THE UNIVERSITY OF MICHIGAN STUDIES IN RADAR CROSS SECTIONS XXVII CALCULATED FAR FIELD PATTERNS FROM SLOT ARRAYS ON CONICAL SHAPES Ro E. Doll, R. F. Goodrich, R. E Kleinman, A, L. Maffett, C. E. Schensted and K. M. Siegel Contracts AF-33(038)-28634 and AF-33(600)-36192 Hughes Aircraft Company Purchase Orders L-265165-FA47 4-500469-FC-47-D 4-5.26406-F C-89-3 February 1958 271.3-1-F Approved 9 /'k Keeve M. Siegel Project Supervisor The University of Michigan Engineering Research Institute The Radiation Laboratory Ann Arbor, Michigan

THE UNIVERSITY OF MICHIGAN 2713-1-F STUDIES IN RADAR CROSS SECTIONS I "Scattering by a Prolate Spheroid", F. V. Schultz (UMM-42, March 1950), W-33(038)-ac-14222. UNCLASSIFIED. II "The Zeros of the Associated Legendre Functions Pm (,) of Non-Integral Degree", K. M. Siegel, D, M. Brown, Ho E. Hunter, Ho A. Alperin and C. W, Quillen (UMM-82, April 1951), W-33(038)-ac-14222, UNCLASSIFIED. III "Scattering by a Cone", K. M, Siegel and H. A. Alperin (UMM-87, January 1952), AF-30(602)-9. UNCLASSIFIED. IV "Comparison Between Theory and Experiment of the Cross Section of a Cone", K. M. Siegel, H. A. Alperin, J. W. Crispin, Jro, H. E. Hunter, R. E, Kleinman, W. C. Orthwein and C. E. Schensted (UMM-92, February 1953), AF-30(602)-9. UNCLASSIFIED. V "An Examination of Bistatic Early Warning Radars", K. M. Siegel (UMM-98, August 1952), W-33(038)-ac-14222. SECRET. VI "Cross Sections of Corner Reflectors and Other Multiple Scatterers at Microwave Frequencies", R. R. Bonkowski, C. R. Lubitz and Co E. Schensted (UMM-106, October 1953), AF-30(602)-9. SECRET - Unclassified when Appendix is removed. VII "Summary of Radar Cross Section Studies Under Project Wizard", K. M. Siegel, J.W. Crispin, Jr., and R. E. Kleinman (UMM-108, November 1952), W-33(038)-ac-14222. SECRET. VIII "Theoretical Cross Section as a Function of Separation Angle Between Transmitter and Receiver at Small Wavelengths", K. M. Siegel, H. A. Alperin, R, R. Bonkowski, J. W. Crispin, Jr., A. L. Maffett, C. E. Schensted and I V. Schensted (UMM-115, October 1953), W-33(038)-ac-14222. UNCLASSIFIED. IX "Electromagnetic Scattering by an Oblate Spheroid". L M. Rauch (UMM-116, October 1953), AF-30(602)-9. UNCLASSIFIED. X "Scattering of Electromagnetic Waves by Spheres", H, Weil, M. L. Barasch and T. A. Kaplan (2255-20-T, July 1956), AF-30(602)-1070. UNCLASSIFIED. XI "The Numerical Determination of the Radar Cross Section of a Prolate Spheroid", K. M. Siegel, B. H. Gere, I. Marx and F. B. Sleator (UMM-126, December 1953), AF-30(602)-9, UNCLASSIFIEDo iii

THE UNIVERSITY OF MICHIGAN 2713-1-F XII "VSummary of Radar Cross Section Studies Under Project MIRO", K. MJ Siegel, M. E, Anderson, R, R. Bonkowski and W. C. Orthwein (UMM-127, December 1953), AF-30(602)-9. SECRET. XIII'Description of a Dynamic Measurement Program' Ko M. S iegel and J. M. Wolf, (UMM-128, May 1954), W-33( 038)-ac-14222. CONFIDENTIAL. XIV "Radar Cross Section of a Ballistic Missile", K. M. Siegel, M, L. Barasch, J, W. Crispin, Jr;, W. C. Orthwein, I. V. Schensted and H, Weil (UMM-134. September 1954), W-33(038)-ac-14222, SECRET. XV "Radar Cross Sections of B-47 and B-52 Aircraft", C, E, Schensted, J. W, Crispin, and Ko M. Siegel (2260-1-T, August 1954), AF-33(616)-2531. CONFIDENTIAL. XVI "Microwave Reflection Characteristics of Buildings", H. Weil', R, R. Bonkowski, T. A, Kaplan and M. Leichter (2255-12-T, May 1955), AF-30(602)-1070M SECRET, XVII Complete Scattering Matrices and Circular Polarization Cross Sections for the B-47 Aircraft at S-band"', A Lo Maffetto M. Lo Barasch, W. E, Burdick, R F. Goodrich, W. C. Orthwein, C. E Schensted and K. M.Siegel (2260-6-T, June 1955), AF-33(616)-2531L CONFIDENTIAL. XVIII "Airborne Passive Measures and Countermeasures", K. M.. Siegel, M. L. Barasch, J.W. Crispin Jr., R F. Goodrich, A.. HaHlpin, A. L. Maffett, W. C. Orthwein, C E, Schensted and CG J. Titus (2260-29-F, January 1956), AF-33(616)-2531, SECRET. XIX "Radar Cross Section of a Ballistic Missile II", K. M. Siegel, M, L. Barasch, H. Brysk, J. W. Crispin, Jr, T, B. Curtz and T A, Kaplan (2428-3-T, January 1956), AF-04(645)-33, SECRET, XX "Radar Cross Section of Aircraft and Missiles", K. MO Siegel, W.V E, Burdick, J. W. Crispin, Jr-. and S. Chapman (WR-31-J, ONR-ACR-10, March 1956), SECRET. XXI "Radar Cross Section of a Ballistic Missile IIIv,' K. M.Siegel, H. Brysk, J. W. Crispin,Jr., and R. E, Kleinman (2428-19-T, October 1956), AF-04(645)-33, SECRET, XXII "Elementary Slot Radiators", R, F, Goodrich, A, L. Maffett, N, E. Reitlinger, C. E. Schensted and K. M. Siegel (2472-13-T, November 1956), AF-33(038)-28634, HAC.PO L-265,165-F47, UNCLASSIFIED. XXIII "A Variational Solution tot the Pi-oblem of Scalar Scattering by a Prolate Spheroid", F, B. Sleator (2591-1-T, March 1957), AFCRC-TN-57-586, ASTIA Document No. 133631, AF-19(604)-1949, UNCLASSIFIED. iv

THE UNIVERS ITY OF MI CH I GAN 2713-1-F XXIV'Radar Cross Section of a Ballistic Missile IV", to be published. SECRET. XXV "Diffraction by an Imperfectly Conducting Wedge", T. B. A. Senior (2591-2-T, October 1957), AFCRC-TN-57-791, ASTIA Document No. AD 133746, AF-19(604)-1949. UNCLASSIFIED. XXVI "Fock Theory Applied to an Infinite Cone", R. F. Goodrich (2591-3-T, January 1958), AF-19(604)-1949. UNCLASSIFIED. XXVII "Calculated Far Field Patterns from Slot Arrays on Conical Shapes", R. E. Doll, R. F. Goodrich, R. E. Kleinman, A. L. Maffett, C. E. Schensted and K.M. Siegel (2713-1-F, February 1958), AF-33(038)-28634, AF-33(600)-36192. UNCLASSIFIED. v

THE UNIVERS ITY OF MICHIGAN'27'13-1-F PREFACE This paper is the twenty-seventh in a series growing out of studies of radar cross sections at the Engineering Research Institute of The University of Michigan The primary aims of this program are1L To show that radar cross sections can be determined analytically. 2, A, To determine means for computing the radiation patterns from antennas by approximate techniques which determine the pattern to the accuracy required in military problems but which do not require the unique determination of exact solutions, B, To determine means for computing the radar cross sections of various objects of military interest. (Since 2A and 2B are inter-related by the reciprocity theorem it is necessary to solve only one of these problems ) 3o To demonstrate that these theoretical cross sections and theoretically determined radiation patterns are in agreement with experimentally determined ones. Intermediate objectives are: 1 A. To compute the exact theoretical cross sections of various simple bodies by solution of the appropriate boundary-value problems arising from electromagnetic theory. B. Compute the exact radiation patterns from infinitesimal solutions on the surface of simple shapes by the solution of appropriate boundary-value problems arising from electromagnetic theory. (Since 1A and 1B are inter-related by the reciprocity theorem it is necessary to solve only one of these problems., ) vi

THE UNIVERSITY OF MICHIGAN 2713-1-F 2. To examine the various approximations possible in this problem and to determine the limits of their validity and utility, 3. To find means of combining the simple-body solutions in order to determine the cross sections of composite bodies. 4. To tabulate various formulas and functions necessary to enable such computations to be done quickly for arbitrary objects. 5. To collect, summarize, and evaluate existing experimental data. Titles of the papers already published or presently in process of publication are listed on the preceding pages. The major portion of the effort in this report was performed for the Hughes Aircraft Company and under Air Force Contracts AF-33(038)-28634 and AF-33(600)-36192. K. M. Siegel vii

THE UNIVERSITY OF MICHIGAN 2713-1-F TABLE OF CONTENTS Section Title Page Studies in Radar:Cross Sections iii Preface iv I Introduction and Summary 1 II Infinitesimal Slots 14 2, 1 Introduction 14 2,2 Notation 15 2, 3 Fock Theoiy 15 2o 4 Tip Scattering 17 2. 5 Plane Waves versus Dipoles 20 2. 6 The Reciprocity Theorem 21 20 7 The Far Field Radiated by an Infinitesimal Slot 23 III A Theory of Low Side Lobe Antenna Arrays 25 IV A Theory of Maximum Gain Antenna Arrays 48 Ap'pendix A The 65-Slot Problem - The Method of Computation and the Results Obtained 54 A. 1 The Method of Computation 54 Ao. 2 Beam Direction 60 Ao. 3 Results of the 65-Slot Computations 62 B A Further Analysis of Tip Scattering 72 C Ideal Current Distribution for a Conical Antenna 78 References 114 o O ol

THE UNIVERSITY OF MICHIGAN 2713-1-F INTRODUCTION AND SUMMARY This is the final report on Purchase Order Numbers L-265165-F47; 4-500469-FC-47-D; and 4-526406-FC-89-3 entered into between The University of Michigan and the Hughes Aircraft Company on Air Force Contracts AF-33(038)-28634 and AF-33(600)-36192. The ultimate purpose of this study is to replace radome-dish combinations (postulated but not yet researched and developed) with arrays of slots on the nose cone of a high speed vehicle. Such a replacement would eliminate the many problems associated with radomes, e. g., rain erosion, thermal stress, boresight error rate, and weight and balance problems. Theoretically, it is always possible to duplicate a pattern produced by one type of antenna using another antenna of equal or larger surface area. For example, the pattern produced by a radome and parabolic dish combination can be duplicated by using any given covering surface as an antenna by correctly distributing the current on that surface. The required current distribution can be calculated by an application of Huygen's Principle. In particular, if the surface is chosen to be a semi-infinite cone, assume that the parabolic dish lies within the conical surface and compute the "dish fields on this surface, this field, then, if induced on a real cone, would reproduce the "dish pattern". Such a calculation serves two purposes: (1) it yields the precise current necessary to duplicate the parabolic-dish pattern, and (2) it yields results which serve as design criteria for arrays of slots, indicating a possible interim slot array design. 1

THE UNI VE RS ITY OF M I CH I GAN 2713-1-F Such a computation is presented in Appendix C. The current distribution required varies as a function of the scan angle and the problems involved in attempting to instrument this distribution on a conical surface appear to be extremely difficult; however, since more promising methods are available, eo g., optimized slot arrays, a serious attempt at instrumentation is not warranted. The value of most theoretical studies on either radiation or scattering problems is that the. theoretical analysis very often has in it the seeds of better methods of design. Experiments can only determine the characteristics of existing designs. The approach -used here was to: 1. Determine the theoretical means of computing the complete pattern of a single slot on a conical surface. 2, Determine the means of computing the pattern for any given array of slots using the. methods of 1. 3. Determine optimization techniques for various design criteria. 4. Given-a design configuration, to compute the pattern, e. g., the 65-slot array chosen by the Hughes Aircraft Company for analysiso 5. Apply the optimization techniques of 3, to the pattern of 4. This report gives the results of 1 through 4. The slot location is illustrated in Figure 1-1 which also indicates the coordinate system employed in the theoretical analysis of the 65-slot problem. Figures 1-2 through 1-5 contain comparisons of results obtained in the theoretical study with results obtained in the experimental program at the Hughes Aircraft Company. * The formulas employed See Appendix A (Section A, 3) for a discussion of these comparisons and additional experimental data obtained by HAC. 2

THE UNIVERSITY OF MICHIGAN 2713-1-F in the theoretical analysis of this 65-slot problem together with all of the theoretical results obtained are presented in Appendix A. This paper reports the first analysis which theoretically determines the pattern resulting from a many slot array on a conical surface. Sketch of Cone Surface Sketch Showing Geometry Sketch Showing Geometry Showing Location of Employed - In "Plane Employed in "Conical Cut" I First Few Slots Cut'" Computations 0=0~ Computations Generator on which slots i _(Extension of generator conare located (in xz'-plane) 1 tai-niy"g 9slots - axis of conical Main Beam Directionz co i Beam A Z \ o Directi \, 2.,. I e ~ " ~/ — —.Receivel ~ -/ ~~;3 ~.~ ~~ I \' ~ ~Located on ~;F/~~ 0 B Direction to Sauface of' I Receiver ne FIG 1- GEOMETRY USED IN ANALYSIS OF THE 65-SLOT PROBLEM 3~Slots FIG. 1-1- GEOMETRY USED IN ANALYSIS OF THE 65-SLOT PROBLEM

