THE UNIVERSITY OF MICHIGAN 5548-5-T AFCRL- 66-124 THEORETICAL AND EXPERIMENTAL STUDY OF THE SCATTERING BEHAVIOR OF A CIRCUMFERENTIALLY-LOADED SPHERE by V.V. Liepa and T. B.A. Senior February 1966 Scientific Report No. 5 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

THE UNIVERSITY OF MICHIGAN 5548-5- T ABSTRACT The electromagnetic scattering behavior of a metallic sphere loaded with a circumferential slot in a plane perpendicular to the direction of incidence is investigated. The slot is assumed of small but finite width with a constant electric field across it, and under this assumption, the analysis of external fields is exact. The scattered field is obtained by superposition of the field diffracted by an unloaded sphere and that radiated from the excited slot, with the radiation strength and phase determined by the loading parameters. Thus the scattering behavior is determined by the loading admittance and the position of this slot. The numerical study is restricted to the frequency range corresponding to 0 < ka< 10.0, and the results presented are primarily for passive loading. The maximum and minimum back scattering cross sections and the loading needed to attain these are presented for different loading positions. The bistatic scattering patterns are computed for a sphere loaded for zero back scattering and from these the back scattering null widths are obtained. The extreme total scattering cross sections are evaluated using the forward scattering theorem and, for small values of ka, the result is compared with that obtained by direct integration of the differential cross sections. Experimental data, obtained using a metallic sphere with an equatorial slot backed by a radial cavity of adjustable depth, is presented for the surface field and back scattering measurements. The results are compared with the computed data. ii

THE UNIVERSITY OF MICHIGAN 5548-5-T ACKNOWLEDGMENTS The authors wish to acknowledge the assistance of Miss A. Maldups and Mr. T. L. Boynton for computer programming and of Mr. H. E. Hunter and Mr. G. Antones for hand computations and technical illustrations respectively.

THE UNIVERSITY OF MICHIGAN 5548-5-T TABLE OF CONTENTS I. INTRODUCTION 1 II. THEORETICAL FORMULATION 4 2.1 Summary of Mie Series 4 2.1.1 Surface Fields 4 2.1.2 Scattered Far Fields 6 2.2 Radiation Problem 7 2.2.1 Surface Fields 11 2.2.2 Radiated Far Fields 12 2.3 Radiation Admittance of Asymmetrically Excited Slot 13 2.4 Complete Problem 15 2.5 Low Frequency Approximation 20 III. NUMERICAL COMPUTATIONS 26 3.1 Computation of Pertinent Functions 26, 3.2 Loading for Zero Back Scattering 35 3.3 Optimum Passive Loading 42 3.4 Surface Fields for a Loaded Sphere 53 3.5 Bistatic Scattering by a Loaded Sphere 54 IV. EXPERIMENTAL STUDIES 71 4.1 Experimental Model 71 4.2 Back Scattering vs. Loading 71 4.3 Back Scattering vs. Rotation 77 4.4 Surface Field Measurements 78 4.4.1 Development of the Surface Field Measurement Facility 78 4.4.2 The New Facility 85 V. STUDY OF TOTAL SCATTERING 90 5.1 Exact Theory 90 5.2 Low Frequency Approximation 96 VI. DISCUSSION 99 6.1 Some Practical Considerations for Obtaining the Desired Loading 99 6.2 Conclusions 103 RE FERENCES 105 APPENDIX A: AN EXTREMUM PROBLEM 108 APPENDIX B: THE IMPUT ADMITTANCE OF AN ASYMMETRICALLY EXCITED RADIAL CAVITY 111 vii

THE UNIVERSITY OF MICHIGAN 5548-5-T CHAPTER I INTRODUCTION When an electromagnetic wave encounters a conducting object, a current is induced which, inturn, reradiates to produce a scattered field. At present, one of the major problems in scattering theory is to develop ways to control this scattering through modification of the shape of the object or by changing the current distribution on its surface. It has long been recognized that minor shape changes can be effective in decreasing (or even enhancing) the scattering cross section, especially at high frequencies, and with the development of absorbing coatings during the past decade, the application of these materials has become one of the most important tools for cross section reduction. However, it is desirable to investigate other means of cross section control, particularly those which are effective in the resonance region and can be used either to decrease or increase the radar cross section. One such method is surface or impedance (admittance) loading and in the last few years it has received increased attention. An additional advantage of this technique is that its application does not require alteration of the body shape or the surface characteristics as do the shaping techniques and the application of microwave absorbers, respectively. In essence, the method is to introduce an impedance over a restricted portion of the surface using a cavity-backed slot, lumped network, or some tther type of microwave circuit, and as such is only a special case of the general theory of surface impedance effects. Mathematically at least, it is similar to the application of absorbers, but in practice differs both in the:localized nature of the region where the loading is employed and in the greater variety of impedances that can be achieved either to enhance or decrease the scattered field. The idea of using loading to reduce the reradiated fields dates back to the 1920's (Meissner, 1929) when it was common practice to use lumped inductors and capacitors to detune the broadcast transmitting antenna supporting structures whenever their lengths were near resonance and interfered with the antenna radiation 1

THE UNIVERSITY OF MICHIGAN 5548-5-T patterns. The first reported application of the loading technique for scatterirg reduction at microwave frequencies was by lams (1950), who used a coaxial loading to decrease the scattering from the metallic posts in a parallel plate pillbox structure. King (1956) investigated the change in the current distribution on a thin cylindrical rod when a central load is introduced, and later Hu (1958) and As and Schmitt (1958) showed that a high reactive impedance can appreciably affect the scattering behavior of such a rod. However, it was not until the recent study by Chen and Liepa (1964a) that the capability of loading for cross section reduction was fully demonstrated. For normal incidence on a thin cylinder of length Q, 0 < t < 2X, the induced current was calculated as a function of an arbitrary central load, and the results were confirmed by detailed current measurements on a model. The back scattering cross section was then determined, and it was found that for every value of V/X within the chosen range, a loading exists for which the cross section is zero. The real and imaginary parts of the corresponding optimum impedances were obtained as functions of 4X, and whereas the required loading was passive when 2 < X, that for X < I < 2X was primarily active. Chen and Liepa (1964b) also considered the scattering in directions other than normal to the surface and extended the analysis to oblique incidence; later Chen (1965a) showed that by using two symmetrically placed loads on a cylinder, its cross section can be reduced with passive loading up to length 2X. Valuable as this work is, however, its usefulness for most applications is limited by the requirement that the cylinder be thin (radius much less than the wavelength). Although Sletten et al (1964) have shown experimentally that reactive loading is still effective when the cylinder is thick, a theoretical treatment of loading of moderately thick cylinders by Chen (1965b) shows that in this case, different loading must be supplied for each circumferential mode, which would be rather complicated to achieve even at a single frequency. A somewhat different and more abstract approach to reactive loading is to represent the body as a one-port (Harrington, 1963; Green, 1963) or n-port (Weinberg, 1963; Harrington, 1964) device, which leads to the expression of the 2

THE UNIVERSITY OF MICHIGAN 5548-5-T scattered field in terms of commonly-defined antenna parameters. However, to use this methodto obtain quantitative results it is necessary to determine the transmitting and receiving properties of the body, and for an accurate treatment this again involves the solution of the boundary value problems. The most simple example of a "thick" body is a sphere, which we shall consider here. A plane wave is assumed to impinge on a perfectly conducting sphere loaded with a narrow slot in the plane perpendicular to the direction of incidence, and in this case only the asymmetric mode is excited. The field scattered in any direction can then be expressed as a superposition of the field diffracted by an unloaded sphere and that radiated by an excited slot at the position of the load. The radiation strength and its phase are determined by the loading characteristics of the slot, and by varying the admittance Y! of the slot, a wide degree of scattering control can be exercised. Even if attention is confined to passive loads (admittance whose real parts are non-negative), substantial increases or decreases in the scattered amplitude -in almost any specified direction can be achieved by an appropriate choice of YI and the slot position 6. Numerical results are presented, primarily for the case of back scattering and surface fields, and these are compared with the measurements made using a model with an equatorial slot backed by a radial cavity of adjustable depth. The agreement is excellent. 3

THE UNIVERSITY OF MICHIGAN 5548-5-T CHAPTER II THEORETICAL FORMULATION 2.1 Summary of Mie Series 2.1.1 Surface Fields Consider first an unloaded perfectly conducting sphere of radius a, whose center is located at the origin of a Cartesian coordinate system (x, y, z). A linearly polarized plane electromagnetic wave is incident in the direction of the negative z-axis, and since there is no loss of generality in tang its electric vector to lie in the x-direction, we choose E A i A ikz Ei ikz and Hi= Ye ikz (2.1) where k is the propagation constant and Y the intrinsic admittance of free space. For convenience, the electric field has been normalized to unity and the time factor e suppressed. If we also introduce spherical polar coordinates (r, 8, p) such that x = rsinecoso, y = rsin9sinO, z = rcos0 with e = O representing the back scattering direction and 9 = Ir the forward one, the incident field may be written in the form (Stratton, 1941) Ei=_7 in 2n+1) (M1) (2.2a) E M -AN (2.2a) l n(n+ 1 —) -oln -e N) 00 Hi= in 2n +1 (N(1) _iM(1) H iy I T(,,,+1) N ln n> (2.2b) n=l non - -eln where M and N(1) are the spherical wave functions (1) Am r-sin m~} ath d m cos mO] M =0- h (ka) Pr(cos () -sin mkO- P (cos)fJ -m-e inO n co Cos j n dO n s inm 0

THE UNIVERSITY OF MICHIGAN 5548-5-T hn(kr) cos m[ krhn(kr)' d m' Ne - rn~ n1) (cs inm+) krkr d P(Cos0) + p(cos ) sinm~ _J ^m [rhn(kr)]' -sn m kr sine n (Co{5nCos MO The prime refers to the derivative with respect to kr. From the requirement that the scattered field represent an outgoing wave at infinity, we are led to assume the scattered field to be of the form o) E = + iB N (2.3a) n=l \ oln n —oeln II iY M(4) M4 (23b) where M(4) and N(4) differ from M(1) and N(1) in having the spherical Bessel function jn(kr) replaced by the spherical Hankel function h n(kr) of the second kind. Application of the boundary condition rx(E +E) = 0 at r=a then determines A and B n n The total field is the sum of (El H) and (Es HS), and the only nonzero components (at the surface of the sphere) are Er Ho and Hp. The last two are of direct concern to us, and they can be expressed, in the notation of Kazarinoff and Senior (1962), as H8 = YsinOT1 () (2.4a) H= Ycos T_(0), (2.4b) where 5

THE UNIVERSITY OF MICHIGAN 5548-5-T T1(H9) =1 in+l 2n +1 1 n + a ( 2j + - (cos (2.5a) 1 = n(n+1)'(ka)" sin) 0 (ka) 80 n n=1 ln n _ D'I P 1(Cos 8 1 2n+1 2n +1 8a pl ~cos)+ i ( T (9) {Iin- as P 1(cos@) + )e)+. (2. 5b) 2 ka = n(n+1)'(ka) 89" n ~(ka) sin9 Here, n(x) = x h (x), and the prime denotes differentiation with respect to the entire argument. P (cos 9) is the Legendre function of degree n and order unity n as defined, for example, by Stratton (1941). 2.1.2 Scattered Far Fields In the far zone the expressions for the scattered field are obtained by replacing h (kr) and its derivatives in equations (2.3) by the leading terms of their asymptotic expansions for large arguments. Using the notation* of Senior and Goodrich (1964), we write -ikr ES = icose - Se () (2.6a) -ikr E = -isine kr S 2(), (2.6b) where S (O) and S (9) are defined as the far field scattering amplitudes and are 1 2 given by co ipInk)(ka) Pl(cOs 0) S()= 2n+l n(ka) a P (cos0) _ D).(c. 1)- (2.7a) 1 n(n + i) n(ka) n C (ka) sin G 2n+ (ka) P(cos9) ( (ka) 5 (9) 1 2n + 1 n an 2 n=1 n(n+1) 1 5'(ka) sinO - (ka 8 ) a Pn(2b Note the change in time convention. 6

THE UNIVERSITY OF MICHIGAN 5548-5-T with bn(x) = x n(x). In the back (9 = 0) and forward (0 = 7r) directions L (cos 9) ] n In(n +1 sin(CO 9 = ep (cos (n) (2. 8a) P (cos0) 7n Ja P 9 )n+l n(n+ 1) = (cose = (-1) (2.8b) Lsin 2' respectively. Hence, from equations (2.7) St(0) = S5(0) (2.9a) 1 2 S (r ) =-S (7r) (2.9b) 1 2 implying that for forward and back scattering the field has the same linear polarization as the incident field. In other directions, however, the field is elliptically polarized. 2.2 Radiation Problem We now consider a separate but related problem of a perfectly conducting sphere with a narrow slot symmetrically placed with respect to the z-axis (and hence, with respect to the incident field direction in the problem just discussed). The slot occupies the region o -2 < 0 o + 2 (see Fig. 2-1) and its angular width 6 is such that ka 6 < 1. Within the gap, the tangential electric field is specified, and in view of our intention to regard the slot as a passive or semiactive* device excited by the incident field, the excitation must be chosen in accordance with the surface field behavior shown in equation (2.4b). It is therefore assumed that for 0e- 0 < 6/2 E=- vcosP (2. 10a) 0 6a E = 0. (2.10b)' By a semi-active device it is implied here that the real part of the loading admittance (or impedance) is negative. 7

THE UNIVERSITY OF MICHIGAN 5548-5-T z 8 -0o FIG. 2-1: SPHERE GEOMETRY. 8

THE UNIVERSITY OF MICHIGAN 5548-5-T This corresponds to a constant (but asymmetrical) voltage v cos 0 applied across the gap. Over the rest of the sphere, EB and Eb are both zero, as is appropriate for a perfectly conducting surface. To calculate the field (E, H ) radiated by the slot, we again postulate a field of the formn shown in equations (2. 3), but with An and Bn replaced by new constants C and D respectively, so that CD Er = (CM(4) + iDN( ) (2. la) H n=i n —-oln n —eln nl ur 1 (4) (4) Hr = iy Nn+iD M( ) (2.11b) When these are substituted into the boundary conditions at r = a, we obtain ina a 1(cos 1 (ka) n.+iD!(ka) a P (cos < n n sine0 nn &O n 6 2 (2.12a) = 0. otherwise from the 0 component, and P (cos 0) {CP(ka) P(cos0) + iD g'(ka) n } = 0 all 0 (2.12b) n=l n n)~ n n n all9 from the 0 component. Moreover, from Bailin and Silver (1956) 7r {P1O(Cos) a P(cos9) + P (cos90) P (cos08) d = 0 P o a 1 1 1 P(08 ) a P(cos98)+ a Pn(cos 9)Pm(co sin d

