Abstract Equivalent currents and incremental length diffraction coefficients are derived for an impedance half plane. These apply to a half plane with unequal face impedances and reduce to the corresponding incremental length diffraction coefficients and equivalent currents derived'by Mitzner and Michaeli for the perfectly conducting edge.

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INCREMENTAL LENGTH DIFFRACTION COEFFICIENTS FOR AN IMPEDANCE HALF PLANE 1 Introduction In the context of GTD, the scattering by a complex body is represented as a sum of contributions from reflection and diffraction mechanisms [1,2]. The last are attributed to any abrupt surface derivatire discontinuity or possibly any abrupt changes in the material composition of the body. To permit the characterization of a specific discontinuity in surface curvature or material composition, it is necessary that an analytical treatment of an associated cononical geometry be found. Unfortunately, this has only been rigorously accomplished for a handful of geometries which primarily include metallic edges and wedges at normal and skew incidences [3-5], second order surface discontinuity [6,7], impedance wedges at normal incidence [8], thin dielectric and resistive edges at normal and skew incidences[9,10], impedance or material discontinuities in a plane [11,12] and the right- angled wedge at skew incidence with one of its faces perfectly conducting[13]. Solutions may be also found in the literature for creeping wave characterizations and for some non-generic configurations. All of the aforementioned cononical solutions are applicable to straight edges formed by planar surfaces, whereas in practice the edges are curved and generally formed by non- planar surfaces. This prompted the development of uniform theories [2, 14] which allowed the treatment of practical edge configurations and removed the non-physical field discontinuities at the geometrical optics boundaries. Two different uniform theories were formalized in the seventies almost concurrently but among these the uniform theory of diffraction (UTD) introduced by Kouyoumjian and Pathak [2] has been most popular. Aside from its geometrical complexity, the uniform GTD permits the treatment of practical geometries in a systematic manner. Nevertheless, the GTD and UTD are still at caustics and cannot handle finite length edges in a rigorous manner due to a lack of appropriate corner diffraction coefficients [15, 16]. The method of equivalent currents (MEC) was developed to overcome these difficiencies by replacing the diffraction effects with

fictitious equivalent currents which are then integrated over the length of the (curved) edge to obtain the diffracted field. Original versions [17, 18] of such equivalent currents were not as accurate for computations away from the Keller cone. This was corrected by Michaeli [19,20] whose equivalent currents (EC) turned out to be identical to Mitzner's [21] incremental length diffraction coefficients (ILDCs) once the physical optics contributions to the diffracted field was removed from Michaeli's equivalent currents as noted by Knott [22]. Since Mitzner's ILDCs were derived in the context of the Physical Theory of Diffraction (PTD) developed by Ufimtsev [23], this comparison provided a rigorous connection between GTD (actually the MEC) and PTD. In practice when treating arbitrary surfaces, it is more convenient to compute the physical optics field (rather than the geometrical optics field) and supplement these with any edge diffracted field. Consequently, the ILDCs are more suited for practical implementation and can be readily derived from the ECs as noted above. The equivalent currents derived by Michaeli [17] and their similar versions proposed by other [24, 25] are applicable only to perfectly conducting edges. Nowadays, though, man-made vehicles are composed of non-metallic material and a need therefore exists to derive corresponding equivalent currents to characterize the diffraction by material edges and discontinuities. The simplest non-metallic edge for which a skew incidence solution has been derived is that formed by an impedance half plane [9, 10]. It is therefore instructive to first derive the ECs for this geometry and such is the subject of the paper. The procedure employed for this derivation parallels that adopted by Michaeli [19] for the perfectly conducting case and in the process we employ Sommerfeld's inversion theorem [26] as suggested by Pelosi etc [27]. Use of this inversion theorem is essential in obtaining a closed form expression for the equivalent edge currents. In contrast to the ECs derived for the conducting wedge, those derived for the impedance edge are more involved because of the presence of electric and magnetic currents on the surface of the faces forming the edge. They are, nevertheless, given here in explicit form which is suited for computer implementation. As expected they involve the tangential spectral components (with reference to the Sommerfeld integral) of the surface fields. These are the components which must be obtained from a function theoretic solution of the cononical problem for the impedance half plane with equal

