THE UNIVERSITY OF MICHIGAN 5548-2-T AFCRL-64-915 MODIFICATION TO THE SCATTERING BEHAVIOR OF A SPHERE BY REACTIVE LOADING by V. V. Liepa and T. B.A. Senior October 1964 Scientific Report No. 2 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 5548-2-T TABLE OF CONTENTS ABSTRACT 1 I INTRODUCTION 2 II FIELD EXPRESSIONS 5 III RADIATION AND LOADING ADMITTANCES 16 IV COMPUTED RESULTS 23 V EXPERIMENT 46 VI CONCLUSIONS 60 ACKNOWLEDGEMENTS 63 REFERENCES 64 APPENDIX A: AN EXTREMUM PROBLEM 66 APPENDIX B: THE INPUT ADMITTANCE OF AN ASYMMETRICALLY EXCITED RADIAL CAVITY 70,-~, iii,

THE UNIVERSITY OF MICHIGAN 5548-2-T ABSTRACT Electromagnetic scattering behavior by a metallic sphere loaded with a circumferential slot in the plane normal to the direction of incidence is investigated. The slot is assumed to be of small but non-zero width with electric field constant across it, and under this assumption the analysis of external field is exact. The field scattered in any direction is obtained by superposition of the field diffracted by an unloaded sphere and that radiated by an excited slot at the position of the load, with the radiation strength of the slot related to the loading characteristics in the combined problem. Thus, there are two parameters that determine the scattering behavior of this object: the loading admittance and the position of the slot. Numerical results are presented primarily for the case of back scattering and these are compared with experimental measurements made using a metallic sphere with an equatorial slot backed by a radial cavity of adjustable depth. __ _ _ _ _ __ _ _ _ _ __ _ _ _ _ __ _ _ _ _ __ _ _ _ _ _1 1__ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _

THE UNIVERSITY OF MICHIGAN 5548-2-T INTRODUCTION A major problem in scattering theory is the development of ways for controlling the scattering behavior of an object through modifications to its surface. It has long been recognized that minor shape changes can be effective in reducing (or even enhancing) the scattering cross section, primarily at high frequencies, and with the development of high performance absorbers during the past decade, the application of these materials has now become one of the most potent tools for cross section reduction. As the frequency is decreased, however, absorbers lose much of their utility. At wavelengths comparable to, or larger than, the overall dimensions of the scatterer, the thickness of any good non-resonant absorber is liable to be intolerable, and to achieve a reasonable degree of absorption the properties of the material must also be tailored to the individual shape parameters of the surface. This last severely complicates the problem of designing the material. It is therefore desirable to investigate other means of cross section control, particularly ones which are effective in the resonance region, and of these new techniques the most promising is that known as reactive loading. In essence, the technique is to change the impedance "seen" by the incident field over a restricted portion of the surface using a cavity-backed slot, lumped network, or other 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 akin 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 first reported application of this technique for cross section reduction was by Iams (1950), who used it to decrease the scattering from metallic posts in a parallel plate pillbox structure. King (1956) investigated the change in current on a thin cylindrical rod when a central load was introduced, and Hu (1958) and As and Schmitt (1958) later showed that loading can appreciably affect the scattering ____________________________ 2 __ ______________________

THE UNIVERSITY OF MICHIGAN 5548-2-T behavior of such a rod. However, it was not until the recent study by Chen and Liepa (1964) that the ultimate capability of loading for cross section reduction was fully demonstrated. For normal incidence on a thin cylinder of length i, 0< -< 2X, the induced current was calculated as a function of an arbitrary central load, and the results 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 fl/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.k/X, and whereas the required loading was passive when < X, that for X. < 2X was primarily active. Chen and Liepa also considered the scattering in directions other than normal to the surface, and in two later papers Chen (1964a and b) has extended the analysis to oblique incidence and to the case of two identical symmetrically-placed loads. 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 wave length), and though Sletten et al (1964) have shown experimentally that reactive loading is still effective when the cylinder is thick (radius comparable with the length), no theoretical treatment of this problem is yet available. 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 scattered field in terms of commonly-defined antenna parameters. However, to use the method to 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 the sphere, and this is the shape that we shall consider here. A plane wave is assumed to impinge on a perfectly conducting sphere loaded in a narrow azimuthal region whose plane is normal to the direction of incidence. The field scattered in any direction can then 3

THE UNIVERSITY OF MICHIGAN 5548-2-T be obtained by superposition of the field diffracted by an unloaded sphere and that radiated by an excited slot at the position of the load, with the radiation strength of the slot related to the loading characteristics in the combined problem. The concept of distributed admittances is introduced, and by varying the admittance Y' of the slot, a wide degree of control over the scattering behavior can be exercised. Even if attention is confined to passive loads (admittances whose real parts are non-negative), substantial increases or decreases in the scattered amplitude in almost any specified direction can be achieved by appropriate choice of Ye. Numerical results are presented, primarily for the case of back scattering, and these are compared with measurements made using a model with an equatorial slot backed by a cavity of adjustable depth. The agreement is excellent. _____________________ _ 4

THE UNIVERSITY OF MICHIGAN 5548-2-T II FIELD EXPRESSIONS Consider first an unslotted perfectly conducting sphere of radius a whose center is at the origin of a Cartesian coordinate system (x, y, z). A plane electromagnetic wave is assumed incident in the direction of the negative z axis, and, since there is no loss of generality in taking its electric vector to lie in the x direction, we choose i ikz A ikz E = xek and y = -yYe where Y is the intrinsic admittance of free space and a time factor e has been suppressed. If we also introduce the spherical polar coordinates (r, 0, 0) such that x= rsine cos0, y= rsine sin, z = rcosO with 0 = 0 representing the back scattering direction and 0 = r the forward one, the above expressions for the incident field can be expanded in terms of the vector wave functions M(1) and N(1) in the form i. n 2n+ (1) (1) n i -iN (1) n=l n(n+ 1) -ln -en Ei o in 2n+ 1 (1) i(1) (2) Hi- iY ~i n(n+ 1) n el) (2) (Stratton, 1941), where,n(kr) P r(cos0) o n(kr) M(1) _ n n sin ) p( ) cos M =+m kr sin moo - P (coso) C O -e kr sin Cos kr n sin mn O

THE UNIVERSITY OF MICHIGAN 5548-2-T on(kr) n(kr) N1) =n(n+l) pm(cos0) COS n ar P (cos0) CSmO 0 -e (kr)2 n sin kr ao n sin mn (kr)'(kr) pm (cos) ) n n sin + kr sin 0 MCO and On(kr) = kr jn(kr), where jn(kr) is the spherical Bessel function of order n. P (cos0) is the n Legendre function of degree n and order m as defined, for example, by Stratton (1941). At the surface of the perfectly conducting sphere the scattered field (E, HS) satisfies the boundary condition r x (E+ ES) = 0, and from the requirement that the scattered field represent an outgoing wave at infinity, we are led to write 00 Os, fA (3) (3) E AM + iBnn)' (3) n-ol~n -e 1n/' implying o fA(3) (3)' /3=iY Q1 (4) Hs-= i Y E'n-oln +i n-eln/) The M() and N() differ from the M) and _l) in having n (kr) replaced by r (kr) = kr h(2)(kr) n n where h (2)(kr) is the spherical Hankel function of the second kind. Application of n the boundary condition at r=a now gives

THE UNIVERSITY OF MICHIGAN 5548-2-T n 2n + 1 On(ka) A -1 (5) n n(n+ ) in(ka) ( n' (ka) and by substitution of (5) and (6) into (3) and (4) the scattered field is then determined. The total field is the sum of (E, Hi) and (E, H ), and at the surface r=a the only non-zero components are Er, H0 and H. The last two are directly related to the components of the induced current J via the equation J= rxH, so that J =-Ho J%, and if we define Ho Ysin T1(0) (7) H0 = Ycoso T2(0) (8) we have T (0) = n+1 2n+1 P (cos O) (9 T1(8)= ka E i+ (n 1nn) {'(ka) sin 0 (ka) -T n 1 k n1 nn1 n i n.n+ 2n+ 1 1 a 1 i n T (0) =5I1 1n l P (cos0) + 2 1a r an(n+1) (k) 0 n n (ka) sin0 n (10) The interpretation of these in terms of creeping waves is discussed in Kazarinoff and Senior (1962). In the far zone the scattered field behaves as an outgoing spherical wave whose properties can be obtained by replacing {n(kr) and its derivative by the leading terms of their asymptotic expansions for large kr, viz. i______________________________i______________ 7

THE UNIVERSITY OF MICHIGAN 5548-2 -T Vn (kr),- -i ~ (kr)~- ikr Since E and H now satisfy the equation Hs =r YEs it is sufficient to confine our attention to the former, and following the notation' in Senior and Goodrich (1964), the components of the scattered electric vector are written as -ikr s e s Es icos kr S1(O) (12) -ikr E.= -i sine0 S (0) (13) kr 2 where (0) n(n+l (ka) (cos) - (ka) P(cos0)l = n(n+) (ka) a- n n(ka) sin( ~~~n=1 n~ (14) 1 2n00l j (ka) P (cos0) (ka) SS(W) - (-1) n 2n+n(n;1) I We note that S (O) = S (0) 1 2 and S S S1 (r) = -S (r), implying that, for back and forward scattering, the field has the same linear polarization as the incident field. In all other directions, however, the field is elliptically polarized and the component cross sections are r= k sl(0)J cos2g, (16) >Note the change in time convention.

