RL 892 Diffraction by half plane junctions Thomas B. A. Senior March 1993 RL-892 = RL-892

Diffraction by Half Plane Junctions Thomas B.A. Senior March 15, 1993 1 Introduction A problem of some interest is the diffraction of an electromagnetic field by the junction of two half planes at which generalized impedance boundary conditions (GIBC/s) are imposed. The conditions simulate a thin dielectric layer backed by a perfect electric or magnetic conductor. Their order M is specified by the highest derivative present, and the conditions are logical extensions of the first order ones corresponding to the standard impedance boundary conditions. By increasing the order, it is possible to improve the accuracy of the simulation, but when applied to a problem where there is a discontinuity in the surface properties, GIBCs in conjunction with the standard edge conditions are no longer sufficient [Senior, 1991] to ensure a unique solution if M > 1. Additional constraints are necessary to produce a well-posed problem, and these have been referred to as contact conditions [Ljalinov, 1992] by analogy with the similar situation that occurs in mechanics in wave propagation across the junction of two elastic solids. Without a knowledge of these constraints, the expressions for the induced electric and magnetic currents contain arbitrary constants associated with solutions of the source-free problem. For the two-dimensional problem of an H-polarized incident field, the general form of a GIBC imposed at the surface y = 0 is f ~ 'k/m) HzY 0 (1) 1

which can be written alternatively as bm Om bm m -=0. (2) m=0 (ik)m 0ymHz The parameters 7ym or, equivalently, bm are determined by the properties of the layer and are typically obtained by expanding the interior fields in terms of a small parameter 6. When this is done, it turns out that to any given order in 6 it is sufficient to confine attention to even values of M if M > 1 with, moreover, bm = 0 for all odd mn > 1. As shown in a recent report [Senior, 1993], it is then possible to develop a uniqueness proof that specifies the additional constraints that must be imposed at any surface discontinuity. To illustrate the application of these constraints, we consider here the diffraction of a plane wave by the union of two half planes having first, second or fourth order GIBCs imposed at them. 2 First Order Conditions Although a first order GIBC does not require an additional constraint, it is convenient to examine this case first. The problem considered is the plane wave Ui(, y) = e-ik(xcosffo+ysinqo) (3) incident on the surface y = 0 at which the following boundary conditions are imposed: ( + -ik U = 0 x < 0 (4) ( + ik^) = 0 x > 0 (5) where a time factor e-iwt has been assumed and suppressed. We seek the resulting field U(x, y) in y > 0 and note that if U = H, then au = -ikYEx. In accordance with the standard edge condition, it is necessary that the singularity of Ex at x = 0 be integrable. 2

If the boundary condition (4) were to apply for all x, then U(x, y) = -ik( cos o+ysin o+o) + Fe-ik(xcos o-ysin o) (6) with 1 - sin o(7) r=- - (7) 71 + sin qo Denoting this field by the superscript '0' we now write U(x, y) = UO(x, y) + US(x, y) (8) and represent U (x, y) as s(x, yP) -= f P(6)ei z+iy-V (9) 1 ' ( (9) On the surface y = 0 the boundary conditions on U8 are a+y ik) Us = 0 x < 0 (10) y + ik7 U = l Me-i~o x >0 (11) where ~o = k cos;0o and sin &o Ml = 2izk(y, - 71) sin 71 + sin 0o From the edge condition it is necessary that PP([)j - 0 as 1-1 - o. When (10) is applied to (9) we obtain 1 - +o 1 kp( --- )P( ei d = O (12) for x < 0. Let 11 + k _ = K1 (~) K1 (-13) (i+ p3';jo (13) where K1() = K+ (,I) (14) 3

is the uper half plane (split) function defined by Senior [1952]. From the expression given there or, alternatively, from Leppington [1983], K1 (~) -= 1 T In + O( )} (15) for large |Il provided 71 75 oo, but if 71 = oo AK,(~0 ~ ( (16) On inserting (13) into (12) we have f_ P(1) e' d - o /1 | Iv (K ) 1\, (- \ for x < 0, and therefore P(') = I () L(() (17) where L(g) is a lower half plane function. Similarly, for x > 0 P7) (o I=() ()-i ox (18) where KI () differs from KI1 () in having 7' in place of 71. Hence P(=) I +(-) UQ() (19) where U(7) is an upper half plane function, and on combining (17) and (19) we can write p(g) =K1() K1 (go) KI (-) K(-) A(() (20) ~ + G0 where A(g) is a function analytic everywhere. It is therefore at most a polynomial in s, and because of (15) and the edge condition, A(g) is simply a constant A1. When (20) is inserted into (18), we have i~IA1 K1 (go) K' (-o) K () + = M1,e-iox c l+ ',o 4

