388967-6-T APPLICATIONS OF A CLASS OF GENERALIZED BOUNDARY CONDITIONS TO SCATTERING BY A METAL-BACKED DIELECTRIC HALF-PLANE J.L. Volakis T.B.A. Senior Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, MI 48109 March, 1988 Rockwell International North American Aircraft Operations P.O. Box 92098 Los Angeles, CA 90009 388967-6-T = RL-2560

Abstract The problem considered is that of a plane wave incident on a perfectly conducting half-plane with a thin dielectric coating on its upper face. The solution is accomplished by introducing a higher order boundary condition to simulate the effect of the dielectric, thereby allowing the structure to be treated as an infinitesimally thin half-plane. Two types of boundary conditions are developed, one applicable to a low contrast dielectric and the other to a high contrast one. The problem is then solved using a generalized version of the Maliuzhinets method in which certain additional constants are introduced to assure the correct behavior of the field at the edge. Although these constants play no role in the final solution, they are needed to cancel out inadmissible singularities in the solution of the homogeneous difference equation. The final solution is expressed in a uniform manner and some numerical results are presented.

TABLE OF CONTENTS 1. 2. 3. 4. 5. Introduction Boundary Conditions Solution for H-Polarization Determination of the Field Numerical Results References Appendix A Uniform Evaluation of the Diffraction Integral Appendix B Solution for E-Polarization List of Figures 2 3 6 12 17 20 21 25 28

1. Introduction A structure of considerable interest in scattering theory is a thin metal-backed dielectric layer, and it is important to develop effective techniques for computing the scattering from plates and other targets formed in this manner. One approach is to simulate the layer using an infinitesimally thin sheet placed at the location of the metal backing, and it is shown that this is possible for a thin homogeneous layer. For materials with low and high dielectric constants, the boundary conditions which must be imposed at the upper surface of the sheet are determined. In both cases these are generalized impedance boundary conditions involving second order field derivatives. For a semi-infinite metal-backed layer the model is a half-plane subject to the appropriate boundary conditions. Based on these, the solution for an H-polarized plane wave incident in a plane perpendicular to the edge is derived. Since the boundary conditions differ on the two sides, the Wiener-Hopf method produces coupled integral equations which cannot be solved using presently available techniques, and though Maliuzhinets' method is effective, the generalized nature of the conditions requires a modification to the method that is customarily employed. The solution is presented in Sections 3 and 4, and numerical data are included to show the effect of the various terms in the boundary conditions. The analysis is much simpler for E-polarization, and the results for this case are given in an Appendix. 2

2. Boundary Conditions The geometry considered is shown in Figure 1, and under the assumption of an electrically thin layer of thickness X (kr << 1), we seek a boundary condition which can be applied at the surface y = 0 to simulate the effect of the metal-backed layer. From a Taylor series expansion of the field in the dielectric Ex (0+) = Ex(c-)- -r Ex('-) ay = Ex( E (-) + ik',r ZHz('-) ax y where we have shown onltty the dependence on y. In these expressions k and Z are and tr are the relative permittivity and permeability of the dielectric coating. A time factor e-i&t has been assumed and suppressed. From the continuity of the tangential field components at the air-dielectric interface we then obtain Ex(0+) = Ex(T+) - _ a Ey(,+) + ikr r ZHz(T+) r ax ) and when the boundary condition at the perfectly conducting surface y = 0+ is imposed, the result is Ex(r+) = -ikr A ZHz(r+) + - -- Ey(l+) (1) Er ax 3

which is an equivalent boundary condition at the upper surface of the layer. We can transfer this to the surface y = 0+ by noting that in free space Ex(x (+) +) + -a Ey(0+) -ikr ZHz(0+) (2) Thus, to the leading order in T, the equivalent boundary condition applied at the surface y = 0+ of a sheet in free space is Ex = pZHz +O ~ (y=o+) (3) where p =-ikc (r - 1), q = ik ( -1) (4) Er Similarly, EZ =-pZHX + (y=o+) (5) Ez+ i z with Ex =Ez =0 (6) on the lower (y = 0-) surface of the sheet. The boundary conditions (3) and (5) are identical to the ones derived by Weinstein [1] and are valid for a low contrast dielectric. Apart from the modification provided by the derivative terms, they are the conditions for an impedance surface with normalized surface impedance p, and the derivative terms vanish if q = 0 corresponding to a pure magnetic material. We also remark that (3) and (5) can be 4

