volume for the arbitrarily oriented metallicflakes considered is larger than that for metallic spheres.In Sec. IV the cross section of the metallic flake is alsocompared with the cross sections of similar flakescharacterized by either small-scale or large-scaleroughness.
The authors are with University of Nebraska-Lincoln, Departmentof Electrical Engineering, Lincoln, Nebraska 68588-0511.
To determine the scattering cross section for arbi-trarily oriented metallic flakes at optical frequencies,it is convenient to use a two-scale model of the roughsurface of the flakes. Thus, iY, the position vector toa point on the surface of the flake, is expressed as follows(see Fig. 1)
F, = F1 (x,hl,z) + ihs (1)
In Eq. (1), y = hi(x,z) is the equation of the surfaceconsisting of the large-scale spectral components, andh, is the height of the small-scale surface measured inthe direction of the normal () to the large-scale surface.It is assumed that the lateral dimensions of the flake LFare much larger than both the wavelength of the elec-tromagnetic waves and the correlation distance of therandom rough surface hs [Eq. (1)]. For a homogeneousisotropic surface height, the spectral density function2 4is the Fourier transform of the surface height autocor-relation function (h(x,z)h'(x',z'))
W(vx,vz) = f (hh') ep(ivxXd + iVzZd)dXddZd, (2)
where the symbol (-) implies statistical average, and(hh') is a function of I d I:
rd = (x - xO)x + (z - z')Nz = Xdax + Zdaz. (3)
The surface h is assumed to consist of the spectralcomponents
kF<k = (V2+V2)112<kd, (4)
where kF = 2/LF is the smallest wave number char-acterizing the surface of the flake, and kd is the wavenumber where spectral splitting (between the large- andsmall-scale surface) is assumed to occur. The surfaceh, consists of the small-scale spectral components
Fig. 1. Plane of incidence, scattering plane, and reference (x,z)plane.
where kc, the spectral cutoff wave number5 is the largestwave number characterizing the flake.
The full wave approach, which accounts for bothspecular point scattering as well as Bragg scattering ina self-consistent manner, is used in this work to deter-mine the scattering cross section of the composite modelof an arbitrarily oriented flake.6 Thus, the total nor-malized scattering cross section/unit area (PQ) is ex-pressed as a weighted sum of two cross sections
(UyPQ) = (PQ)i + (PQ)., (6)
in which (PQ ) is the cross section associated with thelarge-scale filtered surface h, and the cross section(o-PQ )s is associated with the small-scale surface hs thatrides on the large-scale surface h,. The first superscriptP corresponds to the polarization of the scattered wave,while the second superscript Q corresponds to the po-larization of the incident wave.
On deriving the full wave solution it is implicitly as-sumed that the wave number where spectral splittingoccurs, kd, is chosen so that the large-scale surface hlsatisfies the radii of curvature criteria (associated withthe Kirchhoff approximations of the surface fields) andthe condition for deep-phase modulation. The scat-tering cross section (PQ) 1 is given by7 ,8
(UPQ)I = IXs(V.1is)j2 (.P'Q) (7)
in which s is the characteristic function for thesmall-scale surface
Xs(v .ns) = xs(v) = (expivh ),
= f - i = omf - ) = .
(8)
(9)
The unit vectors Wi and f are in the directions of theincident and scattered waves, and ho = o(@yoco)1/2 is thefree-space wave number for the electromagnetic waves(go and co are the free-space permittivity and perme-ability). An exp(icot) time dependence is assumed inthis work. The vector nis is the value of the unit vector1i normal to the surface h(x,z) at the specular points.Thus,
(10)
(11)
(12)
The expression for the physical optics cross section(opQ) for the surface h is
(aQ) -k 2= i D- P2(HHi)p(H)] (13)V2 tIni. li in
in which DPQ depends on the polarization of the inci-dent and scattered waves, the unit vectors fif, nf, andn and the relative complex permittivities and permea-bilities of the flake. The shadow function P 2 is theprobability that a point on the rough surface is both il-luminated by the source and visible to the observergiven the slopes ii(h ,h,) at the point.9 10 The functionp (ii) is the probability density of the slopes h. and h,.The coefficient XsJ 2 that multiples (Q) is aweighting function that accounts for the degradationof the specular point scattering cross section due to thesuperimposed small-scale rough surface hs.8 Thus, ashts - , Xs 12 _1.