and a residue evaluation now gives — 2i7rTA1i 1(A'0) '1 (-o) e-i~0 = M e-i~x. But y1 sin qo I1 (o) I 1 ( -o) 71 sin 0 Y1 + sin qo and therefore Al= K -1. (21) Al ik(1 1)( Hence T " 1ik ( 1 1 ') I ( ' ) ( ) = K ( ( — ) eix+iYvK_( dg) J K I+ (-)0 e/k2 k (2 (22) and we observe that the standard edge condition has ensured a unique solution. It is of interest to examine the behavior of U(x, 0) for small Ixl. From (22) US (x ) = B1 k(-l -l — ) (-) g id (23) J- V00 - + 0 where B1 = ' I (o) K (- o) (24) 7'7171 and the first step is to additively decompose the non-exponential portion of the integrand into functions analytic in overlapping half planes. A simple analysis shows K1 (f,) K K(-g) 1,i ) K'K( -) } and therefore xoo 0oo Us(x, 0)= T+ ex)e d( + T-()eiXdl (25) -oo - 5

where + ) -1 1(- ) 1 1 + is analytic in the upper half plane Im. a > -Im. k, and V-_() -( B - - (27) is analytic in the lower half plane Im. ~ < -Im. 0o. The first (second) integral on the right hand side of (25) represents a function which is zero for x > (<)0, and accordingly, for x > 0, U(x,0)=) - _(e'i x d. (28) When the expression (27) for T_- () is inserted, the contribution of the second term can be evaluated by path closure, giving 0oo f K ( —),-o ( x, 0oo AI( — + o i( —o) and since 27riBl/i$j e- (x 0), Ki (- o) we have U(, 0) - -Bl/ o 1( — d'. (29) /_o KAi-J) + ~0 For large |I1 l(-)0 1 -- k 1 2i -2) and hence, as x -- 0+ (see Appendix A) U(x,) = -2riB {1 - i - -x In + (x) (30) 7r } For x <0 US(x,0) - T+( e'W d -00 d^[K^ ' jj / (31) 0 If K(-t0 ) G + so, 6

and though we cannot now evaluate the contribution of the second term by path closure, when its asymptotic expansion for large 1|1 is inserted, the resulting contribution for small 1|1 is precisely the expansion of -U~(x,0). Thus U(x, 0) Bll ( d, (32) cJ-oo Ai<; < +;o and since IK() 1 +o I0 2k n O(-) '-]-C" 7(01 — T1 )}-~ In for large |~|, U(x, 0) -27riB1l{ 1 - (7 - 7) x In |xl + O(x)} (33) Y - 7r as x -- 0-. Comparison with (30) shows that (33) is also applicable as x -- 0+ and demonstrates that U and tU are continuous at x = 0. If 71 and/or 7 f oo, U(O, O) is finite but au is infinite logarithmically, and as evident from the boundary conditions, oU has a finite jump discontinuity at x = 0 as long as Y1 7Y1i. 3 Second Order Conditions These are the lowest order GIBCs for which additional constraints are necessary over and beyond the standard edge conditions to ensure a unique solution of the boundary value problem. The boundary conditions imposed at the surface y = 0 are HII + ik7ym) = o x < O (34) m=l y m(+ + ky )U = 0 X>0 (35) m —1 which we write as (2 +,ikbl- -k2bo U = 0 x <0 (36) 9y2 ay 7

(42+ikbl - k2bo)U = 0 x>0(37) where bi = 71 + 72, bo = 7172 b = 7y ' Y, bo = 772 (38) If the plane wave (3) were incident on a surface y = 0 having the boundary condition (36) applied for all x, the total field would be as shown in (6) with F - - n 7"-sin o (39) m=1 /m + sin o' Denoting this field by the superscript '0', we again write U(x, y) = U (x, y) + Us(x, y) (8) and represent Us (x,y) as Us(x, y) = f p(Q) 6X+iYk (9) We note the requirement that |P(~)[ -* 0 as 1[1 -4 o from the edge condition, and on the surface y = 0 (- + ikb - k2bo)s = x< 0 (40) + ikbl - kb) Us = Me x> 0 (41) where M2 = k2 { I (7' - sin qo) + F II (7' + sin o)}. (42) m=l m=l Application of (40) to (9) gives - kbo k ( —1k +?)) I e '( d = 71 V/k -- 2 72 v/k _ V( (43) 8