face impedances they have been derived by Senior [9] (see also corrections given in [28]). For the half plane with unequal face impedances the spectral field components have been derived by Bucci and Franceschetti [10]. Unfortunately, the expressions given in [10] and are cumbersome to use and are recast here (Section III) in a convenient form suitable for computer implementation. In Section IV we present the ILDCs for the impedance half plane and discuss how these expressions can be used for edges formed by the surfaces of an impedance wedge. 2 Equivalent Current Expressions Let us consider the impedance half plane, shown in figure 1, which is illuminated by the plane wave. Ei = (*e + ye- + -e ) e-)eik(xsin13~cosOhO+y sin/3osino~+z cos,3o) (1) Hi - Yo (:hz + Mhy + Zhz) e-ik(xsinpocosSo+ysinosin o+zcospo) in which k = 27r/A is the wave number and Yo = 1/Zo is the free space intrinsic admittance. The half plane satisfies the boundary condition. y x y x E = -7Zoy x H (2a) on its upper face and the condition y x y x E = q7,Zo. x H (2b) on its lower face. Thus 77o,n are the normalized surface impedances on the = 0 and the X = nir = 27r faces of the half plane, respectively. In accordance with the method of equivalent currents (MEC), the entire scattering by the half plane can be thought as generated by filamentary electric and magnetic currents placed at the edge of the half plane. Referring to figure 3, and denoting these edge equivalent currents as I(t) and M(e), respectively, the Fresnel and far zone fields can be expressed as ES -ik J [Z.oI(e') x ( x t) + M(e')s x i] G(r, r')de' (3)

where t(=.z) is the unit vector tangent to the edge, de' is the increment along the edge described by C, eikjr-r'I G(r, r') - 4r I rr- r'I (4) with r and r' denoting the observation and integration points, respectively, and s r - r' s - - - ~ O cos 4 sin = + = sin = sin + cos (5) s r-r' I 5 To find expressions for I(e') and M(e') we revert back to the original problem and note that the scattered field from the half plane can be written as Es = -ik { J [Zos x s x jo(r') + s x mO(r')] G(r, r')ds' + j s [Zos x s X jn(r') + x m(r')] G(r, r')ds'} (6) where jo,n(r') and mo,n(r') denote the electric and magnetic currents over the upper (oth) or lower (nth) surface of the half plane. It is required that (3) and (6) must be equal and by following a procedure similar to that in [19], we deduce that p=o, sin2 d [(= - S COS i) - KP + z- (^ x K)] (7) POnp=o,n Zo1(e) = E zoI(e)= z=o,n E ) sin=E3 [-Z' (s x Kp ) + (z - sgcos/3) KP] (8) where Kp and Kp are given by the integrals o00 Kp = sin o mpe-ik CoSydau (9a) Kr = Z sin /3 jpe-ik Co~Syda (9b) 4O

in which cos = a * s = sin 3 sin /o cos b + cos d cos,o (10) implying that & is a unit vector in the xz plane making an angle of /, with respect to the z-axis. Concentrating on the currents associated with the upper surface of the half plane we note that mo= Ex y, jo = y x H (11) and from the impedance boundary conditions (2a) we deduce that 7e = - x Km (12) These relations imply that we can write Mo(e) and Io(e) in terms of Ke or Km. Alternatively, we may choose to express Io(e) and Mo(e) in terms of the x components of Ke and Km. Doing so, gives Zo' - I(in:n) -cot /3 cos -K 1~ [ K ] (13) Mo cotepcos (psip [ A(x sin# J L It remains to evaluate Koem in terms of known quantities found from a solution of the canonical problem. Combining (9) and (11), yields;or = sinPo0 ZoHe-ik" cos da (14a) Km = -sin loj Eze-ik C~SYdo (14b) and by following the procedure employed in [19], we may now introduce the exact integral representations for the surface fields. From the impedance half plane solution we have [10] [ zHJ eikzcos~ Jrl EZ(( + 2r - q)'e-ikpsinPocosada (15) L/ZoUZ 2ri zJr Lz z(a + r - 4) where r is the Sommerfeld contour shown in Fig. 2 and we remark that = _ O and p = x for observations on the upper surface of the half plane. For

