389757-2-T SCATTERING BY AN APERTURE FORMED BY A RECTANGULAR CAVITY IN A GROUND PLANE December 1989 Kasra Barkeshli and John L. Volakis Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, MI 48109-2122 Northrop Corp. Aircraft Division One Hawthorne Ave. Hawthorne, CA 389757-2-T = RL-2572

Technical Report 389757-3;.T Scattering by an Aperture formed by a Rectangular Cavity in a Ground Plane Kasra Barkeshli and John L. Volakis Department of Electrical Engineering and Computer Science University of Michigan Ann Arbor, MI 48109-2122 December 1989 Submitted to: Northrop Corp. Aircraft Division One Hawthorne Ave. Hawthorne, CA

Table Of Contents page I. Introduction 1 II Traditional Solution 1. Problem Definition 2. Formulation 3. Numerical Solution via Galerkin's Method III Solution via Generalized Impedance Boundary Conditions 3 3 4 11 18 29 IV Results

Report Summary This report describes two formulations for computing the scattered field by a material filled depression/cavity in a ground plane. One of the formulations is based on a traditional modal approach where the fields within the cavity are expressed in terms of the cavity's Green's function and those in the open region are written in terms of equivalent magnetic currents placed on the surface of the cavity. A system of equations is then constructed via Galerkin's method to solve for the equivalent currents by enforcing continuity of the fields accross the aperture. The other formulation involves use of first, second and third order generalized impedance boundary conditions(GIBCs) to construct a system of equations for a solution of the equivalent currents. In this manner, by avoiding use of the cavity's Green's function the resulting system of equations can be solved via the conjugate gradient FFT method which has an 0(N) memory requirement. In addition, the GIBC formulation is applicable to non-rectangular cavities whereas the traditional modal formulation is limited to rectangular cavities. The GIBC formulation was found to yield acceptable results in the case of non-rectangular cavities and those rectangular cavities filled with lossy material. In the case of rectangular cavities, though, they were found to require supplementation with additional conditions to be imposed at the periphery of the cavity.

I. Introduction Traditionally, the standard impedance boundary condition (SIBC) [1] has been employed to simulate dielectric coatings on perfectly conducting objects. As is well known, however, the SIBC provides limited accuracy and is particularly applicable to lossy and/or high contrast dielectrics. This is primarily because it cannot model the polarization current components that are normal to the dielectric layer. The SIBC has been found to be best suited for near normal incidences, unless the coating's material properties are such that limit penetration within the material. The SIBC is a first order condition in that its definition involves a single normal derivative of the component of the field normal to the modeled surface. Recently [2], however, a class of boundary conditions were proposed whose major characteristic is the inclusion of higher order derivatives (along the direction of the surface normal) of the normal field components. These were originally introduced by Karp and Karal [3] and Wienstein [4] to simulate surface wave effects, but have been found to be rather general in nature. In fact, they can be employed to simulate any material profile with a suitable choice of the (constant) derivative coefficients. Appropriately, they are referred to as generalized impedance boundary conditions (GIBC) and can be written either in terms of tangential or normal derivatives provided a duality condition is satisfied [2]. Unlike the SIBC they offer several degrees of freedom and allow an accurate prediction of the surface reflected fields at oblique incidences. This was demonstrated in [2] for the infinite planar surface formed by a uniform dielectric layer on a ground plane. It was found that the maximum coating thickness that can be simulated accurately with a given GIBC was proportional to the highest order derivative included in the condition. The GIBCs offer several advantages in both asymptotic and numerical analyses of electromagnetic problems. For example, in the case of asymptotic/high frequency analysis, they allow an accurate replacement of a coating on a layer with a sheet boundary condition amenable to a Wiener-Hopf analysis [5,6] or some other function theoretic approach [7]. In numerical analysis, the profile of a coating can be replaced by a simple 1

boundary condition on the surface of the coating. This eliminates a need for introducing unknown polarization currents within the coating or material layer and thus leading to a more efficient solution. In an earlier report [8] we presented an implementation of a third order GIBC for scattering by a two-dimensional rectangular cavity in a ground plane. Here we present a corresponding implementation for a three-dimensional material-filled rectangular cavity in a ground plane. To examine the accuracy of the GIBC simulation, as before, it would be necessary to compare the derived equivalent currents and scattered fields with those obtained from a corresponding traditional formulation. Therefore, before proceeding with the GIBC simulation, we first present a moment method solution in conjunction with the modal Green's function representing the cavity fields. This procedure is standard and has been extensively employed in corresponding two-dimensional simulations [9]. However, its application to three dimensional apertures formed by cavities, although straightforward, is voluminous, and has only been considered recently in [10] for computing the radiation pattern of loaded rectangular waveguide arrays. Here, a similar pulse basis Galerkin's formulation is employed for computing the scattering by a single rectangular cavity in a ground plane with plane wave illumination. Following a presentation of the traditional moment method formulation, the corresponding integral equations based on the GIBC simulation are derived. These are subsequently placed in a form suitable for a conjugate gradient-FFT (CGFFT) solution having 0(N) memory requirement. In contrast, the traditional integral equation must be solved via a matrix inversion or LU decomposition approach having O(N2) memory requirement. As discussed in connection with the two dimensional implementation[8], the GIBCs are inherently inaccurate near the termination of the cavity and must, thus, be supplemented with additional conditions there[l 1], [12]. A hybrid procedure was then developed to overcome this inaccuracy without appreciable increase in the low memory 2