for x < 0 and hence P( = 2() K2 L() (44) where L(r) is a lower half plane function and K2() - K+ (, 1) K+ (, )(45) Similarly, for x > 0 -kbk IJ(() K'(-) + d = M2e-=~o (46) where K(() differs from K,2() is having 7j and in place of 71 and 72, and therefore P(/) 2= + U(V) (47) V k- + 0o where U(r) is an upper half plane function. The combination of (44) and (47) gives k2 k2 K2(~) K2(~o) K(-O) K'(-o) A) k2 - k2 _ o2 where A(Q) is an analytic function, and since IP(|)I - 0 as I1l -, A(Q) is at most a first order polynomial in g. Hence k2 k2 co + c P(O) = 2 - 2 - 2(g) K2(0o) K2(-) K/ (-o) + (48) V -- - _ s2+ 20 for some constants co and cl. When this is substituted into (46), we obtain -- f 2 CO + CM^ - K i 2(0o) I2'( -go) f i2(g ) C c i( M2e-di~sin do 2 - I o( ) K' +.o and a residue evaluation now gives (71 + sin qo)(72 + sin qo) co -- bOCI = M2 27tkbob' sin Oo 9

i.e. i b l- 2 CO - Cl - (S2 2( 49) 'k bob' - where s2= k2 1 + b - bl (50) b-b (50) From (49) co+gC1 I bi-b fs2-2 1 co + Cl ik bl - b+ f - 2 + 7 + where bob' C2 = -ZTkr- cl bl - b C1 is an arbitrary constant, and thus i bl-bf S P( bob sin o -2(s) K - K2(-o) (-) 2(-O) + C2 ir bobo sinq0o\ /7-1 2 27T (51) For large I1|, P(~) = O(1|1-') if c2 0 and 0(|ll-2) if c2 = 0, and the expression for U "(x, y) is,) b- b K 2(0) K (-o0) _ K2() K( - ) T7 bob' sin o oo C 2 - q2 e+ 2 cV2 d. (52) The presence of the arbitrary constant c2 shows the need for an additional constraint to ensure a unique solution, and we discuss this later. We now consider the behavior of U(x, 0) for small IxI. From (52) 'Do 2() K'(-~) f2 - i2 US(x, ) = B2(bl - bl) 2 (-) - + C- } i d (53) J-oo K2 - 2 +0 + c2 where a 1 A2(0,)A 2(-&o) r b b sin S (54) 0bob sin do 10

Appearances to the contrary, B2 is finite for $o = 0, I. The first step is to additively decompose the first factor in the integrand of (53), and a simple analysis shows that 1 k2 _ 2 I K'( —) - 2 blb ) 2( - bl _ 2 - s2 bb K2(-) O () } Hence K2(~) K (-5) -. _ _ = T1 1 {_ _boblK(-)_ _2 - - -- b 2 _- 2 {bobi(-) + al + Ca2 2[ 2() ] } b1bi r + a1 +a]} (55) -- b b () and since this is true for any a1 and c2, we can choose them to eliminate the poles at = -s (+s) from the first (second) group of terms in (55). Then a,1 b I\ b 2s [ KR(s) 1 K2(8)J =2,l K2(s) K(s implying 2- as2 = boblbobl, and {oo 1- S (2 C2 Us(x,0) -= B2L 2 {s-()- } - +so + C2- e d (56) ' + J oo +2 - Sc 2 s ~ SO where ( bob= + 'a, + a2 (57) A2i(-i) S+() = b1b + a1 + a2. (58) Finally, on eliminating the pole at C = -<o from the upper half plane function, we have 0) -0 T -00 T) e (X~CIOO 11

where T) -B2 [ 2) {f - T+ 0 = —B^ + C2} S+ (- to) + (60) is analytic in the upper half plane Im. ' > -Im. k, and T- (- [ s- (g) { S - go _~f R +*?( G + C2 S+ (-o)] + + (61) is analytic in the lower half plane Im. ' < -Im. go. The first integral on the right hand side of (59) represents a function which is zero for x > 0 and hence, for x > 0, -00 The term involving S+(-'o) can be evaluated by path closure and the residue at = -'o gives 27riB2 S+(-o) e-~ = -U~(x, 0) -iB2(o - ), but instead of doing this, we introduce the asymptotic expansion of T-('t) in total. From (15) and (45) K2(+~) = bo{1 bF 1- In for large |11, and therefore (63) S-(g '2 - S2 Also giving ( { CI' + + ( b2 bo+b1 ) k 2bn -bl, /o(b,- bl ) g2 In k ~ 0 ('t-)} s + - g2 + + - ](2 2)+ o(~-2) (64) = - 2 (2 - 0)1 + 2 + [ (C2 - o0)a2 + Cl s2 + C2b'l - /bob (bl, - b) l -n 2+ 0(r2)} B2 ibK2 ( Go) {1 + -Abl 2(-(_o) - + Jo(-2)} 12