THE UNIVERSITY OF MICHIGAN 2713-1-F Abscis-sae Exp2 of minima. HI4ma2 o Theor - 3 odb ~~~40 0~~~ O~-40 FIG. 1-2 COMPARISON BETWEE:N THEORY AND EXPERIMENT FOR A PLANE CUT 0B 10~o, 0 = 00)

THE UNIVERSITY OF MI CHIGAN 2713-1-F Abscissae of minima (theoretical), IE( I2 Exp. -10db llIH2m-x X Theory m in Theory X -30db I'I/ 6 40 0 -20 -40 0 in degrees FIG. 1-3 COMPARISON BETWEEN THEORY AND EXPERIMENT FOR A PLANE CUT (0B = 300, ~ = 0~)

FIG. 1-4 COMPARISON BETWEEN THEORY AND EXPERIMENT FOR A CONICAL CUT (eB= 100) Power level in db t -10 A.... 20C -20 20 40_ 60_ 80_ 100 120 1401180 0 in degrees -30 ~~jL8) a~-'x Exp. 4 lHmma,2 e Theory -- -50 _ 0 20 40 60 80 100 120 140 160 180 0 in degrees

FIG... 1-5 COMPARISON BETWEEN THEORY AND EXPERIMENT FOR A CONICAL CUT (0B- 300) Power level in db 0 ~~~~~~~~~ -10~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-' -20 - _ __ /~~~~~~~~~~~~~~~~~~~ / i' -30~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~H2 X -— Exp. H~~~~~~~~~~ H 21 Theory. max 2,~ - Exp.0 -40 H.2 TheorymaxAz 0 20 40 680 100.124 140 1601 0 in degrees

THE UNIVERS ITY OF MICHIGAN 2713-1-F In order to obtain theoretically the far field radiated in all directions, it is necessary to employ different methods of approximation to account for different contributions to the field. Geometric Optics suffices for main beam calculations. For the receiver not in the main beam, other techniques are required to describe both the effect of the cone's tip on the radiated field and the effect of energy which appears to have crept along the body before being radiated, since these effects predominate out of the line of sight of the array. Physical Optics was used to determine the tip effect while the method available to treat the creeping waves was that due to V. A. Fock (Ref. 1) and this application used analyses of N. A. Logan (Ref. 2), J. B. Keller (Ref. 3) and one of the authors (R. F, Goodrich). The generalized arguments for Fock's universal functions were obtained for application to a conical surface; this work was performed under another Air Force Contract (AF-19(604)-1949) and is discussed in Reference 4. In the calculation reported on in Appendix A, the Fock technique was employed only when the slots were in the cone's shadow relative to the receiver, and Geometric Optics was used in the lit region. In addition, the tip effect was examined and found (for a 150 half anglecone with the first slot 16 wavelengths from the tip) to be of a lower order of magnitude than the o ther contributions to the field. These three methods for the analysis of a single slot on a cone are discussed in Section II and the regions of their applicability in the 65-slot problem are indicated graphically in Figure 1-6. As mentioned above, although a pattern obtained from a single analysis such as the 65-slot program is, in and of itself, of extreme interest, the seeds of improvement

THE UNIVERSITY OF MICHIGAN 2713-1-F _+ 1800 rrn:lr=rrmlT~po9t-s inid O-O150~ I6=~~-gga I I m// / -..-' —,s 1= 30-0,'= k -: -301'":300 I. III I IIt c e region where Fock Theory was used I region where NOTE: Physical Optics tip result was employed Geometrc Optics everywhere except the neighborhood of was used2 the point (6=150, 0= 1800). FIG. 1-6 PLANAR REPRESENTATION OF (FAR ZONE),ECEIVE- PWITIONS.s

THE UNIVERSITY OF MICHIGAN 2713-1-F contained mi the analysis are of even greater importance. That is, for many arrays of practical interest it is possible to find a method of predicting how to alter the array to produce a better pattemrn for a given purpose from the given (or computed) pattern until we have reached a satisfactory practical solution to many particular radiation problems. eThere are several methods of optimization possible and these are associated with the particular problems present'in a given system~ Various systems might require as criteria: 1o that a'maximum amount of energy be in the main beam for a given amount of power put into the system, 2,, athat there be a maximum gain for a fixed amount of input power, 3, that the ratio of main beam power to side lobe level be a maximum, 4, that there be a certain spread of power in many side lobes as compared with the admissable power in a -few side lobes, etc. In a given application the criteria which are used will depend upon the specific system requirements. Rather than consider a specific system we have selected two more general criteria for analysis here, We feel that the results obtained are highly important contributions to the optimization problem and athat these optimization methods should be applied to slot array computationso The first of these deals with obtaining a low side lobe level for a given maximum beam power; this method of optimization is discussed in Section IlL The second method utilizes a special purpose analog computor which we feel 1-0

THE UNIVERSITY OF MICHIGAN 2713-1-F may yield the excitation for the maximum gain possible in surface arrays. This second optimization procedure is discussed in Section IV. Thus, in summary, vie have established the fact that radiation patterns from slots on cones can be computed, we know what current distributions are required to duplicate radome-dish patterns, and we have methods which give every indication of leading to patterns which will be an improvement over the radome-dish type. One would expect this on theoretical grounds since utilizing the entire surface of a cone as an antenna increases the area available over a radome-dish combination (thus obtaining a larger effective aperture) and also because the bias errors due to the radDme would be withdrawn from the system. Although the results reported here represent a definite breakthrough for the surface antenna problem, considerably more work is required before this program can be considered to be complete. For example, the optimization methods should be explored more fully and applied to particular cases of interest, e. g., the 65- slots on a cone generator. Further basic research is required on the nature of the radiated field in the vicinity of the transition region (that region of space which separates the "geometric optics region" from the "Fock theory region'). It is our belief that the amplitude is a slowly varying function in this transition region although the phase is a rapidly varyiig function. However, this belief is based upon physical principles and should be substantiated by basic mathematical analysis. In addition to the consideration of optimization fcom the viewpoint of a particular array, considerable attention should be given to 11

TH E U NIVE R.S ITY OF M I C H I-GAN 2713-1-F optimization on the basis of a variable array; such an approach (if successful) would lead to a method of optimization which would be applicable to almost any system regardless of what basic input parameter philosophy was inherent in that system. In considering the optimization procedures, it is necessary to keep sight of the final applications. Indeed, -the optimization goal is, given a certain antenna requirement, to reduce to a minimum the complexity, value, and number of the input parameters~. As an illustration, it has been shown that point by point control of the field on the entire -surface will permit -the reproduction of a dish pattern but at the cost of:a greatly complicated excitation mechanism. On the other hand, a series of linear a rrays of the kind considered in this report may be quite sufficient for a given application at; a much reduced- cost in the complexity of the input mechanism. An understanding of this approach to optimization needs to be an aim of- future studyUnder University of Michigan Purchase Order No. 154216 (under the prime Contract No. AF-33(038)-28634) the University of Illinois investigated the problem of establishing a criterion for approximating a continuous source distribution by a discrete source, distribution for line soures under various conditions of excitation. A further investigation considered the approximation of the radiation pattern of an electric dipole in free space by a distribution of dipoles on the surface of a perfectly conducting sphere. This work is reported in Reference 5. 12

THE UNIVERSITY OF MICHIGAN 2713-1-F Under subcontract to The University of Michigan (University of Michigan Purchase Order Numbers 154700, 239327 and E-500183) the Polytechnic Institute of Brooklyn carried out studies concerned with the computation of the radiation patterns from leaky wave and surface wave ring sources distributed over a finite area on a perfectly conducting semi-infinite cone. The theoretical formulas for the far radiated fields (including geometric optics and diffraction effects) were obtained by integration over the assumed source distribution of the ring source Green's functions. Explicit results for various source polarizations were obtained for two types of variation of the source distributed over a finite distance along the radial dimension of the cone. In order to gain a better understanding of the radiation from sources distributed on a cone, the simpler problem of radiation from sources on a semi-infinite wedge was also investigated as a preliminary. The results of these and related investigations carried out at the Polytechnic Institute of Brooklyn under these purchase orders are reported in References 6, 7, 8, 9 and 10. The authors wish to acknowledge the assistance of the following members of the Radiation Laboratory during the computational phase of the study: H. E. Hunter, S. E, Stone, D Way, D.M.Raybin, M.Plonus, T, Hon, L, S.Gregory, T B, Curtz, J. W. Crispin, Jr,. K, Najita and N E.o Reitlingero 13

THE UNIVERSITY OF MICHIGAN 2713-1-F II INFINITESIMAL SLOTS 2, 1 Introduction We wish to fMd an expression for the far field racated by a transverse infinitesimal slot located on the surface of a cone at a large distance (in wavelengths) from the tip of the cone. The infinitesimal slot can be considered as a magnetic dipole. To find its far field we make use of the Reciprocity Theorem (see Section 2, 6). This theorem requires us then to find the field induced on the cone (at the point where we wish to locate the slot) by a magnetic dipole. The dipole is arbitrarily oriented at an arbitrary point in space but has to be located far from both- the cone tip and the point where the slot is to be placed. I the dipole is located sufficiently far from the cone, the wavefronts of the dipole field near the cone are essentially plane. Hence, we approximate the field incident upon the cone by the field of a plane wave, whose magnitude is carefully adjusted so that the dipole nature of the incident field is retained. Thus our problem becomes one of determining the fields induced on te cone surface by a plane wave incident from an arbitrary direction. We treat this field induced on the cone surface as a quantity having two dominant contributions, One of these contributions arises from energy which either impinges direc~tly upon a neighorhood of the pont where the slot is to be -placed or travels to this point along a geodesic from the shadow boundary. We call Weg former part of this 14

THE UNIVERSITY OF MICHIGAN 2713-1-F contribution the "geometric optics" term and the latter the "creeping wave" term. The other of these contributions consists of energy which is scattered by the tip of the cone to the point where the slot is to be located. We call this the "tip"' term. In the succeeding sections we present the methods by which these two contributions are obtained and the procedures for using them to describe the far field radiated by an infinitesimal slot on a cone. The "geometric optics" term is given implicitly in Section 2, 4 and discussed more fully in Reference 16. 2. 2 Notation We adhere to the following notation throughout this section. Let r, 0, 90 be the usual spherical coordinates describing a field point in the radiation problem, a source point in the reciprocal problem. Primed, these variables designate a field point in the reciprocal problem; double-primed, they refer to points on the cone f E"= a, a < r/2, We specify that all electromagnetic fields have a time dependence given by e-ikct where c is the velocity of light and k = 27r/X, X designating wavelength. The unit vector p designates the magnetic field direction for a plane wave and is called polarization by us. 2.3 Fock Theory We obtain the creeping wave contribution to the field on the surface of a cone due to an incident plane wave by suitably extending an analysis due to V. A. Fock (Ref. 1). We can consider the Fock formulation as a modification of geometric optics 15

THE UNIVER.SITY OF MICHIGAN 2713-1-F based on a local analysis of the field in the region of the shadow boundary, In particular, Fock defines two universal scalar functions, F and G, whose argument 5 is a function of distance measured from the shadow boundary. The polarization of the incident radiation determines the linear vector combination of the Fock functions which is approximately proportional to the field induced on the'scatterer. The creeping wave term, i. e., the magnetic field' H evaluated on the surface at apoint 5 which is a function of distance into the shadow region along a geodesic, can be given in terms of the unit vectors 0", T, and 0" x T as {L ad{F( {)T+ ( x G()O "x T (2.3.1) - P has shadow where T is the tangent to the geodesic, 0," is the normal to the surface, p is the polarization of the incident plane wave, and F(g) and G( >) are Fock's functions. It is important to note that in this expression for Hc,, p 0" andp p 0" x T are evaluated at that point on the shadow boundary where the creeping wave is launched, while F(5)T and G(F)0" x T are evaluated at the point of interest in the shadow region, i. e., at the point where the slot is to be placed. We now need to determine the geodesics on a cone, the shadow boundary for a given incidence of a plane wave, and curvature along a geodesic (to find Fock's argument 5 ). 16

THE UNIVERSITY OF MICHIGAN 2713-1-F We find the equation of a geodesic on the cone el" = a to be rIr,= a sins_ r?= asin\Isec [( 0"1 - 0S)sina+-. asn (2.3.2)' 2 s~in q)-(10 sin -(" ) sina where 0 = arc cos tan, 0a 4 s"4 <r, a,<e<7r - a, (2. 3.3) s tan 8' gives the equation of the generator forming the shadow boundary (for incidence in the xz-plane); where \t = arc cos (-cosO 6 0. V) y4 r, (2. 34X cos a is the angle the incident direction forms with the shadow boundary; and where a is the distance from cone tip to point where slot is to be located. The curvature K along a geodesic is given by K= a2 cos asin2 (25) K= (2. 3.5) r"t sin a Finally, 5, the argument of the Fock functions for the creeping wave term, is = =(k2 ani c' ) ( J11 -0k) sin a. (2.3.6) 2 tan- s 2. 4 Tip-scattering We use the expression "Physical Optics field" to denote the scattered far field, Hs, obtained by using Geometric Optics (i. e., the magnetic field on the surface is given by twice the tangential component of the incident magnetic field intthe lit region and zero in the shadow) in the Kirchhoff-Huygens integral representation for a scattered field. 17

THE UNIVERSITY OF MICHIGAN 271341-F Hs.is, for a plane wave of magnitude Ho incident from an arbitrary direction, ikr'.. _o -ke ^ A -4n /N t A Hs, 19g r op)f-(r f) p (2, 41) where ( -kr"o (r+rf) f n ie dS, (2 4.2) S S is the portion of the cone surface illuminated by Wte plane wave and An is the outward normal to the cone, Equation (2,0 4, 2) may be written c: B i ikr laM f = l(cosacos "ix +cosasin lo y-smina i)e r "sina d"dr" (2 4. 3) A and B are generator shadow boundaries cast on the cone by the incident plane wave; they are A - arc cos;/l n B=O- +arc cos (tan 9 when no shadow exists,, eo, the cone is entirely illuminated or 0 > r - a, A=0-7r and B= -r The quantiy M is given by M = q + b cos + c sin " where q = cos a (cos 01 + cos 6} b = sin a (cos 0' sin0 0 + sin 0 cos 0), c = sin a (sin Of smin 0 + sin 0 sin 0), 18