The scattering cross section (PQ)s for the Gaussiansurface hs that rides the large-scale surface h, is givenby the sum6
(IYPQ) = (Q)sm,m=1
(14a)
where
(UPQ).m = 4irk' JfIDPQjP2 (fJni1ff)- a,
X exp (V2(S)) [VI2= Wm(VYVi) p(h,,h,)dh~dh,.
(14b)
In Eq. (14) (2) is the mean square of the surface heighths, and vx, vy, and v7 are the components of [Eq. (9)]in the local coordinate system associated with the unitvectors ii1, ii2, and n3 (at each point on the large-scalesurface hl, see Fig. 2). Thus, V is also expressed as
Y
I' Local Coordinoles
JncidentC cW ot Plane)
r , N mu _ -In,~~2' n \rmal to rough surface
LocalScatter Plane
Fig. 2. Local plane of incidence and scatter and local coordinate
The function Wm (v-,,v)/22m is the 2-D Fourier trans-form of (hjhi) m:
W. (Vv7)= OT sm exp(ivxd + i7zd)dxddzd22m (2r)2
= I f JWm-i(v,v;)Wi(vx - v;,v-z - v;)dv'dv'
1= - Wm-1(V.,V-)0 Wi(V-,V-Z).
22m (17)
In Eq. (17) the symbol 0 denotes the 2-D convolutionof Wm with W1. The surface height h, is measurednormal to the surface y = hl, and I d ffl + Zd 3l is thedistance measured along the large-scale surface hl. The2-D Fourier transforms of the surface height autocor-relation function (hshs) is equal to the spectral densityfunction W1(ui,vj)/4. When the parameter 4ko (h2)
3 << 1 and ay, the first term in (14) accounts forfirst-order Bragg scattering, and the higher-order termsm > 2 may be neglected. In this case, the full wavesolution Eq. (14) is in agreement with Brown's solution5
based on a combination of physical optics and pertur-bation theory. However, since it is assumed using thefull wave approach that the condition for deep-phasemodulation is satisfied, it is necessary to choose =4k2(h2) > 1. Since fewer terms in Eq. (14) need to beevaluated for smaller values of 3, in this work the valueassumed for /3 is 1.0.
While the perturbed-physical optics solutions511 12
for the cross sections critically depend on the choice ofkd (the wave number where spectra splitting is assumedto occur), the full wave approach is insensitive to vari-ations in kd for /3 > 1.6
Since the metallic flakes are randomly oriented withrespect to the fixed observer, in Sec. III the analyticalresults presented are modified to account for arbitraryorientation of the normal to the mean surface of theflake.
Ill. Scattering Cross Sections for Arbitrarily OrientedMetallic Flakes
Let x',y',z' be a fixed reference coordinate system,and let xy,z be a rotated coordinated system so that theunit vector ay is normal to the mean surface of the flake(y = 0) (see Fig. 3). For backscatter it is convenient tochoose the unit vectors nfjf = -i so that
nf = -i =
Nz = (i X liy)/l ni X ay and a, = ay X ,. (20)
The expression for fii in the flake coordinate system is,therefore,
Hi = (i N.)iax + (i - NY)Ny = sinOibi - cosOt , (21)
where
cosob =-ni*ay = cosOF. (22)
The angle ' F between the reference plane of incidence(normal to !G in the fixed coordinate system), and theplane of incidence in the flake coordinate system (nor-mal to ii X ay) is
Co~f = az ( X ay) = -[i'1lY = a, -i a = t,cs/F - IifX 1YI iiii Nyj Ii x N1 OSF
thus
- X (i X ay). isinC = = sinPFr.F I ii X ay
(23)
(24)
Since nf = -iii for backscatter, the angle Q14 betweenthe plane of scatter in the fixed (x',y',z') coordinatesystem and the plane of scatter in the flake coordinatesystem (x,y,z) is
(25)
Thus, the matrix that transforms the vertically andhorizontally polarized incident waves of the fixedcoordinate system to the vertically and horizontallypolarized incident waves of the flake coordinate system
7,8
cos4' sin 1F-sin4 ' cos4'1j (26)
yt
-f
ni
x
(18)
The unit vector ay normal to the mean surface of theflake can be expressed as follows in terms of the fixedreference coordinate system
ay = sinOF CoSOFax + cosOFNY + sinDF sin0Fa-. (19)
For convenience the two orthogonal unit vectors i, andaz in the mean plane of the flake are chosen so that theplane of incidence in the flake coordinate system is thex,y plane (normal to N,). Thus, Fig. 3. Randomly oriented flake.