and hence (see Appendix A) U8(x, 0) =27wiB2 {(C2 - G~)al -1 a2 + iX [(C2- ~O)a2 + as + C2b' Vbobs] ~nk.bk)-2l + O(X2> OO 1 27 2b, sin q0Z'0+ oX2 ('y1 + sin 40)(7y2 + sin 0) -i0 +Ox) for small x. We recognise the last term as the expansion of U0(x, 0) and therefore U(x, 0) =27wiB2 {(C2 - ~O)al + a2 + iX [(C2- ~o)a2 + alS2 + C2~ bo bs] c~b~b~b~b1 - k x + O(X2)}(5 as x — * 0+ Similarly, for x < 0, US(x, 0) = T+(~ e d~, (66) and for large 1~ =+ -L (C2 - 6~)al -j a2 + ~ (C2 - G)a2 + alS2 + CAb bob Y] b)k 2i~ (~2 CAb bobo(bi - bl)2 kn implying U(x, 0) =27wiB2 {(C2- ~o)ai + a2 + iX [(C2 - Go)a2 + Cls2 + CAb bobs] -CAb bobb0(bi - bD)-x2 In IxIj ~x) (67) as x — + 0-. Comparison of (65) and (67) shows that 1 3

(i) U(x, O) is finite and continuous at x = 0 for all (finite) c2, (ii) U(O,O) = 0 if c2 = 0 - a2/c1, (iii) u has a finite jump discontinuity at x = 0 for all (finite) c2, (iv) limXo Iu = limoo- = + - - if = c2 - 0als2/a2, and vb ax nbguarix (v) from the boundary conditions, u has a logarithmic singularity at x = 0. The boundary conditions (36) and (37) are particular examples of the second order GIBC a -y ( - 2) (68) discussed by Senior [1993] with 1I +b0 z ka bk (Z < 0) =ik bkb (x < O) -=ik "' = - ko (x > 0). As shown there, the solution of the boundary value problem is unique if Im. c, Im. 3 > 0 and, in addition to the standard edge condition, U/ ax] 0 (69) The added constraint is therefore U(O, 0)=0 or [/3x]_ =0 (70) and which is appropriate depends on the physical problem. If U = Hz and the boundary condition specifies the electric current J induced in the surface, then J(x) = xi U(x, 0), and since the electric current is 14

perpendicular to the edge, the required constraint is U(O, 0) = 0, demanding (see (ii) above) C2 = a0 - 2/al. (71) On the other hand, if the condition specifies the magnetic current J*, U is not required to be zero at the edge, and the appropriate constraint is then the second of (70), demanding (see (iv) above) C2 = 0 - al s2/a2 (72) In both cases it can be verified that the resulting solution is in accordance with the reciprocity condition. 4 Fourth Order Conditions For many layered structures, fourth order GIBCs are capable of providing a very accurate simulation of the scattring properties, and we now consider the diffraction of the plane wave (3) by the junction of two half planes each subject to the special form of fourth order GIBCs discussed by Senior [1993]. The boundary conditions imposed at the surface y = 0 are i=)x< 00 (73),( + Z') U = xO > O (74) with 4 4 E = -0-= E, (75) m7=l m=l and these can be written as 4 bm OmU (ik)m y = 0 x <0 (76) 4 bm 0m( b - -am u 0 x > 0 (77) m=Ok)m O9ym 15

where b4 = 1, b3 = 0, b2 = 7172 + 7173 + 7174 + 7273 + 7274 + 7374 bl = 717273 + 717274 + 717374 + 727374, bo = Y71727374 with analogous definitions of the bu. If the plane wave were incident on a surface having the boundary condition (73) for all x, -oo < x < oo, the total field would be as shown in (6) with r = - 7m - sin o. m=1 7Ym + sin q0o Denoting this field by the superscript '0', we again write (78) U(x, y) = U0(x, y) + U8(x, y) with where IP(|)I (8) (9) US(, y)= P( J)ei++iy;2 d_' - 0 as I11 -- oo. The boundary conditions on U8 are then Hn(|+ik7.m)U' = 0 m=l ( y f + ik7m) U = M4e m=l ^y ay x < 0 x>0 (79) (80) where M4 =-k2 f(7 -sino) + rI (7 + sin o). m=l m=l When (79) is applied to (9), we obtain k3bo 0 - + p(g) eiP)edc = O ^-00 \ /MC — l 7 (81) (82) for x < 0 and hence P(=) k K4) L( PW A + ^I/(~ ~ (83) 16