requirement associated with GIBC formulation. This is again employed here for the threedimensional application of the GIBC formulation and the pertinent details are given in the appropriate section. II. Traditional Solution 1. Problem definition i H 00 V X --- ground plane Fig. 1. Geometry of the cavity aperture in a ground plane. Consider the cavity geometry in figure 1 illuminated by the plane wave -i + E9 + E e E = o ^EYoY + Ezo e I~Wz (la) (Hxo A+HYoA+AHzo -jko(ki, ) H =Y txE= Hxox +Hy + Hzo *r (lb) where Yo = 1/Zo is the free space intrinsic admittance, ko =27rA is the free space wave number, 3

in = [ ( sin (cos + sin ) + cos 0 ] (2) - A A A) r =xx + y + zz (3) Exo = cos a cos 0o cos qo - sin a sin o (4a) Eyo = cos a cos 00 sin 0) + sin a cos 0o (4b) Ezo = - cos a sin 00 (4c) Hxo = Yo (sin a cos 00 cos o0 + coso( sin o0) (4d) Hyo = Yo (sin a cos 00 sin 40 - cos c< cos o0) (4e) and Hzo = - Y sin a sin 00 (4f) in which a represents an angle specifying the polarization of the incident field as illustrated in figure 2. For example, when a=7/2, Ei = 0, corresponding to E-polarization incidence and when a=0, Hz' = 0, corresponding to H-polarization incidence. 2. Formulation The usual approach for computing the scattering by the cavity in fig. 1 is to employ the equivalence principle and separate the structure into internal and external regions. The internal region is the cavity region below z=0 and the external one represents the open region above the cavity (z>0). From a knowledge of the Green's function associated with each region, their corresponding fields can then be expressed in terms of the equivalent currents at the aperture. A set of coupled integral equations for the equivalent currents are thus derived by enforcing continuity of the tangential fields across 4

the cavity's aperture. In accordance with the equivalence principle, the aperture is closed by a perfect conductor and the equivalent magnetic currents M = E x = E x = + Ey- E (5) are placed on the aperture at z=O+. Referring to fig. 3, by invoking continuity of the tangential E field it is clear that the sources of the fields within the cavity (internal region) are the magnetic currents M'= Ex = x (-)=- M (6) placed just below the aperture at z=0-. It remains to also enforce continuity of the tangential H field across the aperture. Denoting the fields in the external region (z > 0) as (Ea, Ha) and those in the internal region as (Eb, Hb), we have a i b Hx + 2H = H; z=0 (7a) and a i b Hy + 2Hy =H; z=0 (7b) where 2Hxy represent the fields present in the external region in the absence of the magnetic currents. These conditions are seen to represent coupled integral equations for M once Ha and Hb are written as integral expressions in terms of the aperture equivalent currents. The external fields can be expressed in terms of the free space dyadic Green's function, Fo, as ~a - - = H = jk Y JJ2M(r') * rF(r, F') ds' (8) Sa where Sa denotes the aperture surface, 5

r(r, r') = + k 2 vv VjIC/? (9) (10) -jkoR G.(r, r')= e 4/R (11) R = I - r'I = x - x')2 + (y - y,) and the factor of 2 accounts for the ground plane's presence. More explicitly, (12) a jYk 2fJ Sa r 2 ax2 1 Mx(x', y')k + I + My(x', y') xay r G(r, r') dx' dy' (13) and a jiYo Hy= Y ko 2f Mx(x', y') a)2 axy +My(x', y') ax~y r + 2 L a I Gor, F') dx' dy' (14) The internal fields can be written in terms of the TMz and TEz cavity modes. We have b -TM -TE E =E +E -b -TM -TE H =H +H (15) (16) (17) where -TM TM jZbV(T M E =-zjkbZZbT V(V.z ') kb 6

-TM H =Vx (z ) (18) HTE ^. byi TE JYb V(V TE) (19) T — TjkbY (19) and E =-V x ). (20) In (17) - (20) kb = ko (21) b I zo _ (22) Z b= Z in which eb and Pb denote the relative permittivity and permeability of the material filling the cavity. Furthermore, the functions 'irTM and YTE are the electric and magnetic potentials. They both satisfy the wave equation (X2 aky2 aZ2(23) subject to the boundary condition Ex = Ey =0 @ z =-c (24a) Ex = Ez =0 @ y = 0 and y = b (24b) Ey = Ez =0 @ x = 0 and x = a (24c) on the cavity walls. Referring to fig. 1, a suitable set of functions satisfying (23) and (24) is 7

00 m=l 00 YA sin mn x sin nby cosym(z+C) n=l (25) (26) 00 00 T E Bmn Cos x Cos y siny (z + c) a b m=0 n=O where Amn and Bmn are constants to be determined and 2 2 2 (mr n7C) in accordance with the wave equation. Substituting (25) and (26) into (15) - (20) yields 00 00 b ZZmn E = E m=0 n=0 m=0 nrO m4or r4 (27) (28) (29) where Ex = jZb a + bBmn cos m x - b Amn a sin n y sinyn (z + c) <mn Al I-Br s nr Eymn = [JZb ) mm sin -x cos y sinx cn (z + c) a b Eyn m Amn - sin ma7 x sin y cos ym (z +c) kb a b 1(nn} nm7 H x + jY ' i nsin -a x cos b y cos ym (z + c) kb a (30a) (30b) (30c) (31a) 8

