7456-1-F Copy ___ THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL ENGINEERING Radiation Laboratory SCATTERING BY HOLLOW FINITE CYLINDERS AND INFINITE COATED CYLINDERS Final Report J. J. Bowman, E. F. Knott and V. H. Weston January 1966 PO 614572117 AF 33(615)-3166 7456-1-F = RL-2155 Contract With: Northrop Corporation Norair Division Hawthorn, California Adminis tered mrr:lt OFFICE OF RESEARCH ADYIN:ISTRATION. ANN ARBOR.,, ~. J *,;:;..4A:^!., ^'t,0; 4k- i;ot, ^.;::-,,; 1;:; f^^l.:;ii.*';i%

THE UNIVERSITY OF MICHIGAN 7456-1-F SCATTERING BY hOLLOW FINITE CYLINDERS AND INFINITE COATED CYLINDERS by J.J.Bow man, E. F.Knott and V.H.Weston January 1966 Final Report Contract A F 33(615)-3166 PO 614572117 Prepared for Northrop Corporation Norair Division Hawthrone, California

ERRATA TO REPORT 7456-1-F "SCATTERING BY HOLLOW FINITE CYLINDERS AND INFINITE COATED CYLINDERS" Corrections to: "Scattering by Hollow Finite Cylinders and Infinite Coated Cylinders, " J. J. Bowman, E. F. Knott and V. H. Weston, University of Michigan Radiation Laboratory Report No. 7456-1-F. In Section 2.3 concerning the case of two spaced absorbing layers, and in the following section concerning two resistance foils, an error has been made in calculating the asymptotic approximations for large indices of refraction. In particular, the quantities C and C have been erroneously taken to be 'zero asymptotically, and since they are not zero, this leads to the following set of corrections. J. J. Bowman

ERRATA TO REPORT 7456-1-F "tSCATTERING BY HOLLOW FINITE CYLINDERS AND INFINITE COATED CYLINDERSr? P. 18, Eq. (43) should read [incoi + c(21 L kbx j2 (1 + cot 2k 6 1 1 L2)+ cotk 6i [B2+Jcot k 6i]+[ IoI + 0(2 I9L1 j kbX A.2 0 P. 18, Eq. (44) should read (0) (1) (0) D =-(A - A ) (A -B(1) [in cs - c(1] La X 2 P. 19, in Eq. (45), read in csj (3) [kcX2 0 2 instead of -2 in cos 3 kc A. P. 19, Eq. (47) should read 16 7T2(k b) (k c) 0 0 [in co - C (3% Y(2)3 0 - - 2 (2) C [A 3 1 alnY (2)l F(3) 1 aI nY (21 r in cos ~ ( - 3 (k c) j [~ X a(kC)j +L k2 J 0

P. 20, Eq. (50) should read F(3ot] (1 + cot k363) C = Lkdk2o (4)+ otk3 631 B(4)+ cotk36 + in C 3 P. 23, Eq. (59) should read 3. V3 kd kd20 P. 23, Eq. (59) should read in cosj + (2) R 2 (1) S1 kb X R (2) s1 B - 0. P. 23, Eq. (62) should read ~in cos 3 R 2kd s kdXo -= 3 o - - Z R 0 s3 B(4).3 B _ 0

THE UNIVERSITY OF MICHIGAN 7456-1-F ABSTRA CT To assist in the design of computational programs for scattering cross-sections, investigations of cylindrical objects are considered. In particular, measurements are carried out to determine the surface currents induced in the vicinity of the leading end of finite, hollow conducting cylinders, by an incident plane wave. Theoretical analysis is undertaken to obtain the scattered and surface fields produced by a plane wave incident obliquely to an infinite coated cylinder. The particular case of a perfectly conducting cylinder surrounded by two resistive sheets (separated by an air gap) is given ii

THE UNIVERSITY OF MICHIGAN 7456-1 -F TABLE OF CONTENTS ABSTRA CT ii INTRODUCTION 1 SCATTERING FROM COATED CYLINDERS AT OBLIQUE INCIDENCE 3 2. 1 Introduction 3 2.2 Scattering by a Doubly-Coated Cylinder with Impedance Boundary Condition at the Core 4 2. 3 Scattering by Conducting Cylinder Surrounded by Two Spaced Absorbing Layers 17 2. 4 Simplification in the Case of Two Resistance Foils 21 III CURRENT MEASUREMENTS ON FINITE HOLLOW CYLINDERS 25 FIGURES 31 RE FERENCES 77 iii

THE UNIVERSITY OF MICHIGAN 7456-1-F INTRODUCTION In the determination of the radar crogs-section or scattered field produced by a plane electromagnetic wave incident upon a given obstacle of complicated shape and material composition, the use of computer techniques has recently come into practice. The computation involved is a two stage process which first Involves solving a vector integral equation for the various unknown quantities as a function of position, and then utilizing these results to compute the scattered field. For perfectly conducting obstacles, the unknown quantities that are first determined are the induced surface currents which are a function of position on the surface of the obstacle. As in the usual practice of numerical analysis, the surface is approximated by a finite set of mesh points. The integral equation is then approximated by a set of algebraic equations, with the unknown quantities being the currents at the mesh points. For general coated objects, the set of mesh points must approximate not only the surfaces of the coating, but the interior region. The accuracy of the numerical procedure depends upon the maximum number of mesh points which can be handled, and the degree that the set of mesh points approximate the surface or volume region, as the case may be. Thus, for scattering from very large objects, the fundamental limitation to accuracy is the limited number of mesh points that can be used due to storage capacity. The best way to increase accuracy, in such a case, is to reduce the voitme or surface region for which the sought for quantities are unknown. This can be achieved by using known theoretical results to determine the unknown quantities over portions of the surface or volume beforehand. The unknown 1