THE UNIVERSITY OF MICHIGAN 5548-2-T 2 s 2 (12) The complete scattering cross section is, of course, c = o0 + CT Let us now consider the separate but related problem of a perfectly conducting sphere with a narrow slot symmetrically placed with respect to the z direction (and hence, with respect to the incident field direction in the problem just discussed). The slot occupies the region 0 -6/2 0 < 0o + 6/2 (see Figure 1) and its angular width 6 is such that ka << 1. Within the gap the tangential electric field is specified, and in view of our intention to regard the slot as semi-active by coupling the solution for this problem to the one already derived, the excitation must be chosen in accordance with the surface field behavior shown in equations (7) through (10). It is therefore assumed that for 0-0 < 6/2 E- = -v cos0 (18) E0 = Ta s EF= 0, (19) corresponding to a constant (but asymmetrical) voltage v cos 0 across the gap. Over the rest of the sphere, E0 and E are both zero, as is appropriate to a perfectly conducting surface. To determine the field (Er, H r) radiated by the slot, we again postulate a field of the form shown in equations (3) and (4), but with A and B replaced by new constants C and D respectively, so that Er= M3 +iD N (20) - n oln n-eln/ (20) n=lH i + iD - (n-oln n-eln) (21) I__I__ii__iii__i__i__ii__I 9 __i__i

THE UNIVERSITY OF MICHIGAN 5548-2-T z / (,,0~ a // I / )/ / / ~~~X~~~~~~~~~~/ FIGURE 1: SPHERE GEOMETRY

THE UNIVERSITY OF MICHIGAN 5548-2-T When these are substituted into the boundary conditions at r = a, we obtain oDP (cos) 1kv n O,(kv 6 n(ka) +iD n'(ka) - p(cosO) = < o1 nnk nsine n n a n 6 = 0, otherwise, (22) from the 0 component, and OD1P l(cosO).{CC (ka) a Pn(cos0) + i D n'(ka) n = 0 all 0 (23) n=1a n nn sin J from the 0 component. Moreover, from Bailin and Silver (1956) P (cosO) a P (cos 0) + P (cos )O dO = 0 n ae m mam a0 n(coso) and (Cos 1 0) P (cos,0)+ P (coso)p (cose3 sinO dO =A 0 n ao m sin2 0n m nm where 0 n =m 2n2(n+1)2 2n+1 n and hence, by application of these relations to (22) and (23), 0 +6/2 kv 2n+1 1 1o C = -(k) 2 2 6 P (cos )dO'n(ka) 2n2(n+1)2 6 and and

THE UNIVERSITY OF MICHIGAN 5548-2-T 00+ 6/2 Di kv 2n+ 1 D 1 n rn(ka) 2 2 6 sin (cos 6/2O)d n 2n (n+1) jO-6/2 pl(+) - p (_) kvsinOo 2n+1 n n'(ka) 2 2 6 (25) n 2n (n+l) where, for brevity, we have written Pm(+) = pm Cs {0 2) - n o- 2 In evaluating D it was assumed that the variation of sinO over the slot can be n neglected, and consequently the position of the slot is now limited by the condition c< < 7Twith e >> 6. It is also observed that in the limit 6 —0O P (+)-p(-) n n -) ) > -P (cos ) 6 n o and n n P-n )P (cosO) 6 ao n o 0 The expressions for the components of the radiated field follow from equations (20) and (21) on inserting the above formulae for C and D. Two n n particular cases are of interest. On the surface r = a we have, analogously to (7) and (8), r r H0 = Yvsin Tr(0) (26) and H = Yv cos0 T() (27) where 12,,

THE UNIVERSITY OF MICHIGAN 5548-2-T OE 2 +l n(ka) sin (l(+))) r i 2n+ 1 T (0) =2a - P (Cos 1 2 2a n= n1L(ka) ( 2ka) sin ) cos n _ 1 n (n+1) n tn(ka) 1P (28) n n n 1 + (ka) 1 + P ) o ) (c s(2) + n(ka) n 1 1 co ~ (kan s~j~J8P'0 cos 0a i 2n+ n s n=1 n (n+l)2 Ln(ka) n(ka) si0 Pn(+) - ) The rates at which the above series converge are functions of 6 and 0 as r r well as ka. If 6 / 0, the series for T (0) converges for all 0, but for T (0) 2 1 the convergence decreases rapidly as 0 approaches + 6/2, and in the limit the series actually diverges. This behavior can be attributed to the step discontinuity in the surface field introduced by (18). The two series also diverge if 6 = 0, and to effect a numerical evaluation it is therefore necessary to keep 6 non-zero and to retain a number N of terms which is, in fact, inversely proportional to 6. In the far zone, on the other hand, the expressions for the field components are convergent even for 6 =0, corresponding to an infinitesimal slot across which the voltage v cos 0 is applied. For simplicity we shall therefore proceed directly to the limit, in which case -ikr E =iv cos kr S1(0), (30) 0 kr -ikr r dce r (31) E -iv sin ke S (0) (31) (cf equations(12) and (13)) with l_____________________________________ 13 l

THE UNIVERSITY OF MICHIGAN 5548-2-T r k n+1 2n+ 1 S (0) Tsin O on+ 1 2n + ~ P (cosO) P (Cos 1 S1 () sin0 E 2 2{'(ka) a n a3 n o n=1 n (n+l) n o 1 1 P (cos 0) P (cos )} i n n OO (32) + (ka) sin0 sin 0 (32) __ _ _ 2n+ P (cos0) S (8) = -sin (cos 2 =2 i 2 (ka) sinG8 3O n o n= l n(n+ 1) n o (k)6P (c (Cos e ) +|: ak P AdP (Cos 0) i (I }' (33) and from these the components of the magnetic vector can be obtained as before. The final problem to be considered is a combination of the preceding two in which the plane wave given in equations (1) and (2) is incident on the slotted sphere. If it is assumed that the same voltage is excited across the gap, the expressions for the resulting fields can be found by superposition of those associated with each individual problem. We therefore have E= E+ ES+ Er (34) and H = Hi+ H +Hr (35) where (Es, H ) and (E, H ) are as given above. In particular, in the far zone, the components of the total scattered electric field are -ikr e E = icos kr S (0), (36) -ikr E -isi kr S2() (37) with s (0) = s5(e)+ vS(O), (38) 1 v1 4, 14

THE UNIVERSITY OF MICHIGAN 5548-2 -T S2(0) = S(O) + v S2(). (39) 2 2 2 r r The expressions for S1 and S2 are, of course, independent of v, and if this voltage is induced by a loading applied to the slot (using, for example, a cavity at its back), the voltage can be related to the loading admittance. The derivation of this equation is our next task.