= KaJAI +jb bJ Bmn cos ~~x sin 1~ y cosy7. (z+ c) and Hz = kb b a b (3l1b) (31 c) It remains to evaluate the mode coefficients Amn afld Bmn in terms of the equivalent currents M(x, y). This is easily accomplished by enforcing the boundary conditions ) Z =O0 Z =O (32) (33) BE =-M'=M~ in conjunction with mode orthogonality. We have Amn {bb {M aJ(nn - 1 21Fnit mn (mm Imn]M,n> otherwise (34) and FI ab sin Y-c LY a I -1 - n7u I mn + 6 nm + b )i - a ) x rn b ) 'y j m~!O n>O m =n =0 L in which?ninr It JJ y1)si x Cos Sa m y')dx' dy' (36) 9

,n ) fm7r xf fn7c I I Iy = J My(x' y') cos a x J sin [ Y dx dy Sa (37) and Em= 2 m=O m> 1 (38) -b The required components of the magnetic field H to be employed in (7) can then be explicitly written as -bb 2j Yt 0000 F 1 r X 21 m I ab k0b O mn tan jmnC L a 2 mn m a b jn H,(x~y~-0) —[. ^ ^~ a bc ^ ^ j -Iy- J *sin m x cos n a b (39) Hy (x, y, z=0) 2j Yo 0y 1 b kob Lmn tan 'mn C A =O n=O-0 -n a b Ix + m k Iy cos m x sin by. a b As a check, for b->oo b AY 12 2mn lim Hy (x,, z=0) = - y tan y c 2 Ib Substituting for Iym and setting mn = Ym k - ( a)2 then yields Y b a hnyed (40) (41) Hyb (x,y,z=O) b jkbYE m S os x FMy(x) cos mCT x' dx' a Ym tan Ymc r/ 0... (42) where 10

Sy = sin- b -Y sin b Y dy'. (43) n=0 0o and Sy should equal unity for (42) to reduce to the known two-dimensional result. This can be easily verified by invoking distribution theory. Specifically, interchanging the order of summation and integration yields Sy =y sin b y' sin y 00 Sy=J (b2) j dy' = 8J(y- y') dy'=1 3. Numerical Solution via Galerkin's Method Using the method of weighted residuals, from (7) the integral equations to be solved for M(x, y) are a b a b JJH - HJ W(x, y) dx dy = -2 JH W(x, y) dx dy (44) 00 00 where Ha is given by (13) and (14), H by (39) and (40), and W(x, y) is the weighting function. In the case of Galerkin's method this is chosen to be the same as the expansion function. To discretize the integral equation (44) we first expand M(x, y) as Nx-1 Ny-1 M(x, y) = Mq P(x - Xp) P(y - yq) (45) where xp = pAx + = qy + where Xp = pAx + Ax, yq = qAy + Ay, 11

Ixl < A 2 P(x) = (46) 10 otherwise, 1 lyl < 2 P(y) = (47) 0 otherwise, and Mpq = x Mxpq + 9 Mypq represent the unknown coefficients of the pulse basis function. In accordance with Galerkin's technique we also choose W(x, y) = x P(x - x) P(y - yj) + 9 P(x - X) P(y - y). (48) where i = 0, 1,..., Nx -1, and j = 0, 1,..., Ny -1. Substituting (48) into (44) we obtain the set of integral equations Ax Ay Ax Ay Xi+ -- -x+- y.+ 2 2 2 Ij (Hxa-Hdxd) dxdy= (49a) Ax x Ax x i2 Y 2 X 2 2 Ax Ay Ax Ay 2 YJ2 2 x- 2 j (Hy-Hy dx dy - 2Hy dxdy. (49b) -Ax Ax Ax Ay i2 - 2 2 j- 2 When (45) is further introduced into the integral expressions for H and H, we obtain the system 12

r yaxyb YrYQ - Y M I ixx -X Yx Yxy M Ix l = -.I j (50) lyyx yx yy- Yyyj J MY To compute the admittance elements Y x Y a Yya and Yya we refer to (13) and (14). we t Xo a YX yy Introducing the expansion for M(x, y) as given in (45) we obtain expressions in terms of the integral Ax Ay Ax Ay xi+ 2 yjI 2 Xp. 2 Yq+ 2 g.=g( yxi, yj, J e d'd' 4qjR dx' dY' dx dY Ax Ay Ax Ay '2 Yj2 XP2 YP 2 5 —2 Y —T:JJJJ 4 d' d' ddy' (51) S.. Sp and its derivatives in which R= (x-x')+(y-y,)2 (52) Clearly, when ii - pl < 1 and Ij - ql < 1, gijpq requires analytical evaluation. To do this, we rewrite gijpq as e1 gipq 4= | Ri 4- d' dy'dx dy + J R dx' dy' dx dy. 4n Spq S ij SR Sis SJ (53) The first integral has a non-singular integrand for all i, j, p and q and can therefore be evaluated numerically using, for example, Gaussian integration. In contrast, the second integral has a singular integrand when i=p and j=q but can be evaluated analytically as 13

r r dx' dy' dx dy (x - )(Y Y) (x - x')ln [(y - y) + R] sR 2 Sij + (y - y') In [(x - x') + R]} Ax Ay - _ _ A 6 i i Ax I Ay xJ Y=yq_ (x - ')(y -y')[(x - x') + (y - y')] 4 Ax Ay I I I Ax I Ay x-x.- - 2 (54) Unfortunately, the derivatives of gijpq cannot be evaluated analytically. A possible alternative is to evaluate them discretely using the computed values of gijpq, and a convenient way to do this is to employ the discrete Fourier transform (DFT). Proceeding in this manner, for a given (xi, yj), we first define the sample train Nx-1 Ny-1 ij= gijpq (x - Xp) (y -yq) p=O q=0 whose two-dimensional DFT will be denoted by x... We also note that a - A D4 j2tfx sinc (nfx Ax) = jDx Tx Ax a - A D= j2tfy sinc (fy Ay) = jDy 0y '"Ay (55) (56a) (56b) in which fx = pAf = p/(NX Ax); fy = qAf = q/(Ny Ay); p = 0, 1, 2,..., Nx-1 q=0, 1, 2,..., Ny-1 and 14

