Technical Report 388967-8-T Diffraction by a Coated Wedge Using Second and Third Order Generalized Boundary Conditions John L. Volakis and Thomas.B.A. Senior Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, MI 48109-2122 388967-8-T = RL-2561

Abstract Second and Third Order Generalized Boundary Conditions are presented for simulating metal-backed dielectric coatings. A detailed quantitative assessment of their accuracy as a function of thickness and material parameters is first given. The higher order boundary conditions are subsequently used to derive corresponding diffraction coefficients for a coated wedge. The solution is obtained via a modification of Maliuzhinets method requiring the introduction of a particular solution which serves to produce the correct edge behavior and yield a reciprocal result. 2

Contents page I. Introduction................................ 4 II. Boundary Conditions......................... 5 III. Solution....................................10 First Order Boundary Condition.................13 Second Order Boundary Condition................14 Third Order Boundary Condition...............17 IV. Determination of the Field.................20 V. Summary.................................... 22 References.................................. 25 List of Figure Captions....................26 Apendix: FORTRAN Listing of Program GIBCWEDGE.......................... 28 3

I. Introduction Of considerable interest in the computation of the radar cross section by complex targets is the simulation of metallic geometries coated with penetrable material. Such material are usually magnetic and serve to reduce the scattering of an otherwise perfectly conducting surface or junction. For thin coatings the standard impedance boundary condition[l] has been traditionally employed to provide a suitable mathematical simulation. However, the validity of this simulation is generally poor for oblique incidences, particularly in the case of low loss dielectrics. In a recent study[2] it was shown that higher order boundary conditions analogous to those proposed by Karp and Karal[3] can correctly predict the reflection by a dielectric layer at all angles of incidence, including grazing. These boundary conditions involve higher order derivatives of the surface field and as a first correction to the standard impedance boundary condition they account for the presence of currents normal to the dielectric layer. Notably, the maximum layer or coating thickness that can be accurately simulated depends on the highest derivative order kept in the boundary condition and this determines the order of the condition. When only the first derivative is kept they reduce to the standard impedance boundary condition and are, thus, accordingly referred to as generalized impedance boundary conditions(GIBC). This report is concerned with the use of second and third order generalized impedance boundary conditions for computing the radar scattering by a fully coated wedge. The geometry, shown in Fig. 1, often occurs on aircraft structures and is thus of practical interest. An approximate solution to the diffracted field by the subject geometry using the standard impedance boundary condition is well known and has been given by Maliuzhinets[4]. The solution given here is more accurate by virtue of the employed GIBC and reduces to that in [4] for high loss coatings. The second and third order boundary conditions to be employed in this analysis are presented in the next section along with data quantifying their accuracy as a function of 4

the coating's thickness and refractive index. They were derived elsewhere[2, 5]and are analogous to those employed recently [6, 7] for the diffraction by a semi-infinite ferrite/dielectric layer. The diffracted field is obtained via a modification of the Maliuzhinets' method[4] requiring the addition of a particular solution as a direct implication of the employed higher order boundary condition. Such a particular solution is difficult to determine in a closed form for an arbitrary wedge angle. However, it has so far been derived for two specific wedge angles[5, 8] and, interestingly, its function has been to produce a reciprocal scattered field by cancelling out terms of the homogeneous solution that are of unacceptable order. Therefore, here we avoid the derivation of the particular solution by imposing reciprocity to obtain the "correct" diffracted field. As expected, it involves the usual Maliuzhinets' functions for which highly accurate approximate analytical expressions are available[9]. In the following, after presentation of the boundary conditions we proceed with the solution for the H-polarization. The E-polarization solution is then obtained directly from the H-polarization one via modification of the impedance parameters. II. Boundary Conditions For the problem at hand, a plane wave is assumed to be incident on the coated wedge configuration shown in Fig. 1. In the case of H-polarization, the only non-zero field components are Hz, Ex and Ey and for E-polarization the corresponding non-zero components are Ez, Hx and Hy. To proceed with a mathematical solution of the scattered field due to an incident plane wave excitation, it is necessary to replace the coating with an effective boundary condition. Recently [2, 5], higher order boundary conditions were proposed for this purpose. These can be generally written as (an e-i(ot time convention is assumed) m + ikmr EnO n + ikr )Hn= (1) 5

were, En and Hn denote the respective nonnmal components to the coating's surface and ( implies differentiation with respect to the surface normal. They provide an improvement over the standard impedance boundary condition and are thus referred to as generalized impedance boundary conditions (GIBC). For M = 1, these reduce to the standard impedance boundary conditions +ik En=O (a +ikF Hn=0 (2) with = 1 -i tan (Nk) (3) in which ~+ and A+ are the relative permittivity and permeability of the coating on the upper (+) or lower (-) face of the wedge, N = ji, k is the wavenumber and t denotes the coating's thickness. But an alternative and more common form of (2) is given in terms of tangential components as (Ty k =0 =+~ (4) where (p, () are the usual cylindrical coordinates and Fl for H - polarization ~, 1 for E- polarization (5) lr, A solution for wedge diffraction based on the standard impedance boundary condition (1st order GIBC) has been given by Maliuzhinets [4]. However, as is well known, the standard impedance boundary condition is only applicable for high loss dielectric coatings, particularly for H-polarization, since it does not account for the presence of polarization currents normal to the coating. 6

The second order GIBC provides the next best simulation to the standard impedance boundary condition. It takes the form [5] ~2 i yk2 =0,=+I) (6) where Y1,2 are parameter functions of the material properties and specific expressions for these have been derived in [2,5]. A second order GIBC which has been found to yield reasonable accuracy has associated values of Y1,2 given as Fl+F2 al 1 rFr2+ ao+a2 1 a 2 'r (7a) 2=- +r r2 ao+a2 for H-polarization and as F + r+r2 a 0+a2 1 a -2= ---- = — (7b) 1+r1r2 ao+ a2 for E-polarization. In these 7

=(N - 2N[ tan (N) - t( 2N)] a = [ 1 + tan(kTN)tan - 2] a =(2N2 -1)[1 +cot(kN)cot( )J] (8) = i 2Nt cot (kTN) - cot (kt)] = 1+cot (kTN)cot ]+ kr (N.. [ cot (krN) - cot fl Figures 2 to7 show the maximum thickness for which the above second order GIBC is capable of predicting the coating's plane wave reflection coefficient within 10 degrees of its actual phase and/or 10% of its actual magnitude. As seen, in comparison with the standard impedance boundary condition, the second order GIBC provides substantially better accuracy for incidence angles away from normal. Notably, the simulation improves monotonically as one approaches grazing. In making this observation it should be also noted that the given curves are for lossless dielectrics and, therefore, represent a worse case. We may conclude from figures 2 to 4 that for H-polarization, the second order GIBC is capable of simulating coatings having thickness up to 1/5 of a wavelength for incidence angles greater than 35~ from normal (55~ from grazing). This is regardless of the dielectric's properties since the simulation improves substantially as N and/or the loss in the coating increases. In contrast, the 1st order GIBC provides a superior simulation, with respect to the second order GIBC, for the rest of the angular region (i.e. within 35~ from normal). Turning to Figures 5 to 7, one again arrives at similar conclusions for E-polarization. However, it should be noted that for small N the deterioration of the simulation provided by the second order GIBC as normal 8

incidence is approached, is now more rapid. It is desirable for practical purposes to have a single boundary condition simulating the coating's presence for all incidence angles. From the above, the second order GIBC is obviously not adequate if our goal is to simulate coatings of at least 1/4 of a wavelength in thickness. Therefore, it is of interest to examine the suitability of a third order GIBC. The third order GIBC may be written as _ _ a2 -ik 2 FHZ f+J — 7~ ikY k^J ^ O <=+0 (9) (+k + -t k ~ =+ 4 (9) k ap^ ^ap2 p3 J Y (T LEzi and from [2] al+a3 rl+r+2r3+rlr3+ 12 23 13 a+a2 r+F2+3+rrF1F2F3 - a2 ri+r2+r3 (l 2= ao+a-2 (10a) -a+a2 - +r2+3+F1r2r3 a3 1 3 a0+a2 rF+F2+F3+rlrF1F2F3 for H-polarization and, +a3 r;r'+rF2r'+rlr;+ 2 -O+a2 ri+r'+r3+rir'2r3 +2 23 13 a' r +r +r~ (10b) -rl, +r2' +Fr +r1' rF' rF ' a3 1 Y3 +F' a 3 1 3 +a; 2 +r +r13 3 for E-polarization. In these, ao, a1, a2, ao, a; and a2 are as defined in (8). The remaining constants a3 and a3 are given by 9