derived from the transition conditions [2] for an unbacked dielectric layer of thickness 2r by reflection about the middle. For a layer of high contrast material such that NI >> 1, where N = is the complex refractive index, alternative boundary conditions are required, and well-established ones are [3] Ex = rlZH, Ez = -TlZHx (7) with j= -iNtanNkT (8) Er applied at the upper surface y = a+ of the layer. As kr increases with Im. N > 0, tan Nkr -+ i and the surface impedance reduces to that for a lossy half-space occupying y < r. To transfer the conditions to the surface y = 0+, we again expand the field components in Taylor series as shown in (2). When the expansions are inserted into (7) and the terms collected, we obtain q' aE r Mv E = p'ZH+ Y + Z (9) X z k ax ik az (y = 0+) QI DE,Y r H Ez=-P' ik ik x (10) with 5

I + ikI ikI ikni p J k q, - i r'=. (11) 1 + ikrl 1 + ikr 1 + ikr( The conditions (9) and (10) are similar in form to (3) and (5), but we observe that the transfer to y = 0+ has introduced terms involving Hy as well as Ey. In the particular case of a plane wave incident in the xy plane the boundary conditions simplify. For H-polarization such that Hz, Ex and Ey are the only non-zero field components, the conditions can be written as (7 +ik-2 - kH =0 (12) with oY = q'/p', y2 = 1/p' in the high contrast case (9), and yj = q/p, y2 = 1/p in the low contrast case (3). For E-polarization where Ez, Hx and Hy are the only non-zero components, the boundary conditions are also of the form (12) with Hz replaced by Ez and y1 = -r', Y2 = p' for (10) and 71 = 0, Y2 = 1/p for (5). It is, therefore, sufficient to consider the boundary condition (12), and we note that this is analogous to the one originally proposed by Karp and Karal [4] as a means of simulating complex planar structures. 3. Solution For H-Polarization An H-polarized plane wave is incident on a half-plane simulating a semi-infinite metal-backed layer and occupying the portion x < 0 of the plane y = 0. In terms of the 6

cylindrical polar coordinates p, <, z with x = p cos <, y = p sin |, the incident field is assumed to be i -ikpcos (O- 0o) HI=ee (13) According to the development in the previous section, on the upper side () = r) of the half-plane the total (incident plus scattered) field Hz satisfies the generalized impedance condition ( 2 ik2 -k2 H = (14) ap2 P z whereas on the perfectly conducting side () = -x) the boundary condition is oHz =0. (15) ap Following Maliuzhinets [5] we write H, (p,) =-J epcoa s(a+)da, (16) where a is the double loop Sommerfeld path. To satisfy the edge condition it is required that Hz = O{(kp)e} for small kp with e > 0, and this implies that for large Ilm.al, s(a) = 0 {exp(-ellm.al)). When the boundary conditions are imposed and the 7

differentiations performed, the derivative with respect to p can be eliminated using integration by parts, giving | eikpcsa (sina + sin01) (sina + sin02) s(a + i) da = 0 f -ikpcosa Y sina s(a - x) da = 0 where sin 01,2 2= - y72 +4y,(1+yl) (17) with Im. (cos 01,2) > 0. The necessary and sufficient conditions for these to be satisfied are [6] (sina + sin01) (sina + sin02) s(a + i) = (sina - sin01) (sina - sin02) s(-a + t) + sin a(AO + A1 cosa) s(a - n) = -s(-a - n) + Bo + B1 cosa (18) where AO, A1, Bo and B1 are arbitrary constants whose presence is required to achieve the desired order in Ilm.al. To solve (18) let s(a) = g(a) t (a) (19) 8

with f(a, 1, - ioo) (a, e02, O) g(a) = (20) P(~o0 0r, 2 - ico) (o, 02' o0) where Y(a, p1, p2) is a product of Maliuzhinets [5] half-plane functions Ayd which are free of poles and zeros in the strip IRe.al ~<. Since va (~ ioo) = oo (the resulting factor is cancelled by the corresponding one in the denominator of (20)), and 2,( - ( ) (a -)= {v( cos~(a-n) I we have V,(a +%A - O) _i(ia + 2+ 01 )(a + - 2 A )(a + 02+ cos(a - nt) g(a) = 3(4 3+ 1 (~ + T',1) ( + 1)(o + 2 02),(o + + 02)cos4 (To-() (21) Also g(a-2i)= g(a+2 2) (22) with 9