where L(g) is a lower half plane function and /If4() -= /Kj K+(, l 1\ ( 1) 1 2 K+ ( ) + ( 4) * (84) From the condition for x > 0 0 - - -oo k( k) implying F 3 4 1 m I m=l l m + 2 -- ) P( ) ei4d = M4e-i~1 (85) k I )3/2 I (-) ( U() =() - k-~ + Go (86) where U(Q) is an upper half plane function and KA4() differs from K4(g) in having 7' in place of 7m. The combination of (83) and (86) shows k 2 3/2 p(0 = k2 - g2 A(Q) where A(() is an analytic function, and the allowed behavior of P(~) as 1|1 - oo limits A(() to at most a second degree polynomial in <. We can therefore write P() = k22 P() -' k2 3/2 k2 - 2O, IK4(() K4(o0) IL44(- f 4-) C( ) Co + C c2 S +SO (87) for some constants co, cl and c2. When this is substituted into (85) we obtain k3b 14(0o) I4( —0) |x 1K4(Q) Co + C + C22 d - M 0 K4 K'^- d = M,.e-'O, sin3 qo I-3oo ) ~ o and a residue evaluation now gives k3b' 27ri 3- KI4(Qo) K4(-6o) (co - C1O + C20) = M4 sin q>o But ( 4 b -1 If4(~o) K4(-go) = bo sin4 qo { i (7m + sin o) } m=l1 17

and thus ik 4 C2 7r bob sn qn ) si ( sin o) CO - C16 + C 27rbob sin-o ^ 4 - II (T- sin qo)(7m + sin Oo). m=l Moreover { } (b= -b) sin 0o(s _ x1)(s2 - x) where 2 + k 2 2 b - V1 (88) I bisi bl-I SS = k4{ bl (bo + b) - b (bo + b2)} (89) so that bi - bl CO = CLo- C2o + (k3 b 2 SO Hence co + Cl~ + C22 i 1 -b (s - 2 s2 -o) - o- = c + c2( - 1o) + a b2- -- )5-2 ) which can be written as co + c + C2 = i bl -b { (S -0)(s2-02) + c + ) +?;I-rk3 bo + 0 for some constants C3 and c4, at present arbitrary, and the resulting expression for P(Q) is P = k bl - bl K4(g) Ki4(go) IK4(- (-) K(-o) ) 7r b{(k2 - 2)(k2 -_ o2)}3/2 {(s2- 2)(2- ) )} (90) I + o 18

giving U(x y) i b - bi K4(o o) f1 K4() K4(-) (91) " 7' bob sin3 0o -00 (k2 - 52)2 (S12 02 + C3 + C4( -o)} e i k2 d~. We note the similarity to (52), and the presence of two undetermined constants indicates the need for two constraints. On the surface y = 0 U(x, 0) = B4(bl- bl) -4_4() K4(- ) 2 00 (k2 - 2 2)2 { q2( - 0 -+ C3 + C4(( —o) e0 d~ (92) where i 1 K4(0o) K(-o) ( 7r bob sin3 0 To find the behavior for small Ixl, the first step is to additively decompose the first factor in the integrand of (92), and a straightforward but tedious analysis shows 1 (kC2-$2)2 { Z K1i } ( 4() I' 4 - - ( -s - s(2 _ s)(2 s) bo 4(b) (94) Hence K4() K _ (- 1 1 (k2 2)2 - bb (2 S 2)(2 -_ 2) {S+( - S-(0} (95) where S+ (g) = bl bo4 () + K + /1 23 + /232 + /34 (96) S_() 14 + 13 + 2 + l + (97) Since (95) is valid for any /,, i = 1,2,3,4, we can choose the /i to eliminate the poles at - = sl, s2, and -s, -s2 from the factors involving S+(() 19

and S_ () respectively. The resulting expressions for the /3i are given in Appendix B. The final step is to eliminate the pole at ~ = -(o from the upper half plane function, and then Us(x, 0) = T+(Q) e'~x d + T_() e,'x d (98) oo oo where T+() - H (f _ f^ )(J-. _,,,) + (,l o TS (-f R _ 31 -(99) 4+ 0 T2~)= ~B [ s( ~~ ) { (s~ - ~o)~- ~o) +3CQ~) and the term C4/31 has been subtracted to make each integral converge. The first (second) term on the right hand side of (98) represents a function which is zero for x > (<)0 and hence, for x > 0, U_(x,) = J -T(()e _d. (101) T_ + Bo + C3 + C4(~ o 2-S+)(-~ 0 _S2 C(100) and the again, the term involving s(-o) can be evaluach integrd by path closure in The first (second) term on the right hand side of (98) represents a function which is zero for x > (<)0 and hence, for x > 0, us(x, O ) - T_ (~) ~i~x d. (101) oo Once again, the term involving S+(-~o) can be evaluated by path closure in the upper half plane, and the residue at ~ = -<o gives, in part, U~(x, 0). To isolate this contribution we write K4(-~o) 1 r_() T r_(~) -F B4bl boi o -+ 0 where T ) -B4 ([ 2 1{( ( -4 - + ~C+4( - G)} + - (102) C 1 J 43 (102) 20

