%0<$'%&° Br&aoa, ffl. SYNTHESIS OF APERTURE ANTENNAS COLLEGE OF EN< AF33(616)-310 R-112-110-SR-6F2 Technical Report No. 1 3 re :€'-'".' of EL- JU«y»EI IJ ilTY OF ILL 157*1 tm~s y .>- 1301 we r spr ; ; URBAI I iiittOiS 61361 USA ANTENNA SECTION ELECTRICAL ENGINEERING RESEARCH LABORATORY ENGINEERING EXPERIMENT STATION UNIVERSITY OF ILLINOIS URBANA, ILLINOIS If it SYNTHESIS OF APERTURE ANTENNAS Contract No. AF33(616)-310 RDO No. R-112-110 SR-6f2 TECHNICAL REPORT NO. 1 Date: October 1954 Prepared by: C.T.A. Johnk Research Associate Approved by: j R.H. DuHamel Research Assistant Professor E.C. Jorda Professo antenna section electrical engineering research laboratory Engineering experiment station university of illinois urbana. illinois TABLE OF CONTENTS Page 1. Introduction 1 2. Methods in the Synthesis of Aperture Antennas 3 2.1 Experimental Approach to Aperture Synthesis 3 2.2 Analytical Approach to Aperture Synthesis 4 2.2.1 Synthesis by Geometrical Optics 4 2.2.2 Indirect, or Analytical Methods 6 2.3 Direct Synthesis Methods 17 2.3.1 Finite Summation Method for Rectangular Apertures 17 2.3.2 Polynomial Formulation of Woodward's Method 21 2.3.3 Synthesis Methods for Circular Apertures 24 3. Tchebyscheff Polynomial Synthesis of Rectangular Apertures 25 3.1 Equal Side Lobe Formulation of a Pattern Polynomial 25 3.2 The Aperture Distribution 31 3.3 Narrow-Beam, Tapered -Side-Lobe Designs; Supergain Effects 32 3.4 Comparison of Low Side Lobe Designs 37 3.5 The Optimum Side Lobe Problem 38 3.6 Comparison of Tapered Side Lobe and Optimum Side Lobe Designs 44 4. Circular Aperture Synthesis 46 4.1 Circular Aperture Synthesis by Integral Approximation 46 4.2 Synthesis by Means of an Orthonormal Set 50 4.2.1 An Orthogonal Set of Pattern Functions 53 4.2.2 Approximation of the Integral Equation Solution 58 4.2.3 Determination of the Orthonormal System Coefficients 61 4.2.4 Properties of the Orthonormal Set 63 4.2.5 Properties of the Aperture Distributions Corresponding to G n (u) 65 4.2.6 Synthesis Examples for Circular Apertures 65 4.2.7 The Low Side Lobe Problem for Circular Apertures 84 li > TABLE OF CONTENTS (Continued) Page 4.2,8 Summary of Design Method 85 5. Conclusion 87 Appendix I The Qj^(w) Function and Tabulation 90 Appendix II Ordered Zeros and Relative Maxima of T n (x) 91 Appendix III Circularly Symmetric Aperture Distributions and their Corresponding Space Factors 92 Appendix IV Evaluation of Coefficients Related to Development of Orthonormal Set [G] 93 Appendix V Coefficients b„i, of the Orthonormal Functions G n (u) nk 97 Appendix VI Orthonormal G (u) Expressed as Series 98 Appendix VII Values of G n (u)//n 100 Appendix VIII Values of F n (r) 101 References 102 in Digitized by the Internet Archive in 2013 http://archive.org/details/synthesisofapert01john ILLUSTRATIONS Figure Number -Page 1. Rectangular Aperture 8 2 Ranges of Functions Related to the Aperture Problem 8 3- Circular Aperture Coordinate Systems 14 4. Two Conditions of Linear Phase Shift and Constant Amplitude Across a Rectangular Aperture 18 5. Typical Cosecant Pattern Obtained by Woodward's Method 20 6. Function T„(x) Plotted for Six Orders , 27 n 7. Even Pattern Polynomial Pjj(w) 28 g Generation of a Rroadside Pattern Polynomial Pm(w) from T n (x) 3 for the Case N = 2n = 10. 30 9. Comparison of Tapered Side Lobe Designs Rased on P 2 o(w)» for Two Aperture Widths 34 10. Reamwidth and First Side Lobe Level Related to Choice of Polynomial Degree and AP erture Width 36 11. Aperture Distributions Corresponding to Tapered- Side-Lobe Designs 39 12. Functions Related to Optimum Problem for Apertures 40 13. "Ideal" Unity Side Lobe Pattern 41 14. A Discontinuous Space Factor and its Aperture Function 47 15. Result of Confining Aperture Distribution to Finite Aperture Size 48 16. Orthonormal Functions G^Cu) 66 17. Aperture Distributions F n (r) Corresponding to G n (u) 67 18. Sector Pattern 68 19. Approximations to Sector Patterns 70 20. Triangular Pattern 71 21. Approximations to Triangular Patterns 72 22. Approximations to (sin u)/u Pattern 75 IV ' - ILLUSTRATIONS (Continued ) Figure Number fage 23. Curves of g(u)G n (u) Products, for a Gaussian Space Factor 77 24. Approximation to Gaussian Patterns 78 25. Approximations to Displaced Sector and Gaussian Patterns 7 9 26. Secant Pattern Space Factor 80 27. Approximations to Secant Patterns 82 28. Approximation to an Arbitrary Curve 83 PART I 1 . INTRODUCTION This report is concerned with the problem of determining what fields over a plane aperture are required to produce a given radiation pattern. Such a problem is termed the "synthesis of an aperture antenna". It is the converse of the antenna analysis problem, which is concerned with finding the distant fields produced by given current distributions or their equivalents The problem of synthesis is of utmost importance in antenna design work because the usual design problem involves a pattern which is specified beforehand Solutions are available for