TE DIFFRACTION BY A PAIR OF SEMI-INFINITE MATERIAL SHEETS John L. Volakis Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, MI 48109-2122 Abstract The extended spectral ray method (ESRM) is employed in deriving expressions for the multiply diffracted fields by a pair of semi-infinite parallel material sheets/layers. One of the sheets is chosen to satisfy a second order boundary condition simulating a thin dielectric layer and the other is a resistive sheet. Of interest is the computation of the leading edge scattering given as the sum of the singly and multiply diffracted fields with particular attention given to the computation of higher order diffraction fields beyond the'second. Their evaluation required a non-traditional approach and up to fifth order diffraction fields are derived and applied in computing the scattering by a pair of half sheets simulating a metal-backed dielectric half plane as thin as 1/20 of a wavelength. Numerical results for other practical siulations are presented and compared with results based on alternative computational -methods,' where applicable.

TABLE OF CONTENTS Page # I. INTRODUCTION 1 II. SHEET BOUNDARY CONDITIONS 2 III. SINGLY DIFFRACTED FIELDS 4 IV. DOUBLY DIFFRACTED FIELD 7 V. TRIPLY DIFFRACTED FIELD 9 VI. QUADRUPLY DIFFRACTED FIELD 13 VII. QUINTUPLY DIFFRACTED FIELDS 15 IIX. NUMERICAL RESULTS 17 IX. SUMMARY 19 REFERENCES 21 LIST OF FIGURES 22 APPENDIX A 33 APPENDIX B

I. INTRODUCTION Of interest in recent years has been the electromagnetic characterization of material geometries. In this report we consider the problem of diffraction by a pair of parallel material half sheets as shown in Fig. 1. Specifically, the upper half sheet satisfies a second order boundary conditions [1,2] simulating a thin dielectric layer having arbitrary constitutive parameters. The lower half sheet satisfies the resistive sheet boundary condition [3]. Diffraction coefficients for each of these half sheets in isolation are already available and, therefore, a goal in this report is to determine the interaction fields between the two half sheets. The determination of the interaction or multiple diffracted fields is accomplished here via the extended spectral ray method (ESRM) [4-6]. The ESRM technique has been employed, rather successfully, in the past [5-7] for the prediction of the multiple interaction fields among edges at a distance much less than a wavelength. Particularly, in the case of backscatter by a resistive/dielectric strip [5], the third order ESRM solution remained accurate for strip widths down to 1/8 of a wavelength or less. Similar observations hold for the backscatter by a thick impedance edge [6]. In both of these cases, however, the ray paths associated with higher order mechanisms did not traverse along shadow or reflection boundaries. As illustrated in Fig. 2, this is a particular characteristic of the multiple diffraction mechanisms (beyond the third) associated with the subject geometry. Because of it, their treatment within the framework of the ESRM requires a deviation from the usual approach and is the main contribution of this report. Of interest in this report is also the treatment of the diffraction by a thin truncated layer backed by a resistive half sheet as shown in Fig. 3. An exact treatment of this problem leads to a pair of coupled integral equations that cannot be decoupled. It is, therefore, of interest to pursue a high frequency solution. Since the thin dielectric layer can be modeled by a current sheet satisfying a second order boundary condition at the center of the layer [1, 2], the configuration in Fig. 3 can be modeled by a pair of parallel half sheets (see Fig. 1) in close proximity. Based on experience, the ESRM should be capable of providing a good simulation when a sufficient number of multiply diffracted fields are included in the analysis. For this purpose, up to and including 1

quintuply diffracted fields are derived for the pair of half sheets shown in Fig. 1 Our primary attention throughout the report is the determination of the leading edge scattering by the pair of (penetrable) half sheets. Specifically, the observation and scattering directions will be restricted in the range i/2 < 4 3X/2. Since the half sheets are penetrable, this restriction allows one to avoid at this time the treatment of possible contributions from modal fields within the material. In the following, a mathematical description of the half sheets and their diffracted fields in isolation are first given for reference purposes. Next, the development of uniform diffraction coefficients for the doubly, triply, quadruply and quintuply diffracted fields is presented. Except for the doubly diffracted fields, those contributed by the higher order mechanisms require a non-traditional treatment because they include ray paths traversing along reflection boundaries. The last section of the report presents results which validate the uniformity and continuity of the total field as well as the accuracy and limitations of the solution. II. SHEET BOUNDARY CONDITIONS Consider a pair of half-sheets, as shown in Figure 1, illuminated by the plane wave l jk (x cos0o + y sinko) H=e (1) The upper half plane satisfies the boundary conditions [ 1, 2].~i- (Ex+ BH) = - HqI, ( *+ *)(Hz++Hz) * + 11m'le e je where + imply the field values above and below the sheet, Z = 1/Y is the free space intrinsic impedance and 2

-2j * -2j * -2je,le - le, (3) kt (e-l) kx (g-1) e k (-l) As such, (2) represents a simulation of a dielectric layer of thickness X having relative constitutive parameters (e,!t) and centered at y = 0. The lower half sheet at y = -w satisfies the resistive sheet boundary condition [3] E+ + Ex = 2R (Hz - Hz) E+ = E (4) x X where again the + imply the field values above and below the resistive sheet and R denotes the resistivity of the sheet. Of interest is the evaluation of the diffracted fields in the presence of the excitation (1). Restricting the angles of incidence and diffraction in the range xr/2 < 4, 10 < 3cn/2, the total diffracted field is then given as the the sum of the contributions by the single and multiple diffraction mechanisms among the pair of edges Q1 and Q2 formed by the half sheets. For impenetrable sheets, such as a pair of impedance or perfectly conducting half planes, a solution in some spatial range is directly extendable to another. Unfortunately, this is not the case with the penetrable sheets considered here and, therefore, such a restriction is necessary. Below we begin the evaluation of the multiple diffracted fields with particular emphasis in the derivation of expressions applicable to diffraction mechanisms beyond the second. Since our goal is to generate a solution that remains accurate for small sheet separation distances, say down to 1/20 of a wavelength, expressions are derived up to and including the quintuply diffracted fields. 3

III. SINGLY DIFFRACTED FIELDS The diffracted field by the upper half sheet in isolation is given by [7] z1(0100)= j [sec( 2 )+sec 2 [Hz(a+, o)+ Hm (a +, )] e-j osa d; y <O (5) where H (a, o) = ~l K+ (a, Tll) K+ (o', tl) (6) K Hz() [ J+ (x, KI1) K (0, 1)'n msin a sin ~0 qe tan 0 tan 0 K+ ( a,'12) K+ ( Oo'12) (7) in which K+ (4, A) is the Wiener-Hopf split function explicitly defined in [7, 8] (note that in [7] the first argument of K+ is cos a rather than a as employed here), 1 1 1 1 (8)'le'1 ie +-m and 1 1+( (9) 2 q 114

Also, 2Tc - y>O and (, are the usual cylindrical coordinates of the observation point.y< and (p, 4)) are the usual cylindrical coordinates of the observation point. A non-uniform expression (p —oo) for Hzl is e-j;x/4 1 ~1'16jokp Hzl (, -o) 2sec 0 )+sec( 2 ) + se( 2 ))] () 2e z ( e-jkp (10a) since K+ (4, r1) = K+ (27r - 4, rj), where D1(o1, 4O) is the diffraction coefficient for the singly diffracted fields from Q1. The above is, of course, invalid near the reflection and shadow boundaries occuring at 4 = x ~ 4O, where a uniform evaluation [9] of (5) is necessary. In so doing, at the reflection boundary we find 1 e~-J /4 1~'0 HZ1(~ - x - %o ) Ko - s [2 2i e -kp [H: (4, ) + Hm (,, O)] (10b) in which the first term in the brackets is equal to one-half the reflected field by the planar sheet. Clearly, (10b) implies that at 4 = N = i/2, the diffracted field can be decomposed into a plane wave and a slowly varying cylindrical wave. In contrast, the sum of the two waves yields a rapidly varying field whose treatment cannot be handled via the ESRM for the computation of the 5

multiply diffracted fields. The diffracted field by the resistive half sheet in isolation is given by [5] Hz2 (,, 4) = _ rlR J [q( )+sec( 2++ o)l( (x+y, l) K+(++ (O IR) 4ir 2 2 s (O) -jkp2 cos a e da (11) in which 1lR=Z/2R. (12) Note also that in the far zone P2= P + w cost. A non-uniform evaluation of (1 1) now yields Hz2 ('O e - /sec( + o ) + sec ( 4> 20 ) 1R K (4, 1llR)K+ (4o 1R ) e2 2 2.. sec ( 2jR lR....... -D2 ('t e0) (1 3a) P2 where D2(0, 40) is again identified as the diffraction coefficient for this mechanism. As before, when 4 = n - 40, we find that e-n14 f x/4 ____00 - _ - ______ - 2~rkp e + ( o sec e (13b) ~2,~ 2ikp 20-. sec6

allowing a decomposition of the diffracted field into a plane wave and a slowly varying cylindrical wave. IV. DOUBLY DIFFRACTED FIELD Double diffraction occurs when the plane wave after diffraction from the top (bottom) half sheet propagates towards and diffracts from the bottom (top) half sheet as illustrated in Figure 4. The diffracted field from the top edge (Q1) toward the bottom edge (Q2) is, of course, given by the integral (5) with 4 = x/2 and p = w. But in accordance with the ESRM (see Figure 4b) this integral can be interpreted as a sum of inhomogeneous plane waves emanating from Q1 at an angle 3X/2 + a, where a is the angle measured from the stationary ray [10]. Each of the inhomogeneous plane waves will then be incident on the bottom edge at an angle i/2 + a with respect to its top face. For far zone observations, their individual contributions are given by (13a) and, therefore, an integral expression for the doubly diffracted field from Q1 to Q2 is H2i (' )= [e( a - 32 + 0 ) + sec( a - 3 022 - )] S(O) [sec(a+2 + )+sec( 2 +] F( ),s da (14) in which k- _ _jkp2 A= -j n/4 e e-jkwsin (15) A \1 = e e (15) 2n and