THE UNIVERSITY OF MICHIGAN 5548-5-T where ro nfm A /m 1 2 2 2n (n+1) 2n+1 nm Hence, by application of these r'lations to (2.12) 0 +-6 kv 2n+ 1 1 1 C = nP (cos))de n n (ka) 22( 2 6 n(cos)dO n 2n2 (n+1) o 2 (2. 13a) P (+)-P (-) kv 2n+ 1 n n nn(k) 2n2(n+1)2 6 and D i kv 2n+l1 1 1 n kv.... 2 26sin 9 P (cos9)de n-" 2n2(n+ 1) o 2 kvsin O p(+) pl() i _o 2n+ 1 n n (2.13b) i (ka) 25 2 1 n 2n (n+ 1) where, for brevity, we have written 11 Os + n no 2 J/ In the evaluation of D it was assumed that the variation of sine over the slot can be neglected and, as a consequence, the position of the slot is now limited by the condition 0 with E >>6. 10

THE UNIVERSITY OF MICHIGAN 5548-5-T The expressions for the radiated field follow from equations (2. 11) upon (4) (4) inserting the above formulae for C and D, and M and N. The equations n n are rather lengthy, by straightforward. Thus, we proceed directly to formulas for the surface fields and the far fields. 2.2.1 Surface Fields On the surface, r = a, only the magnetic field is of direct concern to us, and for this we write, analogously to equations (2. 4), He = vYsinT (e,eO) (2.14a) Hr = vYcospT2 (e,e), (2.14b) 0 where _2n+1 n(ka) sine P )P (' 2a 2 ( + )2 sine )P n(c ) r1 O i 2n +l~ n OD (ka~~~~nk) =Pi W P() 1 3 n= n (n+1'(ka) sin 6''(ka) P (- 1 Cn(ka) P (+)- n n 1 n __ ___ /P (+)- PT(-) + Pf~a.1 s _n / pl(cose. (2. 15b) The convergence of the above series is determined by the gap width 6. For non-zero values of 6 the series are convergent, but rather slowly, and the number of terms that must be retained for numerical evaluation is in fact inversely proportional to 6. In the limit as 6 -- 0 the series for T (0, 0 ) actually diverges every1 o where, but for T2(, 0 ) diverges only when O -- o. More discussion on the convergence and numerical computations of these series is given in Section 3.1. Af

THE UNIVERSITY OF MICHIGAN 5548-5-T 2.2, 2. Radiated Far Fields In the far zone, on the other hand, the expressions for the radiated field components are convergent even for 6 = 0, corresponding to an infinitesimal gap across which a voltage v cos p is applied. We obtain the expressions from equation (2. la) by replacing h (kr) and its derivatives by the leading terms of their asymptotic expansion for large kr in the definitions for M() and N, and by replacing P (+) p () p (+) pl( ) n n n n 6 — and 6 6 6 in the expressions for C and D by the derivative forms n a -P (cos9 ) and - P (cos ) n o n o o respectively. Then, analogously to equations (2.6), we write -ikr E = iv cos e S1 (00 ) (2. 16a) kr 110 - ikr E = -iv sin e Sr(09 )' (2.16b) 0 kr 2 0 where the radiated far field amplitudes are given by 0 - r k n+i 2n+1 1 1 S1(0, ) = sin... P (cos) P (os 1 on=1 n2(n+ 1) 2 P (cos9) P (cos ) + i n n 0 (2.17a) r (ka) sin 0 sin~ 0( k >in0 _ 2 a _ P (os O ) r k O i n 2n+1 1 P ci 1 S (9,9) =2 sin 2o2 n'(ka) sin9' 89 n o 2 00n-i n (n+1) n o + (ka)a9Pn (cos9)n Pln O )sin9 J (2.17b) 12

THE UNIVERSITY OF MICHIGAN 5548-5-T In the back and forward directions Sro0 eo) = (0,e ) (2.18a) S(r, 0e) = -S2 (7r, 0. (2.18b) 2.3 Radiation Admittance of Asymmetrically Excited Slot It is customary to define the admittance of a slot as the ratio of the current flowing away from the slot to the gap voltage, where the latter is the line integral of the electric field across the gap. This definition, however, is unique only for TEM waves, and for other modes the admittance is usually defined to best suit the particular situation. In this problem of an asymmetrically excited slot, the total instantaneous current is in fact, zero, and we are led to introduce the concept of admittance density, defined as admittance per unit length of the slot. We define the admittance density as twice the ratio of the complex power flow density to the modulus squared of the applied voltage. The complex power radiated per unit length of the slot is 6 6 o 2 W - \ _(E xHr).ade 62 - 46 o 2 where the tilde denotes the complex conjugate, and the radiation admittance density is therefore 13

THE UNIVERSITY OF MICHIGAN 5548-5-T 2w - = -s5 Yr Vc....cos 6 o 2 H de and since is independent of 0, we write 0 6d o 2 Y = -2T rs(e, O de. o 2 The (total)ation aof the Legendre ttance is 227r Y y a sin( dn (2.19a) r r 0r - and since y is independent of 0, we write ( 2 r -27rsin0 Ya T (O 0 )dO (2 19b) r 0 6o _6 2 o' o 2 Using the expression for T2(O, 0 ) given in equation (2.15b), together with the evaluation of thequally aeptablgendre defunction integrals used in the density is the ratio of n theand by n+ 1 In this case the resultant radiation admittance is of a (2.20) An equally acceptable definition for admittance density is the ratio of the

THE UNIVERSITY OF MICHIGAN 5548-5-T The surface current density J is directly related to the tangential magnetic field by J = rxH (2.21a) so that on the surface of a sphere J = -H and J H (2.2lb) The radiation admittance density is then r yv = _Jo; o r vCos 09=e, = -YTr('f o ) and the (total) radiation admittance 2 7r Y' = y a sinO do r r o = -2rYsinO aTer( ) (2.22) o (2 22) The 9' denotes the edge of the slot and is given by either 0' = 0+or 9' = o 2 o 2 o2 Except for o = 7r/2 or when the gap width 6 -- 0, the two values give different results for Y, and although the numerical effect may be insignificant under most practical circumstances, it is undesirable that an expression should have that degree of arbitrariness. Another difference between this result and the-onepreviously derived is that for Y the surface field component Tr2( O ) is integrated across r 2 o the slot (equation 2.1 9b), and as a consequence the convergence of the infinite series is accelerated by a factor 0(1/a). 2.4 Complete Problem The final problem to be considered is the combination of the scattering and radiation problems in which the plane wave given in equation (2.1) is incident on 15

THE UNIVERSITY OF MICHIGAN 5548-5-T the slotted sphere. It is assumed that the same voltage v cos p is excited across the gap, and the expressions for the resulting total fields can be found by superposition of those associated with each individual problem. We therefore have E = Ei+Es+Er (2. 23a) and H = Hi+Hs+Hr (2. 23b) where (E, Hi), (E,Hs) and (Er, Hr) are as defined before. In particular, in the far zone the components of the total scattered electric field are written, similarly to equations (2. 6), as -ikr E= i cos ekr S1(e) (2.24a) -ikr kr 2 (2.24b) Ep= -isin kr S2(e) v (2.'24b) where the total scattering amplitudes are now given by S (e) = S (O) + v Sr(, ) (2.25a) S2(O) = S2() + v Sr( ). (2. 25b) The expressions for S1r(e 0 ) and S2r(0, B ) are, of course, independent 11P0 2 0 of v, and if this voltage is excited across the slot by the currents induced by the incident field, the voltage can be related to the loading admittance of the slot. To calculate this voltage, we apply the same technique that was used for determining Y r. As before, the complex power entering the slot per unit length is obtained by integrating the Poynting vector over the slot. Hence, 6 + w = -2 (ExH) rad 2 o 2 16

THE UNIVERSITY OF MICHIGAN 5548-5-T where (E, H) are given by (2.23). But since v Cos0 6a the s1ot admittance density is simply +6 = -N+H +H r Yl 6vcos o 2 which, by using equations (2. 19), (2.17b), and (2.5b), can be written as y R 2 Y= -Yr+ l T2(O) de o 2 For 6 sufficiently small, the variation of T2(O) across the slot can be neglected, giving = Y+ + T2(9o) and the (total) loading admittance of the slot is therefore Y =-Y +- 27rasine T( ), (2.26) I r v o 2 o where T2(e) and Y are given in equations (2. 5b) and (2.20), respectively. Solving for v from equation (2.26), = + Y 2rasinO T2(O ) (2.27) and, when substituted in (2.25), the component scattering amplitudes for the loaded sphere are Y r S1(8) = S(O)+ y+y 2rsin9 T2(90 )aS (e,e) (2.28a) 1 r 17

THE UNIVERSITY OF MICHIGAN 5548-5-T I r Each scattering amplitude is a sum of the scattering amplitude for an unloaded h\e*re and the modification term accounting for the presence of the slot. This term contains radiation and loading admittances, a term for the value of the surface current at the position of the slot when the slot is not present, and the radiation pattern from the slotted sphere when it is excited by an asymmetric (unit) voltage across the gap. In terms of these scattering functions, the bistatic radar cross section is a(0, 0) = ae(9, ) + a(e, ), (2.29) where the component cross sections are given by af (t,) _A I S ()l2 cos2 2 (8e )sin2. pa(, 0) = -JS2(6)2 n2. In particular, for back scattering and forward scattering directions 2 (0) = -r iSl(0)2 (2. 30a) 2 = x S2(0)I (2. 30b) and 2 Or= L ls1(7r) (2.31a) 2 = S2(7 ) (2.3lb) respectively. 18

THE UNIVERSITY OF MICHIGAN 5548-5-T To make the scattering amplitude S1(8) zero in the direction 0 = 9' the required loading is a sr(e,, eo) = -Y -Y2r sine0T2(eO) s.(,) (2.32a) Similarly, to make S2(9) zero in the same direction, the required loading is aS (9' 0 ) yI =Y - Y 2r sin T (9 2 0 < (2.32b) L - o 2 o 5 and unless a 0 a'Sr( I., 0 1'o = 2 0 (2.33) Ss (') s (9l) 1 2 the scattering cross section cannot be reduced to zero with a single slot loading of the form discussed here. This does not, however, rule out the possibility of significant reduction in scattering by a suitable choice of Y1. Moreover, in many cases of bistatic scattering only S1(9) or S 2() is of interest, and in such a case zero scattering can be achieved. The obvious exceptions to the above are back and forward scattering, in which cases equation (2. 33) is satisfied. (See equations 2.9 and 2.18). Furthermore, aS r( o )= (ka)2 sin T (0) (2.34a) 1o — 4 o 2o and hence the loading for zero S1(0) and S2(0) is 1 2 Y = -Y -Y s2 kasinOoT2(e)}2 {S (O01 (2.34b) Similarly, in the forward direction we get 19

THE UNIVERSITY OF MICHIGAN 5548-5-T 2ka) aS;(r, 80) 4 s ine T2(7r-9 ) (Z.35a) and consequently for zero Sl(7r) and S2 (r) 1 2 Yd -Y - Y (ka sin 8) T2(e )T2(r - e 7sr (2 35b) The fact that for a passive scatterer, zero scattering in the forward direction implies zero total scattering and absorption indicates that the above loading will have a negative real part, corresponding to an active slot for all values of ka and 80. The expressions for the total surface field components can be obtained directly from the far field expressions. Since the technique used in the derivation of the far fields is quite general, the components of the total surface fields are obtained by replacing the far field formulae in (2.28) by the corresponding surface field expressions. Thus, t Y r (2.36a) T (9) = T1() + 2r sinO T2( )aT;(/,/) (2. 36a) r t Y T2(9) = T2 ()+ y 2r sin8 T 2( )aT2(e, ) (2.36b).1 r and, analogously to (2. 4), the total surface field components are given by t H8o= Y sin 0 T1 (8) (2.37a) H = YcosoT 2(9) (2.37b) 2.5 Low Frequency Approximation The characteristics of the above derived formulae can, in general, be determined only by numerical computation of the functions involved. However, an exception is the case of small ka. Taking the leading term of the series expansions of spherical Bessel functions about ka =O (see, for example, Stratton, 1941), 20

THE UNIVERSITY OF MICHIGAN 5548-5-T 00 jn(ka)= 2n(ka)n I (n)+ (m+irn) (ka)2m m-0 -1(2n-2m+l) 2m Yn(ka) = (ka)n+l mt(n-m+1) (ka) 2m we get jn(ka) - 2 (ka)n n, ka n< 1 (2n+ 1)' ka << -1 (2n) 2, (ka) *ka << 1 2 n(ka)n+ 1 n and when these are substituted into equations (2. 7), the scattered far field ampli-. tudes for an unloaded sphere reduce to S1(O) -i (k (2 cos + 1)+ (1 - 4 cos (2.38a) 2 ~s (ka)3 (ka)6 S2 () -i (2+ cos 9)+ 6 (cos 0 - 4) (2. 38b) and the radiated far field amplitudes, from equations (2.17), reduce to aSr(e, )- 3 (ka) sinO (l+ikacosOcosO ) (2.39a) 1 0 8 o o a~r~e 3 2 aS (e, (ka) sin8 n (cos9+ikacoso ). (2.39b)'00 O8 o o Similarly, for the surface field component T (e), we obtain from equation (2.5b) T 2()3 _ -ika 3c 8+ 12 sin) 2 (2.40) 2 (19). — 2 12 sin and for the real part of the radiation admittance we obtain from equation (2.20) Re 7r sin(ka) 21

THE UNIVERSITY OF MICHIGAN y 5548-5-T Its counterpart, Im -, is not so easily expressible in closed form, however, and since there is no need for it in our investigation, no attempt is made to reduce it. The scattering amplitudes for a loaded sphere are obtained by substituting the above small argument expansions into equations (2.28) and, in particular, with a slot at 8 = 90, their magnitudes are 0 Is1() 4| 2 2 j1cos08 +A2cos- +A (2. 41a) l +A x +y 1 2(@) 1 4 2 2 {A3Cos @ + A2Cos 0 + AX, (2. 41b) where A1 = 4(km) x +4(ka) y A = 4(ka) x +4(ka) y +29r(ka)y (ka)22+ 2 2 9 81 2 Ag = (ka) x +(ka) y r (ka)y 1 1 with x = Re (Y +Yr) and y = Im l(Yl+ Y ) The loading needed to make either IS1(8') | or S2(z') 1 zero in the direction 8 = 8' follows directly from equations (2.32). For 8 = 01 the two are identical, and if 0 = 90, the loading is 0 Y Yr 3 1 Y y 4 ka When substituted in equations (2.41) this gives Is(o) 12 = I 2(0) 2 = (ka)6(1 - cos 9)2 22

