037821-1-T Z. Shen, M. A. Abdelmoneum J. L. Volakis Fast Hybrid PO-MM Analysis of Large AxiSymmetric Radomes Science and Applied Technologies 21050 Califa St. Woodland Hills, CA 91367-5103 April 1999 37821-1-T = RL-2515

PROJECT INFORMATION PROJECT TITLE: Dielectric Radome Simulation with Fast Algorithms REPORT TITLE: Fast Hybrid PO-MM Analysis of Large Axi-Symmetric Radomes U-M REPORT No.: 037821- 1-T CONTRACT START DATE: END DATE: September September 1998 1999 DATE: April 1999 SPONSOR: SPONSOR CONTRACT No.: U-M PRINCIPAL INVESTIGATOR: Dr. Odell Graham Science and Applied Technologies 21050 Califa St. Woodland Hills, CA 91367-5103 Phone: (818)887-0844 x107 Fax: (818)887-1447 P.O. SW01073 John L. Volakis EECS Dept. University of Michigan 1301 Beal Ave Ann Arbor, MI 48109-2122 Phone: (313) 764-0500 FAX: (313) 647-2106 volakis @umich.edu http://www-personal.engin.umich.edu/-volakis/ PROJECT PEOPLE: Z. Shen, M. A. Abdelmoneum and J.L. Volakis

Contents 1 Introduction 4 2 Near Field Computation and Decomposition 5 2.1 N ear-Zone Field.................................... 6 2.2 Coordinate Transformation.............................. 9 2.3 Modal Decomposition of the Incident Fields.................... 11 3 PO Approximation 12 4 Integral Equation Formulation 16 4.1 Surface Integral Equations..................................... 16 4.2 M oment M ethod Solution.............................. 19 4.3 Hybrid PO-MM Solution...................................... 22 4.4 Far-Zone Radiation Pattern............................. 23 5 Sample Results 25 6 Description of ABOR, a Computer Program for Antenna Radiation Through Radomes 30 6.1 Acquiring Data........................................... 30 6.2 Computing and Decomposing the Near-Zone Fields................ 31 6.3 PO Currents........................................... 31 6.4 Moment Method Solution...................................... 31 6.5 Main Program......................................... 32 6.6 Standard Auxiliary Subroutines........................... 32 7 Running the Program ABOR (ABOR Manual) 34 7.1 Sample Input Date Files...................................... 36 7.2 Sample Data File for Defining the Radome's Shape................... 37 7.3 Data File Storing the Aperture Field Distribution.................... 38 2

8 Appendix A: Von Karman Radome 40 9 Appendix B: Input Variables Description 41 10 Appendix C: Optimizing parameters for speeding up the Axi-Symmetric Body of Revolution (ABOR) code 43 3

1 Introduction This report details the development of a new hybrid physical optics-moment method (PO-MM) formulation for analysis of nose-radome antennas. The radome is assumed to be a body of revolution (BOR) and this is taken into consideration to reduce the overall problem to a twodimensional one by decomposing the antenna radiated fields into cylindrical modes used as the excitation on the dielectric radome. The primary reason for considering a hybrid PO-MM implementation stems from our interest to modal very large radomes which are possibly 100A long and 30 - 40A wide at the back of the radome where the antenna aperture resides (see Fig. 1). For this analysis, the radome shell is assumed uniform but layered configurations can be readily incorporated. In combining the PO and MM technique for evaluating the transmission properties of the radome, the MM is employed to rigorously model the tip of the radome (up to 3 - 5A long) and account for diffraction contributions. The PO method (i.e. ray optics) is employed to compute the transmitted fields away from the tip region where multiple scattering and tip diffraction effects are not pronounced. Higherorder ray interactions can, however, be included in the PO implementation to account for multi-bounce effects within the radome cavity. In combining the PO and MM, an equivalent current is computed on the radome surface based on the PO fields computed at the interior and exterior surfaces of the radome. These are subsequently used as excitation in the context of the MM along with the direct antenna fields to evaluate the equivalent MM surface currents. The final radiation pattern is then evaluated by integrating the electric and magnetic equivalent currents (both PO and MM currents) on the exterior surface of the radome. The report is organized as follows. Section 2 gives a description for evaluating the aperture radiation on the radome interior surface near-zone fields. These near-zone field components are then decomposed into cylindrical modes to subsequently perform the BOR moment method implementation. Section 3 introduces the physical optics (PO) approximation for both electric and magnetic surface currents on the flatter radome surfaces. The hybrid BOR moment method formulation is presented in Section 4. Here PO currents for the PO-region are incorporated 4

into the surface integral equation as effective sources. Numerical examples and validations (including a very large radome of 100 wavelengths) are given in Section 5. Finally, Sections 6 and 7 describe the Fortran program that implements the hybrid PO-MM BOR. Sample geometry runs are included as examples in using the hybrid code to be referred to herein as ABOR (Antenna-BOR). 2 Near Field Computation and Decomposition The geometry of the problem considered is illustrated in Figure 1. The radome is assumed to be axi-symmetric (body of revolution), but the curvature of the inner and outer surfaces can be completely arbitrary. The aperture antenna (reflector, horn, open waveguide etc.) is placed inside (as illustrated) and its pattern can be arbitrarily selected as well. In this development, a circular aperture antenna is assumed but this is an arbitrary selection. The governing equation and pictorial shape of a Von Karman radome are given in Appendix A. MM-region................................................................................................................................. Aperture Antenna Figure 1: Geometry of the problem considered. The circular aperture antenna is separately shown in Figure 2, with its corresponding coordinate system. 5

z P y x Figure 2: Circular aperture antenna 2.1 Near-Zone Field Before proceeding with either PO or MM analysis of the transmission through the radome it is necessary to first compute the antenna field on the inner surface of the radome. This computation must be carried out with as much accuracy as possible since it will serve as the excitation field for the MM or hybrid PO-MM solutions. The near-zone electromagnetic field can calculated from the following two equations: Ep =-jA - j V(V A)- -VxF (la) Hp = -jF - j- V(V F) +-V x A ( b) WIUE IL where A= - /4 / J(r() p(-j )ds (2a) /ds' (2a) M(') - ds' (2b) 6

and R = Ir- r'|. Here, vecJ(r') and M(r') represent the radiation currents on the antenna aperture or simply some externally provided sources. For reflector or waveguide-type aperture antennas, J(') = n' x Ha(l) M(r) = Ea(r') x n' where Ea(r') and Ha(r') are the aperture fields. To evaluate the integrals, we must carry out all differentiations first. We have VxF=-J f exP( kR) x M(') ds' V(V F) is /exp(-j kR) V./()ds' 47r a ) R and Vex exp(-jklr - rl) I|r- r'l p(-jk lI-r l) _ (1 + jkR) exp(-jkR) / |Ir-r' R3 -_(1 + jkR) exp(-jkR) (3 - k2R2 + j3kR) exp(-jkR) - R R3 R5 R = [(x - x')i + (y - y')y + (z - z')z]. Substituting these into (1) gives / -ijwl47w a k2R3 a.[~ + [f(r).](3 - k2R2 + j3kR) exp-jk)ds' k2R5 where + S [(R x M(r()]exp(-jkR)ds' J(') z2 x Ha(') = -Hay(x', y') + Hax(x', y') M( = E'a (r) x - Eay(x', y')x - Ea (x', y')y (3) If the "far-zone" relation Ha =-z x Ea TF is further employed on the aperture, we get J(r') = — Ea(x' y' - -Eay(x', y) = —Ea(r'),q 'q27 (4) (4') 7

