RL-862 APPROXIMATE BOUNDARY CONDITIONS, PART II Thomas B.A. Senior June 1990 RL-862 = RL-862

Approximate Boundary Conditions, Part II Thomas B.A. Senior Radiation Laboratory Department of Electrical Engineering and Computer Science University of Michigan, Ann Arbor, MI 48109-2122 June 1990 Abstract Approximate boundary conditions are constructed to simulate the scattering from an opaque body covered with a layer of inhomogeneous dielectric whose inner and outer boundaries are coordinate surfaces in an orthogonal curvilinear coordinate system. Two different situations are considered. The first is a low contrast material for which an expansion in powers of the layer thickness r is appropriate. Approximate boundary conditions through the third order are derived and these illustrate the general form of higher order boundary conditions for a non-planar surface. The second is a high contrast material, and the technique developed in Part I of this report is used to obtain boundary conditions accurate to the second order in 4 where N is the complex refractive index of the material. In the special case of a circular cylinder covered with a layer of homogeneous dielectric of uniform thickness, the results are confirmed by starting with the known modal expansion for the scattered field. 1

1 Introduction Approximate boundary conditions are frequently employed to simulate the material properties of a scattering object, and the standard impedance boundary condition has been in use for many years. In principle at least, the accuracy can be improved by including higher derivatives of the field components, and the application of such generalized impedance boundary conditions (GIBCs) is now being considered [Volakis and Senior, 1989]. For a planar surface it is relatively easy to construct a hierarchy of conditions in terms of the normal derivatives of the normal field components, and methods are available [Senior and Volakis, 1989] to determine the coefficients. By tangential integration, the conditions can also be expressed in terms of the tangential field components. For a curved surface, however, a natural hierarchy is less apparent. In Part I of this report, hereafter referred to as I, a method developed by Rytov [1940] was used to derive a series of approximate boundary conditions applicable at the curved surface of a body whose material properties may vary continuously, both laterally and in depth. The approximations are based on the assumption that IN\ > 1 where N is the complex refractive index of the material, and the order of the condition is determined by the highest power of [N -1 which is retained. For a body whose surface is 7 = 7y where a, /, - are orthogonal curvilinear coordinates, boundary conditions through the second order were derived. We now turn our attention to the practically important case of a coated body, and seek approximate boundary conditions to simulate the scattering properties of an opaque (typically metallic) body covered with a layer of dielectric material. In Sections 2 and 3 a method proposed by Weinstein [1969] is used to obtain boundary conditions for a thin layer of "low contrast" material whose complex refractive index is not large in magnitude. Although this is not the situation of most practical interest, the procedure is very simple and leads to a natural hierarchy of conditions. The analogous problem of a "high contrast" material is treated in Sections 4-6 using an extension of the analysis in I. For maximum generality, the dielectric is assumed to be inhomogeneous, and boundary conditions through the second order are constructed. Finally, in Section 7 we consider the case of a circular cylinder with a homogeneous coating of uniform thickness and use 2

the modal solution to derive approximate boundary conditions for low and high contrast materials. These are in agreement with the preceding results. 2 Low Contrast Coating, Second Order A metallic or other opaque body is covered with a layer of dielectric material whose properties may vary laterally as well as in depth, and is illuminated by an electromagnetic field. In terms of the orthogonal curvilinear coordinates a, /, -7 with metric coefficients h,, h,, h, respectively, the region 7o - 7 < y < 7o consists of a dielectric material with permittivity e and permeability /. The lower surface 7 = 7- - is that of the body itself, and to preserve duality it is assumed that a standard impedance boundary condition is imposed here. Thus, at 7 = y7o - 7 x ( x E') = 7mf x H' implying E - -1nmHo, Ed = rmH; (1) where the affix i denotes the field in the dielectric and r/m is specified. The region 7 > 7y is free space having propagation constant ko and intrinsic impedance Zo, and we seek boundary conditions which can be applied to the exterior field components at y = 7o to simulate the scattering properties. Following Weinstein [1969] the fields in the dielectric are expanded in Taylor series in 7. In particular OE' E(o -r) =E - E r +- - a + O(2) (2) for small r, where we show only the y dependence and the fields on the right hand side are evaluated at = 7 - 0. From Maxwell's equation with a time factor e-iwt h f {e(haE) - (hE) a = koZoH giving = -E -(ln h) + ikoZo- h.,H + )(hE) (3) 9y o ho, ( 3

and therefore E(-,- 7-) = { 715r(ln ha,)} Ect -,7' l H ~ Opr2 (4) Similarly E'(-o - 7) 1~r~(In h,3)} E~ + i'k0Zjor h,Hc' — j-(hE,'+ O(72), (5) and by duality H'(yo - 7) = haOQ {I + 7TQ(Inh1,)} H,% -Ik0Y,,7-hlE~ 7+ Q9r2) (6) H(7 - 7-) = (7) die] On applying the boundary conditions (1) at the lower surface of the lectric, we obtain 1~Tjz(ln ha)-Zko7!-Yoq h h ct}E - - hj E { a0 z =-7m jl+r79Y(In h43 k,, l 7 ILh-yJH; - Ti- (hjHI)] + O(,r') (8) 4

00 e h { I + rT (ln h) - ikoT Yolmh E (h ) 0E ) 9 =rl [{ + - (ln h) Z- ikoT h- H - - (h-H) + O(T2) L 7 [o T/m h OaU and by imposing the boundary conditions at the dielectric-air interface 7 = y7, (8) can be expressed in terms of the exterior fields as r O9 1Eo 7 (9 PO Ea ( =E -r & ~- hE3 ( /1 hy )} + O(r ) ha a ho E - h3 p U. (9) E- h ( ) = + 0(r2) h13 dp \ =E h. Oa pJ where 1 + Tr (ln ho) - ikorT Y h 1m + r (ln ha) - ikOrT TYomhy (10) 1+ r (ln ha) - ikoT h Frl = 7m ik.go l 1 + r (ln h3)- ikoTrYo7mh, We note that Fr = 1/r* where the asterisk denotes the dual quantity. In the context of GIBCs, (9) are second order boundary conditions, and correct to the first order in r they can be written as 7V(7 { ( r ( )}) -r77 H-V ( h ) } ((1 1) with - = r + -- ' It can be verified that (11) satisfies the duality condition and its general form is consistent with (I.53). 5