the radia- tion fields of known source distributions having a quite general char- acter 22,43 But because of the complexity with which the source dis- tribution function is contained in the radiation field integrals, the solution for the most general synthesis problem has not been obtained. Synthesis problems concerned with the design of field sources are usually attacked by constraining the source currents to lines, surfaces, or volumes which have relatively simple geometries Then from known fundamental field solutions which apply to either discrete or to differ- ential source elements, and since electromagnetic field effects can be linearly superimposed, it is sometimes possible to combine such field solutions so as to cause convergence to a prescribed radiation field function The required source distribution for producing that radiation field might then be determined from a superposition of source elements which follows a law of combination related to the combination of the field solutions- Appropriate solutions for providing the desired answer or approximation to that answer may take the forms of series, integrals, or possibly some combination of both Success in formulating an antenna synthesis method depends to a considerable extent on the ingenuity with which the geometry of the source is chosen An antenna can be assumed to be either an array of discrete sources or a continuous distribution of sources. We are concerned here with continuous distributions which are considered to exist over closed, plane surfaces commonly termed "aperture antennas" or just "apertures" An aperture surface may be characterized by other than a plane, e g , a spherical surface or some portion of the surface of a sphere might for certain problems be a better choice The aperture synthesis problems considered in this report are confined to plane > apertures having rectangular and circular boundaries. The field dis- tributions assumed for these aperture types are given certain symmetries so as to reduce the complexity of the integrals relating the distant fields to the aperture distributions. Also, the aperture fields are assumed to be entirely transverse, which reduces the expressions for the far zone retarded fields to the familiar Kirchhof f -Huygens diffrac- tion integrals. 39 These restrictions permit, as is shown later, relat- ing the far zone field to the aperture field in terms of integral transforms. The factors which are usually of importance in a preassigned far zone pattern function may be: (a) the shape of the principal lobe (e.g., the beamwidth); (b) the amplitudes of the side lobes adjacent to the principal lobe; and (c) the degree of supergaining inherent in the re- sulting design. By "degree of supergaining" is meant the relative amount of reactive or circulating energy that an aperture distribution must possess in order that pattern characteristics (a) or (b) above be maintained within assigned limits. High degrees of supergaining are usually associated with narrow beamwidth of the principal lobe, with very low side lobe levels, or both, and are discussed in detail in connection with the specific synthesis results of this report. In Section 2 is discussed some of the background work that has been done in connection with the aperture synthesis problem. Section 3 considers the problem of synthesizing tapered side lobe, broadside beam patterns for rectangular apertures. A method for synthesizing circularly symmetric patterns of otherwise arbitrary shape for circular apertures is developed in Section 4. 2 > 2. METHODS IN THE SYNTHESIS OF APERTURE ANTENNAS Not until recently was it realized that an aperture antenna which is uniformly illuminated is not the one which provides maximum gain. In fact, it has been shown that there is no theoretical upper limit to the gain obtainable from an antenna. 510 For pencil-beam antennas, it is usually desirable to minimize the side lobe level at the same time that high gain is achieved. In radar applications, for example, spurious reflections from off-target objects are minimized only if the side lobes are kept to the lowest practicable value. A better insight into the enhancement of the beamwidth and the side lobe conditions is obtained through antenna synthesis methods. The synthesis of the far zone field patterns of aperture antennas can be handled by several methods which provide approximations to a given pattern within varying amounts. Methods which have been applied in the past can generally be divided into two classes: (1) the exper- imental approach, and (2) the analytical approach. These will be dis- cussed in the following paragraphs. 