and thus -0 jkff -f- k2R 2 -1 j kR 4 (3 - k 2]R2 + j3kR) ep-k~s 4IIa a( k2R3 + a~r k 2R?5 ikff(1~j kR) - +4U I~jR [R x Ea(' x 2] exp (-j'kR) ds' We can rewrite this in terms of spherical components by using transformations x= sin 0Ocos O~j -F Cos 0 cos q$0 - sin q$0 sin 0sin Obi + cos 0Osin qO~ + cos q0q zcos Oi~ - sin 00, to get Eo (x, y, z) f= Q0(X, Y, Z, v', y', z') ex(JRds, (5a) E0(,y, z) f= QOs(x, Y, Z, X', YI, Z exp(-3jkR) ds (Fib) J I~aR where Q0(X, y Z7 x', y' z') =[Eax(rf) cosq + Eay ') sinql] cos 0 [k2R 2- kR + x- x') Eax() + (Y - Y') Eay(T)[ (X - Xv') COS 0 COS~ - (Y - Y') COS 0Sinb- (Z -Z') Sin 0] (3 - k 2R2 +-j3kR) k2 R4 /I +jkR) QEa 5(xos0 Ey, z,* Sil, y', (z' [E () Csill +(7 F, r -yEay(r')co "[kR2-1 jkiR X')EaE(x') + (y - )()Kx-x0in —(- ~ -kR jIj kR) Ya kk Jy Rs JkR4 +[(x - xFH)ax(Tr) sin q3-F Eay(') csi 7$] (z - z') (1o-s jkR)- R '3 jI +kR) 8

r Za Xa system origin Xr system origin Figure 3: Coordinate transformation 2.2 Coordinate Transformation One last step is the transformation from the antenna coordinate system (xa7 Ya7 Za) to the radome system (X,, Yr, Zr). After having computed all the field components, we need consider the coordinate transformation as, shown in Figure 3. Referring to Figure 3, the appropriate transformation is Xr cos Q 0 -sin Q Xa 0 Yr = 0 1 0 ya + 0 (6a) Zr sin Q 0 cos Q Zas where s is shift distance between the two coordinate systems alone the radome axis. Alterna tively, the inverse transformation is Xa cos Q 0 sin Q Xr Ya 0 1 0 Yr Za -sin Q 0 cos Q Zr -s Also, note that ra x+Y zI = V r2 + + (Zr - s)2 ra - a + ya + Za /x + Y + (z ) qa = tan-1 Ya = tan- Yr Xa X, cos Q + (Zr - s) sin Q2 (6b) (7a) (7b) 9

a tan1 VX =tan- [Xr cos Q + (Zr -s)sinQ] + (7c) a, -a tan an (7c) Za -Xr sill 2 + (Zr - s) cos 2Q Because of the rotation angle Q shown in Figure 3, the vector field components will be different after the transformation. Specifically, the 0 and X unit vectors in the two different coordinate systems will change as follows: z Z Xa x 0) Figure 4: Unit vectors in two coordinate systems Oa = all8 + a'120 (8a) (8b) Oa - G21 + a22; where (x2 + y2) cos Q + xz sin Q /[(x cos Q + z sin Q)2 + y2] (2 + y2) x2 + y2 + z2 12 - 21 = ySin f [(x cos + z sin f)2 + y2](x2 + y2) 10

2.3 Modal Decomposition of the Incident Fields Having evaluated all field components in the radome coordinate system, we next proceed to decompose the antenna aperture field into cylindrical modes. Based on the well-known body of revolution (BOR), this decomposition is essential to taking advantage of the radome's geometry. Since the cylindrical mode has the same p-dependence for each M-angle and the same is true with the geometry, the resulting scattered/transmitted field will have the same as the incident mode. Thus, it suffices to only evaluate the currents/fields on a single cross-section of the radome. Therefore, modal decomposition reduces the 3D problem to an equivalent 2D problem. However, each mode must now be analyzed separately and resulting currents/fields need to be summed to compute the total field transmitted through the radome. At point P on radome's inside surface, the electric field components are given We begin by considering the original expression Ep - Epr(z, )f ) + Epo(z, 0)O + Epo(z, q) (9) at point P on the radome. By invoking mode orthogonality, each of these components is expanded as follows: N Epr(Z, O) E Epr,n(Z) exp(jno/) (10a) n=-N N Epo(z, )= E Epo,n(z)exp(jnt) (10b) n=-N N Epo(z, ) = E EpO,n(z) exp(jnr) (lOc) n=-N where 1 o27r Epr,n (Z) = Epr (z, ) exp(-jno)do 1 r2" EpO,n()= ( - Z Epo(z, () exp(-jn))dq) 1n J2 1 /27r Epn(z) = - Z Ep{(z, d)) exp(-jnrq)do. 27r IJ A similar decomposition can be carried out for the magnetic field components. The final expressions are the same as those with the replacement Es by H,. 11

3 PO Approximation This section describes the physical optics approximation for computing the field transmitted through radome. A key assumption in this approximation is that the radome is locally planar. The transmitted/reflected fields can then be computed using the place wave reflection/transmission coefficients for the thick dielectric slab. Based on the approximations for the reflected and transmitted fields, the equivalent electric and magnetic surface currents can then be obtained from JPo(r?) = n' x HPO(r) Ao(?) = E (r/) x W,. (lla) (11b) As illustrated in Figs. 5 and 6, the two incident wave polarizations must be computed separately. The reflection coefficients for the perpendicular and parallel polarizations are are Figure 5: Reflection and transmission at a dielectric interface: perpendicular polarization. RL cos Oi - V/er - sin2 (0i) cos Oi + /r - sin2 (Oi) R{/ = r cosOi - ^/r- sin2(0i) r COS Oi + \/er - sin2(i) and these refer to a simple dielectric interface. (12a) (12b) 12

