THE UNIVERSITY OF MICHIGAN 5548-6-T AFCRL-67-0111 5548-6-T - RL-2068 STUDY OF THE SCATTERING BEHAVIOR OF A SPHERE WITH AN ARBITRARILY PLACED CIRCUMFERENTIAL SLOT by S. Chang and T.B.A. Senior The University of Michigan Radiation Laboratory 201 Catherine Street Ann Arbor, Michigan 48108 February 1967 Scientific Report No. 6 Contract AF 19(628)-2374 Project 5635 Task 563502 Prepared for Air Force Cambridge Research Laboratories Office of Aerospace Research L.G. Hanscom Field Bedford, Massachusetts Distribution of this document is unlimited.

THE UNIVERSITY OF MICHIGAN 5548-6-T ABSTRACT The electromagnetic scattering behavior of a metallic sphere loaded with a circumferential slot arbitrarily placed with respect to the direction of incidence is studied. Under the assumption that the slot is of small but nonzero width with a constant electric field across it, the analysis for the external fields is exact. Expressions for the scattered far field components, as well as for the total surface field components, are derived and then used to investigate the extent to which the scattering behavior can be controlled by varying the loading admittance and the slot position. An explicit formula for the loading of the zeroth mode to annul the back scattering cross section is derived, and from this the desired loading is obtained by means of a lumped load at the center of a radial cavity backing the slot. In particular, emphasis is placed on the case where the slot is in the plane of incidence and normal to the direction of the incident electric vector. The numerical study is limited to the frequency range 0 < ka s 3.0, where a is the radius of the sphere, and results are presented primarily for back scattering. To verify some of these results, a comparison is made with experimental data obtained using a metallic sphere with an equatorial slot backed by a radial cavity of adjustable depth. - - -- -~ -- -— ~ -~ — -- iii

----- THE UNIVERSITY OF MICHIGAN 5548-6-T TABLE OF CONTENTS I. Introduction 1 II. Theoretical Formulation 3 2.1 Diffraction by an Unloaded Sphere 5 2.1.1 Surface Fields 7 2.1.2 Scattered Far Fields 9 2.2 Radiation by the Slot 12 2.2.1 Surface Fields 15 2.2.2 Radiated Far Fields 15 2.3 Radiation Admittances 16 2.4 Complete Problem 18 2.5 Low Frequency Approximations 25 III. Numerical Computations 27 IV. Experiment 37 V. Conclusions 41 Appendix A: Expansion of a Vector Plane Wave in Terms of Spherical Wave Functions 43 Appendix B: The Input Admittance of a Radial Cavity 47 Appendix C: Evaluation of C (0 ) and D (0 ) 54 nm o nm o References 57 v

THE UNIVERSITY OF MICHIGAN 5548-6-T I INTRODUCTION In the recent investigation by Liepa and Senior (1964, 1966) of impedance loading applied to a perfectly conducting sphere, the narrow circumferential slot used to provide the loading was restricted to lie in the plane perpendicular to the direction of incidence. Analytically at least this leads to a considerable simplification since only the tesseral harmonics of the first order then appear, but it does create difficulties in any attempt to achieve the required loading. Thus, to control the scattering cross section over a specified bandwidth, it would seem that the loading must be synthemized either by a distributed network (or nonuniform transmission line) or by a large number of lumped two-port networks distributed around the sphere, and the sophistication of both methods does produce some difficulty in the practical realization of the loading. The analytically simple case discussed above is, in fact, the most complicated one from the loading point of view, and in the hope of reducing the loading difficulties, we shall here extend the analysis to the general case of arbitrary incidence. The circumferential slot will now be located arbitrarily with respect to the direction of incidence and, in addition, the polarization will also be assumed arbitrary with respect to the plane of the slot. The analysis parallels in large measure that provided by Liepa and Senior (1964, 1966), the only major difference being the occurrence of doubly-infinite sets of modes. Analytical expressions for the field components and for the loading necessary to produce any desired form of cross section control are presented, and it is shown that if there exists a zeroth order mode excitation across the slot, the cross section control can be achieved by a lumped load at the center of a radial cavity. For the particular case in which the incident electric vector is perpendicular to the plane of the slot, the expression for the loading required to give zero back I _ 1

THE UNIVERSITY OF MICHIGAN 5548-6-T scattering is obtained and computed. To check the theoretically predicted behavior, the back scattered field was measured using a sphere with an equatorial slot backed by a radial cavity of adjustable depth. The measured cross section as a function of depth is in excellent agreement with the predicted values.. -- 2 -- I

THE UNIVERSITY OF MICHIGAN 5548-6-T II THEORETICAL FORMULATION Consider a perfectly conducting sphere of radius a whose center is located at the origin of a Cartesian coordinate system (x, y, z) and loaded with a narrow circumferential slot symmetrically placed with respect to the z-axis as shown in Fig. 2-1. The sphere is illuminated by a linearly-polarized plane electromagnetic wave incident in the direction of the negative z'-axis whose spherical angular coordinates with respect to the unprimed Cartesian coordinate system are 0. and 0i. If we assume that the incident electric field vector is oriented at an angle: to the x'-axis, where the primed Cartesian coordinate system (x', y', z') is obtained by a rotational transformation of the unprimed, namely by rotating through an angle i0. with respect to the z-axis and then through an angle 0. with respect to the y-axis, so that in 1 matrix form cos 0. 0 -sin0. cos0. sin 0 x y = 0 1 0 -sin0. cos 0 0 zin s 0n 0~ cos O L z0 then the incident field can be written as i AA ^ikz E = (x' cosl3 + y sinl3) e (2. la) Hi = -iY(-x' sin: + y' cos 3) e, (2. lb) where k is the propagation constant and Y the intrinsic admittance of free space. For convenience, the amplitude of the electric vector has been taken to be unity and the time factor e suppressed. The scattered field in any direction can be obtained by superposition of the field diffracted by an unloaded sphere and that radiated from an excited slot at the I 3

- THE UNIVERSITY OF MICHIGAN 5548-6-T y XI FIG. 2-1: COORDINATE SYSTEMS AND SPHERE GEOMETRY 4

THE UNIVERSITY OF MICHIGAN 5548-6-T position of the load. The radiation amplitude and phase are determined by the loading characteristics of the slot, and by controlling these a wide degree of scattering control can be exercised. 2.1 Diffraction by an Unloaded Sphere To study the diffraction of a plane wave by a spherical object we must first find an expansion of the incident wave in terms of the spherical wave functions Me, Ne -omn -omn For the field given by the equations (2.1) the representation in terms of spherical vector wave functions is (see Appendix A): CD n i n=1 m=0 - o mn -i sin Ae + cos/ Bem1 N(em (2.2a) em gn 'N m Hi=iY cos 3 Aemn - sin3 Bemn N()n -1 m LL nn mnj — mn n=l m=0 o o o -i sinAecmn+ cos Bemnl Mm (2.2b) where e (2n+l)(n-m). mP (cos.) nsin m n i _ _ Aemn + 1 n(n+l)(n+m)! cos i sin (2.3a) 0mn e (2n+ 1)(n - m)[ apr (cos. ) n n+COS)(n+m) B I -(2.3b) emn= n(n+l)(n+m)! sin 1 '. i a1i.~_~_~~ ~_~,._ _~ _;__ NR_ 5 _..