THE UNIVERSITY OF MICHIGAN 5548-2-T III RADIATION AND LOADING ADMITTANCES In antenna theory it is customary to define the admittance of a slot as the ratio of the current flowing away from the gap to the applied voltage, where the latter is the line integral of the electric field across the gap. In the present problem, however, the total instantaneous current is zero, due to the cos0 variation of the slot excitation; some other definition of admittance is therefore necessary, and it is natural to introduce the concept of admittance density, where this is the ratio of the current density to the applied voltage at a point specified by the azimuthal variable 0. The implications of such a definition can be seen by considering the radiation admittance of the slot in the second of the three problems discussed in Section II. Since the current flowing from the slot is J -H 0 0' the radiation admittance density is Jo Yr v cos0 = -Y T2(0) and hence the (total) radiation admittance is Yr ix~ Yr a sinO dl 0 = -2r a sin 0 Y T(0). (40) o 2 As written above, however, Y does not provide a unique specification of the admittance. Since the slot width 6 must be non-zero in order to compute T2(0),-, 16,___

THE UNIVERSITY OF MICHIGAN 5548-2-T it would be possible to give 0 any value between 0 -6/2 and 0 + 6/2, and though the numerical effect on Y may be insignificant under most practical circumr stances, it is undesirable that the expression for Yr should have that degree of arbitrariness. In an attempt to overcome the disadvantage, we note that an alternative, but equally acceptable, definition of admittance is twice the ratio of the complex power flow across the aperture to the square of the voltage, and for the case in which the voltage is a function of position along the slot, the concept of admittance density is once again appropriate. The complex power radiated per unit length of the slot is w= s0+6/2. (ErxHr). ad0 0 + 6/2 0 r r ( 0+ 6/2 1 vcos0o Hr 2 6 0-6/2 0 where N denotes the complex conjugate. The radiation admittance density is therefore 2w (vcos 0) -1 706/+ 6/2r vH dO, (41) 6vcos0 H 0 -6/2 and using the expression for Ho given in equation (27), together with the evaluation of the Legendre function integrals employed in the determination of C and Dn, we have 0y = iY 2n+ 1/ n ( n(ka)(l(+) _pl(_)3n r 2a ~-N~ 2n+12 L n(ka) 6 siO (n(ka) 17n

THE UNIVERSITY OF MICHIGAN 5548-2-T Hence 2 2 Y i Y sin2 2n+1 (ka) (P(+) - P (ka) N(+) - P r o 2 n 2(n+l)2 n (ka) k 6 sin (42) and in contrast to the expression for Y originally derived, this new result has no ambiguity. The computation of equation (42) is discussed in Section IV. Let us now apply this same technique to the determination of the loading admittance in the case of a semi-active slot (third problem in Section II). As before, the complex power per unit length of the slot is obtained by integrating the Poynting vector, and for power entering into the slot e o+ 6/2 W= - - -(ExT- -rd 0 - 6/2 2 where (E, H) are as given in equations (34) and (35). Thus 1 vcos o 6/2| 6 gO He de 0 and the slot admittance density is + 6/2 1t A (H + Hs + H d Y/ 6vcoso 30-6/2 ( + + which can be written as 00+ 6/2 - + Y /T2(0) de (43) 0 -6/2 by using equations (41), (8), and (10). For 6 sufficiently small, the variation of T2(0) across the slot can be neglected, giving y = Y+ +-T2(0o) 18

THE UNIVERSITY OF MICHIGAN 5548-2-T and the total loading admittance of the slot is therefore Y - -Y + - 27ra sinO T (0o) (44) ) r v o 2 o' where Y and T 2(0 ) are as shown in equations (42) and (10) respectively. r 2o Yj is the significant parameter for control of the scattering behavior. Solving for v, v = y 2ra sin0 T2(0 ) (45) rr r which can be substituted into the expressions for S (0) and S (0) to give the total 1 2 scattering amplitudes S (0) = s (0) + 27ra sin0 T (0 )Sr(0) (46) 1 or S2(0) = S(0) + + Y 27ra sinO T (0O)S (0) (47) 2 2 y~ ~ o2o 2 (see equations (38) and (39)). The right hand sides of (46) and (47) are functions only of ka, 6, 00 and 0, and in particular, are independent of v. To make the scattering amplitude S (0) zero in the direction 0=0', the loading required is s(e0') Y -Y -Y 27ra sin0 T2(0 ) (48) XI r o 2 o s(0') This differs from the loading necessary to make S2(0) zero for 0 - 0' unless srr (0e') 2 1 2 - =, (49) Ss (o ) s (o,) 1 2 and thus in general the complete scattering cross section a = oB + ac cannot be reduced to zero with a loading of the form discussed here. This does not, however, rule out the possibility of significant reductions in cx by suitable choice of Y. Moreover, in many cases of bistatic scattering only S1(0) or S2(0) is of interest,,,19

THE UNIVERSITY OF MICHIGAN 5548-2-T and a zero (component) cross section can then be achieved. The obvious exceptions to the above are back and forward scattering. The back scattering direction is 0 = 0, and here n'(cos0) n(n+1) = 1 [t =nnP =(COs0, Lsin e 2 ncos j from which we have s5(O) = S(0), 1 2 sr(0) = Sr(0) 1 2 Furthermore, S() = ka sine T2 ( ) (50) 1 4 o 2 o and hence the loading for zero S1 (0) and S 2(0) is 1 2 Y -Y r - {ka sin0 T2(0 ) 2 Sl()} -1,2f~~'n(~n ( ka) ~& (ka) - = -Y -Y ~rkasin 0T2(0) { (-1) (2n+1),(k) (ka) o 0 2 0 n= (ka) /n(ka) (51) This corresponds to either an active or passive load depending on the values of ka and 0. Similarly, for the forward scattering direction (9 = 7r) n(s0)1 )n+l n(n+l) (os L- sine-j - 2 Le n _ O' so that S S S (7r) = -S(T), 1 2 and these also satisfy the conditions (49). Moreover, 20

THE UNIVERSITY OF MICHIGAN 5548-2-T ~r k ~0 2n+1 1 31 (08). P1(cos0) 1 4 n1 n(n+) (ka) n o n(ka) sin 0 (52) k = kasinO T (7r-0 ) 4 o2 o and consequently for zero S1 (T) and S2(7) 1 O 2 |Y =-Y YrY (ka sin0 ) T2(O)T2( -0 { I r 2 o 2 o 2 0o) -Y - Yr(ka sin0 ) T (0 )T (7r%0) ~- n o 2 2 j(2n+1) n (ka) n(ka) n=1 n n (53) The fact that for a passive object zero scattering in the forward direction implies zero total scattering and zero absorption (Schiff, 1954) indicates that the above loading corresponds to an active slot for all ka and 0. If 0 f ir the characteristics of the loading admittance for zero S 1(0) or S2(0) can in general be determined only by numerical computation of the functions involved. An exception is the case of small ka. Using the small argument expansions for i n(ka), n (ka) and their derivatives, it is found that the admittance necessary to make S (0) zero is (1) =Y k a+002 (54) -Y 7y01-y11a~t 2,} (54) Q Olf +1 ka + O (ka2)) (see equation (48)), where n n o 9 sin2O0 + E 2n+1 ( (~) nP() 01 4 1+2cosO n=1 n(n+1)2 6 7 1+cos sin cos0 Ni 2 1+2cos0 o o similarly, to make S2(0) zero 21

THE UNIVERSITY OF MICHIGAN 5548-2-T Y(2) ka 702 + 12ka + (ka2 (55) where 2 sin 0 cos 0 Co 9 sin cos 2n+1 ( n n'02 - 4 2+ cosO l n(n+ (+) - (-) 1 (l+cosO) (2+5cosO) 2 Y sin 80 cos0 -2 2 2+cos sin o o As expected, the two admittances are identical for 0 = 0 and 7r, but to the order shown the expansions give no applicable information about the real part of the admittance in the forward direction. For 0 f 7r, however, the dominant contributions to Re.YA are provided by Y11 and y12' and these show a significant change of character as 0 passes th6ough a critical value. Thus, for example, Re.Y(1) has the same sign as 1 + 2 0 corresponding to an active or passive load according as this ratio is negative or positive respectively, and to reduce the back scattering cross section of a small sphere to zero with a passive load it is therefore necessary that 0 < 7r/2

THE UNIVERSITY OF MICHIGAN 5548-2-T IV COMPUTED RESULTS The six functions S (0), S (0), S (0), S (0), T (0 ) and Y involved 1 2''2' 20 r in the expressions for S1(0) and S2(0) (see equations (46) and (47)) were programmed for calculation on the University of Michigan IBM 7090 computer, and because of the practical significance which each of these functions has, it is of interest to examine their individual values before discussing the loading necessary for particular cross section modifications. Except in the case of Y, all of the computations were straightforward. The number N of terms retained in any one series was essentially a machine variable, and was the largest possible consistent with the machine capacity not being exceeded in any of the subsidiary calculations associated with that term. In practice it turned out that N was of order 5+ 4ka, and to judge from the results obtained using a somewhat smaller number of terms, it would appear that this was sufficient for four digit accuracy. S (0) and S (0) are the complex far field amplitudes for the solid (unloaded) 1 2 sphere, and the corresponding component scattering cross sections are given by equations (16) and (17). The case of back scattering (0 =0) is of most concern. The two functions are then equal, and since adequate tabulations are available in the literature (Bechtel, 1962), no further computations were performed. For reference purposes, however, the back scattering cross section X2 cr(0) = - sS(0) (56) normalized to the physical optics value 7ra, is plotted as a function of ka for 0< ka < 10 in Figure 2. T2(0) is proportional to that component of the current on the solid sphere which, in the shadow region, gives rise to the major creeping wave (Kazarinoff and Senior, 1962). The component is also normal to the slot and, as such, will excite the slot and be altered by the presence of an annular perturbation at the surface. To illustrate the variation as a function of 0, T2(0) has been computed for 0<0< 1800 23