Figure 6: Reflection and transmission at a dielectric interface: parallel polarization. For a finite thickness dielectric layer (as shown in Fig.7), we must consider multiple interactions. These can be accomplished for by the composite reflection and transmission coefficients, Transmitted Ray 0 i 0i Incident Ray Reflected Ray Figure 7: Reflection and transmission through a dielectric slab. R1(1 - PaPa) 1- R2P2Pa = (1 -R)Pd 1 - R2P2 (13a) (13b) 13

where -jkort Pd = exp( - ) P / c - sin2 (0i) P = exp( j2kotsin2(i)) refer to the propagation delays through the layer of thickness t. Contributions from multi-bounces within the radome can be evaluated using ray-tracing. Referring to Fig. 8, let us consider the incident ray direction \k I' r X Figure 8: First-order PO solution i - XpX + yp- + ZpZ i v/- + + z + (4 -V/ p (14) where Xp - pp cos(Op), yp = pp sin(op), pp p(zp). To obtain the reflection angle, we note that the tangential and normal unit vectors at P are given by pi(zvp) ( cos Op + y sin Op) + z T- /[p(p)]2 + 1 (15a) 14

and p cos 5p + y sin Op - p' (zp)z n =(15b) [Pi4(Zp)]2 + 1 Thus, Snell's reflection angle is given by i, = cos-'(ki rip) (16) and the z distance between the entry and exit point of the layer is found from sin i 1 Pi(Zp) q Z sin2 /[(p)]2 + 1 / [i(zp)]2 -+ (17) Also, the angle a of the reflected ray with respect to the horizontal is given cp= 20 + p - (18) where Pe(Z) - Pi(Z) Op -= tan-(PP) P [pi(Zp)]2+ p p The following table summarizes the incident, reflected, and transmitted ray fields at an arbitrary point P on the interior surface of the radome. In the table, Epo(Oa, Oa) and Epo(0aa, a) Ray Perpendicular polarization Parallel polarization Incident ray E_ = Ep (Oa, >a) E/ =Epo(Oa, ~a) Reflected ray El = FrEp(Oa, a) E/ = r//Epo(Oa, Oa) Transmitted ray Et = TIEp(Oa, Da) E/ T//Epo(Oa, (a) are given in (9) and (10). The PO currents JPf and Mf~ are evaluated as JP = ni x (Hi + Hr) (19a) M = Ei + Er) X n1 (19b) J2O = n x Ht (20a) MP E = t x2 (20b) where h1 and n2 are illustrated in the following figure. 15

JX n,1r J2 n1 4 Integral Equation Formulation This section develops the combined field integral equation formulation for evaluating the radiation from an antenna enclosed in a dielectric radome. Figure 9 shows the geometry of the problem being considered. In proceeding to the formulation, we identify regions 1, 2, 3, and boundary surfaces S1 and S2, as shown in Figure 9. The dielectric radome is denoted as Region 2 and the exterior free space is denoted as Region 3. The antenna is in Region 1 (interior of the radome) and is illuminating the boundary surface S1, i.e. the interior surface of the radome. The equivalent electric and magnetic surface currents on the interior surface of the radome will be denoted as J1 and M1. Similarly, the equivalent electric and magnetic surface currents over the outside surface of the radome will be denoted as J2 and M2. 4.1 Surface Integral Equations Based on the usual moment method procedure for multi layer dielectrics [1-8] for axisymmetric structures, we begin by introducing equivalent currents on each cross-sectional boundary separating the three regions of interest. Using these equivalent currents, the field in each region can be expressed as follows. 16

Region 3 - / '@ F::.2 Bi MM-region / 1 /ncident 52 PO-region / /lncllentI - Fields J2 Region Aperture Antenna Figure 9: Geometry of the problem considered. Region 1 (interior of radome): 0(E = Ein - L1J1 (i) + KM1 1(F) (21a) 0(6 H1 = ]inc - K1J7l(f) - L2M2(f/) (21b) where (rf) is the Heaviside function and is defined as 1; for r' R2 0( = 1/2; for r' S (22) 0; otherwise. Here, the electric and magnetic surface currents J1 and M1 are related to fields on J1 = n1 X Hills, M 1- Ells x n1 and introduced on the interior surface of the radome. The integro-differential operator Li and Ki (i = 1, 2, 3) are defined as [1] L,X() = jW j [X2() + ~ VV' X(r) G(k\lr- rl)ds' (23a) 17 17

and KiX(r = - X (r') x VG(kilr- r'l)ds' (23b) where the Green's function is given by G(ki] r-r';) - exp(-jki — r- ) (24) G(kilr-'- r^ (24) 47r1~- r ' and k2 = w2/iEi. Similarly, for the other regions, the fields are given by Region 2: 0(rEL2 = L21(r) - K2M1 () + L2J(f) - K2M2() (25a) 0(f)H2 = K2J1( ) + -2L2M1(f) + K2J2(f) + 2L2M2() (25b) 72 112 Region 3: 0(=)E3 =-L3J2(i) -+ K3M2( f) (26a) 0(f)H3= -K3J2(')- T2L3M2(i) (26b) '73 where the electric and magnetic surface currents J2 and M2 are J2 =- 2 x H3S, - M2 = E31S X P2 and reside on the exterior radome surface. Employing the above equations and enforcing the boundary conditions of the total tangential electric and magnetic field continuity across the surface, we obtain the combined field integral equations: (L1 + L2)J1 (') - (K1 + K2)M1(') + L2J2(r) - K2M2(o) = Ec, on S1 (27a) 1 1 -, 1 (K1 + K2)Jl(r-')+ ( L1 + -L2)M1()+ K2J2( F')+ 2L2M2(') =Hi, on S1 (27b) + 1 172 72 L2Ji (r) - K2MI (f) + (L2 + L3)J2(f) - (K2 + K3)M2(r) = 0, on S2 (27c) 1 - - 1 1 K2J1 (') + 2-L2Mi (r) + (K2 + K3)J2(F) + ( 2L2 + -2L3)M(F) = 0, on S2. (27d) 2 whe2 rrr where EJC and H"c1 are the incident fields from the aperture antenna, as given in (9). 18

4.2 Moment Method Solution The integral equations (27a)-(27d) can be solved by the method of moments for the surface equivalent electric and magnetic currents Ji, MI, J2, and M2. To do so, we proceed to discretize the unknown electric and magnetic currents into a finite series of basis functions spanning the surfaces S1 and S2 as follows: J() = (a,nkJk - a,nkJk) n,k M1 () =o E(bnkJnk -bn kJk) n,k f2(2 = Z(a,nk - a2nkJk) n,k M2 (r) = 2,nk ( nk 2,n - Jnk) n,k (28a) (28b) (28c) (28d) where Tk n- = Ua - exp(jn) and Tk is the triangle function spanning the k-th annulus with four segments, as shown in Fig. 10. t t t t t ' 2i- 1 2i " 2i+ 1 ' 2i+2 Figure 10: Triangle function and four impulse approximation. 19

Substituting the above basis functions and using Galerkin method (testing function Wk = Jn), we can derive the following matrix equation. L1 + L2 -(K1 + K2) L2 -K2 ai Einc K1 K 2 + 2 KL2 L2 linc K +K2 K2 bi H 1 02 172 (29) L2 -K2 (L2 zL3) -(K2+ K3) a2 0 K2 L K2+K3 + b2 0 K2 22 73 - The L operator has the form of Ltt Lt 1 (30) Lft L* where 4 Lk = [wjTpTq (sin vp sin vqGcp n + co p cos vqGn) - TpTqGn (31a) p,q= -1 4 Lk - [ — TpTq sin vqGsi + - TpTqGn (31b) p,q= L WEP 4 Lkl = - [/TpTq sin vpGsn + n TqTqGn (31c) p,q=l L WEPP 4 2 L TpTq jw LGcn + '-.n ] (31d) pql 3WPpPq J p,q -l with Jo Gn- / tpAtq g($) cos(nc )d$ /r Gcn -= tpAtq gj (q) cos(nr) cos qbdq Gn = AtpAtq j g((q) cos(nq) sin Oqdq and g() exp(-jkRpq) Rpq V(Pp - Pq)2 + (Zp Z- q)2 + 2pppq(1 -COS ), if p q Rpq - (t/4)2+2pP(1 -cos),ifpq j= ((Atp/4)2 + 2pp(1 - cos q)), i p-= q 20