In the particular case of a perfectly conducting substrate, rm:= 0 and (10) reduce to r = rl = — ikor-Zoh (12) Po to the first order in r. The impedance is now isotropic in spite of the curvature of the surface, and with this change (9) is unaffected, but because of the factor r/rlm in (8), equations (9) and (11) are no longer correct in the limit as r/m -+ 0. Indeed, from (8) we have x (7x {E -rV (e~hE)}) = ikor -Zoh~ x H (13) when 77m = 0. 3 Low Contrast Coating, Third Order The third order boundary conditions can be found by retaining the terms 0(r2) in the Taylor series expansions of the fields in the dielectric. Thus &E 7 r2 &2Ei E(0-r)=E -r + 2 +O() (14) and from (3) cfE(n h) -E) T ((n ha) + ikoZo h h, l a72 = -E-(ln - () - h h)Ha + 9 (9k H + I) + E)a ) + (7 /o a h0H ) + doa { a Also IH' a a h ( = -H nho) +ikoY hE + o (hH) giving a ho /h (in - - hiy l' a ( t hP17 -P a ( h=p )P 2+ 2 a + h h, a +z t oo he ao 'yH 'y 6

and therefore Et(7o - ) = {1 + r (ln h,)- 2 [jI2 (ln h) - { ~(lnh,)} (1 a ) +k2N2h2] }Et - { l+ r a(lnh)} T2 9 { I a 1 9, E;,) )} - ikoT-Zo 0 { T + I hahao \\ 1) (9k h-y p/J 2 ns0 hoa3 i.e. E(7o - T) = {1 + ra(lnha) - [2 (ln h) - { -(ln h)} +koN h Ea -7 1 + 7 (lnhh) E — '(h~E}) T2 1 a2 +ha — (hE') - 2 hctaa~a-y ikor ~ Z {1 + Yo 7 (I h h, \)}hH 2 ^ In - h Ht +ik -Z h/3H) + 0(Tr3). 12 _o zha 0o Similarly E(o-r) = 1 +r (lnhp) - 2 [2 (ln - {Q (lnh)}) +koN2h] }E- {1- + T(ln h)} + (h EI) (15) 2 1 a2 + 2a ( a ha — ') + ik0T-ZEo 2 h^ O ' 1Po ra ( h1 hH\ (n hih Ph) }hoH k h2-~ a 3), -iko OZh, (hH') + 0(3), (16) 7

and for a perfectly conducting substrate the left hand sides of (15) and (16) are both zero. On applying the boundary conditions at the dielectric-air interface and using (A.1) we then obtain +7 n)- 2(lnha) - (lnhA)} + 2N2h2] {E. 1 a e87h,E, 1 7 - -hj ^+ -a 'In - + — OhV,. -E ha e { 2+ a (n h } 2 e 'Vs E) 7. h a 7 7 1 a P0 r +O(T() =kr ----Zoh 1 + - In n h, H, /lo h- ti 2 hp 9 p {+ r-(lnh,)- 2- -[2(ln 3) - {(ln h3) + ko2N2h] }{E 1 0 E -T — - hE 1+ - In + — hV2 (E = -ikor- Zohy { 1+ - In hH — hH al ) +0(r3), and correct to the second order in r the boundary condition for the exterior field is x (x E-rV hE^1 + 7In +2eh2VV (E)]}) =- x{H -(H V ( ) hh) (17) 2 =e 1JJ c2\ / where - = ra + ri3/3 8

with F = -ikor —Zoh{ 1+ (ln h,h,L )} (18) rl = -iko Zoh {1 + - (n ) MO 2 h( h, p / To this order the effective surface impedance is anisotropic, but if terms O(r2) are negligible, the impedance becomes isotropic are (17) reduces to the second order condition (13). For an impedance substrate we require also the expressions for HT and Hg at 7 = 70 -. From (15) and (16) using duality, H(7o- T) = {1+ r7 (ln h) - - [ -2(ln ha) - {-(lnh)2} a{ 2)~} +k 2h] - 1 + T(ln ha)} - (hH) + 2 h_ a a7(hH) + iko-o {1 + a l ), 72 1 92 _ 7 h _ _ 2 h. aaay (h2y 0 -ko2 -Y h Ot-(hE) + O(T3), (19) 2 eo h,3 f H(7o - r) -= 1 + r (ln h) - 2 (lnh) +koN h] - r 1 + r-(ln h,) } (hyH) +2 I i (hHo)-iko-Yo I + 2 n 221 02 f 70 I h E co(.o, +(h H)k-Y 1 + — tin -o hEs 2 ha aoa/3E 207\ (, h,J r2e E h 0a +ik, —Y ho.(hE') + 0(r3), (20) 2 E h,, ha O 9

and from (15), (17) and the boundary condition (1), {1+ 7 (nha,) - - & [2 (In ha,) a {~(Inha)} + k 2N-2h j E0 1 1+ 7O (9 n h, h, ')] E t - -r 1 hl'y Cf + 7Ta_(In h ) ~1a. 2 eo ha&& 2 1 Q2 2 h., aaa- 1I1Ld~ = - 17M 1 + r-a(In hg3) - T2 [ 2 ra 2 — (in h,3) - ~- (inho +k~N~h~]~k zoi 7 a 2&a-y ( h,, h, g ln aH hi P L/I J - T {1 a(Inh,) - k "Zo h L -(hy1' + - 2 no Tim h 2 1 &2 When the boundary condition at the dielectric interface is imposed wve find, using (A.1) and (A.2), { 1 +,a r~(In hc,) 2 7 2 a2 a.2 2 2 -5- N h, 2(In h,,,,) - (in hce) + k' 2 t Y-t- 0 ly i IE -1k oT -Yo Tl/ lEo 1+ 2~ (inhch0o) hl} {Ea ha &-a 2 E -y co hyE.1,. 6 6. -E fo - 'ko —YoT7mh, — IhyEI> 72 a2 19a 2~i - 2 [~(In h13) - {- (1n1hl)} - k2m {1+2T-(knhp ) ~k lhj MkO1-77Mk 10