THlE UNiVERSiTY OF MCIA 5548-2-T MCIA 0 0 z -r2 o 0 C) 0 H~ H C) C) N _5 0~ z ~~~oJL~~o (Q)~~~~~~~~~~o~Y 24~~~~~~~~

THE UNIVERSITY OF MICHIGAN 5548-2 -T in increments of 5~ with ka= 4. 28, and the results are shown in Figure 3. The oscillatory behavior within the shadow is apparent. Computations were also made for ka= 0(0. 1)10. 0 with 0 = 60, 90~ and 120, and the moduli are presented in Figure 4. On passing from the lit region to the shadow there is a fairly systematic reduction in magnitude which is most evident for the larger values of ka. As ka - 0, however, the amplitude tends to 1. 5 regardless of 0. In the back scattering direction S (0) and S2(0) are equal and proportional 1 2 to T (0 ) (see equation (50)), so that separate computation of these functions is 2o not necessary as long as 0 = 0. For other directions, however, the procedure is quite straightforward and is mentioned briefly later on. 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. To facilitate the computations, the first N terms of the series are treated exactly and the subsequent terms are replaced by their asymptotic forms for large n (Plonus, to be published). Since _n(ka) n! J (57) Cl(ka) ka for n~ ka, and P (cos )- )O Pn(cos0}. n(r sin+ -)0 P 1(cosO)- n os n+ 1)+ for n >> cosec 0, substitution of these expressions into the higher order terms of (42) gives 25

2.5 2.0 1.5 < 01b 0 I_ 1.0 0.5 0' I I I I I I I I I 0 30 60 90 120 150 180l e in degrees FIG. 3: AMPLITUDE OF SURFACE FIELD COMPONENT T2(0) FOR ka=4. 28. 2

2 c (a) (b) 0 ka FIG. 4: AMPLITUDE OF SURFACE FIELD COMPONENT T (0 ) FOR (a) 0 = 600, (b) 0 = 900 and (c) 0= 120. 2 0

THE UNIVERSITY OF MICHIGAN 5548-2-T r i20,2n + 1 r n(ka) (+) - nP(-) n (ka) P (+ I ir sin 20 2n+ 1 (n n n n n Y 0 o 2 (ka) 6 sinO i(ka) 6 n (n+ 1) o n n t n6 2 +i 1 2n+ 1 sin! 2 N+1 (+1) 2 n6 2) Lka sin oo11+ sin 2n+0eo kasnoot1-sin2n+1oo}1 n=N-(l n (n+1 (58) 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 0 = 90 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, N= 15, Y /Y= 15.255+i30.385, N = 20, Y/Y = 15. 255+ i 30. 676, N = 24, Y /Y = 15. 255+i30. 748. (N = 24 is the maximum attainable by the machine for ka = 5. 0). The rapid convergence of the series for Re. Yr 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 per cent of the correct value. Since the expression for Yr is symmetrical about 0O = 900, it is sufficient to restrict attention to 0 <90, and the values that were chosen for computation were 0 =600 and 900 with 6=0.0392 (approximately 2. 250). The real and imaginary parts of Yr/Y for the two cases are plotted as functions of ka, 0 < ka < 10, in Figures 5 and 6 respectively, and from these it would appear that a change of slot position does not affect the general character of the curves. The real parts are zero for ka = 0, and rise through positive values with a small but regular _________N_-__10,___/Y__=_15.255+i 28. _______86__________

THE UNIVERSITY OF MICHIGAN 5548-2-T l 0 II =- Q z I I ~\ < \, ~I D -oD O o o I I I 1/-JA \29 O O O O O O O O~~~~~~~~~~~o

50 40 30- /, _ - lo1:L-20- z ~-30,,,.,. — 0 2 4 6 ka do=0 go AND = 0. 0392.0 -10 2 q 6 8 ka FIG. 6: REAL ( —-) AND IMAGINARY (-) PARTS OF NORMALIZED RADIATION ADMITTANCE FOR 00=900 AND 6=0.0392.

THE UNIVERSITY OF MICHIGAN 5548-2-T oscillation as ka increases. The imaginary parts, on the other hand, have a negative singularity for ka = 0, but as a consequence of the asymmetric excitation of the slot, their signs change from negative to positive at a value of ka near to unity. For larger ka, the curves for the imaginary parts lie almost parallel to and above those for the real parts. The value of 6 used in the above computations was determined by the equivalent slot width in the experimental model discussed in Section V, but to investigate the effect which a change of slot width has on the radiation admittance, Yr/Y was computed for 6 = 0. 02(0. 02)0.24 with ka= 4. 28 and 0= 900. As can be r o seen from Figure 7, the real part is essentially constant for the full range of 6 considered, but the imaginary part shows a significant variation. Because of the increasing local capacitance in the region of the gap as 6 approaches zero, the imaginary part has a positive singularity at 6 = 0, and is monotonically decreasing as 6 increases. Having computed or available S1(0), T2(0) and Yr/Y it is now possible to determine the loading admittance necessary for zero back scattering, and in Figures 8 through 10 the real and imaginary parts of YJ/Y based on equation (51) are plotted as functions of ka for 0 = 600, 900 and 1200 respectively. Taking, for example, Figure 8, we observe that the curves for both the real and imaginary parts are quite irregular in structure and, as ka -- 0, Im. Yo/Y- oo. Of the two curves, however, the one for the real part is the more interesting because of its greater practical significance. Bearing in mind that all non-negative values of Re. Y, correspond to passive loads, it is apparent that zero back scattering is achievable using only a passive load for all ka less than (about) 2. 9. At this value of ka the curve for Re. YfI crosses the zero line and stays negative until ka reaches 6. 6(approx.), at which time it becomes positive again. This oscillatory pattern continues out to the largest ka computed, and for every ka such that Re. Y >_ 0, passive loading is sufficient to annul the back scattered field. In view of the later experimental study, it is remarked that the loading is purely susceptive whenever Re. Y = 0. 31 _

50 40. 30)..~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~r (I' 20 ~~~~~~~~~~~~~~~10 ~ ~ ~ ~ 0 1~~~~~0 1 n~~~~~~~~~ O z'0, I I I. I. I, II 0.05 0.1 0.15 0.20 0.25 8 in radians FIG. 7: REAL ( —-) AND IMAGINARY (-) PARTS OF NORMALIZED RADIATION ADMITTANCE FOR ka=4.28AND 0 =900. 0

if"\ 40 I \ I \ I O~~~~~~~~~~~~~ \ **40 moo 0~~~~~~~~~~~~~~ -20 ~ -40 -80 20 12 4- s' / \ / I I / % \ "x I 0... _E -A.....LO FRI BCSTENWT0I t6 II -20 -/ rr 1 ~ / -.~~~~~~~~~~~~~= -40 - I I I / \ I t - ~ ~ ~ ~ ~ ~ ~~~~~4 110 -60 -80 0 z — 1t00' 0 2 4 6 8 10 ka FIG. 8: REAL ( —-) AND IMAGINARY (-) PARTS OF NORMALIZED LOADING ADMITTANCE FOR ZERO BACK SCATTERING WITH 0 -60~. 0

40 / 20 - I'I I /'I I -20 ~ ~ ~ ~ ~ 0 - --— i - -- ---- -— i -- II I I I I I IZ -20 11\I I 1/ I tTI -40 n ~~~~~~p ~ ~ ~ ~ I I a CI) -40 I I II \~~ ~~ I ~ -60 I~T \j~~~~~~ - -80 \ I -100z -loo~~~~~~ 1 I I Iz 0 2 4 6 81 ka FIG. 9: REAL ( —-) AND IMAGINARY ( —) PARTS OF NORMALIZED LOADING ADMITTANCE FOR ZERO BACK SCATTERING WITH 0 = 90O 0

40 200 -20 H -40 ~ ~ //-\ / r r -0 2 I~~~~~~~~~~~~~ / Ig I Q ka~~~~~~~~~~~~~~~~~~v / / "~ Q1 -40r ~c -60 C) -80 ~~ -4 0 2 4 6 8 10 ka FIG. 10: REAL ( —-) AND IMAGINARY(-) PARTS OF NORMALIZED LOADING ADMITTANCE FOR ZERO BACK SCATTERING WITH 0 = 1200.