THE UNIVERSITY OF MICHIGAN 7456-1-F quantities then only have to be determined by numerical means over the remainng region. In regard to this approach, for perfectly conducting objects, Norair Northrup has developed a computer program "S. D. T. -Physical Optics", which employs the physical optics approximation for the surface current on the illuminated portion of the surface. The accuracy of the physical physics approximation at a given point depends upon the local radii of curvature, the distance the point is from the shadow boundary, and from regions of surface disontnauities, such as edges. With regard to the ltter problem, a series of measurements is carried out in Chapter m to determine the surface currents iX th vicinity of the edgs of hllow fanite perfectly4 onducting cylinders, for plane wave incidence. The results can then be used to determine when, and how close to the edge, the physical *tie approximation is useful. No analysis is presented in Chapter HI. At present, Norair is extending numerical procedures to determine the scattered field from coated objects. As a check against such numerical procedures, the exact theoretical expression for the total field produced by a plane wave incident obliquely upon an infinite coated cylinder ai obtained in Chapter II where the coatings are composed of homogeneous layers of different material. The results are expressed in terms of modal series, which are useful for calculations for frequencies around and below the resonance region of the cylinder. Results are also given for the special cases where the coating consists of one or two resistive sheets concentric with the inner perfectly conducting cylindrical core. 2

THE UNIVERSITY OF MICHIGAN 7456-1-F II SCATTERING FROM COATED CYLINDERS AT OBLIQUE INCIDENCE 2.1 IbtroduQtion It is desired to derive the solution for the problem of a plane wave incident obliquely on an infinitely long circular cylinder coated with several uniform layers of material. In considering this problem it is convenient to confine attention to a cylindrical core on which an impedance boundary condition is imposed and surrounding which are two layers with arbitrary complex refractive indexes. The solution, whose form is the exact Mie series solution for incident energy of both TE and TM polarizations, may be immediately extended by inspection to include the case of an arbitrary number of layers. Such a generalization will be employed to discuss the four layer case of a perfectly conducting cylinder surrounded by two impedance sheets, the impedance sheets being separated from each other and from the inner core by air spaces. These results are simplified in the case of two resistance foils with air space separation. The problem of scattering of plane waves obliquely incident on a solid dielectric cylinder has been solved by Wait (1955), who presented the exact Mie series solution and discussed some special cases. This solution is, of course, encompassed by the present investigation and thus serves as a check on the work. The scattering of plane waves incident perpendicularly on composite infinite cylinders has been considered by many authors of which we mention Kerker and Matijevid (1961), who solved the boundary value problem for an arbitrary number of concentric cylinders. 3

THE UNIVERSITY OF MICHIGAN 7456-1-F Simplification of the expressions can be achieved for thin absorbing layers of materials with large complex indexes of refraction. In this case, the asymptotic analysis carried out by Weston and Hemenger (1962) for the absorber coated sphere may be applied to the coated cylinder to simplify the scattering coefficients of the absorbing layer in question. When such a layer, or impedance sheet, is backed by a perfect conductor, the total tangential field components may be shown to satisfy an impedance boundary condition on the outer surface of the layer. As a sufficiently general example, the scattered field produced by a plane wave incident normally on a conducting cylinder coated with two thin absorbing layers has been examined by Bowman and Weston (1965). The present investigation differs from Bowman and Weston not only in the assumption of oblique incidence, but in supposing the two impedance sheets to be separated from each other and from the conducting core. The existence of a transmitted field in the air spaces greatly complicates the expressions for the scattered field. However, simplification results when the thin absorbing layers are non-magnetic and have electric permittivities with predominately large imaginary parts. Such layers are resistive layers. Scattering of normally incident waves by a conducting cylinder surrounded by a single resistance foil has been discussed by Schmitt (1957). 2.2 Scattering b a Doubly-Coated Cylinder with Impedance Boundary Condition at the Core The inner cylindrical core will have radius d and its axis will be the s-axis of the cylindrical polar coordinate system (r, 0, z). Numbering from the outer layer inwards, let E and p1 be, respectively, the relative electric permittivity and the relative magnetic permeability of the outer coating, while E2 and M2 refer to the inner coating. The inner coating will be of thickness 4

THE UNIVERSITY OF MICHIGAN 7456-1-F 6 -b-c, where b is the outer radius of the cylindrical surface formed by 2 this layer. The outer layer will be of thickness 6 1a-b, where a is the outer radius of this cylindrical shell. The source of excitation will be a plane wave incident obliquely on the coated cylinder. The field of the incident plane wave is taken in the form -* -e -~ -' A ik- r-it - o ^ik- r-it (1) ae, H e (1)be o o o where the polarization vectors a, b and the propagation vector k are given by a- x cosacos 3+ysina+ z cosa sinf3, b- sinacos^-ycosa+ z sina sin3, (2) k s: eu (X sin:3 -zcosoS). The direction of the incident plane wave makes an angle 3 with the negative z axis; the special case f=3- results in normal incidence. The angle a determines the state of polarization of the wave: a= 0 corresponds to an incident TM (transverse magnetic) mode while a= - corresponds to an incident TE (transverse electric) mode. Due to the presence of the dielectric coatings on the cylinder, the total electromagnetic field must be expressed as a superposition of both TE and TM modes, even though the incident wave may be a pure TE or a pure TM mode. This coupling effect disappears in the case of normal incidence (03:). 5

THE UNIVERSITY OF MICHIGAN 7456-1-F Since E, p with appropriate subscripts I or 2 denote relative material parameters, Maxwell's equations read -9 -VAH X -i~e oE 0 where e,, refer to the free-space constants. In the region exterior to the 0 0 cylinder (r> a), the z components of the total electromagnetic field may be represented in the form OD nio E - i [si cosa J(kr sin3) + H 1}(kr e,]e z M I L n n n 0n - o (3) Within the outer dielectric layer (a > r> b) these components are written in the following manner co ik z cosn () (1) (1) (2) in E =e- s i H (k lr)+b( ) (kr) e az Ln n 1 n n 'SOD H -e a( H (kr)+b H (kr) e + z ia n n nn nU -oo (4) 6