THE UNIVERS I TY OF M I C H I G A N 2713-1-F The r" integration is. accomplished by an Abelian limit argument (Ref. 11), yielding for f B f A A sf_ in a (cscos M0ix + cos a sin 0"i - sin aiz), (2.4.4) this integration can be expressed in finite terms (Ref. 12); - sina cos c+q q n arctan (-b)tan(/2) + c i - cos a [b+g cos + 2cQ / arctan (q-b) tan (0"/2) +c i. MQ where Qq2_b2 _c2 Care must be taken with the value of the arctangent to note where its value has moved The field Hs given by Equation (2.4 1) is called the the tip-scattered field because in the Abelian evaluation of the integral for f (Eqn. 2, 4.4) the major contribution, for M bounded away from zero, is obtained from a neighborhood of the point r"=0, io e., the tip of the cone. The cases for which M=0 for all values of ~" in Equation (24. 4), correspond to forward scattering situations and these cases are dealt with by Geometric Optics. The cases for which M=0 at only one value of | 2"| 19

T H E UNIVE R S ITY OF M I C HI GAN 2713-1-F in the region of integration either correspond to specular reflection where again Geometric Optics is used or require a re-examination of the Physical Optics approximation. Although the integral in (2. 404) does not exist in this last case we know how to correctly handle this situation. In Appendix B we make a closer examination of the Physical Optics integral (Eqn. 2 4 1) for this case and find that the evaluation given above can be formally extended to include it. Since the major contribution to the Physical Optics integral, when it exists, comes from a neighborhood of the cone tip we now require only that the field point be far from this neighborhood. In this way we obtain a representation of the tip-scattered contribution to the field on the cone surface far from the tip, There are substantial indications that lthe Physical Optics field closely approximates the exact field scattered by a cone. For example, the Physical Optics method produces, for large and small angle cones, a result for the nose-on monostatic radar cross section which is in excellent agreement with the result produced by an exact expansion of the cross section in powers of the cone angle(Refs. 13, 14). In addition, the rigorous bistatic radar cross section for an electromagnetic plane wave incident along the cone axis agrees well with the Physical Optics result (Ref. 14). We know also that it has exactly the right wavelength dependence. 2.5, Plane Waves versus Dipoles In the previous sections we dealt with an incident plane wave, The field of the plane wave on the cone surface is given by A -ikr. r -r r Hi = p H e (25.1) 20

THE UNIVERSITY OF MICHIGAN 2713-1-F Since we are really interested in ani incident field due to a magnetic dipole far from the cone, we relate these fields as follows: Let a magnetic dipole with orientation m, such that mi\ r vanishes, be located at the point (r, 0. 0). Then its far field is - eAikR^ R x (R x m), R where C is the dipole strength coefficient and R is the distance from the point (r,,,0) to the point of observation (r', 0','), R=[r2+rt227r ri/2 = r- r' for r>> r', and R= VR. A For large r, R may be replaced by -r. Since the dipole is far from the cone its field at the cone surface is given approximately by ik(r-. rV') --- - C (2. 5.2) r Thus we see from Equations (2, 5. 1) and (2. 5. 2) that, at the cone surface, we may replace the incident field of a plane wave by the incident field of a dipole placed at a very large but finite distance from the cone simply by. plafigjk p = -m and ikr e Ho Ce- v 2, 6 The Reciprocity Theorem We now present the Reciprocity Theorem in a form whicha with the results of the previous sections, will enable us to obtain the far field radiated by an infinitesimal slot

THE UNIVERSITY OF MICHIGAN 2713-1-F on a cone. The Lorentz Reciprocity Theorem may be stated in the form HI' M2 dV H Ml M dV where H1(1H2) is the field due to the magnetization M1(M2) and the integration is over all space. If MlI2 are of the form ml n2 - rl 2 ) that is, point sources, then we find H(r2) m2 =H-2(rl)H m.M (2, 6.1) A Let m,2 designate the direction of some specifically oriented dipole located by the vector i2o We wish to determine its field. at some far point rio Suppose H2(rl) = Al(r)u1 + A2(rl) where ul o v= 0- Let ul and vl designate in tu-rn the orientation of mn of a dipole located by ri. Its field at r2 is in turn HI(r2,ul) and HIAr2 v)o It follows from (2. 6o 1) that HI(r2 u )~ m2 = [Alr)Ul + A2((rl)vl+ ul} from which Hl(r2 ul) 2 = A1(rl) (2. 6, 2) Similarly, from (2, 6. 1) Hlr2, vI) m2= Al(r )l +A2(rl)V1l Vl, from which HI r2 v) m2 A2(rl) (2Z 6 3) Therefore H2r)-(rl ~ (2r22 6e 4) 22

THE UNIVERSITY OF MICHIGAN 2713-1-F Equation (2. 6. 4) gives the radiated far field of a slot on the surface of a cone if we let -p -~ A A rl,-rr r, 2 -~ r", m2 -- direction of slot, ul, vl -- successive values for polarization, p- H1 -- Hs + Hc and H2- H. 2. 7 The Far Field Radiated by an Infinitesimal Slot The final part of the preceding section explains exactly how to employ the Reciprocity Theorem to obtain a radiated far field due to an infinitesimal slot on a cone. In Section 2. 3 we have, without loss of generality, restricted the direction of incidence to lie in the xz-plane. We may treat the results of Section 2.4 in a similar fashion, again without loss of generality. Now employing the results of Section 2.6 we may express the radiated far field at a point (r, e, 0) due to an infinitesimal slot located at (aa, I") in the direction 0" on the cone as, {[ic(Pl) +Hs(P,)]' P} + {Hc(P2)+s(P2) I" P2, where the results of Section 20 5 must be used to define the quantity Ho appearing in expressions for Hc and Hs given by Equations (2.3. 1) and (2.4,1), respectively. Note — D. that Hc, for a point in the lit region, is given by geometric optics. If the slot has a small but finite length, L, then the expression for the radiated far field obtained above can be adjusted so as to include the length and voltage, V6, across the slot. As long as a radiating magnetic dipole is equivalent to a slot in a plane we can relate the dipole strength coefficient, C, of Section 2. 5 with the voltage 23

THE UNIVERSITY OF MICHIGAN 2713-1-F across the slot as follows. ik VoL C = —(see Section A. 1. 3 po 68 of Reference 15). It is significant to note that the term "Geometric Optics field" is a misnomer when applied to radiated fields obtained via reciprocity from scattered "Geometric Optics fields", since such radiated fields are no longer wavelength independent. 24

THE UNIVERSITY OF MICHIGAN 2713-1-F III A THEORY OF LOW SIDE LOBE ANTENNA ARRAYS In the problem which will be considered in this Section it is assumed that there are a number of current distributions for the array which can be excited independently, and for each of which the radiated field is known. We might assume that the array consists of a number of elements which can be separately excited, and that the field radiated when any one of these elements is excited is known. For example, we have shown in Section 2 how to find the radiated field of an infinitesimal slot on a cone. The problem we set for ourselves is that of determining the excitation coefficients for an array in such a way as to minimize the side lobe level (compared to the main beam) for a beam of a fixed width. To restate the above symbolically let the currents In give the radiated fields, Fno We now want to determine coefficients An which are related to the current, I= -IAnIn, and field, F = _AnFn, in such a way that the largest value taken on by the ratio F(, 0) in a given region, R, which excludes a neighborhood of the point 8 = 0, = 0 F(09 0) is as small as possible (8 and 0 are the polar angles of the field point). We will consider a series of less ambitious problems which will finally lead us to the desired problem. Rather than minimizing the largest value taken on by I F(0, )e F(0, 0) for all points 0, 0 in R we will minimize the largest value taken on by VF(8O ) 0 at a F(0, 0) set of points 8i, 0i in R. We will eventually let the number of points cover R densely so as to solve the original problem. Throughout the discussion we will not insist on any more mathematical rigor than is necessary to put forward a convincing argument for the methods used. 25

THE UNIVERS ITY OF M I CHIGAN 2713-1-F As a matter of notation let Fn(Oi, 0i) = Fin Fn(O, 0) = Fon Then we seek the values of Al, A2,. AN which will give.A F. Min Max In n_ in (3.1) Al, Al2z..AN i |i- AnFon that is we seek the values of Al, A2, o. AN which will minimize the largest value taken on by Z An Fin Let M be the number of values taken on by i (excluding zero). An Fon First of all assume M < No Then we can make h-AnFin =0 for all i 0 while ) AnFon O. This disposes of this case which will not be of interest hereafter. Next consider the case M = No. This case, as it will turn out, is fundamental to the rest of the analysis. We are assuming that the matrix Fin is non-singular which will ordinarily be the case. For simplicity we now let Bi = L Fin An (3.2) We can solve for the An in the form (3-3)'An = L(F)niB (33) n ni 1 Thus we have _ F n i on(F ni i= GiBi (3.4) where 26

THE UNIVE'RS ITY'OF MI CHI GAN 2713-1 -F Now our problem is to determine the values of B19 B29. o BN which will give Min Max B 2, oo*,BN i GjBj (35) It is clear that the B's are not unique to the extent that we can make the replacement Bi - X Bi without affecting the value of Max We will make use of " i' "Gj Bj this later to simplify the form of the problem. Let us now assume that all of the B's except B1 have the correct value (or rather a correct value in view of the non-uniqueness). We will now proceed to evaluate B1 in terms of the other B's. First we note that Max Bi. =Max Bi i Gj Bj i G G 1 Max Bi (3.6) JGII ~i aj B. G1 Next we make use of the non-uniqueness to replace Bj by BjeM (a real) where a is chosen so as to make L G Bj a positive real number, f o Finally we let x = - B1, jfl G1 i Now what we seek is Min Max Max x B -x (37) x i -x f-x Let iM be a value of i (other than 1) for which Bi takes on its maximum valueo Thus we want Min Max M ~ x Ki x K I (3.8) 27

THE UNIVERSITY OF MICHIGAN 2713-1-F Figure 3-1 is a contour plot of Max Bi M x } in the x-plane - x' for Bi M = Y/2. The contour lines are for the indicated side lobe levels. The dotted circle divides the region IBiMI > x from the region Bi <lx i. It is clear that the minimum occurs for x -BiMi' From this we see that IB1 | = BiM - Since we could equally well have chosen any of the other B's in place of B1 we see that the minimum occurs when all of the B's are equal in magnitude. Due to the non-uniqueness we can take the B's to all be equal to one in magnitude. Then (3.5) can be written Max B2ZGa BN jB jI (3.9) B1. B2,... BNI!Bj! = IB,.:... ='B~I = 1.e But zGjBj GjBj. = Z)Gj1 (lBj 1), so that it is clear that the maximum will be attained for Bj Thus we have Gj Min Max Bi | = 1 (3.10) Bj,B2,oonB i Z GjBj ZIGjI is attained for B. Gj. (3.11) J Gj To get the corresponding values of the A's we use (3. 3). This completes the solution of the problem M = N. Next we consider the case co > M > N. Suppose we had a correct set of An. Now order the M quantities K An Fin! starting with the largest and ending with the EAn Fon smallest, Pick out the first N of these. 28

THE UNIVERSITY OF MICHIGAN 2713-1 -F.7 1' —.6.8.9 15,5 FIG. 3-1: CONTOUR PLOTF MAX ( 3M IN THE COMPLEX x-PLANE FOR THE CASE OF BiM = X/2 29

TH.E UNI'VE RS ITY OF M.I CH I GAN 2713-1 -F Suppose, in the first instance, that the N quantities are not all equal for the solution when all M are involved. Then we can make a small change in the A's in such a way as to decrease the largest of the N quantities while, if the change in the A's is small enough, none of the other M - N values will be larger than the largest of the N values. Thus the N values are equal. Next suppose that the N chosen quantities are larger than the next largest of the Mo. Then unless the N A's have the values they would have if the remaining M - N quantities were ignored, we can again make small changes in the A's which will reduce the N quantities uniformly while keeping any of the remaining M - N all smaller than the chosen No Thus we see in this case (when the N largest quantities are larger than the remaining M - N if the A's have been correctly chosen to solve the problem) that we need to consider only N of the M quantities in order to arrive at the correct solution. The question is, which N of the M quantities must we deal with. We will now answer this question. We assert that we must choose the N quantities which yield the largest value of 1 (note that the Gj depend on which N of the M quantities are chosen). IGj I To establish this assertion we show a contradiction in the converse assumption. Let us assume, then, that the solution is obtained for a choice which does not lead to the maximum value of 1. Now let us consider the N quantities which lead to the largest E Gj. value of 1 We know that not all of these N quantities can be made smaller than Gojl the largest 1 _ and thus the assumption that none of them are larger than one of IGjl the smaller 1 is false. This establishes the assertion. jGjj 30

THE UNIVERSITY OF MICHIGAN 2713-1-F We are thus led to the following method of attacking our original problem. First we start with N arbitrary points. We then vary these points so as to maximize the value of 1 We would then hope that the A's which correspond to the N points which maximize 1 would solve the problem. We can easily test to see if they do solve z | Gj I our problem by computing the field resulting from this choice of the A's. If the side-lobe level is nowhere greater than it is at the N chosen points we have solved our problem since the side-lobe level cannot be made any lower at the N points. However, if the side-lobe level is higher at some other point then we have not solved our problem. Now it turns out that in some cases the above method works, and in some cases it does not. We can get some insight into the question of how frequently we can expect the method to work by considering the following analogous problem (it is, in fact, a special case of our general problem but we will not bother to make the exact connection). Suppose that we have a number of points in the complex plane, Zn, and that we wish to find a point z such that Max j z - Znl is made as small as possible. A method of attacking this problem which is exactly analogous to the method described above is the following. Find the pair or points, Zn and zm, from the given set which maximize Izm - znl (this is analogous to maximizing 1 ). Then let z = (zm + zn)/2. This value of z will sometimes ZIGjI solve the problem, and sometimes not. It depends on whether the rest of the points in the set lie inside the circle having z as a center and Zm - znl /2 as a radius. In the figure below (Figure 3-2) we show some sets of points for which the method would work, and some for which it would not work. 31