(sina - sinGe) (sina - sin02) r=. (sina + sin01) (sina + sin02) being the plane wave reflection coefficient for the coated surface. In the two equations (18) replace a by a + i and a - I to obtain (sina - sinGe) (sina- sin02) s(a + 2x7) + (sina + sin0e) (sina + sin02) s(a - 2c) = - sina (Ao - A1 cosa) + (sina + sin01) (sina + sinG2) (Bo - B1 cosa) From (19) and (22) 1 sina (- - C a+2)- t(a-2t) = s(a-2 in) (sinaa+sin) (A-A1 cosa) - (Bo-B1 cosa) (23) and if the right hand side of (23) is denoted by h(a), a particular solution of the difference equation is t(a)=- h(a+ 2 + 4mn) mr)0 Hence, from (22), 00 to (a)=-h(a + 2x) plrem= h(a + 2 r ) ano 1 -r and when the expression forr is inserted, we find 10

t() = 1 -{(sina+sin01) (sina+sinO2) (BO-B1 cosa) -sina (AO-Alcosa) 2g(a)p(a) (24) where p(a)= sin a + sin1 sin02. (25) We note that to(a) has poles at the zeros of p(a), and if ax is such that sin ap i (sine, sin02)112, the four poles which lie in the strip |Re al xI are a = + a, +({ - ap). The general expression for t (a) is t (a) = a(a) + to (a) (26) where a(a) satisfies o(a ~+ n) = o(-a +~ ). It is therefore a function of sin a/2. To reproduce the incident field (13), a(a) must have a pole at a = 4|o with residue unity and, in addition, poles which cancel those of to(a). With this in mind we choose 11

)1/2 f (sin2a)+ in l2(sin2) (a) - + 2 2 2~ 2 O lsin -sin P(a) | 2nfsn2 2 n in) cos (27) where f, and f2 are polynomial functions still to be determined, and from (24) and (27) we then have S(a = g(~a) + 2 22 (^sin 1/2 {fsin 2)+sinaL (sin2 ) ) sin -sin P(a) cos 2 2 2 2p((cx + 2p()in1) (sinain + Sin02) ( - B Cos) - sina (A - Acos)} Given the functions and f2, the constants A, and B must be hosen to eliminate the Poles of s() at a and ( ap). Beyond this, the constants play 4- Determinaton of the Field When the contour a is closed with the aid of two steepest descent paths through the Pol e first term in the expression for s(a + 4) that lie within th rip IlRe. al <,r are captured, and their residues give rise to the incident and reflected waves. In addition, a surface wave Pole may be captured, and the evaluation ofthese 12

residue contributions is given in Appendix A. The non-residue portion is the diffracted field and this can be written as H (P, ) 1 I eipos(a- ) (s(a+) - s(a-x))da z 2iri S(M) where S()) is a steepest descent path through a = ). Since p(a) is a function of sin2a, (29) sin ~ cos 2 s(a ~+ c) = g(a ~ i) 2 2 cosa + cosOo f, (cos2a) + cos Af, (cos2 a) 2 22 2 cos p(a) + L <(sina-sine0) (sina-sine2) (Bo+B1 cosa) + sina (Ao+A1 cosa)} (30) 1 2p(a) g(a + ) = G(a, (o (a - x) (b - x) (-+x) (31) g(a - c) = G(a, ) (a - y) (b - y) ( + y) where G(a, ) = {()} {8 (a, 01' 2 ioo) T (a, 0 o)' ( - oo) 13

-1 1 1 T (0ov 029 11 - 'm) COS T (a - n) COS -Z (00 -,M) 2 and a=cos -( ( -) 2cos(a 2 x=cos (a- ), b = cos 1 - 'n ) ((2 2 Y=si 1 - It). n T(a 2 (32) We note that G(a, 0) is symmetric in a and 4O and O{exp (-3/411m.al )} for large lIm.al. From (30) and (31) we now obtain o ( xsin-'+cos2 s(a+c) - s(a-i) = G(a,eo) cos - [(a-x) (b-x) ( )+ x) { - + 2 cosa+cosOo f1(cos2~ + cos af2(cos 2 sin --- cost - (a-y)(b-y) ( + y) { 2+ p(a) 2 cosa+cos fl (cos2 ) - cos a f2(cos2) 11 p(a) and after much tedious simplification, the coefficient of G(a, v) / (cosa + cos(o) is found to be j2.+ab- - cos -Cos 2 -+ sina sin(o 14

