R.L. Report No. 930 Circular Cylindrical Absorbing Terminations Stephane R. Legault Thomas B.A. Senior Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, MI 48109-2122 December 15, 1995 1 Introduction The need to terminate the computational domain used in various numerical methods such as finite elements (FEM) has lead to the development of a number of absorbing boundary conditions (ABCs) as well as various material absorbers [1-4]. Most of this work. was done in the context of a planar geometry and it is not immediately apparent how well the performance observed for planar boundaries translates to more general curvilinear ones. Indeed, while it has been shown that it is possible to obtain a perfectly reflectionless interface in the planar case by using a particular anisotropic material [4], it remains to be seen if this is also true for a curvilinear boundary. We therefore examine herein the performance of ABCs as well as material absorbers applied to a circular cylindrical (from now on referred to simply as cylindrical) boundary. The absorption of cylindrical wave functions is first examined at the interface of the various truncation schemes, and expressions for the modal reflectionn coefficients are obtained. The results are then used to solve for the scattered field due to a perfect electrical conducting (pec) cylinder surrounded by a given absorbing termination. 2 Absorption of Cylindrical Wave Functions Consider the surface p = p, where p, X, z are cylindrical polar coordinates, with the surface illuminated by a field front within. The geometry is illustrated in Figure 1. The most general 1 RL-930 = RL-930

Figure 1: Geometry used to study absorption of cylindrical wave functions such field can be written as 00 =-oo -U= E amH)(kop)6-ji (I) originating from sources or a scattering body in p < pi, and we seek a termination at p = p, that will completely absorb the field. The terminations considered are ABCs ranging from first to fourth order, various material interfaces, as well as metal-backed versions of these materials. 2.1 Absorbing Boundary Conditions The simplest approach to mathematically terminate a computational domain is to impose an ABC at p = pi. From the corresponding impedance boundary conditions derived using Rytov's [5] method, or, alternatively, from the asymptotic expansion of the Hankel function for large argument (see [1], for example), the following ABCs are found: LU = 0 (2) where the first order ( = 1) operator is 0 1 L1= + jko + (3) ap 2p A poorer version of this valid for large kop is L1=p ko. (4) The higher second order ABC has a. + j (+ 1 2 1\ L2 =- +jko++ - ( 1. -- +1 (5) Op 2p 2ko \jkop \p2 2 4/' 2

and to a lower order (annihilation wise) we have, 1 j 2 L2 a + jo + 2p + 2 )02 (6) We note that (3),(4),(5) and (6) correspond respectively to zeroth, first, second and third order Rytov approximations. For the fourth order ABC we have 0 1 j{ 1 p + + 2p 2ko jkop 1 a 92 25 a 2 l2 1a 4( kp)2 [02 + 4 jkp +2 4 } p2 02 4 ) The extent to which these boundary conditions annihilate the field (1) has been examined by Senior et al [6]. It is found that for small m with kop > 1 all of the conditions are reasonably effective, with the amount of suppression increasing with the order of the condition. As m approaches kop however, the effectiveness decreases, and for rn > kop there is almost no suppression. This is consistent with the fact that the ABCs can be derived from the asymptotic expansion of H(2)(kop) for fixed m and large kop > m. Fortunately, for most fields U the dominant contribution to the infinite series (1) is provided by the terms with Iml < kop, and for Irn > kop the terms are relatively small. As evident from a Watson transformation applied to the series, and in the context of the planar case, increasing rn corresponds to increasing the angle of incidence X on the interface, with m ko(p equivalent to grazing incidence. Alternatively, one may observe that H(2)(x) behaves as a traveling wave for m << x and as an evanescent wave for m > x. Such an interpretation supports the fact that the ABCs absorb traveling waves well, but perform more poorly in the case of evanescent waves, in agreement with results obtained for planar interfaces. For the incident field Ui = H)(kop)e-jmO (8) the reflected field is r = R(m)H()(kop)e-ij (9) and in the case of the ABC given in (2), R(m) = Ri(m) with L~{H- )(kop)c-j"~) Rl() - L{H) (kop)e- } ( ) For fixed m and large kop > m it can be shown that Rl(?n) = O{(kop)-2}, R2(m) = {(op)-4}, R4(m) = {(kop)-6}, (11) and in contrast to the planar case, none of the RI(O) is precisely zero. In Figure 2 the reflection coefficients are plotted vs m for I = 1, 2, 4 and kop = 10 + koAo/2, and the similarity to the planar counterparts is evident. It must also be pointed out that the reflection coefficients 3