THE UNIVERSITY OF MICHIGAN 2713-1-F Among the cases where this method does work is the design of a broadside linear array where the results of this method coincide with those of the Dolph-Chebychev method. We can certainly expect the method to work also in cases which do not differ too radically from this oneS as well as in some of the cases which do differ radically. It is expected that future study will yield methods which work in all caseso The method works for the The method does not work for any, following four sets of points, of the following four sets of points. FIGo 3-2~ THE PROBLEM OF MINIMIZING MAX z - Zn - EXAMPLES OF SETS OF POINTS FOR WHICH METHOD WORKS AND SETS OF POINTS FOR WHICH ME THOD DOES NOT WORK 32

TH E U N IVE R S I T Y OF M I C H I GA N 2713-1-F In order to give an idea of how this method goes we will now consider a very simple numerical example. We will take the case of a linear broadside array of four isotropic elements where the ratios of the currents in the elements is to be chosen so as to produce a low side-lobe level. This problem can also be treated by standard methods so that we have a standard of comparison. Let the inter-element spacing be X/2. We will consider the case where there are only four elements which we take to be located at x = y = z = X/4, 3 X/4. (3.12) The radiated field in the direction (0,.) is given by E = eikr F = A2 expik(r + - cos 0) + A 1 expik(r +cos ] (3.13) r r r +A1 exp ik(r - cos 0)] + A exp ik(r - 3 cos )1 r r Let us first of all consider a standard design of such an array. In the DolphChebychev design we would have A1 = A_1, A2 = A_, A1 and A2 realo In this case we get F =2A1 cos( r cos) + 2A2 cos( 3 os 0). (3.14) 2 2 Now the field broadside (9 = i/2) is Fo = 2(A1 + A2). There will be sidelobes symmetrically situated relative to 0 = wr/2 which are located by the condition c F = 0 or 3cos 0 - IrA1 sin( rcos ) - 3r A2 sin ( — cos ) = O. (3. 15) 33

THE UNIV E RS ITY OF M I CHIGAN 2713-1-F Making use of the relation sin 3x = sin x (3-4 sin2: x) we put (3. 15) in the form A1+ 9A2= 12A2 sin2 ( cos 0) (3. 16) A,+ 9A2 sin( " cos e)= 2 Using this value of (e) in (3. 14) we get 3/ 3A -Al.+3A2-A1 1. A+ 9A2 - (3A2-Ai) (3.7) ~F=2A, 22t- =_ E (3.17) F2A 12A2 3A2 3 f The value of 0 for which the main beam has decreased to the value given by (3. 17) is also of interest. It is given by 2A1 cos( cos e)+2A2 cos(t cos 0)) - (3.A18 A) or (3. 18) 3A,- A cos(E cos ) = IAt- 7 3A2 Figure 3-3 gives 0 as given by Equations (3. 18) and (3. 16) as a function of A,/A2 while Figure 3-4 gives the side-lobe level in db down from the main dobe as a function of A1/A2o The results given above from Equation (3, 14) on are essentially taken from Dolph's ",A Curreiit Distribution for Broadside Arrays which Optimizes thLe Relationship Between Beam Width and Side-Lobe Level". With these results available for comparison we are prepared to treat the same problem with the method discussed previously, 34

THE UNIVERSITY OF MICHIGAN 2713-1-F4 FIGO 3-3 90 End of Main -"-'~. Lobe 60 0 in degrees 30 Center of Side obe 0 1 2 3 A1/A2 FIG. 3-4 50 40 30 20 e in degrees 10 0 1 2 3 A1/A2 35

THE UN I VE RS ITY OF MICHIGAN 2713-1-FIf we take the main lobe to be constrained to 600 0 9 1 1200, then (3. 18) and (3. 17) show that the side lobes will be about 17 DB down from the main lobe. The relationship between the currents is A1 = 3 A2o 2 In our method if we were to take the Oi to be symmetrically placed about 9 = 90~0 we would obtain symmetric current distributions automaticallyo We will avoid this by taking the initial values of Oi to be el 0 02 = 600 03 = 1200 (3.19) 04 =120~+ where by 1200 + E we mean to take the limit as (- 0O Using (3.13) and (3.19) we find Gn Bn = AnFn A2 + A-1 +A1 +A2 (3.20) B1 = E AnFln = -i A_2 + iA_1 - iA1 +iA2 B AnF2n= +i A_2+ —i Al+ li A + i B3 = 7AnF3n li A +iAl++iAl-i A2 B4 = 7AnF4n — i A_2 + A1-i A1 + l+iA1 +-l+i A2 + 3 - + [3 (-1+i)A2 (-1-i)A_l + fff (-l+i)A1+ 3_' (-1. 4 F2" 4-4 4Jv2 42 Rather than solve for the A's in terms of the B's for the special values given by (33 19) we will find it more profitable to carry through the work for general 6i 36

THE:UNI VE RS I TY OF M I C H I GAN 2713-1-F In the general case 37ri,]ri exp X1. A-2 +exp X A_ + exp i Al +exp i A2=B ~L. Ae[.lA1+x{:- A exp x3 A- +exp X3 A-,+exP[-.ix3J AI+exp A2=B3 exp i p x34]A+exp Al+exp [- 3ri x3] A2=B3 where xi cos 8i. The solution to the set of Equations (3, 21) is exp [3 +xlA B1 A-2 = 2 (etrixl-e trix) (etix! _eJix3) (eixl -_eTix4) expw L L1 (exii + eoix3+eix4) ]B A - exp [A1L x1] (ei(%2+x3)+ e$(X+4)+e 7ri(x3+x4))B. (3.22) - — (eTiXerix2) (erierix3) (ei l_erix4) - exp[3 xi] eriX(%2+X3+ X4) + (e7ixleix2)(ex7rix TX3)(e ixl 7rix4 where the dots indicate similar terms involving B, B, B3, B4 37

THE UNI'VERSITY O'F MICHIGAN 2713-1-F From (3 20) and (3.22) we find = (1 eix2)(1 - elrix3)(1 - eix4) exp (3.23) (eiXl e ix,)(e ri- eX _ iX(erixl eix4) with similar expressions for G2, G3, G4 Since we take Bi = Gi we have Gi Gn Bn= ZIGn1 rx2 Irx3 TX4 sin — z- sin ___ sin2 sin sin_ sin 2 2 2 sn (X1 -X2) r(in (X3-x2) (X4-X2) (3.24)x) | sin.x1 sin 3 sin 2, 2. sin (x sin - s in +2 2 2 sin r(xl -X3) sin,(x2-x3) sin (x4-x3) - i 2 2 2 fX1 rx2 fx3 sin sin 2 sin 3 2 2 2 r(X - {(X sin ) sin ) sin T(x3-X4) 2 2 2 In order to restrict the beam in the desired way, we must keep x2 = 1/2, x3 = -1/2. Thus (3.24) becomes 38

THE UNI-VERSITY OF MICHIGAN 2713-1-F Tsin \ nXsin sins 4 cos rxs sin sin I s i cos r1sx n 2 2 4 2 4 (3.25) rx1 rx4 + sin sin — sin2 2+,'Xfinrx + iX | By sin I fxl + t) sin ( X4 + 4) | | cos ~rx4 sin x 4) What we now seek is a minimum of (3. 25) with respect to x1 and x4 (:ixl | x41 > 1/2). It is clear that it will be simpler to vary x4, then xl, then x4 again, etc. rather than computing the gradient of (3, 25) and varying xl and x4 along the gradient. Thus, instead of (3.19) we have initially = 1 x2 = 1/2 (3.26) x3 =- 1/2 where X4 is to be chosen to minimize (3. 25), which becomes due to (3. 26) ZI Gn =tan-4 + __ + 1 + (3 27) i n'cot 1 | cox4 1 + cot Aos cos rx4 2 2 2\ It is clear that (3. 27) is unchanged if we replace x4 by -X4o Thus we will take -1 4 X4 < -1/2. With this restriction we are able to remove the absolute value signs in (3.27) obtaining G I t-tan-4 + rx +..iU. 1 (3.28) 1 - cot. 1 + cot cos cosrx4 - 1_cos x4 + f cosXx4-1 _f_ 1 + cos x4 cos x4 cos x4 1 + cos x 39

THE UNIVERS ITY OF MICHIGAN 2713-1-F Figure 3-5 shows the variation of ljGnl with X4 for x1 = 1. The minimum occurs approximately at x4 = -0. 73. The value of I |GnI at the minimum is approximately 9.43 (19o 5 DB). Since the predicted side lobe level is appreciably below that actually obtainable, the true side lobe level is probably a few DB higher than the ultimate limit. Thus we 25 20 Z GnI 15 10 -1 -0.9 -0.8 -0.7, -0.6 -0.5 x FIG- 3-5 will postpone the calculation of the corresponding current distribution until we have gone through at least one more step. For this step we take 2 = 1/2 x = -1/2 (3.29) x4 = -0.73 and choose xt, so as to minimize (3.25). Using (3, 25) in (3 29) we obtain 40

TH E U N I VE RS I TY' OF M IC H I G-AN 27'13-1-F Irx1 cos rx sin (5. 70+ 1 2 sin Xl _ 450) sin 69 30 sin! sin 65, 7~ sin + + o 0 j sin( +450) sin 20. 70 |cos 48 6 sin ( + 65. 70) (3o 30).9114 _+ 689 r xi g < sin (- - +X 1 65.70)X in — - 450) irx Iax sin 1 sin sin(- l+ 450) sin (rxl + 65 7~) in2 S. 2 Figure 3-6 shows the variation of 1_Gnl with x1 in the allowable region xl 1/2 for x4 =.0o 73. The minimum occurs approximately at x1 =.775. The 25 20 15 10 -1 -~9 - 8 -~7 -~6 -.5 o5 o6 o7 o8 o9 1 X1 FI[Go 3-6

THE UNIVERSITY OF MICHIGAN 2713-1-F value of t 1IGn at the minimum is approximately 7.17 (17.1 DB). Since the predicted side lobe level is now only slightly below that actually obtainable, we can expect the true side lobe level to be only slightly above that actually obtainable. Thus we will compute the actual pattern with X1 =.775 X2 =.5 (3.31) x3 = -. 5 x4 =-.73 We have G1 = -1. 707 G= 1. 553 (3.32) G3 = 1. 896 G =-2.037 4 Since B we have B1 = B4 = -1, B2 = B3 =1. For simplicity in Gi computing the A's it is convenient to rewrite (3.22) in the form 42

THE UNIVERSITY OF MICHIGAN 2713-1-F. 2 2 2 A exp [zi (-l+x2 —x3 )+ exp r-1-x2+x3-x4 +exp[i(-1-x2-x3 +4 A-2 _ _ +'+=A.* 1 si -'''~8 sin __ X_ _ sin nr(xl-x3) sin 2 2 exp (-x +x3+x4)] +exp [2 (1+x2-x3+x4)]+ exp [i1 (l+x2+x3-x4J 8 sin 21-2 sin ar(x-x3) sin i(x1-xI~4) 2 2 2 A= r -exp [- (-+x2+x3 )] B1 2 8 sin x1x) sin X1-X3) in'r(X12 2 2 A --- - - a-:4' I + ooo. (3.33) 8 Sin. sin sin - 2 2 2 where the fact that A-_ = Aj* follows from the fact that the B's are all real. The general expression for F is F = A_2 exp x +A_ exp x + A1 exp 2 x]+ Aexp E 3i x] (3,34) Using the values of the A's given by (3. 33) we obtain sin r(x-X2) sin?(x-x3) sin -r(x 4 2 2 2 F= (x -x2 B1+. (3.35) sin sin sin 2 2 2 Figure 3-7 shows the pattern finally obtained after two steps. Figure 3-8 shows the optimum (Dolph-Chebychev) pattern. Figure 3-9 shows the pattern obtained by the above method after only the first step, Upon comparing the patterns of Figures 3-7 and 3-8 we see that the pattern obtained after two steps cannot be differentiated3 graphically, from the optimum patterno 43

THE UNIVERSITY OF MICHIGAN 2713-1-F In closing, a word is in order about the use of the above method when the fields due to the individual elements are only approximately known. In this case we go ahead with the design procedure as outlined above using the approximate patterns of the elements. We test the result experimentally. In general the side-lobes will not actually be as small as indicated by the design theory. We then use the array and its experimentally determined pattern as one element in an array, with the other elements corresponding to elements in the original array. We then apply the design procedure to these elements. Since we are now only making a small correction to the first design we can expect to get an array whose theoretically predicted pattern agrees well with the experimentally determined one. If necessary, the procedure may be repeated. 44

THE UNIVERSITY OF MICHIGAN 2713-1-F DB 20 10 x — ~~~~~~1.01.0 -5 -10 -15 FIG. 3-7: ILLUSTRATIVE EXAMPLE - PATTERN OBTAINED UPON THE COMPLETION OF TWO STEPS 45

THE UJ N:I VE R:S I T Y OF M ICH I GAN 2713-1-F DB 20 10 5 -1. 0 1.0 -5 FIG. 3-8 ILLUSTRATIVE EX-AMPLE - OPTIMUM (DOLPH-CHEBYCHEV) PATTERN 46

THE UNIVERSITY OF MICHIGAN 2713-1-F DB - 15 // \-10 -1. 0 II_ -15 FIG. 3-9: ILLUSTRATIVE EXAMPLE - PATTERN OBTAINED UPON THE COMPLETION OF THE FIRST STEP 47