THE UNIVERSITY OF MICHIGAN 5548-5-T Since the moduli of the scattering components are equal, the scattering is independent of O, which, for a vector problem, is rather surprising. However, a further examination of equations (2. 38) and (2.39) reveals that this is a special ease and occurs when the imaginary terms in (2. 39) disappear, i.e. when o = 90~ 0 In other words, the radiation field is a "mirror image" in 0 of the scattered field and when the two are combined, the result is independent of 0. For presentation purposes it is convenient to write the results in terms of component cross sections (equation 2.29) and then normalize the resulting values to the back scattering cross section ao of an unloaded sphere. Thus, for an unloaded sphere, - ( 1 2 cos 0+1)2 cos2 +(2cos+1)2 sin (2.42a) 0 or the sphere loaded or zero back scattering, and,, for the sphere loaded for zero back scattering, - ) 4 (1 - cos)2. (2.42b) 2 9 The results for 0 =0 and 0 = 7r/2 are plotted in Figs. 2-2 and 2-3 respectively. 23

THE UNIVERSITY OF MICHIGAN 5548-5-T 1800 2.0 1. 0.5270 90~ I %! \ % I 1.0.e 0~ tEi FIG. 2-2: NORMALIZED SCATTERING PATTERNS FOR LOADED ( —) AND UNLOADED ( —-) SPHERES IN E-PLANE (0 = 0) WITH 0 =90~ AS ka-0. o 24

THE UNIVERSITY OF MICHIGAN 5548-5-T 180~ 2.O 1.5 0. - 270~0 90~ \. - \,/ I 0.0 0 0 I L Hi FIG. 2-3: NORMALIZED SCATTERING PATTERNS FOR LOADED( ) AND UNLOADED (- - -) SPHERES IN H-PLANE (p = i/2) WITH 9 900 ASka-O. 25

THE UNIVERSITY OF MICHIGAN 5548-5-T CHAPTER III NUMERICAL COMPUTATIONS 3.1 Computation of Pertinent Functions The expressions of all pertinent functions derived in Chapter II were programmed for computation on the University of Michigan IBM 7090 computer. These included the surface and far field expressions (equations 2.5 and 2.7 respectively) for a solid sphere, radiated surface and far fields (equations 2.15 and 2. 17 respectively) due to radiation from an excited slot, and the radiation admittance (equation 2. 20) of this slot. Except in the cases of the radiated surface fields, Tr(o 9o) and T2(O 60), and the radliatlon admittance Yr, the computations were straightforward. The infinite series were approximated by the same, but truncated series and the number N of terms retained in any given series was determined essentially by the machine. A criterion was set to terminate the series -7 whenever the magnitudes of successive terms fell below 10. On the average, only about 4 terms were required for ka = 0. 1, but as many as 30 for ka = 10. 0. The spherical Hankel and Legendre functions are contained in all the series and their evaluation was carried out using external subroutines. The function C'(ka) was written in the form V'(x)=2n + 1 -hn- (x) (n+ 1)h + ) +h (x) 2n+ 1'n- 1 (x1 1 n and h (x) was itself broken up according to h (x) = (x)- iy n(x). The spherical Bessel functions of the first kind, j (x), wvr evaluated by numerical integration of the finite integral expression )n r/2 in(X) = (n Scos(xsin )cos2n+ do (see, for example, Adams and Hippisley, 1947). The range of integration was 26

THE UNIVERSITY OF MICHIGAN 5548-5-T subdivided into 40+2n increments, and judging from spot checks, the resulting evaluations were accurate to six significant figures for n < 20 and to five for no< 44. The number n = 44 was the largest value, consistent with machine capacity, for which jn(x) could be evaluated. The spherical Bessel functions of the second Jn kind were evaluated from the finite series expansion (-1)n+ n r n/2r Yn cos(x+W. -2r Yn(X) = + r-O (2r)[ (n- 2r). (2x)2r n-1 - sin(x+) (-1)r(n+2r+1) - sin(x+ ) r=0 (2r+ 1)! (n- 2r- 1): (2x) (Watson, 1948), giving seven digit accuracy for n < 20 and better than five for n < 44. The Legendre functions were computed from the recurrence relations n(cos 0) n+ 1osP (cos8) -D lP (cos ) n- n+l n n+ n- 1 and 2(cn cos0) (cos)(cos- P1 (cos0) n+1 n n n- 1 starting with P (cos ) = 1, Pl(cos O) = cos 0 1. 1 3 and P 1(cos ) = sin, P2(cos ) = sin 28 1 2 2 respectively. The differential form P (cos 0) was then obtained from the relation 1 11 P (cos 0) P (cos 0) a n ( n-i P (cos9) = ncos n - (n+1) sin ae n sin sin with the conditionals 27

THE UNIVERSITY OF MICHIGAN 5548-5-T [-~ Pln(csO]S = n(n+1} 106 n 3=0 2 La P (cos)] (l)n n+1) n 2 The values of the Legendre functions computed in this manner are believed to be accurate to seven significant figures for n < 50, but the subroutine has been used up to n = 300 with no appreciable loss in accuracy. For reference purposes, the back scattering cross section for an unloaded (solid) sphere, normalized to the physical optics value ira, is presented as a function of ka in Fig. 3-1. The formula for computing these values is given either by equation (2. 30a) or (2. 30b), but since there are already adequate tabulations of the back scattering cross sections in the literature (Bechtel, 1962; Hey et al, 1956), no direct computations were performed. The evaluation of the series for the radiation admittance Y is complicated by the slow convergence of its imaginary part for all non-zero 6. This is a consequence of the local capacitance in the vicinity of the gap and, indeed, in the limit as the gap width tends to zero, the series for the imaginary part fails to converge. In contrast, the series for the real part is rapidly convergent even for 6 = 0. The first N terms of the series are treated exactly, and to facilitate the computations, the subsequent terms are replaced by their asymptotic forms for large n. Since C'(ka) n x (ka) ka n for n >> ka, and P (cos )' 2r s {n+ 1 } Pcn ir sine 2 4 28

3 H 2 - t~~~~~~~~~~~~~~~~~~~c~~~~~~c1 I —,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~r 0 b ulCf I -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I — 1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ C) 0 2 4 ka 6 8 10 FIG. 3-1: NORMALIZED BACK SCATTERING CROSS SECTION OF UNLOADED SPHERE.

THE UNIVERSITY OF MICHIGAN 5548-5-T for n >> cosec 0, substitution of these expressions into the higher order terms of (2. 20) gives r 2 2n+ 1 n (n n n ( n n Y i 2 E 2n+ 12 1(ka) pn(+)- pn) I (ka) /pl6+)6 Pn( n=1 n (n+ 1) n o n Oo in6 2 n=N+ l 2n+ 1 2 n ka sin 0 O +sin(2n+1)6}- ka n 1-sin(2n+ 1)0 n=N+l (n+ 1) O 0 (3.1) In all of the computations, N was given the largest value consistent with machine capacity, and 1000 terms were retained in the second series. This last is certainly more than sufficient for our purposes, and the only possible source of error then lies in the use of asymptotic formulae. To get some feeling for the probable magnitude of these errors, Yr/Y was computed for ka = 5. 0, 6 = 0. 0392 and e = 900 using four different values for the upper limit of the summation variable 0 in the first series, and the results are summarized below: N = 10, Y /Y = 15.255+i29.686, r N = 15, Y /Y = 15.255+i30.385, N = 20, Y /Y = 15.255+i30.676, N=24, Y /Y = 15.255+i30.748. r (N = 24 is the maximum attainable by the machine for ka = 5. 0). The rapid cony vergence of the series for Re - is reflected in the constancy of the real parts above, and if the trend of the imaginary parts remains the same as N is increased still further, the computed magnitude of Im -Y with N = 24 should be within one percent of the correct value. Since Y is symmetric about 9 = r/2, it is sufficient to restrict attention to /O 90~0 The gap width chosen for computation was 6 = 0 0392 (approximately 30

THE UNIVERSITY OF MICHIGAN 5548-5-T 2.25 ), and this was determined by the equivalent slot width of the experimental model (Section 4. 1). The sample computations of Y /Y for 0 < ka < 10.0 with slot positions at o = 450 and 0 = 90- are shown in Figs. 3-2 and 3-3 respec0 0 tively, and from these it appears that the change of the slot position does not affect the general character of the loading behavior. The real parts are zero for ka = O and rise through positive values with a small but regular oscillation as ka increases. The imaginary parts, on the other hand, have a negative singularity at ka = 0, but as a consequence of the asymmetric excitation, they become positive at ka around unity, and then remain as such. To investigate the effect of the slot width on the radiation admittance, Y /Y was computed as a function of 6 for ka = 4.28 with 9 = 900. As shown in Fig. 3-4, 0 the real part is essentially constant for the full range of 6 considered, but the Imaginary part shows a significant variation due to local capacitance across the gap. At 6 = 0 the imaginary part has a positive singularity and then is monotonically decreasing as 6 increases. Computation of the radiated surface field components T (9, e ) and T r(, ), 10o 2 o given by equations (2.15), is even more involved than that of the radiation admittance Yr/Y. Whereas the terms in the series for Yr/Y are O(1/n3) for n large, r r 2 for aT l(,0 ) and aT2(,0e ) they*are O(1/n) and O(1/n ) respectively. Fortunately, the terms in the series alternate in sign in groups of 2r /6 terms, and therefore may be treated as an alternating series. The first 43 terms (or less, whenever the machine capacity was exceeded) were computed exactly. From there on, to facilitate the computation, the ratios of the Hankel functions were replaced by their asymptotic form _'(x) 2(n+ 1)(2n- 1)- (n- 1)x2 + nx V(x) 3x 2(2n.- 1)x+ x The factor a has been incorporated to make equations (2.15) functions of ka rather than ka and k. 31

THE UNIVERSITY OF MICHIGAN 5548-5-T rz z I.q I \ z 0 a., 0 ro 0 \,o I~~~~~~~~~~~~~~~~~I \~~~~~~~ I. x~~~~~~~~~~ P o-e'4~~~~~~~~~~~~~~P 0E I I, I,I 32

40 1J13 20 _ - C Yr/Y! /'I I 0 -20 0 2 4 ka 6 8 10 Z FIG. 3-3: REAL ( ) AND IMAGINARY (- - -) PARTS OF NORMALIZED RADIATION ADMITTANCE FOR 0 =900 AND 6 = 0.0392. O

40 - % 30 \ \\ C Z 20 10 Gap width of the experimental model E 0.05 0.1 6i 0.15 0.20 0.25 FIG. 3-4: REAL ( —) AND IMAGINARY -— ) PARTS OF NORMALIZED RADIATION ADMITTANCE FOR ka=4.28 AND 0 = 90~. o

THE UNIVERSITY OF MICHIGAN 5548-5-T and the Legendre functions by their corresponding asymptotic forms, but only wher n > 100. As such, the terms for both series contained the same factor n6 sin -- and approached zero as n6 2 Ems, m= 1,2,3... A comparison of partial sums terminated at 2m7r n -6 showed that for m a 3 the results, for all practical purposes, were identical. To ensure good accuracy for the numerical evaluation, the series were terminated at m = 6, and with 6 = 0. 0392 this corresponded to 962 terms. A sample computation for the amplitude of Tr(0, r /6) with ka = 1.03 and 8 = 300 is shown in 0 Fig. 3-5. The amplitude at the front (8 = 0) is rather large, as compared to that at the back, and becomes even larger (but finite) in the region near the slot. A similar computation for Tr(Or /2) was also performed for ka = 4.28 with 8 = 900. The field, as one would expect, is symmetric about 90, but its amplitude is more uniformly distributed over the surface of the sphere. 3.2 Loading for Zero Back Scattering Once the numerical results for Ss(0), T2(80 ) and Y /Y are available, it is a simple matter to determine the loading admittance Y for zero back scattering from equation (2. 34b). In Figs. 3-6 through 3-10 the real and imaginary parts of the loading admittance are presented for 0 < ka A 10, with 0 = 300, 450, 600, 900 and 1200 respectively. All the curves are quite irregular and, as ka -0, the imaginary part becomes infinite. The real part is, however, of more importance. Bearing in mind that the non-negative values of the real part correspond to a passive load and the negative values to an active load, it is apparent that zero back scattering can be obtained with a passive load only when Re YLt: 0. As we see from the graphs, the loading is quite dependent onthe slot position e, with 35

THE UNIVERSITY OF MICHIGAN 5548-5-T _ Xq o. 0 0 a I Si 0.t < 36

40 20 Lo t~dz 0 - IOr~~~~~~~~~~~~~~~~~ --- c.wo o -20'c._ -40 0 \ t -60 I. I I, I I I I I. I 0 2 4 6 8 10 C FIG. 3-6: REAL ( ) AND IMAGINARY ( —-) PARTS OF NORMALIZED LOADING ADMITTANCE Z FOR ZERO BACK SCATTERING WITH 0 = 30. O

40 20 400~~~~~r / /Y 0 own comm ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~/ 1 -20~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~2 -40~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-601 -460I I r I p I,10 2 ka 6 8 10 FIG. 3-7: REAL (-) AND IMAGINARY (-. -) PARTS OF NORMALIZED LOADING ADMITTANCE FOR ZERO BACK SCATTERING WITH 9 = 45 0

40 - \ I \ - 0I FOR ZEROl, I~ ~ ~' BAKSATRNZIHC=6 C.0o ~ ~ ~ ~ 4C -40' -80 I p _ I _I I I I 0 2 4 6 8 10 FIG. 3-8: REAL (-) AND IMAGINARY (- - -) PARTS OF NORMALIZED LOADING ADMITTANCE FOR ZERO BACK SCATTERING WITH 0 = 600 0

20 I10%~ ~ ~ ~ ~ ~ ~~~~1 I H 0 I moo*~~~~~~~~~~~ 0 /. I, Y YI z' II I I I I I I j I c V~~~~~~~~~~~~~~~~~~~~~~~~~C ci3 I H 0~~~~~~~~~~~~~~~~~ -60 I I -80 / 0 I -d II 0 -100 II I p I I I I a I 0 2 4 6 8 10 FIG. 3-9: REAL (-) AND IMAGINARY (- - -) PARTS OF NORMALIZED LOADING ADMITTANCE FOR ZERO BACK SCATTERING WITH 0 = 900 0

40 -l 20 L \ rJ -60 \ / ~\_ / 0 -60 - \ /'I I I I c~,,,,,',, -80 wI, I. I. I I I' II 0 2 4 ka 6 8 10 Z FIG. 3-10: REAL ( —) AND IMAGINARY (- - -) PARTS OF NORMALIZED LOADING ADMITTANCE FOR ZERO BACK SCATTERING WITH = 1200. 0