sin 7 sine (=()= -i (57) Consequently, the DFT of the sample train (with xi and yj kept constant) N.-I Ny-I ao2 e^R Xx =.Z a 4R dx' dy' dx dy] (x- p) (y- yq) (58) p=O q:O sf "pq can be approximated as xx 2 2(59) ix = DFT{iJ} = - 47:2 2sinc2 (7fx Ax) (59) Proceeding in this manner we may express the impedance matrix elements Ya, Yy, etc. as (Yx = DFT {[kr Dx] D } (60) Yxy) = ~ DFT q-DXDyx iy (61) ij ko 0 (Y.l 2Yo DFT-l-DyD (62) (Yy) - DFT rko2-D j (63) i koJ J where DFT-1 denotes the inverse discrete Fourier transform. The above, of course, apply for a specific test point (xi, yj) and thus fill only one row of the admittance matrix. Because of symmetry, however, it is not necessary to repeat the computations for the remaining rows of the admittance matrix. A simple rearrangement/shifting of the row given by (60)-(63) is performed to obtain the others. 15

b b To evaluate the admittance matrix elements Yx, Y etc. we refer to (36)-(40). Substituting the expansion (45) into (36), we obtain nn, m7uA^x. n7u Ay i fm2 1 fn7r YC I =Ax Ay sinc a 2 sincb 2) Mxpq a os b P q (64) mn Ax Ay y.) f A Y Iy = Ax Ay sinc ma 2 sinc b 2 Mx pqos p sint b q P q (65) Substituting the above IXmn and lymn into (39) and (40) and integrating as called in (49) yields F 1 b = 2 fM7E. im si. mx C Y COS y )jpq mC n L kb a- a sn a T) cos b q cosJb i (66) (yb]) l - rm7yn (m7 * fmf t it i. =-C m ) b n cos a sin Ta xi sin cosb Y xyiij p q m n nb b (67) m: b fmnt; (m; x) (m~: xi YaT fy) (n; yn7, X m nb yx = n b sn a c(68)os cos b (68) and Iyb) b I m2; X2(m xinM nnl y n7ry (yl =Cl mnmk - b j cos [ a P cos a I sin ( sin (b nYYljp (X2n I (x (69) In these 16

sinc1 2a Ax sinc2 2- A d C y2b ) C- 2jYo - (70) Ymn tan YmnC (Ax Ay)2 ab (71) and Cm have been defined in (38). It remains to compute the excitation elements Ix1 and Iy7. These are given by Ax Ay x2 Yj 2 IC = 2 J J Hi(x, y, z=0)dx dy (72) Ax Ay - 2 -- 2 Ax Ay X.+ — yj+1 2 *J 2 1ic 2 Hy (x, y, z=) dx dy Ax Ay -2 Y- 2 (73) Integrating we obtain inc jko sin 00 (xi cos 0 + Yj sin %) I = 2Yo(sin a cos 00 cos 0 + cos a sin 0)) e sinc ko sin 9o cos k sine sin A0 sin s 0 AY inc jko sin 00 (xi cos (0 + y sin o) Iy = 2 Yo (sin a cos 00 sin 0 - cos a cos 0) e (74) 17

.sinc ko sin 0o cos A sinc (ko sin 00 sin Ay (75) This completes the derivation of all elements appearing in the system (50). Upon a solution of M = x Mx + y My via matrix inversion or LU decomposition, the far zone scattered fields can then be obtained by integrating M over the groove's aperture. III. Formulation with Generalized Impedance Boundary Conditions It is desirable to work with a formulation (or an integral equation) that is amenable to a conjugate gradient FFT (CGFFT) implementation. The CGFFT has an 0(N) memory requirement and is thus suitable for treating large size grooves or cavities, particularly when applied to three-dimensional geometries. Unfortunately, the exact formulation, in addition to being restricted to rectangular grooves and cracks, is not suitable for a CGFFT implementation. On the other hand, the integral equation resulting from an application of the SIBC, a first order condition, although amenable to a CGFFT solution, is not of acceptable accuracy. Recently, however, higher order impedance boundary conditions involving field derivatives beyond the first have been found to provide a substantially better simulation for fairly thick dielectric coatings. These are referred to as generalized impedance boundary conditions (GIBCs) and take the form M am am = ~ (76a) M a, am Hz am = 0 (76b) m= (-jko)m azm where am and a'm are constants specific to the surface, layer or coating being modeled. Alternatively, these can be written as 18

i aa jkorm Ez= (77a) r~.. jkoF' = n Jkor H,=O (77b) m=l where Im and F'm are constants that can be expressed in terms of am and a'm, respectively. For M=1, the above conditions reduce to the SIBC provided we set ZN ao al Zb=lbZo=j j tan (kbt) =Zo ZZ ' (78) where N = \b C- A third order GIBC corresponds to M=3. In that case, an accurate simulation of the reflection coefficient for a metal-backed uniform dielectric layer can be obtained by choosing ao=N - r[tan tN -tan (ktN)tan j ao NJ - f - [kttl al = -jeb 1 + t an (ktN) tan (79) ktNkt N-kt 2 = 1 tan (ktN) tan -2N t N an!a a2-2N [N) 2N_ kt [tan (ktN) - tan 2N 2an and 19