2 -j j/4 F(a,, ~) = 4 [ Hez (a + ir/2, o)- Hm (a + /2, o)] lR K+ (, 1R) K+ (a + n/2, R) 2 2 2Ik z ] (16) From (14), it is clear that Hz21 (), o0) can be written as a sum of four integrals i 3n i 3n i 3; Hz21(9,o):II-=-I ( 2 o, —)+ (2 I -'o,+)+I ( 2+ o,- 2-~+ 2 2 2 2 +I I (++ 0o'- +)) (17) where a -jkw cosa I(, j) = F A F(a, o)se( 2 )sec( )e da (18a) sO) whose asymptotic evaluation yields -jkw -jkP2 Ii (ci, cx.) I(ai, ac) F(O, 4,.0) (18b) A uniform expression for I(cxi, aji) that accounts for the case when ai and/or aj are near the saddle point is [9, 10] a a.a. a a. Fk(kwai) FkP (kwa.) i j kp (19a) I(ai, aj)= sec 2! sec 2 - a a. a ] (19a) 2 ~ j 1 provided ai ~ aj. Alternatively, if ai = acj then a suitable expression for I(ai, aj) is ai) = sec2 (-kwai [Fkp (kwai)- 1] + - Fk (kwa) (19b)

In (19a) and (19b), 2 ai a = 2 cos - (20) 2 and or F (z2) = 2j eJz e-J dt (21) is the UTD transition function whose properties are discussed in [9]. The double diffraction coefficient D21 (p, 4O) is now determined in accordance with the relation -jkP2 -jkp H e e e-jkw sine (22) Hz21 ( 0, 2-o ~, D21 ( ) -' e (22) together with (17)-(19). By invoking reciprocity, the doubly diffracted field traversing from Q2 to Q1 (see Fig. 4c) is simply given by 1jkp -jkwsinDo =jlD -jkwsOol2 Hz12 ((a so) eD12 (, Do) e=e (23) and this completes the analysis for the doubly diffracted fields. V. TRIPLY DIFFRACTED FIELD The mechanisms associated with the triply diffracted field are illustrated in Figure 5. 9

Let us first consider the mechanism shown in Figure 5a. In this case the plane wave incident onto Q1 generates spectral waves that are in turn incident upon Q2 at an angle 7/2 + a to subsequently undergo a double diffraction before returning to the observer. Accordingly, an integral expression for the contribution of this mechanism is Hz21 (,o)=A sec( + sec 2 -,o ~2i) 2 2 2 [ HI (a + ir/2, )- Hm (a+ J/2,,o)] D12()', /2 + a)kw cosa d (24) where D12 (4), i/2 + a) = D21 (I/2 + a, 4) is the double diffraction coefficient defined in (17) - (23). Of importance in the evaluation of (24) is, of course, the integrand value at and near the saddle point a = O. We, thus, require D12 (4, i/2). However, expression (19) becomes non-uniform with respect to 4 when n i /2 and is thus invalid, unless w is large. Rewriting D12 (4),:/2) as D12 (, ) = D12 )+ D12 (), () (25) with D12 (,13)= I(- 2- ) +1 2 -- 2 -+ )] F(0,13 (26) 2D2(q,~)= I(3+,,-2- )+I( 2+,,2-~+[)F(0,,,)'-jk (27)w 1D ~, > 3n 3n,4)-j, (27) 10

we observe that D12 ({, p) is the invalid portion of D12 (9, f) when P = 2. To find a valid expression for D (2 It 2) we return to the scenario associated with the triply diffracted field shown in Figure 5a. The spectral plane wave emanating from Q1 is incident upon Q2 at an angle 7/2 + ca. Most of the scattering from Q2 will then be near the specular direction 7c/2 - a, but if a = 0, Q1 will be in the path of the specular return and the diffracted field from Q2 will be given by (13b) with p = w if evaluated at Q1. That is, the diffracted field from Q2 consists of a plane wave portion equal to one-half the reflected field plus a slowly varying cylindrical wave. The plane wave portion of Hz2 (x/2 + a, it/2 - a) is obviously that associated with D12 (', I/2). At Q1, the plane wave portion of Hz2 (n/2 + a, c/2 - a) has the value 2 1R K+ (n/2, 1R) ejkw and makes an angle 3X/2 - a with respect to the top face at Q1-. Thus, an appropriate far zone expression for DI2 (4, i/2) is DI (9'1 + a) 2-f 2 -jkw 3n: D12 ('2 2 + 2C) | 11R + (2 1lR) e D1 (, —— a) ~~~~32 2 2F(0,I1 e a=O sec + sec F(Oj 7 () 2 2' (28) By comparison with (26), we thus observe that 1 1

I(a,-lC+ a) Oej / sec 2 (29) 2 for ac = 0, a result that will prove useful in simplifying the analysis associated with higher order diffraction mechanisms. Substituting (25) along with (27) and (28) into the triple diffraction integral (24) and performing a uniform evaluation as in the double diffraction case yields e-jkp HZ1 21 ( 29, o) ) D121 (9 o) = - 2rw eJ/4I ( 3. - 3- -) I(3 —, 3-~ + o) + 22 +(2 -o ] 2 3) 2 [ (1 with I( ) as define -jkwd in (19), I3 -(~ s' 0) + I 2 )l + F( (3 2 3 0, with I(ap, aj) as defined in (19), Fe =/4 [H ( ~,'o)' Ho'] F(0,, (31, 2, 2n2~k L Z and F(0, i/2, 4) as given in (16). The triply diffracted field whose ray scenario is shown in Figure Sb can be obtained in a parallel manner. We find 12

H)De-jkp -jkw (sin" + sin"" HZ212 (i,,o) ~D212 (o+ +o) X/I p e k [ 2 2 2 2 2 = A- /2kw ej;/a4 I(-2-,- -o) +I(- -,- +o) + I(- + - 2- o) + I(- +, -2 + o)1+[I( —)' O)+I(-i+4o 0 )] I(-1- 0)+I( —+ o 0)]1 2 -) Ff3 (0o ) +/ e (32) in which F3 (O, 4, 4l) = - e; li K+ ( 42o+ I1R) F(,O I 2l) (33) and D212(0, %0) is the diffraction coefficient for this mechanism. VI. QUADRUPLY DIFFRACTED FIELDS The two mechanisms associated with the quadruply diffracted fields are illustrated in Figure 6 and as in the double diffraction case they are reciprocal. Let us first consider the evaluation of Hz2121 (4, 4k). In this case the plane wave incident upon Q1 generates spectral waves which are in turn incident upon Q2 at an angle n/2 + a with respect to the upper face of the lower sheet. Each of the plane waves subsequently undergoes a triple diffraction before returning to the observer. An appropriate integral expression for this quadruply diffracted field then is 13

[He ( +, -Hm (+, o)] D212 (},,-+ C) e-jkw cosa da (34) and in view of (29), the triple diffraction coefficient D212 ( n/2 + a, 4) is given by D212 (2 +a,) = 2kw ej 4 I( —( a) + -2ckw e1/4 [ sec( -:/2 -a )+sec( - i:/2 2 ++ a (35) for a -0. Performing a uniform evaluation of (34) yields Hz212(~' o) =[ kw ei /4 (iI( 3__ ~o', O)+ I( 3_ + ~o'O)) ( I(-~- 2' 0)+ I(-2 +, 0)) +j2 tjkw + 2 2a) +I( —2 -020)+I(-2-+~ 3_. o)+ I(- - -,+ —-+1 +.[2kw ej /4 (i,( 32. ->o) + I( 32+ o)} + )I( 0, ) 3 + I( 2+ o 0) 14

.fi-2 - -2 +j | F4 (0, *, o) e -w sie (36) in which o[. I1 (oai) = sec 2- Fkp(kwai (37) and -j ir/4 F4 (O,, o) 2= / [ 2- Hz' ( s ( (38) where F'3 (0, 4, %0) is defined in (33). Also, by invoking reciprocity, -jkp -j kp Hzl212 ( o) = D1212 (09, o) ze = Hz2121 (Jo' ) = D2121 (0o',) e ek P P (39) where D1212(4, %0) = D2121(0, 0) is the diffraction coefficient associated with the quadruply diffracted field. VII. QUINTUPLY DIFFRACTED FIELDS The mechanisms associated with fifth order diffraction are illustrated in Figure 7. Proceeding as before, the quintuply diffracted field Hz12121 (4, %0) can be expressed as 1 5