the moment we will postpone the definition of the spectral quantities E,(a). and X2z(a). These are explicitly given in [10] and will be expressed later in a convenient form for computational purposes. However, before we substitute (15) into (14), it is necessary to introduce an alternative representation to (15) which will permit use of the Sommerfeld inversion theorem (see Appendix) [26] for a closed form evaluation of K',m. This amounts to replacing ~E(ca) and /z-(a) by some odd functions without affecting the outcome of the integration. Such a replacement is possible by exploiting the properties of r. As seen from figure 2, r is comprised of two contours symmetrically located with respect to the imaginary axis. Thus, ~z(a + ir) and 7'H,(a + ir) may be replaced by their odd parts FE(aC) = 2 [Ez(ce + 7r)- ~z(-a + 7r)] (16a) zo FH(a) = - [7-t(a + ir) - 7,z(-a + 7r)] (16b) without affecting the value of the integral in (15). It is convenient to reexpress these only in terms of the spectra ~z(ac + 7r) and Hz(a + 7r) by invoking an alternative form of the boundary conditions (2). From [12] we find that (2) can be rewritten as (iy + iki7o sin2 /3) ZoHz + cos =o 0 (17a) 09HZ y9 + — sin2,o Ez - Zo cos 0 o =0 (17b) ay =7 09X nlau wllnnl ClC3c conditions are applied to the integrals in (15), the resulting integrand must be even with respect to a. It then follows that [ sin ca + sio cos a cos 0 [ r Z(+a + 7r) tlo 7( + ) - cos a cos Po r/7 sinflo + sin a Zoi',(+a +') [ -sin a + sin.,~ cos a cos [ (-a + )

permitting us to rewrite (16) as - sin a FE((a) = D [(t7 sin o - sin ca)~z(a + ir) - cos, cos aZo0-z(a + 7r)] (19a) - sin a& r FH(a) D [cos co csP(a + ar) +( sin o - si Zo0(Ht(a + 7r)] (19b) where Do is the determinant Do = 1 +sin2lPosin2a + sin a sin - (20) In accordance with the above analysis we may now express the surface fields as ZoHZz = 2'7ri FH(Y) ] (21) and when these are substituted into (14), upon setting p = a sin 3lo, z a cos Po we may invoke the Sommerfeld inversion theorem given in the Appendix to find iKoe = s' FH(ao) (22a) zi sin ao -2 K~- itc sin FE(ao) (22b) where t = k sin /3o (23) and { ossn2 o (24) Q~~~ - CO7

Explicit expressions for the equivalent edge currents Io and Mo can now be. readily obtained in terms of E,(ac + 7r) and H7(-(a + 7r). By substituting (22) into (13) and making use of (19) we have = _2i [f(sin_ 1(77o sink -sin a,) tcDo L sin 77 r/o + cotf lcos/o cos cos o }Ez(ao + 7r) + { ( 1 _ sin jc) oS d cOS a cotcos(sin sino) }ZOZ(ao +7r)] (25) MO [( - cotP cos (o7 sinPo - sinea,) - (sin ) cos Po cos ao}Ez(aO + ir) (~ sin ) + {cot / cos /, cos. cos ao s sin o sin Po - ( n- ) -sin aco }ZoHz((ao +7r)] (26) in which Do is given by (20) with c = ac. We remark that from (25) and (26) it is seen that Io and M, are dual quantities, as expected. The equivalent edge currents In and Mn which are associated with the bottom face of the half plane can be obtained directly from Io and Mo by letting P, -- r -/00, P -, 7r - P, do -- 2,r - 40, — + 27r- 4, E~(ao + 7r) z(Cn, - 7r) and'7-(ao + 7r) E- 7z-(an - ir) with cos Cn = cos C,I sinca = - 1-Cos2 a = sin ao (27) Doing so, we obtain I = Dn [{ sin + sin sin + 1(7n i p -si 8,