a'0 = (2N2 - 1) [1 + cot (ktN) cot 2t a'1 = -j2Nb [cot (ktN) - cot 2N (80) a't = 1 + cot (ktN) cot k + kt N - 2N [ cot (ktN) - cot -N j a3= jkRt [1 + cot (ktN) cot 2kNj The above conditions are applied on the surface of the coating and predict the proper surface wave modes. However, they were derived for an infinite layer without the presence of any terminations. Therefore, when applied to the case of a groove having abrupt material terminations, they are not expected to be as accurate. As a result, the GIBC must be supplemented by additional conditions at the terminations of the coating or in this case the groove. At this point, no standard methodology has been devised for imposing these supplementary conditions, but in any case such conditions will be specific to the geometrical and material properties of the termination. Below, we first pursue a direct implementation of the GIBC without imposing supplementary conditions at the abrupt terminations of the groove. The application of the conditions (76) or (77) over the aperture of the groove requires the introduction of a magnetic current M(x, y) as defined in (5). A surface integral equation for M(x, y) can then be derived by expressing the field quantities in terms of the magnetic currents. Before doing so, however, it is instructive to rewrite the boundary conditions in terms of tangential derivatives. This is expected to directly yield a symmetric set of equations with respect to Mx and My. From [2], we find that (76) or (77) are equivalent to the conditions 20

E =-PZ H +1 QEz +1 Q'z aHz Exj-PZkHY+ x jko P Z ay a1 Ia2z 1 R' aZH + —R _+_ —z 2 axaz p 0-Z k oydz (81) -E =-PZ H Q kEz ~xjk 3Y jkP' a 3x 2 1 ayaz 2pI ' % aHz (82) provided the duality condition t IN am I am M=OZ... ) M=OZ... I I am m=l,3,... ) atm) m=-,3... ) (83) 10 is also satisfied. In these p ao rF + r2 + r3 + rFr2r3 al + 1 l2 +F3 + Frr3 + Q=aO+a2 rl+r2+r3 al + 1 F1F2 + rF2F3 + FF3 + 1 R= 1 - = 1 ad + 1 lFF2 + r2r3 + F1F3 + 1 (84) (85) (86) (87) and R' 1 P a0+a2 F, +F' +F' 1 2 3 F' + r' + +F' rF' r' 1 1 2 13 123 (88) where am = am/a3, anl -= am/a3 with am and a; as given by (79) and (80). 21

As can be expected, an implementation of (81) and (82) will be rather involved and for illustrative purposes it is instructuve to first develop the formulation using a lower order condition. From (77) we observe that letting r3 = r3 -> oo allows a reduction of the third order GIBC to a second order one. The corresponding set of conditions in terms of tangential derivatives is 1 aEz 1 Q Zo Ex = -PZOHY + k Q aZ + Z aHZ (89) 1 Ez, 1 Q' Zo -E = -PZO H,- -4 Q +.- - j Z a- (90) jk0 ay jkO P' ax In these, 1 +FiF2 do+ 1 P F2 =- (91) rl + r2 al Q 1 1 (92) F1 + r2 al and Q' 1 1 (93) P' I +r'2 ato0 + 1 where am = am/a2 and -a = am/a2. The SIBC (a first order GIBC) is obtained by further letting r2 =1'2 -o yielding the conditions 22

E = -PZo Hy (94) -Ey = -PZo H, (95) with P = = a (96) in which Tb is defined in (78) and denotes the normalized surface impedance seen by a plane wave at normal incidence. It is a simple task to derive an integral equation based on the SIBC conditions for solution via the CGFFT. By invoking (5), (13) and (14) in conjunction with (94)-(95) we obtain 2 2 Mx+ k JJ [MX(x', y) ko + a + My(x', y') GCr - i') dx' dy' = +2PZoHy Sa ax ' axay (97a) 2j M +2 J4J Mx(x' y) a( + y(x' 2 ) GOr - i') dx' dy'= +2PZOHX (97b) To discretize these we employ the expansion (45) to yield Sy-1 N-1 2 2 Mx(x,y)+ k+ 0a2 +yp aaj gpq(X, y)= +2PZoH' p(98a) (98a) 23

Nx 4 N-I r 2 a 2 - aX2Yy 2j aI + [+ My(x y) + M aa +Mypq ay) gpq(x, y) = +2PZoHx op= q=0 qL () (98b) in which -p jko(x- x)2+(- )2 gpq(x,y)=J e dx' dy'. (99) S 4 (x- x') +(y-y') When (98) are tested at (xi, yj) where i = 0, 1, 2,..., Nx- 1 andj = 0, 1, 2,..., Ny - 1, we obtain a system of equations to be solved for Mxpq and Mypq. The solution of this system, however, can be facilitated by noting that gpq(i x) = gi-p j-q (100) and invoking the discrete convolution theorem N.-I Ny-1 DFT XX pgi-pj-q |= (101) P=O cq:0 As before, DFT denotes the forward discrete Fourier transform and DFT (Mp) = 1 (102) DFT (gpq) =. (103) From (99) Ax Ay p+ 2 Yq+-2 e _ Jjkx +y gpq=f dx' dy' Ax Ay 4it ~/x' + y, xp~- yq-_. 24