Similarly, the matrix that transforms the vertically andhorizontally polarized scattered waves in the flakecoordinate system back into vertically and horizontallypolarized scattered waves of the fixed coordinate systemis
T-f [C4 -Sin4 f] (27)lsintF cos4' F
Thus, in view of Eq. (25) for backscatter,
T = Ti (28)
The coefficients DPQ in Eq. (14) are elements of the 2X 2 matrix D given by7' 8
D = Ci'TfFTi, (29)
in which Cin is the cosine of the angle between the in-cident wave normal i [Eq. (21)] and the unit vector ffnormal to the rough surface of the flake [Eq. (10)].Thus,
= =- - U = cosY cosOF -siny sinOF cos. (30)
The elements FPQ of the scattering matrix F in Eq. (29)are functions of the unit vectors jji,jjf,fj and the relativepermittivity and permeability of the flakes.7 Thematrix Ti transforms the vertically and horizontallypolarized incident waves of the x,y,z coordinate systemto vertically and horizontally polarized waves of thelocal coordinate system (, 2, 3 ) associated with therough surface of the flake [Eq. (16)]. Similarly, thematrix Tf transforms the vertically and horizontallypolarized scattered waves of the local coordinate systemback into vertically and horizontally polarized waves ofthe x,y,z coordinate system.7 8
To account for the arbitrary orientation of the flakewith respect to the fixed (x',y',z') coordinate system[Eq. (29)] must be postmultiplied by Ti and premul-tiplied by TF. Therefore, in the expressions for thescattering cross sections [Eq. (6)], the elements of matrixD must be replaced by the elements of matrix DF givenby
uniformly distributed in the interval [0,27r] the back-scatter cross section for an ensemble of randomly lo-cated flakes is given by the sum of the individualbackscatter cross sections.1 Thus, accounting for therandom orientation of each individual flake, the averagebackscatter cross section of a single scatterer in theensemble of flakes is
,, fPQ f= r (('Q)P(OF,0bF)d 0FdOF
- (aPQ)P(0F)d OF,
in which it is assumed that the unit vector normal to themean surface of the flake is uniformly directed in thehalf-space 0 • OF ' 7r/2, 0 < OF < 27r and
P(OFF) = SinOF -P(OF)Pwp;-2-r 2ir (38)
Since (PQ) is dependent on OF only through the sinesand cosines of the angles (4 i + OF) [Eq. (36)], it is con-venient to evaluate the average of (ecrPQ) with respectto the angle OF analytically by first evaluating the av-erage of IDPQI 2 with respect to OF. Thus, for P ao
The expressions for FVV and FHH are complicatedfunctions of OF. Thus, the integrals with respect to OFare not evaluated analytically. Instead, (PQ)O [Eq.(37)1, is evaluated numerically using Eq. (14) withI D Pli 2 (p,Q = V,H) replaced by I D 2 [Eqs. (39) and(49)]. To obtain the ensemble average for the scatteringcross sections (SPQ) [Eq. (37)], the result from theprevious integration (PQ) is multiplied by P (OF) =sinOF and integrated with respect to OF. Thus,summing up, the integral (37) is performed analyticallywith respect to OF and numerically with respect to OF.The integrations with respect to hh, [Eq. (14)] are alsoperformed numerically.