THE UNIVERSITY OF MICHIGAN 2713-1-F IV A THEORY OF MAXIMUM GAIN ANTENNA ARRAYS To give a broad base to the method of design of surface antennas we now shall consider a general approach. We let mr(s) be a magnetic dipole distribution on a perfectly conducting surface S. The radiation pattern arising from such a configuration is given by R(k) - SH(k9s) (s) dS, (4.1) S where H can be taken as the field induced on S at s by an incident plane wave of direction k and unit amplitude. In briefer notation we put R = Ho m (4.2) The properties of this transformation will be our present concern. The antenna synthesis problem is that of determining the inverse, i. e. given a radiation pattern R we need to find -1 m = H ~ R. (4.3) This is very difficult for most surfaces S so we seek to characterize some general properties of the transformation. What we have accomplished is no more than some conjectures based on analogy with the better understood case of a line source. The principle properties we need to know are (1) how to maximize the gain and (2) how to control the side-lobe levelso We consider these for a line source. In this relatively simple example the radiation pattern is given by a R = 5i eiz f(5) d5 (4.4) -a 48

TH E UN I VE RS ITY OF M I CH I GAN 2713-1-F where the length of the source is 2a, f is the excitation and z = 2A cos 0 If we take a'> X we see that (4.4) approximates a Fourier transform. Hence the gain is maximized for f = constant and the side lobes are reduced for f approximating a Gaussian. By use of our knowledge of Fourier transforms we can characterize the transformation from the excitation f to the radiation pattern R in such a way as to enable us to modify the pattern without any detailed treatment of the inverse transformation. Ultimately we wish to find such a technique for our general surface S. Up to now we have considered only the first problems that of obtaining the maximum gain configuration. Our progress is yet only a conjecture. By analogy with the long line source we presume that for large surfaces, i. eo the characteristic dimensions much larger than a wave length, the maximum gain configuration results from an excitation which is just the complex conjugate of the field induced on the surface by a plane wave incident from the main beam direction. As an illustration we give the case of a circumferential slot on a large circular cylinder confining our attention to the principal plane. By reciprocity the radiation pattern at an angle 0 due to a source at 0' on the surface is given by the field induced at 0' by a plane wave incident ini the direction.o Let this be denoted by k(W, 0'); then, for an excitation f0) we have the radiation pattern R(0) = k( 0') f(')d 0' or (405) IEt) = K o f(0) 49

THE UNIVERSITY OF MICHIGAN 2713-1-F a The problem is to determine the excitation f such that the side-lobe levels meet the requirements of a particular application. i. e. if the main beam is in the direction 9 = O and the side lobes are required to be down A db, we have for the set { ~n' which are solutions of aR(0i) - 0 0 o, that 2 (4.6) min 10 log R() 1 ) A n R(in) In our example we can express (4. 6) in terms of the Fourier coefficients of the functions R, k, f, Rhn kn f (4.7) where we use the fact that = Zein(-t)kn So the inversion of the operator K results in f(0) = 7e Rn (4.8) 50

THE UNIVERSITY OF MICHIGAN 2713-1-F a relatively simple result. We are interested primarily in the more difficult cases in which S is not such a simple shape. We wish, however, to approach the more difficult problem in terms of the simpler one- so instead of solving (4. 8) we will look for a method of choosing f which can be used in the more general cases. In our example, if we choose the main beam to be in the direction 0 = O we know that we control the side-lobe levels by tapering the excitation, i. e. f(p) should decrease as | increases in order to decrease the side lobes as compared, say, with those resulting from a uniform excitation, f = const. A readily available tapered excitation is just the complex conjugate of the field induced by a plane wave f(o) = k (0, 0) (4.9) where we take the complex conjugate in order that the phase be such that the radiation pattern be maximized for this case. This gives R(0) = K o K'o (0) a (4.10) where i (0) is the Dirac delta. This has been computed (Fig. 4-1) in the case ka = 10 where a is the radius of the cylinder and k = 27r/X, X is the wavelength. This gives a side-lobe level of approximately 18 dbo. If such an excitation were to be used the origin of it suggests a novel means of producing this excitation. Since the excitation is, in fact, the complex conjugate of the field induced on the surface by a plane wave, we can then devise an analog device in which we model the antenna surface, illuminate it by a plane wave, and use the measured surface 51

THE UNIVERS ITY OF MI C H. IGAN 2713-1-F FIG, 4-1 RADIATION PATTERN FOR A CIRCULAR CYLINDER WITH A CIRCUMFERENTIAL SLOT EXCITATION (ka 10) 0 in degrees 0 10 20 30 40 50 Relative p.e of R(0)I___ __ 52

THE UNIVERSITY OF MICHIGAN 2713-1-Ffield after an appropriate phase change of 1800 as the input of the excitation on the actual antenna surface. Alternatively, the measurements could be made in the laboratory and recorded on tape to be used in some sort of digital computer to control the excitation. Since the above scheme uses an approximation to a continuous distribution, we must solve two additional problems. The first is to discover the density of slot radiators necessary for a sufficiently close approximations the second is the method of exciting these elementary radiators. It may be that the second problem is too difficult- i. eo if we need control over each of the elements the scheme may not be useful for the type of antenna we finally wish to construct. To this point little effort has been spent on this problem. 53

THE U NI VE RS IT Y OF MICHIGAN 2713-1-F APPENDIX A THE 65-SLOT PROBLEM - THE METHOD OF COMPUTATION AND THE RESULTS OBTAINED A. 1 The Method of Computation In this Appendix we present the formulas employed in the 65-slot problem together with a discussion of the method of computation and the results obtained. The single slot expressions needed for such calculations have been discussed in Section II (these single slot expressions also appear as Equations (4. 2.4), (4. 2. 5). and (4. 2. 11) for Geometric Optics, Fock Theory, and Physical Optics respectively in Reference 16. In the expressions for the 65-slot calculations given in this Appendix the polarization, p, will denote directions of the magnetic field with Pi = 0 and P2 = The following parameter values are fixed throughout: OB = (main beam direction) = 700, 500, 300. 100 00 = 1650, denotes cone surface do = (distance from tip of cone to first slot = 16X, dj = (distance from tip of cone to jth slot), j = 0, 1, 2,.., 64 d = dj+1 - dj = (distance between slots) = 0O 4X. In performing the calculations for the 65-slot problem, each of the 65 values of H determined according to the methods of Geometric Optics, Physical Optics, or Fdbk Theory (corresponding to j = 0, 1, 2,.., 64) are multiplied by the corresponding phase factor, exp(ikdj ( B - eo) ), and the resulting 65 expressions summed. ~54

TH.E UNIVE R S I T Y OF M I C H I GAN 2713-1-F We now turn our attention to the formulas employed in the 65-slot computations; the derivations of these expressions have been discussed in Section IL A. 1. 1. The Geometric Optics Formulas In determining the magnitudes of the Geometric Optics contributions in the 65-slot problem we employed the following formulas. For a single slot HGO -ikdoD ^ GO = 2 0ikde D (cos 0 sin + cos ikr r where d = distance from tip to slot, 0 D = cos 0 oos 0o + sin 0 sin: 00 cos and 2C = ikVo L/2r, with Vo = the voltage across the slot and L = the length of the slot. In the general case of N slots with dj = distance from tip to thejth slot N GO = 2 (cos 0 sin p1 + cos P2) exp rikd. (-D + cos( - ) )7 C e j=1 r This becomes, for 65 slots, HGO sin (65(04x) ) -57. 6 rix - Ikr = 2 (coS 0 sin 0 p1 + cos ~ P2) e5 C e sin (0.4rx) r where x = D - cos (0B- 0)o 55

THE UNIVERSITY OIF MICHIGAN 2713-1-F A, 1.2. The Physical Optics Formulas In determining the magnitudes of the Physical Optics contributions in the 65-slot problem we employed for the entire set of 65 slots kH = 2iT Lfx sin e cos sin -C sin0 +sin + os + sin cos cos -f cos ocos 0 sin ] [ sin 0 +fz cos 00 cos P 64 where T = X 2 (i/df) exp ikdj + ikdj cos (%B - 0)] j =0 dj = do +jd 2 f =sin Cos x 2 csin x - 2 q sin x (q + b cos x) +2 bA fy = sin 2 b sin x c 2os Csin x cos 0x + 2 cA ( (q+bcosx)2-c2sin2x) Q fz =sin2 c0 -2 2sinxcos x - 2 b sinx(q+b cos X) +2 qA ( (q + b cos X)2 _ C2 sin2 x ) Q and A =Q-3/2 arctan Q1/2 sin x /(qcos x + b)] Q = sq2 b2 2 q = cos i0 (Cos x0 + cos O) + In the one slot case the expression is the same except that the factor iT/4i2 is replaced by the factor ikeikdo with do being the distance from the tip to the slot. 4r2do 56

T HE UNIVERSITY OF MICHIGAN 2713-1-F b = sin 80 (cos 0 sin 800 + sin ) c = sin0sin20o, and x = arccos tan o)A The values of T corresponding to the four values of OB studied are T 0. 11193 +0.34491 i for B = 10, = 0.0273 +0.112 i for 8B 30~ = 0.0211 +0.0209 i for B =500, and = 0.0276 +0.0265 i for B =700~ o The C appearing in the above expression is defined in the same manner as in the Geometric Optics expressions. A. 1. 3 Fock Theory Expressions To determine the contribution for each of the 65 slots as determined by Fock Theory we use the following expressions: H P eikr r ean e {cig( i) sin (2n - ) - [ s ) sno + ) sin 0 r/2 - n=0 + (ic2/mn) A.n) cos [ (2n + 0 - s) sin oo +/2 - n-l + (ic2/mn') f( %n') cos [(2r - ) sin Oo + /Z -Y | 57

THE UNI'VE R S ITY OF MICHIGAN 2713-1-F where cos =s = (tan o0)/(tan 0). cos o = - cos eos %s 0 > 0s; k = Zr/X, An(an,) = - kan (sin 00 sin 0 cos Os +'cos 0o cos 0) + kdj sin 1Znw + - s)sin ao-] sin YJ An' =An (an' -) an() di Cos [(2ni + 0 - s) sin 00 + r/2 - ] sin U) an' an (-) 9 dj = distance from tip to jth slot, * = sums cut off when argument of the cosine reaches ar/2, kdj39tan 03 1/3 mn (an) 2= i tan Zan2 sin2.) mn' = mn an) 9 gn(ann ) = kan Sin }/A (2nr + 0 - Os) sin 0 -n' = yn (an', -0) and 2 C = ikVoL/27r where Vo= the voltage across the slot and L is the length of the slot. A A For p = P2 C - sin 0/sin c2 = cos60 sin5s, and for p = Pi 0 cl - cos 00 sin s c2 = - sin o0/sin 0e The expressions f( ) and g( ) are the Fock functions which are tabulated (Ref. 17). In the conical cut computations, described more fully in A. 3, p was taken in 5~ intervals from i = ~s to 0 = 175~ and the corresponding 8 value was determined from the following 58

THE UNIVERSITY OF MICHIGAN 2713-1-F cos (o -0B) c cos, 0o + sin 800 cos, /cos2 0o + sin2 0 cos20 -cosZ (0o -8] cos2 0o + sin2 0o COS 2 In the limiting case of 0 = Os we have (on the conical cut) cos Os= sin 00 cos (00 - ea -cos2 00 cos 2 (00 - B) Upon multiplying each of the single slot expressions by the corresponding phase factor exp (ikdj (0B - 00) ) and summing, we obtain an expression of the form N / F (j) e j j =o Approximating the function F(j) by F(O) (F(N) \/N we obtain \F(O)/ -N1 ia (N+1) N+i L F(j) e'j ~ F.0) (i 1 -e iF (N)/F(O0) I~j 1 - eia - F(N)/F(0)] N This approximate summation method was employed for all four cases (i. e. for B= 100~ 30~0 500, and 700); the actual sum was obtained only for the 0B = 70~ case. A comparison of the sums obtained in the OB = 70~ case indicated a maximum difference of about 20 per cent between the actual sum and the result obtained using the approximate summation method described above. The nature of this approximate summation process is such as to lead one to expect the greatest accuracy in the middle of the interval between the shadow boundary and 0 = 1800o 59

THE UNIVERSITY OF MICHIGAN 2713-1 -F A. 2 Beam Direction The relative maximum in 0 for a given 0 and main beam direction eB is given, in the illuminated region, by the cone sin B sin e cos +cos B cos = cos (o - B) about the slot array as axis9 while in the shadow region9 the locus of the relative maximum departs from the conical surface and is given parametrically by 0B +a = -(- Sina B - O-S) sina cos = cos cos a cos = -_ tana and tan O a a-Oo Plots of the beam direction for B = 100,300 500, 700 are included here in Figure A.2-1, showing 9 as a function of | o!4 Although the locus of the relative maximum in 9 is continued until it reaches the cone surface, O = 165Q, the field is dominated by that arising at the other shadow boundary when 0 > 1800. The relative maximum in 0 coincides with the conical cut out to about 0 ~ 1350 this is also shown in Figure Ao 2-1. 60

THE UNIVERSITY OF MICHIGAN 2713-1-F FIG. A. 2-1 THE RELATIVE MAXIMUM IN 0 FOR A GIVEN 0 AND MAIN BEAM DIRECTION 0B6 0 j 1500L / / / 0 (0 165 - cone surface) I / / (OB=700) 120 Relative Maxim / in 0 positions A - j, -3 t A _ _ _ _ _ _ _ 7 50 o A _ _ _ _ _ _ _ _ _ _ _ _ _ _ (OB= 300) A (e~-= 1000 OB 100) 600 300 { Conical'Cut pos itions 00 900 1800 2700 3600 61

TH E UNI VE RS ITY OF MICHIGAN 2713-1-F A. 3 Results of the 65-Slot Computations As pointed out at the beginning of this Appendix and in Section I computations were performed for four different beam directions, OB = 10~ 300, 50~, and 70~, for the 65 slots located on a generator of a cone of half-angle 15~0 For each beam angle computations were performed on two "cuts"' (1) a conical cut for which the receiver is restricted to lie on a conical surface whose axis is the extension of the generator containing the slots and which has a half-angle equal to 150 + eB9 and (2) a plane cut for which the receiver is restricted to lie in the plane of the generator containing the slots and the axis of the 150 cone. These two geometries are displayed in Figure 1-1 of Section Io Values of IHI 2e and I H0I2 were computed at appropriate intervals in. 0 (or e) using Geominetric'Optics, Physical Optics, and Fock Theory in their regions of applicability. The space regions of the applicability of these three methods is shown graphically in Figure 1-6 in Section Io The results were obtained in db below the Geometric Optics maximum and are tabulated in Tables Ao 3. 1 through A. 3o4. Only the Geometric Optics and Fock Theory results are given in these tables it was found that the Physical Optics results were.negligible. * Graphical presentations of these Physical Optics results are given in * As will be noted by anl examination of Tables A. 3. 1 through A. 3.4, only values less than 60 were recorded; that is, a value which was more than 60 db down from the Geometric Optics maximum was not included in the final summary. 62