THE UNIVERSITY OF MICHIGAN 5 548 -2-T it, I H~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Z I f 0 I z~~~~~~~~~~~~~~~~~~~~~~~P H r~~~~~~~~ 0 -O 0 c~J 1 6~~~~~~~~~~~~ I~~~ I I 36~~~~~~

THE UNIVERSITY OF MICHIGAN 5548-2-T Since passive loading is of most practical interest, let us consider the maximum and minimum back scattering cross sections that are obtainable in this manner. From equations (16), (46) and (50), the cross section o(0) of the loaded sphere, relative to the cross section o(0) of the unloaded body, can be written as O u(0) - ~~~1+ ~~ ~~~~(59) ao(0) x+ix2 where ir{ka sin 0 T2 ( 2 (60) v1 + i s 2 0 (60) and xl+ix2 = + r (61) with'1,' 2' xl and x2 real. The minimum and maximum values of the above cross section ratio for Y X=iRe. — b>0 are now given by the formulae in Appendix A, and with a passive slot at 0 = 600 0 the results are as shown in Figures 11 and 12. Taking first the minima, the return is zero for ka < 2. 9; then, as Re. YR in Figure 8 swings from positive to negative, the minimum return increases from zero and rises rapidly to a peak value only infinitesimally less than the return from the unloaded sphere, before falling back to zero again. The pattern is repeated, apparently without end. If, on the other hand, we aim for a maximum return (Figure 12), an arbitrarily large enhancement can be achieved by taking ka sufficiently small, but as this is a consequence of the higher order zero of the normalizing function a (0) at ka = 0 the result is somewhat misleading. For ka greater than (say) 0. 5, however, the enhancement is genuine, and Figure 12 shows that an increase of as much as 19. 4db can be obtained for a particular value of ka, with an average enhancement of more than 10 db for the range 1.0<ka < 10.0. 37

25 20 - trJl z -o 15 0 0 1~~~~~~~~~~~0~~~~O bb 0 2 4 6 810 ka FIG. 12: MAXIMUM RELATIVE BACK SCATTERING CROSS SECTION FOR PASSIVE LOADING AT E ~I 5I I I ~~~~~~~~~~~I I'""I "I ka FIG. 12: MAXIMUM RELATIVE BACK SCATTERING CROSS SECTION FOR PASSIVE LOADING AT 0 = 600. 0

THE UNIVERSITY OF MICHIGAN 5548-2-T The analogous data for slots at 900 and 1200 is presented in Figures 13 through 16, and though the qualitative behavior is similar to that described above, the displacement of the slot towards and into the shadow region decreases markedly the control that can be had with passive loading. Thus, for example, with the slot at 1200 a zero cross section is possible only for very limited ranges of ka, with the first range not commencing until ka= 1. 99. The maximum enhancement is also decreased, and averages a mere 4. 7 db for 1. 0< ka < 10. 0. To illustrate the bistatic performance of a loaded sphere, the case ka- 4. 28 and 0 = 900 has been investigated, with a loading Y /Y = -i 63. 45 (purely suscep0 tive) necessary to give zero back scattering. The radiated far field amplitudes Si(0) and S (0) were evaluated at 50 intervals from 0 =00 to 1800~, and the total 1 2 scattering amplitudes obtained from equations (46) and (47). To simplify the computations, the bistatic cross sections were normalized relative to the back scattering cross section O(0) of the unloaded sphere, and for ease of presentation, attention will be confined to E-plane (0=0) and H-plane (0= r/2) scattering. The results are given in Figures 17 and 18 respectively. Because of the choice of loading there is complete cancellation of the return for 0= 0, and the null widths (between 3 db points) in the E- and H-planes are 360 and 600 respectively. Beyond the nulls there is, on average, a slight enhancement of order 1 db in the scattered field, and it is tempting to conclude from this that any power removed from directions near 0 = 0 must be redistributed amongst the other angles. Such a redistribution, however, is by no means essential. For different values of ka and 0 it is possible that the loading for zero back scattering would also produce a reduction in most other directions as well, but if the real aim is to decrease the overall pattern, the choice of loading should be based on a criterion specifically developed for the reduction of the total (i. e. integrated) scattering cross section. To judge from the results found for a thin cylinder (Chen and Liepa, 1964), it is possible that some decrease in the total scattering could be achieved with passive loading at least for the smaller values of ka. u_______________ _.39

THE UNIVERSITY OF MICHIGAN 5548-2-T I Lo o 0 (Y3 ~ ~ ~ ~~0 0 ry~ t11 0 lU 0 0 I I -'~ I IH I~~~~~~~~~~~~~ 0 I 0V It~ ~~~~~~~~~~~~~ IO b-1 l k ~~~~~~~O U 0 i — 0 I0 |~~ ~~~~ IN ~~I~~ ~~~~ IM I F ~~~~~~~~~~~~~~~~~~~~~F I I W( I/ l ~~~~ ~ ~ c M |~~~~~~~~~~~i -- I I I qp' UIW (0).D 40

TNE UN'yERsrTy 55qS_2_TOF MICHIGAN O (3 cn o rj ~CC zr L3 Cd O (L) tj t3 PI 0' a crD u3 Pr; t) P q t t> U r-e cc7 cu 5 ~e' u, cu d d ch) n o r IC) O Ci lo) 09, QP' To-- xDu, C) 91

- ---- -—, — -- ---... -5'2 -10 6 0 ~bb E E -15 -20 - -' I I I.I I I I' I 0 2 4 6 a 1 Z ka FIG. 15: MINIMUM RELATIVE BACK SCATTERING CROSS SECTION FOR PASSIVE LOADING AT 0 = 1200. O

25$ 20 -.0 Z 0 0 _'_ 15b 0 E 0 io0 1 1 1 1 1I II I* I 0 2 4 6 8 10 ka FIG. 16: MAXIMUM RELATIVE BACK SCATTERING CROSS SECTION FOR PASSIVE LOADING AT O=120~.

16 412 do,~~~~~~~~~~~~~ I~~~~I 4 z 4 ~~~~~~~~~~~~~~~~~~~I -8 I 0 06/ Q1~oi5 I~~~~~~~~~~~~~ I ~~~~~CC 4P I C~0 I I 0 30 60 90 1 20 150 180 & in degrees FIG. 17: RELATIVE BISTATIC SCATTERING CROSS SECTIONS IN E-PLANE (0=0) FOR LOADED (-) AND UNLOADED ( —-) SPHERE: ka=4. 28, 0o 900 AND Y1 -1 63. 45 Y.

16/Q 12 -/0 ~~~~-41 ~ ~ ~ ~ ~ ~ ~r H 8 / -t2 | / I' I' I [ l [ l' Iz 0 30 60 90 120 150 180 & in degrees FIG. 18: RELATIVE BISTATIC SCATTERING CROSS SECTIONS IN H-PLANE ( 90) FOR LOADED (-) AND UNLOADED (- -) SPHERE: ka=4. 28, 0 =900 AND Y1 =-i 63. 45 Y. ~~~~~P Oo~~~~~~~~~~~~

THE UNIVERSITY OF MICHIGAN 5548-2-T V EXPERIMENT To confirm the above predictions of the back scattering behavior, a series of measurements were performed using a sphere with a circumferential cavity-backed slot. A photograph and sketch of the model are given in Figures 19 and 20 respectively. The model consisted of two identical solid aluminum caps joined together by means of a partially-threaded shaft at the center. The spacing or gap between the caps formed a radial cavity with input at the outer (sphere) radius and a short at the inner radius. Whereas the outer radius of the cavity was identical to that of the sphere itself, the inner radius was determined by the size of the shorting disc used. In all, a total of 21 discs were available, and with these the inner diameters could be varied from 0. 3125 (the diameter of the shaft joining the caps) to 3. 133 inches. The discs were made from 1/16 inch sheet aluminum. With the exception of an outer rim, each had a slight undercut in thickness to give better electrical contact, and when in place the two caps conformed to a spherical surface of diameter 3. 133 inches everywhere except for the equatorial slot. The surface width of the slot subtended an angle of approximately 2. 250 at the center of the sphere. The back scattering measurements were made at frequencies 2. 808, 3. 605, 3. 709, 3. 838 and 5. 136Gc, corresponding to which ka = 2. 340, 3. 004, 3. 090, 3. 198 and 4. 280 respectively. The equipment was that generally used in cw scattering experiments except that the conventional azimuth-amplitude recorder was replaced by a HP 415 B meter for greater accuracy of reading. A block diagram of the equipment is given in Figure 21. The distance from the antenna to the pedestal was approximately 25 feet, and the sphere was placed on the pedestal with the plane of the slot perpendicular to the direction of the incident, horizontally polarized, wave. At each frequency the back scattering cross section was measured for a series of shorting discs, with calibration relative to the return from the unloaded sphere. The resulting normalized cross sections are listed in Table 1, and in Figures 22 through 26 the data is plotted as functions of the imaginary part of Y,,/Y 46

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THE UNIVERSITY OF MICHIGAN 5548-2-T 3.133 5/8' 1ii'~ A'~~T 2b Materials: 0. 500 Sphere - aluminum 0. 625 Discs - aluminum 0. 750 Shaft - steel 0. 875 1. 000 i. 125 1. 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 -7zZZZT 16 2. 625 I I 2. 7505 2.875 I 3.133 h- — 2b --- FIG. 20: SECTORIAL VIEW OF MODEL 48

Stabilized Oscillator Isolator Pedestal and Transmit-Receive L O cID Antenna Meter Receiver Mixer C) FIG. 21: BLOCK DIAGRAM OF EQUIPMENT

THE UNIVERSITY OF MICHIGAN 5548-2-T TABLE 1: EXPERIMENTAL DATA Shorting disc'I, Relative Return,' 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 50

8 4 0 z 0lO ~3 b@-4 C;'~~~~~~~~~~~~~~~~~~~~~~~~~~~p~ -8 -12 C) C) -16 Z -200 -10o 0 100 20 Im [ Y/] FIG. 22: THEORETICAL (-) AND EXPERIMENTAL (000) RELATIVE BACK SCATTERING CROSS SECTIONS WITH SUSCEPTIVE LOADING: ka=2. 340 AND 0 = 900.