IV. Illustrative Examples
The specific form of the surface height spectral den-sity function [Eq. (2)] selected for the illustrative ex-amples presented in this section is
-B/k 4 , kF < < kcW(vxvr) = 0 k > andk <kF, (41)
in which
B = 0.016
k2 = V2 + V2 (cm)- 2
kF = 2r/LF and LF = 0.002 cm (42)
= 0.45 X 106 cm 1
[For naturally generated surfaces, W can be approxi-mated by kn, where n is between 3 and 4.4] Thewavelength of the electromagnetic wave is
Xo = 0.555 x 10-4 cm = 27- = 1.132 X 105 cm-i) (43)
The relative complex dielectric coefficient for the alu-minum flakes at the assumed optical frequency iS13
f 2, fkd W(k) B 11(h 2)f=- k dkdo 2~i ~
- 7 f 4 2 XF idm
= 0.791 X 10-9 C2. (48)
Thus, ho (h 2) >> 1 and the physical optics treatment ofscattering by the large scale structure is justified. Thetotal mean square slope of the large-scale surface h,is
2 2) g2 fkd WkUt= (hi f fW(k) k3dkd0b0 kF 4
= B In (V-) = 0.298 X 10-1. (49)
The slope probability density is assumed to be Gauss-ian, thus,
1 (hx2+h,2p() = 2 exp 2
7ralt alt(50)
In Figs. 4 through 8, the backscatter cross section(0 .PQ ) o averaged over the range of the azimuth angle OF
is plotted as a function of the polar angle OF (see Fig. 3).This is formally given by
1 f2
(UPQ)0 = 1 (,PQ)d0F.27- J (51)
However, in Sec. IV it is shown that Eq. (51) can beevaluated directly on replacing ID PQ 2 in Eq. (14) byits average value DPQ I 2 [Eqs. (39) and (40)].
In Fig. 4, (PP)o (P = V,H) is plotted as a functionOf OF for the composite surface characterized by thespectral density function [Eq. (41)]. This cross sectionincludes the effects of both the small and large rough-ness scales of the surface of the flake. Thus, it takesinto account both specular point scattering and Braggscattering. In view of Eq. (40), the average cross sectionis the same for vertically and horizontally polarizedwaves. The average cross section of a single scattererin the ensemble of randomly oriented flakes is (PP)= 0.77 [Eq. (37)]. For a conducting sphere of radius aF,
the normalized cross section is 0.92 provided that koaF>> 1 and er is given by Eq. (44).
In Fig. 5, (PP) (P = V,H) is plotted as a functioncr = -40 - i12, (44)
and the permeability of the flake is assumed to be thatof free space ( = 1).
The mean square height for the small-scale surfaceh, (kd <k < kc ) is given by
d 4
B (1 11 I B -_0 = 0.195 X 10-1o cm2
= 2 (45)2 \k k
since in this work: = 4k2(h2) = 1.0. The wave numberwhere spectral splitting (between the small and largeroughness scales) is assumed to occur is
Fig. 6. (HH)o = (aVV)o for the small-scale surface ho as a functionof OF.
b
b
CD W io.00 2a.00 30. W 6O.u 56.W 0 .00 O 7.00 a F
Fig. 7. (a VH) o = (aH V) O for the composite surface h as a functionof OF.