-cot /os os cos a, cos.o} E(an - 7,) + { ( nCo + 1 )o, os a, sin;3 77n +cot cos (sin sin at ) }Zo(n-7r (28a) 2i =Mn + D [( cot " cos q(7n sin/P - sin a,) +Cos Cosa,,( sin +n 77n) }Ez(aln - ) + { coti cos3o cos cos ao (__ \(sinfl + (sin, S ) }Z0 ( sin7(O) }z0i-ir)] (28b) where Dn = 1 + sin2fo sin2 ao - (/n + sin ao sin /3 For the perfectly conducting half plane Io and Mo reduce to 2i sin 0 r k sin sin Bo sin qo (29a) 2i Io = ksin3oE(2-r - a') + ksin sin a' (- cot /o cos a' + cot, cos d) Zo-'L(27r - a') (29b) in which ca' = r - ao and from [3, p.255] Ez(27r - a 2 sin 2 (30a) ( ) = 4 Cos - cos (30a) 2 2 12sin (- ) 2 2 7'/(2r-o) 4cs (9h

It is then readily concluded that these expressions are in agreement with those in [19] provided the definition for p = cos c'o given in [20, equ.(22)] is used. In the next section we present an explicit computationally efficient form for the Sommerfeld spectra associated with the impedance half plane. 3 The Sommerfeld Spectra for the Impedance Half Plane From [10] we find that sin /o EZ(a + 7r)-i (1 - sin2 /o sin2 a) [sin a cos i3o~y(a + 7r) + cos aZo7y((a + 7r)] (31a) H, (a~ + ~) = _sin /3o 1 - sin2 Po sin2 a [sin a cos f0oZo-y,(a + r) - cos caEy((a + 7r)] (31b) where the spectra ~y(a + 7r) and 7'y(ca + 7r) are associated with the y component of the corresponding fields. They can be conveniently expressed as (1.- sin2 fo sin2 40) [U.(a + r, o; o, - -1)ey - V(a + 7r, o+; o, 1 —)hy 32a) ZoH,3C~~~~~~a~To + 7= (1 - sin2 3o sin2 o0) *[U,(c + 7r, 4o; P5o, r0, 77n)hy + V.(a + 7r, O; flo0,7, 7, )ey](32b)

where U,(ar + r, o0;/o30, 7/, 77n) = {[cos2 0 + sin2/0o Cos ~0 cos C] [ 2 Cos2 -COS 2 V,(a] + 7r, /; Xi0, 07 7n) - sin/Ml cos + M3 in os ) (34) In these, M= (a+ cos 2 -4 a(35) I2 2 M2 = - a + a COS a -a+ a l - a cs 2 }j (36) M3 =- (a3 Co -4a) (37) i =(a+_ -a+a+(39) [21 2 a _ a( ) f (40) M4-= A + = a )Sin 2 a r,-a+ ala (cos — 11

a = + +(c ) f+ +(aX ) (42) 3 sin 2 sin - 2 2 0 (7r - ~O0, 0 n t,.,) Ityb(ca, 977~1Onn ) (-a, Osn Onn)1 f~(a) ~i~(*-d,,e~,s ~~) [d~aB,,$,) ~I(-a,9~Bo.)l (43) (7 (- 0',0)~71',o' ) r'k(a, 0770977n) I (07,701o, ~7n)1 aa = r'/2 + ien (tan,o/2) (44) and slin =?lo,n Sin 7on1 sin o,n sin/3o' sinn =" (45) Also, O(, +, -)= =r a + -— + 2 a -- 3 + )0( 2 ( 2(46) in which,r(ca) is the Maliuzhinets [8] function and simple algebraic expressions for this have been given in [29]. We remark that the edge diffracted fields can be conveniently written as Ed ~ (') (1 - sin2d0o sin2 o)[u (~, 0; 1 ) eY - V (0, 0;,, 1, ) h] (47a) ZHd e(pr/4) -i2 2 +Hy~ ei"') (1 - sin2 sn()- [U (, 0o; Xi, ro,q r7n) h ( +V A(b, ao;, o, r.) %] (47b) 12