Hz12121( Ao + )=J - sec( 2o+sec( ")o)] ) 0C 2 2 [HI(a + 2 0)-Hm(a+2 s9O) D 2 +a) ekw cosada (40) s(o) 2 in which D1212 (4, 7c/2 + a) is defined in (36)-(38), but must be modified in view of (29). We have 2 I2(' eT4'- ( 2 ~+sI( 2+O)] +j2 kw e- 4sec( 3),2 -4)+-c) 2 ++)} + ~2nkw d~~ /. 3n +Io, 37r +_ ( e + I - T + 02 ), + I(0,0) I(2- +I1 + 0'~)] [ I (-1r - co)+I(c, -) F4(Oi.+ I (41) for oc = O. Performing now a uniform evaluation of (40), as before, we obtain ['2',(x /i "/4 l3x-~)+(, 3+I(O,O)[I( c 22 2iW d 7rkw e [I(' -5-+)00) +j2rkw -n2;kw ejI( 3, 3_ 2o) I( 3 +_,, q0) +I( 3+ 31-n) + I( + + o) [ I(0, ) + I(0, [ I(0, o + I(, 23...) ]o) 16

2irJejkw n/4 3x 3n + I2C kw LA 2- + I1( 2 +) + I(O,O) 0 0 I( 3+ -, 0) 2ikw ej / I[i( 7 2- ) ( ) + + + IT)+ I(0,0) I(0, 3o o)+I(o, + )o) ) (1) 1 = D 21 (42) ]() in which Fs (0~,, o) is-/ [dHe (3 )H (a 2 o )] F4 (0, ) (43) where F4(0, o, D0) is defined in (38). The evaluation of the diffraction coefficient D212g2 (Z, 00) can be deduced from D12121 (~, 00) by interchanging 0 and ~0 in (42) and by replacing F5 (0, ), %0) with F5s (0, 9, )0) 22/ TIR K+(j iR4 K+ (0' 1R ) F4 (0 2). (44) + 2' R IIX. NUMERICAL RESULTS To test the validity of the derived diffraction coefficients it is instructive to first consider the special case when both half planes are perfectly conducting. The derived diffracted fields are then valid everywhere (unless shadowed) and this should, therefore, allow verification of the continuity of the scattered field which is equal to the sum of the diffracted fields. Every 17

diffraction mechanism has a shadow boundary either at 4 = n/2 or 4 = 3X/2; however, the total field should remain continuous because of appropriate sign reversals in the unshadowed diffracted fields. For example, given that I/2 < %0 < 3ic/2, H,2 is shadowed in the region O < 4 < n/2 and thus discontinuous at 4 = n/2. However, at 4 = i/2 the terms of Hz12 associated with a transition function whose argument vanishes at this boundary, experience a reversal of their sign so that the sum H,2 + Hz12 remains continuous at 4 = i/2. Similarly HZ21 + HZ121, Hz212 + Hz1212 and Hz2121 + Hz12121 are continuous at 4 = n/2. Shown in Fig. 8 are backscatter and bistatic patterns for two parallel perfectly conducting half planes separated by a distance w = 1.55 X. It is seen that the high frequency solution is in good agreement with the exact [11] pattern. We also observe that the primary role of the multiply diffracted fields beyond the second order is to maintain continuity of the scattered field at the shadow boundaries = r/2 and 4 = 3r/2. As noted earlier, of interest in this study was to examine whether the derived high frequency solution remained valid when w was much less than a wavelength. The pattern in Fig. 9 corresponds to the case when w = 0.05 X and it is clear that the derived high frequency solution is still in good agreement with the exact. For the general case when the half planes satisfy the boundary conditions (2) and (4), there is no available exact solution, neither is it possible to generate numerical data of acceptable accuracy. However, when R=O and w is small, the pair of half sheets represent a metal backed dielectric half plane which is traditionally modelled as an impedance half plane [12]. Also, recently [13], an improved solution was obtained using a second order boundary condition to simulate the coated surface of the half plane. Data based on these formulations can then be employed in examining the validity of the presented high frequency solution. Figure 10 presents backscatter patterns for a coated half plane as computed by this high frequency solution and those based on the 18

standard [12] and second order [13] impedance boundary conditions. It is again observed that the high frequency solution derived here is in general agreement with that predicted by using a second order boundary condition to simulate the coating. This was, of course, to be expected because like the second order boundary condition given in (2), the formulation in [13] also allows a simulation of the normal polarization currents within the dielectric. In contrast, the standard impedance boundary condition (a first order condition) lacks such a capability and does not provide an accurate simulation near edge-on incidences. It should be noted, though, that since the solution given in [13] is also approximate, its small disagreement with that predicted by the two sheet simulation is not necessarily indicative of the accuracy of that solution. Echowidth patterns corresponding to the case when R~O are shown in Fig. 11. One characteristic of these backscatter patterns is their independence on the value of R at edge-on incidence. This is particularly true for H polarization, where the resistive half plane is much less observable in that region. As a result, the entire scattering contribution is primarily caused by the dielectric half-plane itself. IX. SUMMARY In this report, the extended spectral ray method (ESRM) was employed to derive the multiply diffracted fields for a pair of semi-infinite parallel sheets (material layers). The top sheet was chosen to satisfy a second order boundary condition simulating a thin dielectric layer and the lower one was a resistive sheet. Of interest was the computation of the leading edge scattering given by the sum of the singly and multiply diffracted fields. Particular attention was given on the computation of the higher order fields beyond the second since their derivation required a deviation from the traditional ESRM. This was because the diffracted fields beyond the second order were associated with ray paths traversing along reflection boundaries. The resulting spectral representation of these fields then consisted of highly oscillatory components which were 19

essentially decomposed into a pair of slowly varying ones before evaluation of the spectral integrals. Up to fifth order diffracted fields were derived and employed in scattering computations demonstrating their validity. For a pair of perfectly conducting half planes, the generated echowidth patterns were shown to agree with exact data for separation distances as small as one tenth of a wavelength. Comparisons were also provided with other solutions simulating a metal-backed dielectric half plane with favorable agreements. In closing, it should be noted that the given solution can be easily modified for the case when the resistive half plane is replaced by a thin dielectric layer similar to the other. Once this is accomplished, a solution for E polarization can be obtained by invoking duality. 20

1. T.B.A. Senior and J.L. Volakis, "Sheet Simulation of a Thin Dielectric Layer," Radio Sci., vol. 22, 1261-1272, 1987. 2. L.A. Weinstein, The theory of diffraction andfactorization method, Golem Press: Boulder, CO, 1969, p. 301. 3. T.B.A. Senior, "Combined Resistive and Conductive Sheets," IEEE Trans. Antennas Propagat., vol. AP-33, pp. 577-579, 1985. 4. R. Tiberio and R.G. Kouyoumjian, "A Uniform GTD Solution for the Diffraction by Strips at Grazing Incidence," Radio Sci., vol. 14, pp. 933-941, 1979. 5. M.I. Herman and J.L. Volakis, "High Frequency Scattering by a Resistive Strip and Extensions to Conductive and Impedance Strips," Radio Sci., vol. 22, pp. 335-349, May-June 1987. 6. M.I. Herman and J.L. Volakis, "High Frequency Scattering by a Double Impedance Wedge," IEEE Trans. Antennas Propagat., vol. AP-36, pp. 664-678, May 1988. 7. J.L. Volakis, "High Frequency Scattering by a Thin Material Half Plane and Strip," Radio Sci., vol. 23, pp. 450-462, May-June 1988. 8. J.L. Volakis and T.B.A. Senior, "Simple Expressions for a Function Occurring in Diffraction Theory," IEEE Trans. Antennas & Propagat., vol. AP-33, pp. 678-680, 1985. 9. 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, 1974. 10. H.H. Syed and J.L. Volakis, "Multiple Diffractions Among Polygonal Impedance Cylinders," IEEE Trans Antennas & Propagat., May 1989. 11. J.L. Volakis and M.A. Ricoy, "Diffraction by a Thick Perfectly Conducting Half Plane," IEEE Trans. Antennas and Propagat., vol. AP-34, pp. 172-180, Feb. 1986. 12. G.D. Maliuzhinets, "Excitation, Reflection and Emission of Surface Waves from a Wedge with Given Face Impedances," Sov. Phys. Dokl., Engl. Transl., Vol. 3, pp. 752-755, 1958. 13. J.L. Volakis and T.B.A. Senior, "Application of a Class of Generalized Boundary Conditions to Scattering by a Metal-Backed Dielectric Half Plane," Proc. IEEE, May 1989. 21

List of Figures Fig. 1. Geometry of the resistive and material half sheets. Fig. 2. Illustration of propagation along reflection boundaries for higher order mechanisms beyond the second. Fig. 3. Geometry of a thin semi-infinite dielectric layer backed by a resistive half sheet. Provided z < 0. 1., this configuration can be simulated by the pair of half sheets in Fig. 1 with t=2w. Fig. 4. Illustration of the double diffraction mechanisms. Fig. 5. Illustration of the triple diffraction mechanisms Fig. 6. Illustration of the fourth order diffraction mechanisms. Fig. 7. Illustration of the fifth order diffraction mechansims. Fig. 8. Hz echowidth pattern for a pair of perfectly conducting parallel half planes seperated by 1.55 wavelengths; comparison of high frequency and exact patterns. (a) Backscatter. (b) Bistatic pattern; 40 = 150~. Fig. 9. Bistatic Hz echowidth pattern for a pair of perfectly conducting half planes separated by 1/20 of a wavelength; comparison of high frequency and exact data. (a) %0 = 30~. (b) ~0 =1500. Fig. 10. Backscatter Hz echowidth pattern for a metal-backed dielectric half plane of thickness 0. 1X and having e=5-j 1; comparison of this solution with those based on simulations of the coating using the standard and a second order impedance boundary condition. Fig. 11. Bistatic Hz echowidth patterns for a dielectric half plane (e=5, t=2w=0. 1 ) backed by a resistive half plane having resistivities R=0O, 1 and 10. 22

x Material Sheet Resistive Sheet Fig. 1. Geometry of the resistive and material half sheets. Fig. 2. Illustration of propagation along reflection boundaries for higher order mechanisms beyond the second.