m Figure 2: Modal reflection coefficient for ABCs at xl = 10 + r: ( —) 1St order ABCs, ( ---) 2nd order ABCs and (...) 4th order ABCs. The poorer versions are also given. for the higher order ABCs actually overshoot unity for m _ 16. This effect is reminiscent of Gibbs' phenomenon and is accentuated with diminishing x1. The overshoot disappears as xc1 increases as we tend towards the planar case, and JR(m)l becomes increasingly similar (albeit on a different scale) to the planar reflection coefficient IR(()|. 2.2 Material Interfaces We examine next the implementation of truncation schemes by using various material interfaces. Although an interface alone does not constitute a practical termination scheme, the effectiveness of a metal-backed layer depends on our ability to eliminate (or at least minimize) the interface reflection, and we therefore consider the interface problem first. To this end, it is assumed that the region p > pi is occupied by a certain material extending out to infinity. As it will ultimately be necessary to terminate the medium at (say) p = P2 with, for example, a pec, the media examined will all be lossy. The material types studied are homogeneous isotropic and anisotropic materials, as well as inhomogeneous anisotropic ones. 2.2.1 Homogeneous Isotropic Medium Consider the case when the outer region p > pi is occupied by a homogeneous isotropic dielectric with Or = fr. For the incident field (8), the reflection coefficient at the interface 4

P =P1 is R[)H4)/(x) - FH47)(2) (12 R(m) _ p~(\/(12) H4 mx) - F Hr ( X ) ( with r= (13) an(1 x = kop1. From the asymptotic expansions of the Hankel functions for fixed m and large x > m, r = - 2 4m () () F282+ 4 I 2jCrX Ser2X giving R(m) ( ) -j(x-mr/2-ir/4) I{ +O(x-)} (15) which is small if x > 1 or if e r~ 1. Note that if we let IcEr > oo, then I' > -j and the behavior of a lower first order ABC is recovered. This also holds for a planar interface with a homogeneous isotropic medium. Figure 3 gives a plot of IR(m)l for various values of c. Given the previously made observations, it is not surprising to see that the behavior is slightly poorer than the first order ABC given in (3). As is the case for the ABCs, good absorption is observed when m < x, with total reflection occurring for the evanescent modes when m > x. 2.2.2 Homogeneous Anisotropic Medium For a planar interface, the uniaxial medium proposed by Sacks et al [4] produces a perfect match for all angles of incidence, and represents a substantial improvement over an isotropic medium. We now seek the analog for cylindrical coordinates and assume Er =- r = apP + boo + czz (16) where a, b, c are constants. If the field in the medium is Hz = f(x)e-jm (17) where x = kop and the q dependence is chosen to match (8), then E jZ~ of -jm ~E4 z0aj (18) b Ox and f(x) satisfies I (axf + 0 (1f ) xax x Oax j \ ax2/ The appropriate solution of this is f(x) = H(2)(xVbc) (20) 5