- THE UNIVERSITY OF MICHIGAN 5548-6-T with n, m=o L2, m=1,2,... and [i (kR) (1) m: n(kR) Pm( )s in A Me + MP-P cose) omn sine kR n cos n a(kR) aP(cos ) - kR- a sinsi N~e = nn+l) pm(cos e).~m R n 2 n sin kR ae sin sin0 n cos '(kR) aP (cos). s Here, n(kR) = kR j(kR) where j (kR) is the spherical Bessel function of order n, and the prime denotes differentiation with respect to the entire argument. P (cos 0) is the Legendre function of degree n and order m as defined, for example, by Stratton (1941). For the scattered field (E, HS), the requirement that it take the form of an outgoing wave at infinity, leads us to postulate the representation o n E = (Cen Mmn+iDe Nen ) (2.4a) n=1 m0 0m 0 0 mn-0mn ao n) H- = iY(Cemn N emn+ i Dm mn-em (2.4b) n=/ M=O 0 0 0 0 1.. -- -—. ~ -- --- 6

THE UNIVERSITY OF MICHIGAN 5548-6-T in which the M and N differ from the M and N in having b (kR) replaced by C (kR) = kR h 2)(kR), where h(2) (kR) is the spherical Hankel function of n n n the second kind. Application of the boundary condition Rx(E + ES) = 0 at R = a then gives O (ka) Ce =- [cos Aemn -sin B7mn r(ka) (2. 5a) f - (ka) D = sing 3Ae+ cosj Be \ k) (2.5b) and by substitution of (2.5) into (2.4) the scattered field is determined. 2.1.1 Surface Fields 5 5 The total field is the sum of (E, Hi) and (E,H ), and the only nonzero components at the surface of the sphere are ER, Ho and Hp. Since the last two are related to the surface current density J via the equation J = RxH, Ho and HE are of direct concern to us. They are written, for convenience, in a notation similar to that of Kazarinoff and Senior (1962) as oD J = H0(a,,0) = Y ~ {cos3 sinm(0 -i)T1m(0, 0.) *m=0 i m -sin3cosm(O- )T (0T0j) (2.6a) i lM i 7

~~THE UNIVERSITY OF MICHIGAN 5548-6-T J0= H0(a. 0, ) =Y I ~Osf pCos m(o - 0)T 2 (0,0i) + sinj3sin m(0 -O~)T%(0.0 (2. 6b) where T (0 0)- 1 lm:i ka 00 T'.n+l 1 n=m+6 m m n(n + 1)(n +m) t 1 t(ka) n ap (cos 0.) fl 1 ao. MP (Cos 0) n sin 0 + ____mP (cos 0.) ap (cosO0) n(ka) sin 0.a (2. 7a) TT ( 00) 1 l m 'i ka n=m+6_ En+ e(2n+l)(n-m),. MP1ImP(Cos 0.) MPM(Cos 0) n(n +1)(n +m)! ~ (a sinoi sin 0 m apm (Cos 0.) ap (cos 0) f 1 f E (2n +1)(n -m) i apm(co.) m J 1 Iin 1 n~n+ 1(n m L fl 1 k) (2. 7b) T (00. e) -1 2m i ka OD Z.n+l n=m+6 m n ___ MP In(Cos0.) mP (cos 0) 11 1 flI n n(ka) sin 0. sin 0 (2. 7c +Note that the order of double summation has been changed from II I to m=0 n~m+6m 8

THE UNIVERSITY OF MICHIGAN 5548-6-T 2m 1 2m'i ka 00.n+l j i n=m+6 m e (2n+ )(n- m) F mP (cos 0.) aP(cos) m )( - 1 n 1 n ( ) n(n+ l)(n+m)! 1?(ka) sin0. 80 L n aP (cos 0.) mP (cos e) + 1 n 1 n (2.7d) (ka) W0. sin0 ' n i1 with 1' 6 = 1 m -O m=O m= 1, 2, 3,... In the special case 0. = 0, the fact that apm(cos ) mPm(cos) 0) lim n = lim 0 - ae0 6 0 sin n(n+ 1) sin0, ml1, m=1 implies T (0 0) = T' (0,0) = Tm (e'=Tlm(o T2 (0 0)=T' (0e0) = 2m 2m 0 ' T 2(0), m 1 m=1 m=1 where T 1(0) and T2(0) are identical to the current components employed by Liepa and Senior (1964). 2.1.2 Scattered Far Fields In the far zone the expressions for the scattered field can be obtained by replacing ~ (kR) and ~' (kR) by the leading terms of their asymptotic expansions for k n large kR, viz. - ~ -~ --- 9

THE UNIVERSITY OF 5548-6-T MICHIGAN.n+l -ikR (kR) ~ 1 e n ' (kR) in e n The transverse components of the scattered electric field in the far zone then become E = i kR cos cosm(-)Sm(0, )+ sin sinm( - 0im(0 i (2. 8a) E; -ikR OD I "= -i-kR; cos sinm(- p)S m(,.) - sinf cos m(o - 0i)St (0 Oi (2. 8b) where s (O, O.) = Sm 1i () e (2n+l)(n-m)' (-1)m6 n(n+ )(n+m)' n=m+6 m L n(ka) <n I(ka) n aP (cos.) aP (cos e) n 1 n ao. ae 1 i (ka) mPm(cos 0.) n n 1 r (ka) sinO. n 1 mP (cos 0) n sin0 J (2.9a) S'm (0,0e) = im i OD I e (2n+l 1)(n-m)! ff (ka) r (-1) n(n+ l)(n+m)! ] I'(ka) n=m+6 n m mP (cos Oi.) aPm(cos 0) sin0. a0 1 n (ka) anP m(cos0.) mPm(cos0) n n I n. n 1 (2.9b) 10

0-THE UNIVERSITY OF MICHIGAN 5548- 6-T S S (O.,O.)= 2m '1 E(2n +1)(n -m)! nm+ n(n +1)(n +m)! m Ir m VAlka) OP (cosO0.) mP (cosO0) fln 1 ___ __ I j(ka) 00. sin 0 n 1 n) ~ (ka) n mp(cos 0.) OP (cos )0 1 n sinO0 00 P 1 - (2. 9c) 5'S (O,00.)= 2m ' n~m+6 n (-) n(n + 1)(m +m) f a fV~ka) n MIn(CosO0.) mP (cos 0) n 1 fl sinO0 sine0 i On(ka) n(ka) ap (c Os 09.) n 1 00. 1 OP (cos 0fl n 00 (2. 9d) In the back (0= o, 0 = 0) and forward (0 = 7T +~ Op0 = wf- 0.) directions it can be shown that 1 co co 2~S7(V0 0) = S5S (0 o90) = ss(O) M~o m i mO 2m 1 i (2. l Oa) and (-l~m(7r- 0. 00 0.) = - 2m(7r0l 0 ) = s(r 1 m0o 2 i 1 (2. l Ob) where s I n(2n+ ) I (ka) n=1l ~ +1 n OP1(Cos 0) n 00 V'n(ka)P p1(cos 0)2 ~ (ka) sminO n is the far field amplitude defined by Liepa, and Senior (1964). The component cross sections are therefore. I m 11

THE UNIVERSITY OF MICHIGAN 5548-6-T (00 )=.. cos=PS (0) (2.11a) 7r k 2 u (O.(!,5.) = - sinf3S5(O) (2. lib) 11 i 71 E1 1 (2.11b)1 for back scattering, and a (7r-0., T+) = - cosP S(7r) (2.12a) 7T i a(7r-0i, 7+0 inp S (r) (2.12b) ab(1-sin3+Si) -- 7r for forward scattering. 2.2 Radiation by the Slot Let us now consider the separate but related problem of a perfectly conducting sphere with a narrow zonal slot situated at 0 =0 as illustrated in Fig. 2-1. 6 6 The slot occupies the region 0 - - 6 0 0 + and its angular width 6 is assumed o 2 o 2 very small (such that ka6 << 1). Without any loss of generality, the direction of the incident field propagation can be assumed to lie in the x- z plane (0i = 0). Since the 0-component of the unperturbed current density at the surface can be represented as a summation of cosine and sine harmonics of 0 (see equation (2.6b)), the following expressions for the tangential components of the electric field at the surface are assumed: - - cos P Cos po+Vp sinp sin po - 0- < _ E (a0a P0=) 0, otherwise (2.13a) E (a,,0) =. (2.13b) 12

-- THE UNIVERSITY OF MICHIGAN 5548-6-T V cosf and V' sin 3 are the amplitudes of the gap voltage due to the pth mode conP P tribution corresponding to the cosine and sine variations, respectively, and they will be determined later. Let the field radiated by the slot be oo n Er (4) (4) E Fe Me +iGe N (2.14a) n=1 mn=0 mn- mn mn —mn / iGe1a ao n Hr = iY (2.14b) -= Y emn mn mn-mn-) ' (2.14b) n=l m=O o o o o where the coefficients F and Ge are to be determined from the above boun%mn 8mn dary conditions. From the 0 component of the electric field we have, at R = a, 0' m (ka) mP ncos ) s' (ka) aPm(cose) n n sin n n Cos ^ ne m=O mn ka sinO cos emn ka a sin n=l m=O 0 ~ - {vcoscosp0+V' sin sinp} |0-0 < 6a p=0 P P o 2 O, otherwise (2.15a) and from the 0 component Z r { (ka) aP(cos0) '(ka) mP (cose) s F n, n CoS i n n sin emn ka a sin + ka sinO cos = 0. (2.15b) But - -- -- -- -- 13 -- I --