= 2w Resistive Sheet Fig. 3. Geometry of a thin semi-infinite dielectric layer backed by a resistive half sheet. Provided X < 0.1 X, this configuration can be simulated by the pair of half sheets in fig. 1 with t = 2w.

22 wa Q2 ~~~~~~(a) ~~~~(b) (c) (d) (CFig. 4.) (d) Fig. 4. Illustration of the double diffraction mechanisms.

p 0Q2 0Q (a) p... Q2 (b) Fig. 5. Illustration of the triple diffraction mechanism.

2 02~~~2 2 (a) Q (b) Fig. 6. Illustration of the fourth order diffraction mechanisms.

p QI Q2 (a) Q; p2 Q2 (b) Fig. 7. Illustration of the fifth order diffraction mechanisms.

PERFECTLY CONOUCTING HALF PLAtES wl.,5,X PwoHigho Frequency,, C(up to 5th ider) 0 I~ m~Fuv(up to 2nd om" E.x [111 GA, um " 12Lo " ANGLE IN DEG PERFECTLY CONOUCTING HALF PLANES High Frequncy a High FIeOeCY (up to 52ndh orld 0' Ex= (I 1 ANGLE IN DEG (b) Fig. 8. Hz echowidth pattern for a pair of perfecdy conducting parallel half planes seperated by 1.55 wavelengths; comparison of high frequency and exact patterns. (a) Backscatter. (b) Bistatic pattern; %o = 1500.

PERFECTLY CONOUCTING HALF PLANES w.O.05X\.. High Frequncy (up to 5th ordW)'r a ~ ~ Exact 11 f' l WIM.E smaE 12N.E 2 L 15. ISL. "NGLE IN DEG (a) ai:'~: / \PERFECTLY CONDUCTING HALF PLANES w-O.O5X O-1 50", High Frequency (up to Sth order) Ex (111 i'l 213 O.E N. 1215 5. I. ANGLE IN DEG (b) Fig. 9. Bistatic Hz echowidth pattern for a pair of perfecty conducting half planes separated (b) 4n =150~.

C3! METAL-BACKED DIELECTRIC HALF PLANE Pair of Half Sheets (up to 5th order) -e —- 2nd Order Boundary Condition [13] \A -- Standard Impedance C' X~ Boundary Condition [12] =2w 0.1/ Q3 5 L)'90.00 120.00 150.00 180.00 210.00 240.00 270.00 RNGLE IN DEG Fig. 10. Backscatter H echowidth paen for a metal-backed dielectric half plane of thickness O.1, and having e=5-jl; comparison of this solution with those based on simulations of the coating using the standard and a second order impedance boundary condition.

o!1 Ln 0 BISTATIC PATTERN =2w=O.1 k o =5,g=1 I ~fR=O -e —R=10 0 u c. C, -i6 Ho 0 ANGLE IN DEG Fig. 11. Bistatic Hz echowidth patterns for a dielectric half plane (E=5, t=2w0.1) baIed by a resistive hald plane having resistivites R=O, 1 and 10.

APPENDIX A DERIVATION OF BOUNDARY CONDITIONS FOR A DIELECTRIC LAYER ON A RESISTIVE SHEET Consider a dielectric/ferrite layer residing on a resistive sheet as shown in Figure 1A. Below our goal is to derive boundary conditions to effectively replace the composite effect of the dielectric/ferrite layer on the resistive sheet. Two approaches are considered in accomplishing this. One involves (approach A) transferring the effect of the dielectric/ferrite layer to the location of the resistive sheet. Another, shifts the resistive sheet condition to the center of the layer. In the following we derive the appropriate boundary conditions for Hz-incidence (Ez = Hx = Hy = 0) followed by a similar analysis for Ez-incidence (Hz = Ex = Ey = 0). H-polarization - Approach A Referring to Figure 1A, at y = 0 the boundary conditions due to the presence of the resistive sheet are 2RZ [Hz (0+) - Hz (0-)] = Ex (0+) + Ex (0-) = 2Ex (0-) Ex (0+) = Ex (-) (1) 2=~ t (cgL) Resistive Sheet Fig. 1A. Geometry of a dielectric/ferrite layer on a resistive sheet for Approach A. where R denotes the normalized sheet resistivity, Z is the free space intrinsic impedance and Hz (0~) refers to the field value at y = 0~ (that is, above or below the resistive sheet). To account for the presence of the dielectric/ferrite layer we may now expand Ex (0+) and Hz (0+) using the first two terms of a Taylor Series expansion giving 33

+aE (25) E (+) = E (28-)- 28 ay 2 aEy (2+) (2+) ) EX )26 -. + i 26kgZ Hz (28+) (2a) e ax a Hz (28) 2~ke Hz (O) = Hz (26) - 2 = H (28-) + i z E (206) (2b) in which e and g are the relative permittivity and permeability of the dielectric/ferrite layer, respectively, k denotes the free space wave number and an e-i)t convention has been assumed and suppressed. Substituting (2) in (1) we obtain R Z Hz (2) i2k E (2) - H (0)] = E ( + 28 aEy(26+) Ex (~2) = Ex (2a+) - ( + i25kgZ Hz (286) (3) ~ ax We may again transfer the fields back to y = O+ through another application of a Taylor Series expansion to find H+ -H 1 E- +__ E Z Z R Z x e 2 aE+ 2z E+- E =+ -+ H (4) ikX ay m m z with * 2ie * 2i 2i e' = m (5) 2k6 (~-1) 2k6 (g-1) 2k6 (e-1) and we have set H = HZ (0). Similarly E: = Ex (0O) 34

The boundary conditions (4) are applied at y = O and represent an approximate replacement of the configuration in Figure 1A. However, they can be shown to be most accurate for small R. In addition, their derivation implies that 6 is small with respect to the wavelength within the dielectric/ferrite layer. Obviously, they represent co-planar electric and magnetic sheet currents, but unlike previously encountered ones, these are now coupled, except when R = 0. In this case the electric currents vanish and (4) reduce to those given earlier. E-polarization - Approach A Referring again to Figure 1A, the boundary condition with Ez-incidence at y = O due to the presence of the resistive sheet are -RZ [H ()- H ( 0) ()]=E (0) Ez (O+) = EZ (0-) (6) As before, to account for the presence of the dielectric/ferrite layer we expand Hx (0+) and Ez (0+) using the first two terms of the Taylor Series expansion to obtain -RZ [H (28+) a x -( ) i2k E (28+) H = E (0-) (7) g Dx Z E (0) = Ez (286) - i28kZg Hx (28+) Transferring now the fields from y = 28+ back to y = 0+, we finally obtain H+ -H- Ez+ Y E+ (8) X X RZ ikmax Z Z 2+ E+ -E =- -H z z * X Tm with 35

2ig 77 m =(9) 2k6 (4t-l) Similarly to the H-polarization case, (8) is best for small R and results to coupled integral equations for the determination of the electric and magnetic currents defined as H+-H =-Jz + ( Ez -E =-Mx (10) H-polarization - Approach B Under this approach the resistive sheet boundary condition is transferred to the center of the layer to be combined with the equivalent sheets of the dielectric/ferrite layer. It is, therefore, convenient to reposition the coordinate system to have its origin at the center of the dielectric/ferrite layer as shown in Figure 2A. 2 8 Resistive Sheet Fig. 2A. Geometry of a dielectric/ferrite layer on a resistive sheet for Approach B. Referring to this new coordinate system, let us now assume that the field at y = O is Ex and attempt to bring in the effect of the material in the regions y > O and y < O by employing the usual two term Taylor Series expansion. We obtain a Ex(-) J E (~') Em:: = EE()-) - E() ik~... ZHz (i) (11) and by invoking the boundary conditions relating the fields at y = 6+ and y = 6& we have 36

aE,(8 + =Em = (+> E- Y - ikg ZHz(6+) (12) Transferring now the fields at y = 26+ back to y = O+ we obtain tm=E+ - — 1) Y -ik (g -l)H (13) x x ax where, as usual, we have set H+ = Hz (0+) and E+ = Ex (O+) A similar expansion of the fields in the region y < 0 yields a Ex(-6+) 1 a E (-8-) Em = Ex(-8+) + E(-8-) + 8- ikg Z H (-8) (14) ay e ax where consistent with the previous notation Ex (-6+) denotes the field's value at y = -6 just inside the dielectric and, of course, above the resistive sheet. Similarly, Ex (-6-) denotes the field's value outside the dielectric's surface at y = -6 and above the resistive sheet. To account for the presence of the resistive sheet we now recall the resistive sheet boundary conditions 2R[H (-)- H (-8) E- E (-6) = 2 Ex(-) Ex(-8-) = Ex(-S-) (15) where Ex (-6=) implies the field's value on the lower side of the resistive sheet. Thus, on the assumption that R 0 O we have Ex(-6-) H (-6) = Hz(-8) + R Ex(-6) = Ex(-6=) (16) and the first of these also implies 37