1 + -( n P hl} f ia[7 09 Ho- r-1-a-Ph-fHf Ta+-.(In h-h 7L 2 V y S1L P H (PH) - o2 p o m hl 1 &9 YohH +-'3) p and correct to the second order in7T, Ece - 7 (i + io 2 6 7M [IEOh-yEj~ 7Ta 1 + — 2 a1~ (n in h~ hl JJ6 2 e 1k. c 02 [Lo 7rn I h,3 9fl 'o hH, 1, P T a (ln h ct,h0p 7I.L0h2V7 - Sy. (~H)]} ('21) with r' = T/rn{1+ 7y(Inlh,3Vz- ko 'loh L a2 (In h,3) - { —(In h3)} + k2N 2h 2+i2k0~ i ih kI (In h~~ o~ 0 oT/rn 77M&y1- h ~j { 1 + -r nh - Ik~2 Y Ilnhy [a2 (nh,, { -(In h,,)} Similarly (22) 11

E- r 7(1 + koYmh) ( hah )} 2 - - - -o ) l H-r (1 + ikor h Zo h) 1 a [ - H \ I { + -i (n h > -) } + H)] (23) [2 7 y hY P)o 2 p P o,/ with 2 F&2 ra 2 rl = rm{l+r [(lnho)-iko h] - (lnh{^lnh)} + 2Z+o - (9in huh 1+)]}{+ T (hn) - io-Yo h i - h I n h) - { (1n h) o h + koN h, + ko —Yomh in )}' (24) More significantly, Fi - l/rF where the asterisk denotes the dual quan-x (x E-r (1+ -< (In hh), (nh)} \2o 2 0[e 2d7 +2 e V(e.E)]}) =-.x H-r (l+ ikO 222 2 ofh h~ /2 *+ {-hyth + i -y i hh V - - +(H)] } (25) o h2i. with ra1 + -/3l. (26) - raa +:. (26) 12

The boundary condition (25) is a third order one which satisfies duality and reduces to the second order condition (11) when terms 0(r2) are neglected. However, as was true with (11), we cannot recover the result for a perfectly conducting substrate (see (17)) by simply putting r7m = 0. A case of particular interest is a circular cylinder with a homogeneous dielectric coating. Putting a = X, = z, y = p where p, X, z are cylindrical polar coordinates so that ha = p, hp = h- = 1, (25) becomes Px { [x E-rV [- + io,-o )}e P + E}) 7 2 — q.) H-rV[{1+~(+iko. - )} + V..-H] (27) with F - 7m 1 7 1+ - )iko Z - (koTN)2 Po (7m 2p )7 2 2p IE 2 ko 2 J 2 Finally, we note that (25) has the general form 7 x x {E - V [A ehE + BV, EJ}) =-(raa + r:) I x {H-V [A* h..H. +B*V, Hs } (29) where F, A and B are geometry and material dependent, and as befits a third order boundary condition, there are three quantities at our disposal to simulate the scattering properties of the surface. 4 High Contrast Coating, Zeroth Order We now consider the problem of a high contrast coating consisting of a dielectric having INI > 1. The treatment is based on the method of Rytov [1940] and makes use of the analysis in I, but in addition to the inward 13

propagating field considered there, we must also include an outward propagating field produced by reflection at the dielectric-substrate boundary 7 = 7o - r. The properties of such a field are given in Appendix B. For generality, we again assume that the standard impedance boundary condition (1) is imposed at 7 = 7o - r, and we start by considering the zeroth order approximation. For the inward propagating field = Aeik/q - Beiko/q (30) with Aa = -K1B,, AO = K2BO at any point in the dielectric, and to the zeroth order KA = K2 = 1. Since 4 = 0 at =70, (7) = - vciy = -q j Nd7 A~o YO (31) (32) where the integral write is simply the optical path length, and for brevity we I YoT Ndf = -L so that (7o-7) = qL. (33) In the particular case of a homogeneous dielectric, L = Nr. Similarly, for the outward propagating field, r = AeikO'/ t' = Beik'/ (34) with = -K B (35) 14

and to the zeroth order K' = K2 =1. (36) Also +'(7) = q +| )Ndd7,o-T yo-T = qL+q Ndso that (7o - T) = qL (37) and V/(7o) = 2qL. (38) The total field in the dielectric is n 1- 1 (9) E = ( + ~'), = + (39) At 7 = 7o- r the boundary condition (1) requires that a + ~ = -rvmY('H +;) (40) Ep + = 7mY(H(+Ha +) and therefore Ae'"o/ + A, e'oi' /q - -.rY (Bgeio'/ + Be,,,o,''/) Ape'ikq + Ae*'l = rmY (Beiko/ + Bikoi,'/q) where the phases are evaluated at 7 =-70 - r. Hence, from (31)-(33), (35), (36) and (38), I 1- l mY 1 + rmY (41) B 1-mY BmY 1 + nmY 15

At the upper surface -y = 7o E = (A, + A e2,koL) and using (31), (32), (34), (35) and (41), ln_ 1 I — 77__Y 2ik~ BoL. = I1 + rnY } Similarly v/ _ L 1 - l./ye D E = { 1 + 71r 2ikoL} B and in + l rmY e2ikoL}B\ "1 + 1 { 1 —Y2ikoL} }B. ai { 1+ 71,Y } On matching to the exterior fields we then have Ea Ein tan koL + Zi7omY H = H n iZ 1 -iv tan ko (42) and E_ EYn _ tan koL + ii7mY -iZ (43) H Hin 1- i-7mY tan koL ( and the resulting boundary condition can be written as x (?y x E) = -r7y x H (44) with tan koL + il7mY 7 1 - 7mY tan koL (45) We observe that the boundary condition is entirely local in character and independent of any transverse variation in the properties of the dielectric. Any normal variation enters in only through the optical path length L, and (44) is identical to the boundary condition obtained by considering a. plane wave at normal incidence on a locally flat structure. 16

5 High contrast Coating, First Order The procedure is very similar to that for the zeroth order case. To the first order K1 =1+ 2ikoNh y (n Z) (46) x, 1 0 ( h_) 2 =1+ 2ikoNh 7 ( h, ), 1 a han K = 1- 2i koN hy O (n hZ) (47) AK. 1 O (9nho) 2= -2koNh, y~ ( h ) and from the boundary condition at 7 = 0 - r -K1Beiko+ I Biko~'/q -iTmY (Beik~o/ + Beik' ) K2B ' aeik/- KB iko'/ = mY (B eiko/ + B' ik '/q) giving ' K2 - rl7,Y B -K2 + 7mY (48) Ki?7mY, Kj - rmY At the upper surface 7 = y =n = {-_iB + KB, 2ikoL} and when the expression (48) for B, is inserted, we obtain iEn, - e rY 2ik L} B oL of r K 1 K' 7/nY 17