THE UNIVERSITY OF MICHIGAN 5548-5-T o increasing irregularity as 0 increases. For instance, with 0 = 30, the real part is non-negative for almost the entire range of ka considered, but as 0 goes to 900, or even 120, both the real and imaginary parts become rapidly varying about the zero axis. This variation can be directly associated with the spacial separation between the slot and the equivalent phase center for the returned signal from an unslotted sphere. This phase center is located somewhere near the front of the sphere. An increase in separation makes the load more frequency sensitive, while a decrease has the opposite effect. Unfortunately, as the slot is moved towards the front, its total circumferential length decreases, and the amount of energy available for "reradiation" is not sufficient to cancel out the far field. In such a case an active load is required. In Fig. 3-6, for 0 = 300, we see the cross-over appearing at about ka = 1.5 with the values of Re YI/Y being just 0 slightly negative. When the slot was moved to 0 = 25, an exploratory calculation showed that the real part is even more negative. 3.3 Optimum Passive Loading Since passive loading is of most interest, let us now consider the maximum and minimum values of back scattering cross sections that are attainable with such a loading. From equations (2.28), (2.30) and (2.34) we can write the back scattering cross section o(0) for a loaded sphere, relative to the back scattering cross section a for an unloaded sphere, as = 1+ a+ b2 (3.2a) a- x+ iy 0 b2 2ax 1+b + 2ax + 2by (3.2b) =1 2 2 2 2 2 2' (3.2b) x +y x +y x +y where lr kaslineT(()1 Y a+ib = --. 2S5(0) YI+Y 42

THE UNIVERSITY OF MICHIGAN 5548-5-T x+iy= y + y with a, b, x and y being real quantities. The maximum and minimum values of the above cross section ratio, as well as the passive loading that is required to attain them, are determined in Appendix A. By passive loading we here mean only that Re 0y >/0 and, by no means, that the resultant loading admittance is always physically realizable. A discussion of synthesis methods is presented in Section 6. 1 The computed results for the extreme cross sections and for the corresponding passive load are presented for various slot positions, 8, in Figs. 3-11 through 3-19. First, taking the minima, we observe that there is a complete reduction in the cross section wherever Re rY/YI > 0. For example, with 9 = 30~ (Fig. 3-11) I 0 0 the cross section is zero for almost all ka, 0 < ka, 10. 0. (At ka = 1.5, the real part does cross the zero axis, but even then the reduction is over 40 db.) Next, let us consider the result with 9O = 450 in Fig. 3-12. Here the return is zero for o ka<4.25, but as Re Y/Y~ in Fig. 3-7 swings from a positive to a negative value, the minimum return increases from zero and rises to a peak value at ka _ 8. 0 which is only infinitesimally less than for an unloaded sphere. It then appears to fall back to zero somewhere around ka = 12.0. The pattern apparently is repeated without end. Any further displacement of the slot towards and into the shadow region (9 > 90 ) compresses this "stop and pass band" pattern and decreases the 0 total ka range over which the scattering can be appreciably reduced. For instance, with 0 = 300 almost the entire band 0 < ka < 10 is covered, whereas with 0 = 600 o o the coverage has decreased to 53 percent and with o = 1200 to only 33 percent. 0 If, on the other hand, we aim for the maximum return, an arbitrarly large enhancement can be achieved in all cases by taking ka sufficiently small, but since this is only a consequence of a higher order zero in the normalized function a, the result is sonmewhat misleading. To consider an enhancement that is more relistic, we shall confine our attention values of ka > 1.0. The maximum return, in such a case, appears to be close to 20 db for a particular value of ka and with 43

THE UNIVERSITY OF MICHIGAN 5548-5-T 20 10 0 - -20 I I i I, I, I, I I 0 2 4 6 8 10 FIG. 3-11: MAXIMUM AND MINIMUM RELATIVE BACK SCATTERING CROSS SECTIONS FOR PASSIVE LOADING AT 9 = 300. o 44

THE UNIVERSITY OF MICHIGAN 5548-5-T 20 10 Q -10 -20 I I I I I, I, I 0 2 4 6 8 10 ka FIG. 3-12: MAXIMUM AND MINIMUM RELATIVE BACK SCATTERING CROSS SECTIONS FOR PASSIVE LOADING AT = 450. o 45

150 I I 50 1 00 1 a-50L - ka 0 FOR MINIMUM BACK SCATTERING WITH = 45 0 I I \ I I, I,, I 0 2 4 ka 6 8 10 Z FIG. 3-13: REAL ( —— ) AND IMAGINARY ( —-) PARTS OF NORMALIZED PASSIVE ADMITTANCE FOR MINIMUM BACK SCATTERING WITH 0 = 45 O

60 I I 40 I 20 - Fr1 0 -20,' -40 I I. I, I, I 0 2 4 6 8 10 FIG. 3-14: REAL ( ) AND IMAGINARY ( —-) PARTS OF NORMALIZED PASSIVE LOADING ADMITTANCE FOR MAXIMUM BACK SCATTERING WITH 0 = 450. 0

THE UNIVERSITY OF MICHIGAN 5548-5-T 20 10 la II -10 -20 I,I. I I I.1. I 0 2 4 6 8 10 ka FIG. 3-15: MAXIMUM AND MINIMUM RELATIVE BACK SCATTERING CROSS SECTIONS FOR PASSIVE LOADING AT 0 = 600. o 48

150 I \ ~~~~~~~~~~~~~~~~~~~~~~~~1~~~~ I I 100 1 100 H 50 0 -100 -100 \ \ -150 I - I a' I I a 0 2 4 6 8 10 Z FIG. 3-16: REAL ( —) AND IMAGINARY (-.-) PARTS OF NORMALIZED PASSIVE LOADING ADMITTANCE FOR MINIMUM BACK SCATTERING WITH e = 600 0

60 I 1 40 - r 20 - ~~~~~~~\ m~~~~~c -20 -40 - _. I * I I I, I' I 0 2 4 6 8 10 ka FIG. 3-17: REAL (-) AND IMAGINARY ( —-) PARTS OF NORMALIZED PASSIVE LOADING ADMITTANCE Z FOR MAXIMUM BACK SCATTERING WITH 0 = 600. O

THE UNIVERSITY OF MICHIGAN 5548-5-T 20 10 -10 I, I. I, J 0 2 4 6 8 10 ka -20

THE UNIVERSITY OF MICHIGAN 5548-5-T 20 10 -20 L, I I I I I a I 0 2 4 6 8 10 ka FIG. 3-19: MAXIMUM AND MINIMUM RELATIVE BACK SCATTERING CROSS SECTION FOR PASSIVE LOADING AT e = 1200 o 52

THE UNIVERSITY OF MICHIGAN 5548-5-T 0 somewhere in the neighborhood of 60. As 0 is changed to 30,0. this value 0 0 decreases to 15, db and, for 0 = 120, it is a mere 12.8 db. The maximum average enhancement for the range 1 ( ka < 10 also appears to take a similar pattern. With 0 = 600 it is 11.2db, but with = 300 and 1200 it decreases to 0 0 7.8 and 4. 7 db respectively. Figures 3-13, 3-14 and 3-16, 3-17 show the passive loading admittance required to attain the given extremum scattering with 0 = 450 and 600 respectively. In the case of a minimum return, the optimum passive loading LY/Yi is given by when Re 0 when Re Y/Y] 0,t but when the Re Y/ < 0, the optimum real part is set zero and the corresponding imaginary part rapidly approaches a large value (Figs. 3-13 and 3-16). For a maximum return the real part of the optimum loading [Y/Y] is zero and the imaginary part is a slowly varying function max with a negative slope behavior (Figs. 3-14 and 3-17). 3.4 Surface Fields for a Loaded Sphere The expressions for the surface fields on a loaded sphere are given by equations (2. 36), and from these the numerical results were computed for ka = 1.03 and ka = 4.28 with 0 = 300 and 900, respectively, when the sphere was loaded 0 for zero back scattering. The amplitudes are presented in Figs. 3-20 through 3-23, along with those for the unloaded body. The behavior of the surface fields on a loaded sphere is somewhat surprising. In the light of the results by Chen and Liepa (1964a) on the loading of thin cylinders, it was expected that for a sphere loaded for zero back scattering the amplitude of the surface fields would also decrease when ka is unity or less, but as the computed values indicate, this is not true. In fact, the fields are Actually enhanced. An examination of the equations for the total scattered fields for the cylinder and the sphere show that they can be treated as a sum of the fields scattered by an unloaded body and the radiated fields whose strength and phase are determined by the loading parameters. In the case of a cylinder with central loading and of length X/2 53

THE UNIVERSITY OF MICHIGAN 5548-5-T or less, the two fields are similar. When the loading is (say) chosen for zero back scattering at broadside, an overall reduction of the far fields as well as the surface fields results. For a sphere' loaded with a slot in a plane perpendicular to the direction of incidence, the unloaded scattered field and the radiated field are siddimilar, and when such a body is loaded for zero back scattering the total surface field is modified, but not necessarily reduced. This is demonstrated by the numerical computations presented here. For ka = 1.03 the loading at 0 = 300 0 increases the fields at the front of the sphere by a factor of two, and even more in the region of a load. Over the remaining surface (say, 0> 60 ) the fields are slightly reduced. In the other case, ka = 4.28 with 0 = 900, the loading does 0 not appreciably change the surface field amplitude, but induces in it periodically varying oscillations. 3.5 Bistatic Scattering by a Loaded Sphere The general expressions for the scattering by a loaded sphere were derived in terms of the far field scattering amplitudes 81(0) and S2(0) in section 2.4. We normalize the bistatic cross section by the back scattering cross section of an unloaded sphere for suitability in computation, and for ease of presentation, restrict our attention to the E-plane (~ = 0) and the H-plane ( 7 = r /2). Thus, from (2.29) and (2.30), in the E-plane (0!, O0) 2 (3.3a) 0 and in the H-plane o(0, 7r/2) = IS2(0) 12 (3. 3b) o To study the bistatic behavior, the two equations were programmed and computed at 0 = 0(5)1800 for ka = 4.28 and for those values of ka at which the as well as for a cylinder with central loading of length X or longer. 54

THE UNIVERSITY OF MICHIGAN 5548-5-T oscillations in the back scattering cross section for an unloaded sphere (see Fig. 3-1) have the extremum values. These correspond to ka = 1.03, 2.33, 3.55, etc for the maxima and ka = 1.75, 2.96, 4.16, etc. for the minima. Two slot positions 0 = 300 and / = 90~ were considered and the loading for zero scattering in O O the back direction (9 = 0) was chosen. With 8 = 300 the loading is always passive, 0 but with 9 = 900 there are values of ka where active loading is required and such, o for example, is needed at ka - 2.33. Sample calculations for ka = 1.03 with 80 30~ and 90 are shown in Figs. 3-24 through 3-27 and for ka = 4.28 with = 900 in Figs. 3-28 and 3-29. Included, 0 for comparison purposes, are the scattering patterns for the corresponding unloaded cases. From these patterns, and the others that were computed, we can now estimate the beamwidths for the minima in the back scattering direction. For example, for ka = 1.03 and with 9 = 300 we find from Figs. 3-24 and 3-25 (dashed curves) 0 that the total beamwidths for 20db reduction are about 550 and 560 in the E-plane and H-plane respectively. Similarly, the values were estimated from the other patterns and are presented for 0 < ka < 10 with 9 = 300 and 0 = 900 in Figs. O O 3-30 through 3-33. The limiting value as ka - 0 is obtained from the asymptotic analysis in Section 2. 5. The beamwidths are always maximum at ka = 0 and then decrease in an oscillatory manner with an increasing ka. The maximum and minimum of these oscillations occur at the same values of ka as those in the back scattering cross section for an unloaded sphere (Fig. 3-1). In general, the beamwidths are wider in the H-plane than the E-plane and overall there is about a 50 percent improvement with the slot at 300 as compared to that at 0 = 900. 0 In the above discussion, there has been no reason to restrict the loading to zero back scattering. For instance, if we choose a loading for zero scattering in the direction 9 = 300 (in the E-plane) rather than 9 = 00, a wider beamwidth in the back scattering direction can be obtained. As shown m Fig. 3-24 (long dashes) there is almost a 30 percent improvement in the beamwidth in the E-plane, but the 55

8.0 6.0 <c, P 4.0 \3 0 4-4 H I O 2.0o _ _ ~ ~ ~ ~ ~ ~ ~,ram ~,m ~ ~, ~ ~~ ~ - - -l 0 L I,, 0 30 60 90 120 150 180 0 in degrees FIG. 3-20: AMPLITUDES OF SURFACE FIELD COMPONENTS FOR SPHERE LOADED FOR ZERO BACK SCATTERING ( —) AND UNLOADED ( —-) SPHERE: ka = 1.03, [YI = 0. 845+il11.952, = 30. 0 0

8.0 6.0 C,1 cd.0 2.0 00 0 30 60 90 120 150 180 0 in degrees FIG. 3-21: AMPLITUDES OF SURFACE FIELD COMPONENT FOR SPHERE LOADED FOR ZERO BACK SCATTERING (-) AND UNLOADED ( —-) SPHERE: ka = 1.03, =Y/Y] =0.847+ill.952, 0 =30. 0

2.5 2.0 -W%0 _\ til 1.5 \ & a. 1.0v4/ 0, I! I I I I 0 0 30 60 90 120 150 180 9 in degrees z FIG. 3-22: AMPLITUDES OF SURFACE FIELD COMPONENT FOR SPHERE LOADED FOR ZERO BACK SCATTERING (-) AND UNLOADED ( —-) SPHERE: ka = 4.28, [Y/Y] = -i63.742, =90 0 ~00 co 10

2.5 2.0 - C) 1 1. I! in degrees~~i1 0 30 60 90 120 15O0 0930 60 90 120 150 1 0 in degrees S CATTERING ( —-) AND UNLOADED ( —-) SPHERE: ka = 4.28, [YI/Y]= - i63. 742, 0 9

0 / / / / Q| >20 ) / o~2 C -30 - I{c I I I I I I 0 30 60 90 120 150 180 0 in degrees FIG. 3-24: RELATIVE BISTATIC SCATTERING CROSS SECTION IN E-PLANE (p=0) FOR LOADED ( —, — -) AND UNLOADED ( ) SPHERE: ka = 1.03 AND O = 30 0

_lo z / / 0 // x 15 b -20 l/ i1 I 0 I 0 -30 L / I I I,I I I I Z 0 30 60 90 120 150 180 0 in degrees FIG. 3-25: RELATIVE BISTATIC SCATTERING CROSS SECTION IN H-PLANE (= 7r/2) FOR LOADED ( —-, - — ) AND UNLOADED ( -) SPHERE: ka = 1.03 AND 0 = 30 0