z aE (-7-) E (-6) = E (-6=) + (17) Since iZ aH Y k ax Substituting (16) - (17) into (14) we now obtain mlaEy(-) 1 a2E( —) Ex(-_=) BE~=E (-8-)+6[- + 2 ik)~Z Hz(-=-) -ik.Z HR- (18) x ~x C ax ikeR ax and by transferring the fields back to y = 0-, (18) becomes 1 aE 1 a2E- EEm=E + 1 + —ZH1 - 6ik+ (19) ikl* ax * Z ik~R ax2 R valid to 0(6). Equations (13) and (19) now represent two expressions for E each involving components of the fields above or below y = 0. Eliminating Ef yields the boundary condition E+ E- = l. (E++E-) +Z (H+ -H) Z a E (20) Zx -X axx Y'Y ikYYR axax Following a similar procedure we may now expand Hz in the y - O region of the layer to obtain aH' E+ + + Y + + Hm =H- 1aH+ - E H+ (21) Z klm az 71z kz 1eZ where Hm is the field's value at y = 0. Also, z Hm =Hz(-_6).m Ex(-6_) (22) 38

and again we may invoke the resistive sheet conditions (15) giving E (-8-) ike8 Hm =Hz(-=) + x - z Ex( ) (23) R Z Transferring the fields in (23) back to y = 0- and retaining terms to 0(8) yields a Hz()(O Ex(O) 5 a Ex(o) ik E - Hm =H(0)-8 ay R R. Z Hz z ay -6,yR R ay Z X ik E [aE 1 ik&6 =H + _X Y ikZ H_ E z R RR ax Z x or E- E a EHm=H-+ x + Rx 8 x (24) z z ZTle R R ay Equations (21) and (24) may now be combined to eliminate Hm giving H - H = -.(E8+Ex)+ 8 E- 6 (25) zne R ay J The above boundary condition along with (20) form a complete set for the simulation of a layer backed by a resistive sheet. As noted earlier they are valid for R ~ O and we also observe that for R -00o they reduce to the iknown boundary conditions for the isolated dielectric/ferite layer [1, 2]. Clearly, (20) and (25) represent a co-planar pair of magnetic and electric current sheets. Unfortunately, they result into coupled integral equations for the solution of the sheet currents. So far, our attempts to decouple them have not been fruitful precluding us from obtaining an exact solution of the relevant half plane problems. An alternative, though, is to consider a simulation of the geometry using a pair of parallel sheets (see fig. 3) whose solutions in isolation are known. Referring to Fig. 1A, the obvious choice is a resistive sheet at y=O, and another sheet (supporting electric and magnetic currents) placed at y='/2 to simulate the dielectric layer. The boundary condition associated with the last has been derived in [1]. A high frequency solution of the diffraction by a pair of such sheets is now possible and in view of previous experience, it is expected that the inclusion of a sufficient number of higher order diffraction effects should allow an accurate characterization of their scattering. 39

APPENDIX B COMPUTER PROGRAM LISTING

1 C DIELRHP: Program for computing the scattered field by a C 2 C a pair of half planes, one simulating a resistive C 3 C half sheet and the other a thin semi-infinite dilectric C 4 C layer. C 5 C Includes up to fifth order diffraction terms C 6 C Compile with HZ1M,HZIR,DOUB21,TRIP1,TRIP2,QUAD,QUINT1,SEC C 7 C FFUN,FI,FIP,FI 1, KPLUSM,PSIPI,HEE,CSQRC,BTAN2,FFCT,GENPLO C 8 C C 9 COMPLEX ER,UR,CJ4,CJ,DENOM1,DENOM2,RES,RSTAR,RSTARE,RSTARM 10 COMPLEX ETAE, ETAMS, ETAES,ETAS, ETAl, TEMPC, ETAM1, ETAM2, ETAR 11 COMPLEX KPLUSC, CON1, CON2, KPLUSM, TEMP1, TEMP2, RU, RL 12 COMPLEX HZ1,HZ2,HZ21,HZ12,HZ121,HZ212,HZ412,HZ421,HZ51,HZ52 13 COMPLEX TH1,TH1S,TH2S,HEE,THO,THN,THR,AA,BB,CC,RR,TT,RR1,TT1 14 COMPLEX DOUB, TRIP, DUBM, TRIPMG, PHASI, PHASS, DUBE, TRIPE 15 COMPLEX TT1I,TT1S,TTI,TTS 16 DIMENSION ANG(361),HE(361),HM(361),HZ(361) 17 DIMENSION HZF(361),HZS(361), HZT(361) 18 COMMON /BLK1/ETA1, ETAR, ETAS, ETAES, ETAMS, ETAM1, ETAM2, CON1, CON2 19 COMMON /BLK2/CJ,CJ4,PI, P 12 20 COMMON /REFL/RU,RL,TT,TT1,PHASS,PHASI 21 COMMON /THETAM/TH1,THR,TH1S,TH2S 22 COMMON /THETAS/THO,THN 23 PI=3.141592 24 PI2=PI/2. 25 cJ=(0.,1.) 26 CJ4=CEXP (-CJ*PI/4.) 27 PRINT *,'NUMBER OF PLOTS,IPRINT,#OF RAYS:' 28 READ(5, *) NPLOT,IPRINT,M,M2 29 DO 2000 IPLOT=1,NPLOT 30 PRINT *,'LAYER REL. PERMITT.,PERMEAB. AND THICKNESS(WL)' 31 READ(5,*) ER,UR,THICK 32 PRINT *,'IS THIS SIMULATING A THIN DIEL.H.P. ON A RES SHEET?' 33 READ(5,*) ISIM 34 IF(ISIM.NE.1) THEN 35 PRINT *,'SEPARATION BETWEEN THE DIEL. AND RES. H.P.s:' 36 READ(5,*) D 37 ELSE 38 D=THICK/2. 39 ENDIF 40 PRINT *,'NORM. RESISTIVITY OF THE RES. HALF PLANE:' 41 READ(5,*) RES 42 ETAR=1./(2.*RES) 43 PRINT *,'ETAR:',ETAR 44 C ENSURE RIGHT BRANCH FOR LATER SQUARE ROOTS 45 ER=ER-CJ* 1. E-6 4 6 UR=UR-CJ*1.E-6 47 DENOM1= (ER-1) *2. *PI*THICK 48 DENOM2= (UR-1.) *2. *PI*THICK 49 C COMPUTE ETA/IMPEDANCE PARAMETERS FOR USE IN DIFFR. COEFFICIENTS 50 RES=-CJ/DENOM1 51 RSTAR=-CJ/DENOM2 52 RSTARE=-CJ*ER/DENOM1 53 IF(CABS(ER).GT.1000.) THEN 54 RSTARE=-CJ*ER/(2.*PI*(ER-1.) *.001) 55 ENDIF 56 RSTARM=-CJ*UR/DENOM2 57 ETAE=2. *RES 58 ETAMS=2. *RSTAR 59 ETAES=2.*RSTARE ~60 ETAS=ETAES*ETAMS/ (ETAES+ETAMS)

61 ETA1=1./ETAE 62 C PRINT *,'ETAES,PHAS:',ETAES,BTAN2(AIMAG(ETAES),REAL(ETAES)) 63 TEMPC=ETAS*ETAES 64 C PRINT *,'ETES*ETAES,PHAS:',TEMPC,BTAN2(AIMAG(TEMPC REAL(TEMPC)) 65 CON1=ETAS*CSQRT(1.+(4./TEMPC)) 66 PRINT *,'OLD TEMPC:',CON1 67 TEMPC=ETAS*CSQRC (1. + (4./TEMPC)) 68 C PRINT *,'NEW TEMPC:',TEMPC 69 ETAM1=. 5* (ETAS+TEMPC) 70 ETAM2=. 5* (ETAS-TEMPC) 71 TEMPC=KPLUSM((.5, 0.),ETAM2, 0,1.)/SIN(.25) 72 PRINT *,'ETAES,ETAMS,ETAS:,',ETAES,ETAMS,ETAS 73 C PRINT *'ETAEETAM1,ETAM2:',ETAEETAM1, ETAM2,-1. /ETAM1 74 PRINT *,'INITIAL INCIDENCE AND SCATTERING ANGLES(DEG):' 75 READ(5,*) PHI,PHS 76 PHI=PHI*PI/180. 77 PHS=PHS*PI/180. 78 PRINT *,'INCREMENTS IN INCIDENT AND SCATTERING ANGLES,# OF PTS:' 79 READ (5, *) DPHI,DPHS,NPTS 80 DPHI=DPHI*PI/180. 81 DPHS=DPHS*PI/180. 82 C TH1=PI2-HEE(ETA1,0, 1.) 83 C THR=PI2-HEE(ETAR, 0, 1.) 84 C TH1S=P I 2 -HEE (ETAM1,0, 1. ) 85 C TH2S=PI2-HEE (ETAM2,, 1. ) 86 CC THO=P I2-HEE (ETAMS,0, 1. ) 87 C THO=TH1 88 C THN=THO 89 C PRINT *,'THO,TH1S,TH2S:',THO,TH1STH2S 90 C PRINT *,'CSIN(TH2S),1/ETAM2:',CSIN(TH2S),1./ETAM2 91 DO 1000 I=1,NPTS 92 CPHI=COS (PHI) 93 CPHS=COS (PHS) 94 SPHS=SIN(PHS) 95 SPHI=SIN (PHI) 96 CPHI2=COS (PHI/2.) 97 CPHS2=COS (PHS/2.) 98 SPHI2=SIN (PHI/2. ) 99 SPHS2=SIN (PHS/2. ) 100 C Compute reflection and transmission coefficients 101 C Incident ray refl & transm coef. 102 AA= (1. /ETAMS) + (CPHI*CPHI/ETAES) 103 BB=SPHI 104 CC=SPHI/ETAE 105 RR=- (AA/ (AA+BB) )+(CC/ (CC+1. )) 106 TTI=RR-((CC-1.)/(CC+1.)) 107 CC=SPHI*ETAR 108 RR1=CC/(CC+1. ) 109 TT1I=1.-RR1 110 C Scattered ray refl. & transm. coeff. 111 AA= (1. /ETAMS) + (CPHS*CPHS/ETAES) 112 BB=SPHS 113 CC=SPHS/ETAE 114 RR=- (AA/ (AA+BB) ) + (CC/ (CC+1. )) 115 TTS=RR-( (CC-1.)/(CC+1.)) 116 CC=SPHS*ETAR 117 RR1=CC/(CC+1. ) 118 TT1S=l.-RR1 119 C NORMAL INCIDENCE REFL & TRANSM COEFFICIENTS 120 RU=(1. /(1.+ETAE))-(1./(1.+ETAMS))