with Then us (x, 0o) =LTZQ(~) e,~ d~ + 2wizB4bi b'$~~ i and since bKI4(-o) -~ 2 bi K4(~o) K4(-~o) -_ 2iriBb1 0b(o)sin3 - we have U(x, 0) =JTI (~) eru d~. (104) We seek the behavior of TI (~) for large and, hence the behavior of U(x, 0) for small x > 0. From the definition of IK4Q) -I ()If4bo(k2-~2)2__\ TYm 12 2 k 3bi k4bo bo(2~) 2( b) 2++ k2- 2 since =3 0, and therefore IK4(~) bok k~I{+(2n~) for large 1~, implying =4~ b0 {1~1( 1+ b2) + I( n0} (105) Then S~ 3{l~+. ~A( 4~ bl bobs) - b' bob' 1~+b2 - +0 5i 21

and since ~ i7 +s) ~ /hs -s ~ 2 -2) ~ -kb 4 bob( 12 ~b2- 1+bSl)] +O2 1~} A f0 +2 1 A[3+0 Ax0 Axgj) +c3+4(~o) 04 + ~c -2( 24)+ - 2 ___b _ (+ 0)3(2)'1 2 o + C32 + o(4+S2 +S4 443 C/1+C(3 - 2i1 1 2)} + k b~{Co [/33I+ /)-(S ~ sf)] + 4 /3 +/3(S +s~ -/3~o- 3i~OS~+ 1 + b~bb]+ (Y2 /i~) + ~C3 [/4(~G + 3(1 4 + b(C bobs] + 30 +24 / (2 ))+/3(s 2ss + sC ) - C 34( 0 - 36 (S) 2 22

+ A[0/33+ /1(s 2 + S 2) + /3142 - /324]} + Q(45 In4) Finally, since C~4 o1_C0 42 + 3 44. ( the expansion of 'T' () is 134 134 B4rj 4 2 34 F2 ALoo 134 {F3+b b' ~boY 1~- I+b2 - 1 +b/)] + O(4-5ln) (106) where =O C3/1 + C4Q32 - i~i0) + C = (C4 4 -0)(/32 /h40) + C3f31 +/0330 -/34 (107) 1 C3/32~+c4 [/3 +~i1 (s, ~sS)- 026 +/31A -C4o = (C4 _ 42) ['3 + /31(S2 +I s2) - 34]+C/2+ /044 + O (108) IF2 = C3 [33 + i (S 2 ~ S2)] ~ C4 [/4 + /32(S 2 + S2) - 3 1o /i(S2 + S2)] + A(/32 - /3i~o) + C42 = (C4 - 4(2) 0/4 + 32 (S2 + S52) - /3340 - 13i(o52 + S2)] + 3[3 +/S +2f) + (/32 - 34)1S2 (109) F3 C3[/34 +fl2(S2+S 2)] + C4 [/3(S 2 + S 2) + /31(S4 + S2S2 + S4) - 34 34(S2 + S2)] + A/ [/3 ~0 3(S~ 2+ s2) + /341-/3402 C43 = (C4 - 42) 0/1(S4 + S2S2 + S4) + (3-/2)(S2 + s52) _ /340] + C3 [/4 + /32(S 2 + s2)] + [/3 + 31 (S2 + S 2) - 320 2s2 (110) 23

and hence (see Appendix A) U(x,O) = -2riB4 {Fo + ixfz - 2x 2 + c41b -b 3 -+Jb bobo {c3 - k + b2 - \1 ) }] + O(x4lnx) (111) for small x > 0. For x < 0 -oo U(x, 0)= / T+()e'x dE, (112) and since the analysis is similar to the above, we will omit the details. It is found that U(x,O) = -27riB4 {Fo + ixF - x 2 [2 + c4bl O - x 3 + bl bob {c3- c4k ( / 2- \ + b)}] + O(4 ln x)} (113) for small x < O. Comparison of (111) and (112) now shows: (i) U and a- are finite and continuous at x = 0 for all (finite) C3 and c4, (ii) U(O, 0) = 0 if c3 and C4 are such that Fo = 0, (iii) U = 0 at x = 0 if c3 and c4 are such that F1 = 0, (iv) LI and U3 have finite jump discontinuities at x = 0 if c3 and c4 / 0, (v) Iim o- b ou = lim -o+ 1 a2U if C3 and c4 are such that F2 = 0, (v) limm _o_ (vi) lim o- 1aU = limxo+ b1 3U if C3 and c4 are such that F3 = 0, b(._o3, - x (vii) ~4 and (from the boundary conditions) a have logarithmic singularities at x = 0. 24