a3-i [tan(ktxN)-tan( ] (11) a3 2N a3 = -ikxtP[ 1 + cot (kxN) cot [kxJ Clearly, (10) reduce to (7) when a3 = a3 =0 or F3 = r3 - oo. From Figure 8 to 10, it is now seen that the above third order GIBC provides an acceptable simulation for coating thicknesses of at least 0.4k regardless of material properties, angle of incidence and polarization. This is, of course, a conservative statement since the simulation improves for lossy coatings having large refractive indices and for angles of incidence away from normal. III. Solution Consider a wedge whose faces are located at = + ~ = + n The wedge is illuminated by the plane wave z -ikpcos(o-%).= Ui(p, )=e (12) and each face is subject to an Mth order boundary condi ition with impedances = sin ~ for H-polarization and Fm = sin m0 for E-polarization (m = 1, 2,..., M) on the upper (positive) and lower (negative) face with Re sin Om > 0. It is required that, for small kp, the total field u(p, ~) satisfy the edge condition f Hz(p,)]) E )= u(p,) = 0 {(kp)E} LEz(P,9) with E > 0. Following Maliuzhinets [4] we write 10

u(p, ) e-ikpcos s(a+4) da (13) where y is the double loop Sommerfeld path. Application of the boundary conditions then gives Je ikpcosaI (sina~ sinO )s(a ) da =0 (14) and the necessary and sufficient conditions for this to be satisfied is [10] M M M n(sina~sina s )s(a~_i)-n(-sina~sin )s(- a~)=sin a A cosma m =1 m m=l m m (15) for appropriate constants Am As the solution of (15) we write s(a) = g(a) (0(a) + h(a) (16) where M T(a,,+ ) g(a)= nI (17) vm em and P is a product of four Maliuzhinets functions [4]. For large Im. al g(a)= {exp( lIm.ai)} where n = 2D/c, and for brevity we shall refer to this as "order M". To satisfy the edge condition it is necessary that s(a) be of order -ne. In order to reproduce the incident field (12) we choose Ic (a)= (18a) S-So 11

where S =sin a* S =sin -, n C=cos-, n CO n (18b) and thus the first term in (16) is of order M- 1. The second term on the right hand side of (16) is a particular solution of (15) which is free of singularities in Re. a I < 4 and cancels all terms of excess order in g(a) cYO (a). In addition to these three requirements on h(a), there is a fourth one as we shall show later. Provided such an h(a) can be found, the resulting solution will satisfy the correct edge behavior and reciprocity. When the expression for the the diffracted field is reduced to an integral over a steepest descent path through the origin, the non-exponential part of the integrand is where and we have s(a+n) - s(a-c) = f(a) + h(a-;) - h(a-rc) f(a) = g(a+JL) 0O(a+n) - g(a-i) Oo(a-i) M g(a+7) = XM(a, 4O)I l(am-x) (bm +x) M g(a-R) = XM (aX, 0)mI l(am -y) (bm +y) (19) (20) (21) where am = cos-(0 - /2), x = qS +pC, b = cos ( - 2) y=qS-pC (22) (23) (24) with and p 7si n q =cos 2 12

XM (a, ~0) = 4-N {I(l/2) } { { (a 0o )(0 m ) (25) V~. ^ )=40 ()y( 0)) j' XF^ ^. e, w,, m.e } (25) Thus, XM(a, 0o) is symmetric in a and 4o and of order -M. Further, C ao (a~+C (2) S= 2pq C) (26) where D 2 _4222(q2 D =S2 2 (2q2-1) SS + S4pq (27) and D is also symmetric in a and (o. It follows that f(a)= XM(a D'0) { 2q2-)S-SO}{n (am-x) (b m +X)- m (a -y) (b+y)} M M -2pqCI {l(a -x)(b +x) +L(a -y) (b Y) ( 28) In general this is neither symmetric in a and o0 nor of acceptable order, and to see this we consider the cases M = 1, 2 and 3. First Order Boundary Condition (M=1) This is the simplest case. When M=1 it can be shown that the constants A4 in (15) are all zero and hence h(a) = 0. Since x-y=2pC, x+y=2qS, (29) - y2 = 4pqSC, x2+y2= 2(2q21) S2+2p2 we have 13

f(a) I X (a 0)- (2q2-1)S-S0} {a1-x)(b1+x) - (a,-y)(b,+y~l n 1'X 00 DL -2pqC { (a,-x)(b,+x)+(a1-yXb,+y)] — X, (a, 00)"~ C0 { (2q2-1) S-S 0 { (X2_y2) + (a,-b1)(x-y)} - 2pqCC0 (X2+y2) - (a,-b,) (x+y) + 2a~b1} which reduces to f(aX)-njX1(a, 0) 2p-DbQ- [2q (SS0+p2) -(a1-b1)(S+S0) -2qa~b1] (30) This is symmetric in a and %0 and of order -1, implying u(p,400) = 0 ((kp)1/nl} for small kp. Consequently, there is no need for any particular solution h(a). Second Order Boundgar Condition (M=2) The constantsA andA in(15) are no longer zero and hence h(a) ~0. From (28) fa)-I X2 (a, O~ fb- (2q2_1) SS0} I (a1-xXa2-x)(b1+)(2+)- (a,-y)(a -y)(bl+y)(b2+) -2pqC f{(a1-xXa 2-x)(b 1+x)(b2+x) + (a I-y)(a 2-y)(bl+y)(b2+y) } x C [ f (2q2-1) S-S0} { x 4-y4 _ (A I-B I(x3-y3) + (A2+B2-Al B1)(X2_y2) + (A2 B1 - A1B2) (x-y) I-2pqCC0 {x4+y4 - (A1-B1 Xx 3+y3) + (A 2+B 2- A1B1)(x 2+y2) +(A2 B1 -A1B 2)(X+y) + 2A2B21] (31) where 14

Al = al+a2, B =bl+b2 A2= ala2, B2 = bb2. (32) The first order term (involving x + y) in square brackets is - 2p CC0 (S+S), and the second order term is -4pq CC0 (SS +p2). Both are symmetric in a and 4o and of allowed order in UIm. al. Since X3 y3 =2pC {(4q2-1) S2+p2, x3 + y3 =2qS {4q2- 3) S2 + 3p2} x4- y4 8pqSC (2q2 1) S+p2, (33) x4 + y4 = 2 (8q4 - 8q21) + 1) S 4(4q2) p2S2 + 2p4 after much tedious trigonometric manipulations the third order term is found to be 2pCCS3-(4q2-1) S2 (4q2+1)S -p2S0 and this is not, obviously, symmetric. Moreover, the first term is not of allowed order, but the entire expression can be rewritten as 2p CCoSD- 2p CC0(S+So)(SSo+p2), and the second term of this is symmetric and of allowed order. Finally, after much simplification the fourth order term in square brackets is 4pq CCO S4 - 2(2q2-1) S3S -4p2q2S2 - 2p2SS0 - p4} =4pq CCoS2D - 4pq CC(SS+ p2)2, and the second term of this is both symmetric and of allowed order. Note that the "elimination" of the term S4 has also eliminated the unacceptable one involving S3. Thus 15

f(a) =2X2 (a, o) CCo { 2qS2- S(A - B) + K -2p X2(a o) CC 2q (SSO+p2)2 - (A-BI)(S+So)(SSo+p2) + 2q (A2+B2 - AB 1) (SSO+p2) + (A2B1-AB2)(S+So) (34) + 2q A2B2 + K { (S+S)2 - 4q2(SSo+p2) where K is an arbitrary constant independent of o0. The second group of terms is symmetric in a and 0o and of order -1; however, the first group violates both the symmetry and order conditions, and it is therefore necessary that the particular solution h(a) be chosen such that h(a+7) - h(a-7) - X2 (a,0) CC {2qS2 S(A-B) + K. (35) This is the fourth and final condition to be satisfied by h(a). To complete the solution it now remains to find the constant K and to show that h(a) exists as defined above. The first can be determined by examining the edge behavior of the solution for a given o0 and from this it can be conjectured that K = 0. However, the existence of a solution for h(a) has not yet been rigorously established for an arbitrary wedge angle (2-n)7, although at present we believe that a proof of its existance can be established for = n = for integer values of L 2 2 2L2+1 1,2 16