121 RL=ETAR/ (1.+ETAR) 122 c PRINT *'CPHI,CPHS,SPHI2,SPHS2:','CPHI,CPHSSPHI2,SPHS2 123 C IF (CABS (ER). GT.1000.)THEN 124 TT1=1-RL 125 TT=1.-RU 126 C TT1=(0.,0.) 127 C TT=(0.,0.) 128 C ENDIF 129 C PRINT *,'RURLTTTT1',RU,RLCABS(TTS),CABS(TTI) 130 C CALC FIRST ORDER DIFFRACTION 131 CALL HZ1M(HZ1,PHI,PHS) 132 PHASI=CEXP (-CJ*2. *PI*D*SPHI) 133 PHASS=CEXP (-CJ*2. *PI*D*SPHS) 134 CALL HZ1R(HZ2,PHI,PHS) 135 HZ2=HZ2*CEXP(-CJ*2.*PI*D* (SPHI+SPHS)) 136 C PRINT *,'HZ2 SEC EDGE:',HZ2 137 C DOUBLE AND QUADRUPLE DIFFRACTION 138 SGNS=1. 139 SGNI=1. 140 CALL DOUB21(HZ21,PHS,PHI,D,M, SGNS,SGNI) 141 CALL QUAD(HZ421,PHS,PHI,D,1,SGNS,SGNI) 142 HZ21=HZ21*CEXP(-CJ*2.*PI*D*SPHS) 143 HZ421=HZ421*PHASS 144 C CALC DOUBLE DIFFRACTION FROM RESISTIVE TO MATERIAL H-P 145 SGNS=1. 146 SGNI=1. 147 CALL DOUB21(HZ12,PHI,PHS,D,M,SGNI,SGNS) 148 HZ12=HZ12*CEXP (-CJ*2. *PI*D*SPHI) 149 CALL QUAD(HZ412, P PHS, D, 2, SGNI, SGNS) 150 HZ412=HZ412*PHASI 151 SGNI=1. 152 SGNS=1. 153 CALL TRIP1(HZ121,PHS,PHI,D,1, SGNS,SGNI) 154 CALL QUINT1(HZ51,PHS,PHI,D, 1, SGNS,SGNI) 155 SGNS=1. 156 SGNI=1. 159 CALL TRIP2(HZ212,PHI,PHS,D,M, SGNI,SGNS) 160 CALL QUINT1(HZ52,PHI,PHS,D,2,SGNI,SGNS) 161 HZ212-HZ212*PHASI*PHASS 162 HZ52=HZ52*PHASI*PHASS 163 11 ANG (I) =PHS*180./PI 164 PHI=PHI+DPHI 165 PHS=PHS+DPHS 166 TEMPC=HZ1+HZ2+HZ21+HZ12 167 IF(M2.EQ.5)TEMPC=TEMPC+HZ51+HZ52+HZ421+HZ412+HZ121+HZ212 168 IF(IPRINT.EQ.1) THEN 169 PRINT *, (PHI-dphi), (PHS-DPHS)*180/PI,HZ1,HZ2 170 PRINT *,HZ2+HZ12,HZ21+HZ121,HZ212+HZ412 171 PRINT *,HZ421+HZ51,HZ52 172 C PRINT *,'HZ421,HZ51:',HZ421,HZ51 173 PRINT *,HZ21,HZ12,HZ121,HZ212,HZ421,HZ412 174 ENDIF 175 TEMPC=2.*PI*TEMPC*TEMPC 176 HZ(I)=10*ALOG10 (CABS (TEMPC)) 177 PHASE=ATAN2(AIMAG(TEMPC),REAL(TEMPC)) )*180./PI 178 PRINT *,'ANG,HZ:',ANG(I),HZ(I),TEMPC 179 WRITE(2,100) ANG(I),HZ(I),PHASE 180 C100 FORMAT(F10.5,'*',F10.5,'*',2F10.5,'*',F10.5) 181 100 FORMAT(2F8.3,F8.2) 182 1000 CONTINUE

183 IEND= 0 184 IF (IPLOT. EQ.NPLOT) IEND=1 185 CALL GENPLO(ANG,HZ,NPTS, IPLOT-1, IEND) 186 2000 CONTINUE 187 CALL EXIT 188 END 189 C **************** 190 SUBROUTINE HZ1M(HZ1,PHI,PHS) 191 COMMON /BLK1/ETA1, ETAR, ETAS, ETAES, ETAMS, ETAM1, ETAM2, CON1, CON2 192 COMMON /BLK2/CJ,CJ4,PI,PI193 COMPLEX KPLUS1,KPLUS 2KPLUS3,KPLUS4,KPLUS52,KPLUS3,KPLUS4,6KPLUS5, 194 COMPLEX KPLUSM,KPLUSP,CON1, CON2 195 COMPLEX ETAl, ETAS, ETAR, ETAES, ETAMS, ETAM1,ETAM2 196 COMPLEX CJ,CJ4, HZE, HZR, HZM1, HZM2,HZ1, TU,TL,RU,RL 197 COMMON /REFL/RU,RL,TU,TL 198 KPLUS1=KPLUSM(CMPLX(PHS, 0.),ETA1,0,1.) 199 KPLUS2=KPLUSM (CMPLX (PHI, 0.), ETA1, 0, 1.) 200 KPLUS3=KPLUSM(CMPLX(PHS,0.),ETAM1,0,1. ) 201 KPLUS4=KPLUSM(CMPLX(PHI, 0.),ETAM1,0,1.) 202 KPLUS5=KPLUSM(CMPLX(PHS, 0.),ETAM2, 0, 1.) 203 KPLUS6=KPLUSM(CMPLX(PHI, 0. ), ETAM2,0,1.) 204 TEMP1=1. /COS ( (PHI+PHS) /2.) 205 TEMP2=1. /COS ((PHI-PHS) /2.) 206 TEMP=TEMP1+TEMP2 207 SPHI=SIN(PHI) 208 SPHS=SIN(PHS) 209 CPHI=COS (PHI) 210 CPHS=COS (PHS) 211 HZE=-CJ4*KPLUS1*KPLUS2*TEMP*ETA1/( 4. *PI) 212 KPLUSP=KPLUS3*KPLUS4*KPLUS5*KPLUS6 213 HZM2=-CJ4*ETAS*KPLUSP*TEMP*CPHI*CPHS/ (4. *PI*ETAES*SPHI*SPHS) 214 HZM1=CJ4*ETAS*KPLUSP*TEMP/(4. *PI*SPHI*SPHS*ETAMS) 215 HZ1=HZE+HZM1+HZM2 216 RETURN 217 END 218 C 219 SUBROUTINE HZlR(HZ1, PHI,PHS) 220 COMMON /BLK1/ETA1, ETAR,ETAS, ETAES,ETAMS, ETAM1, ETAM2, CON1, CON2 221 COMMON /BLK2/CJ,CJ4,PI, P 12 222 COMMON /REFL/RU,RL,TU,TL 223 COMPLEX KPLUSM,KPLUSP,KPLUS7,KPLUS8,CON1,CON2 224 COMPLEX ETA1, ETAS,ETAR,ETAES,ETAMS,ETAM1, ETAM2 225 COMPLEX CJ,CJ4,HZR,HZ1,RU,TU,RL,TL 226 KPLUS7=KPLUSM (CMPLX (PHS, 0. ),ETAR, 0, 1. ) 227 KPLUS8=KPLUSM (CMPLX (PHI, 0. ), ETAR, 0,1.) 228 TEMP1=1. /COS( (PHI+PHS) /2.) 229 TEMP2=1. /COS ((PHI-PHS) /2.) 230 TEMP-TEMP 1+TEMP2 231 SPHI=SIN (PHI) 232 SPHS=SIN (PHS) 233 HZR=-CJ4*KPLUS7*KPLUS8*TEMP*ETAR/( 4. *PI) 234 HZ1=HZR 235 RETURN 236 END 237 C ************************** 238 SUBROUTINE DOUB21(HZ21,PHS,PHI,W,M, S1,S2) 239 C DOUBLE DIFFRACTION FROM MATERIAL TO RESISTIVE HALF PLANE 240 COMMON /BLK1/ETA1,ETAR,ETAS,ETAES,ETAMS,ETAM1,ETAM2,COM, COM2 241 COMMON /BLK2/CJ,CJ4,PI, P I2 242 COMMON /REFL/RU,RL,TU,TL, PHASS, PHASI