2.1 Experimental Approach to Aperture Synthesis The experimental approach to aperture synthesis consists essential- ly of selecting a desired pattern from a collection of available, ex- perimentally determined patterns. In the event that a minor modi- fication of such patterns is desired, various intuitive schemes may be employed to effect the desired modification. For example, if an angular displacement of the principal beam of a paraboloidal antenna is desired, this displacement may be secured by shifting the primary feed position with respect to the focal point. In this example, the quasi-optical properties of a reflecting surface are intuitively employed. In many cases of purely experimental approach to the pattern problem, the pro- cedures will reduce to cut-and-try processes, primarily because of the large number of parameters which are usually at the command of the observer. It must be agreed that there is a point of departure at which analytical results of investigations of antenna patterns and their relations to aperture parameters become a necessary adjunct to the investigator's art. -3 2.2 Analytical Approach to Aperture Synthesis The analytical approach to the aperture synthesis problem can be divided into the following three broad groups: (1) A method depending on the laws of geometrical optics which apply approximately to the aperture problem under consideration. (2) An indirect method, involving first the substitution of classes of aperture field functions into an appropriate integral equation which relates the far zone field pattern to the aperture distribution. The resulting class of field patterns might, then, for some cases be superimposed in certain ways so as to provide an approximation to a desired field pattern. (3) A direct method, which depends on satisfying directly the appropriate integral equation mentioned in (2). The first method has found specific application in the design of curved reflectors or lens antennas which serve to direct or disperse the power derived from an appropriate primary source having a known field pattern. The second and third methods are useful in specifically de- signing the aperture field distribution without specifying how such a distribution might be obtained practically. In the following is dis- cussed some of the background of these analytical methods, together with some of their limitations. 2.2.1 Synthesis by Geometrical Optics Early applications of the principles of geometrical optics were investigated in this country and abroad during World War II, and were primarily concerned with the focusing effects of parallel-plate lenses on microwave beams. Results of the developments during that period were not published until 1946 or later. 25,36,44 Aberration problems connect- ed with spherical reflectors also found early treatment by means of optical principles. 2 A microwave scanning problem, that of preserving the shape of an electrically tilted pencil beam over a relatively wide angle, was provided with a solution by means of a thick dielectric lens. 1 ' The formulation of the Luneberg lens offered interesting variations to the solution of the scanning problem. 16 • 3 *■ > 32 > 3 5 A ray- tracing method, described originally in terms of vector notation by Silberstein 38 with a view to its application in optical design work, was extended by Kelleher 24 for use in the design of microwave reflectors. 1 The work in polar molecules of Debye 13 was given a macroscopic interpre- tation by Kock in his development of a dielectric lens based on a large scale model of molecular structure. Lens and spherical reflector combinations which had been designed originally for use at optical fre- quencies and which found application at microwave frequencies are the Mangin reflector 19 and its more elegant wide -aperture refinement, the Schmidt reflector. 7 The major part of the development work which has been done in the study of aperture antennas from the geometrical optics viewpoint has been done on the basis of existing optical designs whose performance at optical frequencies had been well established and for which equiv- alent wavefront results could be obtained at microwave frequencies with perhaps additional experimental considerations . A geometrical optics design method having sufficient generality to provide a technique for synthesizing a far zone field pattern was given by Chu 9 in connection with designs of cylindrical reflectors fed by line sources for producing cosecant-squared patterns useful in airborne navigational systems. This method, using a metallic reflector and a line source having a known pattern, permits synthesizing a prescribed field pattern by applying simple laws of geometrical optics and the concept of the conservation of radiated energy. Dunbar later extended this method to include reflecting surfaces having two planes of curvature, 15 and this in turn was modified by Marston 29 to account for the flaring of the primary source fields for such reflectors. The particular hypotheses of geometrical optics upon which the synthesis of a given pattern is based, when using the method proposed by Chu, are essentially: 1. Straight line propagation in a homogeneous medium. 2. Independence of the "rays", or solid-angle segments, of pro- pagated power. 