THE UNIVERSITY OF MICHIGAN 7456-1-F while for the inner layer (b> r> c) OD E elZco8s1 'n (2) (1) (2) (2) 1e in E ZZ L n n k2r) + b H (k2r e nx -co - kz con (2)H(1) r)~(2)(2 r) en Hz ej i a n (k2r)e n.n k2 jr n= -Co) (5) The parameters k1, k2 appearing in the arguments of the Hankel functions H(, H(2) are defined as follows: n n 2 /2 00 00082~ (6) Analogously, it is convenient to define k =X k, X =(1-cos23/) = sin3, (7) o o O where k is merely the x component of the free-space propagation vector k. The total electromagnetic field is completely determined upon specifying the ten unknown coefficients a, an, etc., which occur in the summations n n above. These unknown coefficients are obtained, of course, by imposing the boundary conditions on the general solution. In particular, the tangential fields must be matched across the dielectric boundaries r= a and r= b. However, in 7

THE UNIVERSITY OF MICHIGAN 7456-1-F addition to these usual continuity requirements, we shall impose an impedance boundary condition at r= c, namely Ez 7173 CH, (8) riyu where n3 represents a constant surface impedance. At each of the dielectric interfaces, the continuity conditions give rise to four boundary equations, making a total of eight equations, while the impedance condition on the inner core contributes two more equations. There is, then, a set of ten linear, inhomogeneous equations for the ten unknown coefficients. The equations are complicated'by the coupling between TE and TM modes. In enumerating the ten boundary equations we shall drop the summation index n; no confusion should arise. Beginning with r: c and applying the impedance condition (8), one obtains the two equations 22 (2 2 a ) )(c)-i.3 X H (kc)] 2 +b )[H()(k2c)- n3 2 (Hk2 = kn 3 H'2)(1) C'b"' (k2c2 r3cs/ [2 H (k2c) +bH)(k2] (9) 2 8

THE UNIVERSITY OF MICHIGAN 7456-1 -F 2 fl73 '-2 2 nX 2 - w ~2!a~f a(2)nH(1) (kc b(2)H()(,3 kc\ 2 L 2kc+ 2 (10) At r= b the continuity of the z components requires thle following two equations a(2H(1 (k2b)+b (2H (2 k b) =a H( (kb)+ b( H( (k b), 2 1 (11) (12) while continuity of the 0 components gives I1 ()(11t2 +Incs - 1 -M (1) (1) (2 2 a H (k Ib)+#-b* H k1b c2Fa(2)H (1) (2) (2)1b H L (k2 b) +b H (kb 2 (13) (1 in co3 kb ( 1) 11 2 A2 A., (2) (1).. (2) t (2) x a Hl (k2b)+b H (k2b). X2 (14) 9

THE UNIVERSITY OF MICHIGAN 7456-1 -F At r= a one obtains from z component continuity a()H (1)(kla)+b(1)H(2)(k a)=a H(1)(k a)+sinicosaJ(k a), (15) 1 n 0 o0 a()H((kla)+b( )H(}(kl)=a H()k oa)+si8si J(k a), (16) and from 0 component continuity a H' (k a)+b (ka)]- (k a)+b)H(2)(kla)] 0 1 1 (1)\ =- La H' (k a)+sin3cosaJ'(k a)l, (17) n o o o cidence 2 in ten e t dcp i (1) (2) fa Ht c n(k)s a)+insin Jt(k a) (18) ) n o o A perusal of the boundary equations shows that in the case of normal incidence (=f ) the ten equations decouple into two sets of five equations, one set for the electric coefficients and one set for the magnetic coefficients. 10

THE UNIVERSITY OF MICHIGAN 7456-1-F As might be expected, there are many ways to solve the boundary equations; however, it Is wise to look before leaping and to attempt to preserve the natural duality inherent both in Maxwell's equations and in the boundary conditions. The method by which we proceed is the following. From the first four Eqs. (9) to (12) we obtain the coefficients of the second dielectric layer in terms of the first layer coefficients. The resultant expressions are then substituted into (13) and (14) to obtain two homogeneous equations for the four coefficients of the first layer. Two inhomogeneous equations for these coefficients are obtained from the last four boundary Eqs. (15) to (18) upon eliminating a and a. We now have four equations n n for the four first layer coefficients; these yield expressions which, when substituted into (15) and (16), give a and a explicitly. n n The final solution in the case of TM incidence (a=0) may be written in the form J(koa) a )) (19) a X -s 0 -B (19) n Odl)(k ik a LH(1 )k2 1) an -k- 1 1 in cos ( _)c()]- (20) ark ta [H(1\ )]2 D ka 2 ) ( o a k in the case of TE incidence (a: ) one obtains 2 21 1 1 in co- 1 (1) (21) n br:a [1, 2 D ka 2 0 X 1 11

THE UNIVERSITY OF MICHIGAN 7456-1 -F J(k a) 2 a =-).'. - oz (1) ka0!H (k a)~ k a 0 1 1 [ t a 2 D (A(0)-A1) (22) where (0) A X H' ()(kasin ) sinS H(1 ka sin) (23) D= (A0)-A(1) (A - B1)) [rincooa ( 1 ka 2 O _C(1)]2 (24) (24) The remaining quantities A(, B(, C) are related to the parameters of the coatings by the following set of relationships A(1)As 3-(2) akl) 51 x1 L (kla) X1 a2y(1) 1r2)1) i ll) 1 (kla)(kb) - 1 a(k b) fincoaJ (s 1 2)12 Ikb 1 2 22 Cr- -TT_l~~~* [ o 2_ C d(k a) (25) 12

THE UNIVERSITYOFMHIA OF MICHIGAN 7456'-1-~F j'B ' tiB (2)kla);- a (k adk 1) 1 11 1 Mk1;x 1 d(k b) I klb a(k a) ' (26) (1) (1) = I' l [in 08/3 1 -k 1q2 7 '-16 i 2(k a)(k b) (27) x~a(k~b)E 0 1) a(k1b)J [in cosf I (.. A. 2)] 2((1) 2 X1X (28) I where similar expressions are obtained for the second layer, namely -aA______-i r __ % La~ik2~b j. I k2b)ia(k2C) J L -2 ~2 c)i 2 u2co2L (2).a2 A~k2b) (29) 13