243 COMPLEX ETA1, ETAS, ETAR, ETAES, ETAMS, ETAM1, ETAM2, COM1, COM2 244 COMPLEX CJ,CJ4, HZE, HZR, HZM1,HZM2,HZ21,RU,RL,RM,RM1,RM2,RMM 245 COMPLEX DEL,FFUN, FI, tt, TU, TL, PHASS, PHASI 246 C DEL=CJ4*CJ4/(16*PI*PI) 247 DEL=(1.,0.) 248 RM=.25*CEXP (-CJ*4. *PI*W) 249 RMM=(1.,0.) 250 C GO TO 200 251 C M=10 252 DO 100 N=4,M,2 253 N1=N-2 254 RM1=CEXP (-CJ*N1*2. *PI*W) / (2. **N1) 255 N2=(N-2+.0001)/2. 256 RM2=(1., 0.) 257 DO 10 I=1,N2 258 10 RM2=RM2*RU*RL 259 100 RMM=RM1 *RM2+RMM 260 200 CONTINUE 261 C PRINT *,'RMM IN HZ12:',RMM, RM*RU*RL, N1,N2 262 C RMM=(1.,0.) 263 DEL=DEL*CEXP (-CJ*2. *PI*W) /SQRT (W) 264 AL1=1.5*P I-PHI 265 AL2=1.5*PI+PHI 266 AL3=-PI2-PHS 267 AL4=-PI2+PHS 268 C TEMP=(1. /COS (. 5*AL))+ (1./COS (. 5*AL2)) 269 C TEMPC=(1. /COS (. 5*AL3))+(1./COS (.5*AL4)) 270 HZ21=S1*FI (AL1, AL3, W)+FI (AL1, AL4, W) 271 HZ21=HZ21+S1*S2*FI (AL2, AL3, W) +S2*FI (AL2, AL4, W) 272 HZ21=HZ21 *DEL*FFUN (PHS, PHI, 0 ) 273 HZ21=HZ21*RMM 274 RETURN 275 END 276 C ************************* 277 SUBROUTINE TRIP1(HZ121,PHS,PHI,W,M, Sl,S2) 278 C TRIPLE DIFFRACTION FROM MATERIAL TO RESISTIVE HALF PLANE 279 COMMON /BLK1/ETA1, ETAR,ETAS, ETAES, ETAMS, ETAM1, ETAM2, COM1, COM2 280 COMMON /BLK2/CJ,CJ4,PI,PI2 281 COMMON /REFL/RU,RL,TU,TL 282 COMPLEX ETA1, ETAS, ETAR, ETAES, ETAMS, ETAM1, ETAM2, COM1, COM2 283 COMPLEX CJ, CJ4, HZE, HZR, HZM1, HZM2, HZ121, RU, RL, RMM, RM1, RM2, RM 284 COMPLEX DEL, FFUN, FI, FIP, T1, T2, TT, FFCT, TT1, DEL1, TU, TL 285 DATA RT2/1.414213562/ 286 DEL=CEXP (-2. *PI*CJ*W) /SQRT (W) 287 DEL1= (4. *16*PI*PI*PI*PI) / (CJ4*CJ4*CJ4) 288 IF(M.EQ.1) THEN 289 HZ121=-FFUN (PHS, PHS, 1) *FFUN (PHI, PHI, 1) *FFUN (PI2, PI2, 2) 290 ELSE 291 HZ121=-FFUN (PHS, PHS, 2) *FFUN (PHI, PHI, 2) *FFUN (PI2, PI2, 1 ) 292 ENDIF 293 HZ121=DEL*DEL*DEL*HZ121 294 AL1=1. 5*PI-PHI 295 AL2=1.5*PI+PHI 296 AL3=-1. 5*PI+PHS 297 AL4=-1. 5*PI-PHS 298 TT=FI (AL1, AL3, W) +S1*FI(AL1, AL4, W) 299 TT=TT+S1*FI (AL2,AL3, W) +S1*S2*FI (AL2,AL4, W) 300 TT1=-2. *PI*SQRT (W) *CJ*CJ4*TT 301 TT= (FI (0., AL1,W) +S2*FI (0., AL2,W) ) 302 TT=TT* (FI (0., AL3,W)+S1*FI (0.,AL4,W))

303 C TT=(O.,O.) 304 IF(M.NE.1)PRINT *,'ALS,TT1,TT:' ALl,AL2,AL3,AL4,TT1,TT 305 HZ121=HZ121*(TT1+TT) 306 RETURN 307 END 308 C ****************************** 309 SUBROUTINE TRIP2(HZ212,PHS,PHI,W,M,S1,S2) 310 C TRIPLE DIFFRACTION FROM MATERIAL TO RESISTIVE HALF PLANE 311 COMMON /BLK1/ETA1, ETAR, ETAS, ETAES, ETAMS, ETAMl, ETAM2, COMl, COM2 312 COMMON /BLK2/CJ,CJ4,P I, P I2 313 COMMON /REFL/RU,RL 314 COMPLEX ETAl, ETAS, ETAR, ETAES, ETAMS, ETAM1, ETAM2, COM1, COM2 315 COMPLEX CJ,CJ4,HZE,HZR,HZM1,HZM2,HZ212,RU,RL,RMM,RM1,RM2,RM 316 COMPLEX DEL, FFUN, FI, FIP, T1, T2, TT, FFCT, TT1, DEL1 317 DATA RT2/1.414213562/ 318 DEL=CEXP (-2. *PI*CJ*W) /SQRT (W) 319 HZ212=FFUN (PHS, PHS, 2) *FFUN(PHI, PHI,2) 320 HZ212=DEL*HZ212 321 TTl=HZ212*DEL*FFUN(1.5*PI, 1.5*PI, 1) 322 RM=. 5*RU*CEXP (-2. *PI*CJ*W) 323 RMM=(0., 0.) 324 RM1=(0., 0.) 325 C M=11 326 DO 100 N=3,M, 2 327 N1=N-3 328 RM1=CEXP (-CJ*Nl*2. *PI*W)/ (2.**N1) 329 N2=(N-3+.0001)/2. 330 RM2=(1., 0.) 331 DO 10 I=1,N2 332 10 RM2=RM2*RU*RL 333 100 RMM=RM1*RM2*RM+RMM 334 c PRINT *,'RMM:',RMM,.5*RL*CEXP(-2.*PI*CJ*W) 335 AL1=-.5*PI+PHI 336 AL2=-.5*PI-PHI 337 AL3=-. 5*PI+PHS 338 AL4=-. 5*PI-PHS 339 TT=FI (ALl, AL3, W)+Sl*FI(ALl,AL4,W) 340 TT=TT+S2*FI (AL2, AL3, W) +S1*S2*FI (AL2,AL4,W) 341 C PRINT *,'FIS IN 121:',TT,HZ212, (CJ4/(4.*PI))*(CJ4/(4.*PI)) 342 HZ212=HZ212*TT*RMM 343 TT=FI(0.,AL1,W)+S2*FI(0.,AL2,W) 344 TT=TT*(FIP(0.,AL3,W)*SEC(AL3/2.)+Sl*FIP(0.,AL4,W)*SEC(AL4/2.)) 345 C PRINT *,'AL,TT:',AL1,AL2,AL3,AL4,TT 346 TT=TT1*TT 347 HZ212=HZ212-TT 348 RETURN 349 END 350 C ************************* 351 SUBROUTINE QUAD(HZ4,PHS,PHI,W,M, S1, S2) 352 C QUADRUPLE DIFFRACTION FROM RESISTIVE/MATERIAL HALF PLANE 353 COMMON /BLK1/ETA1, ETAR, ETAS, ETAES, ETAMS, ETAMl, ETAM2, COMl, COM2 354 COMMON /BLK2/CJ,CJ4,PI, P I2 355 COMPLEX ETAl, ETAS, ETAR, ETAES, ETAMS, ETAM1, ETAM2, COM1, COM2 356 COMPLEX CJ,CJ4, HZE, HZR,HZ4,KPLUS7KPLUS8,KPLUSM 357 COMPLEX DEL,FFUN,FI,FI1,FIP,T1,T2,TT,FFCT,TT1,DEL1,IA 358 DATA RT2/1.414213562/ 359 DEL=CEXP (-6.*PI*CJ*W)/ (W*SQRT (W)) 360 KPLUS7=KPLUSM(CMPLX(PI2,0.),ETAR,0,1.) 361 HZR=CJ4*ETAR*KPLUS7*KPLUS7/(4.*PI) 362 HZ4=FFUN(PHS,1.5*PI,0)*FFUN(PHS,PHI,1)