where V(q, ho; P/o, 7o, t7n) = Us(-q, Oo; 3ow, r7o, Tn) - U,(27r -, q0o; Po, 77o, T7n) (48a) U(S, qo; /3o, 770, 7n) = Vs(-S, ho; /3o,,7o, 7n) - V,(2r - q, qo; /3, 77o, t77) (48b) and these are in a form compatible to that given by Senior [9] for the case of a half plane having equal face impedances (770 = r). The solution procedure followed by Senior [9] is completely independent of that employed by Bucci and Franceschetti [10]. It is therefore of interest to compare (47) with the diffracted fields given in [9] when 77 = 77 = r77. This should provide some validation of (47), particularly since these expressions differ (only in the sign at two locations) from those given in [10, equ.(82)]. To compare the U and V functions given above to those derived in [9, 28] for the half plane of equal face impedances we proceed as follows. First the U and V functions for the half plane with equal face impedances given in [9] are rewritten as U (4 o;/o, r)= [cos2/3o - sin2 f-cos q cos q - sin0 PoOs L= [ (cos q + cos(0o) 2 2co h-, (aa, a* 7,~7)o p- (aa, a*,,'7)] + sin Po coS Po - i2q7 sin Pf cos cos - - h+ (a,, a ) 2 2 p+ (c,Ar, a )J t+ (-; cos d; /0o, 7).K+ (-K cos qo; o0, 77) (49a) 4 cos ( 2 cos o sin/ocosfl 77sin P, 2 2 h+ (ca,', ) p+ (aa, ) * K+ (-Ks cosqI;/30,r17) K+ (-/cncos 5o;Ido' I1) (49b) 13~~~l

in which h~A la or~sin 2-, =( ) ((501 ) p ca u 7)gc, lsin SAx 2 S-2 ) 1 ~(a., 8., 0 (2).(a', 0(O (52) 4 K+ (-K cosa;/3o, {) = 2n (53) where sin O" = 1l snl. and aa was defined in (44). These were obtained from those in [9, 28] by replacing the parameter 7 employed in [9, 28] with its explicit form in terms of the split functions. We must now show that (49) reduce to those given by (33)-(45) when r7o = f. = fl. To do so we note that /(a,9,9) = 0b(-a,9,9) and for ro= r77 = 77 the function f~(a) simplifies to f+(a) = 2 t (( -,0,,. 07) b (a, ", 0") (54) ~f+(c~a ) 7 =-2/, _ I, 07, 07)?k (a, O, IOn)(54) f_(a) = 0 where sin,7 = 7/ sin.,o. Consequently, 1 M = 2717 (7 - hon, 0 (, ) (55) M2 = -i cos - ( ) (56) M3 = 2r/cos 2,b(1r (57) 14 VI'r - (" )1 14

and M = -i h ( ) (58) h+ (aA a*1,7) In addition, from [30] we find that, (7 - p F 970, l9t71n) {r(7r/2)}8 ___r - _r):', 8"~-, 8n) - 4_ (xr -, 8.o, 8,n) 1 + B sin ( sin + sin ) sin (59) where B( = cos (- 29) cos 2 () (60) B = ~c2sin, o Cos - 20))n (61) -' 2 sin7' 4 and when ro7 = 77= l,(r - tqn j: r, 0" ~, 8""n) simplifies to Or (r - c:: 7r, 0", 0") 2 + sin.... 40 (r - 0, 97, 917) [2' sin 2 ) J2 Using (55)-(58) and (62) in (33) and (34), it is then readily shown that the U and V functions given in (48) reduce to those in (49) when.7o = 7tn = 7. 4 Fringe Equivalent Currents In PTD implementations the total scattered field is obtained as the sum of the physical optics fields and fringe wave contribution. The last is computed by integrating the fringe wave equivalent currents along the edge or surface discontinuity given by Zoo,n Zolo,n- Zoo,n (63) Ms = Mo, - MP~ on Mon 15