3. The law of reflection at a perfectly conducting surface, as applied to the rays which constitute the primary pattern of the line source. These hypotheses, which are customarily applied to optical problems involving dimensions much greater than the wavelength of the radiation, neglect diffraction effects entirely > Certain limitations of such a geometrical optics approach to the synthesis of a far zone pattern will necessarily exist. Diffraction phenomena, which the ray approach ignores, cause the appearance of side lobes outside the given range of the specified pattern and permit only an approximation to the idealized pattern within that range. A minimi- zation of these effects is largely a matter of the experience of the designer. Another disadvantage of this method is that it may utilize aperture area uneconomically; that is, there frequently exist designs which provide approximations to the same pattern with less aperture height. The method has an advantage, however, in that it is a complete design method, for it specifies how the aperture field distribution is to be produced physically, i.e., by means of the curved reflector whose shape is determined by the design. Other analytical synthesis methods do not possess such an advantage; they lead to an aperture field dis- tribution without giving any information as to how the distribution may be established physically. But some of the other analytical methods to be discussed do provide better control over side lobe and beamwidth conditions by particularly considering the diffraction effects estab- lished by the properties of the aperture field itself. The synthesis method of Chu has been extended to the case of doubly curved reflecting sheets, whereby the pattern is focused to approach an idealized uniform phase front wave in one plane and controlled to approximate a desired pattern in the other plane. Such methods are readily extended to the design of shaped lenses. The principal differ- ence between reflector and lens designs lies in the replacement of the law of reflection, in the geometric optics of the problem, with an equivalent Snell's law of refraction appropriate to the kind of lens structure chosen for the design. 2.2.2 Indirect, or Analytical Methods In the following paragraphs are discussed relationships between certain aperture field distributions and their corresponding far-zone field patterns which, while not providing explicit solutions to the general antenna synthesis problem under consideration, do, on the other hand, provide possibilities of synthesis techniques from the basic de- signs which are made available. The discussion is restricted to the far zone, or Fraunhofer region, of arbitrary apertures for which the familiar simplifications regarding the ratio of transverse electric and magnetic field components is permissible. The relation of the field pattern to an arbitrary field distribution over an aperture of finite size can be conveniently carried out using the Kirchhoff method of integration 30 Whereas the field integrals pertaining to this method were derived originally by Kirchhoff for the scalar wave equation, they also apply to any aperture field consisting only of transverse electric and magnetic field components (i.e., a plane-wave field), or to any generalized aperture field which can be expressed as the superposition of a finite or infinite number of plane waves of the same frequency and polarization but traveling in different directions. Solutions of this latter type which are shown to lead to a Fourier transform pair as applied to this aperture-pattern problem, are indicated from a generalized approach by Stratton. Woodward and Lawson, following the technique of superposing an infinite number of plane waves over a rectangular plane aperture to provide a solution for the distant field pattern, also pointed out substantiation of the earlier results of Bouwkamp and de Bruijn. Ramsay has also discussed solutions from the Fourier transform pair viewpoint in considerable detail and lists many examples. 2.2.2.1 Analysis : Rectangular Apertures Since the Fourier transform approach is significant to the aperture pattern synthesis problem, a discussion is given here of some of its ramifications. It is recalled from the definition given in Section 1 that by "aperture antenna" is meant that surface or area defined by a closed curve, which is a part of the surface of an arbitrary closed volume from which radiated power is emanating, such that all the sig- nificant fields on the surface of that volume are restricted essentially within the closed curve or aperture. 