The K operator has the form of Ktt Kt (32) Not Koo where 4 K3 = -jro TpTqHsn [sin Vp sin Vq(zp - q) + pq sin p cos Vq - p cos Vq sin vq] (33a) p,q=l 4 Kk= -o 70 TpTq [(sin q(Zp - zq) + pq COS q)Hcn - P COS vqHn] (33b) p,q=l 4 Kk = -o y TpTq [(- sin Vp(zp - Zq) + pp cos Vp)Hcn - Pq cos VpHn] (33c) p,q=l 4 k = k j (Zp - Zq)T) TTqHsn (33d) p,q=l with Hn -= Atp\Atq j h() cos(no)do r7 Hcn -= tpAtq f h(b) cos(nq) cos Oqdq$ Jo r Hn - AtpAtq j h(o) cos(no) sin Odqo and (1 + jkRpq) exp(-jkRpq) h(-) 3 Pq The right-hand side of the matrix equation (29) can be written as follows. E nC = 2 Tk(t) Etn dt (34a) t,nk k-thsegment Ek = 27Tf Tk(t)E cdt (34b) -thsegment Htnrk = Tk(t)Hn dt (35a) -thsegment H,,k = 27r0o Tk(t)H~n dt. (35b) -thsegment Solution of the matrix equation (29) gives the surface equivalent electric and magnetic currents on the radome surface. This moment method solution is accurate and suitable for relatively 21

small radomes. For electrically very large radomes (such as 100 wavelength long radome), the MM solution is a formidable task. In order to alleviate this difficulty, next subsection introduces a new hybrid PO-MM technique for the analysis of very radomes. 4.3 Hybrid PO-MM Solution As shown in Fig. 9, the entire radome surface is divided into two parts: MM-region and POregion. The moment method described in the previous subsection is applied to the MM-region, and the PO-region is modeled by the physical optics approximation introduced in Section 3. The corresponding equivalent surface currents are denoted by J1MM, f1IMM, Jf2M,.fM" and JPO, IPo, JPO, Mf2 for the MM-region and PO-region, respectively. Having invoked the PO approximation presented in Section 3, we obtain the PO currents J1~, MlPO, JPo and M2P~ These known currents are then incorporated into the matrix equation (29) in the following manner. LMm + LMN -(KMM+ KIM) LM -KMM a 1 K__M + KM L M L2| 1 2 2 2 KM LM KMM M LMM b LMMM ^l 1 2 2 2 LM -KiM (L M + LIM) -(KMM + KMM) aM K2 M + K3M LMM L L bMM Enc - (LMP + LP )afP + (K1MP + K2P)bPO - LmPa2 + KMPbPO H n - (K '+ - K2P - ( )+ )bP0 - KMPaPO - 2 bPO 2 1 ~ 2 2 2(36) -LMPaf0P + K~mPbP0 - (LmP + L3m'P)af2 + (K2MP + KmP)bfP -KMPapo - L O - (KMP + KMP)aPO L- L+ bPO 2 1 1 -2 2 2 ( 2 —)b2 where the operators L.Ik and Ki (i = 1, 2, 3 for regions 1,2,3, respectively, and j = M, P for MM-region and PO-region, respectively) are defined in (30) and (32) for different parts of the boundary and different associated regions. are coefficients aP~, bf~, aO~, and bP~ are related to PO currents P, M~, JfP and 2~ on af~ = J~p, (37a) bP = M~ pi (37b) i I A 22

-PO jPOp b2 = M2 Pe bo= M2p0Pe. (37c) (37d) Here pi and Pe are the radii of points on the interior and exterior surface of the radome, respectively. Equation (36) is the hybrid PO-MM matrix equation and can be solved for the JM ac j iM M M 2 - I -E bmm M b' MM currents JMM =, MMM = -- JMM = and MM = 2 Having MM Pi Pe Pe currents J2M and M2AM and using the PO currents J2~ and M2~ on the exterior surface of the radome, we can then proceed to compute the far-zone radiation pattern of the radomeenclosed antenna. 4.4 Far-Zone Radiation Pattern Once J2 and M2 (from MM and PO) are found, the computation of the far zone field is obtained from the radiating integral following standard steps. For the far-region radiated field, we have -D 2/2 0 D2/2 x Figure 11: Calculation of far-region radiation pattern. E = kexp kr) j ( x [k x x E~(,r)] +r x [n x E(r')]) exp(jkor. t)dS' (38) 4-Fr' 23

where rTHs = ks x E, is understood. The 0 and f components can be decomposed from this to get E - x j = jL j ) x [ksh x x Es()]o - r x [n x E,(r )]) exp(jko.r')dS' (39a) 47rr E- -jk exp(-jkr) LFo 2)7 ( \ = jkexp( r x [kr x E (H)] [ + P x [ii x Es(rj)] ) exp(jko.r')dS' (39b) where the integrands can be explicitly expanded to give n x [k, x Es()] - r X [n X Es(E)]|l E[p'z') (cos 0 + cos 0') - sin 0'] cos(f - E') - Eso sin 0 V1 + [P'(zs)] ] Es,[p' (z) - sin 0' cos + P' (z) cos 0 cos 0'] sin(f - ') / + [P'e (zs)] and x [ x x [k En x Es(rl)]lo Eso[-p'e(') + cos 0 sin e' - p'(z') cos 0 cos 0'] sinQ(O - ') /1 + ['e (zs)]2 Es [p (z) (cos0 + cos 0') - sin 0'] cos( - ') - sin 0 + v1 + [P'e(s)] The double integral in (39) can be reduced to a single integral by introducing the modal expansion N Es = -E(z s, ') = 3 E. (z[) exp(jn4') (40) n=-N and the the identities r r' = Z cos 0 + ps sin 0 cos(& - q') j2 exp(jkr r')d' exp(jckz cos 0 + jnc)2rjnJ,(kps sin 0) r27 2 exp(jkr ) cos(q - ')dq' = exp(jkzs cos 0 + jno)27rj-1J'n(kps sin 0) exp(jkr r') sin( - q')do' = - exp(jkzs cos 0 + jnf)2nrj n s kp, sin 0 The final expressions for the far-zone radiated E-field are then given by 'k -jexp (-j'kr)2~ i inslp (z')dz' /- ~ \ E -jk exp(-jkr) >iE j i pe(Z[)dz (EsoI1l + EsoI2n + EsI3a) exp(jkz, cos 0 + jn) (41a) 24