THE UNIVERSITY OF MICHIGAN 5548-6-T P ) apm( apcos) ) ap c ) and and raP (cos ) apm(cos ) 2 0 0n n' +m Pm ( cos0)Pnm(cos0j sinO d L sin2 0 0, n f n' 2n(n+ l)(n+ m). (2n+ l)(n- m)! n = n' (Bailin and Silver, 1956). Hence Fe Omn + (2n+l)(n-m)! - 2n(n+ l)(n+ m)! D (0 ) nm o C (e ) nm o (2.16a) (2.16b) G (2n+ l)(n- m)! ik Gmn 2n(n+l)(n+m).! '(ka) omn where 1 C (0 )= nm o 6 1 D (0 ) = nm o 6 0+6 o 2 apm(cos0) n n sin0 d, o -2 o 2 (2.17a) (2.17b) 0 6 o 2 0 2 o 2 mPm(cos0)dO, n and the required expressions for the components of the radiated field now follow from equations (2.14) upon inserting the above coefficients. -- ~ ~ -- 14

THE UNIVERSITY OF MICHIGAN 5548-6-T 22.21 Surface Fields At the surface R = a the components of the radiated magnetic field can be written in forms analogous to those in equation (2.6), viz. Ho =Y E V cossinm-V sin3cosm02 T (0,0 ) (2.18a) O0 H r Y E V { coscosmo+Vt sinsinm} T rm(0,0 ) (2.18b) where r _ _ _ 7 1 (ka) mP M(cos0) Tr (. 9 ) =L (2n+ )(n-m) C ( n n Im o 2a n(n+1)(n+m)' nm o I'(ka) sin0 n m+6 U n m ' (ka) aPm(cos 0) -D (0 ) n n -- (2.19a) nm o (ka) 00 n D (ka) P m(c (os 0) Trme e ) = - i > (2n+ 1)(n - m):. I ) n- nn 2m(0, 2a E n(n+l)(n+m)' anm o '(ka) 0a n=m+6 o n m m (ka) mPm(cos 0>) -D (0O) (ka) sin - ~n (2.19b) nm o (ka) sin j n For 6 =0 the above series expressions for T (0, 0 ) and Tr (0 0 ) are convernm o 2m o -2 gent, albeit slowly, with their nth terms for n > m being O(n ) and 0(n ) respectively. The terms in the series alternate in sign in groups of 27r/6 terms, and therefore the series may be treated as an alternating series. 2.2.2 Radiated Far Fields In the far zone, expressions for the radiated field components can be obtained from the equations (2. 14) by replacing r (kR) and I(kR) by the leading n n --- - -- L 15 ~

THE UNIVERSITY OF MICHIGAN 5548-6-T terms of their asymptotic expansion for large kR: Ee i kV cospcosm+V' sinpsinm0 S (0 0) (2.20a) m=O LM 1 -ikR OD kR m m 2m o =i kR I VmosBsinm -Vmsincosm0 S0 ) ( ) (2.20b) Sr ( k E n+l (2n+l)(n-m)! nm o n im o0 2 n=^ n(n+l)(n+m)! I n(ka) a8 n=m+6 n m D (0 ) mP( cos0) n + (ka) Pn 3 (2.21a) Sr (k 1 n+ (2 n+ 1)(n-m)! nm o n 2m ' O 2 n(n+ l)(n+m)! i'(ka) sin0 n=m+6 L n D (0 ) aPm( os 0) + i 0 n (2.21b) 2.3 Radiation Admittances It is customary to define the admittance as the ratio of the current to the voltage for each given mode. This definition, however, is not well suited for our purpose since we have a nonuniform current density across the slot that does not permit a unique specification of the admittance. We shall therefore use an alternative but equally acceptable definition of admittance, namely, twice the ratio of the complex power flow across the aperture to the square of the applied voltage. 16 - ~ — — ~ — ~

ft THE UNIVERSITY OF MICHIGAN 5548-6-T From equations (2.13) and (2.18) the mth mode power radiated per unit length of the slot is 6 2 2 o 2+ W r 2^V cosfcosmO+V'nPs sinm T 0 0)d rm 26 m P o6 o 2 where - denotes the complex conjugate. The mth mode radiation admittance density is therefore rm 2W CVmcos cos m +V sin3 sinm0 m P 6 0 +- 6 o 2r Y ~ T. (0, 0 )dO, 6 6 _ 2m o o 2 o 2 (2.22) and the total mth mode radiation admittance is Y = y aasin0 d rm rm o 0 = -iY7r sine (0 n=m+6 m (2n+ 1)(n- m)! n(n+ 1)(n+m)'. c (0 )E (0 ) nm o nmn o yn(ka) ' (ka) I(ka) -D (0 )F (0 ) n nm o nm (ka) n (2.23) where - --- ~ - - - - -- 17

THE UNIVERSITY OF MICHIGAN 5548-6-T E (0 )= 1 nm o 6 F (0 )= nm o 6 6 0 0 - 6 o 2 mP (cose) n \ n deo 6 sin0 2e - o 2 The above expressions simplify somewhat if we assume e 0 < 7T- e with >> 6, allowing us to neglect the variation of sin0 over the slot. We then have C ( ) E () nm o nm o sin0 o D ( ) F ( ) nm o nm o sin o which lead to 2 ~n(ka) 2 rn(ka) -iY Mr (2n+ l)(n - m): Ji 2 n(a) Dka rm -Yr ' n(n+ l)(n+m)' inm(0oi J '(ka) -D (nm (ka) n=m+6 nL n I m (2.24) 2.4 Complete Problem For the complete problem in which the plane wave given in equations (2.1) is incident on the slotted sphere, the total field scattered in any direction can be obtained by superposition of the field diffracted by an unloaded sphere and the field radiated by an excited slot. If we assume that the same voltage 00 V = ZZ~V cos3 cos m0+V sin3 sinmj M=O m / m -~ -- 18

THE UNIVERSITY OF MICHIGAN 5548-6-T is excited across the slot by the currents induced by the incident field, the transverse components of the total electric field in the far zone are the sums of those given in equations (2. 8) and (2.20) and are E=i kR { Sm (O,)+ Vm Sl (ee cosocosmo 0 kR Ir- 1 - 1M E =2 i m=0 Li cos 3 si s (0, 0 )+V VmSm(0,0 o) sin3 sin m (2.25a) L 1m i M 1m -ikR Oi j 8 e s r E =-ie I ~ O., (0,.0.)+V S2 (0,0 0 cos t sin m0 0 kR -! L1O 2m 1 m 2m oJ " ' -SS (0, O) + VI Sr(0,0 0 sin cosmo.(2.25b) L2m i m am Ij nI The components cross sections are therefore 2 2 (2.26a) 2 o2 <0) = 1 2m(O, i)+V S0 cos3 sinme - s (e0ei)+V S2r(0 )sincosm 2 - [~2m i m m s (2. 26b) The V and V' can now be related to the loading admittance of the slot for m m the corresponding mode. To derive these in terms of the loading admittance, we require the expression for the mth mode loading admittance density, y. Applying 2m I 19

THE UNIVERSITY OF MICHIGAN 5548-6-T the same technique as used above, we have +6 YIm = ' +"+H 6[V cosPcosmo+ Vf sinpsinm \:Im m _0 - 6=a o 2 dO which can be written as m rm + 6V m m rm b6V' m ( + )d o 2 \ T2m(8, i)dO 60 + (, 0 )d (2.27a) or (2. 27b) by using equations (2. 6b) and (2. 22) and, in addition, the relation V T (0, 0.) m _2m = (2.28 V' T' (0,0.) * m 2m 1 This last can be obtained from the linear property that the excited voltage is proportional to the current induced by the incident field. Since the variations of T2 (0 0.) and T' (0, 0 ) across the slot can be 2m i 2m i neglected if 6 is sufficiently small, equations (2.27) can be approximated as m = Ym + V 2mo, m ) or m =-Yrm + V T (0 0.) ym rm 2m o 1 m 20

THE UNIVERSITY OF MICHIGAN 5548-6-T The total loading admittance for the mth mode is therefore Y =-Y + 27rasin0 T2m(o,.) (2.29a) An rm V o2m o i m or Ym =- V+ 27rasinO Tm(O Oi.) (2. 29b) n rm V o 2m o0i m from which we have V = - 2~r asinO T2m(O0 0 (2.30a) im rm or V' = - 27rasinO T'm(9 Oi) (2.30b) m Yl +Yo 2moi(2.30) (m rm When the above equations are substituted into equations (2.26), the explicit expressions for the component cross sections are obtained as functions of ka, e6, 0i and Ym. Thus we expect some degree of scattering control-especially the reduction of the cross section which is of main interest to us-by varying the loading admittance. To make the 0-component cross section zero in the direction 0=0' and 0 =' it is required that 00 - Zf Ss ((', _)+ Y 27rasin0 T (O,O)Sr (O, ) cos3cosmx' Im rm +s' (',. ) Y 27rasinO T (o)s(2. sin3sinm (2.31) 21 I