Similarly in 1 - 1K2 -imYe 2koL} B Ho K2 - i 1 + v {K 2 K 2 + rmY } E E 2 1 +,mY) - (K2 - )e H H- Kh + l7mY + (K2 -mY)e and on matching to the exterior fields we find Eo 2K 1 + -mY(K + - K)tan KL - + imY(Kr + K which can be written as E 2 KK + Y(K Y - )} tan kL + i (K + K IK1 + Kt' + i(Ki - K - 2rmY) tan koL (49) {2K2K2 +,mY(K2 - K2)} tan koL + i7imY(K2 + K) H K2 + KK + i(K2 - KI - 277mY)tan koL But K1 + K = K2 + K2 = 2, and to the first order in INI-1, K1K = K2K = 1 The boundary conditions are therefore Ea = -rHi, E = rFlH (50) 18

where r = -i FI = -i 1 + 2ikoNh, (i hIn Z tan koL + i ZmY i 2ikoNh- O 7 (h ht ) J L z {1 ik+ -N (ln h Z tan koL + i.1Y CmY{ 2ikNho h (n h h) } an 1i.1 2ikoNh -f \ hO? (51) and (50) can be expressed as 7 x (7 x E) = -. -y x H with - = ra' + ri/3. (52) (53) Since Fr = l/r* where the asterisk denotes the dual quantity, the boundary condition (52) satisfies duality, and reduces to (44) when terms O(INI-') are neglected. Here again, any lateral variation in the dielectric properties has no effect. In the particular case of a planar surface z = constant we choose a = x, i = y, - = z and then =riZ + Oz (In Z) tan kNoL + )i7mY r = -i, ()}tan 1 177l+mY 1Z/77m Ol (nZ)}tankil L 1- i"Y 1-12i'koN O' (54) but for the cylindrical surface p = constant (a = X, d = z, - = p implying ho = p, h = h = 1) 19

{ 2ikN + f(ln Z) tan koL + ZijmY r = -i + 2ik~ N p+O 1 - mY{1- Z/7 (1 + (1n Z)) } tan ko L (55) l= -1 1 + - p(lnZ) tankoL + i7mY rP,= __{Z 2i'koN P 1) 1iY Z/77m (1 + a (n Z) tan koL 1mY1 2ikoN \p 9 p ) Finally, for an arbitrary surface with a perfectly conducting (im = 10) substrate, F = -iZ cotkoL+ -2kNh (n Z} { 2koNhw 3aj ( ha )} (56) F, = -iZ cotkoL+ 2koNh- h )} 6 High Contrast Coating, Second Order This is important in order to demonstrate the practical advantages of a higher boundary condition, and it is unfortunate that the analysis is significantly harder than in the previous two cases. For the inward propagating field the expressions analogous to (31) but accurate to the second order are 1 9B I1 A A = -K= B -L L- +M --- he, da ho op (57) I 1B 1 QA A - K2Ba - L +MhTO 9l h, Oa 20

(see (1.43) and (1.47)) where K1 = 1+9ikNh 9 (In hcZ) K2 = 1 + 2Ik0N a& (In Z) I 1 1 (9 ha 2 In -Z (2koNh..y)l L. a-t h,3 laz - a In hotZa(In h, N) o 5-7- ho (97 )i (58) (2k0. 1~h~) 2 ho 1 (9 In z 2 ay ha 02 + ay2 (ln oZ) - ~T- (lnhZ)5 -(ln hN)] OB m aA v/Iaf 92k2N h at [ 1h#. V X ].VxI7P] (5F-9 ) 2k2Nh at [ZN Similarly, for the outward propagating field lOaB' 1 aA' =l - KIBI+L- -M of 1 0 TCi O a (60) (see (B. 14) and (B. 16)) where = 1 a K I koNh&- - = 1 a 2 ikoNh & (ln cZ) (lnZ) (61) 21

and to the required order it is sufficient to insert the zeroth order approximations to the fields in the derivative terms in (57) and (60). At any point in the dielectric i n {Aa,-e'~Q / + ' A eo'q/ ' - 1 {B kok/3 q + B eiko/ } and when the boundary condition (1) is imposed, we obtain 1 KB l B l OAN (Aol q 1 B' (-AiB - Lh + M M + KI- + L K2B~ - 9M 90A e + L ho Ofa + M 1- e'k~"lq =- rAY (Boeikq + Beiolq) he ha &Q j K2 - Mh - ) e = mBa+ B' L) giving (K + 7mY)B; -(K - 7mY)Ba + L (B - B') - M (A + A) (62) (K2 + 7mY)B = (K2 - 7mY)Ba + L (B- B')- Mh ~ (A + A') (63) At the upper surface 7y = 7 E, = {Aa,3 + A e2ikoL} (64) Ha, - HL' = v {Ba,, + B',e 2 } 13 VP 22

Hence, from (57), (60) and (62), VII-; a - KBa -L1 o9B T", 'ga 1 aA + MT- + hp dp 1 3 + L IiB T", 'ga +M aM e2s'k,L ho 90 = -KIBO -- LI 9 T", go, +L I da (B - Bl) +M I O T, (9M ~i K, + 77,Yy 11Y) 3 Al)] I a - MT OT - (A - +LI B T,,, 9aL TO a dp e~~Y I' r~ 2ikoL Kl' +71,mY l(i - 7,Y- K, ( K + 'qrn Y Bo + KII e 2kL - (K + i my) I aB1 9A +77Ye29k L1 c9B/ Ta a a +M 1aA'\ T4 q o giving KI'(KI - qmy)e2"".L -K,(:+ )Y and similarly i(Kl +77m Y) VcE, -M d aA herda (65) B = 2 - r kat- h 2 + 77my) 23