The boundary conditions (73) and (74) with (75) are particular examples of the fourth order GIBC au (a - 2 +a4 ) ay - 2 ax4 (114) discussed by Senior [1993] with a k 31 + bo + b2 a = lk -3 - bl 2 + b2 bl 2 + — b' /= -ik 2 "1 i i Y= bb;' (x < 0) (x >0) The additional constraints necessary for a unique solution value problem are of the boundary [U 3u9 -Ua3ux -0, a9U a2U + 7* 'a2_ =- 0 Ox ax2 J that is, U =0 at x= 0 or Ox x 9 93U] (115) (116) and OU = 0 ax o 2U a + at x = 0 or 7Tx2J =-0. Ox 2X\ On inserting the expressions for / and y and using (88), the constraints reduce to Fo = or r3 = (s2 + s2)rF (117) and F1 = 0 or 2 = 0. (118) As was true for second order GIBCs, the appropriate constraints depend on the physical problem simulated, and this is discussed by Senior [1993]. In particular, if U = Hz and the conditions specify the induced electric current, then Fo = 0, F2 = 0 (119) 25

whereas for the induced magnetic current r3 = (s + S2)Fi, r2 = 0. (120) The corresponding values of the constants c3 and c4 are derived in Appendix C, and the resulting expressions for P(~) are in accordance with reciprocity. References Abramowitz, M., and I. A. Stegun (1964), Handbook of Mathematical Functions, National Bureau of Standards, p. 258. Leppington, F. G. (1983), "Travelling waves in a dielectric slab with an abrupt change in thickness," Proc. Roy. Soc. (London), vol. A386, pp. 443-460. Ljalinov, M. A. (1992), "Boundary-contact problems of electromagnetic diffraction theory," to be published in Radiophys. (USSR). Senior, T. B. A. (1952), "Diffraction by a semi-infinite metallic sheet," Proc. Roy. Soc. (London), vol. A213, pp. 436-458. Senior, T. B. A. (1991), "Diffraction by a generalized impedance half plane," Radio Sci., vol. 26, pp. 163-167. Senior, T. B. A. (1993), "Generalized boundary and transition conditions and the uniqueness of solution," University of Michigan Radiation Laboratory Report RL 891. 26

Appendix A: Initial Value Relations Let f(x) = f(x) + f2(x) where fi () = f(x)u(x), f2(x) = f(x) {1 - u(x)} and u(x) is the unit step function. Then fi(x) = 0 for x < 0 and f2(x) = 0 for x > 0. Consistent with the representation (9) we define the Fourier transform pair as 1 rfO F(=) = {f()} = (x) fx) e-x dx 27r 0-oo C00 f(x) = -F1 {F(~)} = F(0) eix dl, -00oo so that 00 1 0~i[ 00 - Fl(= fl(x) 6 dx, F2() = f2(x) e dx. We seek to connect the behavior of Fi(() as - oo with the behavior of fi(x) as Ix - 0,o i = 1,2, and vice versa. If fi(x) x~ x as x -+ 0+, integration by parts shows that 1 (1! F1() ( as - -00. 2,x (i7)c+, We now differentiate with respect to a. Since da! = a! V(a + 1) where 4(a) is the digamma function [Abramowitz and Stegun, 1964], we have aq 9 a (ecn)=xlnx and 0 a! a! Oa (i~)>+' (i~)~+{ 27

In the particular case when a is a non-negative integer n - 1, we have the following results: F1 () as - oo fi(x) as x — 0+ 1 (ix)n-1 21i (n-i1)! n -2i ( - 1) {ln x + O(1)} -li (n - 1)! For the function f2(x) it is convenient to write y = -x so that 1 roc )=2- f2(-y) e dy. Then if f2(-y) y' as y 0+ 1 a! F2() ) s as -00 27r (-i~)+l and if f2(-y) ~ yc In y, F2() ~ 27( +{ln(-i) - ( + 1)} -2r ) {lni + (1)}. In the particular case when a is a non-negative integer n - 1, the results are as follows: F2(g) as - oo f2(x) as x -, 0 -n1 -27rwi (n -i! 1 (iX)n-I,n (n - 1)! -ln In i~ 27ri (n ) 1 )! {ln |x| + 0(1)} 28