1 a+b 1 2v2 2 + 2 Co2csa Co\ s 22 2 2 22 + (a2cos 2 sno+ cos ~sina+ cosa cos 2cos 2 ( sin -sin All of the terms in the above coefficient are of an allowed order in Ilm.al except the last one, but since Cosa Isin sin- =- cosa sin - + cos 0 s + sin + sin ( cosa + cos) we can write s(a +.)-s(a-7)= os-+c + 2 ab- ICos %Cos2T cosa+cos0 L2 2 2 sina sino + 1 + ab- os sin o + o s -ina (1 a+b) ( b G(aa)() 0) -ocos 2 cos cosa sin - + coso sin +- cos q() (33) 2 2 2 p() 2 where + X) fj (Co 2 2) + co q(a) = (a-x) (b-x) ( + x) (cos2 ) + os -2 (cos2 )} 1 r 2 2 2 -(a-y) (b-y)( a - (a-y) (b-y) + y) f (b-y) y) (cos2) - cos (COS ) + sin - cos a p(a) The function q(a) must be zero to satisfy the order conditions, and this requires f (cos2 a) ~ cos 2 f (Cos2 ) = 4 {(a~ b~ xbx) ( + x) + (a+ y) (b+ y) ( ~ y)} 2 2 2 15

implying fl (sin2 ) + sin f (sin2 ) = 4 { (a+x) (b+x) ( - x) + (a+y) (b+y) (_ - y 2 2 22 Thus, (sin2 )4 ( 1 + ab a+b f,(sinn2).4152'(l+ab-/-) (34) f (si2 ) - ~ (3 a+b 2 f2 (sin 2x-)=4 (- + ab - sin2 ) A and the right hand sides are functions of sin2 as required. The resulting expression 2 for s(a+i) - s(a-i) is symmetric in a and o and O exp( - 4 m ) for large Ilm.al. 1 Hence, for small kp, H = O{(kp) in accordance with the desired edge behavior, and the above choice of f1 and f2 is the only one that achieves this. From (29) and (33) the diffracted field can be written as H1 r eikpcos(a- ) (c o 0' (35) Hz (P, J) = H(a,)) ses 2(35) s5() where H(a,d ) ==.G(ad,sin sin 4 sin sin + +ab1 + sin + sin ) v22 2 2 2 +si 2 21 16

+ 2 1 + +2sin.sin - +sin + sin }, (36) and for kp >> 1 a uniform asymptotic representation of the total field, including optical and surface wave contributions, is given in Appendix A. We note that if Er = pr = 1 and = 0 or if Er = 1 + ioo, i.r = 1 and T = 0, H(a,o) = cocot cot and (35) then reduces to the known expression for the diffracted field of a perfectly conducting half-plane. 5. Numerical Results Using the uniform expression (A.2) for the diffracted field, scattering patterns were computed for a number of dielectric coatings. The patterns correspond to the low and high contrast boundary conditions, and are compared to those obtained with the standard impedance boundary condition (7). Figures 2(a) - (c) show the total Hz field patterns for a plane wave incident on a perfectly conducting half-plane whose upper surface is coated with a dielectric layer of thickness T = V20 (kx = 0.314). The field is incident at the angle 0- = 150 degrees and the curves correspond to coatings having er = 2,.r = 1; Er = 5 + i0.5, gr = 1.5 + i0.1; and er = 7.4 + il.1, lr = 1.4 + i0.67, the last being a commercially available radar 17