I Pe 2 sinc -Ly -ic/2+v2 2, ~ ' 2 i Xp=Yq=0 (104) I e'r - M - a - e Ax Ay 47 /Xp2 + yq2 XpY otherwise in which po = iAx Ay/7. Based on (100) - (104), the discrete system (98) can now be compactly rewritten as F 1 +4C ko2 KJ} DFT1 --Dx Dy g <9 -DX Dy 1+ i - DyJ kM} I ^0 FHY1 = 2PZ0o LHJ (105) in which Dx and Dy have been defined in (56). The system (105) is now directly amenable to solution via the CGFFT. To derive a corresponding system based on the second order GIBC we return to (89) and (90) and proceed with the usual substitutions. As before, E, = -M,, E, =M, (106) and i -r a go -a H=H +H +H =H +H with Ha given by (8) and (107) 25

H = Hxo +Hyoy-Hzo e = (Hxo + Hyo 9 - Hz ) e+k (xsin 0 os + y sin 0,sin -zcose (108) being the reflected magnetic field when the aperture covered by a perfect conductor (note that the z component of the reflected field is the negative of the incident). From (8), jYo 1i 2 1 Hx- DFT. Dx2) - DxDy yJg} H a2jYo DFT- I x by x + (ko - Dy ) mg ko L (109a) (109b) (109c) and where we have employed the identity fAgm A ff e +Y2+Z2 DFT( Pe = DFT( A e dx' dy' As~ "} J F2 2m LAzj zz 4s42 y2+z2 As a result of (109), we may also write,1 I l = 1 Iz=j 2 (110) aHa ax AH - Yo DFT-1'D IDxx + D Ax oDF xIDL 1 (Ilia) and similarly a AHz Ay -io DFT- )D IDx + Dylb y I ko% YL i (lib) In addition, 26

[F = DFT x Dy 1 =-2j DF1-1 Y I - Df xg1X aEx ax DX a f = 2 DF Ax I2DTl XLi 1 g (112) (113a) a a I F I aE; 2E = 2DFT-1DyD Ix L - Dy[ (113 ay Ay ( L and by making use of (110) or Maxwell's divergence equation V * Ea = 0, we also have b) aE AEa az Az az Az F = j DFTI[Dxy- Ly XI (114) The last is, of course, to be employed in conjunction with the third order GIBC condition. Introducing (106) - (113) into the second order boundary condition (89) and (90) yields the system DFT- 1l k P(ko-Dx ) jk QDy g 12 Q P]2j A r 2j 2 2 + j 2p Dx DY = -2PZoHxo - 2P sin 0o sin ~ Eo (115) DFT-1 {P ko + Q jk-og.1 r 1 Q' I DI j Pk2 ) 2 Q + - - Dx by, %+ (k - Dy ) - jQDx g 1 Q' -2 r2 y ly = -2PZoHyo + 2Q sin 0o cos 0o Ezo which is directly applicable for solution via the CGFFT. (116) 27

Proceeding in a similar manner, we may also derive the system corresponding to the third order GIBC given in (81) and (82). To do so, in addition to (106) - (114) we further note that a -- DFT-1 (k -D,-Dy)(D, I +DDy^ g (117a) axaz ko L J 2Hz -j2Yo 2 2 2 2 DFT I D(117a) DFT-1 [(k2 - DX - Dy )()x Dy Kx + xy My) g (I17b) ayaz _2 Lo 2E r 2 1 Z-=-DFT x D I -DI D MI (118a) a2F 2 1 -2 k- 2j P(k -D g 2 - QDY g+ p D, + 2 R Dy 2 0 2 ' y g ] + DFT-1 A x g - D +2 g1 2o - Djk 2< 2 2j R' 2 - 2 2 Q2 1 ko koD +2j pR (k- 3x- y ) g D) D y = -2PZo Hxo -2Q sin 00 sin 0o E, cos sin -2 - Zocos 0o sin 0o cos 0o Hm (119) P' 28

2j 2 R2 p g2A 1 Q' 1 R+ _ R_(ko - Dx - Iy, DFT i^^ko -ko ko kg 2j { P(ko2 ) 2 Q + 1 Q 2 1 R, 2 Pt DXD2PZo Hy + 2Q sin 0 cos 0o - 2 - Zo cos 0o sin 0o sin q0 Hz. (120) This is the system corresponding to the third order GIBC with the constant coefficients as defined in (84) - (88). As stated earlier it can be solved in a straightforward manner via the CGFFIT method to obtain the current components Mx and My. In this implementation it should, of course, be noted that the size of the DFT must be at least four times that of cavity aperture as demanded by the convolution theorem. In addition, the usual sampling requirements must be satisfied to ensure accuracy and avoid aliasing. Clearly the system (119) - (120) is associated with a higher spectrum and this often results in a slower convergence in comparison with the system (115) - (116). In the next section we examine results based on the traditional and GIBC implementations that have been presented so far. IV. Results In this section we present data generated via the traditional (to be referred to hereon as the moment method or MM solution) and the GIBC formulations. The implementation of these formulations is indeed cumbersome as is usually the case with all three dimensional solutions. Also, a lack of available reference data coupled with the excessive CPU time required for the execution of the codes complicated the testing of the 29