THE UNIVERSITY OF MICHIGAN 7456-1-F (2)B(2) [ a 2 - i a2 (2) (2 2 a(kL2b) (3 k2 ^Yb) ) L 3X2 C) 2) c)(k 2 C) p2 n2cos 2 (2) 2 (kc '2 (2) (2) e2#2 incos.L -16 2 4 k w2(k2b- ) (30) (31) (2) - (2) 4 = Y 22 [(() 2 ) 3i 2 dk2) 1 1a 3 dkaC)j +n(kc (32) In the above, Y(1) and Y(2) represent combinations of Hankel functions given by Y()=H( 2(k a)H( )(klb) - H (kla)H (2)(kb), Y(2)= H(2(k2b)H((k2) - H(1)(2b)H(2)(k2C). (33) This completes the specification of the scattered field. We note the identity 14

THE UNIVERSITY OF MICHIGAN 7456-1-F (2) a2 1) a l) aY'>). 16 ( ka)a(k b) a(kla) a(kb) 2 1 (34) which derives from the Wronskians of the Hankel functions. Of course, an analogous identity holds for y(2. The z components of the total field on the outer surface (rza) of the cylndrical structure have a particularly compact form. In the case of TM incidence (a x 0) one finds 0D Esurf. 2i - ikcos3 o zka I "' - e / n= - o burf. 2i -ikzcos~ I z MArka n= -ao and in the case of TE incidence (a s2 i ^ Ao-B1) e n n i1 (ka) Dn n o (35) eino 1 rnco 1 1 c 1) H1' k a)D ka n2 2 n J' n o o l (36) 00 Esurf. 21 e-ikzcoos )' e 1 incosL_ l ) C1] nz k a o o k (37) co I A^(1) lsurf. o 2i - 2i ikzeos8 ei np - n 0 z e rk_ Hlka) n D n= -a> n (38) 15

THE UNIVERSITY OF MICHIGAN 7456-1-F The other oomponents of the field may be obtained by reverting to Maxwell's equations which are, ea*4Iliity Er) r ~E a z 1 Eo y -o E - -3 j* OH 0 0 0 0 H o ~r l H dOE -) (39) r dH 1 T z H -- [cos + 0 ] Z X L d(kr r)rk a)J zH I o-' as It is well to point out that the basic symmetry of the boundary equations has indeed been retained in our solution. Thus, with the transformations c -M-, A.-, I 1 77 one easily verifies that TM.,TE a a — n n..TM TE a -- -a n n 16

THE UNIVERSITY OF MICHIGAN 7456-1-F and this is a result of the natural duality of Maxwell's equations and the boundary conditions. The form of the solution is also quite amenable to generalization. In particular, it is clear that the solution for a larger number of layers is obtained by generalizing the quantities A, B, C, A of the second layer to the form of the first layer quantities A, B ( C ( A ) and introducing further such quantities to correspond to the added layers. Finally, we remark that the solution has been checked in several special cases by comparing it to solutions derived independently in these cases. 2.3 Scattering by Conducting Cylinder Surrounded by Two Spaced Absorbing Layers. A special scattering structure of interest consists of a perfectly conducting cylindrical core surrounded by two absorbing layers, the core and the layers each being separated by an air gap. The problem of scattering from this structure is, in essence, a four layer problem: two thin absorbing layers and two air spaces comprise the four layers. The scattered field may be obtained as a straightforward generalization of the solution we have presented above. We shall further suppose, however, that the indices of refraction of the two absorbing layers are very large. The asymptotic analysis of Weston and Hemenger(19862)may then be employed to simplify the expressions. A particular instance ob lhe general absorbing layer would be that of a thin resistance foil discussed by Schmitt(1957). This ease is easily derived from the ensuing equations. Consider first the outer absorbing layer of thickness 61= a-b and index of refraction N= (E11/2. The procedure of Weston and Hemenger is centered around the leading terms in the asymptotic approximations to the 17

THE UNIVERSITY OF MICHIGAN 7456-1-F coefficients corresponding to A(, B() and C(1) in (25) to (27). For a thin layer with very large index of refraction, the function y(1) has the following asymptotic form y(l) 4 ink1 6 Y n~ P r~b)(40) (1) (1) where kl N k. With this approximation, the coefficients A, B (1) 1 and C take the form 2 (2(2) 1 - 2) -C' incL ) AA(ctan k 6 B +o-cotk 6 - +L JL Lkb2 (41) (1) ~- A(2)t +2 (2) B (1) ame thing with e p1 and A 2) — B()},(42) rinc_ + C (1+ cot2 k6) <(1) LkbXo J 11 [ 0 C (2) 1 (+ +incos3 + C21 2 A + -cotkl 6 B + cotk 6 + — cs + C( A L. 0 (43)' and the quantity D in (24) now has the form 2 Dz-(A0 -A*') (A^-B^) [2 - C'' 1 (44) kaX2 -C 0 18

THE UNIVERSITY OF MICHIGAN 7456-1-F In deriving the above equations, we have taken the index of refraction of the second layer to be that of free space (N2 1). The next layer is the free-space layer b > r > c with coefficients A(2) B() C given by A- 3) a a )( 1. i) 1 a2 - in coi.(3) a. 'c k) B d)(kjc) 2 ( o (45(kb) C -- --. (47) [ (3) 1 al L 1 ik 3)1 ao tnyc)) - B-_ -JLkck2 o The thi ae ithrd layer i another absorbing layer with thickness 6c-d nd index of refracton N2. The coefficients of thi layer are and index of refraction N3= (E3)/2 The coefficients of this layer are 19