12~." V" " —---------. -—.___ ___ _ _ -8 | F-81 — -— 6. C) -200~~~[ly 00 Z SUCETVE00DNXP=AD0 FIG. 23: THEORETICAL () AND EXPERIMEToo SUSC EPTIVXERI ~NTAL (0 ) RELATIVE BACK SCATTERING CROSS SECTIONS WITH SUSCEPTIVE LOADING. k-3.o004 ND 9. ~,-. u~A~= 900. RSECON

128 4~~~~~~~~o........................... 01 0 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0 010 -8-12 ~C) —---------------- -200 -1oo 0 100 200 Im [Y1/y] FIG. 24: THEORETICAL (-) AND EXPERIMENTAL (000) RELATIVE BACK SCATTERING CROSS SECTION WITH SUSCEPTIVE LOADING: ka=3. 090 AND 0 = 900. 0

12 8 4 0 0 or] -8 -12 -( -16- I' I -200 -oo00 0 100 200 Z Im [YI/Y] FIG. 25: THEORETICAL (-) AND EXPERIMENTAL (000) RELATIVE BACK SCATTERING CROSS SECTIONS WITH SUSCEPTIVE LOADING: ka=3.198 AND 0 = 900. 0

12 0 0I 0~~~~~~~~~~~~~~~ -8 l 04 -i1 " Im y~~~~~~~~~~~~~~~~~~~~~~~~~~~~T bib U, —--------- -7 ~ ~O I -, L ~ ~ ~ ~ ~ RIM NT L ( 00 RE ATI E ACK SCA TE ING SE 0 0'HOR T C L( ) N X EqD 0 =g o 0~~~~ ~US EP IV.4 -~'\ C) -12 0 I) FIG.~ ~ ~~~~~~~~~~~~~0 26 HOEIA ADEPRMNA 00)RLTV AKSATE0 ETOSWT 200~~~~~~~~~~~~ SUSCEPTIVE LOADNG: kia=4. 280 AND 0 900.

THE UNIVERSITY OF MICHIGAN 5548-2 -T For this purpose, the disc diameters were converted to equivalent susceptive loadings using the formula in Appendix B. Also shown for comparison in Figures 22 through 26 are the theoretical curves computed from equation (46) with 0 = 0 and 0 = 900, and the agreement between theory and experiment is extremely gratifying. The frequencies used in the above study were chosen to provide a reasonable sampling of the effects obtainable with susceptive loading only. Thus, for example, when ka = 2. 340 zero back scattering would require a loading with large negative real part (see Figure 9) and this is, of course, unrealizable with a passive slot. In consequence, Figure 22 indicates no substantial cross section reduction (0. 6 db at most), but an increase of 8. 7 db is achieved for a particular Im. YA, and since the maximum possible enhancement for this ka demands a susceptive loading, the peak level in Figure 22 is in agreement with Figure 14. Figures 23 through 25 show the results of 3 per cent changes in ka and span a range of ka within which a complete suppression of the back scattered field with susceptive loading occurs. In Figure 24 (ka = 3. 090, corresponding to the second crossing of the zero line in Figure 9) the theoretical cross section for some small positive Im. Yi goes to zero. Experimentally, this particular loading was not obtainable with the available shorting discs, but for the disc which was nearest to this, the observed reduction in cross section was 14.7db. With a loading slightly less than this, the normalized cross section rose to a peak value of 10. 3 db, and a similar peaking before the minimum is also apparent in Figure 23. On the other hand, at a frequency just greater than that for which a zero cross section can be obtained, the peak return occurs for a loading larger than that appropriate to the minimum (see Figure 25). The final set of data in this group is presented in Figure 26 and is for ka = 4. 280. This is yet another value for which susceptive loading gives complete cancellation, and the results are similar to those shown in Figure 24 except for the presence of the peak on the opposite side of the null. The changeover is attributable to the fact that the corresponding zero crossing in Figure 9 is now negative to positive as ka increases. 56

THE UNIVERSITY OF MICHIGAN 5548-2-T Although the analysis in Sections 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, and it therefore seemed worthwhile to carry out a sample measurement of the back scattering cross section as a function of the rotation of the slotted sphere. The frequency selected was 5. 136 Gc, corresponding to ka = 4. 280, and to obtain the loading appropriate to the null in Figure 26 a new shorting disc of the requisite diameter was cut. The sphere was again mounted on the pedestal with the slot in a vertical plane, and measurements were made for both horizontal and vertical polarizations. The results are presented in Figures 27 and 28 along with the curves for the unloaded sphere. The large cross section reduction for incidence in the direction normal to the slot is clearly evident. For both polarizations the reduction is of order 20 db and though theoretically it should be infinite, the minor peaking at the center of each minimum could be due to the shorting disc being fractionally smaller than required. However, in view of the slightly differing magnitudes of the cross sections for zero rotation, a more likely source of the residual contribution is a sphere-pedestal interaction or a room effect. 57

0 ni z <~ 20 m ijQ -t0 1 X On U) I I I - R IN I a 0AND UNLOADED ( —-) SPHERE: ka=4.280, HORIZONTAL POLARIZATION. 0 -90 -60 -30 0 30 60 90 ROTATION ANGLE IN DEGREES FIG. 27: EXPERIMENTAL BACK SCATTERING CROSS SECTIONS OF LOADED (-) AND UNLOADED ( —-) SPHERE: ka=4. 280, HORIZONTAL POLARIZATION.

cl)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~r V) 0 (00 (0 co~9 -0 30030609 ROTATION ANGLE IN DEGREES 0 u,~~~~ I I I Id 0~~~9 -6 -3 0 09 FIG. 28: EXPERIMENTAL BACK SCATTERING CROSS SECTIONS OF LOADED(- AND UNLOADE SPHERE: ka=4. 280, VERTICAL POLARIZAT.ION.

THE UNIVERSITY OF MICHIGAN 5548-2-T VI CONCLUSIONS In the preceding sections we have considered the scattering behavior of a metallic sphere loaded with a slot in the plane normal to the direction of incidence. The slot was assumed to be of small but finite width, with the electric field constant across it, and under this assumption the analysis for the external fields is exact. Expressions for the scattered far field components were derived, which can then be used to investigate the modification to the scattering cross section produced by various types of loading. The loading admittance necessary for some particular modification is in general complex, with positive or negative real part corresponding to an active or passive load respectively. If attention is confined to the back or forward scattering direction or, for b0 u 0 or 7r, to a single"' component of the scattered field, it is, perhaps, obvious that there is no limit to the amount by which the cross section in some specified directions can be varied when active slots are allowed. The effectiveness of passive slots, however, may come as more of a surprise, and because of the immediate practical application of passive loading, emphasis has been placed on this case. For slots at 600, 900 or 1200 the loading required to produce a zero back scattering cross section was computed for 0 < ka < 10 at intervals of 0. 1. The variations of the real and imaginary parts are quite complex, but taking just the real part of the loading admittance, we observe that ranges of ka in which Re. Ye is positive (passive slot) or negative (active slot) alternate with one another. Thus, with a given slot position, passive loading can nullify the return only for certain ranges of ka, but since their limits are functions of 00, it is feasible that two (or more) slots could be used to cover much larger ranges within which a null can b achieved. Even outside such a range a passive loading may still provide a significant reduction in cross section. The maximum and minimum cross sections I''With the chosen type of slot there is, in general, no loading either active or passive which will reduce a(O) to zero for 0 =O or w. 60