In these Io, n and Mo,, are the edge equivalent current expressions (26). whereas IP~ and MP~ are the corresponding edge equivalent currents which result when the current on the faces forming the edge is set to its physical optics value. They can thus be obtained from (13) in conjunction with (9) upon setting mon -- moP. [Ei+ Er] x joPO = x (Hi' + Hr) (64) where (Er, Hr) denote the reflected fields from the corresponding face. As can be expected, jPO and M'PO are set to zero when the upper face is not illuminated and the same holds for jn and Mn when the lower is not illuminated. On applying the boundary condition (2a) we can readily obtain (Er, Hr) from which we find that mPO~ = -2r sin o, {xi [(sin O, + t70 sin,3l) ez - cos /o cos Coh] -z70 [cos o cos oe + (sinn e + (s hZ e] [ s- co 0] (65) where (1 + %70 sin,o sin qo)(i77 + sin Po sin O,) (66) When (65) is substituted into 9(a), upon setting x = a sin /3,, z = a cos P, and integrating we obtain (lower limit only) 2r sin os a ik sin o, [cos' + c os o] [(sin k, + 7-o sin,o) ez - cos /,cosoh,] (67a) ore __ Kox = — K' ik2sin/3c sin + o + - sino h 7) 16

where again a' = r - a, with cao as given in (24). The corresponding physical optics equivalent currents IP~ and MPO are now obtained from (13), we have Z IPo ( - (-1 sin) K- co (68) MP~ = - cot P cos qo + (T sin K (69) sin # o9 and these can be shown to reduce to the known expressions for the perfectly conducting case. The physical optics equivalent current for the lower face can be obtained from (68) and (69) upon letting o30 - r- o30, q0 -- 27r - 0o, l —+ 7r-/3 and ~ -- 27r - q. 5 Appendix: Sommerfeld's Inversion Theorem [26] Let f(x) = 12 i ec~osaF(a)da (Al) where I arg(u) I< 7r/2 and f(x) satisfies the inequality I f(x) 1<1 A I-'l+ eblzl for A, a and b positive real numbers. Also, 0 <ix 1< oo and f(x) is analytic in this region. Then, there exists one and only odd function F(r) which is regular on r and within r (except possibly at infinities), and which satisfies the inequality I F(a) j< Ale(1-a')lIm(a)l. This function is represented by the integral F(a) MIusing l(z)e-~'dz (A2) (A2) and for this function al = a, whereas Al is positive real. Substituting (Al) into (A2) yields F(ao) = - sinA | ~F( )eF e"(c~os-c)daodx and for the particular case of interest tt = -ik

6 References 1. J.B. Keller, "Geometrical Theory of Diffraction,"J. Opt. Soc. Am., Vol.52, pp. 116-130, 1962. 2. R.G. Kouyoumjian and P.H. Pathak, "A Uniform Geometrical Theory of Diffraction for an Edge in a Perfectly Conducting Surface," Proc. IEEE, Vol.62, pp. 1448-1461, Nov. 1974. 3. J.J. Bowman, T.B.A. Senior and P.L.E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes, Hemisphere Publishing Co.: New York, 1987. 4. A. Sommerfeld, "Mathematische Theone der Diffraction," Math. Ann., Vol. 47, pp. 317-374, 1896. 5. H.M. MacDonald, Electric Waves, Cambridge Univ. Press, Cambridge, England, 1902. 6. L. Kaminetsky and J.B. Keller, "Diffraction Coefficients for Higher Order Edges and Vertices," SIAM J. Appl. Math. Vol.22, pp. 109134, 1972. 7. T.B.A. Senior, "Diffraction Matrix for a Discontinuity in Curvature," IEEE Trans. Antennas and Propagat., Vol.AP-20, No. 3, pp. 326333, May 1972. 8. G.D. Maliuzhinets, "Excitation, Reflection and Emission of Surface Waves from a Wedge with given Faces Impedances," Sov. Phys. Doke., Vol.3, pp. 752- 755, 1958. 9. T.B.A. Senior, "Diffraction Tensors for Imperfectly Conducting Edges," Radio Science, Vol.10, pp. 911-919, Oct. 1975. 10. D.M. Bucci and G. Franceschetti, "Electromagnetic Scattering by a Half Plane with Two Face Impedances," Radio Science, Vol.11, 212218, Jan. 1976.