I I i I I I I I I 0.8 - E0.6 /0.4 - 0.2 - O C ', ' ' 0 2 4 6 8 10 12 14 16 18 20 m Figure 3: Modal reflection coefficient for homogeneous isotropic interface at x1 = 10 + Tr: ( — ) er- = 1-jl, ( ) r- = 1-j2, and (...)c, = - -j3. with I f H x2)'(Vbc) (21) bf =x -V b H!(2) i(X.bc) where p = pi and v - m b/a. To minimize R(m) for large x, choose 1/a = c = b as in the case of a planar surface. Then ~=,(2) (22) with b complex to produce attenuation, and for large x > m we have R(m) -g - (-) e-2j( m/2-/4) (1 + (O(x-)). (23) Compared with the isotropic material, some of the higher order terms in the braces are eliminated, but overall the performance is no better. Nevertheless, as seen in Figure 4, the homogeneous anisotropic material provides some improvement over the isotropic one up to m _ 13, where the reflection coefficient behaves quite peculiarly. This erratic behavior is apparently attributable to phase reversals of the Hankel function of the second kind with complex order and argument. For a given complex order z, it can be shown that as 21 increases the function H(2)(z) behaves as an incoming wave for small Izl, switching to the typical outgoing wave when Re(v) cv Re(z). For fixed x and varying m, H( (bx) is initially 6

m Figure 4: Modal reflection coefficient for a homogeneous anisotropic medium at x1 = 10 + ir: ( —) b = 1 -j, (- -)b 1-j2, and (..) b = 1 - j3 outgoing for low values of m but the sharp overshoots above unity indicate values of m where H(2(bx) is incoming instead of outgoing. It will be shown in Section 3 that good results can be obtained nevertheless, at least when considering scattering from a pec cylinder. Comparing IR(rn) for the homogeneous isotropic and anisotropic materials as m -> 0 reveals that they behave almost identically. This parallels the case of normal incidence for the planar interface since rn = O is the order of the mode which is normally incident on the cylindrical boundary. The relatively poor performance of the layer (compared to the planar case) is disappointing, since a perfectly matched interface is not achieved. In an attempt to improve the absorption, we next consider an inhomogeneous anisotropic material interface. 2.2.3 Inhomogeneous Anisotropic Medium We now allow the quantities a, b and c in (16) to be functions of x = kop. Consider the field (17) with f() = H(o (boy) (24) where bo is a constant and y = y(x). Since f(x) must satisfy I a ( 9f m2bo f- (Oa\ +(bombo~f~O 25) ydy Yby (bo y2 )= ( and O (f z)f (26) 7

we have 1 a 7y(x)y'(x) ax in agreement wit (1)If bo'y'(x) ax) - m2bo~f 22( - f (27) b (x) = bo~x)11/ c7(x) x (28) a(x) - 1 y (x) box 'y'(x) Thus a(x) - bx'c(x) with b(x) given in (28). Then 2 (X) b(x) x (29) 1 = I f = bf ax b H (2)(boy) (30) so that _ =yx H7 W (bo-y) x H Ib bo) (31) For large Ib07 > m H (2) (bo,-y) = -1 I + 1 2j 'bO 4m 2 _ -b S.y 0 1 I) + Of{(bo-~y<4}) (32) Hence. 7 F=-d -+ x 1 4m' - 1/b2 2j'box 8x-y ( -IbO and therefore R jm 2A2J(x-m-/2 —/4) {1 + O(x1)} (3:3) (34-) (35) where A H4'(x) _X F. If -y x the medium is again isotropic, and since A = - I I 2x (36) 8

we recover the reflection coefficient (23). Alternatively, if y(X) = ( +a-) (37) with al- (38) then A 8 (- {i + O(') (39) giving R(m)= 162 1- I 1 e-2-j(-/2-/4) {1 + 0(x-1)}, (40) and as judged by the order in x, the performance is now that of the first order ABC (3). More accurately still, if 7(X) = X1 + + 2) (41) with a1 given in (38) and 2 8( - b2 (4) then 1 2( 1\ 1 i A - 16x{42 1 —) -2 + l +-)} (43) giving {4 () - 2 + (1 + ) } -2J(m/24) {1 + (-)}, (44 and the performance is now that of the poorer second order ABC (6). Unfortunately, it is not possible to continue further in this manner. To reproduce the best second order ABC (5), and thereby come close to a perfectly matched interface, would require 7(X)= a(1+ 2 3(45) '+,, ) (45) with 3 16j {4m2 ( - ) - 2 + + ) I (46) and this is unacceptable since it implies a medium whose properties depend on m. However, it does show that the profile of the inhomogeneous medium can be specified to absorb to a higher degree a particular mode by fixing m. 9