363 HZ4=HZ4*HZR 364 HZ4=DEL*HZ4 365 DEL1=HZ4 366 AL4=-PI2-PHS 367 AL3=PHS-PI2 368 AL11=1.5*PI-PHI 369 AL12=1. 5*PI+PHI 370 IF(M.EQ.1) THEN 371 IF(PHS.GT.PI2)AL=-PI+.01 372 IF(PHS.LT.PI2)AL=-PI-. 01 373 ELSE 374 IF (PHI.GT.PI2) AL=-PI-. 01 375 IF (PHI. LT.PI2)AL=-PI+. 01 376 ENDIF 377 IA=FI (AL3, 0., W) +S1*FI (AL4,0.,W) 378 T2=-2. *PI*SQRT(W) *CJ*CJ4 379 T1=T2*IA 380 TTI (AL11, 0.,W) +S2*FI (AL12, 0.,W) 381 T12 *T1*TT 382 TT=FI(AL3,ALll, W)+S2*FI(AL3,AL12,W) 383 TT=TT+Sl*FI(AL4,ALll, W)+Sl*S2*FI(AL4,AL12,W) 384 T2=T2*T2*TT 385 HZ4=DEL1* (T1+T2) 386 TT=FI (AL3, 0.,W) +S1*FI (AL4, 0.,W) 387 T1-=FI (0., 0.,W) * (FI (0.,ALll,W) +S2*FI(0.,AL12,W)) 388 C PRINT *,'QUAD:',TT,T1 389 Tl=Tl+FI(AL, 0.,W)*(FI1(ALll, W)+S2*FI1(AL12,W)) 390 C PRINT *,'QUAD:',AL,AL11,AL12,AL3,AL4,T1 391 HZ4=HZ4+DEL1* (TT*T1) 392 RETURN 393 END 394 C 395 SUBROUTINE QUINT1 (HZ51,PHS,PHI,W,M, S1,S2) 396 C QUINTAPLE DIFFRACTION FROM RESISTIVE/MATERIAL HALF PLANE 397 COMMON /BLK1/ETA1, ETAR, ETAS, ETAES, ETAMS, ETAM1, ETAM2, COM1, COM2 398 COMMON /BLK2/CJ, CJ4, PI, PI2 399 COMPLEX ETAl, ETAS, ETAR, ETAES, ETAMS, ETAMl, ETAM2, COM1, COM2 400 COMPLEX CJ, CJ4, HZE,HZR,HZ51, KPLUS7, KPLUS8,KPLUSM 401 COMPLEX DEL,FFUN,FI,FF1,FIP,T1,T2,TT,FFCT,TT1,DEL1,IA,IB,IC,ID 402 DATA RT2/1.414213562/ 403 DEL=CEXP (-2. *PI*CJ*W) / (SQRT (W)) 404 C KPLUS7=KPLUSM(CMPLX(PI2, 0.), ETAR, 0, 1.) 405 C HZR=CJ4*ETAR*KPLUS7*KPLUS7/ (4.*PI) 406 Ql=l. 407 Q2=1. 408 Pl=l. 409 P2=1. 410 IF(M.EQ.1) THEN 411 HZ51=FFUN(PI2,1.5*PI,0)*FFUN(PI2,PHS, 1)*FFUN(PI2,PI2,2) 412 HZ51=-HZ51*FFUN(PI2,PHI, 1) 413 Ql=Sl 414 Q2=S2 415 ELSE 416 HZ51=FFUN(1.5*PI,.5*PI, 0)*FFUN(PI2, PHS, 2) *FFUN(PI2, PI2,1) 417 HZ51=-HZ51*FFUN(PI2,PHI,2) 418 P1=S1 419 P2=S2 420 ENDIF 421 DEL1=CJ4/(4.*PI) 422 HZ51=DEL*DEL*DEL*DEL*HZ51

423 AL1=1. 5*PI-PHI 424 AL2=1.5*PI+PHI 425 AL3=1.5*PI-PHS 426 AL4=1.5*PI+PHS 427 IF(M.EQ.1) THEN 428 IF(PHS.GT.PI2)AL=-PI+. 01 429 IF(PHS.LT.PI2)AL=-PI-.01 430 ELSE 431 IF (PHI.GT.PI2) AL=-PI-. 01 432 IF (PHI. LT.PI2) AL=-PI+. 01 433 ENDIF 434 IA=Pl*FI (0. AL3,W) +Ql*FI (0.,AL4,W) 435 T2=-2. *PI*SQRT (W) *CJ*CJ4 436 IB=FI(AL, 0.,W)*(P2*FI1(ALl,W)+Q2*FI1(AL2,W)) 437 IB=IB+FI(0.,0.,W)*(P2*FI(0.,AL1,W)+Q2*FI(0.,AL2,W)) 438 T1=T2*IA*IB 439 C PRINT *,'T1:1',TlIAIB,FI(0., 0.W) 440 IA=T2* (P2*Pl*FI (ALl, AL3, W) +P2*Ql*FI (AL1,AL4,W)+ 441 &Q2*Pl*FI (AL2,AL3,W)+Q2*Ql*FI (AL2,AL4,W)) 442 IB= (P2*FI (0., AL1, W) +Q2*FI (0., AL2, W) ) 443 IB=IB* (Pl*FI (0., AL3,W) +Ql*FI(0.,AL4, W)) 444 Tl=Tl+T2*T2* (IA+IB) 445 C PRINT *,'Tl:3:',T1,IA,IB 446 C HZ51=HZ51*T1 447 CC PRINT *,'HZ51:',HZ51 448 IA=FI(AL, 0.,W)*(Pl*FI1(AL3,W)+Ql*FI1(AL4,W)) 449 IB=FI(0., 0.,W)*(Pl*FI(AL3, 0,W) +Ql*FI (AL4, 0.,W)) 450 IC=FI (AL, 0.,W) * (Ql*FI1 (ALl,W) +Q2*FI1 (AL2, W) ) 451 ID=FI (0., 0.,W) * (Ql*FI (ALl, 0.,W)+Q2*FI (AL2, 0.,W)) 452 Tl=T+ (IA+IB)*(IC+ID) 453 C PRINT *,'T1:4:',TlIA,IB,IC,ID 454 HZ51=HZ51*T1 455 RETURN 456 END 457 C ************************* 458 REAL FUNCTION SEC(X) 459 SEC=./COS (X) 460 RETURN 461 END 462 C ************************** 463 COMPLEX FUNCTION FFUN(PHS,PHI,IC) 464 COMPLEX ETAl, ETAS, ETAR, ETAES, ETAMS, ETAM1, ETAM2, COM1, COM2 465 COMPLEX KPLUSM,KPLUS1,KPLUS2,KPLUS7,KPLUS8,KPLUSP 466 COMPLEX KPLUS3,KPLUS4,KPLUS5,KPLUS6 467 COMPLEX CJ,CJ4,HZE,HZR,HZM1,HZM2,HZM 468 COMMON /BLK1/ETA1, ETAR, ETAS, ETAES, ETAMS, ETAM1, ETAM2, COMl, COM2 469 COMMON /BLK2/CJ,CJ4,P I, P I2 470 DATA RT2/1.414213562/ 471 IF(IC.EQ.2)GO TO 110 472 KPLUS1=KPLUSM (CMPLX (PI2, 0. ), ETA1,0, 1. ) 473 KPLUS2=KPLUSM (CMPLX (PHI, 0.), ETA1, 0,1.) 474 KPLUS3=KPLUSM(CMPLX(PI2,0.),ETAM1,0,1.) 475 KPLUS4=KPLUSM(CMPLX(PHI, O.),ETAM1,0,1.) 476 KPLUS5=KPLUSM(CMPLX(PI2,0.),ETAM2,0,1.) 477 KPLUS6=KPLUSM(CMPLX(PHI, O.),ETAM2,0,1.) 478 SPHS=SIN(PI2) 479 SPHI=SIN(PHI) 480 HZE=CJ4*ETA1*KPLUSl*KPLUS2/(4. *PI) 481 KPLUSP=KPLUS 3 *KPLUS 4 *KPLUS 5 *KPLUS 6 482 HZM2=CJ4*ETAS*KPLUSP*COS (PHI) *COS(PI2) / (4.*PI*ETAES*SPHI*SPHS)

483 HZM1=-CJ4*ETAS*KPLUSP/(4. *PI*SPHI*SPHS*ETAMS) 484 HZM=HZM1+HZM2 485 C PRINT *,'IC:',IC 486 HZR=(1.,0.) 487 IF(IC.EQ. 0)THEN 488 KPLUS7=KPLUSM(CMPLX(PHS, 0.),,ETAR, O,1.) 489 KPLUS8=KPLUSM (CMPLX (P I2, 0. ), ETAR,, O 1.) 490 HZR=CJ4*ETAR*KPLUS7*KPLUS8/ (4.*PI) 491 ENDIF 492 FFUN= (HZE-HZM) *HZR 493 GO TO 100 494 110 KPLUS7=KPLUSM(CMPLX (PHS, 0.),ETAR, O,1.) 495 KPLUS8=KPLUSM (CMPLX(PI2,0.),ETAR, O, 1.) 496 FFUN=CJ4*ETAR*KPLUS7*KPLUS8/ (4. *PI) 497 100 RETURN 498 END 499 C 500 COMPLEX FUNCTION FI(AL1,AL2,W) 501 COMPLEX FIP 502 TEMP=1. / (COS(.5*AL1)*COS(.5*AL2)) 503 FI=TEMP*FIP (AL1,AL2,W) 504 C fi=temp 505 RETURN 506 END 507 C 508 COMPLEX FUNCTION FI1(ALl,W) 509 COMPLEX FFCT 510 DATA RT2,PI/1.414213562,3.141592654/ 511 TEMP=1. /COS (. 5*AL1 ) 512 A1=RT2/TEMP 513 Al=Al*A1 514 FI1=TEMP*FFCT (2. *PI*W*A1 ) 515 RETURN 516 END 517 C ****************************** 518 COMPLEX FUNCTION FIP(AL1,AL2,W) 519 COMPLEX FFCT, fipl 520 DATA RT2,PI/1.414213562,3.141592654/ 521 A11=RT2*COS(0.5*ALl) 522 A22=RT2*COS (0. 5*AL2) 523 Al=All*All 524 A2=A22*A22 525 IF(A1.EQ.A2) THEN 526 TEMP=2. *PI*W*A1 527 FIP1=FFCT(2.*PI*W*A1) 528 FIP=-(0.,1.)*TEMP*(FIPl-1.) 529 FIP=FIP+.5*FIP1 530 C PRINT *,'FIP:',FIP 531 RETURN 532 ENDIF 541 END 542