THE UNIVERSITY OF MICHIGAN 7456-1-F A4)- tank3IB +1cotk36 + 2 ] Ao &- --- Lk —2 -A otk363 L 3 r IA 1 r ~,A 1 r 2 (48) B( 4 ame thing with --, 3and A() — B(4) L 3 (49) ncos t ( LkdX2J fm 0 t + cot2 k33) (3) C =,,(4)l3 I 3.. I L kI 2 [A(4J +cotk6J [B(4+ c cotkA 2 I3LkdX2 I (50) Finally, the last layer is the free-space layer d > r > e, where e is tie radius of the perfectly conducting core. The coefficients of this air space are A(4) 1 o A X (k'd) 0 0 4) 1 1 F ( — B \ kd) ^I e)n lbcd(kod) LJ(k e)j. 0 0 (51) (4)=. In the equations are given by concerning the two air gaps, the functions 2) and 4) 20

THE UNIVERSITY OF MICHIGAN 7456-1-F Y(2)= Hz (k b)H(1)(k c) - H )(k b)H 2)(k c) 0 0 0 0 (52) y(4) H(2)(kod) H(1)k) - H(1)(kod) H(2)koe) This completes the specification of the scattered field. In view of the above results, we can immediately obtain the field scattered by a conducting cylinder surrounded by a single absorbing layer which encloses an air space. In this case, the coefficients of the outer absorbing layer are given by (41) to (43); however, the coefficients of the inner free-space layer are identical in form to those in (51), namely A(2) 1 ata2) A a(k b) o o 0 0 0 C(2) 02 C:0. and these are the only coefficients that appear since the problem is now a two layer problem. 2.4 Simplification in the Case of Two Resistance Foils. The equations pertaining to thin absorbing layers may be further specialized to the case of thin resistive foils as considered by Schmitt.(1957). To 21

THE UNIVERSITY OF MICHIGAN 7456-1 -F accomplish this, the absorbing layers are assumed to be non-magnetic and to be characterized by a very large imaginary electric permittivity. Thus, for the outer layer we take 1 1, e1 (54) 0 where K1 is the conductivity of this layer. In addition, the wall thickness 61 is allowed to vanish at the same time the conductivity K1 is increased so that the product K 6 T (55) 1R B1 remains constant. We may then write -tank61,- I, i cotkR- -, (56) 1 R 11 k '1 where Z is the free-space impedance Z = e 5 377 ohms, and take the o o e limit K 1 - > oD in the expressions (41) thru (43) for A(1) B(1) C1) The quantity R represents the surface resistance of the outer foil. The following resl are obtained lowing results are obtained 22

THE UNIVERSITY OF MICHIGAN 7456-1-F 81 L 2 + 0 R (2) I' B Ii~. -- 2: 0 (57) (58) 1) 0 (2) RS, incs (2) S1) 1 kb0 z.1 i-~ — zo B ()A A- __ 0-% o (59) 2)(2) (2) The expresjsions for A 2P,B C ri (3) (3) A 3) A, B, C are similarly given by -emain unchaned. The coefficients (3) F 4) z 1 A =L -jy (3) 1 1 o (4) R0 83 coo2 "kdX2 + 0 R (4) 83 0 (60) (61Y 4mm ftjrq to (3) C =W-i Rs kdX2 z R 0 (4) 83 B -i z 0 (62) 23

THE UNIVERSITY OF MICHIGAN 7456-1-F while the expressions for A4) B4) 4) remain unchanged. We note that duality has now been sacrificed, since the layers have been taken to be non-magnetic. The specialization to a single-foil structure is accomplished, as previously indicated, by employing (53) in place of (45) thru (47). The scattering of a plane wave normally incident on such a single-foil structure was derived by Schmitt.(1957). Our results, when reduced to normal incidence (= 2), agree with his. 2i 24

THE UNIVERSITY OF MICHIGAN 7456-1-F Il CURRENT MEASUREMENTS ON FINITE HOLLOW CYLINDERS A series of measurements have been carried out to determine the currents induced on finite, hollow, conducting, circular cylinders by an incident plane electromagnetic wave. The measurements were restricted to obtaining the magnitude and phase of particular current components in the vicinity of the leading end of the cylinders, both inside and outside, where the term leading end is used to refer to the end of the cylinder directed toward the transmitter. Two orientations of the cylinders and two different cylinders were involved -in the measurements which were performed at 3.03 GHz. The orientations correspond to end-on incidence (incident propagation vector along the axis) and 30~ from end-on (where the propagation vector makes an angle of 30 with the cylinder axis). The two cylinders, both the same diameter, were distinguished from each other in that one had both ends open, whereas the other had its rear end capped or closed. The inner diameter, outer diameter and length of the open cylinder were 4. 375", 4. 5" and 33.125", respectively. The length of the closed-end cylinder was 33. 375". The measurements were carried out in an anechoic chamber, with the current measured by a small loop probe, which was suspended vertically from the celing. The incident field was horizontally polarized, thus minimizing a any interference effects of the probe lead. Slots were cut into the cylinders to permit insertion of the probe into the cylinder, so that the internal currents may be probed. As shown in Fig. la, the probe was inserted only far enough so that the measurements essentially 25

THE UNIVERSITY OF MICHIGAN 7456-1-F were of the inside currents. Two kinds of slots were required: very short circumferential slots and a long axial slot. These were oriented along the expected direction of current flow in order to minimize radiation from the slots. The internal circumferential and axial currents are measured only at the appropriate slots; i. e., the axial currents are not probed through the short slots. The external surface fields were measured with all slots concealed by conducting tape. For oblique incidence, the cylinders were tipped sideways, so that the probe lead still remains perpendicular to the direction of incident polarization. The axis of the cylinder thus lies in the plane of incident polarization. This is shown in Fig. 1, where the axis of the cylinder is taken to be the z-axis of a rectangular Cartesian coordinate system. The leading end of the cylinder lies on the xy plane, with the xz plane being the plane of polarization. For Fig. 1, the angle a is the angle of incidence, thus, for the end-on case, a is zero, and for the other series of measurements a is 30. The longitudinal current component (directed parallel to the z-axis) will be denoted by j, and the circumferential component by j. In referring to a position on the cylinder, the parameter z and 9 will be used, with z referring to distance along a generator measured from the leading end, and 0 referring to angular position measured from the x-axis. The following series of measurements were performed: 26