T:H E UN I VE RS ITY OF MICHIGAN 2713-1-F Figure A. 3-i for the conical cut cases and a brief examination of the data contained in Figure A.o 3-1 indicates that the Physical Optics contribution is negligible in comparison with the "Fock Theory" contributions in the shadow region. In the lit region where the Geometric Optics results were used, only the relative maxima were computed except in the vicinity of the main beam. At the time of the writing of this report experimental results were available for the two beam directions eB = 10~ and eB = 30o~ The theoretical results of Tables A. 3o 1 and A. 3,2 are compared with the experimental data in Figures 1-2 through 1-5 in Section IL The theoretical data is superimposed upon copies of the actual patterns obtained from the Hughes Aircraft Company. For the plane cuts the experimental data was not normalized and thus the peak value (which would be 0 db) appears on the experimental curves as approximately -4. 5 db. Thus, in making the comparisons between theory and experiment, the theoretical data has been normalized to the experimental peak value; iL e. each theoretical value, x, is plotted in Figures 1-2 and 1-3 as the quantity, -(x + 4. 5). The slight angular variation between the theoretical and experimental patterns which can be observed in Figures 1-2 and 1-3 is due to a slight' missalignment in the experimental 0 tests (in fact, the main beam direction in the 30~-experimental-test was actually 29'); no attempt has been made to normalize angularly the results in the plane cut. On the "conical cut", this normalization is partially accomplished since the comparisonn between theory and experiment show in Figures 1- and 1-5 is actually a comparison between theoretical values for receiver positions on the conical cut while the experimental values are for "relative maxima in e" positions. Examination of Figure Ao 2-1 63

TH E UNI VE RS ITY OF M ICHIGAN 2713-1-F indicates that such a comparison loses validity only near 0 = 1800. Also, since the summation technique employed to obtain the Fock results was expected to be least accurate near = 1800, we expect the greatest deviation between theory and experiment in Figures 1-4 and 1-5 to occur near = 180~0 * This program of experimentation and theoretical calculation was set up primarily as a feasibility study and thus one would not expect "extreme" precision in the results. We feel that the excellent agreement between theory and experiment displayed in the comparisons of Section I definitely establishes that radiation patterns for slot arrays on conical surfaces can be theoretically determined. During the final editing stage, the experimental data for eB=500 became available. These results are compared with the theoretical estimates in Figures A. 3-2 and A. 3-3. For this case in addition to the slight angular misalignment, it was found necessary to locate the slots in a manner slightly different from that used in the eB —600 and e =30 cases; that is, the slots were tilted a few degrees from the transverse position. TUis slight tilt in the positioning of the slots could be expected to lead to larger differences between theory and experiment due to the polarization effects and thus the relatively poor agreement between theory and experiment for the H~ case shown in Figure A. 3-3 is not unexpectedo 64

THE UNIVERSITY OF MICHIGAN 2713-1-F TABLE A, 3 1 Results Obtained for 6B= 100 Conical Cut Plane Cut 0 0 IHI * l* 00 0 0 10o >60 0 0 150 >60; 33, 3 30 10, 5 6e 2 1o 2 0 135 >60 34, 3 60 14 1o5 6 0 0 120 >60 35 4 90 20 3 o 6 >60 0 105, 60 36, 1 100 23, 1 9 15, 2 0 90 7 60 36o 2 105 24, 6 1o 1 1 1'7 i 0 75 >60 35 4 110 26,1 1.5 9,3 0 60 >60 33. 5 15 2707 19 7,6 0 45,60 30o. 0.o, I 29.2 5.8 9.,6 0 28 ~,60 22.5 125 30, 7 6,.6 11.1 0 24 ~60 20, 8 130 32.2 7,.8 12, 6 0 22.5,60 > 60 135 33. 5 9. 1 14.3 0 21 7 60 17,9 140 34.8 10,7 16, 2 0 19 760 >60 145 36 12,5 18e 3 0 17 >60 13,5 150 37 13,9 20,2 0 15 ~60 >60 155 37, 9 15. 6 22, 0 12,5 60 4.0 160 38.6 17.8 24,4 0 10 >60 0 165 39. 2 21,4 28~ 2 0 7 - 60 4.0 170 39, 7 30.3 37. 1 0 4 >60 >60 175 39,9 28, 7 35. 8 0 0 > 60 13 5 180 40 32,3 39,1 1 180 8 > 60 >60 180 16 ~60 17,9 180 20 22, 2 20, 9 180 30 34.9 39.0 180 40 32, 3 39,1:180 50 23. 9 32. 6 180 60 47. 5 57 1 180 70 55. 1 >60' 180 80 59o2 >60 * Measured in db down from the Geometrics Optics Maximum *S Geometric Optics result only 65

THE UNIVERS 7T3Y OF MICHIGAN TABLE A. 3. 2 Results Obtained for OB= 300 nia i Plane Cut O I BO (Ag~a~l Hdla 1 0o0 I 0 * 0 0 30 > 60 0 0 150 >60 * ~ 35 30 31L4 7.4 1 2 0 135 >60 35.6 60 35.8 3.1 6,0 0 120 >60 36.1 90 42 9 2Z 7 ~60 0 105 > 60 36.3 100 45.7 3. 2 15.2 0 90 > 60 35. 6 105 47 6.2 14.4 0 75 >60 34.1 110 48 4 7,9 15.6 0 60 >60 30.7 115 49. 8 9.2 17 0 43 >60 22.5 120 51. 1 10o 6 18.6 0 40 >60 20.8 125 52, 4 12 22 20.4 0 38.5 >60 > 60 130 53. 6 13.9 22 0 37 >60 17 9 135 54. 7 15.9 24.5 0 36 >60 >60 140 55. 8 18. 3 27.6 0 35 > 60 13 5 145 56.7 21.4 30.4 0 33 >60 >60 150 57. 6 25. 8 35.1 0 31.5 >60 4 155 58, 3 32.2 41,7 0 - 30 >60 0 160 58, 9 41. 1 50.1 0 28. 5 > 60 4 165 59.4 37.4 46.9 0 27 >60 >60 170 59. 7 38. 1 47. 8 0 25 > 60 13 5 175 59.9 >60 >60 0 23 >60 >60 180 60 40.9 50.7 0 21 >60 17.9 0 19.5 >60 >60 0 18 >60 20,9 0 0 >60 26 180 15 >60 27,3 180 20 32.3 31,0 180 30 38.4 42,7 180 40 41.8 48.6 180 50 40.9 49,6 180 60 40.9 50,7 180 70 33.5 44.2 180 80 55,4 >60'D Measured in db down from the Geometric Optics Maximum s Geometric Optics result only 66

THE UNIVERSITY OF MICHIGAN 2713-1-F TABLE A. 3. 3 Results Obtained for 0= 50~ Conical Cut Plane Cut 00 00 o IH I 012 0~ I Hel2 * H2 0 50 >60 0 0 150 >60 36.1 30 51o 6 11.4 1,2 0 135 >60 36. 3 60 56. 6 6.4 6 0 120 >60 36.1 90 64 7.2 >60 0 105 >60 35. 5 100 66, 7 12, 2 18. 7 0 90 > 60 33.9 105 68.1 13 1 198 0 75 >60 30. 3 110 69, 4 14, 4 21, 6 0 61 >60 22.5 115 70.7 16.1 23 6 0 58 >60 20.8 120 71. 9 17.8 250 7 0 57 >60 > 60 125 73.1 19 9 28,1 0 56 >60 17 9 130 74, 2 22. 1 31.6 0 55 >60 > 60 135 75. 3 27.4 36,5 0 54 >60 13.5 140 76.2 28.5 38.7 0 53 >60 >60 145 77.1 35.6 39.1 0 51.5 >60 4 150 77.,8 46.4 55, 3 0 50 260 0 155 78. 5 > 60 > 60 0 49 >60 4 160 79 >60 >60 0 48 >60 >60 165 79.5 52,4 > 60 0 46 >60 13,5 170 79. 8 52 >60 0 45 760 >60 175 79.,9 58, 5 >60 0 44 760 17,9 180 80 51, 3 > 60 0 42.5 >60 >60 0 41 >,60 20,8 0 30 760 26 9 0 15 760 30. 7 0 0'60 32.1 180 15 760 32,7 180 20 54,8 58.8 180 30 52,3 46.5 180 40 46,6 53,3 180 50 56,7 >60 180 60 51,4 >60 180 70 52.9 60 180 80 51.3 > 60 s Measured in db down from the Geometric Optics Maximum ~ ~ Geometric Optics result only 67

THE UNIVERS ITY OF MI CH I GAN 2713-1-F TABLE A, 3o 4 Results Obtained for OB= 700 Conical Cut Plane Cut 00 0 1 |- 0 | IHl | IH 0 2* 1 —4 H 1 0 70 160 0 0 150 - >60' 36 30 71,9 16.2 1,2 O 135 > 60 35, 6 60 77,2.14, 4 6 0 120 >60 34, 7 90 84. 8 20, 9 >60 0 105 >60 32, 8 95 86. 2 24,7 19, 7 0 90 >60 28.6 100 87. 5 26,8 22, 6 0 80.5 > 60 22 5 105 88. 8 28.8 25 8 0 78 >60 20.8 110 90ol 30, 7 29 0 77 >60 > 60 115 91, 3 32,6 32,5 0 76 >60 17, 9 120 92, 5 34.5 36,2 0 74, 5 >60 | >60 125 93, 6 36, 5 39. 9 0 73 i >60 13 5 130 94, 7 39, 3 44, 6 0 72 >60.>60 [ 135 95o 6 43,4 50, 2 0 71 >60 4 140 96.5 51,3 58,4 0 70 >60 0 145 97, 3 58 5 760 0 69 >60 4 150 198 53,1 >60 0 68 > 60 >60 155.98, 6 58.9 >60 0 67 >60 13,5 I 160 99 1 >60 760 0 66 >60 60 J 165 99, 5 >60 60 0 65 > 60 17 9 170 99. 8 >60 >60 0 64 > 60 ~ 60 1 175 99. 9 >60 60 0 62. 5 60 20, 8 0 45 > 60 30. 2 0 30 > 60 33, 2 180 15 > 60 35, 5 180 20 423 41'/:| i I I a 180 30 44,8 49, 180 1 40,49,3 56, 2 180 i 50 53,7 60 180 i 60 56,3,60 180 70 58,3 4 60': Measl'ed in db down from the Geom 1etric Optics Maximum * * Geometric Optics res-At only 68

:THE UNIVERS ITY OF MICHIGAN 2713-1-F FIG. A. 3-W1PHYSICAL OPTICS RESULTS OBTAINED FOR THE CONICAL CUTS eGO(max)1 IN DB db - -40 -60 100) (H;eB= 10~) - -80 0.:(H0;B=300) 699 (HO;eB70o) AO 1100 1,00 130~ 1700 (,0B=500) 69

THE UNIVERSITY OF MICHIGAN 2713-1-F x -10db 12 r_ Exp. Abscissae of min-& a Ter (theoretical) l Thery ] He2 2 Exp -20db j l H ax 2 (<-40db) Theory -30db FIG. A. 3-2: COMPARISON BETWEEN THEORY AND EXPERIMENT FOR A PLANE CUT (0B 50~0 0 = 0~0) 70

FIG. A. 3-3- COMPARISON BETWEEN THEORY AND EXPERIMENT FOR A CONICAL CUT (eB=500) (see footnote on page 64) Power level in db -10 I I I I I 1 -| cll | Hma~xl 2 6 Theory )2 / 4 0 80 100 120 140 160 180 0 in degrees in degrees

THE UNIVERSITY OF MICHIGAN 2713-1-F APPENDIX B A FURTHER ANALYSIS OF TIP SCATTERING It will be recalled that in Section 2.4 we obtained an expression for the field scattered by the tip of a cone. It was noted at that time that the approximation which was employed gave rise in some cases to divergent integrals. In this Appendix we shall examine these cases more closely. We start with the Kirchhoff expression for the scattered magnetic field A-, ),eikR H (n x H) x -- d S, (B. 1) 4r RS where n is the unit outward normal R is the distance from the integration point (x", y" z") to the field point (x',y', z') k. 2,r S is the entire surface areas and time dependence e -iwt has been assumed. Our problem concerns a plane wave incident on a conical surface. The incident field at any point (x',y',z') is given by H A'< H -ik. rr' Hi pH0 e where p is the unit vector in the polarization directions r is a unit vector toward transmitters r' position vector for field point. 72

THE UN I VE R-S ITY -OF M I C H I GAN 2713-1 -F Points on the surface will be denoted by r". (, will denote unit vectors while - will denote vectors of arbitrary magnitude). As before, we employ the Physical Optics approximation, i. e. (1) the magnetic field on the surface is taken to be H = 2 H e-ikro r Pt where Pt is the tangential component of p and (2) S is the illuminated area. Thus, we may rewrite Bo 1, remembering that n x p n x Pt as follows -H nH0 x(p) x ) R_ ei d S (B. 2) illuminated area ikR = ikR ikR R However, + ke F+ Rike ikR +ike R ikR'kRl R R and (nxp) x (r' r") - r - n) p r'-'- ^ hence, (B. 2) becomes HO (W W-r"f)' n~ (a ri-r)pn / e'ikR _ikR _ik;i ~PHs= Ho j r"-) ) p) ike ie ikS. s 2_ r — R2 R illuminated area (B. 3) 73