Third Order Boundary Condition (M=3) This is similar to the previous case but just a little more complicated. From (28) f(e)= X (a, 30 ) -- [ {(2q2-1) S-So} {(al-x)(a2-x)(a3-x)(bl+x)(b2+x)(b3+x) n D3 2 3X (a 1-y)(a2-Y)(a3-y)(bl+y)(b2+y)(b3+y) } - 2pqC { (al-x)(a2-x)(a3-x)(bl +x)(b2+x)(b3+x) + (al-y)(a2-Y)(a3-y)(bl+y)(b2+y)(b3+y) } ] I x 66,So 6) 1 n 3 (oX D[ Co (2q2-1)S -S } +- 1(x6-y6) + (A-B)(x-Y) + (AB2-A2B)(x4-y4) + (A3-B3+A1B2- A2B 1)(x3-y3) + (A3B1 + A1B3 - A2B2)(x2-y2) + (A3B 2 - A2B3)(x-y) }- 2pq CCO { - (x6+y6) + (A -B )(x+y 5 + (A1B2 -A2B )(x4+y4) + (A3-B3+A1B2 - A2B l)(X3 +y3) + (A3B 1+AlB3 -A2B2)(x2+y2) + (A3B2 - A2B3)(x+y) + 2A3B3} 1 (36) 13 2232 2 ~ 3 (36) where now A1 = al+a2+a3, B1 =bl+b2+b3 A2 = ala2+a2a3+a3a1, B2 = blb2+ b2b + b3b, A3 = ala2a3, B3 = blb2b3. (37) The first and second order terms in square brackets are, apart from the multiplying constants, - 2p CCO (S+SO) and - 4pq CCO(SSo+p2) respectively, as in the previous case. Similarly, the third and fourth order terms are 17

2p CCO SD- 2p CCo(S+So)(SSo+p2) and 4pq CC S2D - 4pq CC (SS0+p2)2 respectively, but there is now a subtle point that should be noted. In the third order case even the first term is of allowed order, and its separation out is forced by symmetry considerations and not by order. Thus, for a third order impedance boundary condition, specification of the edge behavior is not sufficient to ensure reciprocity. Since x5- y5 = 2pC (16q4- 12q2+1)S + 2(6q2-1)p2S2 -p4} x5 + y5 = 2qS { (16q4 - 20q2 + 5)S4 + 10(2q2-l)p2S2 + 3p4} x6- y6 =4pq CS {(16q4 - 16q2+3) S4 + 2(8q2-3)p2S2 + 3p4 x6 + y6 = 2(32q6-48q4+1 8q2_)S6 + 6(16q4-12q2+ )p2S4 (38) + 6(6q2-l)p4S2 + 2p6 we again find after much trigonometric manipulations that the fifth and sixth order terms in square brackets of the expression for f(a) are 2p CCO S ( (4q2-) S2 + SS + 2p2} D - 2p CC0 (S+SO) {(SSo+p2)2 2p4} and -4pq CCO {2 (2q2-1)S4 + S3S + 3p2S2 D +4pq CCO SSo (S2S + 3p2 SS0 + 3p4) respectively. Clearly, in each case the second group of terms is symmetric and of allowed order. The resulting expression for f(a) now is 18

f(a) - X3 (a, 0) CC[ - 2q {2(2q2-1)S4 + S3S + 3p2S2} + (A-B) S {(4q2-1)S2 + SSO + 2p2} + 2q(A1B2- A2B 1)S2 + (A3-B3 + A1B2 - A2B1) S + K(S+So) + K2 ] - X (a, o) D~ 2q { (SS + p2)3 p6 +(A-B)(S+S) {(SSO+p2)2 -2p4} + 2q (A1B2-A2B1) (SSO+p2)2 + (A3-B3 + A1B2-A 2B )(S+SO)(SSO+P2) + 2q (A3B1 + AB3 - A2B2)(SSo+p2) + (A3B2-A2B3)(S+So) + 2q A3B3 + { Kl(S+So) + K2} {(S+S)2- 4q2 (SS0 + p2) } (40) where K1 and K2 are arbitrary constants independent of 0o. The second group of terms is symmetric in a and 0o and of order -1, but the first group violates both of these requirements. It is therefore necessary to choose the particular solution h(a) such that h(a+7) - h(a-7n)= - 2p X3(a, 0) CC [ -2q {2(2q2-)S4 + S3+ 3p2S2} n + (A1-B1) S {(4q2-)S2+SSo+2p2} + 2q (AiB2-A2B1)S2 + (A3-B3+A1B2-A2Bi)S + KI(S+So) + K2 ]. (41) Comparison with (35) shows how rapidly the complexity of h(a) increases with the order of the imposed boundary condition. Similarly to the second order case we may again conjecture that K1 = K2 = O0 but the existence of h(a) for an arbitrary 4> remains to be shown. 19

IV. Determination of the Field To determine the field u(p, 4) given by (13) we may close the contour y by two steepest descent paths through a = ~H. In this process, we may capture the geometrical optics poles located at a = al = - ) + +0 and a = a2= - )- 0 + 20 as well as possible surface wave poles located at a = a4 = - ~ + + + ~ + 1 and a = a4 = - - ( D + D +. locaedat a= a = a - + + 0 + O and a = a = - (rC + ( + 0. If captured, the residue of these poles must be added to the total field u(p, )). The remaining contribution (the non-residue contribution) is the diffracted field given by u(p,)=- 1 eikp cos (){s(a+I )-s(a- L)} da (42) s(M) where S(4)) is a steepest descent path through a = >. For the first order GIBC {s(a+t) - s(a-n7)} = f(a) = fl(a, 0o) (43) with f(a) as given in (30). In case of the second order GIBC we have from (19), (34) and (35) {s(a+t)- s(a-i)} =f2(a, 0o) =2p cc ('2 0C) =- -n X2(a,^) Do [2q (SS+p2)2 - (A1-B1)(S+So)(SSo+p2) + 2q (A2+B2 - A,1B)(SS0+p2) + (A2B 1 - AB2)(S+So) + 2q A2B2 ] (44) with X2(a, 0o) as defined in (25), and B as defined in (32), p and as defined in (32), and q as defined in (24), S, So, C and C0 as defined in (18b) and D as given in (27). Finally, for the third order GIBC we obtain from (19), (40) and (41) 20

(S(a+t) -s(a-x)} = f3(a, %o) 2p X3(a o) [-2q{(SSo+p2)3-p6}+(AI-B)(S+So){(SSo+p2)2-2p4 + 2q(A1B2-A2B )(SSo+p2)2+(A3-B3+AB 2-A2B1)(S+So)(SSo+P2) + 2q(A3B,+A1B3-A2B2)(SSo+p2)+(A3B2-A2B3)(S+So) + 2q A3B3 (45) with X3(a, 4o) as defined in (25), Ai and Bi as defined in (37), p and q as defined in (24), S, So, C and Co as defined in (18b) and D as given in (27). A non-uniform evaluation of (42) now yields ikp u( 0) ~e - D(' 0),(46) where D(, 40o) is the associated diffraction coefficient given by D(<,<>) = - -- ei'4 {S((+-)-S(<-_)} (47) = 1 eix/4 fr(q, q0) (47) for the Mth order GIBC. The corresponding echowidth is given by =2 WM(,0)1 (48) G = 2x D(0, 0 )I2 = (48) Figures 11 to 14 include several backscatter patterns for coated wedges with a variety of material coatings. As a simple verification of the validity of the derived second and 3rd order solutions, figures 11 and 12 present the echowidth patterns for a wedge with an electrically (e = 1+i 103, g=1) or a magnetically (e=1, jl=l+i 105) conducting coating. It is seen that all solutions, regardless of order, are in agreement and predict the 21

traditional known patterns. The patterns in figure 13 correspond to a wedge with an internal angle of 30 degrees (n=1.833) and coated with a uniform lossless dielectric layer having e=4 and,g=l. It is now observed that although the first and second order solutions are reasonably close, the third order one deviates from both of these, particularly for the thicker coating. Interestingly, the three solutions do not display substantial variance near edge-on incidences, but in contrast, for incidences near grazing, the third order solution (which provides an accurate simulation of the reflected field) predicts much higher echowidth. This is surprising and is currently investigated on whether it is inherent to the employed simulation and not necessarily related to a physical phenomenon. The patterns in figure 14 correspond again to a similar wedge which is now coated with an absorber layer having e=7.4+il.1 1 and p= 1.4+i0.672. As expected, the general level of the echowidth patterns is now lower, particularly for the thinner coating. However, we again observe a similar disagreement among the solutions for grazing incidences. This, of course, demonstrates the inadequacy of the first and second order solutions in simulating the coatings. V. Summary The problem considered was that of diffraction by a coated wedge of arbitrary angle. To obtain a solution for the diffracted field a simulation of the coating was first developed using higher order impedance boundary conditions, referred to here generalized impedance boundary conditions(GIBC). A qualitative assessment was then given on their accuracy. In particular, the second and third order GIBCs were examined in some detail. This examination amounted in comparing the coating's plane wave reflection coefficient predicted by the employed GIBC with the exact. It was found that if a phase error of 10 degrees (or 10% in magnitude, but the phase error criterion is usually 22