THE UNIVERSITY OF MICHIGAN 5548-6-T It is readily seen that for the left-hand side of equation (2.31), each mth term of the series may be made zero individually by choosing the proper value of the loading admittance for the corresponding mode if the loading for each mode is independently available. Unfortunately, this is almost a non-realizable case since all the mode admittances are, in general,dependent upon each other. If there exists only one mode excitation such that equation (2.31) is independent of all the other mode loading admittances, it is, of course, always possible to satisfy equation (2.31) by the proper loading of this mode. On the other hand, if there is at least one mode whose loading admittance can be arbitrarily chosen without producing any significant change in the sum of all the other mode terms in equation (2.31), it is possible to select the loading admittance for that particular mode in such a way as to cancel out the effect of all the other mode loading admittances unless, in equation (2.31), the sum of the terms involving Y/(Ym+ Y ) for that particular mode vanishes. This can be achieved rm by means of a radial cavity (see Appendix B) with a load, Yb at the inner radius b when kb is very small (kb < 1). Here, Ye alone can be controlled by varying Yb without creating any significant change in the loading admittance of the higher order modes. If we assume that cos3 T2 (0o, 0)S(0' 0 ) 0 (2.32) then, by solving equation (2.31) for the zeroth mode loading admittance, me have -~ -- -- -~~ ~ -- ~ - 22 -— ~

THE UNIVERSITY OF MICHIGAN 5548-6-T L YoJ ro 27raYsin0 cosT20(0 0 )S ' 0-e ) o0 20 0 1 2 l0 S(0', I0.)cos cos mo + 1{V Sr1 (00 o)cos 0cos ml+ s m(0l.0 )+V Si (0, 0i)1sin3sinm0p ^m~ m 1m o 1m i m 1m i (2.33) If the condition (2.32) is not satisfied, a cannot be controlled using Yf. The obvious example of this is the case when the direction of the incident wave is perpendicular to the plane of the slot so that there is no zeroth mode excitation (in fact, only the first mode excitation exists). Similarly, if we assume that s T' 0 ) er 0, (2.34) sinT20(o' i 200) (2.34) to make the — component of the cross section zero in the same direction (0 =0' and 0 = 0'), we have from equation (2.26b) { o} ro 27raYsin0 sinl3 T0(0o 0 i)S2( 0 ) OD m ' S m(0, 0.)sin3 cos m0' + rVSr ( 0', 0o)sin' cos m -S (0,0i)VmS (01', ocos 03sin mDM -m z m mi m m o (2.35) Of course, the 0-component and the 0-component cross sections cannot be reduced to zero simultaneously unless equations (2.33) and (2.35) yield the same value of i7,J0. L 23

THE UNIVERSITY OF 5548-6-T MICHIGAN For the case of back scattering (0 =0i and 0 =i = 0) the equations (2.26) become x2 o=0 77 O0 m=0 2 V Sr (0i.0 ) cos 3 m im 1 o (2.36a) x= a = - S (0)+ 77* 1 vt sr (0., ) m 2m 1 o 2 2 sin 2/ (2.36b) and by using the relations aS0r (Oo ) 10< 0 r aSr (0i, 0 ) aS20(i, 0) 2 (ka) 2 sin 0 T 20(,0.) (ka2 = k sinO T' (o,0) ) 2 o0 0 i (2.37a) (2.37b) valid for E <0 7Tr-e, e > 6, it follows that under the essential assumption that 0 T20(o, 0) 0 the zeroth mode loading required to make c0 zero is Y kasin 0T20 (0,0i2 -O O - 0 r (2.38a) ro ao S1(0)+ ~ VS ( lm ) m=l and it also follows that under the assumption 20( o' i) the zeroth mode loading required to make o0 zero is _ 24

THE UNIVERSITY OF MICHIGAN 5548-6-T 7rY[kasin 0 T' (O ) 0) 2 } = Y - - s 20 ~0 i. (2.38b) L j ro 0 0 S(O)+ VI S1m(0i'0o) 1 m1 m 1mi o Again, the total back scattering cross section cannot be reduced to zero except in certain selected cases. When the incident magnetic field is parallel to the plane of the slot (I = 0), a0 becomes zero regardless of the loading, and zero back scattering can be achieved by the loading given in equation (2.38a) if T20(0,0.) ~ 0. Similarly, when the incident electric field is parallel to the plane of the slot (/3 = 7r/2), it can be achieved by the loading given in equation (2. 38b) if T2(0, i) 4 0. 20 ' i 2.5 Low Frequency Approximations The low frequency characteristics of the above formulae can be obtained by expanding each expression in a series of increasing positive powers of ka. If we consider only the case when 0 = 0 = 7r/2, 0. =0, and = 0, then from equation (2.38a) the expansion for the zeroth mode loading admittance required for zero back scattering is { -- Yr{f[ 2n+2 C2 (7T/2) ka (2.39) and for the corresponding loading admittance Y at r = b we have, from (B. 12), Re Yb (2.40a) 25

THE UNIVERSITY OF MICHIGAN 5548-6-T o - n=3 n 2(n+1) no Furthermore, when the sphere is loaded to give zero back scattering, the equations (2.26) can be reduced to 2 2 6(2.41a) 0(0',) ~ r(ka) (sin0-cos)2 (2.41a) 2 X 6 2 2 a (0,0) - (ka) cos &sin2 (2.41b) and hence the total cross section is 2 - (a6 F2 +2 o.2] c(0, 0) = re(, 0)+(0, 0), - (ka)6 sin - cos)2+cos sin2. (2.42a) If we now express this in terms of the primed coordinate by transformation, we hav 2 6( o<(0, ') (ka)6(1-cos )2 (2.42b) 47. or, by normalizing with respect to the back scattering cross section (a of an unloaded sphere, o(, (1 -) 0os 0')2 (2.42c) o 9 0 Comparison with the results given by Liepa and Senior (1966) shows that the scatter ing pattern is the same as when the slot is positioned in the plane perpendicular to the direction of incidence apart from a reduction in amplitude by a factor 4. 26 -~ ~ -

I THE UNIVERSITY OF MICHIGAN 5548-6-T III NUMERICAL COMPUTATIONS A computer program was written to calculate either the loading impedances, Zb (= 1/Yb), required for zero back scattering, or the relative back scattering cross section, o(0)/ao, for a given value of the loading impedance when the slot is in the plane of the incident propagation vector and normal to the incident electric field vector (O = = =90~, 0=0 =0=0 and j =0). In the formulas for computation, the functions Y, Y S(0), Im rm' 1 T20(7r/2, iT/2), V and Sr (7/2, nr/2) are involved through equations (2.30a), 20 m m (2.36a), (2.38a), (B.10) and (B. 12). In all cases the infinite series were approximated by neglecting all the terms for m and n greater than M and N respectively, with the numbers M and N chosen sufficiently large for four digit accuracy. In practice the computation was restricted to the frequency range corresponding to 0 < ka< 3.0, where the maximum value of N was 35 for S1(0) and T20(Ir/2, r/2), 1 20(/ ) 100 for Y (mO0), V and Sr (7r/2, 7r/2), and 1080 for Y; and throughout rm m lm ro the computation the value of M was 5. The coefficients C (7r/2) and D (7r/2) nm nm in the expressions for Y and S (7r/2, 7r/2) were approximated by using equations (C. 7) and (C. 8) for n 15 and by equation (C. 9) and (C. 10) for n > 15. To see how fast the series o Z VmS (7r/2, 7r/2) m=l converges, some typical values of Vm, S lm(r /2, ) and V S (7, /2) were m' 1m In(/2M 7/2) wre computed and these are shown in Table 3-1. It is seen that the summation of the terms only through m = 5 gives a good approximation to the infinite series in this case. More terms are, however, required for ka > 3.0. If Fig. 3-1, the real and imaginary parts of Y /Y are plotted as functions of ka, 0 < ka, 3.0 for m =O to 5. The real parts are zero for ka = 0, and rise I... _. _ __'___ - - 27 I I -