/N _ ftoft idge dolpo~~Co 0 -20 0 30 60 9020 150 180z e in degrees FIG. 3-26: RELATIVE BISTATIC SCATTERING CROSS SECTION IN E-PLANE (~-=0) FOR LOADED ( —-— ) AND UNLOADED ( —-) SPHERE: ka = 1.03 ANDO = 900. 0O

0 I L / -20 - I I I I i 1z 0 30 60 90 120 150 180 0 in degrees FIG. 3-27: RELATIVE BISTATIC SCATTERING CROSS SECTION IN H-PLANE (0 = w/2) FOR LOADED (-) AND UNLOADED (- - -) SPHERE: ka =1.03 AND 0 = 900

15 10 -# st ~~~~~~~~~~~~~~~~// 5 T 1op, r~~~~~~~ 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~4o -5 -10 O I ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C) - 15 P~~~~II I I I 0 30 60 90 120 150 180 0 in degrees z FIG. 3-28: RELATIVE BISTATIC SCATTERING CROSS SECTION IN E-PLANE (=0) FOR LOADED ( —)AND UNLOADED (.-)SPHERE: ka = 4.28 AND 9 =900 0

15 // 10 i/ H./,/,. 5 0 ---— ) rS b ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~co0 -10 -15 0 30 60 90 120 150 180 Z 0 in degrees FIG. 3-29: RELATIVE BISTATIC SCATTERING CROSS SECTION IN H-PLANE (0 r/2) FOR LOADED ( —) AND UNLOADED ( —-) SPHERE: ka = 4.28 AND 0 =900 0

60 45- Z C) 0 2 4 6 8 10 ka FIG. 3-30: EXPECTED BEAMWIDTH FOR 20db REDUCTION IN E-PLANE (~ =0) WITH WADING FOR ZERO BACK SCATTERING AT e = 300. O

60 45 e Z CC 30 o 0 2 4 6 8 10 ka FIG. 3-31: EXPECTED BEAMWIDTH FOR 20db REDUCTION IN H-PLANE (p = 7r/2) WITH LOADING FOR ZERO BACK SCATTERING AT 0 =300. O

60 45 z bG -8~~~~~~~~~~~~~~~~~~' 0 C) 150 LI * E. I I 0 2 4 6 8 10 W ka FIG. 3-32: EXPECTED BEAMWIDTH FOR 20 db REDUCTION IN E-PLANE (p = 0) WITH LOADING FOR ZERO BACK SCATTERING AT 9 = 90. 0

60 45. 30 15 0I I. a. I 0 2 4 ka 6 8 10 FIG. 3-33: EXPECTED BEAMWIDTH FOR 20db REDUCTION IN H-PLANE ( = t/2) WITH LOADING FOR ZERO BACK SCATTERING AT e = 900. 0O

THE UNIVERSITY OF MICHIGAN 5548-5-T reduction at 0 0=0 is 18 db as compared with the previous value of -aD db. In the H-plane there is no change in the beamwidth. If there is a need to increase the beamwidth in the H-plane, rather than the E-plane, one simply must choose the loading accordingly. 70

THE UNIVERSITY OF MICHIGAN 5548-5-T CHAPTER IV EXPERIMENTAL STUDIES 4.1 Experimental Model To confirm the theoretical predictions, a series of measurements were carried out on a sphere with a circumferential cavity backed slot. The model consisted of two identical solid aluminum caps joined together by means of a partially threaded shaft at the center (Figs. 4-1 and 4-2) but spaced 1/16 inch apart to form a radial cavity of the same width. The cavity was shorted at the center and the diameter of this short was determined by the size of the disc used. In all, there were 21 such discs available, and with these the inner diameter could be varied from 0.3125 inches (diameter of the shaft) to 3.133 inches (diameter of the sphere). The discs were cut from 1/16 inch aluminum sheet and except for the outer edge, had a slight undercut in thickness for positive metallic contact. At the surface of the sphere, the slot subtended an angle of approximately 2.25 or 0. 0392 radians. 4.2 Back Scattering vs. Loading The back scattering measurements were made with the above model at several S- and C-band frequencies corresponding to values of ka = 2.340, 3. 004, 3.090, 3. 198 and 4.280. Conventional cw equipment in an anechoic room was used, except that the azimuth-amplitude recorder was replaced by a HP 415B meter for greater accuracy in reading. At each frequency, the back scattering was determined for a number of shorting discs of different diameters and the values were calibrated with respect to the back scattering from the unloaded sphere. The results are summarized in Table 4-1. The resulting normalized cross sections were then related to the cavity loading by converting the disc diameters for a given frequency to equivalent loading admittances using the formula derived in Appendix B. The results were then compared with the theoretical values computed from equation (2. 28) with 0 = 900 and 0 = 0. The agreement was extremely gratifying. Two sets of results are plotted in Figs. 4-3 and 4-4 for ka = 2.34 and ka = 4.28 respectively. The choice of ka is such as to present a reasonable description of the susceptive loading. When ka = 2.34, zero back scattering requires 71

THE UNIVERSITY OF MICHIGAN 5548-5-T -_. _ - 3.133 T' /\T 5/8! 2 _, -T 2b I I Materials: 0. 500 Sphere - aluminum 0. 625 Discs - aluminum 0. 750 Shaft - steel 0. 875 1. 000 1.125 i. 250 1. 375 (Dimensions in inches) 1. 500 1. 625 1. 750 1. 875 2. 000 2.125 3 2. 250 64 2. 375 _ 2.500 nZZZZ - 2. 625 I I 2. 750 16 8 2. 875 M - 3.133 FIG. 4-1: SECTORIAL VIEW OF MODEL 72

......!.. z'i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~........ FIG..... EXPERIMENTAL MODEL.......... ~~~~~~~~~ ~o~................... ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1~..........~~~.....................~~........................ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ C.......................... ~ ~ ~ ~ ~ ~ ~ ~ FIG 4 2: EXPERIMENTAL MODEL~~~~~~~~~~~~~~~~~~~~~......

THE UNIVERSITY OF MICHIGAN 5548-5-T TABLE 4-1: EXPERIMENTAL DATA Shorting disc Relative ReturnS' db diameter 2b, inches 2. 808 Gc 3. 605 Gc 3.'709 Gc 3. 838 Gc 5.136 Gc.3125 -0.7 -1. 5 -0.7 -0.8.500 -0.8 -1. 7 -1.7 -1.1. 625 -0.7 -1.9 -1. 8.750 -0.7 -2.3 -1.3.875 -0. 6 -2. 5 -2.5 -1.6 1. 000 0.0 -3. 3 -2. 5 1.125 2. 3 -3. 8 -3. 0 -2.0 1. 250 7. 5 -5. 6 -3. 5 -2. 4 1. 3 1. 375 6.1 -8.0 -5. 6 -3. 4 1. 500 3. 2 -13. 4 -9. 4 -4. 6 1. 9 1. 625 2. 2 -0. 8 -14. 7 -9. 6 2. 3 1. 750 1.6 10.4 6.4 -5.4 2.8 1.875 1.3 7.0 7.0 7.0 3.7 2.000 1.0 4.5 4.4 4.3 5.3 2.125 8.2 2. 250 -2. 5 2. 375 -11. 7 2.500 0.5 1.4 1.0 1.1 -4.8 2. 625 -2. 8 2. 750 -1.8 2. 875 -1.0 3.133 0.0 0.0 0.0 0.0 0.0'relative to the unslotted sphere 74

15 10 - 10 -5 O -10 -200 -100 Ye 0 100 200 Imy FIG. 4-3: THEORETICAL (-) AND EXPERIMENTAL (~ o o) RELATIVE BACK SCATTERING CROSS z SECTIONS WITH SUSCEPTIVE LOADING: ka= 2.34 AND 0 = 900 O

10 5H "0 0 qbo 10 C -15 I -I I I p -200 -100 0 100 200 Im y FIG. 4-4: THEORETICAL (-) AND EXPERIMENTAL (o RELATIVE BACK SCATTERING CROSS ~~~~~~~~-20 w~~~~~~0 SECTIONS WITH SUSCEPTIVE LOADING: ka = 4.28 AND 8 = 90. o

THE UNIVERSITY OF MICHIGAN 5548-5-T a loading with a large negative real part (Fig. 3-9) but this, of course, is unattainable with this or any other passive cavity or network. As Fig. 4-3 indicates, there is no substantial cross section reduction W. 6 db at most), but an increase ye of 8.7 db is achievable for a particular value of Im Since, according to the results of Section 3.3, maximum enhancement always demands a susceptive load and maximum reduction requires a susceptive load whenever Re y (0) < 0,. the peak and the minimum measured here are the extreme values attainable with a passive load. They agree with those predicted in Fig. 3-18. When ka = 4.28, the required loading for zero back scattering is susceptive (Fig. 3-9), and this is attainable from the cavity. The theoretical results in Fig. 4-4 show a complete cross section reduction, but since a disc for this particular susceptance was not available, the one nearest to it gave 11.7db reduction in cross section. (For the experiment described in the next section, a shorting disc was especially made to supply a susceptance for zero back scattering, and with this over 20 db reduction in cross section resulted. ) For maximum return, an enhancement of 8.8 db was expected for a particular value of susceptance and with a disc nearest to it, 8.0 db was attained. Another value of ka for which a complete reduction in back scattering cross section can be attained with a susceptive load is ka = 3. 090. The results are sinmilar to those for ka = 4.280 (Table 4-1) except for the reversal of the peak and a null about YI= 0. This reversal is attributed to the sign change of S (0) in equation (2.28) as ka changes from 3.090 to 4.280 while the values for T2(7r/2) and a Sr(0, 7r/2) remain essentially unchanged. 4.3 Back Scattering vs. Rotation Although the analysis in Chapters II and III was limited to the case of a field incident in a direction perpendicular to the plane of the slot, no such constraint existed in the experimental study. Therefore it seems a worthwhile effort to carry out a sample measurement of a back scattering cross section as a function of rotation of a slotted sphere. For this a frequency corresponding to ka = 4.28 (5. 136Gc) was chosen. To obtain the loading required for the null in Fig. 4-4, a new shorting 77

THE UNIVERSITY OF MICHIGAN 5548-5-T disc of the requisite diameter was cut. The sphere was mounted on a pedestal with the slot in the vertical plane, and back scattering measurements were made for both horizontal and vertical polarizations. The results, along with the measured values for the unloaded sphere are presented in Figs. 4-5 and 4-6. The large cross section reductions are clearly evident. For both polarizations, the reduction is on the order of 22 db and although theoretically it should be infinite, the minor peaking at the center of the minimum could be caused by a small deviation in the optimum dimension of the shorting disc. However, in view of the slightly different magnitudes of cross sections for zero rotation, a more likely source of the residual contribution could be sphere-pedestal interaction, room effect, or both. Within, say, 30 of zero rotation there are obvious similarities between the back scattering patterns measured here and the bistatic patterns computed in Section 3.4. The horizontal polarization pattern (Fig. 4-3) is similar to the bistatic E-plane plot (Fig. 3-24) and the vertical polarization pattern (Fig. 4-4), to the bistatic H-plane plot (Fig. 3-25). The measured null widths are in good agreement with the corresponding bistatic values and the 5 db enhancement that was measured for the horizontal polarization is also evident in the E-plane bistatic plot. 4.4 Surface Field Measurements To supplement the back scattering data, the surface fields were measured on a loaded, as well as an unloaded, sphere. The facility and equipment required to perform such measurements are not as common as those generally encountered in routine back scattering work and therefore a short discussion of the current measurement facility is in order. 4.4. 1 Development of the Surface Field Measurement Facility About three years ago, a limited program to study the feasibility of measuring surface fields on three dimensional objects was initiated at the Radiation Laboratory. The primary purpose of this project was to establish the basic requirements for designing a surface field measurement facility and to gain experience in the measurement techniques involved. Up to that time, there was little 78

or,,4 l -10 00 C.O 0 -20 C) I I I I I I I I -90 -60 -30 0 30 60 90 z Rotation angle in degrees FIG. 4-5: EXPERIMENTAL BACK SCATTERING CROSS SECTIONS OF LOADED (-) AND UNLOADED ( —-— ) SPHERE: ka = 4.28, HORIZONTAL POLARIZATION.

-2 c -90 -60 -30 0 30 60 90 ( —) SPHERE: ka = 4.28, VERTICAL POLARIZATION.- I I, I 1,,

THE UNIVERSITY OF MICHIGAN 5548-5-T information on surface field measurements available in the literature and any research that had been reported dealt, for the most part, with current measurements on radiating elements rather than bodies illuminated by plane electromagnetic waves. (See, for example, Reynolds, 1948; Wetzel and Brick, 1955; or Row, 1953.) For our study, a rather crude setup, shown in Fig. 4-7, was constructed. The facility was housed in a large room, but the entire operation was confined to a region 25 by 14 feet. The experimental apparatus consisted of a signal source, an absorbing screen, and a probe with its traversing mechanism. Their arrangement is shown in Fig. 4-7. To make measurements, the model was placed on a styrofoam pedestal located directly in front of the absorbing screen, which served to shield the traversing mechanism for the probe. The model was illuminated by a signal from the horn antenna located some 10 feet from it. The choice of frequencies was influenced by two main factors: the desire to measure models whose dimensions are comparable to or longer than a wavelength, and the necessity of having probes very small in comparison with wavelength if they are not to disturb the field unduly. The first of these forces one to higher frequencies if the models are to be of reasonable dimensions, but this in turn requires the use of extremely small probes and more accurate positioning equipment if the measurement accuracy is to be retained. Ultimately, the limit is determined by the difficulties in construction and handling of probes only a few millimeters in dimension, as well as accurate positioning of them along the required path along the surface of the body. For this reason, all the measurements were carried out in the 1 to 4 Gc frequency range. The type of probe used depends in part on the field component to be measured. A number of different types was investigated, including a self-rectifying dipole, a two-diode balanced loop, and a simple shielded loop. The last proved most convenient for our measurements, and several versions of it, differing only in size, were constructed from miniature 50 ohm rigid coaxial line. The top 81

probe positioning high performance carriage pyramid type absorber current probe model K co 3m K horn antenna h hairflex absorber Top View Side View FIG. 4-7: ROOM LAYOUT.