E, -kexp(-jkr) [ i e(Zs)Z (EsIln + Esq32n - Es~I3n) exp(jkz8 cos0 + jnq) (41b) where Iin = [p' (z'')(cos0 + cos 0') - sin 0'] J'[kpe(z') sin 0] I2 -j sin OJn [kp (z) sin 0] 3 [P' (z) - sin 0' cos 0 + p' (z/ ) cos 0 cos 0'] n kp () sin 0] k p, (z~) sin 0 and the well-known integral identity r27r exp(jz cos 0 + jnO)dO 27rn Jn (z) was employed in deriving (41). 5 Sample Results In this section, we present numerical results for three illustrative examples. The first example is a sanity-check of the developed program. By setting the dielectric constant of the radome material to 1, the computed radiation pattern is compared to that directly calculated by integrating the equivalent surface currents over the antenna's aperture. In this example, the lengths of the interior and exterior Von Karman radome surfaces are respectively L1 = lA0o L2 = 10.2Ao. The interior and exterior diameters at the radome base are assumed to be D1 = 5Ao and D2 = 5.2o0, respectively. The distance s and steering angle Q are all set to zero. For the antenna, a circular aperture is used carrying a uniform y-polarized E-field. As shown in Figure 12, the radiation pattern of the aperture antenna through a transparent radome is in very good agreement with that from direct integration (no radome). The moment method solution is used as reference for the hybrid PO-MM and as seen the two methods are also in good agreement. The second example shows the computed radiation pattern for the same aperture through the same radome shape and size except that the dielectric constant of the radome material is now Er = 2.0. In Figure 13, we show the radiation patterns for the three different methods. The pure MM solution refers to the case when the moment method is applied everywhere on the radome's surfaces (no PO approximation is invoked); the pure PO solution refers to the 25

case where almost the entire surface is modeled by the physical optics currents (except for a very small region around the tip). The hybrid PO-MM solution represents the results obtained by using MM formulation up to 3A from the tip and the rest contour currents replaced by the PO currents. Again, we observe that the hybrid solution agrees fairly well with the pure MM solution. However, the pure PO solution is substantially in error in the side lobe region. In fact, the PO result shows peaks where the MM gives nulls. The CPU time needed by these three methods for this relatively small problem is listed in the following table. Method CPU time (in seconds) Pure PO 133 Pure MM 224 Hybrid PO-MM 148 The third example is that of radiation through a very large radome. The Von Karman radome is now assumed to be of length L1 = 100Al (inner), L2 = 102Ao (outer); the diameters at the radome base are D1 = 30A0 and D2 = 34A0, respectively; and r, = 4.0 for the radome material. The aperture antenna is of radius 2Ao and a uniform y-directed field distribution is again assumed. The computed radiation patterns by the hybrid PO-MM and the pure PO are shown in Figure 14. No pure MM solution is shown due to the excessive computer resources needed to carry out this analysis. For the hybrid PO-MM solution, the MM arc was 10 long from the tip and the rest was assigned to the PO-region. It should be mentioned that the CPU time for the hybrid PO-MM is about 5.5 hours. 26

L7J u qiWrnoiij4 9jn4ide uv Jo u-1044ud uoiwiepe-i aq aoj s~qnsai p~4ndtuoo jo uosixedmoD:Z{ 0.0i Radiation Pattern IBI 2(dB) I (-.001 I (-Al) I I OC 6N, C:) 1 4z CD q'Ic? C'D CD CZ) C C =i ar" =.. 4: "- 4 - ) 0 ), CD 14=.-4 00

I I OC 00o - - - - - - - -- - '-S: O~lO OCr CNC3 CIAI (HP l Ujold UOiwipUFigure 13: Radiation pattern of a radome-enclosed aperture antenna (L2 1O.2A, L1 10A, D2 = 5.2A, DI 5A, er - 2). 28

I: o; I - -0 (up) 13I UJOlcWd UOtlipeNp Figure 14: Radiation pattern of an aperture antenna through a very large radome (L2 = 102A, LI = 100A, D2= 34A, D1 = 30A, Cr = 4). 29

6 Description of ABOR, a Computer Program for Antenna Radiation Through Radomes Our computer program for evaluating antenna radiation through radomes is called "ABOR" and can be divided into six sections. The first section of the program needs the user-provided parameters,including radome's length, diameter, shape and dielectric constant, aperture size as well as the location and orientation and field distribution on the aperture surface. The second section of the program calculates the near-zone radiated field from the antenna aperture on the insider surface of the radome. This section of the program can of course be changed by different antenna apertures. The third section of the program determines the PO equivalent surface currents on both the interior and exterior radome surfaces. Another section of the program carries outht l the moment method solution for thehybrid implementation and uses the antenna radiated fields and the PO currents on the radome's surfaces as excitation. The fifth program section combines all available information to calculate the far-region radiation pattern through the dielectric radome. The latter section of the program is a collection of standard subroutines, such as LU decomposition, spline interpolations for curved arcs and surfaces. The following subsections give all the six sections of the program mentioned above. 6.1 Acquiring Data The subroutines that fulfill the task of acquiring the input data are: * radome inpu Subroutine radome-inpu gets the information pertaining the radome shape, radome length and bottom diameter, dielectric constant or. * apertureinpu Subroutine aperture inpu acquires the information for the aperture antenna, including the radius rap of the circular aperture, files storing the field distribution on the aperture surface, distance s, and steering angle Q. 30

* input(mode, ang, angl, ang2, nang, npir, npor, nplo, np2o) This subroutine reads-in the following parameters: mode, ang, angl, ang2, nang, npir, npor, nplo, np2o, bk, and 1mm. These variables are explained in Appendix B. * DBORIN This subroutine calculates the required parameters such as information for the discretization of the radome arc and the basis functions based on the entered parameters. 6.2 Computing and Decomposing the Near-Zone Fields * subroutine nfint(r,phi,xa,ya,za, sph, cph,sth, cth, efr, eft, efp,hfr,hft,hfp) Subroutine nfint defines the functions for all six near-zone field components (three electric components and three magnetic field components) at a given point (r, phi) for a known aperture field distribution. * subroutine nearfd(nip, zp, rop, some, come,fepr,fepa,fepe,fhpr,fhpa,fhpe) This subroutine calculates the near-zone electric and magnetic fields. * subroutine ecomp(mode,npir) Subroutine ecomp decomposes all six field components into modes and stores all the computed results into six two-dimensional array, which are called later for computing the PO fields (interior and exterior of the radome) and the MM excitation fields. 6.3 PO Currents Subroutine pof(n, mode, npir, npor,nplo,np2o,cfv, cfi, cfo) calculates the PO equivalent surface currents on the interior and exterior surfaces of the radome, for a given mode. 6.4 Moment Method Solution * SUBRO UTINE LOP Subroutine LOP(Z, ISYM, N, C, MU, EPS, NPI, RSI, ZSI, DSI, SVI, CVI, TI, TPI, 31