of OF for the large-scale filtered surface only (kF < k <kd). Thus, only specular point scattering is consideredhere. In this case (PP) = 0.47. In Fig. 6, (PP) (P= V,H) is plotted as a function of OF for the small-scalesurface only (d < k < kc). Since d -t 0 for this case,to evaluate O-PQ), using Eq. (14), p(h,,h,) is set equalto the product of the Dirac delta functions b(h.,)b(h,).Thus,
(a~Q)sm 4r7k2 IDPQI2
2 M ! YmVV)
and the leading term in the sum (PQ), (i.e., m = 1)reduces to the first-order Bragg scattering cross section.For this case, (OPP), = 0.5. The corresponding co-herent scattering term for a smooth flat flake3 is (o.PP)
= 0.32. A flake with a two-scale roughness has, on theaverage, a larger backscatter cross section than a cor-responding flake with no small-scale roughness. Since,for the randomly oriented flakes, (KNPP) = 0.77, theirbackscatter cross sections are on the average smallerthan the cross sections for spheres with a cross-sectionalarea ra 2 = L2 ((&'') = 0.92). However, consideringthe fact that a volume of nine flakes each of thicknessd LF112 is approximately equal to the volume of asphere of radius aF, for a given volume of particles thebackscatter cross section for the flakes is 7.5 times largerthan the cross section for the spheres. In Fig. 7 thecross-polarized backscatter cross section (averaged overOF) (IVH)L a -HV)O is plotted as a function of OF. Thevalue of (o- VH) (0.522 X 10-2) is significantly smallerthan the average backscatter cross sections for the likepolarized case (PP). In Fig. 8, (PQ) 0 (P # Q) isplotted as a function of OF for the small-scale surfaceonly (kd < k < kc) [Eq. (52)]. At near-normal incidencethe cross section ( a VH) 0 is very small since I D VH l 2 isvery small for backscatter when OF << 1. Furthermore,since W, = 0 for k < kd, therefore, (u VH)s 1 -> 0 for OF- 0. For grazing angles, higher-order Bragg termsbecome very small and (oVH)- ) (y VH)Si in Eq. (14).For the flake with the small-scale roughness, (o- VH)
0.52 X 10-2.
V. Concluding Remarks
The average normalized backscatter cross sectionsfor arbitrarily oriented metallic flakes are derived usingthe full wave approach. Thus, the total backscattercross section is expressed as a weighted sum of two crosssections. The first is associated with the filtered surfaceconsisting of the large-scale spectral components of thecomposite rough surface, while the second is associatedwith the surface consisting of its small-scale spectralcomponents that ride on the large-scale surface.14-16
These backscatter cross sections are compared withthose of similar flakes having either a large- or a small-scale surface roughness.
It is shown that the average normalized backscattercross section (per unit volume) for the flake with thecomposite surface is larger than the backscatter crosssections for metallic spheres.
Fig. 8. (a VH)o = (aH V)o for the small-scale surface h. as a function
of OF.
This paper was sponsored by the U.S. Army ResearchOffice, Contract DAAG-29-82-K-0123. The manu-script was typed by E. Everett.
VI. References
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book (Plenum, New York, 1970), Chap. 9.4. A. Ishimaru, Wave Propagation and Scattering in Random
Media in Multiple Scattering, Turbulence, Rough Surfaces andRemote Sensing, Vol. 2 (Academic, New York, 1978).
5. G. S. Brown, IEEE Trans. Antennas Propag. AP-26, 472(1978).
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7. E. Bahar, Radio Sci. 16, 331 (1981a).8. E. Bahar, Radio Sci. 16, 1327 (1981b).9. B. G. Smith, IEEE Trans. Antennas Propag. AP-15, 668
(1967).10. M. I. Sancer, IEEE Trans. Antennas Propag. AP-17, 577
(1969).11. T. Hagfors, J. Geophys. Res. 71, 279 (1966).12. G. L. Tyler, Radio Sci. 11, 83 (1976).13. H. Ehrenreich, IEEE Spectrum 2, 162 (1965).14. J. W. Wright, IEEE Trans. Antennas Propag. AP-14, 749
(1966).15. J. W. Wright, IEEE Trans. Antennas Propag. AP-16, 217
(1968).16. G. R. Valenzuela, Radio Sci. 3, 1051 (1968).
': AIK. Kompa of Lambda Physik Gottingen photographed by W. J. Tomlinson of Bell Laboratories