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TH E UNI VE RS ITY OF MI CH I GAN 2713-1-F Now we rewrite(B. 5)using (B. 61 kikr';iA 2 21r" 2L(' r2 = Ho ike, _ntp(r'-'p) n exp [ik(-r"fr'+r)+ r -(r.r))J dS illuminated area (B.7) EquatiQn(Bo 5) gave rise, in Section II, to the integral 1f f n e-ikr"e r' +) r"sin d r"d " illuminated area which proved troublesome for values of 0" such that r"~ (r' + r) = 0. It is true that, -4 A A as written, f diverges if there is a 0" in the range of integration for which r"(r' +r) = 0. If at the same point (x", y"if Z") we also have n x (r' + r) = O, then of course, we have a specular reflection situation and we would use Geometric Optics to describe the scattered field. However, let us assume that there is such a p"9 say pi', where r" (r' + r) = 0 but n x (r' + r) A 0 and examine the following three casesD using 0. 7)instead of(B. 5) (1) r". r' = 0 (2) r", r'= r" cosy where 0 < cos2) 2 1 (3) r "r' + r In the following we will treat B.o 7) as consisting of two contributions H == + illuminated S area where S2 contains 0o' and S1 + S2 = illuminated areao We will concern ourselves only with f since f presents no problem. S2 si 75

THE UNIVERSITY OF MICHIGAN 2713-1-F CASE I In the neighborhood of 0o' we have r"{r' +r) = 0 and r".r' = 0 hence we may write(B. 7)as (B. 8) Hs Hoikeikr'A00 Hs + Hoike' n) p-(r' n exp (ikL~ r" sinadr" d " 2 r r' 2r' Si 00 - 0 The r" integration can be performed by an Abelian limit argument as follows lim exp ri2 r =k e x p {(JI )rf2 j r" dr " 0 0 lim exp [( ik -_ )r"2] L (2 (i) 2r' ik Thus (Bo. 8 becomes Applying the mean value theorem, we obtain ( Heikr' oA A A A HS + __, esina I(r' n) p (r p n d S S S Applwhere can be ean valuated by theoreme meod of Section I. Si 76

THE UNIVERSITY OF MICHIGAN 2713-1-F CASE II ^ A In the neighborhood of 0" we have r"-(r' + r) = 0 and r"r' = r" cos 0 where cos2 is bounded away from 0 and 1. In this case, we may write (B. 7) as ( Hi ikr' Af 2 2 H= j + o_ sin a (r' n)p - (r' p)n exp I sin r r" dr" d 27rr' ) L' 2r') -S1 - 0 (B. 9) Since sin2 f is bounded away from zero, we proceed exactly as in Case I obtaining Hs= + 0 sin (r n)p - (r' p)nJ =, i HIr Sin2 mn Si for n =n(fl 0 6 o Thus again we see that the contribution from the neighborhood containing 00 vanishes with E and Hs SI CASE III A A In the neighborhood of p" we have r"( r' + r) = 0 and r" r' = + r". Thus 0 either (a), r" r' r" and r = -r" or (b), r".r =-r" and r". r = r". In (a) we have the receiver on a cone generator and the transmitter on the extension of that generator while in (b) the transmitter is on the generator and the receiver is on the extension. These are the transition cases which we cannot handle by these techniques and must exclude from our analysis~ 77

THE UNIVERSITY OF MICHIGAN 2Z713-1-F APPENDIX C IDEAL CURRENT DISTRIBUTION FOR A CONICAL ANTENNA It was pointed out in Section I that it was theoretically possible to duplicate a pattern produced by one type of antenna with another antenna of equal or greater aperture. It was pointed out there that the problems involved in attempting to instrument the current distribution required appeared to be of such complexity that a serious attempt at instrumentation would not be warranted since this is not an optimal array. In this Appendix we consider this problem of duplicating a given pattern. Specifically, for illustration, we consider the pattern produced by an ideal parabolic dish and calculate the current distribution which, if activated on a cone surface in any manner whatsoever, would make the cone yield a pattern identical with the dish pattern. Consider a conical surface behind which a dish antenna is pivoted about the point 0. By application of Huygens Principle the field at a point P outside a surface S can be obtained from knowledge of the tangential field produced by the source at the surface. In this case the source is a dish and the surface a cone. The field of a dish is approximated by a plane wave; the tangential component on the cone surface will be studied. An i, j, k coordinate system is considered to be affixed to the cone as in Figure C-1o A plane wave is considered to be produced from the dish with polarizations as follows in the two cases to be considered hereo 78

THE UNIVERSITY OF MICHIGAN 2713-1-F A k A r cos a / / / sin Sin/ FIG. C-1 79

THE UNIVERSITY OF MICHIGAN 2713-1-F =-~ ^ k^ ik1 r E1 = (icos,- ksin3) e (C. 1) - ik. r E2 = j:e (C. 2) in which the eiwt time dependence is assumed. E1 lies in the y-plane and E2 is.J to the y-plane. In each case the beam is considered to take the orientations 3= 0~, 20~40~ 60~, 800 In order to graph the amplitude, phase and direction of the field, the cone is unrolled and forms a two-dimensional cone of apex angle = 2 7r sin a. The tangential compolnent of the field on the illuminated part of the cone is respectively.,\ -+ ik-r^ nxEI= e L(-cosasin,3sin0)+j(sina co s,3+cosasin3cos0) 3) +k(-cos a cos sin 0)] nxE= eik. r -sina,)+k(cos a cos 0) (C. 4) and,in spherical coordinates on the cone surface, nxEl== eik rT(cos (3 sin 0)+ 0(sina cos cos sincos a) (C, 5) nxE2=e r(-c os 0)+(sinasin 0) (C. 6) The amplitudes and directions of the field on the cone surface are E,= /cos2 3 sin2 0 +(sin a cos1 Bcos 0 + sin cos a) (C.7) E2 Jcos2 +sin2 a sin20 (C. 8) tan 01 sin a cos + tan cosa (C. 9) sin 0 tan8 sin a sin (C. 10) n2~= -cos ( 80

TH'E UNI VE R.S ITY OF MICHIGAN 2713-1 -F A FIGo C0-2 Since propagation takes place in y = const. planes, the phase in each case is given by k or = kzcos + kxsin, (C. 11) We are.interested in the phase over the illuminated region of the cone's surface. From the figure we see that z = h - r cos a and x = r sin a cos p. Therefore, the phase angle is given by 2r [sin a - r cos a cos B + r sin a sin f cos 0 (C. 12) To find the minimum distance between two points along a cone generator at which the radiation is in phase we equate-the phase angle above to zero and 2r, we solve for r (obtaining ro and r2,), and then subtract the values of r obtaining the distance. = ro-r 2ir 3 (cm.) (C. 13) (cos a cos f - sin a sin, cos ~) 81

TH E UNIVERSITY'O F MICHIGAN 2713-1-F Equations for the direction lines The equations for the direction lines on the cone surface may be obtained in the following manner: Consider a 5, 0 coordinate system on which the unrolled cone is placed as shown in Figure C-3. Then dn tan (0 + 1) d7 n = r sin 01 dn = sin1 =dnr cos1 5 rcos o - cos 0' -= r sin " dr do1 then sin 0 dr + r cos 01 d 1 = tan ( + 1) = tan + tan cos dr - r sin'd 0l 1I- tan 0 tan - plane FIG. C-3 82

THE U N I VE-RS ITY OF M I C H I GAN 2713-1-F (sin p dr + r cos 01 do1) (1 - tan 9 tan O)-(cos 0t dr - r sin 01 do') (tan 9 + tan 01) = 0 - tan 0 dr + rdo1 = 0 but dpl = sin a do dr = sin-a d 0 (C.14) r tan 8 For the two cases considered we have r r = 0 r = integration const (C.15) (cos tan I tan a 82: r = r CsC 0 r = integration const. (C. 16) The constant ro is a parameter which takes on all values on a generator of the cone. Due to the discontinuity of the trigonometric functions, the integrations must be performed piecewise resulting in a sign change in the ro at the points of discontinuity. Determination of illuminated region We will now determine 9 = (r); the envelope of the illuminated region on the cone: We will first find the maximum and minimum intercept points of the beam on the cone. It can be seen from the geometry that the cylinder generators in the y = 0 plane intersect the cone at rmax and rmin 83

THE, UNIVERSITY OF MICHIGAN 2713-1-F On 0i = 0 the range of r is determined by rma.,X = a ( sin _ sin (.17) mmn sin a sin(a, + ) s is determined as a function of r by solving the equation of the intersection of the cone and cylinder. In a spherical coordinate system passing through the cone apex, the intersection equation is r2 sin a sin2 (1 - cos p)2 + 2r sin a sin r cos (a + A) - h cos: (1 - cos 0) - r2 sin2 (a + 2) + 2rh sin 3 sin (a + f) + h2 (sin2 a - sin2 3) = O whose solution is (C. 18) cos 0 = (r cos a - h) cos +. r - h cos (C.19) r sin a sin B the change of sign would occur at r = h cos a which is the point of maximum possible 0 In order to obtain the boundary of the illuminated region on the cone we use expression (C. 19) cos r cos a os h cos -h+os+r - -hcosa r sin a sin f having already obtained the range of r. For 0max the phase is zero since the dish touches the cone at that point. Knowing the phase along the boundary of the illuminated region, we can, for given 0, keep laying off the distance X along the generators starting at a boundary point until we attain some pre-assigned phase. The phase on the boundary is obtained from (Co 12). For laying off successive curves we subtract 84

THE UNIVERSITY -OF MICHIGAN 2713-1 -F Phase given minm in terms of 2Z(e. g.. 2r) / = perio Line of constann zero phase. ~ min + V(l) as a function of 0 FIG. C-4 FIG. C -5 F F' A.0 B FIG. C-6: The cone in the y = plane 85

THE UNIVERSITY OF MICHIGAN 2713-1-F some multiple of 27r from the line of constant zero phase where,=,-(I0) is used for the distance equivalent to 27r. Discussion of Resultso In order to see, physically, what occurs, we will consider a sphere of radius "all equal to the radius of the dish to be inserted into the cone as shown in Figure C-6. The points of contact will constitute a circle whose two-dimensional projection is BC. The dish is focused at 0. and has a point of contact C when in the position DC and two points of contact F, F' when in the position EF. The graphs in Figures C-8 through C-12 show the variation in amplitude of the tangential component of El over the unrolled cone surface. The field along the 0=0 generator of the cone would be small (for / small) increasing to a maximum near -900 and becoming small again for 1800. As /3 increases toward 900 the amplitude will increase on the 0=00, 1800 surfaces and remain fairly constant on the surface near ~90. Therefore, for large / the amplitude is fairly constant with 0. The graphs in Figures C-13 through C-17 show that the amplitude variation of the tangential component of E2 is independent of 3 which is clear from Figure C-6. Figures C-18 through C022 illustrate lines of constant phase within the illuminated region of the cone. The fact that these lines are ellipses for /3=0 to 800, parabolas for /= 800~ and hyperbolas for 3= 900, can be seen by observation of Figure C-7. 86

THE UNIVERSITY OF MICHIGAN 2713-1-F Parabola I / Hyperbola I "'Elipse!I 0o<f3 800 P= 800 I=90o FIG. C-7 87

THE. UNIVERSITY OF MICHIGAN 2713-1-F The direction of tangential El is shown in Figures C-23 through C-27. It may be noted that, along the 0=-O cone generator, as E1 points more and more in the dn(0=0) direction, the direction lines take a more acute change in direction. The direction lines would be reversed on either side of 0=0 if the case 13=100 had been calculated. In the case of 13=800, the fact that the direction lines go along lines of constant r near 0-0~ is seen from the fact that El vector points along the cone near 0=0; the dip near 900 occurs since El is not tangent near 90~. The direction of tangential E2 is shown in Figure C-28; only in the case of 13=0~ is shown sincen x E2 is independent of /3, The discussion of this appendix is illustrated through the application of the following parameter values, a = 100 a = 20" X = 3cm. The results are presented in tabular form in Tables C. 1 through Co 3. 88

THE U NI VE R S ITY OF M I C H I G AN 2713-1-F TABLE C. 10 /EI (a = 10~) Amplitude 0 0 o o o 3= 00 3=20 13=40 - 60 3=80 00 0.17363 0o 500 0 1766 0,940 1. 000 1.000 100 0.:524 0. 776 0. 942 1O000 0. 985 20~ 0O 37893 0O 586 0. 802 0. 950 1. 000 0. 942 30~ 0. 670 0. 841 0. 961 1.000 0. 870 40~ 0. 65641 0O 760 0. 885 0. 974 0. 999 0. 774 50 0.845 0.928 0. 986 0. 998 0.656 60 0o 87037 0. 915 0. 964 0. 995 0. 996 0,522 700 0o 966 0. 989 1. 000 0.994 0. 379 800 0O 98527 0.995 1. 000 0. 998 0o 990 0. 244 900 0. 998 0o 994 0O 989 0. 985 0. 174 100~ 0. 98527 0. 975 0. 970 0, 972 0, 980 0. 244 110~ 0. 927 0. 929 0. 948 0. 973 0. 379 120~ 0.87037 1 0.853 0. 872 0. 918 0. 967 0. 522 130~ 0 756 0O 803 0 0 884 0. 960 0. 656 140~ 0O 65641 0. 640 0. 724 1 0O 850 0. 953 0O 774 150~- 0. 870 160~ 0O37893 values not needed 0. 942 170~- 0. 985 1800~ 0. 17363 1. 000 89

TABLE C. 2:~ Boundary of Illuminated Region and Phase on Boundary: = 20~ Period for the r in cm. 0 Phase given 0, in cm, 98.9 0 63.26066 (2r) 3.464 100 160~ 33' 62.805 (2) 3.454 125 580 2' 54.488 (2r) 3. 354 150 760 50' 46.151 (2r) 3.289 175 890 6' 37.813 (27r) 3.245 200 -980 15' 29.475 (2r) 3.212 225 1050~ 30' 21. 138 (2r) 3. 187 250 1110 29' 12.8 (2r) 3. 167 275 1160~ 32' 4.463 (2r) 3.1515 288. 38 1180 58' = ~ max 0 300 370~ 48' 3. 875 (2r) 39415 302.9 0~: 400 Period for the r in cm. 0 Phase given 0, in cm, 179. 9 0 36. 3488 (2r) 4. 6672 200 490 37' 29.475 (2r) 4.479 225 640~ 38' 21. 138 (2r) 4.246 250 800~ 23' 12.8 (27r) 4. 077 275 920 46' 4.463 (2r) 4.005 288. 38 1020~ 08' = j max 0 300 620 12' 3.875 (2r) 4.291 313 00 90