NPJ, RSJ, ZSJ, DSJ, SVJ, CVJ, TJ, TPJ) calculates the L operator for a given boundary and a specified region. * SUBRO UTINE KOP Subroutine KOP(Z, ISYM, N, C, MU, EPS, NPI, RSI, ZSI, DSI, SVI, CVI, TI, NPJ, RSJ, ZSJ, DSJ, SVJ, CVJ, TJ) calculates the K operator for a given boundary and a specified region. * SUBROUTINE CZGEN(N,Z,Z1) Subroutine CZGEN assembles the computed L and K operators into a matrix, which is then factorized into two triangular matrices (LU decomposition). * SUBROUTINE RCRGEN(N,NPIR,NPOR,NPIO,NP20, CFV, CFI, CFO, CBR) Subroutine RCRGEN generates the right-hand side of the hybrid matrix equations. This subroutine also calls another one named LKOP(N, C, MU, EPS, NPI, RSI, ZSI, DSI, SVI, CVI, TI, TPI, NPJ, RSJ, ZSJ, DSJ, SVJ, CVJ, TJ, TPJ, QLTT, QLTP, QLPT, QLPP, QKTT, QKTP, QKPT, QKPP) for the L and K operators of the PO part. 6.5 Main Program The main program utilizes all available information and control the execution. The far-zone radiation pattern is computed at the end of the main program by calling subroutine RCSPAT(N,NPOR, TH,RBT,RBP). Matrix dimension check is also done in the main program. It should be mentioned that after gathering all entered information, the main program calls a subroutine named RECOVE(NPIR,MODE,ANG, ANG1,ANG2,NANG, rsum) to calculate the radiation pattern directly from the field distribution on the aperture antenna. 6.6 Standard Auxiliary Subroutines The following standard auxiliary subroutines are used in the program 'ABOR'. * SUBROUTINE GAUSS(WT,ASC,N,AA,BB) Subroutine GA USS calculates the weights and integration points of coordinates for performing Gaussian quadrature of a given order. 32

* SUBROUTINE BESJ(X,N,BJM,BJ, BJP) Subroutine BESJ computes the J, Jn+', and J,+1 Bessel functions for a given argument x and order rn. * SUBROUTINE CGECO(A,LDA,N,IPVT,RCOND,Z) This standard subroutine from LINPACK does the LU decomposition for a complex matrix A. Some other similar subroutines such as SUBROUTINE CGEFA(A, LDA, N, IPVT, INFO) and CGESL(A, LDA, N, IPVT, B, JOB) are also needed to solve a linear matrix equation. subroutine curvl (n,x,y, sip l,slpn, islpsw, yp, temp, sigma, ierr) Subroutine crvl determines the parameters necessary to compute an interpolatory spline under tension through a sequence of functional values. Another function curv2 (t,n,x,y,yp,sigma) is used to interpolates a curve at a given point using a spline under tension. * subroutine surfl Subroutine surfl(m, n, x, y, z, iz, zxl, zxmn, y, z y zn, zxyll, zxyml, zxyln, zxymn, islpsw, zp, temp, sigma, ierr) determines the parameters necessary to compute an interpolatory surface passing through a rectangular grid of functional values. A follow-up function surf2(xx, yy, m, n, x, y, z, iz, zp, sigma) interpolates a surface at a given coordinate pair using a bi-spline under tension. * Functions bessjd(n,x) and BESSJ(N,X) These two functions calculate the Bessel Jn(xc) and its derivative J(x) using double precision. 33

7 Running the Program ABOR (ABOR Manual) The program ABOR consists of two parts: ABOR.f and ABOR-SUBS.f. On unix, one can use the command f77 -o ABOR ABOR.f ABOR-SUBS.f -O for compiling, or just simply type make to generate the executable file "ABOR". A detailed description of the input data required to execute ABOR is given below. The definition of each input variable is also provided. * read(*,*) BK -- Input the free-space wave number (2w7/A) * read(*,*) MODE -- Read number of modes considered. The recommended number of modes is the closest integer to kpmax + 1. * read(*,*)IRS - Specify the radome shape IRS = 1, for Von Karman Radome IRS = 2, for Radome shape defined by the user as data files. * Read(*,*) LI, D1, L2, D2, and s if IRS = 1 - Give values of Von Karman radome's interior length, interior diameter, exterior length, exterior diameter, and shift s along the z-axis. * Read(*,*) radin.dat if IRS = 2 -- Input the file name storing the interior curve of the dielectric radome (see 7.2 for definition of format). * Read(*,*) radout.dat if IRS = 2 - Input the file name storing the exterior curve of the dielectric radome (see 7.2 for definition of format). 34

* Read(*,*) real(Er), imag(Er) -- Read the real and imaginary parts of the complex dielectric constant of the radome material. * Read(*,*) Lmm - Read the length along the z-axis defining the radome arc over which moment method will be performed. * Read(*,*) Phi _ ---_-__ — Define the X angle (in degrees) at which the radiation pattern will be computed. * Read(*,*) Angl, Ang2, Nang -— _- - _ — Input the starting angle and stop angle (in degrees) of 0 and the number of sampling points between Ang1 and Ang2. * Read(*,*) Omega -- Define the angle Q for antenna's rotated direction. * Read(*,*) Rapa, Rsubr, Nrad, Nphi -- Read the radius of the circular aperture antenna, radius of the circular sub-reflector, number of sampling points in the radial direction, number of sampling points in the azimuthal direction. * Read(*,*) Ex.dat ---- Input the name of the data file for reading the x component of electric field on the aperture surface (see section 7.3 for definition of data file format). * Read(*,*) Ey.dat -- Input the name of the data file for reading the y component of electric field on the aperture surface (see section 7.3 for definition of data file format). To familiarize the user with the input data file, two sample data files are given in Subsection 7.1. Subsection 7.2 demonstrate the structure of the data file defining the radome shape. Subsection 7.3 details the structure of the data files storing the aperture E-field distribution. 35

7.1 Sample Input Date Files This appendix gives two examples for the input data file. The first is for a Von Karman radome, and the other is for a radome specified by the user. Example One: Von Karman Radome 6.2832 5 1 15.0, 5.0, 16.0, 6.0, 0.0 2.2, 0.0 3.0 0.0 0., 90., 91 0.0 2., 20, 20 Ex.dat Ey.dat Example Two: User-Defined Radome 6.2832 5, 0.5 2 radin.dat radou.dat 2.2, 0.0 3.0 0.0 0., 90., 91 0.0 2., 20, 20 Ex.dat 36

Ey.dat 7.2 Sample Data File for Defining the Radome's Shape This subsection gives a sample data file for defining the radome's interior and exterior surface. Both interior and exterior use the same data structure. The number of sampling points will be determined by the program from this input data file. Sampling points can be non-uniformly distributed over the radome arc. z p(Z).0000000 1.5000000.5000000 1.4859140 1.0000000 1.4604454 1.5000000 1.4277130 2.0000000 1.3891178 2.5000000 1.3454080 3.0000000 1.2970309 3.5000000 1.2442598 4.0000000 1.1872478 4.5000000 1.1260552 5.0000000 1.0606601 5.5000000.9909589 6.0000000.9167565 6.5000000.8377454 7.0000000.7534657 7.5000000.6632326 8.0000000.5659961 8.5000000.4600385 9.0000000.3421973 9.5000000.2050837 10.0000000.0000000 37