THE UNIVERSITY OF MICHIGAN 7456-1-F Measurement Order 1 2 3 4 5 6 7 8 9 10 11 Surface Measured Outside Inside Angular Position 0 0 0 Current Component Jz jz Jo jz Jz Outside Inside Outside Inside Inside Outside Inside Outside Inside l/2 0 0 Incident Angle a 0 0 0 0 30 30~ 30~ 30 30~ 30~ 300,/2 1/2 1/2,/2 jz jz jz Jo Each measurement order consisted of measuring the indicated current component, at discrete intervals along a generator of the cylinder starting at the leading edge. This measurement was performed for both the open and closed rear end cylinders. The angular position given by 0 indicates that the measurements were either along the top of the cylinders ( =I) or the sides (0= 0 or f), see Fig. 1. 27

THE UNIVERSITY OF MICHIGAN 7456-1-F Each measurement was calibrated by measurements upon a sphere of known size. Different probes were required for measurements along the sides and tops of the cylinders, see Fig. la. In order to have a meaningful calibration, the same probe used for the cylinder measurement was used for the sphere calibration. Since the probe must be held normal to the surface of the sphere, and the orientation of the plane of the probe with respect to the probe lead cannot be changed once it is set, only certain points of the sphere can be used for calibration. Using computed values for the currents on a sphere (Ducmanis and Liepa, 1965}, the current component j along the sides of the Z cylinder were calibrated with the measured current component in front of the sphere. The current components on top of the cylinders were calibrated against the measured current component on the topof the sphere. The resulting current measurements, both phase and amplitude, are presented in Figs. 2 to 45. The magnitude of the current has been normalized with respect to the incident field, which as calibrated as indicated above, for each series of measurements described by the pair of figures giving phase and amplitude of the current along a specific generator of a particular cylinder. An exception to this is the measurement of current magnitude given in Fig. 10. Here it was normalized with respect to the calibrated incident field associated with the measurement series given by Fig 12. In addition, it should be recalled that the phases of the current components are relative. A block diagram of the basic receiving system is shown in Fig 46. In this arrangement, the probe signal is compared to a controllable reference signal in a hybrid tee, after the reference signal has passed through a coaxial switch. When the switch is closed, both signals are present and the operator commences tuning both the attenuator and the phase shifter in the reference arm, seeking a null indication (no signal) on the receiver. When he has obtained a 28

THE UNIVERSITY OF MICHIGAN 7456-1-F null, signifying that reference and probe signals have the same amplitude but opposite phase, he records the phase shifter setting. The switch is then thrown open, permitting only the probe signal to enter the receiver circuits; the signal level is then displayed at the receiver output and the operator records the amplitude. Thus, the phase measurement is performed with the switch closed and the amplitude is measured with the switch open. During the measurement, of course, the probe is held at the desired point on the surface to be probed; it is moved to the next position of interest only when the above sequence is completed. This scheme is the one adapted in lieu of a more conventional system in which both phase and attenuation values are read from the dial settings of the respective devices. Here, the receiver linearity over a 40 db range is utilized in the place of the attenuator. All the phase angles determined this way are relative. The phase shifter settings are directly proportional to the relative phase angle measured. The relative phase is obtained by multiplying the phase shifter settings by the frequency used (in GHz). A phase shifter setting of 32. 0 at 3.00 GHz, therefore, implies a phase angle of 96. 0 degrees. As the measured phase progressively advances, the required phase shifter setting for the nulling process approaches the limit of its capability. Then a new setting must be sought which is electrically 3600 from this limit and the raw data shows a sudden shift in the dial settings. When this occurs, the operator records both values so that the remainder of the readings may be related to those previously obtained. These two values, when multiplied by the frequency in GHz, are usually very nearly 360 apart. 29

THE UNIVERSITY OF MICHIGAN 7456-1-F During the course of a measurement run, the range operator plots the amplitude values as they are recorded, showing the trend of the measurements. This technique helps him spot any anomalous behavior due to recording or positioning errors at or near the time they occur, permitting him to go back and re-examine the datum point in question. 30

THE UNIVERSITY OF 7456 -1-F MICHIGAN 04 0 an 0 I 0 0 '.4 0 04?5 "4 'SI

+4 +3 z 0 +2 4-'c-fl Hz 0 -— Ij Ci) 0 z +1 0 0 2 4 6 z 8 10 12 FIG. 2:, MAGNITUDE OF" CURRENT COMPONENT j OUTSIDE OF OPEN CYLINDER; (0 =0, a=0) z 14

1500 1250 1:4750 -J Al 1 -< P-o cri CD z 0 2 4 6 z 8 10 12 14 FIG. 3: PHASE OF CURRENT COMPONENT j OUTSIDE OF OPEN CYLINDER. (8 =0, a=0) z

+5 +4 +3 +2 +1I 0 -1I Hz -11" Cl) zI 0 2 4 6 z 8 10 12 14 FIG.- 4: MAGNITUDE OF CURRENT COMPONENT j - OUTSIDE OF- CLOSED CYLINDER.- (e = 0,1 a = 0) z

1500 1200 c~o00900 P4 6Q0 300- I —II I I I I I 0 2 4 6 z 8 10 12 14 z (7) -el 0 z~ FIG. 5: PHASE OF CURRENT COMPONENT j OUTS]IDE OF CLOSED CYLINDER. (e =0, a=0)

-1 ~D -2 H z -3 -4 0-i z C) 0e z 0 1 2 3 4 z FIG. 6: MAGNITUDE OF CURRENT COMPONENT j INSIDE OPEN CYLINDER. (9 =0, a=0)

THE UNIVERSITY OF MICHIGAN 7456-1-F \ 1 \ z \ O z z 0 \ ~0 \ VH ocu to 1r asv~kI 37