+ [Ke2:o -(K~ ~Y](+ + M+ A +7,Y 2k(L IlOB' 1OaA'\ (66 +Q~eikL M a — J '6 Also — HaBo (K2 77Y)Bt- L —(B- B') VI1~~a+ A+ lmYjh / - (A+ A')} giving (K2-rnmy)eltikoL + (K' +?lmY) + iW2 /ic + e,2ikoL [LI —(B - B') + M+ (A + Al)] (67) and (K1 - rimY)e2i'kkL+ (K' + 7im Y) {(K + 7imY)iI 1 ( - e 2ikcoL [L-hc B & B') -Mj-? (A +A/)] (68) On equating the expressions for B,0 and rearranging the terms, we obtain E - (Kl' + TimY)(e ikoL_ 1) 1 L1 O9B (K iY~~kL+ (K{ + ijm Y) fi hcv a (K1 -?7mY)(e 21koL - )e 2ikoL 1 1 OB' (Kli -7mY)e2ikcL +(K'+T/Y) \/che& KK - 7imY)e 2ikoL - K, (K' + TmmY)J (K1 rqm Y)e2ikoL + (K; + tjmY) 24

(KI +?7jmY)(e2ikoL - 1). K"i- rim Y)e2ikoL - K1(K1 + iY),~ hg + (K1 lY(eio 1)e2?koL 1 1M &AI 69 +K;(K1 - rimY)e2i-koL - K1(K; + iimY) h (9 and from the analogous expressions for Be,, (K+,qmy)(e2ikoL) (K2 1 Ye2kL(K+im)f hBf (K -2 -rm)(kL-1)ikL1 18B (K2 - ri Y)e2lkoL (K + (K7+7mY) fie ha0/ W2- rim y)(e 2%koL - K~2('k, + TmY Z1H (K2- imY~2i'koL + (KI + 77mY) ae (K'2 +rimY)(e2 hL1 1 OA KK2- rim y)e 2ikoL _ K2(AK+rmY /7 'd + (K2 - rim Y)(eL-le t2kL 1 1OA 12/(2- rimy)e2i'koL - 2K rm)J In the terms involving L and M is is sufficient to replace all quantities by their zeroth order approximation, and since K1 K2 = K' = K2'= to this order, (69) and (70) reduce to E-ae2ikoL - 1OB 1 o 2kL1 aB' Ea- ~e 1)LT~ - abe io(e~io =+a(e2ikoL - O 1 1 OA/~ + a*b*e 2ikoL (e 2ikoL - 1)~~-Al->(71) (2 tk0L- 1 lOB 1 o 2kL1 aB' EO -a~e 1)7L-5~- abe io(e - 1koL le - f h,0/ JE h 90f 25

2t'k,1 1 aA = rl {H, + a*(e2koL-) M - 1 a1 1A'" + a*b*e2ikoL(e2ik~- 1)-Ma1 a- } (72) where -i- K(K - rm Y )e2ikoL - AK ( + 77mY) (K1 - r7mY)e2ikoL + (Kh + 7mrY) (73) _zK(~2 -.r)(e2ikoL - K2(Ixt + r ) Z (K2 - )lY)e2kL + (K2 + imY) a- + 1Y a + {l -rlm Ye2ikoL} (74) 1 - 7/mY b = 1 +?7mY and the asterisk denotes the dual quantity. Thus = {1 - 7_mY 2ikoL} (75) a - (75) and b*= -b. We remark that the admittance Y appearing in these quntities is evaluated at the surface - = 0- r of the substrate. To this same order (67) and (68) give B = aH,,. (76) Also, from (31) and (32) Aa = -Ba, A, = Ba 26

and since (see (65) and (66)) Bo= -~r C we have Bar =a*Vf EI Moreover, from (41) ____- = aE. BI = bBa, -fo abH,3 (77) implying (7T8 ) and (see (35)) Al= BI, A/ =-B so that -r, a*b*Ea,,3O. (" 79) Thus I1 l9OB V\/- ht Ot 1 1 a Lhy 2k2Nh~ ht &t LN. V x(aH)] 1 1 a { -^ -ikoYoaE - H x 2k2Nh. ht at IN Val}] 1 1 a c 2i-koNhly ht&.fIk and similarly 1 laB'fi- h at 1 1 aA' v'iM h~= 1 i& a I0chE- (1ZH- y 2ikoNhlyhOt 1abyhfI k V(ln ab) 1 10a 2ikoNh-f ht at 1 la 2i'koNhl ht at ~o -y y+ 1YE - ~ x V(ln a'b)} a ~ i k0 27

Equation (71) then becomes a 112ikL 181 Eofe -1- Iaht' 22koNhly hcr (a L iko ab-%O ( 1e2ikoL I a 0 h l- Z V(1n ab)} 2i Zk oN h l e h'1.a Z'ko =-rFH - *kh(e 2ikoL -1+ a*I' hfI - 21k yh [a -h f{H + 1 *XVl&} a* b*-1& Z'ko ~22-koNhlyhaa [a*b*!MO h,~ {H + -Yo x V(1na~b')}} with an analogous result for (72), and these can be written as lx (lx{E -9 h(e ik 1)'Vra'Oh{ E - Z0 H.Ix V'(lna)} ab -~h ko2ioV ab0N [ah -i0-E0v1a~ji, - ~liX{H~2ikNh ( 1e2ikoLv 1)V [aEkhy -ii (e(e ikoL7 1)eikaLV ikYE~xka f]-2ikoL - Valn*b*} -Y0 -l(80) with 77=F&a&+F1r303. (81) It can be verified that (80) satisfies duality and FI' = 1/P'. 28

For a homogeneous dielectric coating of constant thickness, a, b, a* and b' are independent of a and /, and the boundary condition reduces to ik \ 2ikjNhs J x( { E -Nw V ( pE_ O h EH) }) H 2 k VH V ( f~h H } (82) 2ikoNH hH (82) where w = a2 (i + b2e2ikoL) (e2koL - ). (83) We observe that if koL = koNr = mr where m is an integer, then P = rl = rim (see (73)), and the boundary condition becomes y x (7 x E) = -77m x H. (84) The coating is now invisible, and as we shall see later, this is a consequence of ignoring the thickness r in all amplitude factors. To illustrate the boundary condition (82), consider the special case of a circular cylinder of outer radius po. Putting a = -, f = z, y = p so that ha = p, h3 = h, = 1, the expressions for K1 and K2 computed from (58) are 1 1 K, 1+ + 2ikoNpo 2(2koNpo)2 1. 3 K2 = 1 -2ikNpo 2(2koNpo)2 Hence, from (61), 1 1 K' =1- 1 + 1 l 2ikoNpo 2(2koNpo)2 1 3 K' + 2 +2ikoNpo 2(2koNpo)2 29