I I I I I I I 0.8- - EO 0.6 - 0.4 0.2 -/ 0 0 2 4 6 8 10 12 14 16 18 20 m Figure 5: Modal reflection coefficient for a inhomogeneous anisotropic medium at x = 10 + r: ( — )b-= 1 -j,( — )b = 1 - j2, and( )b=1- -j3. Thus, it appears that the nearest counterpart in cylindrical coordinates to a planar perfectly matched layer is the inhomogeneous anisotropic medium (16) with a, b and c given by (28) and (29) and 7(x) = x 1 + 2 1-b )I+ - -2 1-b (47) where x = kop1 and bo is an arbitrary complex quantity. As indicated by the reflection coefficient (44), the match is significantly superior to that provided by a homogeneous medium for any finite x, but as x -- oo c(x), b(x) -- bo, a(x) -~ 1/bo as in the planar case. The reflection coefficient IR(m)\, using (31) and (47), is plotted as function of m for x1 -= 10 + k0oo/2 and a sequence of bo in Figure 5. The increased absorption for all modes having m < 13 is quite pronounced. Note that the oscillatory behavior of |R(m)l is still observed and is explained in the same manner as in the previous section. 2.3 Absorbing Material Layers Keeping in mind that our objective is to efficiently terminate a given computational domain, we now examine the performance of metal-backed layered versions of the materials presented 10

above. The geometry is basically as shown in Figure 1, except that we now have a pec located at x = kX = kop2 = ko(pi+ T). The result is a metal-backed layer of absorbing material of thickness T located at x1. Naturally, the behavior as bo varies differs from the case of the infinite medium. For example, letting bo - 1 leads to IR(m)I -- 1 as the layer becomes more transparent. An actual FEM implementation of such a geometry requires a finite value of bo, and we select here values which are typical of those used in the planar case with similar thicknesses. That is, for a thickness of T = 0.15Ao, we select an imaginary component in the range of 2 to 3. 2.3.1 Homogeneous Isotropic Layer It is a straightforward task to show that the reflection coefficient for a metal-backed layer of homogeneous isotropic material with ec = Lr = bo is NmH(2)(Xl) - M,,H )(iZl ) R(m) -NH (48) NmH( (Xi) - MmHI(1) (Xi) where Mm = H( )(boxl) + rH()(boxi), (49) Nm = H()'(boxl)+ FH)' (bol), (50) and r H)'( ) (51) H4 (box2) We observe that as X2 -> oo then F -- 0 and the solution for the infinite medium is recovered. As for the infinite medium we also recover the behavior of the lower first order ABC as bl -- oo. Figure 6 illustrates IR(m)l for various bo when xl = 10 + koAo/2 and x2 -= x1 + ko0.15Ao. The distinction between the infinite medium and the layer progressively disappears as either T or bo increases, in which case both layered and infinite versions of the medium tend to behave as the first order ABC given in (3). 2.3.2 Homogeneous Anisotropic Layer The analogous solution for a homogeneous anisotropic layer as presented in Section 2.2.2 may be written in the form (48) with Mm = H( o(bol) + rH(l) (boxi), (52) Nm (= (b) o + rbom) (ol), (53) and H(2) ib0X2) bom (bo) -- /_/(1e) 11