TABLE 3-1: NUMERICAL VALUES OF V S1 (7/2, r/2) rn Im Re S(0 Im Re lT20(7O/27/2 Im R 20(7r/2, 7/22 0.738229 -0.139549 -0.492003 -0.536931 -3 For: ka = 1.9074 Zb= 0 b/a = 0.05 6 =0.0399 41:;d CT) m Re[V] Im [V Re[Sm (7/2, /2) Imlm (7/2, 7/2 ReFv Sl Im[V S m lm Cn 0n o Ul CC) 0^9 1 0.552345 0.068610 0.628829 2 0.026019 0.010616 0.183804 -4 3 -4.317376x10-4 0.012744 0.419609 4 -3.438531 x103 -7.298828x 10 -3.87993x10 5 5.779219x108 -6.995535x 104 -0.045191 -1.048052 0.419237 0.535755 old -0.430829 0.009356 -0.009258 0.015120 -0.000373 0.005341 0.169496 0.000026 -0.000582 0 -6 -9 -4.071185x10 -5.5x10 0.000031 - C) z Z4

\ \ \I\ \ \ \ THE UNIVERSITY OF MICHIGAN 5548-6-T CD ~-i co - i ~~P-4 II I1 II c II II Y ) 0 \ \ \ III II 0 H I\ \ \ x\\ x x ' \ \\ \ \ 0 ~ I 0 Nbk ' ~ o \\ 1 \Y \ \I \\ 1 \ \ \ I-" % \ \ c \ \^- ~~ \ ~'~~. "x 0 0) C~j r'I o r-4 I 29

THE UNIVERSITY OF MICHIGAN 5548-6-T through positive values with a small but regular oscillation as ka increases. On the other hand, the imaginary parts are O(ka) for m =0 and 0 ka) 1] for m ^ 0 when ka is small, and also increase with a small regular oscillation as ka increases while their signs change from negative to positive at a value of ka near 0.95 m for the mth mode excitation. In Table 3-2 the numerical values of Y for ka = 2.0 with 6 =0. 0349 rm (approximately 2 ) and 0 = 90 are given for comparison with the results obtained 0 by Mushiake and Webster (1957). Their values for m > 0 have been multiplied by a factor 2 to account for a difference in the definition of admittance and are shown in parentheses. Although the real parts are very close to each other, the imaginary parts differ by as much as 10 percent. It is believed that the present values are much closer to the exact ones, and that the discrepancies in Mushiake and Webster's data are attributable to their retention of an insufficient number of terms in the slowly converging series for Im [Y. The real and imaginary parts of the optimum loading impedance, Z b at R=b for zero back scattering, are shown in Figs. 3-2a and 3-2b, respectively, as function of ka for the range 0 2 ka s 3.0. The real part is zero for ka = 0 and increases slowly with ka up to ka = 1.6 (approx.). Above ka = 1. 6 the curve for the real part is quite irregular in structure. It crosses the zero axis at ka = 1.685, 1.710 and 1.907, and as ka increases further above ka = 1.907, it decreases through negative values out to at least ka = 3.0. In the region where Re LZb < 0 it is apparent that zero back scattering is achievable only by using an active load. The curve for the imaginary part of Z has a singularity of order (ka) at ka= 0, and decreases rapidly as ka increases, crossing the zero axis at ka = 0.845. This curve is somewhat similar to that obtained by negatively inverting the impedance of an L-C parallel circuit or transmission line. Finally, the relative back scattering cross sections were computed for a certain range of Z within which a complete suppression of the back scattering 30

THE UNIVERSITY OF MICHIGAN 5548-6-T TABLE 3-2: RADIATION ADMITTANCES FOR ka =2.0, 6=2~ AND =90~ o m 0 1 2 3 4 5 ReY x103 16.60434 (16.601) 17.96643 (17.9600) 5.55220 (5.5502) 1.58845 (1.5876) 0.20727 (0.2072) 0.01362 (0.0136) Im ~xxl 3 43.88532 (38.629) 37.42239 (34.310) 4.74812 (5.080) -44.52784 (-39.516) -103.57280 (-93.082) -147.56936 (-134.756) ( ): Values given by Mushiake and Webster (1957). 31

0 0 -0.4 Re[Zbo ohms-0.8- \A c.\CT N - \ -1.2 - -1.6 I I I I ' I 0 1.0 k 2.0 3.0 FIG. 3-2a: REAL PART OF LOADING IMPEDANCE FOR ZERO BACK SCATTERING WITH 0 = 0 = 90, 0 = = 0 AND = 0. (b/a = 0.04, 6 = 0. 0399) -- ci t2 - 0 0 z ~n

30 - Im[ZbJ ohms CAD 20 10 0 -10 -20 -30 0 FIG. 3-2b: Ci" Cio H til z P!1 H 0 0 z -lF 1.0 ka 2.0 -1F 3.0 IMAGINARY PART WITH O = 0.=900, 0 1 OF LOADING IMPEDANCE FOR ZERO BACK SCATTERING 0=,0 0 0ANDf = 0. (b/a =0. 04., 6 =0. 0399)

o - - - - - - - - - - - - - - - - - -10 um/u0)c in db -20 --3.0 -2.0 -1. 0 0 1.0 ImLb] ohms FIG. 3-3: RELATIVE BACK SCATTERING CROSS SECTION WITH ReL ]=0. 0129 Q (-) AND Re z]=0 (O ce0) FOR ka = 1. 0 b/a = 0. 04, 6= 0.03 49 and 90 __ bo =0. 0129 - i. 5219 7~) H z 0 -z-e

THIE UNIVERSITY OF 5548-6-T MICHIGAN I I I I I I I I I I I I I I I I I I I I e I I 0 0 01 cnC) 0 0O m z 0 OOL 0) (12 0 Lo 0 0 0 II 0 O V-4 I C;o C1.1 I 0 b.01 - 35

THE UNIVERSITY OF MICHIGAN 5548-6-T field occurs. The curves are plotted in Figs. 3-3 and 3-4 for ka = 1.0 and ka =1.5, respectively, as functions of the imaginary parts of Zb while the real parts are fixed. The cross sections are so sensitive to the loading that only a one or two percent deviation of the loading impedance from the optimum, Zbo, is sufficient to decrease the cross section reduction to only 20 db. Figure 3-3 shows that for ka = 1.0 neglecting ReZb] = 0.0129, which is about 0.8 percent of |Zbo, has little effect. On the other hand, for the case ka = 1.5, only a 5 db reduction would be possible (see Fig. 3-4) if a purely reactive loading were used; that is, neglecting Re[Zb] = 0.1263, which is about 2 percent of IZbo. 36 - -- -- -- ~ ~ ~

THE UNIVERSITY OF MICHIGAN 5548-6-T IV EXPERIMENT To confirm the theoretical predictions, the back scattering cross sections were measured using the model constructed for the earlier experiments (Liepa and Senior, 1966). The model consists of two identical solid aluminum caps joined together by means of a partially threaded shaft at the center, but spaced 1/16 inch apart to form a radial cavity of the same width. The cavity is shorted at the center and the diameter of this short (2b) can be varied from 0.1253 inches to 3.133 inches (the diameter, 2a, of the sphere) by changing the size of disc employed. The measurements were carried out using conventional cw equipment in an anechoic room. The distance from the antenna to the support pedestal was 24 feet and the model was placed on the pedestal with the plane of the slot perpendicular to the incident electric field (which was horizontally polarized). The back scattering cross section was measured for a series of shorting discs at a frequency 2.2873 GHz, corresponding to ka = 1.9074, and the results, normalized relative to the cross section of the unloaded sphere,are given in Table 4-1. A comparison with the theoretical values, computed from equation (2.35) for 0. = 0 =90 and f3 =0 is shown in Fig. 4-1 and it will be observed that the agree1 0 ment is excellent. By using the above described model, shorted at the center of the cavity, Ym is always purely susceptive and consequently it is not, in general, possible to obtain a complete cross section reduction at any frequency. For the experimental verification, however, it appeared reasonable to choose frequencies at which the back scattering could be made quite small using only susceptive loading even though perfect cancellation was not attainable. From Fig. 3-2a, b it is seen that four such choices of frequency are 1.0133, 2.0206, 2.20506 and 2.2873 GHz, corresponding to ka = 0.845, 1.685, 1.710 and 1.9074, respectively. At ka = 0.845 the imaginary I 37

THE UNIVERSITY OF MICHIGAN 5548-6-T TABLE 4-1: 2b, inches 0.1253 0.3125 0.445 0.500 0.657 0.750 0.875 1.000 1.125 1.250 1.370 1.500 1.625 1.750 2.000 2.250 2.500 2.874 3.133 EXPERIMENTAL DATA, BACK SCATTERING CROSS SECTION NORMALIZED WITH RESPECT TO THE UNSLOTTED SPHERE. (2a = 3.133", ka = 1.9074, f = 2.2873 GHz) b/a 0. 0399 0. 0991 0.1420 0.1596 0.2097 0.2394 0.2793 0.3192 0.3591 0.3990 0.4389 0.4788 0.5187 0.5586 0.6384 0.7182 0.7980 0.9177 1.000 cy(O)/uc (O), db -8.1 -10.3 -15.3 -13.0 +2.8 +7.1 +6.4 +5.5 +5.5 -4.3 -0.7 +0.1 +0.5 +0.5 +0.3 +0.3 +0.2 +0.2 0 _ __ 38

n do)/ in dl 5 -0 --1b -15 - -15 -1 Ul 01 I 01 a) 00 H z C1 -cn 0 z I 0 0.2 I 0.4 0.4 b/a 0.6 l 0.8 I 1 1.0 FIG. 4-1: THEORETICAL (-) AND EXPERIMENTAL (e oa ) RELATIVE BACK SCATTERING CROSS SECTIONS WITH VARIOUS CAVITY DEPTHS: ka= 1.9074, 0. = 0 = 90~ and 6= 0.0399.