THE UNIVERSITY OF M I C H I GAN 2713-1 -F TABLE C.2: Boundary of Illuminated Region and Phase on Boundary: (Continued) = 60 2 Period for the r in cm. Phase given 04 in cm. 216.4 0 24.'207 (20 ) 8. 74 225 340 53' 21.138 (2r) 8. 129 250 660 22' 12.8 (2ar) 6. 9405 275 860 42' 4.463 (2wr) 6.202 288.38 950 50' = ~ max 0 (2r) 300 730 21' 3.875 (21r) 6.676 325 0~ 12.212 (27) - =800~, Period for the r in cm. p Phase given p, in cm. 238.2 0~ ka = 16.95 (2a) 250 440 54' 12. 8 (27) 60.105 275 780 17' 4.463 (2w) 22. 018 288.38 910 47' (0 max) 0 300 760 23' 30" 3.875 (27) 22.94 325 410 56' 12. 21 (27r) 68. 09 340.2 00 91

THE UNIVERSITY OF MICHIGAN 2713-1-F TABLE Co 3, Direction of -ta = 100) 0 01 1 -I _0 ~ | = 200:=40~ B = 60 = 800 02 00 9 9090~ 90~ 90~ 90 0 10~ 71, 8~ 80 1~ 84~ 7 88 3~ 178 2~ 20 ~ 25~ 5 56~ 7~ 70, 9 79, 6~ 86 60 176o4~ 300 4550 562~ 90 74~ 90 85~ 00 174, 30 40~ 11, 70 3740 4~ 56 20 70, 70 83~ 60 171o 70 50~ 31, 5~ 50, 8~ 67 1~ 82, 30 168, 3~ 600 5, 70 27~ 20 46~ 5~ 64, 20 81.30 161o 30 700 24 00 43, 30 62 00 80~ 5~ 154, 50 80~ 1 8~ 21.50 41 00 60.40 80 10 135, 5 90~ 19, 70 39 6~ 59, 60 79,80 900 1000 - 1 8~ 18 40 39Q 00 59~ 6~ 79~ 90 44~ 50 1100 17o 7~ 39,2~ 60, 3~ 80~ 30 25~ 50 1200 - 1,80 17o40 40, 5~ 61, 90 81o0~ 16'.7o 1300~ 11 7~ 140~ -11 7~ 8.30 150~ values not needed 5o 70 160~ -25 5~ 3.6~ 1700 1o 80 1800 -90~ 0~ 92

THE UNIVERSITY OF MICHIGAN 2713-1-F i25050 180 -160 93 20 m Omdegrees 93

THE UNIVERSITY OF MICHIGAN 2713-1-F 50 15 20 0 in degrees FIG. C-9 A:'~.PLi'UDE OF El ON UNROLLED CONE SURFACE FOR =, 20~ 94

THE UNIVERSITY OF MICHIGAN 2713-1-F 35 11 1 -0 25?, 95 350 -180 180 -100 -20 0 20 0 in degrees 95

THE UNIVERSITY OF MICHIGAN 2713-1-F 50 150 200 4 -0 - 40 0 FIG. C-ll AMPLITUDE OF E. ON UNROLLED CONE SURFAC.~R FOR' -.:0c 96

THE UNIVERSITY OF MICHIGAN 2713-1-F 50 20 25,,= 4 0 in degrees FIG. C-12 AMPLITUDE OF El ON UNROLLED CONE SURFACE FOR 3= 80~ 97

THE UNIVERSITY OF MICHIGAN 2713-1-F 50 _9155 150 300 350 0 in degrees FIG. C-13 AMPLITUDE OF E2 ON UNROLLED CONE 980 98

U~i VE R. Sl T yOF lC'27'13.1_- ~. MGAAN 50 10 150 200 250 300/' 35^ -i8 — 1` "1 ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~2 -'"' I. "'C~ c-i, ~ ~ la degre,: o'o 60~ PIGC-14AMPLITUDE.j degre'2 120''~2~ S"BCEB -`200 -VLE CN t~.e i~,ters. ~~ e,,~ enedt the~.efo~.o~yte ect~n Chazgeswe mith 1~. ) n h envelope of 99

THE UNI VE RS ITY O F MICHIGAN 2713-1-F 50 10 150 / J 200 250 i.! J 350 / / - / ~-f~a~I~jjll II!!1!1 Illllllfillllil lW111~1\1~\~ \ IUUU~yZY~St~ ) 180 0 in degrees FIG. C-15 AMPLITUDE OF E2 ON UNROLLED CONE SURFACE FOR 400 the intersection changes withi ) 100

THE UNIVERS ITY OF MICHIGAN 2713-1-F 5O 100 20O 25O ~~ 7"Uuru lell~~tl0-1 180 -16 00 -12i 120 -1 100 -40' 20 20 0 0 in degrees FIG. C-16 AMPLITUDE OF E2 ON UNROLLED CONE SURFACE FOR 3 = 600 (IE2t is not -dependent therefore only the envelope of the intersection changea.s with, ) 101

THE UNIVERSITY OF MICHIGAN 2713-1.-F 50 ~Qo/ /'I 1 10 i degrees 300SURFACE FOR 80 ~-180~- / / ( E is not,i3dependent therefore only the envelope of 10102 i mn degrees FIG. C-17 AMPLITUDE OF E2 ON UNROLLED CONE SURFACE FOR: 80~ 102

THE UNIVERSITY OF MICHIGAN 2713-1-F 60 ~ —--- ri-Pthase. in degrees 40 18FIG 18 PSE OF E ON UNROLLED ONE SURFACE 103 FIG Ct- -18 -IPHMXASOFZ -%1 EW ON%'KTTXTl% UNOLEDT' T CONETSURFACEln

THE UNIVERSITY OF MICHIGAN 2713-1-F 50 60(25) r 63. 261(21) -62. 805(4)) 1~~~~~~~~~~~~~~~8 $~~OQ2a 50( 254. 488(20r) 46~ 151(2w) 30(2-r) / 1\2~ ~G~I I ~d/Y~ ~6~1~5~ \ \ ~37.813(2z,) 20(27r) 29. 475(2r) 20 0,. 13803 () FIG. Cz~-1 PHASE OF E ON UNROLLED CONE. 8(Fr) 0% 25 4. 463(2 wr) 300 0 3511 180 -12 20 -10, ~ O0 -$Q 80 -'.D 3,875(27 ina degrees FIG. C-19 PHASE OF E ON UNROLLE D CONE SURFACE FOR P:- 20~ 104

TH E, UN'I V:Ei"RS I TY OF M I-C H I'GAN 2713-1-F 50 150.36o 349.(27r) r" 29~ 475(;~) 200 iog~1) o 138(21r iz. 8(z~r) 25.463(27r) 350. -asb~ H~~~~~~~~~~~~~~~~~~~80 -11 8 -40'n-"0 ZO 40 9Om- degrgaees FIG. C~-20 PHASE OF E ON UN~ROLLEaD CONEE SURFACE VAI)R = An$g </~~~~~~~0

THE UNIVERSITY O.OF MICHIGAN - a 13-1-F 50 -8 26140-oo 40 0 80 3875(27r) 10(27120 in degrees 12 212(2) ~~~~~FOR:I3= ~ ~ ~~ 6063(2 106 l 3.875(2r) 16 0.0 in degrees 12. 212(27a) FIG. C-21 PHASE OF E ON UNROLLED CONE SURFACE FOR/3 = 60~ 106

THE UNIVE.RSITY OF MICHIGAN 2713-1-F 503 21 2(r) r 0 —v"11(zm-) II 55\ I 200a 1 / i16. 95(2f) lit 12o 8(z7) 25 o463(2wr) /~~~~~~~~~~~~~~~~~~~~ 30 875(z~) 35O -180. 180 / I I I 1 It~~~~~~~e ~ \~~~1 -82~~~~~~~~~~ 60. ~-!o 140 -101 -{' 2 "~I, 8 0 60 12o 212(.:r).0 -20 0 20 % Bi ~. in degrees "' FgG.'C-22 PHASE OF E ON UJNROLLED CONE SURFACE FOR P =- 80~ 107

THE U N I V:ERSITY OF MICHIGAN 2713-1-F 50 10108 200 25 80 0 in degrees FIG. C-23 DIRECTION OF nxx El ON UNROLLED CONE SURFACE FOR f = 0o 108

TH E UNIVER SI TY OF MICHIG A N 2713-1- F 50 30 950- -6 -4 -2 0 o 4 6 802 0 OB 109 g rvrA%~ #86, - %00

THE UNIVERSITY OF MICHIGAN 2713-1-F 50.150~~ ~110,0 40 -280 0 i m' degrees SU.RFAC E FOR P ~- 40~ 110

THE UNIVERSITY OF MICHIGAN 2713-1-F 50 10 r 150 200 250 300 -8~0 -60~ -~40 -20 0 206080~180 -180 SURFACE FOR = 60060 / I I I ~111~t~ $ ~140 -1280 -100 100 -80 -6 80 -40 40 -20 0 0 Om' degrees FIG. C-26 DIREQna OF wx El ON UNROLLED CONE SURFACE FOR?' 600 ~111

THE UNIVERSITY OF MICHIGAN 2713-1-F 50 100 150 350 -8SURFACE FOR j3= 800~f 18 112 /~~~~~~~~1

THE UNI VE RS ITY OF MI CHI GAN 2713i-4-F 50 10 150 20 300 ~350 180 -180 1130 -a~raa~ / I I i 1 1240 100 20 0in degrees F'GI C-2V IRECTION OF aa x E2 oN' UNROLLED CONE 113

THE UNIVERSITY OF MICHIGAN 2713-1-F REFERENCES 1o, V. A. Fock, Journal of Physics, X, 399(1946). 2 N. A. Logan,, "The Role of Fock Functions in the Theory of Diffraction by Convex Surfaces", Air Force Cambridge Research Center, Paper presented at the URSI Meeting, May 22 - 25 1957, in Washington, D. C. 3. J. B. Keller, "Diffraction by a Convex Cylinder", Transactions of the Institute of Radio Engineers, Vol. AP-4, 312(1956). 4. R. F. Goodrich, "Studies in Radar Cross Sections XXVI - Fock Theory Applied to an Infinite Cone", The University of Michigan, Report No. 2591-3-T (January 1958)o 5. P. E, Mayes and W, D. James, "Pattern Synthesis With Small Radiating Slots in a Prescribed Conducting Surface", The University of Illinois, Prepared under Subcontract with The University of Michigan, Purchase Order 154216 (1957). 6. L, B. Felsen, "Field Solutions for a Class of Corrugated Wedge and Cone Surfaces", Polytechnic Institute of Brooklyn, Microwave Research Institute, Memorandum No. 32 (19 July 1957). 7. CO No Campopiano, "Summary of Asymptotic Expansions for Bessel Functions", Polytechnic Institute of Brooklyn, Microwave Research Institute, Report No, R-582-57, PIB-502 (28 May 1957). 80 L, B, Felsen, "Radiation Patterns of Leaky and Surface Wave Distributions on a Wedge", Polytechnic Institute of Brooklyn, Microwave Research Institute, Memorandum No, 33 (23 September 1957). 9, L, B. Felsen, "Radiation of Sound From a Vibrating Wedge", Polytechnic Institute of Brooklyn, Microwave Research Institute, Repo-t:No, -:R-613-52, PIB- 541-..(1:1 Octbber 1957). 10. C, N. Campopiano and L. B. Felsen,, "Radiation Patterns of Two-Dimensional Leaky and Surface Wave Distributions on a SemiInbfinite Cone", Polytechnic Institute of Brooklyn, Microwave Research Institute, Memorandum No. 35 (18 November 1957), 11, K. M. Siegel, H. A, Alperin,: R. R. Bonkowski, J. WO Crispin, Jr., A, L. Maffett, Co E. Schensted and I. V. Schensted, "Studies in Radar Cross Sections VIII - Theoretical Cross Sections as a Function of Separation Angle Between Transmitter and Receiver at Small Wavelengths", The University of Michigan, Report No, UMM-115 (October 1953). 114

THE UNIVERSITY OF MICHIGAN 2713-1-F REFERENCES (cont' d) 12. Wolfgang Grobner and Nikolaus Hofreiter, "Integraltafel, Erster Teil, Unbestimmte Integrale", Springer-Verlag (1949). 13. K. M. Siegel, H. A. Alperin, J. Wo CrispinJr.,, H. E. Hunter, R. E, Kleinman, W. C. Orthwein and C. E, Schensted, "Studies in Radar Cross Sections IV - Comparison Between Theory and Experiment of the Cross Section of a Cone", The University of Michigan. Report No. UMM-92 (February 1953). 14. L, B. Felsen, "Plane-Wave Scattering by Small-Angle Cones", Transactions of the Institute of Radio Engineers Professional Groujp on Antennas and Propagation, Vol. AP-5 (January 1957). 15, R. F. Goodrich, A. L. Maffett N. E, Reitlinger, C. E, Schensted and Ko M. Siegel, "Studies in Radar Cross Sections XXII - Elementary Slot Radiators", The University of Michigan, Report No. 2472-13-T (November 1956). 16, R. F. Goodrich, R. E. Kleinman, A, L, Maffett, N. E. Reitlinger, C. E. Schensted and K, M, Siegel, "Radiation and Scattering from Simple Shapes II", Presented at Congres International Circuits et Antennas Hyperfrequences, Paris, France, 21 - 28 October 1957. 17. Tables of Fock Functions, Air Force Cambridge Research Center, Antenna Laboratory (to be published). 115

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