m Figure 6: Modal reflection coefficient for a homogeneous isotropic layer at x1 = 10 + 7r with r = 0.15Ao: ( —) bo = 1 - j, (- -) bo = 1- j2, and (. ) bo = 1 - j3. Figure 7 shows IR(m)l for various bo. Apart from the previously encountered overshoots in the 13 < m < 18 range, the behavior is seen to be somewhere between a first and a second order ABC. We note that as for the material interface of the previous section, the anisotropic layer behaves similarly to the isotropic one for lower orders m but provides improved absorption as the order increases. This is particularly true for the higher values of losses used. 2.3.3 Inhomogeneous Anisotropic Layer In the case where the anisotropic layer is inhomogeneous with a profile as discussed in Section 2.2.3, the reflection coefficient is (48) with Mm H= Hom(boTy(x)) + FHb (bo(xl)) (55) Nm Hbom [ (boy(xi)) + FHbOm (bo(x ))], (56) and H(2) )) r- o (bo, X2)) (57) H(1) (bo7(X2)) The material profile -y(x) is given in (47). Figure 8 shows the behavior of R(m)| as a function of in. As for the material interfaces, an inhomogeneous anisotropic layer provides the best 12

10 12 m Figure 7: Modal reflection coefficient for a homogeneous anisotropic layer at pi with r - 0.15A(: ( — ) bo = 1-j, (- -) b0 = 1-j2, and (..) bo = 1-j3. performance overall, especially so when a higher loss is used. Consistent with the previous results, the sharp fluctuations can still be observed. 3 Application: Scattering from a pec cylinder To better illustrate the performance of the various mesh termination schemes, we now app]ly them to the problem of scattering from a pec cylinder. The geometry is basically that of Figure 1, save that the arbitrary central body is now specified as a pec circular cylinder of radius x = xo < x1 on which impinges an H polarized plane wave incident in the positive x direction. The cylinder is surrounded by an ABC or a material absorber located at x1 = xO -+ d. The scattered field may be written as 00 H s = [amH~)(kop) + bmH~)(kop)] e-jm( (58) in the range po < p < p1 [7,6]. The coefficients bm take into account reflections from the imperfect absorber. If the absorber is ideal then bm vanishes and am = am, recovering the exact solution given by am - -J H), (xo ) (5.) Hm()'Xo) 13

20 m Figure 8: Modal reflection coefficient for a inhomogeneous anisotropic layer at P1 with r = 0.15A0: ( — ) bo= 1-j, ( -) bo = 1- j2, and (..) bo = 1-j3. The various terminations are easily applied and it can be shown that am= am 1 (60) + R(m) ) (xi) and bm = R(m)am. (61) We confine our attention to a cylinder of radius xo = 10 with an absorbing termination located at a distance of d = Ao/2 such that xl = xo + w. Typical results for lai, are shown in Figure 9 which compares the exact solution with the ones obtained using the ABCs. We note that the smallest errors are obtained for lower values of m and that the highest error occurs at m ci 10. Such a presentation is of limited value and we focus instead on the behavior of Ibml as shown in Figures 10 and 11. This is more useful for comparison purposes as 1bm, ideally goes to zero. Figure 10 compares the performance of the various ABCs with the material interfaces given in Section 2.2. As previously noted, the homogeneous isotropic material follows the poorer first order ABC performance quite well. It is also seen that the homogeneous anisotropic material is between a first and a second order ABC, its performance improving with m relative to the 1st order ABCs. Lastly, as expected, the inhomogeneous anisotropic medium provides the best performance among the material interfaces, being almost as good as a fourth order ABC. Figure 11 presents the same results for layers of 14

10 m Figure 9: &m with x = 10 and d = Ao/2: ( — ) Exact, (- ) 1st order ABC, (- -) 2nd order ABC, and (...) 4th order ABC. various thickness, and similar conclusions can be reached. We note that the inhomogeneous anisotropic layer with bo 1 - j3 and T = 0.18A is comparable to a second order ABC. The effect of these errors is illustrated in Figures 12 and 13, which show the magnitude of the scattered field on the surface of the cylinder for various terminations. FFror Figure 12, which shows the performance of the ABCs, we observe that the relatively large errors associated with the first order condition lead to spurious oscillations similar to those for an enhanced creeping wave. With the second order condition the agreement is much better, and better still for the fourth order. Figure 13 provides a similar picture for a selection of metalbacked layers. It is interesting to note the similarity between the field for the homogeneous isotropic layer and the poorer first order ABC. Note also the relatively good performance of the inhomogeneous anisotropic layer. One way to quantify the accuracy is to compute the percent RMS value S of the magnitude of the relative error in the surface scattered field, viz. 6 = 100 ( E 1 - Uappro (PO ) (percent) (62) IM kl exact(PO, k) where M is the number of angles at which the error is computed, Uxact is the exact scattered field and tjspro is the scattered field computed using a given termination. For the data that follow, the errors were computed at one degree increments in Ok, and to appreciate the significance of 6, Tables 1 and 2 give the errors computed using the ABCs and the various materials absorbers, respectively. 15