7.3 Data File Storing the Aperture Field Distribution This subsection provides a sample data file for storing values of the electric field components on the aperture surface. Both x- and y-directed components have the same data structure and the same sampling points. For the example considered here, we assume Nrad = 4 and Nphi = 6. The location of these sampling points is shown in Fig. 15. (1.0, 0.0) (1.0, 0.0) (1.0, 0.0) (1.0, 0.0) (1.0, 0.0) (1.0, 0.0) (1.0, 0.0) (1.0, 0.0) (1.0, 0.0) (1.0, 0.0) (1.0, 0.0) (1.0, 0.0) (1.0, 0.0) (1.0, 0.0) (1.0, 0.0) (1.0, 0.0) (1.0, 0.0) (1.0, 0.0) (1.0, 0.0) (1.0, 0.0) (1.0, 0.0) (1.0, 0.0) (1.0, 0.0) (1.0, 0.0) 38

Here Nrad=4, Nphi=8 Reflector covers the shaded area 5 Data file format List the value Ex in Ex.dat fi Ex at phi = 0 Ex at phi = 45 deg Ex at phi = 90 deg Ex at phi =135 deg Ex at phi = 180 deg Ex at phi = 225 deg Ex at phi = 270 deg Ex at phi = 315 deg Ex at phi = 360 deg 8 7,, 7 le for the inner circle Extra data circle outside the reflector needed for derivative (set data equal to those of the next interior circle or as computed) Figure 15: Von Karman Radome 39

8 Appendix A: Von Karman Radome The governing equation for the Von Karman radome is p(z) =- [ - 0.5 sin(21)]1/2 2JVwhere = cos-'( - 1). L Differentiating (Al) gives -D sin P ( ) rL ^-0.5 sin(2) A typical Von Karman radome is shown in Figure 16. f, Z....;' S A..... 4!-il ^^ ^^^^ (Al) (A2) /6;/ / / / i /:.-D, /2 -D I /2 Figure 16: Von arman Radome Figure 16: Von Karman Radome D /2 D2 /2 X 40

9 Appendix B: Input Variables Description ANG Fixed q angle (degrees). ANG1 Starting 0 angle (degrees). ANG2 Stop 0 angle (degrees). BK Wave number (27r/Ao for the problem (meters-1). DI Diameter of the interior bottom surface. D2 Diameter of the exterior bottom surface. Er Dielectric constant of the radome material. Ex.dat File name that stores the x-component of E-field over antenna aperture. Ey.dat File name that stores the y-component of E-field over antenna aperture. IRS Index for radome shape. IRS==1 for Von Karman radome; IRS=2 for radome shape defined by the user as data files. LI Length of the interior radome surface. L2 Length of the exterior radome surface. Mode Number of modes considered. Maximum number can be set to kpmax + 1. NANG Number of varied 0 angles. NPIR Number of sampling points on radome's interior surface. NPOR Number of sampling points on radome's exterior surface. NP10 Number of Sampling points for the PO-part on the interior surface. NP20 Number of Sampling points for the PO-part on the exterior surface. 41

Nphi Number of sampling points in the azimuthal direction for aperture field distribution. Nrad Number of sampling points in the radial direction for aperture field distribution. radin.dat File name storing the interior curve of the radome (IRS=2). radou.dat File name storing the exterior curve of the radome (IRS=2). Rapa Radius of the circular aperture antenna. Rsubr Radius of the circular sub-reflector. s Distance from radome's bottom center to antenna's location (meters). 2 Steering angle of the antenna (degrees). 42

10 Appendix C: Optimizing parameters for speeding up the Axi-Symmetric Body of Revolution (ABOR) code Introduction The Axi-Symmetric Body of Revolution (ABOR) code used the new hybrid physical optics-moment method for analysing nose-radome antennas. The code demonstrated excellent accuracy and speed for radomes of large length as 100A with an aperture antenna of radius of 2A. Motivated by the trials to run the code for such large radomes with large apertures of 23Aradius extensive work were done to optimize the code so that accurate results are produced in a reasonable amount of time. The purpose of this report is to present the different optimizing parameters for the code and show how can these parameters be used for different kinds of problems. In the first part of this appendix the blockage effect of the antenna subreflector is introduced. In the second part the parameters governing the speed and the accuracy of the code are investigated. We spent quite a lot of time in investigating these parameters. We ran diffirent radome geometries combined with different aperture andblockage sizes. We have concluded the optimum parameters settings for a large radome with a large aperture. Aperture Blockage The antenna subreflector blockage was implemented in the code by altering the integration interval over the aperture. Instead of integrating from 0 to rap, the integration now is now modified to be from rsubr (the sub-reflector radius) to rap (the aperture radius). The aperture subreflector radius should be entered in the input file after the apperture radius. The old format was Rapa, Nrad, Nphi. The new make file the format of this line should be Rapa, Rsubr, Nrad, Nphi. Explicitly Old Format: Read(*,*) Rapa,Nrad, Nphi New Format: Read(*,*) Rapa, Rsubr, Nrad, Nphi Fig. 17 shows the results of the farfield pattern for the exact solution and that generated by the modified code. As seen the agreement between the modified ABOR code and the exact solution is excellent. The exact solution for a constant X-directed aperture field is given by 43

Far field Pattern for L=1 OX, L2=10.2X, D1=5X, D2=5.2L2, ~r=1 o 0 2.................................................................... o 0o 0, - 5 0 o -................................................................... 0 ' ' ' ' o Exact Solution: -40 0 --- — 0 00: -100..-80 -60 -40 -20 0 20 40 60 80 100 -50 C' "................0...................0 0.. 0 Exact Solution PO-MOM solution -70 -100 -80 -60 -40 -20 0 20 40 60 80 100 0 (degrees) Figure 17: Exact solution and the farfield pattern through a radome with Or = 44

E(r) = f(t) [cos s 0 - sin q cos 0 ] (1) (t) a2 Ji (ako sin 0) b2 J1 (bko sin 0) (2) k0a sin 0 k0b sin 0 where a is the main reflector radius, b is the subreflector radius, ko is the free space wave number, J1 is the Bessel function of the first kind and of order one, and 0 and ( are the spherical coordinates. Parameter Optimization for speeding up the code: The parameters that can be varied to speed up the code are: 1. NGUASS: number of Gaussian integration points used between sampled points on the aperture. Fig. 18 shows the results for Ngauss = 2,3 and 4. It can be concluded that NGAUSS can not be degraded when integrating over the aperture as it will severely affect the accuracy of the solution. 2. NPIR: number of sampling points used on the radome's internal surface. NPOR: number of sampling points used on the radome's outersurface. The formaulas for calculating NPIR and NPOR in the code are: NPIR=nint (FACTOR* 15*tll *bk/6.2831) + 1 NPOR=nint(FACTOR*15*tl2*bk/6.2831)+1 Fig. 19 shows the results for different values of FACTOR when L1 =- 100A, L2- = 100A, D = 30A, D2 = 30A and = 4. This parameter is a key to speed up the code. Table 1 shows the CPU time and the corresponding factor value for the NPIR and NPOR. From these results it can be concluded that the optimum value for this factor is 7/15. Let us now consider a much larger reflector. For a large radome (LI = 100A, L2 = 102A, D1 = 60A, D2 = 63A), but with much lareger reflector Rapr = 23A. The corresponding pattern for er = 1 and Rsubr = 0 is given in Fig. 20. Since er is unity, itis expected that this pattern will gree with the analytical integration. It is clear that in spite of the CPU time reduction from 20 days (estimated) down to 54 hours and 43 minutes, the accuracy 45