THE UNIVERSITY OF MICHIGAN 5548-6-T part of Zbo crosses the zero axis and the real part is very close to zero, while at ka = 1. 685, 1. 710 or 1. 9074 Zb is purely reactive and there is a possibility that Z bo can be made zero (or almost zero) by choosing the cavity depth appropriately. Since the preferred frequencies for the available experimental facilities were above 2 GHz, the frequency f = 2.2873 GHz was selected for the experiment, although the minimum cross section obtainable at this frequency was only 15.2 db below that of the unslotted sphere. To achieve this requires b/a = 0.1420, as shown in Fig. 4-1. The incomplete cancellation of the cross section that then results is attributable to the fact that when the cavity depth increases in the region b/a > 0. 05, not only Y changes, but also Ym, m >0. jIm 40

THE UNIVERSITY OF MICHIGAN 5548-6-T V CONCLUSIONS This report has been devoted to the study of the scattering behavior of a metallic sphere loaded with a narrow circumferential slot arbitrarily placed with respect to the direction of incidence. General expressions for the scattered far field components, the total surface field components and the scattered cross section components were derived as functions of the loading admittances and the position of the slot, and it was shown that the modification of the scattering behavior, primarily for zero back scattering, can be achieved by means of a lumped load at the center of a radial cavity backing the slot. For simplicity, emphasis has been placed on the case where the direction of the incident wave is such that its electric vector is perpendicular to an equatorial slot backed by a radial cavity, and the numerical computations have been limited to the frequency range corresponding to 0 < ka a< 3. 0. To confirm the theoretical predictions, the back scattering cross sections were measured using a metallic sphere with an equatorial slot backed by a radial cavity of adjustable depth. When the measured data were compared with the numerical data, excellent agreement was found. Unfortunately, complete reduction in cross section was not possible using this model since the optimum loading impedanc ance, Zbo' at the center of the cavity for zero back scattering is, in general, complex, while the model was shorted there. The minimum cross section obtained with the model was, however, about 15 db below that of the unloaded sphere. In the realization of the required loading for zero back scattering over a certain frequency band, it is clear that simple passive loading alone will not suffice, and that more sophisticated loading techniques must be developed. The loading impedance required tends to have an imaginary part whose behavior as a funttion of frequency is more or less the reverse of what we would expect from a passive impedance, while its real parts are very small in comparison with its imaginary parts (see Fig. 3-2a, b). This suggests the use of active loading using negative impedance ---. - - -.w 41

THE UNIVERSITY OF MICHIGAN 5548-6-T converters (NIC's). Another variational method would be to use an appropriate frequency-dependent medium in the cavity so as to modify the frequency-dependent characteristics of the optimum loading, and, hopefully, make them similar to those of a passive impedance. Some materials composed of dipolar molecules have their dielectric constant decreasing as the frequency increases in the region we are interested in (Smyth, 1955; M. I. T., 1945). The use of this kind of material medium would make the propagation constant (k = i) not directly proportional to the frequency, but less dependent on it. Although it is obvious that the realization of the required loading by means of NIC's or by a material medium will be rather difficult, these approaches appear to have sufficient promise to warrent further investigation. - ~ -~-~~- ~ - ---- 42

THE UNIVERSITY OF MICHIGAN 5548-6-T APPENDIX A EXPANSION OF A VECTOR PLANE WAVE IN TERMS OF SPHERICAL VECTOR WAVE FUNCTIONS Consider a plane wave represented by i A ikz' Ei = 'e (A.1) ir,, ikz' H = -y'Ye (A.2) where a time factor e has been suppressed and the coordinate system is as shown in Fig. 2-1. (We assume 3 =0 for the time being.) Since the field must be finite at the origin, the expression for the above vector plane wave can be written in terms of (1) (1) the spherical vector functions of the first kind Me,e N( as mn EiA Me -iB Om (A. 3) 0o n H iY- II ZZAe N -iBe (m ) (A.4) (1) (1)1 where M(en and N n are given in Section 2.1. -emn -emn To determine the coefficients A mn and B n we multiply equation (A. 3) by Me mn' and Nemn?, respectively and integrate over the surface R = constant. By expressing equation (A. 1) in terms of the spherical polar coordinates (R, 0, 0) (Stratton, 1941): -~ -~ -- -— ~ — -— ~~ -- -- 43 ~- ' ----- ~

THE UNIVERSITY OF MICHIGAN 5548-6-T E= cos.sinecos(0 - )-sin.cos 0 R + [os.cos cos(0 - 0i)+sineOsin 0 n" - cos 0.sin(0 - i (2n+ )j (kR) P [.(cos i.)P (cos ) z i i \ n n 2 (n- m) m m(5) +2 1 (n+m) P (cos0.)Pn(cos0)cosm(0-p), (A. 5) m=l and by using the orthogonality relations for the trigonometric and Legendre functions followed by the recurrence relations for the latter, we obtain 7t 27r 0 E - m'n sinO dpdO 0m'pn' 0 0.n' nt i sin = + 47r - i cs ms O m [j,(kR) (A.6) sin0. cos i n 1 \ 0 EN, sin d0d - -emm ' O O0 ap i c(cos 0.) =in 4 } sin mI * 2nm + {(n +1)Ln-l(kR)i a. sinm+l On the other hand, from the right hand side of equation (A. 3), by using the orthogonal relations of M, N functions, we have - -- 44

THE UNIVERSITY OF MICHIGAN 5548-6-T 7r 0,27r oo n \ Z I emn-Memn 0u -iBe Ne ),sinOdodO emn-0mn/ -emn' n'+ ) 2 ') (n' + 1 ) [j) (kR) Aemln(1 o6 (2n, +l)(n - m)'(n + 1)Lnt (A.8) S7T 0 -2ir oo n \e (AemnM-mn iB8emnNmnm }0 *N mnsin0d0d ^emIn'? 0 -= B e,(1+6 ) -n (n '+1) n'+l) jn,(kR +n tn+l(kR o (2n' + 1) (n' - m?); (A. 9) where 0, 6 = - 0 L-, m40 m=O By equating equations (A. 6) and (A. 8), and (A. 7) and (A. 9), we then have A emn Ben 0 e (2n+l)(n-m:n m n(n+ 1)(n+ m). e (2n+l)(n- m)'..n m 1 n(n+ )(n+ m)' ):. mP (cos 0.) sin n I cos mi sin 0. 1 (A. 10) (A.11) where l, m=O m L2, mm O Now, for an arbitrarily polarized plane wave whose electric field vector makes an angle 3 with respect to the x'-axis, its field can be expressed as 45

THE UNIVERSITY OF 5548-6-T MICHIGAN Ei (' cos + ' sin:) e1k i = iY(x sin+ cos ikz' H -iY(-= ' sin + y' cos P) e (A. 12) (A.13) Hence, using the relations given in equations (A. 1) through (A. 4), we finally obtain E = [os Ae - sinp Be mn n= m=O o mn i in Aemn cos mn-Nemnj - 0 0 0J (A. 14) oo n( H l miY os Amn - sin Bem Nem n=l m=O mn 0 -i sin Aemn + cos 3 Bemn- 7Momn L 0 0 0 (A.15) -- 46