absorber. In all cases the pole at ix + 82, whose location is a primary function of the thickness, is far from the path S(4), but the pole at -x + 01 has a strong effect on the diffracted and total fields. The latter pole is associated with the propagation factor of the TEO mode in the layer, and for a boundary condition to produce the correct reflected field it is necessary that -ikcosH. equal the attenuation factor of the TEO surface wave mode. The three curves in each figure were computed using the low contrast, high contrast and standard impedance boundary conditions, and since the curves differ, it is necessary to determine which provides the most accurate picture of the field for a coated half-plane. We do this by examining the reflected field recovered by each solution. As expected, in the case er = 2, Pr = 1, the solution based on the low contrast boundary conditions accurately reproduces the reflection coefficient of the metal-backed layer and is therefore best. At the other extreme, when 8r = 7.4 + iH.1, Pr = 1.4 + iO.67, the solution obtained using the high contrast boundary conditions is accurate to within 6 percent in amplitude and 6 degrees in phase, whereas the low contrast solution is in error by 35 percent in amplitude. The high contrast solution is therefore better, and because of the large refraction index, the solution based on the standard impedance boundary conditions is almost as good. The intermediate case is the coating having 8r = 5 + iO.5, Fr = 1.5 + iO.1 in Figure 2(b). Compared with the exact reflected wave, the low contrast solution is high in amplitude by 4.5 percent and low in 1 8

phase by 8 degrees, and the high contrast solution is low in amplitude by 2 percent and high in phase by 7 degrees. The two solutions are comparable in accuracy, and the differences between the curves can be attributed to the opposite signs of the errors. The backscatter echowidth patterns for the three half-planes computed using the three boundary conditions are shown in Figures 3(a) - (c), and the above comments are also applicable here. With boundary conditions such as those presented, a matter of concern is the accuracy as a function of the thickness and material properties of the layer, and the above comparisons merely illustrate the type of accuracy achievable with the boundary conditions described. Nevertheless, the comparisons suggest that some combination of the low and high contrast conditions could prove accurate for all values of cr and pr, and this will be addressed in a future paper. Acknowledgements This work was supported in part by the U.S. Army under contract DAAA 15-86-K-0022 and by Rockwell International - NAAO under purchase order L6XN-395803-913. 19

References 1. A.L. Weinstein, The Theory of Diffraction and the Factorization Method, Golem Press: Boulder, Co., 1969. 2. T.B.A. Senior and J.L. Volakis, "Sheet Simulation of a Thin Dielectric Layer," Radio Sci., vol. 22, pp. 1261-1272, Nov. - Dec. 1987. 3. T.B.A. Senior, "Approximate Boundary Conditions," IEEE Trans. Antennas Propagat., vol. AP-29, no. 5, pp. 826-829, 1981. 4. S.N. Karp and F.C. Karal, Jr., "Generalized Impedance Boundary Conditions with Applications to Surface Wave Structures," in Electromagnetic Wave Theory, Part 1, ed. J. Brown, pp. 443-483, Pergamon: New York, 1965. 5. G.D. Maliuzhinets, "Excitation, Reflection and Emission of Surface Waves from a Wedge with Given Face Impedances," Sov. Phys. Dokl, Engl. Transl., vol. 3, pp. 752-755, 1958. 6. G.D. Maliuzhinets, "Inversion Formula for the Sommerfeld Integral," Sov. Phys. Dokl, Engl. Transl., vol.3, pp. 52-56, 1958. 7. J.L. Volakis and M.I. Herman, "A Uniform Asymptotic Evaluation of Integrals," Proc. IEEE, vol. 74, pp. 1043-1044, July 1986. 8. P.C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields, Pergamon Press: New York, 1966. 20

Appendix A Uniform Evaluation of the Diffraction Integral To perform a uniform evaluation of the diffraction integral (35) it is necessary to take into account the geometrical optics poles at a = a = ~+ C + 0 and a = a2 = + TC - 0, as well as the surface wave pole at a = a3 = X + 01, or a = a2 = 7+ 02. This last is associated with the function r,(a + - - 12 ) appearing in (21), and its presence is apparent when the identity 3ic 1 1 V (a + - n ) = sin (a - - on ) cosec 4 (a- 2- n ) (a- - on ) (A. 1) is employed in (31). A straightforward method that assures a uniform evaluation of the integral is the additive procedure discussed in [7]. The method regularizes the integrand by the addition and subtraction of secants. Each secant has an appropriate singularity and a multiplying constant chosen to produce the desired residue. The integrand can then be split into two parts, the first of which is a slowly varying function of a and the second a sum of secants. The latter integral can be evaluated exactly [8], leading to a uniform result. Following this procedure, we obtain 4i f F41 A Dr Hd e- e kp H(,) sec -(o)+ sec( J 1-(- -F 1- (2kpcn)] z C(A.2) (A.2) 21