1[7 l1st order 0.06 - 0.04- 2nd order 0.02 4. lrder 0 0 2 4 6 8 10 12 14 16 18 20 m Figure 10: Ibml for interfaces with xo = 10 and d = o/2 and bo 1 - j3: ( —) ABCs, (* *) homogeneous isotropic, (- -) homogeneous anisotropic, and (- ) inhomogeneous anisotropic. 0.12 l l ll 0.1 0.08 - T=0.15. l~lrnI 0 2 4 6 8 10 12 14 16 18 20 m Figure 11: Ibml for layers with Xo = 10, d = Ao/2 and bo = 1 neous isotropic, (- -. ) homogeneous anisotropic, and ( -j3: ( — ) ABCs, (...) homoge-) inhomogeneous anisotropic. 16

1. C, k.......v 1. 1. I 0. I'\.4 I I..2 i *\ I. 1* 4; I I ~~\ t., I\,2 0. 0. 0. 20 40 60 80 100 120 ) (deg.) 140 160 180 Figure 12: IHzS(xo)l using ABCs (- -) 1St order ABC (poorer), ( when xo = 10 and d = 0.5A0: ) 2nd order ABC, and (. *) 4th order ABC. ABC. ) exact, 1. el 1. 1. b~ I II I' I I ~/ 2t I 0. 0. 0. O. U. I 0 20 40 60 80 100 120 140 160 180 q (deg.) Figure 13: IHz(xo)l using layers when xo = 10, d = Ao/2, r = 0.18Ao and bo 1-j3: (- ) exact, ( -) homogeneous isotropic, (- ) homogeneous anisotropic, and (* ) inhomogeneous anisotropic. 17

ABC RMS Error 1st order (poorer) 31.80 1st order 31.21 2nd order (poorer) 5.81 2nd order 4.55 4th order (poorer) 2.26 4th order 1.30 Table 1: Percent RMS error of surface field for ABCs Type bo 1-j | bo1- j2 bo 1- j3 Homogeneous Isotropic Infinite 35.12 35.80 34.69 r = 0.15Ao 52.60 37.92 35.05 T = 0.18Ao 48.99 37.41 34.88 Homogeneous Anisotropic Infinite 6.60 8.56 9.24 r = 0.15Ao 74.90 21.64 12.19 r =0.18Ao 54.21 15.76 10.47 Inhomogeneous Anisotropic Infinite 1.52 2.02 2.21 T = 0.15Ao 68.55 14.90 5.21 T = 0.18Ao 47.94 9.00 3.48 Table 2: Percent RMS error of surface field for material absorbers Theses errors are presented graphically in Figures 14 to 16 which compare the performance of the terminations in the context of the scattering problem considered. We see, for example, that a second order ABC is bested by an inhomogeneous anisotropic layer of thickness r = 0.18Ao and material parameter bo = 1 - j3. 18

80 70 - 60 +. 50 - 50 +.. '. 40- - -, 4- - ------- E - - - — + --- 30 20 -10 -I v 0.5 1 1.5 2.5 3 3.5 Figure 14: Error 6 for ABCs and an( d =.5Ao: ( --- ) ABCs, (.. and ( —) infinite medium. homogeneous isotropic material *) layer with T = 0.15Ao, (- - with bo = 1 -- -, xo =10 -) layer with T — 0.18Ao, 8O 70 - 60 - 50 " ~ 40 30 X \. 20 '+ 10 - II 0.5 1 1.5 2.5 3 3.5 Figure 15: Error 6 for ABCs and homogeneous anisotropic material and d = 0.5Ao: ( ---) ABCs, (...) layer with r = 0.15Ao, (- - and (- -) infinite medium. with bo -) layer -- j/, xo=:10 with T = 0.18A3o, 19