43 An aperture antenna may have any simple geometric shape, but for the purpose of discussion from the Fourier transform viewpoint, a rectangular aperture of length " a" as shown in Fig. 1, having transverse fields varying in only the x direction, is considered here. Its pattern in the xz plane is the one here of interest, so its height is not important. The distant field pattern may be determined in several different ways. The technique of ; FIGURE I. RECTANGULAR APERTURE, -I A(u) I U invisible L^ Visible , Range-v J ... ^ invisible -l (a) Range, aperture distribution. (b) Visible and invisible ranges of space factor. FIGURE 2. RANGES OF FUNCTIONS RELATED TO THE APERTURE PROBLEM, I superposing plane waves traveling in different directions, employed by Woodward and Lawson and mentioned previously, may be used. The field may also be derived as a limiting expression for a linear array of discrete sources whose separations approach zero but whose total width remains fixed. Integration using Kirchhoff's theorem and a component of the Herzian rt-vector may be used, and Schelkunoff applies his induction theorem using equivalent electric and magnetic current sheets to find the field. ' The techniques differ to the extent that they do or do not include the effects of the line charge and current distributions in which the fields must terminate at the edge of the aperture in order that surface field components satisfy boundary conditions. The distant field intensity of the aperture electric field distri- bution A(x) of Fig. 4, from any of these methods, is proportional to a E y (0) = (1 + cos 9) \ A(x) eJ^ x sin e dx, (1) where (1 + cos 9) is the familiar "obliquity factor" of a Huygens source, |3 is the phase-shift constant 2rc/X of free space, and the aperture dis- tribution is A(x), generally complex. The integral of Eq. 1 is the important factor in beam-shaping discussions, since the obliquity factor is essentially a constant for the important region near the forward aperture axis of the positive half of the x-z plane. In the terminology of Woodward and Lawson, this integral is called the "space factor" of the aperture distribution A(x). If the integral is denoted by a S(9) = C T A(x) eJ^ x sin 9 dx, (2) C A(x) eifr sin and it is defined that: sin 9 = YI s = a/X = aperture width in wavelengths; u = x/X = unit of aperture measure in wavelengths; then Eq. 2 may be written as a function of a parameter y: g(y) ■ \ A(u) eJ 2n Yu du „ (3) s C A(u) eJ *-' s The space factor g(y) defines the Fourier transform of the aperture dis- tribution A(u), which is zero outside of the aperture limits |u| > s/2. Since for physical field problems, A(u) satisfies the Dirichlet condi- tions and for such problems the integral s ~5" |A(u)| du (4) converges, then the inverse transform exists, i.e., g( Y ) e-J 2rcu Y d Y , ~ s - < u < & (5) oo Z Z , |u| > |. This transform pair, (Eqs. 3 and 5) is of interest in the problem of synthesis, i.e., that of determining what aperture distribution will produce a prescribed space factor g(y)r but the restriction that A(u) = for |u| > s/2 prohibits using the integral of Eq. 5 directly to de- termine A(u) for an assigned arbitrary function g(y)- If Eqs. 3 and 4 are to be satisfied exactly, then only an indirect solution is at- tainable through integration of Eq. 3, from which one can only find the space factor for a prescribed aperture field; this is of course the problem of analysis rather than synthesis. However, results of such analyses of assumed aperture distributions have been used success- fully in certain approximation problems of synthesis of rectangular aperture antennas which are described in Section 2.3. The integral of Eq. 5 for the aperture distribution A(u), as the discontinuous Fourier transform of g(yK has interesting interpretations with regard to the interval of integration (- 00 , °°) . First, the interval of y integration over the entire real y ax i s is significant insofar as that integral representation for A(u) is concerned. But as pointed out 10- by Woodward and Lawson, because y = s i n 9> the physically measurable portion of the total range (- 00 , °°) of the space factor g(y), sometimes called the u visible range" of g(y), is confined to -n/2 £ 6 £ tc/2, or -1 £ y £ 1. This condition is represented in Fig. 2b. Hence many functions obviously can be assigned to the " invisible range", for which |y| > 1, without affecting the visible pattern. So there is a con- siderable choice of aperture distributions A(u) which will provide, in the interval -1 £ y £ 1, an approximation to a preassigned g(y) within an arbitrary degree of exactness. Bouwkamp and De Bruijn showed that a necessary condition that an aperture distribution exist such that a preassigned g(y) be arbitrarily exactly approximated is that g(y) be continuous and single- valued over the interval -1 £ y £ 1. But the transform equations (Eqs. 3 and 5) do not themselves provide the in- formation as to how such approximations may be accomplished. In the next section are described methods which provide varying degrees of success in this problem. It will then be noted that it is not always desirable from the standpoint of efficiency or practical physical realizabi li ty to approximate a prescribed space factor arbitrarily closely. Much investigation has already been done from the analysis view- point with the transform pair (Eqs. 3 and 5) in