THE UNIVERSITY OF MICHIGAN 5548-6-T APPENDIX B THE INPUT ADMITTANCE OF A RADIAL CAVITY Consider a radial transmission line with input at its outer radius (r = a) V = V cos m, (m integer) (B.1) and with load admittance at its inner radius (r = b) where the load admittance is assumed uniformly distributed about the circumference as shown in Fig. B-1. If we assume kd < 1 and d << a, the field components are H =E =E =0 z r and EH(r,) = A J(kr)+ B Nm(kr)} cosm0 (B.2) H(r,0) )=-iY4IAJ' (kr)+BN' (kri)}cosm0 (B.3) Hr(r,) -J m(kr)+ B (kr )sinm0 (B.4) r kr - m (Ramo and Whinnery, 1944) where J (kr) and N (kr) are cylindrical Bessel functions of the first and second kinds respectively. To determine the constants A and B, the boundary conditions at the inner and outer surfaces must be considered. 1. Inner Boundary Condition: By using the same definition of the admittance as given in section 2.3, the load admittance density at r = a is expressed as 2Wb HOlr=b Yb vb1)2 = dE B.5) bh z r=b where - -~ ~ - 47 ~

THE UNIVERSITY 5548-6-T z OF MICHIGAN V y FIG. B-l: GEOMETRY OF A RADIAL CAVITY. --- -- ~ ~ ~ 48 - --

THE UNIVERSITY OF MICHIGAN 5548-6-T d/2 Wb 2 (Ex H) rdz J-d/2 r=b and V(b) =-dE r=b from which we have YbJmb)+J(kb)+ kb)A+ [YbN (kb)+Nm(kb)B = 0 (B.6) where Yb = 2rb b is the total load admittance at r = b, and = iY7b d 2. Outer Boundary Condition: At the outer radius, r = a, equation (B. 1) gives Ez = E cos m r=a where E =-V /d. a mn Hence, from equation (B. 2), AJ (ka)+BN (ka)=E (B.7) m m a After determining the coefficients A and B from equations (B.6) and (B. 7), substitution into equations (B. 2) and (B. 3) gives 49 - --- - ~ -- ~ --- - -~ ~ --

THE UNIVERSITY OF 5548-6-T MICHIGAN J m(kr) LYb N (kb) + P3N'I (kbjj - N (kr) LYbJm(kb) + f3J I(kb) V; Z J(ka) L~N (kb) + N I (kb)1 - Nm(ka)tYbJm(kb) +I3J t(kb) a =J I(kr) L~N (kb) +1 N I (kb)] - N I (kr)L J (kb) + 3J I(kb] Cos o EL (B. 8) Jm(ka)L Nm(kb) + N I (kbj - Nm(ka)LYbJm(kb) + f3Jj (kbj E acos mo (B. 9) Analogously to equation (B. 5)., the input admittance density at r = a is H1r=a Im d E and thus the total input admittance is Ym = 27ra yI= -a J I(ka)FY N (kb) +j3N I (kbil- N I(ka)FYJ(kb) +f3J I(kbl mL b m m JmLYb m m J (ka)[YN (kb) +f3N I (kb~ - N (ka4 YbJ (kb) + OJJ'(ktb m m m m m m B. 10) where 2wra a = dy~ When kb is very small the asymptotic expression for Y is as follows: NF 1 _ _ _ _ _ 27rY 7 -a Jka1J ka J (ka) ka In(kb) N (ka)~ 2 -ikb kd Y In(kb)j Ye for m=0 kbYb +i2ir mY ( b2 T9I'(ka) +(2 /ka)[] 4 -a ka m M )2 kbYb - i2 7mY 2~b for m.1 50

THIE UNIVERSITY OF MICHIGAN 5548-6-T The above expression shows that when kb approaches zero, the term involving Yb approaches zero as [1/inkbj and (kb)2m for m =0 and m > 1 respectively.* Therefore the proper choice of kb would make it possible to change Y within some range by varying Yb without changing significantly the higher mode admittance values. Moreover, if kb is sufficiently small, we may consider the loading admittance Yb at r = b as a lumped one. Solving equation (B. 10) for Yb in terms of Y1o we obtain N'(kb)Y, J (ka) + aJ(ka -JI(kb)Y N (ka) +aN (ka2) Y -~ (B. 12) b N (kb)[YJ J (ka)+ aJ' (kal - J (kb)FY, N (ka) + aN' (ka 0 oo o -J o L oo o - i and in particular, for kb small, 7"" Y N (ka) +aN1 (ka) 7 Y iY 2 1+ o ~0 ) (B. 13) b -kdi(kd) 2on(kb) Y, J (ka)+aJ'(ka) while the higher mode input admittances are given by J' (ka) Y, '" c m.(B 14) m -aJ (ka) (B. m To see the feasibility of the above approximations when kb is small, the input admittances for several modes have been computed using the exact formula (B.10) for (b/a) = 0.05(0.05)0.95 with ka = 1.9074, Y/Yb= -2- +iO and b 120,7 6 = 0. 0399,and are plotted in Fig. B-2. It will be observed that the higher the modal Note that (1/lnkb) approaches zero more slowly than kb; x lim -x = lim 2x = 0. x — > (1/Inx)2 x -— > 0 51 - - ~ -~ ---- --- - ~-~ - --

100 Im )j/Y -100 - m =0 01 01 00 I r1 -z Ci2 H 0 -Cl) 0 z -200 - -300 - m =4 I 1 I I I I 0 0. 2 0.2 I 0.4 0. 6 0. 8 1.0 b/a FIG. B-2: NORMALIZED INPUT ADMITTANCE OF A RADIAL CAVITY FOR ka = 1. 9074, y/yb =1o-5/120ow+ iO, AND 6 = 0. 399

THE UNIVERSITY OF MICHIGAN 5548-6-T number, the slower the variation of the admittance as a function of b/a. In particular, for b/a near 0. 05, Y1o changes rapidly while the others remain almost constant, and when the cavity depth approaches zero (b/a -- 1. 0) Ym approaches -ico for all m, as expected. If the cavity were filled with a medium of refractive index n, the expressions for the corresponding Ym and Yb would follow immediately from the above jIm b equations on replacing k by nk and Y by the intrinsic admittance of the medium. Thus, for real n, numerical values can be obtained by scaling those for an airfilled cavity. 53

THE UNIVERSITY OF MICHIGAN 5548-6-T APPENDIX C EVALUATION OF C (0 ) AND D (O ) nm o nm o From equation (2.17a) we have C ( )= nm o 0 +6 o 2 apm (cos) sine n de 0J - o 2 (C.1) If we limit the position of the slot to be such that e 0 < 7- c, with e > 6, an adequate approximation to the above is adequate approximation to the above is sin C ( ) = nm o 6 frm os( + 6 - c(0-} n o 2i- n L-o 2JJ (C.2) For the case in which n > m, each Legendre function can then be replaced by the leading term of its asymptotic expansion, viz. m 2 7 1 7r 7 P (cos0) ^ (-n) n nrsin cos -n+2 +)e - n n7rs0 L 2 4 2 to give 2nsin0 sinm o 1 7r m 7r 2 (8 ) -_(-n) s (n+)0 — + nm o r 2 o 4 2 n6 2 (C.3) Similarly, from equation (2.17b), 0 + nm o 6 6 n 0o 2 -0 2 (C.4) 54

THE UNIVERSITY OF MICHIGAN 5548-6-T but unfortunately an approximation analogous to that given for C (0 ) in equation nrm O (C.2) is not possible here. There are, however, the following asymptotic approximations to D (0 ): nm o D (0 ), mPm(cos ) nm o n o (C.5) valid for n6 << r, and si n6 D (0 ), (-n) m 2 cos in+ )0 - +] — 2 nm o n sin e 2 o 4 2 n6 __ 2 (C.6) valid for n > m, and these are quite effective for most numerical purposes. Indeed, for n< 15, the maximum error in using equation (C. 5) is only about one percent, and this is true also for equation (C.6) when n > 15 with m 4 5. Not surprisingly, the error is greatest in the crossover region. In the particular case 0 = 7r/2 equations (C.2) and (C.5) reduce to 0 0 c (T/2) = 2 m ros( r +- 6 n n2 (n- m) even (n- m) odd, n6 << 7 (C.7) 0 D (7r/2) = nm n-m (-1) 2, (n- m) odd (C.8) (n+m- l)(n+m- 3)...3.1 m (n-m)(n-m-2).. 4.2 (n-m) even, n6 << r respectively, and equations (C.3) and (C. 6) reduce to. I..... 55 ~ ~