10 0 wA ft w Q t z 0 4, Ul -all I I It z ci: H 0 — 0 z -1.0 -20 -30 0 z FIG. 8: MAGNITUDE OF CURRENT COMPONENT j INSIDE CLOSED CYLINDER. (0 =0, a=O)

400 X z 4-.4 300 - 20 C0 1 2 3 4 Z FIG. 9: PHASE OF CURRENT COMPONENT j INSIDE CLOSED CYLINDER. (0 =0., ao) Z

-15 z< Q, I - -20 ~* -20 - -25 -30,, U 4 8 z 12 16 FIG. 10: MAGNITUDE OF CURRENT COMPONENT j OUTSII)E OPEN CYLINDER. (0-, -J)

1500 1000 500 0 -ir-11 V. W.-O..04 A 1Ct 0-4 —4,...e all,, W-61 rl "I" 'oz 0 4 8 12 16 FIG. 11: PHASE OF CURRENT C"OMPONENTj OUTSIDE OPEN CYLINDER. (~aU 2

THE UNIVERSITY OF MICHIGAN 74Th-i -1 0 0 Q Cl) 0 U 0 Cl) N H 0 H z z 0 0 U H z U N 0 H N 0 iC) 1r 0 N 7 I qpZ'3GflhINDWA0 42

TH E U NIVLFR SVILY' 01 IMIC1HLIG A N coc CD4,:D LO U LnU 3SVIzd 43

I l -2 -4 0 -3 - 0 1 2 3 4 FI. 14: NITUDE F CUREN COMPONENT INSDE OPEN CYLIN. FIG. 14: MAGNITUDE OF CURRENT COMPONENT j INSIDE OPEN CYLINDER. [6 j' ' a=0)

600 v - 450 < 300 150 4,..^ - z L?0 0 1 2 z 3 4 PH~~~~~~~~~~~~ASEO URN OPNN 0ISD PNCLNE (0 —2

THE l NIVERSITY OF M I C HIGAN 74..... i- i *I9 z Q +,, I. ' L-4 Cl CY) + --- —------------------ o 'qp 'aa3fjlNoViM~I 46

i I I I lw 300 wI -1 1 4 -i i ,Tl z rni CL.H 0 -0 -z 150-i 0 0 1 2 z 3 4 FIG 17: PHASE OF" CURRENT COMPONENT ja INSIDE CLOSED CYLINDER. (a:7i o (&-. U

THE UNIVERSITY OF MICHIGAN 7 i 4f; i - I.. -,: ts is r aa3 ' I H ~S iH (C 0 H t;3 Z.. - t;2., o z 00 0 XF O Z 3 Ip ' No I IE C —4 qP {anzIdov 48

1200 900 44 600 300 II I 0 2 4 6 8 10 12 z FIG. 19: PHASE OF CURRENT COMPONENT j ON OUTSIDE OF ILLUMINATED SIDE z H rj z ". ~=...e -> /C) z OF OPEN CYLINDER. (0 =0, a=30~)

5 -4 0 2 4 6 z 8 10 12 FIG. 20: MAGNITUDE OF CURRENT COMPONENT j ON OUTSIDE OF ILLUMINATED SIDE C 0 2 4 6 z 8 10 12 FIG. 20: MAGNITUDE OF CURRENT COMPONENT j ON OUTSIDE OF ILLUMINATED SIDE OF CLOSED CYLINDER. (e=0, a=30~) C: rt IrZ. P-4 cri.. i p —q Z

THE UNT'IVWER S ITY Ou ~ F MIC ~=H IGA N CN1 S~ 0 H~ 01 S-] cJ2 N 0 z O Z6 X Q: 4 0W 0 i4 qSV'Hd 51

THE UNIVERSITY OF MICHIGAN 74 5F - I 11 I I I i I i I I I I I I I I i I I i I I I -- I -1 z H 0 A 2 In N Ca - z 0 N H z Up C z I 0 k V - I: 0 r j Z H Z P '.',-4 " z 0 > C QN -- ro ' 6 -I I I I I - I - I I I I I I i - I - - -. qp 'saJIflINoDV 52

THE UNIVERSITY OF MICHIGAN 7456-1- F Q z 0 0 z H 0 i N -4 S z z 0 N 0 H z z 0 0 0 H z U 0 0 0D 0 0 0 cl- r-~~~~4 c Oc - 'Nc -4 '-.4 '-4. aSVHd 53

THE UNIVERSITY OF MICHIGAN z H 0 z 0 N H 0c z C,Z LOC qp 'lIcjfljLINqVW 54

T-rHE U7N IV ERIS ITY Or aF MIC ~HIrG A N?4~-1 I 2 U7 0 -2 U -4 H 0 Cl) N H z z 0 0I U H Ci) r4 3S~Hd 55

4 z 0 0 12 z 3 4 FIG. 26: MAGNITUDE OF CURRENT COMPONENT jz ON INSIDE ILLUMINATED FACE OF OPEN CYLINDER. Hz (7~ H 0-4 <01 z (O =r, a=30 0)

420 390 360 330 uw 300 -1< 270 240 210 I I II? — 3 UP I 0 -< -r 0 To -O Z ' I, I 1 i 0 1 2 z 3 4 FIG. 27: PHASE OF CURRENT COMPONENT j ON INSIDE ILLUMINATED FACE OF OPEN CYLINDER. ( a z30) (0=r, a=30~)

TH fIE U JI VER~~S I T Y OF MlI C H IG(AN 7456-i -F Qei 0I Qd 07 Q H4 S N Z 0) N Hl zi zI~ ~ 0 0 Q~i Hl zj Q 0 PI 00 -4 0) 0 tO if) qp 9HRjflLINqWVj 58

I I I I i I 450 420 390 CD X 360 C: 330 300 270 rZ 0 -cr 0 -~-,-O 0 Tl Se I I1L -I 0 1 2 z 3 4 FIG. 29: PHASE OF CURRENT COMPONENT jz ON INSIDE ILLUMINATED FACE OF CLOSED CYLINDER. (0 =, a=30~)

0 -2 -5 -10 Hi5 z -20 -25 0 2 4 z 6 8 10 FIG. 30: MIAGNITUDE OF CURRENT COMPONENT jON TOP OF OUTSIDE OF OPEN CYLIN~DER. 20 HZ! z (.) — z..