more limiting) was acceptable, the second order GIBC was capable of simulating coatings up to 0.25 wavelengths in thickness regardless of the coating's material composition provided the incidence was away from normal. This last limitation prompted the need to consider simulations using a third order GIBC. It should be noted, however, that the second order GIBC is probably sufficient for most practical materials with some loss and under the same conditions the maximum allowed thickness for a 10 degree error could also be relaxed. The employed third order GIBC was found far superior to the second order one. In particular, the third order GIBC allowed simulation of coatings as thick as 0.4 wavelengths regardless of their composition and the angle of incidence for the same error criteria. Similarly, to the second order GIBC the maximum allowed thickness for a 10 degree error can again be relaxed for high contrast and/or lossy coatings. The solution to the scattered and diffracted fields by the coated wedge were obtained via a generalization of Maliuzhinets 'method. This required the introduction of a particular solution h(a) to the difference equations which result upon application of the subject GIBC imposed on the wedge faces. Such a particular solution is zero for the standard impedance boundary condition, but is required for higher order GIBCs and causes substantial complexity in the analysis. It serves to correct the order of the spectral function under the Sommerfeld integral and allows for a reciprocal solution in the last phase of the analysis. Fortunately, it does not appear in the final result and, therefore, the only requirement (for a valid solution of the diffracted field) is a proof of its existence. Once established, the particular solution can be ignored and the solution for the diffracted field can then be obtained by retaining the terms of the homogeneous solution satisfying reciprocity as well as the edge condition. The second part of this report presents solutions of the diffracted field based on the second and third order GIBCs. In both cases the necessary difference equations satisfied by h(a) are stated, but not solved. Essentially, the diffracted field is derived by ensuring 23

reciprocity. As mentioned above, this amounts to the tedious task of partitioning the homogeneous solution into reciprocal and nonreciprocal terms. Since the last are to be cancelled by the particular solution h(a), the diffracted field is the contribution of the remaining reciprocal terms. Finally, using the derived diffraction coefficients for the first, second, and third order GIBCs a number of backscatter echowidth patterns were presented. These included several coating configurations and demonstrated the strong variance of the third order solution from the first and second order ones for grazing incidences. The FORTRAN program used for generating the patterns is given in the Appendix. 24

References 1. T.B.A. Senior, "Approximate Boundary Conditions," IEEE Trans. Antennas and Propagat., Vol. AP-29, No. 5, pp 826-829, 1981. 2. T.B.A. Senior and J.L. Volakis, "Derivation and Application of a Class of Generalized Boundary Conditions" submitted to IEEE Trans. Antennas and Propagat.; see also University of Michigan Radiation Laboratory 388967-7-T. 3. S.N. Karp and F.C. Karal, Jr., "Generalized Impedance Boundary Conditions with Applications to Surface Waves Structures," in Electromagnetic Wave Theory, Part 1, ed. J. Brown, pp. 479-483, Pergamon: New York, 1965. 4. 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. 5. 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," Proceeding oft he IEEE, May 1989 (in press); see also University of Michigan Radiation Laboratory Report. 388967-6-T. 6. 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, Dec. 1987. 7. J.L. Volakis, "High Frequency Scattering by a Material Half-Plane and Strip," Radio Sci., Vol. 23, pp. 450-463, May-June 1988. 8. T.B.A. Senior, "Diffraction by a Right-Angled Second Order Impedance Wedge," submitted to Electromagnetics. 9. M. I. Herman, J.L. Volakis and T.B.A. Senior, "Analytic Expressions for a Function Occurring in Diffraction Theory," IEEE Trans. Antennas and Propagat., Vol. AP-35, Sept. 1987, pp. 1083-1086. 10. G.D. Maliuzhinets, "Inversion Formula for the Sommerfeld Integral," Sov. Phys. Dokl., Eng. Transl., Vol. 3, pp. 52-56, 1958. 25

List of Figure Captions 1. Geometry of the coated wedge. 2. Maximum allowed thickness vs. INI for a metal-backed layer modelled using the 1st and 2nd order GIBC with a 10-degree phase (and/or 10 percent amplitude) error. Curves shown are for E=2 and ~=7 with incidence at 30 degrees from grazing (a) H-polarization 2nd order GIBC. (b) H-polarization 1st order GIBC. 3. Maximum allowed thickness vs. INI for a metal-backed layer modelled using the 1 st and 2nd order GIBC with a 10-degree phase (and/or 10 percent amplitude) error. Curves shown are for ~=2 and ~=7 with incidence at 55 degrees from grazing (a) H-polarization 2nd order GIBC. (b) H-polarization 1st order GIBC. 4. Maximum allowed thickness vs. INI for a metal-backed layer modelled using the 1st and 2nd order GIBC with a 10-degree phase (and/or 10 percent amplitude) error. Curves shown are for ~=2 and ~=7 with incidence at 70 degrees from grazing (a) H-polarization 2nd order GIBC (b) H-polarization 1st order GIBC. 5. Maximum allowed thickness vs. INI for a metal-backed layer modelled using the 1st and 2nd order GIBC with a 10-degree phase (and/or 10 percent amplitude) error. Curves shown are for ~=2 and ~=7 with incidence at 30 degrees from grazing (a) E-polarization 2nd order GIBC. (b) E-polarization 1st order GIBC. 6. Maximum allowed thickness vs. INI for a metal-backed layer modelled using the 1st and 2nd order GIBC with a 10-degree phase (and/or 10 percent amplitude) error. Curves shown are for ~=2 and ~=7 with incidence at 55 degrees from grazing (a) E-polarization 2nd order GIBC. (b) E-polarization 1st order GIBC. 7. Maximum allowed thickness vs. INI for a metal-backed layer modelled using the 1st and 2nd order GIBC with a 10-degree phase (and/or 10 percent amplitude) error. Curves shown are for e=2 and ~=7 with incidence at 70 degrees from grazing (a) E-polarization 2nd order GIBC. (b) E-polarization 1st order GIBC. 8. Maximum allowed thickness vs. INI for a metal-backed layer modelled using 3rd order GIBC with a 10-degree phase (and/or 10 percent amplitude) error. Curves shown are for ~=2 and E=7 with incidence at 30 degrees from grazing (a) H-polarization (b) E-polarization. 9. Maximum allowed thickness vs. INI for a metal-backed layer modelled using 3rd order GIBC with a 10-degree phase (and/or 10 percent amplitude) error. Curves shown are for E=2 and ~=7 with incidence at 55 degrees from grazing (a) H-polarization (b) E-polarization. 10. Maximum allowed thickness vs. INI for a metal-backed layer modelled using 3rd order GIBC with a 10-degree phase (and/or 10 percent amplitude) error. Curves shown are for E=2 and E=7 with normal incidence grazing (a) H-polarization (b) E-polarization. 11. Backscatter Hz echowidth for a plane wave incident on a perfectly conducting half plane coated on both faces with 0.1 wavelengths thick electrically (e=l+i 103) or magnetically (g=l.+ilO5) perfectly conducting layer. (a) Magnetically perfectly conducting layer. (b) Electrically perfectly conducting layer. 26

12. Backscatter Hz echowidth for a plane wave incident on a perfectly conducting wedge having n=1.5 and coated on both faces with 0.1 wavelengths thick electrically (= 1+i103) or magnetically (g=l.+105) perfectly conducting layer. (a) Magnetically perfectly conducting layer. (b) Electrically perfectly conducting layer. 13. Backscatter Hz echowidth for a plane wave incident on a perfectly conducting wedge having n=1.833 and coated on both faces with a layer whose e=4 and pg=1. (a) Coating thickness T=0.1 wavelength. (b) Coating thickness =-0.2 wavelengths. 14. Backscatter Hz echowidth for a plane wave incident on a perfectly conducting wedge having n=1.833 and coated on both faces with an absorbing layer whose e=7.4+il.l and [p=1.4+i0.672. (a) Coating thickness T=0.1 wavelengths. (b) Coating thickness =-0.2 wavelengths. 27