Appendix B: The Constants Pi In (96) and (97) the functions S+(() and S-(~) are defined for the fourth order conditions, and for these to be analytic in the upper and lower half planes respectively, it is necessary to choose the constants /fi, i = 1, 2, 3, 4, to eliminate the poles at - = sl and s2 from S+(() and the poles at < = -51 and -s2 from S-(~). When this is done, the equations that result are p(sl)++/1 '+3 2 l2 331 +/34 -= 0 p(51) + 3 + s028 + 0351 + 04 = 0 p(s2) + 3183 - +-/32s22+ 3 4 = 0 (1) q(sl)- _3813 - /232 - 331 -+ /34 = 0 q(s2) - 3183 -+ /s2 - /352 + /4 = 0 where p(si)= bi q(si) = bob (si) (B-2) K4 Si) K4(Si) implying p(si) q(si) = boblb'b for i = 1,2. The solutions of (B-1) are 1 = 2( -[P(2)- q(s2)] - [p(sl)- q(sl)]} 2 - (2 S2 S1 1 2 2 2(s 2s2 { [P(S2) + q(S2)] - [p(s) + q(5sl)]} (3 Z 2) {[p(si) + q(s)] - [(S2) - q(s2)]} 13 2 2) 04 -=S22\[P() + q(.sl)] - [S 2 [ )- q(s2)]} There is a relation connecting the /3 that turns out to be important. From (B-l) p(si) = -2(322 + /4)- q(s) p(si) -2i((3ls2 + /3) + q(si) and hence, on using (B-2), 2 {p(s)}2 = -2(/21s + /34)p(Si) - boblbob' {p(Si)}2 = -2sI(/l2 + /3)p(s)+ b0b bb 29

When p(si) is eliminated from these we obtain (/2S2 + /43)2 - 2(/,S2 + /3)2 = boblbob, and since this is true for i = 1,2, (/2S + )2 - (s2 + = S( + /3) - S( (/312 + 2- S +3)2 which simplifies to give (4 + S2S2+ S + s) i = (S + S)(V/ - 2/V3) - (/32 - 2/4). (B-4) Appendix C: Admissible Expressions for C3 and C4 The reciprocity condition concerning the interchange of the transmitter and receiver requires that P(g) be symmetrical in ~ and 0o, i.e. unchanged under the transformation g - 0o. This restricts the admissible expressions for the constants C3 and c4 in (90). A simple analysis shows that the most general expressions for c3 and c4 consistent with reciprocity are C3 - a+{2b+ (S2 + 2), +cO ( C2 C1 2 0 (c-i) c4 - + c0 + {20 for any a, b and c independent of 0o, and then -— + _ + C3 + 0C4(~ - <0o) =,+g {S22 + (S_ + 2)o + 2 2 + a + b( + o0) + cg(o. (C-2) Application of either of the constraints (115) and either of the constraints (116) specifies c3 and c4 in accordance with (C-l), and the values of a, b and c obtained are as follows: Fo = 0, F1 = 0 3 /4 + 1 2S2S2 2+ 1/34(fS12 -+ S2) a 3-2 -+ /( 1 +2 003 2 1 ul\1 2 3 30

2 8 22 /2/4+ /3ls1s2 /3//2 +321 (S + S2) 0103 - 02 T l\1 2 01/4 - /2/3 C o/o3 - 2 + MA +,s2), /3lr1/3/2 ~/1 (S1 +S2 Fo = 0, F2 = 0 /32 ~ 32 S1S2 + /32/342 S +s) 04 + 02 1 2 + 'L244 1 2 5 a = 0304 + 0102S 2 2 + 0104(8fc2 S2) b 12 2 /3134 - /2/3 /3 /3 i2 /3i2 / s + /lo 31/42 + + s C 3 1 1 12 rY\1 2 0104 - 02P3 b3 = (S /31/4-/32 2 + 2/2s2 + /3;3S2 ~ ) s 22 a 1=3- {/3 ~ 2 1 + s) sls2 b r/ = (/3/33/3134) S /3a/3 4 + / 1/ 2S 21S 2 + / 1/34(S 2 + + S) 1pp pp) 2 1 2 b 1 2 2 2 + + / (S + 2) 0,30 + /1/32SlS2 + /31/34(4 + s2) 02 28122 S2 (' 2 + S2 a + 4 + -2 1 2 + 02 )4 31 2 /3 fi3s~~+ /2304 + /302s 2s2+1} (S2 + S2) b {/3+ - /32/34 2 / 3( + s ) s s F3 (S 2 + S2 )Fjj F2 = 0 1 2 f /33s4 + {/3 S 2/3 2+ 02/3 (Ss2 ( +\ s1 2) a 21PP3 1 R /2 1 21 o 1 22 42 /4 ( + 3S2S2 ~ s3 + (/32/S232 +1 /32)) + 1 2 f 02 R 04 1 1 2 1 2 b 102 00 1 2 02 S2S,2 2 + R r 2 2 21 (S2 2), ~4 + 0103 1 12 f O2O4 + OIIS2 \1 + 32 C - /31 +4 S 232/3 54) + /2/ + /3?8 (s + s2) C / 1 1S 3 2 2 + s 3 Tr2 121 2 + (03r-4 + 0102 1S 2)( 2 1 2,1 02 82 8 2 + n 02 21 21 S2 2) r4 + 7r;/3103 1 2 f /204 + ISISS2 \1 + 3 In some of these cases, the verification that the expressions for c3 and c4 do have the form (C —i) requires the use of (B-4). 31