THE UNIVERSITY OF MICHIGAN 5548-5-T photograph in Fig. 4-8 shows some loop probes at different stages of construction. The first representation shows a rigid coaxial line bent in a coil of several turns to ensure a circular bend. In the second and third, the excess turns have been cut off and the end, including the center conductor, soldered to form a closed, symmetric loop. The last is a completed loop in which a gap in the outer conductor has been cut and a bead of dielectric (epoxy) applied at the outer edge of this gap. This serves to keep the gap width constant and prevents the loop from "shorting" when placed against a metallic or conducting surface. Unfortunately, the reproduction is too small to show the gap or the dielectric bead. A detailed drawing of a typical loop is given in the center and in the bottom illustration are shown two different loop probes with slip-on miniature connectors. The upper probe has been bent about 5 mm from the end so that the plane of the loop is perpendicular to the lead. Because of the relatively high loss and the flexibility in a line of such small dimensions, only about a 6-inch length of miniature coax is used. This is then connected to a larger diameter cable which together with a styrofoam beam constitutes the support for the probe. The mechanism for positioning the probe consists of two cross-coupled, horizontal motion lead-screw carriages and an associated elevating device. The coverage in the horizontal plane is about 15 by 36 cm, and the probe can be located within 0.2 mm of its intended horizontal position. The vertical adjustment or motion, however, is not as accurate due to the oscillations of the beam, and it is therefore usual to place the probe in physical contact with the model. For detection of the signal, either a heterodyne receiver or a tuned crystal detector together with a VSWR meter was available. The receiver, naturally, gave a better sensitivity,but because of its varying gain and frequency drift, the conventional crystal detector was generally used. While this setup was in operation, the current distribution on a number of different models was investigated. These models included thin cylinders, long wires measured for near end-on incidence, spheres, cone-spheres, and cylinders with impedance loading at the center for cross section reduction. For these 83

THE UNIVERSITY OF MICHIGAN 5548-5-T 0.5- 0.75 mm 0.2mm 2-3mm T- - 50Q Coax. Cable FIG. 4-8: CONSTRUCTION OF CURRENT PROBES. 84

THE UNIVERSITY OF MICHIGAN 5548-5-T loaded cylinders especially, the current measurements were extremely helpful. In a theoretical study, Chen and Liepa (1964a, b) used approximate techniques as is done for all cylinders of finite length, and therefore it was essential to verify experimentally the accuracy of these results as well as to predict the frequency range over which the solutions are valid. The current measurements did exactly that. In addition, they were helpful in determining the equivalent shunt capacitance of the gap which was to be added in parallel with the input impedance of the cavity. 4.4. 2 The New Facility As mentioned earlier, the main purpose of the above study was to gather information for the construction of a permanent, large scale surface field measurement facility. Such has now been constructed and is being used for measuring the amplitude and phase (when required) on bodies of various shapes. The anechoic chamber is 45 feet in length and 10 feet in height. It is of a tapered design with the illuminating antenna placed at the small aperture and the model near the other end of the chamber. A remotely controllable mechanism for positioning the probe is mounted above the chamber and the probe, with its coaxial lead and wooden support tower, extends vertically through the ceiling. The motion of the probe is controlled from the console by the range operator; the probe can be moved in a direction specified by either rectangular or cylindrical coordinates. A detailed description of this surface field measurement facility has been given by Knott et al (1965). The surface field measurements for the loaded and unloaded spheres were performed in the new chamber. The model was the same as that used for back scattering measurements, and the frequency was 5.136 Gc, corresponding to ka = 4.28. For the unloaded case, the model was merely loaded with a disc of the same diameter as the sphere, while for the loaded case, the disc which gave over 20 db reduction in cross section was used. The photographs in Fig. 4-9 show the model in position for measurements. In (a) is shown the overall view from the incident signal direction, and in (b) a close-up, detailed view of the loading slot and the probe. The loop of the probe is 85

THE UNIVERSITY OF MICHIGAN 5548-5-T 0 i8: --- O s =:::::::::..,:....

THE UNIVERSITY OF MICHIGAN 5548-5-T in the horizontal plane and is in physical contact with the surface, spaced only by a small epoxy bead to prevent any conduction current flow. The incident wave is horizontally polarized and the loaded model is oriented with the slot normal to the direction of incidence. The data was taken at 10-degree intervals as the probe was traversed along the horizontal great circle. The measurements, normalized to coincide with the theoretical value at — 00 for an unloaded sphere, are presented in Figs. 4-10 and 4-11 for the unloaded and loaded case respectively. The theoretical values from Chapter III are presented here for comparison. There were two reasons for measuring the fields on the solid sphere. One was to obtain the reference level for the recorded relative data and the other to determine the performance of the ohamber aridd associated equipment. A sphere rather than some other three dimensional body was chosen because, at present, it is the only body for which the fields are known exactly and numerical data have been extensively tabulated. When the experimental results from a sphere are in good agreement with the theoretical values, it is a not a guarantee, but rather a good indication of the accuracy of measurements for other shapes. For example, if we compare the two sets of data presented here, we see that the errors or discrepancies in one are of the same order as in the other, and in both cases the measured values are slightly lower in the shadow region. This difference, which is about 0.5 db, can be attributed to the interaction of the model with the styrofoam support pedestal. 87

2. 2.0 1.5 1T2(0)I cc.n ~o c ~1.0 ~1 0 c~ 0 30 60 90 120 150 180 9 in degrees FIG. 4-10: THEORETICAL ( ) AND EXPERIMENTAL ( ) AMPLITUDES OF SURFACE FIELD COMPONENT ON AN UNLOADED SPHERE FOR ka = 4.28.

2.5 20 0 o 1 1.0 3 6 0 0 30 60 90 120 150 180 (05 SCATTERING AT 30 60 90 120 150 180 0

THE UNIVERSITY OF MICHIGAN 5548-5-T CHAPTER V STUDY OF TOTAL SCATTERING 5.1 Exact Theory The total scattering cross section is a measure of the total energy scattered by a target and is usually defined as (Jackson, 1962) as 4= a(1, )dM (5.1) JS where S is a surface enclosing the object, d/l an element of solid angle, and a(9, p) the bistatic radar cross section for the object. An equivalent expression is the ratio of the total power scattered by an object to the incident power density (King and Wu, 1959), viz., O = P/P (5. 2a) Similarly, the total absorption cross section is defined as aa= Pa/Pi (5. 2b) where P is the power absorbed by the object, and the sum a +aa is known as extinction cross section (van de Hulst, 1957). A useful relation between the extinction cross section and the scattering amplitudes in the forward direction (0 = ir) is given by the forward scattering theorem. This was first discovered in quantum mechanics and since then its equivalent in electromagnetic theory has received a number of different proofs, as, for example, Schiff (1954) and Bolljahn and Lucke (1956). In the latter reference tbhe theorem has been derived for the case of a plane wave incidence and in terms of our notation it may be written as a = - ReS (i) 8 k 1 for an unloaded body and 90

THE UNIVERSITY OF MICHIGAN 5548-5-T 4ir a +as = - Re S(r) a k2 for a loaded body. Ss(ir) and Sl(pr) are the scattering amplitudes defined by equations (2.6a) and (2. 24a) for the unloaded and loaded spheres respectively. There are two methods of evaluating the total scattering cross section. One involves integration of the differential cross section a(e, 0), and the other, computation of the forward scattering amplitude and calculation of the power absorbed by the object. For a loaded sphere, this is then the power absorbed by the load. Due to the complexity of a(6, 0) for a loaded sphere, we proceed to evaluate the total cross section by the second method, which is relatively straightforward. For forward scattering (8 = yr), just as for back scattering, there is no cross-polarized component and the field can be represented by a single scattering amplitude S (r() S ( Y)+ 27 sine T2(6 )aSl(r ) 1!(1 Yi+Y oy + o 2' o r To find a, or the power absorbed by the object, we employ the definition of input admittance introduced in Section 2.4. It is assumed that the surface of the sphere is lossless and that the power is absorbed only by the cavity-backed slot. For an asymmetrically excited slot, the average complex power input per unit length of the slot is YI w = 4 a sin vcos vcos, (5.3) 0 where Y! is the input admittance of the slot and the tilde denotes the complex conjugate. The peak voltage that is excited across the slot is, from equation (2. 27), y +y 2rasin9 T2(e) 91

THE UNIVERSITY OF MICHIGAN 5548-5-T and when it is substituted in (5.3), the total absorbed power becomes 2,r Pa = Re wasineodo 22 = ( rasin 9) T2(O) ReY1 1+ r 20] and the absorption cross section 2 2 Y a = 2(wrasinO) T2() Re a o Yi+Yr2 Y2 The total scattering cross section, normalized to its physical optics value ira2 is then 2= 2ReS (r)-2ir sin | +y ()9 Re T(5. 4) a2 r2 e1 ()-2ino Yl+ Y 2'o Y ir a (ka)' r We now want to find the extremum value of (5.4) subject to passive loading, y y i.e. Re 0ao. Im-I, on the other hand, has no restriction and may take any Y y value. For simplicity, we now introduce the following definitions: T2(0 ) = tl+ it2 Y +Y y r = x+iy Yr/Y = gl+ig2 YI/Y= 11+u2 S((r) = sl+is2 aS(r, ) = rl + ir. (5.5) 92

THE UNIVERSITY OF MICHIGAN 5548-5-T When these are substituted in equation (5.4) we find s= + 2 B3x B4y B + - (5.6a) 2 1 2 2 2 2 2 2 7ra x +y x +y x +y where 4 B - 2 s (ka)2 B2 = 2r sin2 e(t2 +t2)gl 2 0 n 221 B 27rsi 44 2 21 B3:=~ 2rsin { (tlrl-t2r2)- +t2 (ka)...2 sine (t r2+t2r1) (5. 6b) (ka) The extremum values of (5.6), for gl 0, are given in Appendix A. The maximum value is U ~ B2 a 4 1 max. 2 B12 2 (5.7a) 7a2 2(B+B3g)+ (B 2+B3g1) +B4g2 and the minimum value, c B s 4 min. 2 = B1+ 2 (5.7b) 7r a2 1 2 B )-(B2+B3g) +g for gl >Xmin. 2 2 B3+B 1 4BB for g1 xmin (5.7c) 2 where 2B2B3 Xmin. 2 2 3 4 93

THE UNIVERSITY OF MICHIGAN 5548-5-T The results simplify considerably when 8 = 900. From equation (2.35a) 0 42 or r1 ir 4 (t +lt2) and with this condition we have 4 1 -- S1 (ka)2 2 B= 2-r (t +t2)g B = -4r t2 3 2 B4 = 4rtlt2. (5.8) The numerical calculations of (5.7) with 9 = 900 and the constants defined 0 by (5.8) are presented in Fig. 5-1. To emphasize the enhancement or reduction, the values are normalized relative to the total cross section of an unloaded sphere, namely 80 4 ra2 (ka)2 1 B1 Let us consider first the maximum total scattering. If we allow an active slot, it is obvious that there is no limit to the maximum scattering attainable. Of more practical interest is the maximum scattering attainable with passive loading. As shown in Fig. 5-1, a large enhancement occurs for small values of ka, but decreases rapidly as ka is increased. Whereas at ka = 0. 5 the enhancement is 20db, at ka = 1.0 it is 8 db and at ka = 5.0 it is already less than 1 db. 94

20 15 10 10 Z1 o 0 2 4 6 8 10 ka z FIG. 5-1: NORMALIZED MAXIMUM AND MINIMUM TOTAL SCATTERING CROSS SECTIONS FOR PASSIVE LOADING AT e = 900

THE UNIVERSITY OF MICHIGAN 5548-5-T The lower curve in Fig. 5-1 gives the minimum total scattering for a passive load. As we see, there is practically no reduction for large values of ka, and even when ka is around unity, the maximum reduction is a mere 0.5 db. From this we deduce that the loading of the sphere does not reduce its scattering, but rather redirects it from one direction to another. And this, of course, is evident in the bistatic scattering patterns in Figs. 3&24 through 3-29. The question now arises as to whether an improvement can be achieved if active loading is allowed. In such a case there would be no restriction on 11, the condition gl+il = Xmin. would always be satisfied, and the total cross section would then be given by equation (5.7c). However, when the results for Fig. 5-1 were calculated, the condition 1 1 i xn. was met for all values of ka, implying that the same minimum total cross section is achieved with active or passive loading. 5.2 Low Frequency Approximation In Section 2.5 the expressions for the low frequency scattering behavior of a loaded sphere were obtained. They are rather simple, and we can easily find the total cross section by direct integration of the differential cross sections. Substitution of equation (2.29) into (5. 1) leads to a normalized total scattering cross section 7r 27r s2 (k)1 2 50 (Sl(e) I Cos2 + IS2(0)1 sin2) sineOddeO 7r a X (ka)2% where, for low frequencies, the scattering amplitudes are given by equations (2.41). Upon integration, we get! 10 4 27 2 1k. lo(ka) 27 _ 2 + 37r(ka) (2 + (ka) 2 3 8 2 2 2 2 ira -x +y x +y where x and y are the loading parameters defined in equation (5.5). The first 96

THE UNIVERSITY OF MICHIGAN 5548-5-T term on the right hand side represents the scattering by an unloaded sphere, and the other two are direct consequences of the loading and disappear when the cavity is shorted (y -a,-oo). It is of interest to compare this result with the formula previously obtained from the forward scattering theorem. For small values of ka, and using the expressions derived in Section 2.5, the constants for e = 7r/2 in equation (5.8) reduce to =- (ka)4 1 3 B' - 27 2 2(ka)2 2 8 (ka) B = - r (ka)6 3 B' = 3r (ka)3 4 and the total scattering, from (5.6a), becomes s 10 4272 2 1. 6 x 3. 2 () (4+ r2(ka) - r(ka) 2 2 2 ra x +y x +y x +y Except for the third term on the right hand side, this expression is identical to equation (5. 9). The absence of this term in (5. 9) is due to the assumption (in Section 2.5) that x << 1 and theretore its presence or absence, for all practical purposes, has not effect on numerical results. The extremum values for (5.9) are given again in Appendix A, and upon substitution of the corresponding values we find as 10 4 6 max. 2 "(ka) + -_6 2ax. (ka)2 ira 3 9+(ka) 97

THE UNIVERSITY OF MICHIGAN 5548-5-T or, when normalized to the total scattering cross section of an unloaded sphere, max. ~ 1+ - - (5.10a) ua5 6 so (ka) s 9 1 min. a N1-5 6 (5. 10b) so0 9 + (ka) Numerical comparison of this result with the exact computed values in Fig. 5-1 indicates that the above equations are accurate within 10 percent up to ka = 1.0. 98