connection with rec- tangular apertures. The extensive treatment by Ramsay has been men- tioned. In the same series of papers he also considers a synthesis possibility involving "mutilated functions" as a means for determining the aperture distributions which will give arbitrarily close approxi- mations to certain space factors having discontinuities or discontinuous derivatives such as sector or sawtooth patterns; this method usually requires uneconomical aperture sizes for a given wavelength, however, and may also involve difficult integrals. Brown has considered the effect of superposing cosine variations on the aperture distribution of a broadside antenna, and determined that high superposed side lobes are obtained, a result which follows from P. M. Woodward's synthesis method discussed in Section 2.3. Finally, an elaborate automatic calcu- lating machine method for analyzing the pattern integrals and other characteristics of rectangular aperture antennas has been worked out by Allen. 1 -11- Examples of typical space factors analyzed from prescribed rec- tangular aperture distributions from Eq . 3 are given here for con- venience of reference later: (1) Constant-amplitude, constant-phase aperture distribution: Given: A(u) f C 1, for -^ < u < i - 2 ~ ~ 2 for |u| > |. (6) j2rt Y u du - sin K y s /<7\ 71 y (2) Exponentially -varying amplitude and linearly-varying phase: _pu Given: A(u) = e -1 ^, for -3- < u < § - 2 ~ ~ 2 = , for |u| > |, (8) where G - a + jnk = propagation constant of attenuated travel- ing wave across aperture; a = attenuation constant; k = number of 2ti radians phase shift occurring across aperture. The space factor is: r a s - jn ( Y ~|)u du s I e -a S + jTI = sin hT-a/2 + jn (y - k/s) ul (q) -a/2 + jn ( Y - k/s) u Equation 9 is applied to a synthesis method described in the next sec- tion for the case a = 0. Note of course that for G = 0, this result reduces to Eq. 7. ■12- 2.2.2.2 Analysis: Circular Apertures. In the foregoing, analysis of space factors for aperture antennas has been limited to rectangular apertures, the viewpoint being that certain restrictions of the assumed aperture distribution lead to a Fourier transform interpretation of their aperture-pattern consider- ations. Such a transform approach has led to methods of synthesis of rectangular apertures proposed by Woodward and others and are described in Section 2.3. An analogous theory holds for circular aperture antennas whose aperture distribution symmetry is restricted in a certain way, and forms a basis for a synthesis method for circular aperture antennas proposed by the author in Section 4. In the analysis of the space factor of circular aperture antennas, a plane-wave field distribution over the aperture having arbitrary amplitude and phase is assumed, and so yields to analysis by methods pointed out in Section 2.2.2. Thus for the configuration shown in Fig. 2, f^Q-Tt r* a S(9, 1. (13) -14= This result, together with Eq. 12, is analogous and directly related to the Fourier transform pair of Eqs . 3 and 5 which applies to the rec- tangular aperture. Observe that the restriction that the aperture distribution f(r) = for r > 1 prohibits the use of Eq . 13 directly to determine f(r) for an assigned arbitrary space factor g(u). As for the case of the rectangular aperture, only an indirect solution of Eq. 13 is available through the integration of Eq . 12; again this is the problem of analysis rather than synthesis, since Eq. 12 permits only finding g(u) for a prescribed aperture distribution f(r). It is further noted that similar interpretations of "visible" and " invisible" ranges, represented in Fig. 2 for the space factor of the rectangular aperture antenna, hold for the circular aperture by virtue of the relation of the parameter u to the direction measured from the aperture axis, or u = (3a sin 9. Hence the visible range for the circular aperture is confined to — tc/2 £01 rc/2, or -pa < u < (3a. A class of space factors provided by a corresponding class of aperture distributions, i.e., an example of the analysis problem, is given by the discontinuous integrals of Sonine and Schafheitlin: 28 J u+1 (at) J v (bt) t v ~» dt = ( (a 2 - b 2 )V-v b v ( for a > b 2 u-v a Li+l r (n - V + 1) 0, for a < b; (14) if a, b are real and positive; Re v > — 1; Re (m> — v + 1) > 0. The cases of interest relating to Eq. 13 are determined from setting v - 0, a = 1, and b = r. Hence, Eq. 14 becomes Jo " u J u + 1 00 Jo (ur) du „ f (1 „ - r 2 )^ for 1 > r > 0, for r > 1; (15) if Re (|i) > -1. The correspondence of Eq. 15 to Eq. 13, and the Hankel transform of Eq . 13, or Eq . 12, shows that Eq. 15 is also equiva- lent to ^ + 1 ! U) - C rd-rV Jo (ur) dr . (16 ) J 2^ r (n + i) u. + 1 u^ - 1 - 15 Hence, the aperture distribution given by the right hand side of Eq. 15 will produce a pattern space factor g(u) . Ll±jJ!!