and (73) now give imY t- tan kNi + i}iY 2 koN 2(2koNpo)2 tan ko Nr + i71mY = -iZ { + - } tan (85) and therefore w1 = -i tan koN7r - (- kN- + ki-Y)2 (87) (tan koNr + irmMY)2 In the particular case of a perfectly conducting substrate (77m = 0) 3 1 r =-iZ { + 2(2NkoN+Npo) {2 tn 2kopo} (88), fro Lr = -iZ {1 2(2kNp)2 ot N 2koNpo and =- i tan koNr. For a first order boundary condition, terms O(IN1-2) are neglected, and the expressions (88) for r and Fr reduce to (56). 7 Coated Circular Cylinder From the modal series solution for a circular cylinder covered with a homogeneous dielectric layer of constant thickness, it is relatively easy to derive 30

approximate boundary conditions of almost any order corresponding to low and high contrast coatings, and these can serve to check the preceding resuits. A circular cylinder of outer radius po consists of a core (or substrate) of radius pi where an impedance boundary condition is imposed and a homogeneous dielectric coating of thickness r = po - pi. If the cylinder is illuminated with an H polarized plane wave incident in a plane perpendicular to its axis, the non-zero components of the field are H,, Ep, E0, and these can be expressed as: P > Po: 00 Hz =- e,(-i) (Jn(kop) + RnH(1)(kp)} cos n n=o Ep -i - e(-in J,(kop) + RnH()(kop)} sinnO koP n=o 00 EQ = -iZo (-i) {J7(kop) + RnH(l)(kop)} cos n n=0 Po > P > P1: 00 n=O Z 00 EZ= -i Nko n(-i)n {anJn(Nkop) + bnH()(nkOp)} sinn N ^oP n=0 E6 = -iZ efn(-i)n {anJ(Nkp) + bnHl)'(Nko p)} cos n n=O where the prime denotes the derivative. At p = pi the boundary condition is E = -7mHz and this gives bn = Qan 31

with Q — H()'(s) + izY7mH(() (89) and s = Nkopl. By enforcing the continuity of Hz and Ep at p = po we then obtain Jn'(kopo) + iYoPJn(kopo) H(),(kopo) + iYoPH()(kopo) (90) where P =iZ J'(t) + QH(1)'(t) 91) Jn(t) + QH1)(t) and t = Nkopo. The quantity P can be used to specify an approximate boundary condition which is imposed at p = po and reproduces Rn to some desired accuracy. Consider, for example, the third order boundary condition (29) which can be written as = - + )+ P. x H - V A* Hp + B', H]. (92) For H-poalrization this reduces to P2 o02 -p =- _ (93) and when the expressions for the field components in p > po are inserted, we find that R, is as shown in (90) with r + 2 An -2An2 p koPO (94) n2 1 +B 2 Po 32

A third order boundary condition suffices to the extent that (91) can be written in this form. For a second order condition B = 0 and for a first (or zeroth) order, A = 0 as well. In the latter case P is independent of n. We consider first a low contrast coating for which 161 is small where 6 = t - s = NkoI. Then 62 Jn(s) = Jn(t) - Jt) + -J (t) + 0(63) and since j"(t) = - - Jn(t)- 7J'(t) ( n2 1 from Bessel's equation, we have Jn(s) - - ( } (t) 6 (1 + J(t) + o(6). 2 2 2tAlso 52 J,(s) = Jn(t) - J"(t) + 2-J"'(t) + 0(63) and because 1 3n2 ( -+2 J'n (t) = ) we have t2 2t t2 t -2 n2 - 2 J'(t) + o(63). 2 t2 ) There are similar expressions for H,')(s) and H,()'(s), and when these are inserted into (89) and then into (91) we obtain 33

P=i 2~i { T2( -~ + 6 {1 - + 2~ ( - 3n2) =1 6 62(n2 +\2/6\ +Q0(63) To identify the coefficients in (94) we rewrite (95) as (5 P = imi&j2 ~) n2i6Yim 3)n62 +O6 = T m1)&2(. 2I +Z fl26 1- r ( + +L~ k0Li~- - +(0N) B=M 12+ (98) Thee areewit threult inScto 3 (se (27), an28) n lordc to he oluionfora erfctl coducin susrae(se(1)an 18)o putn Tim = 0 34

To include terms in 63 leads to an expression for P involving n4 and this requires a fourth (or higher) order boundary condition, but it is worth noting that we can actually reproduce the terms in n2 using only a second order boundary condition. This is evident since B is proportional to 62 and if we put B = 0 correpsonding to a second order condition, the expression (96) for r is unaffected, but (97) is replaced by A 1 + r p( o + iko - Yo rlm (99) We now turn to the more difficult problem of a high contrast coating for which Isi, It| > 1. For large arguments Irt H(w)(t)= V-e'ifi with 7r 7r t = t - t — - - 2 4 where fi is a series in powers of 1/t. Similarly H(1)'(t) - e gl (t) = 2ia H(l)(s) = -ei sf H~l)'(s) - seia g2 7rs with 7r 7r = s - n24 2 4 where fi and g2 differ from fi and gl in having t replaced by s. Since n(t) = 1H()(t) + )() 2 + 35

it follows that Jn(t) = I (fiea + fleiat) 71 fe1+t 2 where the bar denotes the complex conjugate. There are analogous expressions for J'(t), Jn(s) and J'(s), and when these are substituted into (89) we find Q - { 1+ g2 + iYmf -2 } 9 g2 + iY 77f2 and hence P 1iZ (2 + iY72)e' - (g2 + iYlmf2)e-'6 (100) f i(2 + Yi'mf2)e" - f1(g2 + iY1mf2)e-s where, as before, 6 = t - s = at - a,. From Bowman et al [1987, p. 53] 4n2 - 1 (4n2 - 1)(4n2 - 9) 8it 128t2 (4n2 - 1)(4n2 - 9)(4n2 - 25) t3072it3 and 1 4n2 + 3 (4n2 - 1)(4n2 + 15) 1 - it 128t2 (4n2 - 1)(4n2 - 9)(4n2 + 35) + O(t4) 3072it3 ' If we ignore the difference between s and t except in the phase, 9192e6 - g2e = 2i 1- 2 (4n2 3) + O(-4)sin g1f2e6-glf2e^- = 2i cos6-2 t [1 + 8t(4n2-1)] sin6 + 0(t-4) flI2e -fg2e- -2i cos + 2t + 8 (4n2-1)] sin6 + O(t4)} 36