I THE UNIVERSITY OF MICHIGAN 5548-6-T C (Or/2) z nm 0 n-m+l (-1 2 (n- m) even m 2n n - V 7T. n6 sin 2 n6 2 (C.9) (n-m) odd, n>> m D nm Or /2) - n-m (-l) 2, (n-m) odd m T din rn n6 mn 2n6 2 (C. 10) (n —m) even, n>> m respectively. W 56

THE UNIVERSITY OF MICHIGAN ---- 5548-6-T REFERENCES Bailin, L. L. and S. Silver (1956) "Exterior Electromagnetic Boundary Value Problems for Spheres and Cones", IRE Trans. AP-4, 5-16. Kazarinoff, N.D. and T.B.A. Senior (1962) "A Failure of Creeping Wave Theory", IRE Trans. AP-10, 634-638. Liepa, V. V. and T. B. A. Senior (1964) "Modification to the Scattering Behavior of a Sphere by Reactive Loading", The University of Michigan Radiation Laboratory Report No. 5548-2-T. Liepa, V.V. and T. B.A. Senior (1966) "Theoretical and Experimental Study of the Scattering Behavior of a Circumferentially-Loaded Sphere", The University of Michigan Radiation Laboratory Report No. 5548-5-T. Massachusetts Institute of Technology (1945) Tables of Dielectric Materials (M.I. T. Laboratory for Insulation Research, Cambridge, Mass.). Mushiake, Y. and R.E. Webster (1957) "Radiation Characteristics with Power Gain for Slots on a Sphere", IRE Trans. AP-5, 47-55. Ramo, S. and J.R. Whinnery (1944) Fields and Waves in Modern Radio, (John Wiley and Sons, Inc., New York). Stratton, J.A. (1941) Electromagnetic Theory (McGraw-Hill Book Co., Inc., New York). Smyth, C. P. (1955) Dielectric Behavior and Structure (McGraw-Hill Book Co., Inc, New York). - - - -— ~ r — - ~ 57 --- - --

Unclassified Security Classification DOCUMENT CONTROL DATA - R&D (Security clsassfication of title, body of abstract and indexing annotation must be entered when the overall report is classified) 1. ORIGINATIN G ACTIVITY (Corporate author) 3a. REPORT SECURITY C LASSIFICATION The University of Michigan UNCLASSIFIED Radiation Laboratory 2b GROUP Department of Electrical Engineering. 3. REPORT TITLE Study of the Scattering Behavior of a Sphere with an Arbitrarily Placed Circumferential Slot 4. DESCRIPTIVE NOTES (Type of report and inclusive dates) Interim Scientific Report 5. AUTHOR(S) (Lost name, int^al) Chang, Seichoong Senior, Thomas B.A. 6. REPORT TdAT. 7a. TOTAL NO. OF PAGES 7b. NO. OF REFS February 1967 57 8 8a. CONTRACT OR GRANT NO. *^. ORIIN ATOR'S REPORT NUMBER(S) AF 19(628)-2374 5548-6-T b. PROJECT NO. 5635 -E 02 Scientific Report No. 6 c. 9b. OTHiER R PORT NO() (An oyther nmbers tht may be lassigned DOD Element No. 61445014 his rort d DOD Subelement No. 681305 AFCRL-67-0111 10. AVA IL LABILITY/I!MITA TION NOTICES Distribution of this document is unlimited. 11. SUPPI-EMENTARY NOTES 1 '. -PONSORiNG MIL-TACY ACTIVITY Hq. AFCRL, OAR (CRD) United States Air Force ___L.G. Hanscom Field BedfoGd Mass 13. ABSTRACT The electromagnetic scattering behavior of a metallic sphere loaded with a circumferential slot arbitrarily placed with respect to the direction of incidence is studied. Under the assumption that the slot is of small but nonzero width with a constant electric field across it, the analysis for the extermal fields is exact. Expressions for the scattered far field components, as well as for the total surface field components, are derived and then used to investigate the extent to which the scattering behavior can be controlled by varying the loading admittances and the slot position. An explicit formula for the loading of the zeroth mode to annul the back scattering cross section is derived, and from this the desired loading is obtained by means of a lumped load at the center of a radial cavity backing the slot. In particular, emphasis is placed on the case where the slot is in the plane of incidence and normal to the direction of the incident electric vector. The numerical study is limited to the frequency range 0 < ka < 3. 0, where a is the radius of the sphere, and results are presented primarily for back scattering. To verify some of these results, a comparison is made with experimental data obtained using a metallic sphere with an equatorial slot backed by a radial cavity of adjustable depth. D J. U a f DD I JAN 64 1473 Unclassified Security Classification

Unclassified Security Classification 14. LINK A LINK B LINK C KEY WORDS ROLE WT ROLE WT ROLE WT Sphere Arbitrary Slot Impedance Loading Cross Section Control INSTRUCTIONS 1. ORIGINATING ACTIVITY: Enter the name and address of the contractor, subcontractor, grantee, Department of Defense activity or other organization (corporate author) issuing the report. 2a. REPORT SECURITY CLASSIFICATION: Enter the overall security classification of the report. Indicate whether "Restricted Data" is included. Marking is to be in accordance with appropriate security regulations. 2b. GROUP: Automatic downgrading is specified in DoD Directive 5200. 10 and Armed Forces Industrial Manual. Enter the group number. Also, when applicable, show that optional markings have been used for Group 3 and Group 4 as authorized. 3. REPORT TITLE: Enter the complete report title in all capital letters. Titles in all cases should be unclassified. If a meaningful title cannot be selected without classific:tion, show title classification in all capitals in parenthesis immediately following the title. 4. DESCRIPTIVE NOTES: If appropriate, enter the type of report, e.g., interim, progress, summary, annual, or final. Give the inclusive dates when a specific reporting period is covered. 5. AUTHOR(S): Enter the name(s) of author(s) as shown on or in the report. Enter last name, first name, middle initial. If military, show rank and branch of service. The name of the principal au.Athor is an absolute minimum requirement. 6. REPORT DATE: Enter the date of the report as day, month, year; or month, year. If more than one date appears on the report, use date of publication. 7a. TOTAL NUMBER OF PAGES: The total page count should follow normal pagination procedures, i.e., enter the number of pages containing information. 7b. NUMBER OF REFERENCES: Enter the total number of references cited in the report. 8a. CONTRACT OR GRANT NUMBER: If appropriate, enter the applicable number of the contract or grant under which the report was written. 8b, 8c, & 8d. PROJECT NUMBER: Enter the appropriate military department identification, such as project number, subproject number, system numbers, task number, etc. 9a. ORIGINATOR'S REPORT NUMBER(S): Enter the official report number by which the document will be identified and controlled by the originating activity. This number must be unique to this report. 9b. OTHER REPORT NUJMBER(S): If the report has been assigned any other repcrt tfumbers (either by the originator or by the sponsor), also enter this number(s). 10. AVAILABILITY/LIMITATION NOTICES: Enter any limitations on further dissemination of the report, other than those imposed by security classification, using standard statements such as: (1) "Qualified requesters may obtain copies of this report from DDC." (2) "Foreign announcement and dissemination of this report by DDC is not authorized." (3) "U. S. Government agencies may obtain copies of this report directly from DDC. Other qualified DDC users shall request through (4) "U. S. military agencies may obtain copies of this report directly from I?DC. Other qualified users shall request through (5) "All distribution of this report is controlled. Qualified DDC users shall request through If the report has been furnished to the Office of Technical Services, Department of Commerce, for sale to the public, indicate this fact and enter the price, if known. 11. SUPPLEMENTARY NOTES: Use for additional explanatory notes. 12. SPONSORING MILITARY ACTIVITY: Enter the name of the departmental project office or laboratory sponsoring (paying for) the research and development. Include address. 13. ABSTRACT: Enter an abstract giving a brief and factual summary of the document indicative of the report, even though it may also appear elsewhere in the body of the technical report. If additional space is required, a continuation sheet shall be attached. It is highly desirable that the abstract of classified reports be unclassified. Each paragraph of the abstract shall end with an indication of the military security classification of the information in the paragraph, represented as (TS), (S), (C), or (U). There is no limitation on the length of the abstract. However, the suggested length is from 150 to 225 words. 14. KEY WORDS: Key words are technically meaningful terms or short phrases that characterize a report and may be used as index entries for cataloging the report. Key words must be selected so that no security classification is required. Identifiers, such as equipment model designation, trade name, military project code name, geographic location, may be used as key words but will be followed by an indication of technical context. The assignment of links, rules, and weights is optional. T Tn n 1 nqq f i or Security Classification