THE UNIVERSITY OF MICHIGAN 5548-2-T obtainable with single passive load have been computed for the above values of 0O and ka, and enhancements by as much as 20 db are not uncommon. To verify some of these conclusions, a model was constructed with a circumferential slot at 0 = 900 backed by a cavity whose depth could be varied. 0 Since the corresponding loading is susceptive, attention was concentrated on those frequencies at which a change in susceptive loading would produce a marked variation in the back scattering cross section, and for each such frequency the cross section was measured for a series of different shorting discs. Using the theory for an asymmetrically-excited radial cavity, the disc size was related to the loading, and when the resulting values for the modified cross section were compared with the theoretical curves for cross section versus susceptance, excellent agreement was found. In addition to the choice of loading, we also have at our disposal the location of the slot, and the "optimum" in this regard depends on the type of cross section modification desired. If, for example, it is required to reduce the back scattered returns from small spheres to zero, a displacement of the slot from 900 to 600 increases the upper limit of the ka range which can be continuously covered with passive loading, and it seems probable that further reductions in 0 would increase the range still more. One of the intriguing questions yet unanswered is the value of 0 at which this improvement in control ceases. In any application of reactive loading, some of the parameters of practical importance are the bandwidth, the sensitivity of the cross section modification to small changes in the loading, and the angular width in either back scattering or bistatic operation over which the desired reduction or enhancement is achieved. All of these are, of course, functions of ka, 0 and 0. An indication of the loading sensitivity can be obtained from the afore-mentioned curves of cross section versus susceptive loading, and some estimates of the beamwidth have been determined for the particular case ka = 4. 280 and 0 = 90~. With a loading such as to give zero back scattering, the calculated width (between 3 db points) of the minimum under

THE UNIVERSITY OF MICHIGAN 5548-2-T bistatic operation was 360 in the E-plane and 600 in the H-plane, and for back scattering (which involves a rotation of the direction of incidence) the corresponding measured widths were 200 and 360 respectively. It is almost certain, however, that these widths could be substantially increased if the loading were selected for maximum beamwidth rather than for a null at 0 =0. To provide more information about the above parameters, and to obtain mor complete bounds on the cross section modifications that are possible, it is necessary to pursue further the computations based on the theoretical solution derived in this report. As part of this continuing study it is our intention to investigate the maximum reduction and enhancement of the total scattering cross section, as well as giving increased attention to modifications in directions other than 0 = 0. The solution for arbitrary angles of incidence will also be considered. 62

THE UNIVERSITY OF MICHIGAN 5548-2-T ACKNOWLEDGEMENTS The authors wish to acknowledge the assistance of Mr. J. Ducmanis and Mr. Prakash Sikri who carried out the digital programming and Mr. H. E. Hunter who did the required hand computation. 63

THE UNIVERSITY OF MICHIGAN 5548-2-T VIII REFERENCES o As, B. 0. and H. J. Schmitt (1958), "Back Scattering Cross Section of Reactively Loaded Cylindrical AnteLnnas, " Harvard University, Cruft Lab. Scientific Report No. 18. Bailin, L. L. and S. Silver (1956), "Exterior Electromagnetic Boundary Value Problems for Spheres and Cones, " Trans. IRE-PGAP, AP-4, pp 5-16. Bechtel, M. E. (1962), "Scattering Coefficients for the Backscattering of Electromagnetic Waves from Perfectly Conducting Spheres, " Cornell Aeronautical Lab. Report No. AP/R1S-1, Chen, K-M (1964a),"Reactive Loading of Arbitrarily Illuminated Cylinders to Minimize Microwave Backscatter," submitted to Can. J. Phys. Chen, K-M (1964b), "Minimization of Back Scattering of a Cylinder by Double Loading, " submitted to IEEE Trans. Ant. and Prop. Chen, K-M and V. V. Liepa (1964), "The Minimization of the Back Scattering of a Cylinder by Central Loading, " IEEE Trans. Ant. and Prop., AP-12, pp 576-582. Green, R. B. (1963), "The General Theory of Antenna Scattering, " The Ohio State University, Report No. 1223-17. Harrington, R. F. (1963), "Electromagnetic Scattering by Antennas, " IEEE Trans Ant. and Prop., AP-11, pp 596. Harrington, R. F. (1964), "Theory of Loaded Scatterers,?" Proc. IEE (London), 11l,pp 617-623. Hu, Y-Y (1958), "Backscattering Cross Section of a Center-Loaded Cylindrical Antenna, " Trans. IRE-PGAP, AP-6, pp 140-148. Iams, H.A. (1950), "Radio Wave Conducting Device, " U. S. Patent No. 2,578,367. Kazarinoff, N. D. and T. B.A. Senior (1962), "A Failure of Creeping Wave Theory," Trans. IRE-PGAP AP-10, pp 634-638. King, R. W. P. (1956), "The Theory of Linear Antennas, " HarvardUniversity Press, Cambridge, Mass. 64

THE UNIVERSITY OF MICHIGAN 5548-2-T Plonus, M. A. "On the Impedance of a Finite Slot, " to be published. Schiff, L. I. (1954), "On an Expression for the Total Cross Section, " Prog. Theor. Phys., 1_1, pp 288-290. Senior, T. B. A. and R. F. Goodrich (1964), "Scattering by a Sphere," Proc. IEE (London), 111, pp 907-916. Sletten, C. J., P. Blacksmith, F. S. Holt and B. B. Gorr (1964), "Scattering from Thick Reactively Loaded Rods, " appearing in AFCRL Report No. 64-727. Stratton, J.A. (1941), Electromagnetic Theory, McGraw-Hill, 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. 65

THE UNIVERSITY OF MICHIGAN 5548-2 -T APPENDIX A AN EXTREMUM PROBLEM In Section IV the following extremum problem arises: given'y +iy2 2 =y+i 1 2 (A-1) X1 +i x2 where y1, y2, xl and x2 are real, find the maximum and minimum values of r subject to the condition xl > b > 0. From Equation (A-l) ('Y + x1) + (xY2+ x2) p i=A 2 (A-2) 2 2 X1 + X2 and since there is no restriction on the allowed x2, we can obtain one condition connecting the x1 and x2 for which r is a maximum or a minimum by equating ar/ax2 to zero. Hence Yl x1 +x 2 2x) (x2+x 2 (-y (A-3) 2 {( + Xl) + (Y2 + x2) = (+ x2)(Y2+ x2) (A-3) and the extreme values of r are therefore given by F=1+ 2 (A-4) X2 If x1 is unrestricted it is obvious that r =o mmin corresponding to x= -Y2 and, from (A-3), x = -1 Similarly, 66

THE UNIVERSITY OF MICHIGAN 5548-2-T P- = oo max corresponding to X2 = 1 On the other hand, if it is required that x1> b, the above extremes may not be achievable. This is true of the minimum if -71 < b, and of the maximum if 0 < b. To investigate these cases, equation (A-3) is solved for x2 as a function of x1 and the solution inserted into (A-4) to give _1+i 2 2 22 2 2 rmin 1+ 2 {y1+y2+ 2Y1x1 (T1+'2+2-y1x1- + 4~2x x, (A-5) [r 1 2 +2+ + /(2 2 +2y x 2 + 422 (A-6) max 2 +jjl12 2yY1 + 2 + 11 +2Y2x. (A-6) Note that as x1l + OD, 2 2 r l+ Y++42 y1Ix (A-5, min x1 x1 1 2 2 m x 1 j + 2 1+O1x11) r0 -1+1 + +O(x max x1 1 Also, for x = 0, 2 =,Y1 min 2 2 N + Pmax OD and for x1 -=0, ____________________________ 6

THE UNIVERSITY OF MICHIGAN 5548-2-T min 0, 2 2 y 1+ 72 max 2 T1 A complete schematic of the behavior of. and r as functions of min max x1 is as shown in Figure A. 1, and since ax is monotonically max min imroam mmmmmm mmin FIGURE A. 1: MAXIMUM AND MINIMUM VALUES OF r AS FUNCTIONS OF x (DRAWN FOR yI> 0). -__ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 68

THE UNIVERSITY OF MICHIGAN 5548-2-T decreasing as Ix1l increases, whilst rmin is monotonically increasing as xl+x +l increases, it is now a straightforward matter to specify the extreme values of r for xl> b. Thus, if -Y> b, min = (A-7) mmn for x1 = -1 X2 = -2', but if -yl<b r = 1+, (A-8) min x2 where x1 = b and 1 2 2 2 X2= 2y2 {1 + T2 + 27b f+(TY +'2 + 2l1b) + 4y2 b (A-9) Similarly, if 0 > b, P = o (A-10) max for x1 =x2=0, but if 0<b,'max =1 (A- l) max x2 where x1 =b and - 1 2 2 b1+2 + 2b) 2 2b (A-12) x2 +i- Y +-Y2+2y b- 71 f -2 + 2 Jb) 2b (A-12) ___ ___ ___ ___ __ ___ ___ ___ ____2 69 __ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _

THE UNIVERSITY OF MICHIGAN 5548-2-T APPENDIX B THE INPUT ADMITTANCE OF AN ASYMMETRICALLY EXCITED RADIAL CAVITY In order to use the experimental model of Section V 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 and, in particular, to the radius b of the inner conductor. Bearing in mind that the slot is of small width centered on 00= r/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 = r cos 0, y = rsino, z=z, the situation is now as shown in Figure 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 (18) is a consequence of the fact that z= - at 0=-r/2) and since it is assumed that d << X, the components E and E of the electrical field within the cavity can for all practical purposes be neglected. The only remain ing E component is then E, and this must satisfy the wave equation which, in cylindrical coordinates, is 13a~ 2 1 a2 1 (rE) + (k + - )E = r ar z 2- 2 Z The general solution for 0< b < r < a is E = Zn_ E J (kr) + FN (kr)1 ein (B-) where J (kr) and N (kr) are cylindrical Bessel functions of the first and second n n kinds respectively, and E and F are constants to be determined. The boundary n n conditions on the sides of the cavity are satisfied automatically by (B-i). At the inner and outer surfaces, however, the conditions are 70

THE UNIVERSITY OF MICHIGAN 5548-2-T z d,,."i- - - -,V"- y FIGURE B.: GEOMTRY OF THE RADAL CAVITY FIGURE B. 1: GEOMETRY OF THE RADIAL CAVITY 71

THE UNIVERSITY OF MICHIGAN 5548-2-T E =0 for r=b z V = cos0 for r=a, and on applying these to (B-1), we obtain J (kr) N (kb) - N(kr)J (kb) E = - Os ~(B-2) z d Jl(ka)Nl(kb) - N (ka) J(kb) 0 (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)J (kb) v 1 1 1 1 H i Y v 11 Cos ~(B-3) d J (ka)Nl(kb) -Nl(ka)Jl(kb) cos (B) Since both E and H0 are functions of 0, we shall again employ the concept of admittance density. The power flow across the aperture and into the cavity is (Id/2 w=- 5 (E x H) r dz -c/2 I vcos 2 d 2 d /2H d z = vcos 0 H from which we have r=a Yk1 vcos The total input admittance YL follows on integrating this around the circumference of the cavity, and hence 72

THE UNIVERSITY OF MICHIGAN 5548-2 -T J' (ka)N (kb) -N' (ka)J (kb) a 1 11 1 1 Y - i y 2,, a 1 1 (B-4) 1 d J (ka)Nl(kb) Nl(ka)Jl(kb) (B-4) As b-.a a 1 y -iY 2 7r d k(a-b) and the admittance therefore approaches -i o with decreasing cavity depth. We also remark that if the cavity were filled with a medium of refractive index P1, the expression for the admittance would follow immediately from equation (B-4) on replacing k by pk and Y by the intrinsic admittance of the medium. Thus for real i, numerical values can be obtained by scaling those for an air-filled cavity. The expression for YQ 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 x 3. 133' 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 admittance Yl /Y are shown in Table B. 1 for ka=2. 340, 3. 198 and 4. 280, corresponding to model frequencies 2. 808, 3. 838 and 5. 136 Gc respectively. 73

THE UNIVERSITY OF MICHIGAN 5548-2 -T TABLE B. I Im. Yq/Y x ka=2. 340 ka=3. 198 ka=4. 280 0. 3125 5. 11660 x 10 2. 11267 x 102 -4. 54191 x 102 0. 5 4. 27339 x 10 1. 78511 x 102 -9. 35062 x 102 0. 625 3. 56367 x 10 1. 56155 x 102 -3. 99939 x 103 0. 75 2. 75690 x 10 1. 34651 x 102 1. 66094 x 103 0. 875 1. 86384 x 10 1. 14320 x 102 6. 68595 x 102 1. 0 8. 89309 9. 51331 x 10 4. 07230 x 102 1. 125 -1. 67804 7.'68998 x 10 2. 83760 x 102 1.25 -1. 31419 x 10 5. 93523 x 10 2. 09659 x 102 1. 375 -2. 56254 x 10 4. 21876 x 10 1. 58491 x 102 1.5 -3. 93243 x 10 2. 50748 x 10 1. 19542 x 102 1. 625 -5. 45215 x 10 7. 64651 8. 75723 x 10 1.75 -7. 16175 x 10 -1. 05244 x 10 5. 96227 x 10 1. 875 -9. 11829 x 10 -2. 99670 x 10 3. 37720 x 10 2. 0 -1. 14047 x 102 -5. 13812 x 10 8. 55957 2. 125 -1. 41456 x 102 -7. 57552 x 10 -1. 73586 x 10 2. 25 -1. 75359 x 102 -1. 04581 x 102 -4. 55074 x 10 2. 375 -2. 18995 x 102 -1. 40283 x 102 -7. 79833 x 10 2. 5 -2. 78155 x 102 -1. 87146 x 102 -1. 18196 x 102 2. 625 -3. 64329 x 102 -2. 53616 x 102 -1. 72612 x 102 2. 75 -5.03990 x 102 -3. 59103 x 102 -2. 55854 x 102 2. 875 -7. 74912 x 102 -5. 60545 x 102 -4. 10564 x 102 3. 0 -1. 54720 x 103 -1. 12879 x 103 -8. 39197 x 102 74 J

Unclassified Security Classification DOCUMENT CONTROL DATA - R&D (Security classification of title, body of abstract and indexing annotation must be entered when the overall report is classified) 1. ORIGINATING ACTIVITY (Corporate author) 2a. REPORT SECURITY CLASSIFICATION University of Michigan Unclassified Radiation Laboratory 2. GROUP 3. REPORT TITLE Modification to the Scattering Behavior of a Sphere by Reactive Loading. 4. DESCRIPTIVE NOTES (Type of report and inclusive dates) Scientific Report No. 2 5. AUTHOR(S) (Last name, first name, initial) Liepa, V. V. and Senior, T. B. A. 6. REPORT DATE 7a. TOTAL NO. OF PAGES 7b. NO. OF REFS October 1964 74 19 8a. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S) AF 19(628)-2374 5548-2-T b. PROJECT NO. 5635 c. TASK 9b. OTHERPORT N(S) (4ny other numbers that may be 563502 assigned this report) d. AFCRL-64-915 10. AVAILABILITY/LIMITATION NOTICES U. S. Government agencies may obtain copies of report from DDC, Other persons and organizations should apply to U. S. Dept of Commerce, Office of Technical Services. 11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Air Force Cambridge Research Laboratori ns L. G. Hanscom Field, Bedford, Mass. 13. ABSTRACT Electromagnetic scattering behavior by a metallic sphere loaded with a circumferential slot in the plane normal to the direction of incidence is investigated. The slot is assume to be of small but non-zero width with electric field constant across it, and under this assumption the analysis of external field is exact. The field scattered in any direction i obtained by superposition of the field diffracted by an unloaded sphere and that radiated by an excited slot at the position of the load, with the radiation strength of the slot relate to the loading characteristics in the combined problem. Thus, there are two parameters that determine the scattering behavior of this object: the loading admittance and the posi tion of the slot. Numerical results are presented primarily for the case of back scattering and these are compared with experimental measurements made using a metallic sphere with an equatorial slot backed by a radial cavity of adjustable depth. DD FORMA 1473 1 JAN 64 Security Classification

Unclassified Security Classification 14. LINK A LINK B LINK C KEY'WORDS ROLE WT ROLE WT ROLE WT Sphere 9 1 Circumferential Slot 10 2 Electromagnetic Scattering 8, 5 1 Reactive Loading 10 1 Sphere 9 1 Experimental Data 2 2 Theoretical Computations 2 2 Electromagnetic Scattering 8 5 1 Reactive Loading 10 1 INSTRUCTIONS 1. ORIGINATING ACTIVITY: Enter the name and address 10. AVAILABILITY/LIMITATION NOTICES: Enter any limiof the contractor, subcontractor, grantee, Department of tations on further dissemination of the report, other than those Defense activity or other organization (corporate author) imposed by security classification, using standard statements issuing the report. such as: 2a. REPORT SECURITY CLASSIFICATION: Enter the over- (1) "Qualified requesters may obtain copies of this all security classification of the report. Indicate whether report from DDC." "Restricted Data" is included. 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UNIVERSITY OF MICHIGAN 3 9015 03483 1647111111 3 9015 03483 1647