APPENDIX FORTRAN Listing of Program GIBCWEDGE 28

1 C PROGRAM GIBCWEDGE C 2 C THIS PROGRAM COMPUTES THE DIFFRACTED/SCATTERED FIELD FROM C 3 C A DIELECTRICALLY COATED WEDGE C 4 C IF FAR ZONE IS CHOSEN ONLY DIFFRACTED FIELD IS GIVEN C 5 C IF NEAR ZONE IS CHOSEN THE TOTAL UNIFORM FIELD IS COMPUTED C 6 C PROGRAM WRITTEN BY J.L. VOLAKIS, AUGUST 1988 C 7 C C 8 COMPLEX A0,A1,A2,A3,A4,G1,G2,G3,C0,C1,C2, C3,RI 9 COMPLEX B1,B2,B3,All,Bll,CC,DD,EE,ETA4,DDC 10 COMPLEX*8 AU,BU,AL,BL,QQ,PP, RR, XM, THX,CPOWER,GEE,PSIPHO, PSIPH 11 COMPLEX PSIPI2,PSI,PSIPHI,AC2(3),BC2(3),AC3(3),BC3(3),CONST 12 COMPLEX THP1(3),THP2(3),THP3 (3),THM1(3),THM2(3),THM3(3) 13 COMPLEX REFL1,REFL2,REFL3,REFL4,HZ1,HZ2,HZ3 14 COMPLEX TH1,TH2,TH3,ER,UR,CF1,CF2,CF3,AA,BB,AB,APB,CFF,DFC 15 COMPLEX CI,CI4,ETA,ETA1,ETA2,ETA3,HEE,RINDX,DEN,TEMP,ETAN 16 COMPLEX CSQRC,LGEE,PHC,CFFG,ARG,FFCL,HA,GINCB,GINCT,REFLN 17 COMPLEX GUP,GBOT,FKP1,FKP2,FKP3,SUM,SGEE,TH1L,TH2L,REFLH 18 COMPLEX SWRES1,SWRES2,HAP,FKP4,FKP5,COSW1,COSW2,SWDl,SWD2 19 COMPLEX GAM1,GAM2,TEMPC,ROOT,Rl,R2,EXREFL,CTAN,REFLL,PHASE 20 DIMENSION HZDB1(361),HZDB2(361),HZDB3(361),ANG(361) 21 COMMON /CONS/AA,BB,AB,CF3 22 COMMON /PS/PSIPI2 23 DATA CI,PI,CI4/(0.,-1.),3.1415926,(.707107,-.707107)/ 24 PI2=PI/2. 25 TPI=2.*PI 26 PRINT *,'ENTER WEDGE ANGLE IN DEGREES:' 27 READ(5,*) WA 28 DTR=PI/180. 29 WA=WA*DTR 30 WN=2-(WA/PI) 31 PHIW=WN*PI/2. 32 PRINT *,'WEDGE CAP PHI AND N:',PHIW/DTR,WN 33 PSIPI2=PSIPHI(CMPLX(PI2,0.),PHIW) 34 C PRINT *,'PSIPI2',PSIPI2 35 PRINT *,'ENTER PH,PH0,DPH,DPHO(IN DEG):' 36 READ(5,*) PH,PH0,DPH,DPH0 37 C PRINT *,'NUMBER OF PATTERNS:' 38 C READ(5,*) NPLOTS 39 DTR=PI/180. 40 NPTS=WN*PI/DPH/DTR 41 PH=PH*DTR 42 PH0=PHO*DTR 43 PHI=PH 44 PHOI=PH0 45 DPH=DPH*DTR 46 DPH0=DPH0*DTR 47 C BEGIN PLOT LOOP 48 C DO 2000 J=1,NPLOTS 49 PH=PHI 50 PH0=PHOI 51 PRINT *, 'ENTER PERMIT.,PERMEAB.(EXP(+JWT)) AND THICK:' 52 READ(5, *) ER,UR,THK 53 C EXACT REFLECTION COEFFICIENT 54 SPH0=SIN(PH0) 55 CPHO=COS(PHO) 56 CPHOS=CPHO*CPHO 57 SPHOS=SPHO*SPHO 58 C ENSURE THE CORRECT BRANCH

59 ER=ER+CI*1.E-6 60 UR=UR+CI*1.E-6 61 C COMPUTE EXACT REFLECTION COEFFICIENT 62 RINDX=CSQRC(ER*UR) 63 ROOT=CSQRC(RINDX*RINDX-CPHOS) 64 ARG=2.*PI*THK*ROOT 65 CTAN=CSIN(ARG)/CCOS(ARG) 66 R1=ROOT*CTAN 67 R2=CI*ER*SPHO 68 EXREFL=- (R1-R2)/ (R1+R2) 69 CTAN=CSIN(2.*PI*THK*RINDX)/CCOS(2.*PI*THK*RINDX) 70 ETA=-CI*RINDX*CTAN/ER 71 C PHASE=CEXP (-2. *CI*2*PI*THK*SPHO) 72 C EXREFL=EXREFL*PHASE 73 C "A" COEFFICIENTS OF GIBC 74 AR1=2.*PI*THK 75 RI=RINDX 76 Cl=CSIN(AR1*RI)/CCOS(AR1*RI) 77 C2=CSIN(.5*AR1/RI)/CCOS(.5*AR1/RI) 78 C3=C1*C2+1. 79 C1=C1-C2 80 A0=(RI-(.5/RI))*C1 81 A1=CI*ER*C3 82 A2=(C1+AR1*C3*(RI-(.5/RI)))/(2.*RI) 83 A3=-CI*AR1*ER*C1/(2.*RI) 84 A4=AR1*C3/ (4.*RI*RI) 85 REFL4=-(A4* (SPHO**4)-A3* (SPHO**3)+A2* (SPHO**2)-A1*SPH0+A0) 86 TEMPC=(A4* (SPHO**4)+A3* (SPHO**3)+A2* (SPHO**2)+A1*SPHO+AO) 87 REFL4=REFL4/TEMPC 88 REFL2=-(A2* (SPH0**2) -A*SPH0+A) / (A2* (SPHO**2)+A1*SPHO+AO) 89 REFL3=-(-A3*(SPH0**3)+A2*(SPH0**2)-A1*SPH0+A0) 90 TEMPC=(A3*(SPHO**3)+A2*(SPHO**2)+A1*SPHO+AO) 91 REFL3=REFL3/TEMPC 92 C FIRST ORDER BOUNDARY CONDITION 93 THP1 (1) =PI2-HEE (ETA, 1,1.) 94 THM1(1)=THP1(1) 95 PRINT *,'1ST ORDER TEST on THETA:',ETA,CSIN(THP1(1)) 96 PRINT *,'lst ORDER THETA:',THP1(1) 97 REFL1=- (ETA-SPHO) /(ETA+SPHO) 98 C SECOND ORDER BOUNDARY CONDITION 99 G1=A1/(A0+A2) 100 G2=-A2/(AO+A2) 101 TEMPC=CSQRT(Gl*G1+4.*G2*(1.+G2)) 102 ETA1=0.5* (-G1+TEMPC)/G2 103 ETA2=0.5*(-G1-TEMPC)/G2 104 PRINT *,'2ND ORDER ETAS:',ETA1,ETA2 105 C REFLECTION COEF. USING ETAS 106 REFL2=-(ETA1-SPHO)*(ETA2-SPHO)/((ETA1+SPHO) *(ETA2+SPHO)) 107 THP2(1)=PI2-HEE(ETA1,1,1.) 108 THP2(2)=PI2-HEE(ETA2,1,1.) 109 IF(REAL(THP2(1)).LT.0.)THP2(1)=PI-THP2(1) 110 IF(REAL(THP2 (2)).LT. 0.)THP2(2)=PI-THP2(2) 111 C THP2(2)=THP2(2)-REAL(THP2(2)) 112 THP2(2)=THP2(2) 113 THM2(1)=THP2(1) 114 THM2(2)=THP2(2) 115 PRINT *,'TH1,TH2:',THP2(1),THP2(2) 116 PRINT *,'CHECK:',CSIN(THP2(1)),ETA1,CSIN(THP2(2)),ETA2