corresponding numerical algorithms. The presented data are for a few cases representative of the results generated by the codes. Figures 4 to 6 present backscatter patterns for a rectanfular cavity 2.5X long, 0.25X wide and 0.25? deep. The data in figures 4 and 5 correspond to the case where the cavity is filled with material having Eb=7-jl.5 and gb=l.8-j0.1 whereas those in figure 6 correspond to an air filled cavity. All patterns are in the principal planes ()=0 and )=90 degrees) and were generated using a sampling interval of 15 to 20 cells per wavelength. Also, in the case of the MM solution the mode sums were computed using the scheme discussed in [13]. Up to 100 to 150 modes were used for the self-cell and those adjacent to it and one third of this number for the admittance elements far from the self-cell. Unfortunately, though, there were no available data to fully validity the results at this time. However, since the cavity is long, one would expect some agreement with the two dimensioanl solutions for the p=90 degrees cut. This is demonstrated in figure 4a for the filled cavity and in figure 6a in the case of the empty cavity. A comparison of the curves in these figures also reveals that the radar cross section of the filled cavity is generally lower (by several dB) than that associated with the empty cavity. The curves in figure 5 correspond to data based on the GIBC formulation for the filled cavity. Because of the lossy filling, the first order(SIBC), second order (GIBC-2) and third order(GIBC-3) simulations generated results that are almost identical in the 4=0 and 4=90 degree cuts. They are also in agreement with the MM data and in that respect these curves provide a limited verification of the pertinent codes. It should be noted, though, that for the empty cavity the GIBC simulations did not yield satisfactory results(not shown). This was, of course, expected in the case of the SIBC simulation, but was also found to hold for the higher order GIBC simulations. The higher order GIBC can indeed be shown to provide an improved simulation of the cavity's depth, however, they require correction or supplementation at the termination/periphery of the cavity if they are to yield 30

accurate results. This is particularly true for the empty cavity where the currents at some of the cavity edges can be large or more precisely when the the normal electric field componets are rapidly varying near the periphery of the cavity. When the cavity is filled with lossy material, the normal electric field components are not as dominant and this is attributed to the agreement between the MM and GIBC solutions as observed in figure 5. Similar observations also hold for the curves in figure 7 which correspond to a diagonal cut ( 045 degrees) and for a cavity l.Xxl.X in size and 0.2X deep. For the same reason, the higher order GIBC are also expected to yield accurate results for non-rectangular cavities that form smooth junctions with the ground plane and this has already been demonstrated for two dimensional simulations. At present we are pursuing several tasks intended to correct the GIBC simulations and to improve the efficiency of the codes based on the traditional and GIBC formulations. Some the tasks in progress are: 1. Development and generation of termination conditions to correct and improve the GIBC simulations. 2. Development and implementation of GIBC applicable to multilayered fillings and coatings. 3. Validation of the solution by comparison with measured data. 4. Development and implementation of a finite element-boundary element (FE-BE) formulation as applied to the computation of the cavity scattering. The last of the above tasks involves a non-traditional formulation which is attractive from several points of view. It will permit the treatment of cavities having inhomogeneous fillings and cross-sections other than rectangular. (Of course, the GIBC simulations when properly implemented will also be effective for non-rectangular cavities.) Furthermore, the FE-BE formulation is expected to yield a system amenable to a solution via the conjugate 31

gradient FFT method. As a result, it will have an O(N) memory demand which is of major importance for three-dimensional simulations. At the moment, the development of the FEBE formulation has already been completed and is being implemented. 32

References [1] T.B.A. Senior, "Approximate Boundary Conditions," IEEE Trans. Antennas and Propagat., Vol. AP-29, pp. 826-829, 1981. [2] T.B.A. Senior and J.L. Volakis, "Derivation and Application of a Class of Generalized Impedance Boundary Conditions," accepted in IEEE Trans. Antennas and Propagat.; see also University of Michigan Radiation Laboratory report 025921-1-T. [3] S.N. Karp and Karal, Jr., "Generalized Impedance Boundary Conditions with Application to Surface Wave Structures," in Electromagnetic Wave Theory, part I, ed. J.Brown, pp. 479 -483, Pergamon: New York, 1965. [4] A.L. Weinstein, The Theory of Diffraction and the Factorization Method, Golem Press: Boulder, Co., 1969. [5] J.L. Volakis and T.B.A. Senior, "Diffraction by a Thin Dielectric Half Plane," IEEE Trans. Antennas and Propagat., Vol. AP-35, pp. 1483-1487, 1987. [6] J.L. Volakis, "High Frequency Scattering by a Thin Material Half Plane and Strip," Radio Science, Vol. 23, pp. 450-462, May-June 1988. [7] J.L. Volakis and T.B.A. Senior, "Application of a Class of Generalized Boundary Conditions to Scattering by a Metal-Backed Dielectric Half Plane," Proceedings of the IEEE, May 1989; see also University of Michigan Radiation Laboratory report 388967-6-T. [8] K. Barkeshli and J.L. Volakis, "Scattering by a Two-Dimensional Groove in a Ground Plane," Univsersity of Michigan Radiation Laboratory Report 025921-2-T, February, 1989, 31 pp. [9] D.T. Auckland and R.F. Harrington, "Electromagnetic Transmission through a Filled Slit in a Conducting Plate of Finite Thickness, TE Case," IEEE Trans. Antennas and Propagat., Vol. AP-26, No. 7, pp. 499-505, July 1978. 33