OU [ --- —------— I ------- I ------ I ------ I — ] --- 70 60 50 40. '\ 30 20 \ 10 \ \ +... \101-.... 0.5 1 1.5 2 2.5 3 3.5 Figure 16: Error 6 for ABCs and inhomogeneous anisotropic material with bo 1 -j, xO = 10 and d = 0.5o: (- ) ABCs, (..) layer with r = 0.15Ao, ( -) layer with T 0.18Ao, and (- ) infinite medium. 20

A Numerical Evaluation of Hankel Functions The numerical evaluation of special functions is often a daunting task and the Hankel functions H(1)(z) and H(2)(z) prove to be no exception, especially when complex quantities are involved. In the case where both the order v and the argument z are real, a number of commercially available packages such as Matlab are well suited to evaluate these functions. There are also numerous function libraries, most of them written in FORTRAN 77, which provide routines for similar purposes. The number of adequate packages is, however, narrowed if the argument is complex. One of the better libraries is the AMOS function library. It provides a wide range of routines for Bessel related functions valid for real order and complex arguments. The package is easily obtained through Netlib and the interested reader is referred to [8] for additional details. We note here that simply computing the Hankel function from the Bessel and the Neumann functions using, for example, H( )z) = J- (z)-jN,(z) (63) is often numerically inaccurate for some combinations of z and v and specialized routines to compute Hankel functions such as those provided by the AMOS library become necessary. When the Hankel functions have complex order and argument, there are no widely available routines, and we must instead resort to using the commercial package lfathematica. Once again, as (63) may be numerically inaccurate for some combinations of v and z, and since Mathematica does not provide specific definitions for the Hankel functions, we must instead rely on Bessel functions of the third kind. The formulation used is thus H1)(z) = 2e-C2K(jz), (64) 71 H))(z) = 2i ej -jz). (65) The Wronskians were computed in order to verify the results. An interesting discussion about the evaluation of Hankel function with complex order and argument is given by Paknys [9] and his results were successfully duplicated. 21

References [1] T.B.A. Senior and J.L. Volakis, Approximate Boundary Conditions in Electromagnetics. Stevenage, UK, IEE Press, 1995. [21 S.R. Legault and T.B.A. Senior, "Matched planar surfaces and layers," University of Michigan Radiation Laboratory Report No. 929, 1995. [3] T. Ozdemir and J.L. Volakis, "A comparative study of an absorbing boundary condition and an artificial absorber for truncating finite element meshes," Radio Sci. 29, pp. 1255 -1263, 1994. [4] Z.S. Sacks, D.M. Kingsland, R. Lee and J.F. Lee, "A perfectly matched anisotropic absorber for use as an absorbing boundary condition," to appear in IEEE Trans. Antennas Propagat. [5] S. M. Rytov,"Calcul du skin-effect par la methode des perturbations," J. Phys. USSR 2, pp. 233-242, 1940. [6] T.B.A. Senior, J.L. Volakis and S.R. Legault, "Higher Order Impedance and Absorbing Boundary Conditions", to appear in IEEE Trans. Antennas Propagat. [7] R. Mittra and 0. Ramahi, "Absorbing boundary conditions for the direct solution of partial differential equation," Progress in Electromagnetics Research, M.A. Morgan, ed., Newark, Elsevier, pp.133-173, 1990. [8] D.E. Amos, "ALGORITHM 644, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order", ACM Trans. Math. Software 12, pp. 265-273, 1986. [9] R. Paknys, "Evaluation of Hankel Functions with Complex Argument and Complex Order", IEEE Trans. Antenn. Propagat. 40, pp. 569-578, 1992. 22