117 C 3RD ORDER GIBC 118 G1= (A1+A3)/(A0+A2) 119 G2=-A2/(A0+A2) 120 G3=-A3/(AO+A2) 121 C find cubic roots 122 PP=-G2/G3 123 QQ=-(G1+G3)/G3 124 RR=(1.+G2)/G3 125 AL=(3.*QQ-PP*PP)/3. 126 BL=(2. *PP*PP*PP-9.*PP*QQ+27.*RR)/27. 127 XM=2.*(0.,1.)*CSQRC(AL/3.) 128 ARG=3.*BL/(AL*XM)+CI*1.E-6 129 THX=HEE(ARG,1,1.)/3. 130 ETA1=XM*CCOS(THX)-PP/3 131 ETA2=XM*CCOS((PI/1.5)+THX)-PP/3 132 ETA3=XM*CCOS((4.*PI/3. ) +THX) -PP/3 133 PRINT *,'3RD ORDER ETAS:',ETA1,ETA2,ETA3 134 C reflection coefficient using etas 135 REFL3=-(ETA1-SPH0) *(ETA2-SPHO) *(ETA3-SPHO) 136 REFL3=REFL3/((ETA1+SPH0) *(ETA2+SPH0) *(ETA3+SPH0)) 137 THP3(1)=PI2-HEE(ETA1,1,1.) 138 THP3 (2) =PI2-HEE (ETA2,1,1.) 139 THP3(3)=PI2-HEE(ETA3,1,1.) 140 DEF1=CABS (THP3 (1)-THP1 (1)) 141 DEF2=CABS (THP3 (2)-THP1 (1)) 142 DEF3=CABS (THP3 (3)-THP1 (1)) 143 IF((DEF1.LT.DEF2).AND.(DEF1.LT.DEF3)) THEN 144 TH1=THP3(1) 145 TH2=THP3(2) 146 TH3=THP3(3) 147 ENDIF 148 IF((DEF2.LT.DEF1).AND.(DEF2.LT.DEF3)) THEN 149 TH1=THP3(2) 150 TH2=THP3(1) 151 TH3=THP3(3) 152 ENDIF 153 IF((DEF3.LT.DEF2).AND.(DEF3.LT.DEF1))THEN 154 TH1=THP3(3) 155 TH2=THP3(2) 156 TH3=THP3(1) 157 ENDIF 158 THP3(1)=TH1 159 THP3(2)=TH2 160 THP3(3)=TH3 161 THP3(2)=PI-THP3(2) 162 THP3(3)=PI-THP3(3) 163 THM3(1)=THP3(1) 164 THM3(2)=THP3(2) 165 THM3(3)=THP3(3) 166 PRINT *,'3RD ORDER THETAS:',THP3(1),THP3(2),THP3(3) 167 PRINT *,'CHECK:',CSIN(THP3(1)),ETA1,CSIN(THP3(2)),ETA2 168 C PRINT REFLECTION COEFFICIENTS: 169 PRINT *,' REFLECTION COEFFICIENTS:' 170 EXMAG=CABS (EXREFL) 171 EXPHAS=BTAN2(AIMAG(EXREFL),REAL(EXREFL)) 172 ELMAG=CABS(REFL2) 173 ELPHAS=BTAN2(AIMAG(REFL2),REAL(REFL2)) 174 EHMAG=CABS (REFL3)

175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 EHPHAS=BTAN2(AIMAG(REFL3),REAL(REFL3)) ENMAG=CABS(REFL4) ENPHAS=BTAN2(AIMAG(REFL4),REAL(REFL4)) E1MAG=CABS (REFL1) E1PHAS=BTAN2(AIMAG(REFL1),REAL(REFL1)) PRINT *,'EXACT,4TH,3RD, 2ND & 1ST ORDER:' PRINT *,EXREFL,EXMAG,EXPHAS*180/PI PRINT *,REFL4,ENMAG,ENPHAS*180./PI PRINT *,REFL3,EHMAG,EHPHAS*180./PI PRINT *,REFL2,ELMAG,ELPHAS*180/PI PRINT *,REFL1,E1MAG,E1PHAS*180/PI PRINT *,'CHOOSE 1ST, 2ND OR 3RD ORDER B.C:' READ(5,*) M GENERATE A & B CONSTANTS C C C All=CCOS((THP1(1)-PI2)/WN) B11=CCOS( (THM1(1)-PI2)/WN) PRINT *,'THETA:',THP1(1) DO 10 MS=1,2 AC2(MS)=CCOS((THP2(MS)-PI2)/WN) 10 BC2(MS)=CCOS ( (THM2(MS)-PI2)/WN) DO 11 MS=1,3 AC3 (MS)=CCOS ( (THP3 (MS)-PI2)/WN) 11 BC3 (MS) =CCOS ( (THM3 (MS)-PI2)/WN) P1=SIN(PI2/WN) Q1=COS(PI2/WN) C BEGIN PATTERN LOOP DO 1000 I=1,NPTS Al=All B1=B11 SPN=SIN(PH/WN) SON=SIN(PHO/WN) CPN=COS(PH/WN) CON=COS (PHO/WN) CONST=CI4/(2.*PI) CONST=CONST/WN P2=Pl*P1 Q2=Q1*Q1 DENOM=SPN*SPN-2.*(2.*Q2-1. ) *SPN*SON+SON*SON-4.*P2*Q2 DNOM1=SIN( (PH-PI)/WN)-SON DNOM2=SIN ((PH+PI)/WN)-SON DNOM3=DNOM1 *DNOM2 DI=DNOM1 HZ1=CI4/(2.*PI) HZ1=(HZ1*CON/WN) CONST=HZ1/DENOM C FISRT ORDER STADARD MALIUZHINETS DIFFR. COEFF. C PSIPHO=PSI(CMPLX(PHO,0.),PHIW,THP1(1),THM1(1)) C PSIPH=PSI(CMPLX(PH,0.),PHIW,THP1(1),THM1(1)) C HZ1=HZ1/PSIPHO C TEMPC=PSI (CMPLX(PH-PI,0.), PHIW, THP1 (1), THM1(1))/DNOM1 C TEMP=PSI(CMPLX(PH+PI,0.),PHIW,THP1(1),THM1(1)) HZ1=HZ1* (TEMPC-TEMP/DNOM2) C 1ST ORDER IMPEDANCE WEDGE DIFFRACTION COEFFICIENT TEMPC=(2*Q* (SPN*SON+P2)-(Al-B) * (SPN+SON)-2.*Q1*Al*B1) TEMPC=TEMPC*CPN*2.*P1 XM=GEE(CMPLX(PH,0.),PHO,PHIW, THP1, THM1,1) HZ1=-XM*TEMPC*CONST C 2ND ORDER IMP. WEDGE DIFFRACTION COEFFICINET

233 A1=AC2 (1)+AC2(2) 234 B1=BC2(1)+BC2(2) 235 A2=AC2(1)*AC2(2) 236 B2=BC2(1)*BC2(2) 237 ST=SPN*SON+P2 238 SPON=SPN+SON 239 TEMPC=2 *Ql* (ST**2) - (Al-Bl) *SPON*ST+2 *Q* *(A2+B2-Al*B1) *ST 240 TEMPC=TEMPC+ (A2*B1-Al*B2) *SPON+2 *Q1*A2*B2 241 TEMPC=TEMPC*2. *Pl*CPN 242 XM=GEE (CMPLX (PH, O.),fPH0, PHIW, THP2, THM2,2) 243 HZ2=CONST*XM*TEMPC 244 245 C 3RD ORDER IMP. DIFFR. COEFFICIENT 246 A1=AC3 (1)+AC3 (2)+AC3 (3) 247 B1=BC3(1)+BC3(2)+BC3(3) 248 A2=AC3(1) *AC3(2) +AC3 (2)*AC3 (3)+AC3(3) *AC3 (1) 249 B2=BC3(1)*BC3(2)+BC3(2)*BC3(3)+BC3(3)*BC3(l) 250 A3=AC3(1) *AC3 (2)*AC3(3) 251 B3=BC3(1)*BC3(2)*BC3(3) 252 ST=SPN*SON+P2 253 SPON=SPN+SON 254 TEMPC=-2.*Q1*(ST**3-P2**3)+(A1lB1)*SPON* (ST*ST-2 *P2*P2) 255 TEMPC=TEMPC+2.*Q1* (A1*B2-A2*B1) * (ST*ST) 256 TEMPC=TEMPC+ (A3-B3+Al*B2-A2*B1) *SPON*ST 257 TEMPC=TEMPC+2. *Q1* (A3*B1+A1*B3-.A2*B2) *ST+ (A3*B2-A2*B3) *ST 258 TEMPC=TEMPC+2 *Ql*A3*B3 259 TEMPC=TEMPC*2 *Pl*CPN 260 XM=GEE(CMPLX(PH,0.),PHOPHIWTHP3,THM3,3) 261 HZ3=CONST*XM*TEMPC 262 HZA=CABS(HZ1) 263 HZDB1(I)=10.*ALOG10(2.*PI*HZA*HZA) 264 HZA=CABS(HZ2) 265 HZA2==10*ALOG10 (2 *PI*HZA*HZA) 266 HZDB2(I)=HZA2 267 HZA=CABS(HZ3) 268 HZA3=10*[ALOG (2. *PI*HZA*HZA) 269 HZDB3(I)=HZA3 271 ANG(I)=PH/DTR 272 C PRINT *,IPHHZ:IANG(I),HZ1,HZ2,HZ3 273 PH=PH+DPH 274 PHO=PHO+DPHO 275 1000 CONTINUE 276 C IEND=0 277 C IF (J. EQ. NPLOTS) IEND=1 278 CALL GENPLO(ANGHZDB3,NPTS,0,O) 279 CALL GENPLO(ANG,HZDB2,NPTS,1,0) 280 CALL GENPLO(ANGHZDB1,NPTS,2,1) 281 2000 CONTINUE 282 CALL EXIT 283 END 284 C 285 COMPLEX FUNCTION GEE(ARGPHOPHIW,THP,THM,M) 286 COMPLEX PSIARGTHP(3),THM(3),PSIPI2 287 COMMON /PS/PSIPI2 288 DATA PI,PI2/3.1415927,1.5707963/ 289 M8=8.*M 290 GEE=PSIPI2**M8/(4 **M) 291 DO 10 MM=1,M