> . (17) u m- + l Special cases of this result, certain ones of which may be found in the literature, are cited for convenience of future reference: (1) Constant-amplitude, constant phase aperture distribution (H " 0): Given: f(r) [1, for r < 1 0, for r > 1. (18) The space factor is g(u) - , (19) u (2) A class of tapered aperture distributions: Given, for m = 0, 1, 2, ... : f(r) f (1 - r 2 ) m ~ l , for r < 1 0, for r > 1. (20) The space factor is g(u) ■ 2 m " l (m - 1)! ia—l (21) u (3) An aperture distribution having an edge singularity: Given, for m = -%, f(r) r (1 - t 2 )~ V \ for r £ 1 0, for r > 1. (22) I The space factor is i<.) ■ & r (b ^4^ ■ s -^ ^-L = sin u ( 23) These examples of space factors pertaining to circularly symmetric aperture distributions over circular apertures are discussed in Section 16 4 with respect to a synthesis method for such sources. The investi- gations of space factors of circular aperture distributions, as con- trasted with those for rectangular apertures, are relatively few. One of the more significant of these is by Jones, who treats also the effects of other than plane-polarized aperture distributions set up by several types of primary feed sources. 2.3 Direct Synthesis Methods. Some of the results described in the previous section have been used in the synthesis of aperture antennas. The synthesis of the space factors of finite apertures .has been considered by several authors, although such work has been confined largely to apertures of rectangular shape. The work of Ramsay was mentioned in Section 2.2.2 in connection with the Fourier transform and " mutilated functions". The most signi- ficant contributions to aperture synthesis have been made by Woodward and by Taylor, and their related methods are outlined here because of their connection with the polynomial approach to low side lobe con- siderations in Section 3. 2.3.1 Finite Summation Method for Rectangular Apertures A method contrived by Woodward gives an approximation to a pre- scribed space factor for rectangular apertures. It is based on a theorem stated by Woodward and Lawson, to the effect that specified values can be assigned to the radiation pattern of a finite rectangular aperture in a finite number of directions by suitably distributing the field over that aperture. This method is based on the transform pair of Eqs. 8 and 9 for which a = 0, i.e., aperture distributions in- volving a linearly varying phase but constant amplitude across the x-direction of the aperture of Fig. 1 , or, for some specified real number, k, A k (u) = A k e J K s , where: A k = a real constant; k = number of 2k radians phase shift across aperture; s = aperture width in wavelengths. 17 Then the corresponding space factor is proportional to (y) . A sin rcs( Y - k/s) Tts(y _ k/s) (24) The method bases its simplicity of application on the property that the nulls of the space factor functions such as Eq. 24 are spaced by a cons- tant amount, 1/s defined as the "standard beamwidth". The aperture amplitude and phase conditions defined by Eq. 23 together with corre- sponding pattern space factors given by Eq. 24 are typified in Fig. 4. I |A(u)| 2 2 U n Arg A(u) U (a) No phase shift (k = 0) " IA(u)| U 3TT Arg A(u) I u — *» ,r FIGURE 4 (b) 6rc radians phase shift (k = 3), TWO CONDITIONS OF LINEAR PHASE SHIFT AND CONSTANT AMPLITUDE ACROSS A RECTANGULAR APERTURE. The effect of a linear phase variation of the constant amplitude dis- tribution over the aperture is seen to produce a deflection of the main 18 *) beam of the space factor away from the normal (or y = 0) position. If k is an integral multiple, then the position Yo of the main beam of g(u). given by y s = k, coincides with one of the nulls of all other such patterns g(u) for which k is any other integer. E.g., the pattern of Fig. 4a is moved in the positive y-direction by three standard beam- widths by letting k = 3. If for convenience a new variable, w, is chosen such that w = sy, then w becomes a variable appropriately having the units of "beamwidths" . This has the effect of scaling the y-axis °f Fig- 4 from a visible range of ±1 to a new range of ±s. So g(y) of Eq. 24 becomes gk(w) ■ A k sin^riw^Jil (25) K K n(w - k) Now the device invoked by Woodward is to insist that N + 1 functions such as g k (w) of Eq. 25 pass through N + 1 points g k = (k, A k ) with- in the visible w-range, i.e., for k = -§, - *» + 1 , . . . , ^. So since each maximum beam value of successive g k ( w ) functions, used in the approximation to some function R(w), is located at null points of all other g k (w) functions, then the values of A k required in this synthe- sis of a specified R(w) are simply A k = R(k) (26) for k ■ -Vjj-, -§■ + 1,..., 2- Hence, since the approximation pattern is given by g(w) = V g k (w), (27) k = -8 then the aperture distribution required to approximate R(w) will be I A(u) = Y A k e-i 2 ^ (28) k = -8 K 2 •19- where the Ak are determined from Eq. 26. This completes the deter- mination of an aperture distribution A(u) to approximate some pre- scribed pattern R(w). A number of examples of synthesizing the cosecant 