with 00 e lu F () = + 2iez Je du = ~ 2iz Fc(~z) (A.3) +z where the upper sign is employed if - < arg z < L and the lower sign otherwise. The 4 constants Dn are the residues of the respective poles: ikp cos (-o ) ikp cos (-eo ) D. = H(_+ + I)_. ()e =e ikp cos (++o ) (40 o<0) where (sin01 - sino ) (sinO2 - sin4o ) H(t - 0o, (0) = -.n) (sin01 + sinlo) (sinO2 + sin(o ) t H(-~-(L~oo)=1; 1 -ikp cos (-,) [co 1 s 1 ) D 3=:-)H)H ([c + 01c9)e)+ cosec O1 cosec,-j,0 1I -ikp cos W02) 1 1 ) -ikp o., -) +[o) cosec 2 (92 + Io ) + cosec- (02 (j)o ) D4 = -~.H+ (~+ e, Io) e 22

In the above H+ (at 00) differs from the expression (30) for H(a, 40) in having G(a, 40) replaced by G+~ = g 1 (a-x) (b-x) (-r + x) where ~U C1=oX-4-(+)= Os~( A+- 0 ) 0 0 A- )Co c2 = cos (a,2 - 4~ic ) = -cos (4 4) 1 - 1 1 1 C4 = Cos 2- (a4 - ~)=+ Cos 0-( 2) 23

where the upper (lower) sign corresponds to positive (negative) angles of incidence. It should be noted that a uniform evaluation of the surface wave contribution must be performed only when ) > 0 and provided Re. (i + 01,2 - ) < O, in which case the surface wave pole is in the vicinity of the saddle point. To obtain the total field Hz, the residues must be added to the diffracted field (A.2) whenever the corresponding poles are captured in the closure of the contour y. Alternatively, (A.2) represents the total field when the lower signs are used in the computation of the transition function (A.3). 24

Appendix B Solution for E-Polarizatlon If the incident field is E-polarized with i -ikp cos(e - 0o) the boundary condition on the upper side (4 = -) of the half-plane is again (12) with Hz replaced by Ez, and on the lower side (4 = -X) the condition is Ez = 0. Superficially at least, the problem appears almost identical to that for H-polarization, but in fact the analysis is much simpler. If s(a) is the spectral function for Ez (see (16)), the first of the equations (18) is unaffected, but the second is replaced by s(a - n) = s(-a - n) + sina (Bo + B1 cosa) We again write s(a) = g(a) t (a) where now f(a, 01 -io) Y(a, 02, -ioo) g(a) = T(ol 01, 2- i) f'I( 021 2-100) 25

and this is O{ exp ( Im.a| } for large Im.a|. The function g(a) satisfies (22) with [ reversed in sign, and the difference equation for t(a) is a + 2) - t(a- 2) = - cos (BoB1 osa) g(a - 2i) (sina + sine1) (sina + sin02) A particular solution constructed in the same manner as before is 1 1 to(a) = - {(sin + sin0e) (sina + sine2) (Bo - B1 cosa) - (Ao - A1 cosa)}, 2g(a) sinO1 + sinO2 and since this is free of poles in the strip IRe.al < x, it is sufficient to take AO = A1 = Bo = B1 = 0, implying to(a) = 0. It follows that f1 and f2 must also be zero, and hence 1 n0 s(a) = g(a) Co 2 sin A- sin When the contour is closed an expression for the total field is obtained in the form (29), and because g(a + t) = G(a, 4o) (a-x) (b-x) g(a - ) = G(a, () (a-y) (b-y) with 26

G(a, 0) = {,( 2)} {4'(a, 01, 2- io) ( Q, 01, ioo) P(a, 02 2- ioo) 4(ol 029, iso)} we have cos ( s(a+i) - s(a-7t) = G(a,4o) 2 (a-x) (b-x) sin - + cos a cosa+cosO { 2 2f -(a-y) (b-y) sin - -cos J = G )T2 sina sin4o + 2ab cos a cos - cosa+cos ~ 2 2 a-b ( io -dr~. cos -sino + cos- 2sina. This is symmetric in a and and exp - lm.a )for large lm.a|, implying 1 Ez = 0 {(kp) } for small kp in accordance with the required edge behavior. Thus, in spite of the generalized boundary condition imposed on the upper side of the half-plane, Maliuzhinets' method is applicable in its standard form. With only minor modifications, the analysis in Appendix A is also applicable for this polarization, and, in particular, the diffracted field is again given by (A.2) provided H(a, Jo is replaced by H(, G(, {sinsin a-b(sin sin )} H(a, 0) = G(a, (dsin asin -+ ab sin+ sin. 2 2 r 2 2 27