and flf 2e - flf2e-6 = 2i {1 + (4n2 - 1) + (t-4)} sin, giving tan 6 1 - (4n2 - 3)- Y7m [ + 8(4n2 - 1)]} + ZY1m 1 -iYm tan 5 1 + 8 (4n2 - 1) + 2Y (4n2- )]} +0(t4). (101) Comparison with (94) now shows that to the third order in 1/t, Z tan {1 + 2(2Nkoo)2 - 2Nkl [ - 2(2Nkopo)2] } + iYrtm 1 - i ta2(2Nkopo)2 + Y7m 2Nkopo 2(2Nkopo)21} A 1 tan I 1. tan 6 \ 10N2 2Nko 1- iYTm tan6 2Nkopo 3iY 1 Y t (102) B — 1 iY7jmtan { 1 ( 3i tan5 6 2(NkV)2 1- ZTYm tant + 2Nkopo VYrIm 1 - iYm tan.J J with 6 = Nkor, and these complete the specification of the third order boundary condition (92). To include terms in 1/t4 would require a boundary condition of fourth (or higher) order, but even a second order condition (having B = 0) is sufficient to reproduce terms through the third order in 1/t. Putting B = 0 in (94) the expression for A becomes A t 1- m tan 2Nko (1 - iY71m tan 6)2 1 + (Ytm)2 - 2iY1 tan 5 +L iY - m(2 - iYm tan 6) + 1 - iY tnm } (103) 2Nl kopo 1 - Z im tan J 37

with r as before, and obviously the same boundary condition is sufficient to match terms through the second order in 1/t. When the third order terms are omitted from (102) and (103) we obtain F=-iZ tan6{1- 2N + }2(2nkp)2 + Y7 1- iYrm tansl{ + 2 Nkopo 2(2Nkopo)2 } 1 1 + (Y/?7)2 - 21iYim tan 6 A = tan S 2Nko, (1 - iYrm tan S)2 and the boundary condition is then identical (see (82), (85) and (86)) to the one derived in Section 6. For all of the above contrast conditions, if 6 = mir where m is an integer, then tan 6 = 0 and F = rim with A = B = 0. In other words, the incident field sees only the substrate and the coating is invisible. As we shall now show, this is a consequence of ignoring the difference between s and t in the amplitudes. If the coating thickness is too large to be ignored, 1 1 ' -(1 +mA) Sm tm to the first power of A = r/t. Since A is of the zeroth order in 1/NV, the retention of terms through the third order in 1It now leads to an expression for P involving n4 that demands a fourth (or higher) order boundary condition. On the other hand, through the second order in l/t, 9192e- D e 2t 1 (4n'2- 3)] sin -- (4n2+ 3) cos 6 91f2e' - 2 = 2i 1[+ 16t2(4n2- 1) cos 1 [1- 4 -l(4n- 1)]sinb} g -J~gle-6 = -2i 1 6t (4n2 - ) cosS 38

+ 1 [+ (4n2+3)]sinj} flf2e - ff2e-6 = 2i {[ + St2 (4n2- )] sin - 8(4n2-l)cos6} and when these are substituted into (100) we find P = -iZ tan -1 1- (4n2-)]- (4n2-3) A 3A 2 _ 1) + iYrm -(4n 2 - 1)( 1 t2-1) it 16t t *Y 1 + mtan {1+y 1+ (4n + 3) ++ (4n21) iYa7m 2t 1 4 8t2 - (104) If A = 0 this reduces to (101) with the terms in r omitted, but if S = mw (so that tan 6 = 0) 'A 3A 1 + -- ~(4n2 + 3) + (4n2 - 1) 'Y77, 6t~2 - P=T-m 3 — a (105) A 2 3A_ 21) 1 + iYm (4n2- )- (4n-1) 8t 16t2 and the coating is no longer invisible. Equation (104) can be used to construct second and third order boundary conditions analogous to those given by Senior and Volakis [1989] for a planar layer. 8 Conclusions A problem of considerable practical interest is the scattering from a body covered with a layer of dielectric, and one way to simplify it is to simulate the surface using an approximate boundary condition of appropriate order. For an inhomogeneous dielectric whose inner and outer boundaries 39

are coordinate surfaces in an orthogonal curvilinear coordinate system, two different situations have been considered, and in each instance, boundary conditions of three different orders have been derived. At least for a homogeneous dielectric the results are sufficiently simple to be usable, and in the special case of a circular cylinder with a coating of uniform thickness, the boundary conditions have been confirmed by starting from the known modal expansion for the scattered field. References Bowman, J.J., T.B.A. Senior and P.L.E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes. Hemisphere Pub. Corp.: New York, 1987. Rytov, S.M., "Computation of the skin effect by the perturbation method," J. Exp. Theor. Phys., vol. 10, p. 180, 1940. Translation by V. Kerdemelidis and K.M. Mitzner. Senior, T.B.A., and J.L. Volakis, "Derivation and application of a class of generalized boundary conditions," IEEE Trans. Antennas Propagat., vol. 37, pp. 1566-72, 1989. Weinstein, A.L., The Theory of Diffraction and the Factorization Method. Golem: Boulder, CO, 1969. 40

Appendix A: Matching -0-(h.E7) At the boundary of the layer the boundary conditions give E- = Eo, IE-, E = E, IE = E and we must use these to match - (hH}) to the exterior field. Since V - 1 { a - hjhh (hoh Et)+ (hchyE') + Q(hthE)}) and -(hahpE;) a~(, aE} a hahs i\ = - h hEy h= hia + a( hah) akh f Y E, we have 1 a ) + a (n ah - E' - - - (h- ) + In - h 2(9 ha- hI 1 hahh.yh f{ a 9(hrhnEt + &(hE,(h ahEt)+ 5 (ahaE;). But V. (eE) = 0 and therefore V.E'- Ei -EV. f co Hence 41

0 (h - ( eE0 (e 076 ~hcaa&kE.h00/39k6oI E 0 (e\ } -h E. ( hah) h__ r ),-o hag + a h- ii hohai0 <o ' ' p = -e0h2 fEadJ (9 \ 6 \ h1a 9a \Co Ea ( }6 + ho l- c ~e )2 0 l foJ e 0 ( hah3) hy h 9 a -7ln J - -(h + ) hah + 6h ~hhhh j (hh-!Ea + h (hhyE)} e hahh [ ( ) ( a 60 and if the surface divergence is defined a and if the surface divergence is defined as VI P ={-(hh-hPa h~hoth oa then k(hEt ) = - h {V. ( E) + -- In -- h^, OY hy p7o (A.1) Similarly 'y - {VhH ) ' ko (A.2) 42