900 - ct 600 300 r.-1 " C. Z lI I I 0 2 4 z 6 8 10 FIG. 31~ PHASE OF CURRENT COMPONENT j ON TOP OF OUTSIDE OF OPEN CYLINDER. (o. 6~300 ) a - 2, a=o )

I I, I I l I i 0 -5 1-1 -10 -1 -15 -20 l, I I I I ] I I 0 2 4 z 6 8 10 FIG. 32: MAGNITUDE OF CURRENT COMPONENT j ON TOP OF OUTSIDE OF CLOSED CYLINDER. Z (0, 0 (-,300 ) 2 ' H rt z C or} z 1 -4 H C"r 'DT n: >e

I H 900- -: 600 0 0-4 300 - > I I lI 0 2 4 6 8 10 z FIG. 33: PHASE OF CURRENT COMPONENT j ON TOP OF OUTSIDE OF CLOSED CYLINDER. ( =-, a= 30~) 2

I'' -5 ~-10 z 0-15 -20 0.5 1 1. 5 2 2. 5 z 3 3.5 4 FIG. 34: MAGNITUDE OF CURRENT COMPONENT j ON INSIDE TOP OF OPEN CYLINDER. (e- aIT300) 2' Hz mi cI 04 PC4 ze

450 300 CD, rt -, H rz m -4 H P0 pTI s n rf) z 150 0 1 1. 5 2 2.5 z 3 3.5 4 FIG. 35: PHASE OF CURRENT COMPONENT j7 ON INSIDE TOP OF OPEN CYLINDER.

THE UNIVERSITY OF MICHIGAN 7 4 -1 i- F CY) ~4 z 0 cqi z — z 0O 0 CY kI I zO 0 nc 0 LO 0 qp 'a~lfLLNOXTW CD 66

330 300 270 240 ) 210 - 180- - 150 > I I I I 0.5 1 1.5 2 2.5 z 3 3.5 4 FIG. 37: PHASE OF CURRENT COMPONENT j ON INSIDE TOP OF CLOSED CYLINDER. Z (O = a30 )

THE UNIVERSITY OF MICHIGAN 7456 -1 -F' r-4 z 0 04 H 0 00 0 H z 0 04 z w 00 0 L 0 LO 0 LO III I I qp 'aciflJINQvw 68

1200 900 - Z -1 600 ~ - 300 -,~, 0 z 0 2 4 6 z 8 10 12 FIG. 39: PHASE OF CURRENT COMPONENT j ON TOP OUTSIDE OF OPEN CYLINDER. (0=, a=30 )

THE UNIVERSITY OF MICHIGAN Q 0 0a 0o 0 H~I 0 ~ N 0 N4 z 0 -0 0 Hl z 01 0 0) 0 H -4 zuFr 0 tO 0 t '-4 & t - qp~~~1 'aclIoT 0 tO I 70

1200 - 900r -1 300- - Z 0 2 4 z 6 8 10 FIG. 41: PHASE OF CURRENT COMPONENT j) ON OUTSIDE TOP OF CLOSED CYLINDER. O (3 —, a=30 ) 22

\z -5 l I -I -10 \ / I I I I I l l I I 22 0.5 1 1.5 2 2.5 z 3 3.5 4 FIG. 42' MAGNITUDE OF CURRENT COMPONENT j0 ON INSIDE TOP OF OPEN CYLINDER. (0 -5, 13o~)

450 300 150 0.5 1 1.5 2 2. 5 z 3 3. 5 4 FIG. 43: PHASE OF CURRENT COMVPONENTj ON INSIDE TOP OF OPEN CYLINDER. (0=9 a=30 ) T1~ 0~z-'

-.5 z -15 -20 0.5 1 1. 5 2 2. 5 z 3 3. 5 4 F IG. 4 4: MAGNITUDE OF CURRENT COMPONENT j ON INSIDE TOP OF CLOSED CYLINDER. (=! I a=300) 2' c 24 m Pd 4-1 -1 - II 0-4 r) Z Po2g C) 3:

300 150 60 -l I..... I I I I 0.5 1 1.5 2 2.5 z 3 3.5 4 FIG. 45: PHASE OF CURRENT COMPONENT j0 ON INSIDE TOP OF CLOSED CYLINDER. e=2- a=30~ 2' 0-1 CA z, -

Signal Sample from R F Source Coaxial Switch Matched Load 0-2 — A 4 — l 0 —l i - T, z r"l 0 -0 -z Probe FIG. 46: ELEMVENTS OF THE RECEIEVING SYSTEM.

THE UNIVERSITY OF MICHIGAN 74156- i -F REFERENCES Bowman, J.J. and V.H. Weston (1965), "The Effect of Curvature on the Reflection Coefficient of Layered Absorber," (Submitted for publication). 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. Kerker, vIM. and E. Matijevic (1961), "Scattering of Electromagnetic Waves from Concentric Infinite Cylinders," J. Opt. Sc. Amer., 51, 506-508. Schmitt, H.J. (1957), "Back-Scattering Cross Section of Circular Metallic Cylinders Surrounded by a Resistance Foil, " Cruft Laboratory Scientific Report No. 11, Harvard University, Cambridge, Mass., (ASTIA Document No. AD-133636). Wait, J. R. (1955), "Scattering of a Plane Wave from a Circular Dielectric Cylinder at Oblique Incidence," Can. J. Phys., 33, 189-195. Weston, V. H. and R. Hemenger (1962), "High-Frequency Scattering from a Coated Sphere, " J. Res. NBS, 66D, 613-619 (Sept. -Oct. ). 77