[10] F. Arndt, K-H. Wolff, L. Briinjes, R. Heyen, F. Siefken-Herrlich, W. Bothmer and E. Forgber, "Generalized Moment Method Analysis of Planar Reactively Loaded Rectangular Waveguide Arrays," IEEE Trans. Antennas and Propagat., Vol. 37, pp. 329-338, March 1989. [1 1] T.B.A. Senior and M.A. Ricoy, "On the use of Generalized Impedance Boundary Conditions," University of Michigan Radiation Laboratory Report 389741-4-T, November 1989, 22 pp. [12] J.L. Volakis, "Numerical Implementation of Generalized Impedance Boundary Conditions," URSI International Electromagnetic Wave Theory Synposium, Stockholm, Sweden, 14-17 August 1989, Symposium Digest pp. 434-437. [13] D.B. Seidel, D.G. Dudley and C.M. Butler, "Aperture Excitation of a Wire in a Rectangular Cavity," Interaction Notes, Note 345, June 1977. 34

List of Figures 1. Geometry of the cavity aperture in a ground plane. 2. Definition of the polarization angle x 3. Illustration of the application of equivalence and image theories. 4. Backscatter elevation patterns for a material filled cavity computed via the traditional modal formulation (MM); a=2.5X, b=0.25X, c=0.25X, Cb=7-jl and gb=1.8-j0.1 (a) 0=00=900. (b) (=)=00 5. Comparison of results based on the MM and GIBC formulations for a filled cavity corresponding to the data in Figure 4. (a) 4=0o=90~, H-pol (b) 0=o0=00, H-pol 6. Backscatter elevation patterns for an air filled cavity computed via the traditional modal formulation (MM); a=2.5X, b=0.25X, c=0.25X. (a) 4=4o=900 (b) =lo=00~. 7. Comparison of results based on the MM and GIBC formulations for a filled cavity having a=l.X, b=l.k, c=0.2 X. Pattern is in the 0=4)=450 cut with a=0 (H-pol) 35

- A E. 11t 1 Fig.2. a=0 H-pol. a= 2 E-pol. A Definition of the polarization angle a E. I Equivalence Principle: M= A Z=n \ perfectly conducting Image Theory: \ 2M 0~ 0 0 0 - y- y-lw B ByVY C-, Vy, < I ______M I___ Fig.3. Illustration of the application of equivalence and image theories.

Scattering from a Filled Cavity a=2.5X, b=0.25, c=0.25X, eb=7-jl.5, /Pb=1.8-jO.1, Cut p=90~ 20 — '0 0 C.) Sa co Uo 0 (c "0 10 -0 --10 - -20 - -30 --40 --50 - -- -- mmmO %m % I % % I b,^ I' H-Pol * H-Pol(2D) E-Pol I o E-Pol(2D) % I % I I I I I I! I i I! I I I I I I I 1 II 1' I' 0 -60.. I!mommb I -- - -- 0 I I30 30 I I I 60 I I 90 Angle of Incidence 0, deg. Figure 4(a)

Scattering from a Filled Cavity a=2.5X, b=0.25X, c=0.25., eb=7-jl.5, /,b=1.8-j0.1, Cut p=O0 m -o b0 20 10 0 -10 -20 -30 -40 -50 -60 0 30 60 90 Angle of Incidence 8, deg. Figure 4(b) 4. Backscatter elevation patterns for a material filled cavity computed via the traditional modal formulation (MM); a=2.5X, b=0.25X, c=0.25X, Eb=7-jl and gub=1.8-jO.1 (a) (=~o=900. (b) =~4o=00

Scattering fromn a Filled Cavity a=2.5A, b=.25A, c=.25A, vb —7il.5, -Ub=L8-j0.l, CUt jo=90', H-Pol. '0.2G (I) cri 10. IV 20-. 10 -0 --10 - -20-:...30. -40 - -50 - m -Adk Ah a - —, a a mm tA SIBC o GIBC-2 * GIBC-3.y -60 -t5 0 30 60 90 Angle of Incidence e, deg. Figure 5(a)

Scattering from a Filled Cavity a=2.5A, b=.25A, c=.25A,,b=7-j1.5, b'=1.8-j0.1. Cut y=0Q, H-Pol. 20 m 10 MM oGIBC-2 t a SIBC 0 GIBC-3 * ~ 0 -lo -40 -50 -60 0 30 60 90 Angle of Incidence 8, deg. Figure 5(b) 5. Comparison of results based on the MM and GIBC formulations for a filled cavity corresponding to the data in Figure 4. (a) )=0)=90~, H-pol (b) (=(o=0~, H-pol

Scattenring from an E py Cavity a=2.5;\, b=0.25X, c=0.25X, Cut cp=90' 20 1 0 ~00 0 C.) Q1) CU) CU) 0 SLi 04 0 -10 -20 -30 -40 -50 -60 0 30 60 Angle of Incidence 6, deg. 90 Figure 6(a)

Scattering from an Empty Cavity a=2.5X, b=0.25X, c=0.25X, Cut p=0~ 20 10 0 -10 -20 mP Cd v f -30 -40 -50 -60 0 30 60 90 Angle of Incidence 8, deg. Figure 6(b) 6. Backscatter elevation patterns for an air filled cavity computed via the traditional modal formulation (MM); a=2.5X, b=0.25X, c=0.25X. (a) =00o=900 (b) =O0=0~.

Scattering from a Filled Cavity a=lA, b=l, c=.2,,b=7-jl, z,,=1, Cut (=45~, H-Pol. 10 m 0 < -10 0 -20 C. - 30 U) -40 -50 - -60 -70 0 30 60 90 Angle of Incidence 0, deg. 7. Comparison of results based on the MM and GIBC formulations for a filled cavity having a=l.X, b=l.X, c=0.2 \. Pattern is in the =-)o=450 cut with oc=0 (H-pol)