List of Figures Figure 1. Geometry of the dielectrically coated half-plane. Figure 2. Figure 3. Total Hz patterns for a plane wave incident on a perfectly conducting half-plane coated with a V/20 thick dielectric layer; comparison of solutions based on the low contrast, high contrast and standard impedance boundary conditions. (a) er = 2, Ir = 1 (b) er = 5 + i0.5,.r = 1.5 + i0.1 (c) ~r = 7.4 + il.11, lr = 1.4 + i0.672. Backscatter Hz echowidth for a plane wave incident on a perfectly conducting half-plane coated with a X/20 thick dielectric layer; comparison of solutions based on the low contrast, high contrast and standard impedance boundary conditions. (a) Er = 2, Pr = 1 (b) Er = 5 + i0.5, gr = 1.5 + i0.1 (c) r = 7.4 + il.11, pr = 1.4 + iO.672. 28

A y T, I r, Aa \ I s perfectly conducting Figure 1. Geometry of the dielectrically coated half-plane

4o =1500 kp= 10, t=0.05k ~r=2v r 1 Low contrast B.C. 0- - High contrast B.C. -— A — Standard Impedance B.C. N:3 LU I L, ac tM i C-180.00 -120.00 -60.00 0.00 60.00 120.00 180.00 RANGLE IN DEGREES (a) Fig. 2. Total Hz patterns for a plane wave incident on a perfectly conducting half-plane coated with a X120 thick dielectric layer; comparison of solutions based on the low contrast, high contrast and standard impedance boundary conditions. (a) er = 2, r = 1 (b) E = 5 + i0.5, g = 1.5 + i0.1 (c) er = 7.4 + il.11, Lr = 1.4 + iO.672.

cm 0 4 I CZ Ul) N o = 150~. kp= 10, t =0.05X Er:= 5 + i0.5 /'r = 1.5 + io.1 r 5+i0.5, r=i.1 Low contrast B.C. ~- High contrast B.C. Standard Impedance B.C. LUL N I LU LU) I-fo CZi -120.00 -6 0.00 60.00 120.00 RNGLE IN DEGREES 180.00 (b)

0o = 150~ kp = 10, t = 0.05k IU r4; r= 7.4+ i1.11, ir = 1.4 + i.672 Low contrast B.C. High contrast B.C. - Standard Impedance B.C. Jco UlLLJ -1 -ILU' II — -60.00 0.00 60.00 ANGLE IN DEGREES (c) 120.00 180.00

er=2, rr= 1, t=0.05k -- Low contrast B.C. High contrast B.C. Standard Impedance B.C. CD, I — (a) Fig. 3. Backscatter Hz echowidth for a plane wave incident on a perfectly conducting half-plane coated with a XJ20 thick dielectric layer; comparison of solutions based on the low contrast, high contrast and standard impedance boundary conditions. (a) s, = 2, pr = 1 (b) er = 5 + i0.5. P r 1 5 + iO.1 (c)er=7.4 +i1.11,r=1.4+iO.672. C2 ANGLE IN OEGREES (a) Fig. 3. Backscatter Hz echowidth for a plane wave incident on a perfectly boundary conditions. (a) er = 2,.r = 1 (b) Er = 5 + i0.5, 9r = 1.5 + i0.1 (c) Er = 7.4 + il.1 1, gr = 1.4 + i0.672.

= 5+ i0.5, ccy CD z w — IZc a C"6 '-10.0 g.r =1.5 +10O.1 Low contrast B.C. High contrast B.C. Standard Impedance -120.00 -60.00 0.00 60.00 120.00 RNGLE IN DEGREES 180.00 (b)

CD* er=7.4 + MA 1 Lr =1.4 + iO672 tr=O0.05X Low contrast B.C. 9 High contrast B.C. A- Staindard IMnedance E Wq,%Ao 1%0%.Oo %a 4... rw ---. --- CZLU. N.8 C,, I i8.0 -120.00 -60.00 0.00 60.00 120.00 ANGLE IN DEGREES 180.00 (c)