11. R.G. Rojas, "Electromagnetic Diffraction of an Obliquely Incident Plane Wave Field by a Wedge with Impedance Faces," IEEE Trans. Antennas and Propagat., Vol.AP-36, pp. 956-970, July 1988. 12. T.B.A. Senior, "Solution for a Class of Imperfect Wedge Problems for Skew Incidence," Radio Science, Vol.21, pp. 185-191, March-April 1986. 13. T.B.A. Senior and J.L. Volakis, "Scattering by an Imperfect Right Angled Wedge," IEEE Trans. Antennas and Propagat., Vol.AP-34, pp. 681-689, May 1986. 14. S.-W. Lee and G.A. Deschamps, "Uniform Asymptotic Theory of Electromagnetic Diffraction by a Curved Wedge." 15. R. Satterwhite, "Diffraction by a Quarter Plane, the Exact Solution, and Some Numerical Results," IEEE Trans. Antennas and Propagat., Vol.AP-22, pp. 500-502, May 1974. 16. J.N. Sahalos and G.A. Thiele, "The Eigenfunction Solution for Scattered Fields and Surface Currents of a Vertex," IEEE Trans. Antennas and Propagat., Vol.AP- 31, pp. 206-211, Jan. 1983. 17. R.F. Millar, "The Diffraction of an Electromagnetic Wave by a Large Aperture," Proc. lEE, Vol.104C, pp. 240-250, Sept. 1957. 18. C.E. Ryan and L. Peters, Jr., "Evaluation of Edge-Diffracted Fields including Equivalent Currents for the Caustic Regions," IEEE Trans. Antennas and Propagat., Vol.AP-17, pp. 292-299, May 1969. 19. A. Michaeli, "Equivalent Edge Currents for Arbitrary Aspects of Observation," IEEE Trans. Antennas and Propagat., Vol.AP-32, pp. 252-258, March 1984. 20. A. Michaeli, "Elimination of Infinities in Equivalent Edge Currents, Part I: Fringe Current Components," IEEE Trans. Antennas and Propagat., Vol.AP-34, pp. 912-918, July 1986.

21. K.M. Mitzner, "Incremental Length Diffraction Coefficients," U.S. Air Force Technical Report No. AFAL-TR-73-296 (AD 918861), April 1974. 22. E.F. Knott, "The Relationship between Mitzner's ILDC and Michaeli's Equivalent Currents," IEEE Trans. Antennas and Propagat., Vol.AP33, pp. 112-114, Jan. 1985. 23. P.Ya. Ufimtsev, "Method of Edge Waves in the Physical Theory of Diffraction," U.S. Air Force Foreign Technology Office Document No. AD-733203, Wright-Patterson Air Force Base, Ohio, 1971. 24. R.A. Shore and A.D. Yaghjian, "Incremental Length Diffraction Coefficients for Planar Surfaces," IEEE Antennas and Propagat., Vol.AP36, pp. 55-70, Jan. 1988. 25. T. Gokan, M. Ando and T. Kinoshita, "A New Definition of Equivalent Edge Currents in a Diffraction Analysis," JAPAN-CHINA Joint Meeting on OFSET'90, Fukuoka, Japan, Oct. 1990, Proceedings, pp. 31-40. 26. G.D. Maliuzhinets, "Inversion Formula for the Sommerfeld Integral," Sov. Phys. Dokl., Vol.3, pp. 52-56, 1958. 27. G. Pelosi, S.Maci, G. Manara, R. Tiberio and A. Michaeli, "Equivalent Edge Currents for an Impedance Wedge," 1988 National Radio Science Meeting Digest, p. 271, Syracuse, New York. 28. J.L. Volakis, "A Uniform Geometrical for an Imperfectly Conducting Half Plane, " IEEE Trans. Antennas and Propagat., Vol.AP-34, Feb. 1986. 29. J.L. Volakis and T.B.A. Senior, "Simple Expressions for a Function Occuring in Diffraction Theory,"IEEE Trans. Antennas and Propagat., Vol.AP-33, June 1985 pp. 678-680. 30. R. Tiberio, G. Pelosi and G. Manara, "A Uniform GTD Formulation for the Diffraction by a Wedge with Impedance Faces," IEEE Trans. Antennas and Propagat., Vol.AP-33, Aug. 1985, pp. 881-885.

— v, I l " [ " -.wtl ^' z razing diffracted ray foce 1 face 2 Figure. 1. Edge Geometry and angle definitions \1 \ 0 Figure2Theintegration \u ih m \u 2Te tri cn r Figure 2. The integration contour:1 in the Complex plane

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