292 GEE=GEE/PSI (ARG,PHIW,THP (MM),THM(MM)) 293 GEE=GEE/PSI(CMPLX(PHO,0.),PHIW,THP (MM),THM(MM)) 294 c if(M.eq.3) print *,'gee:',gee 295 10 CONTINUE 296 RETURN 297 END 298 C 299 COMPLEX FUNCTION PSI(ALPHA,PHI,THETAO,THETA1) 300 COMPLEX PSIPHI 301 COMPLEX THETAO,THETA1,UJ,CJ,ALPHA 302 COMPLEX CN1,CN2,CN3,CN4 303 DATA PI2/1.5707963/ 304 DATA CJ,UJ/(0.,1.),(1.,0.)/ 305 CN1=ALPHA+(PHI+PI2) *UJ-THETAO 306 CN2=ALPHA-(PHI+PI2)*UJ+THETA1 307 CN3=ALPHA+ (PHI-PI2)*UJ+THETAO 308 CN4=ALPHA-(PHI-PI2) *UJ-THETA1 309 c print *,'cnl,2,3,4###:',cnl,cn2,cn3,cn4 310 CN1=PSIPHI(CN1,PHI) 311 CN2=PSIPHI(CN2,PHI) 312 CN3=PSIPHI(CN3,PHI) 313 CN4=PSIPHI(CN4,PHI) 314 PSI=CN1*CN2*CN3*CN4 315 c print *,'cnl,cn2,cn3:',cnl,cn2,cn3,cn4 316 RETURN 317 END 318 C 319 COMPLEX FUNCTION PSIPHI(CANG,PHI) 320 C WRITTEN BY MARTIN HERMAN, UNIVERSITY OF MICHIGAN 321 C BASED ON THE SUBMITTED ARTICLE BY HERMAN, VOLAKIS 322 C AND SENIOR. 12/1/86 323 C 324 C CANG: Complex Argument of the function 325 C 326 C THIS CALCULATES THE MALUIZHINETS FUNCTON 327 C FOR ANY ARBITRARY WEDGE ANGLE PHI 328 C 329 C IF THE IMAGINARY PART IS LESS THAN 10 THEN A 330 C REIMANN SUM IS PERFORMED, OTHERWISE THE LARGE ARG 331 C FORM IS USED 332 C 333 C PSIPI2 IS THE COMPLEX MAL. FUNCTION OF THE IDENTITY 334 C GIVEN IN HERMAN, VOLAKIS, AND SENIOR 335 C 336 COMPLEX CANG,CANG1,COEF 337 COMPLEX CN1,CN2,CN3,CN4,CN5 338 COMPLEX PSISQ,U,UJ 339 DATA PI,PI2/3.141592654,1.5707963/ 340 DATA UJ/(1.,0.)/ 341 C 342 C CALCULATE PSISQ USING REIMAN SUM 343 C 344 CIll 345 ClII MIDPOINT METHOD 5 POINTS (INTERVAL 0,1.5) 346 Clll 347 UR=PI/2. 348 UI=0. 349 SUM=0.

350 SUM1=0. 351 FH=1.5/5. 352 FH2=FH/2. 353 DO 10 I=1,5 354 S=FLOAT (I-l) *FH+FH2 355 FS= (COSH (UR*S) *COS (UI*S) -1.) 356 DENOM=S*COSH (PI*S/2.) *SINH (PHI*2. *S) 357 FS=FS*FH/DENOM 358 FS1=(SINH(UR*S)*SIN(UI*S)) 359 FS1=FS1*FH/DENOM 360 SUM=SUM+FS 361 SUM1=SUM1+FS1 362 10 CONTINUE 363 CN1=-.5*CMPLX(SUM,SUM1) 364 CN1=CEXP(CN1) 365 PSISQ=CN1*CN1 366 C 367 AR=REAL (CANG) 368 AI=AIMAG(CANG) 369 ITTT=0 370 IF(AR.GT.0)GO TO 30 371 AR=-AR 372 ITTT=1 373 30 ITT=0 374 IF(AI.GT.0)GO TO 40 375 ITT=i 376 AI=-AI 377 40 IT=0 378 CANG1=CMPLX (AR, AI) 379 IF(AR.LT.PI2)GO TO 90 380 AR=AR-PI 381 C 382 CNS=UJ* (PI*PI/(8.*PHI)) 383 CN4=UJ*PI 384 RS=PI/(4. *PHI) 385 C 386 COEF=PSISQ*CCOS (CANG1*R5-CN5) 387 IT=I 388 IF(AR.LT.PI2)GO TO 90 389 AR=AR-PI 390 COEF= (CCOS (CANG1*R5-CN5))/CCOS ((CANG1-CN4) *R5-CN5) 391 IT=2 392 IF(AR.LT.PI2)GO TO 90 393 AR=AR-PI 394 COEF=PSISQ*CCOS (CANG1*R5-CN5) *CCOS ( (CANG1-2. *CN4) *R5-CN5) 395 COEF=COEF/CCOS ((CANG1-CN4) *R5-CN5) 396 IT=3 397 IF(AR.LT.PI2)GO TO 90 398 AR=AR-PI 399 COEF=CCOS ( (CANG1-2. *CN4) *R5-CN5) *CCOS (CANG1*R5-CN5) 400 COEF=COEF/(CCOS((CANG1-3.*CN4)*R5-CN5)*CCOS((CANG1-CN4)*R5-CN5)) 401 IT=2 402 IF(AR.LT.PI2)GO TO 90 403 AR=AR-PI 404 COEF=CCOS ( (CANG1-2. *CN4) *R5-CN5) *CCOS (CANG1*R5-CN5) 405 COEF=COEF/(CCOS((CANG1-3.*CN4)*RSCN5)*CCOS((CANG-CN4)*R5-CN5)) 406 COEF=PSISQ*COEF*CCOS ( (CANG1-4.*CN4) *R5-CN5) 407 IT=3

408 90 CONTINUE 409 IF(ABS(AR).GT.PI2)PRINT *,'AR > PI2 ',AR 410 C 411 U=CMPLX(AR,AI) 412 UR=AR 413 UI=AI 414 IF(UI.LE.10.)THEN 415 C lI 416 CI I SMALL ARG APPROACH USING 417 C I I MIDPOINT METHOD 5 POINTS (INTERVAL 0,1.5) 418 Clll 419 SUM=0. 420 SUM1=0. 421 FH=1.5/5. 422 FH2=FH/2. 423 DO 100 I=1,5 424 S=FLOAT(I-1)*FH+FH2 425 FS= (COSH (UR*S) *COS (UI*S) -1.) 426 DENOM=S*COSH(PI*S/2.)*SINH(PHI*2.*S) 427 FS=FS*FH/DENOM 428 FS1= (SINH(UR*S)*SIN(UI*S)) 429 FS1=FS1*FH/DENOM 430 SUM=SUM+FS 431 SUM1=SUM1+FS1 432 100 CONTINUE 433 CN1=-.5*CMPLX(SUM,SUM1) 434 PSIPHI=CEXP (CN1) 435 ELSE 436 C ll 437 C ll large APPROX 438 Cl II 439 CN1=U*PI/(4. *PHI) 440 CN2=CCOS(CN1) 441 AMP=CABS(CN2) 442 AMP=SQRT(AMP) 443 R1=REAL(CN2) 444 R2=AIMAG(CN2) 445 PH=ATAN2(R2,R1) 446 IF (PH. LT.0. ) PH=2. *PI+PH 447 PH=PH/2. 448 R1=AMP*COS (PH) 449 R2=AMP*SIN (PH) 450 CN1=CMPLX(R1,R2) 451 B=2.556343 452 C=-3.259678 453 D=1.659306 454 E=-.3883548 455 F=.03473964 456 PSIPHI=CN1*EXP(-(B*PHI+C*PHI**2+D*PHI**3+E*PHI**4+ 457 1F*PHI**5)/PI) 458 IF(REAL(PSIPHI).LT.0.)PSIPHI=-PSIPHI 459 END IF 460 C 461 IF(IT.EQ.1)PSIPHI=COEF/PSIPHI 462 IF(IT.EQ.2)PSIPHI=COEF*PSIPHI 463 IF(IT.EQ.3)PSIPHI=COEF/PSIPHI 464 IF(ITT.EQ.1)PSIPHI=CONJG(PSIPHI) 465 IF(ITTT.EQ.1)PSIPHI=CONJG(PSIPHI)