THE UNIVERSITY OF MICHIGAN 5548-5-T CHAPTER VI DISC USSION 6.1 Some Practical Considerations for Obtaining the Desired Loading We have seen in the chapter on experimental studies that when the optimum loading is purely susceptive, it can be supplied by a simple radial cavity of variable depth. For zero back scattering there are only a few values of ka where the optimum loading is susceptive, and these depend on the slot position 0. For instance, with the slot at 0 = 45, there is only one value of ka (in the range 0 O < ka < 10 considered) where the real part of the optimum loading for zero back scattering crosses the zero value, but with 0 = 90~ there are six such frequencies. 0 For maximum return, however, the optimum loading is susceptive for all ka (cf. Figs. 3-14 and 3-17) and the loading requirement can be met with a lossless radial cavity at any value of ka. Unfortunately the variation of susceptance with frequency for the radial cavity (or any other lossless network) is the direct opposite of that required to reduce or enhance the back scattering cross section. As a result, the loading can be satisfied only where the two loading curves intersect, and narrow bandwidths are therefore to be expected. To determine the actual bandwidths encountered with susceptive loading, we examine the case when the sphere is loaded for zero back scattering with ka = 4.28 and 0 = 900. The required optimum loading and that 0 supplied by the radial cavity, such as used in the experimental model, are shown in Fig. 6-1. The depth for the cavity was chosen so as to satisfy the optimum loading at ka = 4.28. It is evident that the corresponding curves, especially for the real part, have completely different behaviors and, as a result, a minute deviation in frequency from the critical value results in a large difference between the desired and the supplied loading. The cross section. corresponding to the cavity loading is shown in Fig. 6-2. As expected, a complete reduction is achieved at ka = 4.28 and this reduction is highly frequency sensitive. For example, for a 10db reduction in the cross section the bandwidth is 6.5 percent and decreases to only 2 percent when a 20 db reduction in the cross section is desired. 99

20 0 I\~~~~~~~~~~~~ XIt'rl (2) I 0~~~~~~~~~~~~ -20 z \(1) - y1/y~~~~~ I I -40 - I I -40 -10 -60 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~C" o o!A o 0o -] P4. 0 -60 - /' -80'(2) I, -20 i I I I I -20 0) 0 3 4 ka 5 6 ka - I, I I, I i 1 FIG. 6-1: REAL (-) AND IMAGINARY (- - -) PARTS 0 3 4 ka 5 6 OF (1) NORMALIZED LOADING ADMIT- z TANCE FOR ZERO BACK SCATTERING FIG. 6-2: REDUCTION IN BACK SCATTERING WITH 9 =90 AND (2) LOADING SUPPLIED WITH SUSCEPTIVE LOADING AT WITH LeSSLESS RADIAL CAVITY. 6 =900. 0

THE UNIVERSITY OF MICHIGAN 5548-5-T For wider bandwidths and increased scattering control, it is clear that such simple susceptive loading alone will not suffice and that more sophisticated loading techniques must be developed. From a practical viewpoint there appear to be two different methods of approach. One method would be to use a number of lumpedtwo-port networks, whose frequency response is determined by the optimum loading, and distribute these around a sphere to simulate a distributed loading. The resultant admittance YI in such a case would be Y= nYL+Y' I L I where n is the number of loading points or elements used, YL the admittance of each element, and Y' the inherent admittance of the slot and the back-up cavity. This cavity could be similar to the one used in the experimental model. The number of loading points will depend on the accuracy to which the optimum loding must be attained. A rough estimate is that this number may be as high as 10 ka, where ka is the maximum value in the frequency range considered. Once n and Y" are known, YL can be found from the above equation and from then on the problem is strictly one of network synthesis. It would involve the study of the realizability conditions and synthesis procedures for functions such as YL when both the real and imaginary parts are specified over a finite frequency range. In addition, a network to generate this function would need to be determined. The other method would be to consider the cavity as a distributed network or a non-uniform transmission line loaded in a prescribed manner to achieve the requisite loading. For small bandwidths, of order 5 to 10 percent for 20db reduction, a cavity such as used in the experimental model could be filled with a lossy dielectric, or even ferrite materials, to supply the required complex loading for certain frequencies. The formula for the input admittance derived in Appendix B still applies, but it must be modified slightly to accept the complex values of k and Y. For wider bandwidths, however, the loading of the cavity must be considered as a synthesis problem. The work by Sharpe (1962) and Youla (1964) on the analysis and synthesis of non-uniform transmission lines could probably be extended and applied to the synthesis of radial transmission lines and cavities. 101

THE UNIVERSITY OF MICHIGAN 5548-5-T Perhaps a loading that is easier to realize physically may result from either of two related problems that should be considered in a future study. One problem would be to examine the effect of coplanar multiple slot loading with the slots perpendicular to the direction of incidence, and the other would be to solve the boundary value problem for a single slot loading, but with the slot located arbitrarily with respect to the direction of incidence. In the case of multiple slot loading the analysis wouldblesanxension of the results for a single slot presented here. A cursory study has shown that double loading does have certain advantages. For example, by using two slots we could make either S1(8) or S2(0) zero for two adjacent values of 8 (as compared to one value for a single load), or make both S1(0) and S2(0) zero for a single value of 0. Correspondingly,it is expected that with the proper choice of the slot positions, loading admittances whose frequency characteristics are more suited for physical realizability would be obtained. Overall, it appears that with double loading, twice as much controllabilityis obtained as with a single load, but the use of multiple loading adds to the difficulty of adequate numerical treatment. Encouraging as it may be, the slots would still be excited by an asymmetric voltage requiring a distributed (or lumped distributed) loading which is rather ca-nrsometoachieve. However, this is not the case in the problem of loading a sphere with a single slot arbitrarily located with respect tahe direction of incidence. Whereas in the problem that we have considered only a single asymmetric mode was excited in the slot, now the slot would be, in general, excited by an infinite set of modes. In particular, when the direction of the incident wave is such that its electric vector is perpendicular to the plane of the slot, a zeroth mode would predominate and by independently controlling this mode it sioul be possible to control the scattering behavior of the sphere. For such a loading, the slot should be backed by a radial cavity of the type used in our experimental model, but with a lumped load at the center. Such a load would affect only the zeroth mode and would appear nonexistent to all the higher modes. It is expected that the frequency characteristics for this loading would be similar to those for a thin cylinder, and that the scattering control would be less effective than that 102

THE UNIVERSITY OF MICHIGAN 5548-5-T for an asymmetrically excited single slot loading. Nevertheless a possibility of using a single lumped load would alone justify an investigation of a single slot loading with arbitrary illumination. 6.2 Conclusions In the preceding sections we have considered the scattering behavior of a metallic sphere loaded with a slot in a plane perpendicular to the direction of incidence. The slot was assumed to be of small but finite width with a constant electric field across it, and under this assumption, the analysis for the external fields is exact. Expressions for the scattered far field components, as well as the total surface field components, were derived and then used to investigate the modification to the scattering cross section produced by various admittances of the slot. All investigation was limited to the frequency range 0 < ka < 10. The loading admittance necessary for a particular modification is in general complex, with negative or positive real part, corresponding to an active or passive load respectively. The loading required to reduce the back scattering cross section to zero has been examined in some detail. For any given slot position 00, the ranges of ka in which the real part of the loading admittance is positive or negative alternate with one another, and the location of these active or passive bands are functions of the slot position. For example, with the slot at the shadow boundary (0 = 90 ), there are three different active bands, but as the slot is moved 0 toward the front, these bands appear to move towards increasing ka. With 0 = 450 0 the first band does not commence until ka = 4.25, and with 0 = 300, it appears 0 somewhere beyond ka = 10.0. However, any further movement of the slot towards the front introduces a new active band at ka = 1.5 and due to this, it seems that for back scattering reduction with passive loading, the load should be somewhere between 0 = 30~ and 0 = 60~ The exact position of the slot will depend on the O O limitations imposed by the realizability and synthesis requirements of the loading. In contrast, an increase of almost 20 db in back scattering cross section can be achieved for a particular value of ka with passive loading, and maximum 103

THE UNIVERSITY OF MICHIGAN 5548-5-T enhancement occurs when the loading slot is near 600. When a sphere is loaded for zero back scattering, the angular beamwidth for 20 db reduction in bistatic operation is maximum (640) as ka -- 0. It is then independent of the loading position 00 and has the same value in either the E-plane or the H-plane. At ka = 2. 0, the beamwidth is about 50 percent of the initial value and at ka = 10. 0 it has decreased to only about 25 percent. Quantitatively the beamwidths are wider with e = 300 than 8 = 900 and also wider in the H-plane as compared to the E-plane. O O An improvement in the beamwidth can be obtained by selecting a loading for maximum width (with a null between 15 and 30 ) rather than for a null at 9 = 0. The bistatic scattering patterns indicate, in addition to the beamwidths, that whenever the sphere is loaded for back scattering reduction there is enhancement in the forward direction. It appears that the loading, such as studied here, does not in general reduce the total cross section, but rather modifies the scattering pattern by redirecting the scattered energy from one direction to another. Studies of the total scattering behavior do indeed confirm this. To verify some of the theoretical conclusions, a spherical model was constructed. The model had a circumferential slot at 90 = 900 backed by a radial 0 cavity whose depth could be varied by using shorting discs of various diameters. Since the loading was purely susceptive, emphasis was placed on the frequency where complete reduction in back scattering can be obtained, namely ka = 4.28. The back scattering from this model was masurad for different loading discs which, in turn, were related to the corresponding loading admittance by an input admittance formula for the asymmetrically excited radial cavity. Agreement between the measured and computed back scattering cross sections was excellent, and with the loading disc corresponding to the optimum loading for zero back scattering, over 20 db in cross section reduction was achieved. The theory was further verified by measuring the surface fields on the same model loaded for zero back scattering and comparing the data with the computed values. Again the agreement was excellent. 104

THE UNIVERSITY OF MICHIGAN 5548-5-T REFERENCES Adams, E.F. and R. L. Hippisley (1947) Smithsonian Mathematical Formulae and Tables of Elliptic Functions, Smithsonian Miscellaneous Collections, 74, No. 1. As, B. 0. and H. J. Schmitt (1958) "Back Scattering Cross Section of Reactively Loaded Cylindrical Antennas", Harvard University Cruft Laboratory Scientific Report No. 18. Bailin, L. L. and S. Silver (1956) "Exterior Electromagnetic Boundary Value Problems for Spheres and Cones", IRE Trans. Ant. and Prop., AP-4, pp 5-16. Bechtel, M. E. (1962) "Scattering Coefficients for the Backscattering of Electromagnetic Waves from Perfectly Conducting Spheres", Cornell Aeronautical Laboratory Report No. AP/R1S-1. Bolljahn, J. T. and W. S. Lucke (1956) "Some Relationships between Total Scattered Power and the Scattered Field in the Shadow Zone", IRE Trans. Ant. and Prop., AP-4, pp 69-71. Chen, K-M (1965a) "Minimization of Back Scattering of a Cylinder by Double Loading", IEEE Trans. Ant. and Prop., AP-13, pp 262-270. Chen, K-M (1965b) "Minimization of Back Scattering of a Cylinder with a Moderate Radius by Loading Method", Michigan State University Technical Report No. 1, Contract AF 33(615)-1656. Chen, K-M and V. V. Liepa (1964a) "The Minimization of the Back Scattering of a Cylinder by Central Loading", IEEE Trans. Ant. and Prop., AP-12, pp 576-582. Chen, K-M and V. V. Liepa (1964b) "The Minimization of the Radar Cross Section of a Cylinder by Central Loading", The University of Michigan Radiation Laboratory Report No. 5548-1-T. Ducmanis, J.A. and V. V. Liepa (1965) "Surface Field Components for a Perfectly Conducting Sphere", The University of Michigan Radiation Laboratory Report No. 5548-3-T. Green, R.B. (1963) "The General Theory of Antenna Scattering", The Ohio State University Report No. 1223-17. Harrington, R. F. (1963) "The Electromagnetic Scattering by Antennas", IEEE Trans. Ant. and Prop., AP-11, p 596. Harrington, R. F. (1964) "Theory of Loaded Scatterers", Proc. IEE (London), 111, pp. 617-623. 105

THE UNIVERSITY OF MICHIGAN 5548-5-T Hey, J.S., G.S. Stewart, J.T. Pinson, and P.E.V. Prince (1956) "The Scattering of Electromagnetic Waves by Conducting Spheres and Discs", Proc. Phys. Soc. B, 69, pp 1038-1049. Hu, Y-Y (1958) "Backscattering Cross Section of a Center-Loaded Cylindrical Antenna", IRE Trans. Ant. and Prop., AP-6, pp 140-148. vande Hulst, H. C. (1957) Light Scattering by Small Particles, John Wiley and Sons, Inc., New York. Iams, H. A. (1950) "Radio Wave Conducting Device", U. S. Patent No. 2, 578, 367. Jackson, J. D. (1962) Classical Electrodynamics, John Wiley and Sons, Inc., New York. Kazarinoff, N. D. and T.B.A. Senior (1962) "A Failure of Creeping Wave Theory", IRE Trans. Ant. and Prop., AP-10, pp 634-638. King, R. W. P. (1956) The Theory of Linear Antennas, Harvard University Press, Cambridge, Mass. King, R. W. P. and T.T. Wu (1959) The Scattering and Diffraction of Waves, Harvard University Press, Cambridge, Mass. Knott, E. F., V.V. Liepa and T. B. A. Senior (1965) "A Surface Field Measurement Facility", Proc. IEEE, 53, pp 1105-1107. Meissner, A. (1929) "Transmitting Antennas for Broadcasting", Proc. IRE, 17, pp 1178-1184. Reynolds, D.K. (1948) "Surface-Current and Charge Measurements on Flat Metal Sheets", Harvard University Cruft Laboratory Report No. TR-53. Row, R. V. (1953) "Microwave Diffraction Measurements in a Parallel Plate Region", J. Appl. Phys., 24, pp 1448-1452. Schiff, L. I. (1954) "On an Expression for the Total Cross Section", Prog. Theor. Phys.,11, pp 288-290. Senior, T.B.A. and R. F. Goodrich (1964) "Scattering by a Sphere", Proc. IEE (London), 111, pp 907-916. Sharpe, C. B. (1962) "An Alternative Derivation of Orlov's Synthesis Formula for Non-uniform Lines", Proc. IEE (London), 109, pp 226-229. Sletten, C.J., P. Blacksmith, F. S. Holt, and B.B. Gorr (1964) "Scattering from Thick Reactively Loaded Rods", in "The Modification of Electromagnetic Scattering Cross Sections in the Resonance Region", A Symposium Record, 1, USAF Cambridge Research Laboratories Report No. AFCRL-64-727(I), pp 67-87. 106

THE UNIVERSITY OF MICHIGAN 5548-5-T Stratton, J.A. (1941) Electromagnetic Theory, McGraw-Hill Book Co., Inc., New York. Watson, G. N. (1948) A Treatise on the Theory of Bessel Functions, The MacMillan Co., New York. Weinberg, L. (1963) "New Technique for Modifying Monostatic and Multistatic Radar Cross Sections", IEEE Trans. Ant. and Prop., AP-11, pp 717-719. Wetzel, L. and D. B. Brick (1955) "An Experimental Investigation of High Frequency Current Distributions on Conducting Cylinders", Harvard University Cruft Laboratory Report No. SR-4. Youla, D.C. (1964) "Analysis and Synthesis of Arbitrary Terminated Lossless Nonuniform Lines", IEEE Trans. Circuit Theory, CT-11, pp 363-372. 107