Appendix B: Outward Propagating Fields To apply the analysis in I to the problem of a high contrast coating, it is necessary to determine the properties of an outward propagating field in the dielectric, accurate to the second order in INI-1. For such a field V' = vy and therefore An + f x Bn = -/-V x n-I '-fVxi ik~v V// _ An-1 Bn- x An= V - ZfcoV v/ (B.1) for n = 0, 1, 2,... with A-1 = B-1 = 0. Hence, to the zeroth order, Ao, = Bo0, Ao3 =-B,,, Ao, = B = 0 For the first order fields (B.2) Ala - B1l A1o + Bic, Bic, + A10 Blo - A1a ikovhph J07 v /- J ikovhh ay (h h JB v/i a (haBoa ikovhoh, hy v ) ikovhah, 0^y v / (B.3) on using (B.2). Also A1 iko=hah aa (hB ) a ) B ikhhf =Q { hpAo) - 8 (hAAoa 17 ikovh~h,3ap fAJ ao V JJ } (B.4) 43

The expressions for Al, - B13 are identical if -ao-:= -IDoft-9 In -— h B0- 2 1 & (h h ) and then 1 a ( h ASim - Bio =- -f iBoo In -Zr 2ikuhe e re 7 ho ) Similarly, the expressions for A13 + Blc are identical if (B.5) (B.6) OBo__ 1B o - -~ - B a - 1 y (n h ho) LW} (B.7) and then 1 0 Al\ + Ba -- o2ikvhoO 2izkvh, 97 (ln h Z). h" of (B.8) Thus, to the first order in q 4Ac = Aoa + qA1a = Bo, + qB - _2ikovh a In hZ 2iE h, v h.0 ho -z i.e. A {1 - 2ik Nh.y 0 ( h Z Bo, 1..Zk~ - 09 ( ho \ (B.9) and Ap = Aop + qA3 = -Bo - qBla + 2ik h B-r (in Z) 22kovh 7 he, i.e. A - 2ikoNh,1 (n hZ )}B. (B.10) 44

For the second order fields A2, - B213 - — Vfp-. (I hiBiy Z'k,,vhoh, aO VI-y a h)3B10) } ik~vh, { - ao,"/ h13 &jJ ( - hB )} A203+ B2a ik- vh- ah {; (harBla) a hihj} 1 f Bia a ik,,vh-,l9y + Bifa, (nha) in - Jjvtah.,aO (h-Yj~ } B2a c+ A213 ve- (I h"YA1, ik,,vhoh, ao vle - a hOA1,3 (97 V/re (Bli1 ) 1 f _ _ _ _ ik v h-y a9 + a (h0 ln7=) Vf 6 a h__A,_ h13 af V~f: Jj B213 - A2ao _ 1 f Aiaj ik v h-, { y (hatAia, ____hAI + Ala+ (in) VEa (hlA1,,, ha ica vfi Hence A2a - B213 = 2ik~v~y.a(Bl - Ala) + B14-a (in - Al4 (I In ) IjV1-ah13 a/3 (l h~Biy +77 ( h NA~ __ 1 2ik,,vh-, 2ik0 a-7 _vhI 013a khKIJ B013 a 2ikovh, a9Y (lfhl Z) (h~t~) a ha a ha. - In B1,3- In -Z a- 76: - 09- ho - //i, a- (h-B<, ho3t K/0 IP) ve- 0 h,,, cl a 45

giving A~a -B2 -- Blo a In h of 2i-kovh-fa^~ ho/ (2k~vh,)2 [2 {a Inha z) a2 /h at + ~In n —Z a9-~ h1/ a h at (9 - - ln - Z - (ln v h,,) a- h 0 a7 + k -ah h,.V AO ""' V-16 +2k'vh hc, aa B) (B. 12) and A203+ B2a = - 2i k, v hl {a-(Bia + Ail3) +Bia a- (In ~, + Al4o (in -&7 jjiLahot a (- hB 1) h13Q30 (hYA<\l VIJf 1 2ikolvh-~ 1 a[1 Bak lho~I { ika[v- a-B - Ina +2k oo h (InhaZ) giving - Biaya In ho - vIt-L 49 -h., aa h,'By iJ7aV YAIf A203+ B2a = Bia a 2i'kvh, a-y (, i Z + Boa [1 ha/ (2kovh,)2 L2 { aY (ino)2 a2 Inh13Z in. 2 h of - j-t (ihz cl (ivhY)] 2kvh~yha~k{Z e- a hf hZv +2k~vh h1 /3{v (B. 13) It follows that to the second order in q Act = AOca +qAi,+ q 2A2c 46

= Boo + qBl,,3+q 2B2,3-. -2iavh Inyfhe Z) (Boo3 + qB,,3) q2 Fiat hc 12 02 h qn Z + I " In - (jf-~ Q(Invh,)] + q2fi a 2k kvhl h4,3 hlvZ-. AOi~ so that q2vl', a9 r yz I +Qk2vhhor Ia V Ac 1 KB3 + 2k2Nh h~h3/ 34 ii { 2I~ \/ 0 Of h Bo} *Vx ~ (B. 14) where K!= - 1 0 =1 -2i'koNh 0y (h\ 1 (29koNh.-)2 In - 092 0y9'2 Inh\, TO-~Z In - (nZ) ~-(In hN)] I (B. 15) and = -Boar-qBi -q B2ct+ -jk ~ In -Z) (Bo,, + qB,,) + (2kh)2 [1 { 0 h,3 2 a2 h,3 In Z + In Z Ta aY2 Ta - -(In -Z)57 ~(Invh,)] - q2 'i-, 01 2k~vh.1hC. 5a vZ. Vx.-} 47

q2 a +2k2vh h, a: { V h._zzv Bo so that Vfg { a 1 Aoh 2kNhh9a, Nzhc'. v x& /ih + 2kVNhh 3 {Nh< V x I VINC a N - Bo (B. 16) where K'=1- 1 0 2 1- 2ikoNh. -Y (n ha hor 1 (2koNh,)2 2 ( ha \ } + ( lnZ) - (ln hZ) (lnhN). 9,-y2 Tor ^ TY To 9i 'J (B.17) 48