466 RETURN 467 END 468 COMPLEX FUNCTION HEE(ETA,IUD,SBO) 469 Cl I NEW VOLAKIS VERSION 470 CIII COMPUTES THE INVERSE COSINE OF COMPLEX NUMBER 471 COMPLEX ETA,ETA1,CJ 472 DOUBLE PRECISION RE,AE,REP,REM,AA,BB,SGN,RAA 473 DATA SRT2,FPI,CJ/1.414213562,12.56637061,(0.,1.)/ 474 DATA PSIPI2,PI/.9656228,3.14159265/ 475 ETA1=1. / (ETA*SB0) 476 IF(IUD.EQ.1)ETA1=ETA/SBO 477 RE=REAL(ETA1) 478 AE=AIMAG(ETA1) 479 REP=RE+1. 480 REM=RE-1. 481 AA=.5*(DSQRT(REP*REP+AE*AE)+DSQRT(REM*REM+AE*AE)) 482 BB=.5* (DSQRT (REP*REP+AE*AE)-DSQRT (REM*REM+AE*AE)) 483 IF(AE.NE.O.DO)THEN 484 SGN=AE/DABS(AE) 485 ELSE 486 SGN= 1.DO 487 ENDIF 488 RAA=AA*AA-1. 489 IF(RAA.LT.1.E-6)RAA=0. 490 HEE=DARSIN(BB)+CJ*DLOG(AA+DSQRT(RAA)) *SGN 491 HEE=.5*PI-HEE 492 300 RETURN 493 END 494 COMPLEX FUNCTION CSQRC(Z) 495 COMPLEX Z 496 ZR=REAL(Z) 497 ZI=AIMAG(Z) 498 PHAS=BTAN2(ZI,ZR) 499 CSQRC=SQRT(CABS(Z))*CEXP(.5*(0.,1.)*PHAS) 500 RETURN 501 END 502 REAL FUNCTION BTAN2(Y,X) 503 DATA PI/3.1415926/ 504 IF(ABS(X).GT.1.E-6) GO TO 20 505 IF(ABS(Y).GT.1.E-6) GO TO 10 506 BTAN2=0. 507 RETURN 508 10 BTAN2=.5*PI 509 IF(Y.LT.0.)BTAN2=-BTAN2 510 20 BTAN2=ATAN2(Y,X) 511 RETURN 512 END

p Fig. 1. Geometry of the coated wedge.

w 0 1 -z I-. 2.0 1.8 1.6, 1.4 1.2, 1.01 0.8, 0.6 0.4, 0.2 0.01 0 2 4 6 8 MAG REFRACTIVE INDEX 1 0 1 2 (a).0" Ix 2.0 -1.8 - 1.6 - 1.4 - 1.2 - 1.0 - 0.8 - 0.6 - 0.4 - 0.2 - H-POLARIZATION 30DEG 1 ST ORDER BOUNDARY CONDITION -a- 6=2, arg. J~=O -*- S=7, arg. g=O A A I U 11 I 0 2 4 6 8 MAG. REFRACTIVE INDEX 1 0 1 2 (b) Figure 2

oc VI-" z Y. - 2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0 2 4 6 8 MAG. REFRACTIVE INDEX 10 12 (a) po Cc IU en VW 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0 2 4 6 8 MAG REFRACTIVE INDEX (b) 10 12 Figure 4

2.0 a1 LU I~x 1.8 - 1.6 - 1.4 - 1.2 - 1.0 - 0.8 - 0.6 - 0.4 - 0.2 E&POLARIZATION 30DEG 2ND ORDER BOUNDARY CONDMON -in- 8,=2, arg. 0t= -*- 8-7, arg. g= ~~am* a-( I I4 0 I - I - I - 2 4 6 8 MAG REFRACTIVE INDEX (a) 1 0 1 2 2.0 I-. xt 1.8 - 1.4 -, 1.2 - 1.0 - 0.8 - 0.6 - 0.4 - 0.2 " E-POLARIZATION 30DEG 1 ST ORDER BOUNDARY CONDM~ON 8-wC2, arg. L-*- 8-7, arg4L-O 0.0 I4 0 2 4 6 8 MAO REFRACTIVE INDEX 1 0 1 2 (b) Figure 5

2.0 I1 A oc LJ 0 I-2 I,, 1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 -n - E-POLARIZATION 55DEG 2ND ORDER BOUNDARY CONDITION - ~=2, arg. o 0 -.- ~=7, arg. =0 ~ '.. i.... * %.P 0 2 4 6 8 MAG REFRACTIVE INDEX 10 12 (a) 1 -U 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0 2 4 6 8 10 12 MAO. REFRACTIVE INDEX (b) Figure 6

9n. 1.8 -AU * 1.4-. 12 -0 0 1 - 1.0 -L0.8 -o 0.6 -X 0.4 -2 0.2 -ff - E-POLARIZATION 70DEG 2ND ORDER BOUNDARY CONDITKON - S-2, arg.:0 S- C7, arg. L-0 I ItCaaBrfV UU......... 0 2 4 6 8 MAG. REFRACTIVE INDEX 10 12 (a) A: cc LU ICi) 31 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0 2 4 6 8 MAG REFRACTIVE INDEX 10 12 (b) Figure 7

10 0.1 I — 8 6 4 2 0 0 2 4 6 8 1 0 1 2 MAG REFRACTIVE INDEX (a) 10 Z4 cc I. 0 0 2 4 6 8 1 0 1 2 MAO REFRACTIE INDEX (b) Figure 8

W 1.6 O 1.4 1.2 wO -S-2 i 0.8- -- ~=7 ' 0.6 0.4 a H-POLARIZATION 55 0.2 - 3RD ORDER BOUNDARY COt 0.0 * I - I 0 2 4 6 8 MAG REFRACTIVE INDEX (a) 2.0 -1.8 w 1.6 -1.4; 1.2.~. E= 0.8 0.0- I * I 0 2 4 6 8 MAG REFRACTIVE INDEX (b) 10 12 [ 10 12 Figure 9

w a ITIJ o I CJ 0 2 4 6 8 MAG. REFRACTIVE INDEX 10 12 (a) rw o a ILU Vs 2.0 1.8 -1.6 -1.4 -1.2 -1.0 -0.8 0.6 0.4 0.2 -0.0 0 2 4 6 8 MAG REFRACTIVE INDEX (b) 10 12 Figure 10

C1 d w R A rt -A 3rdt order GIBC mr NGLE IN DEGREES 105 \ c3, coating thickness = 0.1X V oo / 3rd order GIBC \ I t ' 2nd order GIBC 1 / A 1st order GIBC \ '-180.00 -120.00 -60.00 0.00oo 60.00 120.00 180.00 RNGLE IN DEGREES Figure 1 la

CZJ Wedge Angle = 0 C.-i I g=l +i i 05 coating thickness = O. 1 3rd order GIBO 2nd order GlBC 1 st order GIBC -ft -A — A CZ CZ) 6 v.-. I:z CZ uci C1) d '-10.0 -120.00 -60.0 0.00 60.00 120.00 RNGLE IN DEGREES 180.00 Figure 1lib

CMJ A C3 cm OD'1 z3 OiQ~ LLJIR ~. D '-135.00 We dge Angle = 900 S=j9,1, g+i 105 coating thickness = 0. 1?X _________3rd order GIBO 0 2nd order GIBO -A 1 st order GIBO -90.00 -45.00 0.00 45.00 90.00 RNGLE IN DEGREES 135.00 Figure 12a

d o I | Wedgcoati ngless = 90.1 Ll ___0 3rd order GIBC CV E=I +i103, Il=1. CZo- 32nd order GIBC 2nst order GIBC '-135.00 -90.00 -45.00 0.00 45.00 90.00 135.00 RNGLE IN DEGREES, Figure 12b

A= m 0D:z 0 CD) NJ Wedge Angle = 300 8=4, gi=1 coating thickness = O. 1 _________3rd order GIBO 0 2nd order GIBC A. ist order GIBO ' -180.00 -120.00 RNGLE IN Figure 13a

Ij a Wedge Angle = 30~ Z o W g coating thiclkness = 0.2X Lu. ____11 3rd order GIBC z I _ _ 2nd order GIBC A11 1st order GIBC '-180.00 -120.00 -60.00 0.00 60.00 120.00 180.00 RNGLE IN DEG RNGLE IN DEG' Figure 13b

co 03 I Cl z('3 H - ', 3 Cl CD Uco -180. I -180.00 Wedge Angle = 30~ &=7.4+i1.11, =1.4+iO.672 coating thickness = 0.1X ______ 3rd order GIBC o__ j 2nd order GIBC.A 1 st order GIBC - I a -120.00 -60.00 0.00 60.00 120.00 RNGLE IN DEG' 180.00 Figure 14a

Wedge Angle = 30~ =s7.4+i1.11, L=1.4+i0.672 coating thickness = 0.2k 3rd order GIBC 2nd order GIBC -— st order GIBC 1st order GIBC | C3 -A A A m c-180.00 -120.00 -60.00 0.00 60.00 120.00 RNGLE IN DEG, 180.00 Figure 14b