THE UNIVERSITY OF MICHIGAN 5548-5-T APPENDIX A AN EXTREMUM PROBLEM In Chapters III and V, the following extremum problem arises: given r= A+B 1 +C x +D y (A.1) 2 22 2 2 2 x +y x +y x +y where A, B, C and D are real constants, find the maximum and minimum values of r subject to the condition x > gl > 0. In addition, A- and B are positive. From equation (A. 1) 2 2 r = Ax +Ay + B + Cx+Dy (A. 2) 2 2 x +y and since there is no restriction on y, we can obtain an extremum of (A. 2) by setting the derivative of r with respect to y equal to zero. Thus 2 2 Ax +Ay +B+Cx+Dy 2Ay+D (A. 3) 2 2 2y x +y and when substituted in (A. 2), the extremum values are given by r =A+ -D (A. 4) ext. 2y The variables x and y are not independent now, but are related by equation (A. 3). Solving explicitly for y in (A. 3) y = 1 -f(B+Cx) + (B+Cx)2+D2x2 and then substituting it in equation (A. 4), we find D2 1 max -(BA+ (A.5a) max 2 2 2 (B+Cx)+ (B+Cx) + Dx 108

THE UNIVERSITY OF MICHIGAN 5548-5 — T Investigating the behavior of these two equations as functions of x, we observe that as x - 0 max D2 F' — >A -- min 4B and as x — +O C C2 +D2 max 2xA+ + JT r A C -C2 + D2 rmin 2x We can obtain the least value of r by setting its derivative with respect to x mmin equal to zero. However, the same result is obtained much more easily if we minimize the denominator in equation (A. 5b). Thus, 2BC Xmin 2 C (A. 6) D2 + C2 and the corresponding minimum value is C2 +D2 r A C +D (A. 7) min 4B A sketch for the extremum values of r with C < O is shown in Fig. A-1. In case C > 0, the curves are similar, but reversed about the r-axis. The Xmin then is on the negative axis, but gl remains on the positive side. Recalling the restriction. x ) gl > 0 placed on x, it is easily seen from the diagram that the maximum value is given by D 1. 2 I2 22 (A.8) -(B+Cgl)+ (B+Cgl) +D g 109

THE UNIVERSITY OF MICHIGAN 5548-5-T r A x xmin gl FIG. A-1: EXTREMUM VALUES OF r WITH C < 0. and the minimum value by D2 rmin =A-D (A.9a) min 2 (B+Cg )+ (B+Cg 2 +D g when Xmin < g C2+D2 = A- 4B when xmin gl (A.9b) where xmin is given by equation (A. 6). 110

THE UNIVERSITY OF MICHIGAN 5548-5-T APPENDIX B THE INPUT ADMITTANCE OF AN ASYMMETRICALLY EXCITED RADIAL CAVITY In order to use the experimental model of Chapter IV to verify the theoretically predicted scattering behavior of the slotted sphere, it is necessary to relate the input admittance of the cavity to its dimensions, frequency, and, in particular, to the radius b of the inner conductor. Bearing in mind that the slot is of small width centered on 0e = 7r /2, it would appear sufficient to regard the cavity as a radial one, and in terms of the cylindrical polar coordinates (r, 0, z) where x = rcosp, y = rsin, z = z, the situation is now as shown in Fig. B-1. The cavity is of width d = a6 and is shorted at r = b. At the outer edge r = a it is excited by a voltage -v cos 0 (the sign difference with respect to the voltage implied by equation (2.10a) is a consequence of the fact that z = -0 at e = ir/2) and since it is assumed that d <<X, the components E and E0 of the electrical field within the cavity can, for all practical purposes, be neglected. The only E component remaining is then E, and this must satisfy the wave equation which, in cylindrical coordinates, is Ora a 2 1 a2 r (r E )+ (k2+ a )E = o0. rr The general solution for 0 < b < r < a is E = E {JEJ (kr)+F N (kr), (B. ) z n are constants to be determined. The boundary n=-ao itions on the s'i n nf th ci s au-Aialy(.1. conditions on the sides of the cavity are satisfied automatically by (B. 1). At the 111

THE UNIVERSITY OF MICHIGAN 5548-5-T z d - - - /6 I FIG. B-1: GEOMETRY OF THE RADIAL CAVITY. 112

THE UNIVERSITY OF MICHIGAN 5548-5-T inner and outer surfaces, however, the conditions are E = 0 for r=b z v = -cosp for r=a, d It and on substituting in (B. 1), we obtain J1 (kr) N (kb) - N (kr) J (kb) z d J (ka)Nl(kb)-Nl(ka)J (kb) os (B.2) The corresponding circumferential component of the magnetic field can be found from Maxwell's equations, and is J, (kr) N (kb) - N' (kr) Jl(kb) H =-iY- 1 Cos (B. 3) I= -iY- d J (ka) N(kb)-N (ka)J cos (B3(kb) Since both E and H0 are functions of p, we shall again employ the concept of admittance density, as for the radiation admittance in Section 2.3. The power flow across the aperture and into the cavity is d/2 w =-S (ExH)' rdz -d/2 d/2 -=!vcos0 Hdz 3-d/2 = ~v cos OH0, from which we have [H0r=a 113

THE UNIVERSITY OF MICHIGAN 5548-5-T The total input admittance YL follows on integrating this around the circumference of the cavity, and hence J r(ka) N (kb) - N (ka) J (kb) Y -iY2a 1 1 (B. 4) 1 d J (ka) N (kb)-N N(ka) J1(kb) As b-)a. a 1 YI - -iYgrd k(a-b) and the admittance therefore approaches -i a with decreasing cavity depth. On the other hand, as b —. 0 (a condition which is equivalent to an asymmetrically excited radial cavity open at the center) a J;(ka) Y -e -iY2 dd J (ka). (B. 5) We also remark that if the cavity were filled with a medium of refractive index n, the expression for the admittance would follow immediately from equation (B. 4) 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 air-filled cavity. The expression for Y! has been programmed for an IBM 7090 computer to give data for any ka and a/d as a function of kb. For the sphere used in the experimental study the diameter was 3.133 inches, the gap width 0. 0625 inches, and the spacing discs enabled 2b to be varied in 22 steps from a minimum of 0.3125 inches to 3. 133 inches. In order to have the computed data directly applicable to the experimental model, kb was written in the form kb = ka 3 and the data was printed out for the first 22 values of the inner diameter x (in inches) appropriate to the shorting discs. Because of the infinity when b = a, the largest x computed was 3.0. Typical values of the relative admnittance Y/Y are 114

THE UNIVERSITY OF MICHIGAN 5548-5-T shown in Fig. B-2. The value 2b = 0. 3125 inches corresponds to the diameter of the center screw in the experimental model, and 2b = 2.322 inches is the diameter of the loading disc which supplies the optimum loading for zero back scattering at ka = 4.28. 115

I I 300 I I I I I I I I l ~~~~/!,4z / ~ /I I/ I / I~~~~~~~~~~~~~~ I I I 200 I I H -300; I / / I - / / 100 / I / / z C I /... i/ / / Y 0 I/ ~~~~~~~/ ~~~~// 10 2 / / I / 0r ~/// /II I -200 I I I I I - I I I 01 / I I -300 / I I I I I 0) I I I I I I I i I 0 2 4 6 8 10 Z ka FIG. B-2: NORMALIZED LOADING ADMITTANCE OF ASYMME TRICALLY E XCITED RADIAL CAVITY FOR 2b = 0. 3125 INCHES (- - -) AND 2b = 2.322 INCHES ( -): 2a = 3. 133 AND d= 0.064 INCHES.

Unclassified Security Classification DOCUMENT CONTROL DATA- R&D (Security claaasifcation of title, body of abstract ancd indexing annotation must be entered when the overall report in classified) 1. ORIGINATIN G ACTIVITY (Corporate author).. REPORT SECURITY C LASSIFICATION The University of Michigan Department of Electrical Engineering 2b GROUP Radiation Laboratory 3. REPORT TITLE Theoretical and Experimental Study of the Scattering Behavior of a Circumferentially- Loaded Sphere 4. DESCRIPTIVE NOTES (Type of report and inclusive datee) Technical Report 5. AUTHOR(S) (Last name, first name, tnftiil) Liepa, Valdis V. Senior, Thomas B.A. 6. REPORT DATE 7a. TOTAL NO. OF PAGES 7b. NO. OF REFS February 1966 116 34 ea. CONTRACT OR GRANT NO. 4a. ORIGINATOR'S REPORT NUMBER(S) AF 19(628)-2374 5548-5-T b. pROJECT NO. 5635 c. 9b. OTHF-R R.EPORT NO(S) (Any other nrmbera thet may be assigned Task 563502 this report) d. 10. A V AV IL ABILITY / MITAT!ION N OTICES U. S. Government agencies may obtain copies of the report directly through DDC. Other qualified users shall request through CFSTI 11. SUPPI..EMENTARY NOTES. 1.-PPONSORiNG MIl.TARY ACTIVITY Air Force Cambridge Research Laboratories L.G. Hanscom Field Bedford, Massachusetts 13. ABSTRACT The electromagnetic scattering behavior of a metallic sphere loaded with a circumferential slot in a plane perpendicular to the direction of incidence is investigated. The slot is assumed of small but finite width with a constant electric field across it, and under this assumption, the analysis of external fields is exact. The scattered field is obtained by superposition of the field diffracted by an unloaded sphere and that radiated from the excited slot, with the radiation strength and phase determined by the loading parameters. Thus the scattering behavior is determined by the loading admittance and the position of the slot. The numerical study is restricted to the frequency range corresponding to 0 <ka 10. 0, and the results presented are primarily for passive loading. The maximum and minimum back scattering cross sections and the loading needed to attain these are presented for different loading positions. The bistatic scattering patterns are computed for a sphere loaded for zero back scattering and from these the back scattering null widths are obtained. The extreme total scattering cross sections are evaluated using the forward scattering theorem and, for small values of ka, the result is compared with that obtained by direct integration of the differential cross sections. Experimental data, obtained using a metallic sphere with an equatorial slot backed by a radial cavity of adjustable depth, is presented for the surface field and back scattering measurements. The results are compared with the computed data. D JD JN~64 1473 Unclassified Security Classification

Unclassified Security Classification 1 W LINK A LINK B LINK C KEY WORDS ROLE WT ROLE WT ROLE WT Metallic sphere Scattering and Diffraction Cross section control Impedance loading INSTRUCTIONS 1. ORIGINATING ACTIVITY: Enter the name and address imposed by security classification, using standard statements of the contractor, subcontractor, grantee, Department of De- such as: fense activity or other organization (corporate author) issuing (1) "Qualified requesters may obtain copies ofthis ~~~~~~~~~~the report. ~report from DDC." 2a. REPORT SECURITY CLASSIFICATION: Enter the over2a..REPORT SECUTY CLASSIFICATION: Enter the over-,(2) "Foreign announcement and dissemination of this all security classification of the report. Indicate whether "'Restricted Data" is included. Marking is to be in accord- report by DDC is not authorized ance with appropriate security regulations. (3) "U. S. Government agencies may obtain copies of.2b. GROUP: Automatic downgrading isspecified. in this report directly from DDC. Other qualified DDC 2b. GROUP: Automatic downgrading is specified in DoD Directive 5200. 10 and Armed Forces Industrial Manual. Enter users shall request through the group number. Also, when applicable, show that optional.. markings have been used for Group 3 and Group 4 as author- (4) "U. S. military agencies may obtain copies of this ized. report directly from IDC. Other qualified users 3. REPORT TITLE: Enter the complete report title in all shall request through capital letters. Titles in all cases should be unclassified., If a meaningful title cannot be selected without classification, show title classification in all capitals in parenthesis (5) "All distribution of this report is controlled. Qualimmediately following the title. ified DDC users shall request through 4. DESCRIPTIVE NOTES: If appropriate, enter the type of,* report, e.g., interim, progress, summary, annual, or final. If the report has been furnished to the Office of Technical Give the inclusive dates when a specific reporting period is Services, Department of Commerce, for sale to the public, indicovered. cate this fact and enter the price, if known. 5. AUTHOR(S): Enter the name(s) of author(s) as shown on 11. SUPPLEMENTARY NOTES: Use for additional explanaor in the report. Enter last name, first name, middle initial, tory notes. If military, show rank and branch of service. The name of the principal.athor is an absolute minimum requirement. 12. SPONSORING MILITARY ACTIVITY: Enter the name of the departmental project office or laboratory sponsoring (pay6. REPORT DATE.- Enter the date of the report as day, ing for) the research and development. Include address. month, year; or month, year. If more than one date appears on the report, use date of publication. 13. ABSTRACT: Enter an abstract giving a brief and factual 7a. TOTAL NUMBER OF PAGES: The total page count summary of the document indicative of the report, even though oT N O a The total page count it may also appear elsewhere in the body of the technical reshould follow normal pagination procedures, i.e., enter the port. If additional space is required, a continuation sheet shall number of pages containing information be attached. 7b. NUMBER OF REFERENCES: Enter the total number of It is highly desirable that the abstract of classified reports references cited in the report. be unclassified. Each paragraph of the abstract shall end with 8a. CONTRACT OR GRANT NUMBER: If appropriate, enter an indication of the military security classification of the inthe applicable number of the contract or grant under which formation in the paragraph, represented as (TS), (S), (C), or (U). the report was written. There is no limitation on the length of the abstract. How8b, 8c, & 8d. PROJECT NUMBER: Enter the appropriate ever, the suggested length is from 150 to 225 words. military department identification, such as project number, 14. KEY WORDS: Key words are technically meaningful terms subproject number, system numbers, task number, etc. or short phrases that characterize a report and may be used as 9a. ORIGINATOR'S REPORT NUMBER(S): Enter the offi- index entries for cataloging the report. Key words must be cial report number by which the document will be identified selected so that no security classification is required. Identiand controlled by the originating activity. This number must fiers, such as equipment model designation, trade name, military be unique to this report. project code name, geographic location, may be used as key 9b. OTHER REPORT NUMdBER(S): If the report has been words but will be followed by an indication of technical conassigned any other report ffmbers (either by the originator text. The assignment of links, rules, and weights is optional. 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 Unclassified Security Classification

UNIVERSITY OF MICHIGAN [1111513111 83[[[I [111[ 1111113 3 9015 03483 1373