Far field Pattern for L1=10 X, L2=10.2X, D1 =5X, D2=5.22X, Er=1 -40 - u*1 I ' " -50 - Ngauss=4 x Ngauss=3 -6o- i Ngauss=2 i -- - Exact Solution -70I,I I I, I -100 -80 -60 -40 -20 0 20 40 60 80 100 0 (degrees) Figure 18: Results for different npir and npor values 46

Far field Pattern for L1=100Q, L2=102X, D1=30X, D2=34L, Or=4 1.I I I -TIII I I i -10 -20 - Vx vI x x I' x X ( x\ x >k x -30 x -40 - -50 F x Factor=l Factor=2/3 Factor=7/15 Factor=1 /3::: -60 1 I I1 I I I -1 -100 -80 -60 -40 -20 0 0 (degrees) 20 40 60 80 100 Figure 19: Results for different npir and npor values 47

Factor CPU time 1 10 hours and 47 minutes 2/3 5 hours and 56 minutes 7/15 3 hours and 16 minutes 1/3 1 hour and 52 minutes Table 1: Values for the npir factor and the corresponding CPU time of the main and side lobe levels is unacceptabl. This is revealed more in Fig. 21 where we see that the main lobe is 6db down. the 0 [,,, [ Exact Solution -10 — Improved CPU time solution -10 -20 - -60 -70 -80 -100 80 -80 60 40 -60 40 -20 0 20 40 60 80 100 0 (degrees) Figure 20: Results for the large radome with large reflector with the parameter factor=7/15 So this optmization parameter FACTOR can not be used alone for a large radome with a large reflector. 3. NR: number of integration pointson the aperture.Fig. 22 shows the results for a small apperture.Table 2 shows the nr values and the corresponding CPU time. It can be observed that for a small radome and a small aperture, NR makes a great difference. However, for a large radome and a small aperture the reduction in cpu time is negligable. The effect of NR on a large radome with a large aperture is expected to be significant (It will be demonstrated later). 48

Main Lobe 0 -1 -2 -3 -4 m -5 -6 -7 -8 -9 I I I I I I, I I I I - Exact Solution Improved CPU time solution I -10 I I. 1. I I -100 -80 -60 -40 -20 0 20 0 (degrees) 40 60 80 100 Figure 21: Results for the main beam with the parameter factor=7/15 nr CPU time 7 10 minutes 4 6 minutes 3 3 minutes Table 2: Values for nr and the corresponding CPU time 4. NIP,number of integration point from 0 to 27r for modal decomposition. Setting NIP to 10 and Factor to 10/15 gives exact results and CPU time of 26 hours and 12 minutes the results are shown in Fig. 23. The combined effect of NIP, NPIR,NPOR and NR is now considered where NIP is set to 10, Factor is set to 2/3 and NR is set to 15 points instead of 60 points. This will lead to a CPU time of 26 hours. Fig. 24 shows the results which is in excellent agreement with the far field pattern.Table 3 shows the values for NIP and NR and the corresponding CPU time. It should be noted that the reduction of the NR points will work fine with apertures that have no rapid tapering. In case of field tapering NIP and Factor are the only parameters to be optimized. Conclussion After the detailed study of optimizing the various parameters for speeding up 49

L1=10k, L2=10.2k, D= D5, D=5.2k, Er=, rapr=2?,rsubr=0 -30 - e 1 /I' V V -40 - -50 -100 -80 -60 -40 -20 0 20 40 60 80 10 8 (degrees) Figure 22: Results nr=7,4 and 3 L =86., D1=61/lambda, L2=88X, D2=63k, Rap=23X and Rsubr=4kr=1 -10 -20 - -30 -40 / -60 - -70 - -80 t v ' 1 -90 - -I I -100 - -100 -80 -60 -40 -20 0 20 40 60 80 10 8 (degrees) Figure 23: Results for NIP=10 and Factor=10/15 the ABOR code, we conclude the following recomended values: NIP = 10 FACTOR= 10/15 50

L1=86X, L2=88L, Dl=61x, D2=63X, rpr=23X, rsub,=4X C" -50 -1 -70 -80 - -90 - -100 c -100 -80 -60 -40 -20 0 20 40 60 80 100 0 (degrees) Figure 24: Results for NIP=10, Factor=10 and NR=15 points nip nr CPU time 10 60 26 hours and 7 minutes 10 15 7 hours and 30 minutes Table 3: Values for nip, nr and the corresponding CPU time For constant aperture fields and slow tapered aperture fields an extra reduction in CPU time is achieved by reducing NR by a factor of 4. 51

References [1] L.N. Medgyesi-Mitschang and J.M. Petnam, "Combined field integral equation formulation for axially inhomogeneous bodies of revolution," MDC Report, 1988. [2] J.R. Mautz and R.F. Harrington, "H-field, E-field, and combined field solution for bodies of revolution," Interim Technical Report, RADC-TR-77-109, March 1977. [3] A.W. Glisson and D.R. Wilton, "Simple and efficient numerical techniques for 3 bodies of revolution," Technical Report No. 105, The University of Mississippi, March 1979. [4] J.R. Mautz and R.F. Harrington, "H-field, E-field, and combined-field solutions for conducting bodies of revolution," Arch. Elek. Ubertragung., vol.32, pp.157-164, 1978. [5] J.R. Mautz and R.F. Harrington, "Electromagnetic scattering from a homogeneous material body of revolution," Arch. Elek. Ubertragung., vol.33, pp.71-80, 1979. [6] L.N. Medgyesi-Mitschang and J.M. Putnam, "Electromagnetic scattering from axially inhomogeneous bodies of revolution," IEEE Trans. Antennas and Propagat., vol. AP-32, no.8, pp.797-806, 1984. [7] K. Umashankar, A. Taflove, and S.M. Rao, "Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects," IEEE Trans. Antennas and Propagat., vol. AP-34, no.6, pp.758-766, 1986. [8] P.L. Huddleston, L.N. Medgyesi-Mitschang, and J.M. Putnam, "Combined field integral equation formulation for scattering by dielectrically coated conducting bodies," IEEE Trans. Antennas and Propagat., vol. AP-34, no.4, pp.510-520, 1986. 52