9 patterns re- quired for radar search work are included in the original paper by Woodward, and one of these is reproduced in Fig. 5. Note that the equality of the desired pattern R(w) and the approximation g(w) is obtained for this example at N + 1 = 21. or the number of crossovers g (w) ; (a) Desired and approximation patterns. (b) Corresponding aperture distribution. FIGURE 5. TYPICAL COSECANT PATTERN OBTAINED BY WOODWARD'S METHOD % equals 21 points of coincidence. Generally speaking, the method will yield an acceptable approximation to patterns which do not demand too much of a given aperture width in the way of rapid pattern changes or very low side lobes. There is no way of predicing the degree of fluctuation that will occur between assigned points of coincidence. The method does have the advantage of automatically excluding the possi- bility of supergain, so long as the process of matching to points on a preassigned space factor R(w) is restricted to the visible region. Furthermore, there is nothing in the theory of this method which will let the designer know just how to insert energy into the invisible region so as to produce a desired effect in the visible region. An extension of Woodward's method to do just the latter has been developed by Taylor, and is described in the next section. 20 2.3.2 Polynomial Formulation of Woodward's Method Before considering the extension of the synthesis method of Wood- ward, consider first the conditions for uniqueness of a space factor defined by the integral process of Eq. 3. Consider the complex func- tion s S(z) \ A(u) e~J 2Ttuz du , (29) V A(u) e~J S 1 where we let the complex variable z - y + j a - Th e visible part of the pattern space factor S(z) is given by the value of S(z) on the seg- ment M of the real axis in the z-plane defined by -1 £ y £ 1. Now if any acceptable aperture distribution function A(u) is piecewise con- tinuous, single-valued, and bounded on the interval -s/2 £ u £ s/2 and is zero outside this interval, then the complex function (Eq. 29) is entire, as Bouwkamp and De Bruijn have proved. Hence the values of S(z) on the segment M determine S(z) for all z. So if S(z) is equal to a given function R(y) defined on M, then an exact, unique solution to synthesizing R(y) by means of S(z) is possible only if (a) R(y) can be analytically continued throughout all z, and (b) the inverse Fourier transform of R(y) together with its analytic continuation (i.e., the inverse Fourier transform of R(z) constitutes an acceptable g(x) to the extent of the Dirichlet conditions upon it. Hence, so long as the prescribed R(y) is continuous and single-valued over M = (-1, 1), it is possible to approximate it arbitrarily exactly by an S(z) whose inverse transform is an acceptable g(x). If R (y) does not satisfy (a) and (b), only an approximation of R(y) in any synthesis technique is possible. The method of Taylor, as an extension of Woodward's synthesis technique, provides a way of deriving, from the properties of the constituent functions of Woodward's summation (Eq. 27), a space factor function whose form is recognizable as having an analytic continuation for the entire z plane, and whose inverse Fourier transform is already known from Eq. 28. This result expresses the Woodward summation as the product of a factorial function and a polynomial, of which the latter can readily be subjected to the various well-known devices for synthesizing polynomials. 21- Taylor considers Woodward's finite sum of Eq. 27, for which N is assumed an even integer, and k is given only the indicated integral values: g(w) Ai sin k(w - k ) k 7i(w -- k) k = -4 K 2 (30) From the periodic property of sin 7t(w - k) for k an integer, g(w) sin rcw K (-1) 2 An (-1) 2 " A_n + 1 ( w + H.) ( w + & - 1) A - 1 + Aa A^_ + w + 1 w w - 1 ,2 X N (-1)' A^ (-1)' A^ 2_i + 2_ (w-J + 1) (w-f) (31) But the bracketed quantity can be expressed as a rational quantity, whence g(w>r sin itw w 1=11 (w* - l)(w' - 2*)--"[ w - (^) 2 ] Bo + BtW +•••+ BjyjW N (32) But from the relation of the unbracketed quantities to a gamma function, Eq. 32 may be simplified to become proportional to g(w) [(N/2)!] (N/2 + w)!(N/2 - w)! [Bo + B x w + •••■ + B^], (33) which is abbreviated as the product of a normalized transcendental 22 = *) function Q N (w) times a polynomial P N (w), the B-coef ficients of which are functions of the A^'s of Eq. 30, as follows: g(w) f- Qn(w) P n (w). (34) The transcendental functions (^.(w), which are the even functions whose values of interest (for integral w) are listed in Appendix I, exhibit tapers similar to Gaussian functions, except that they have equispaced nulls beginning at the ±(N/2 + 1) points, as pointed out by Taylor. So it can be qualitatively discerned from the broadness of tapers of the Q (w) functions, that for wide-aperture broadside antennas, the polyno- mial P N (w) represents the actual pattern space factor, except for the tapering effect which the Q N function imposes primarily on the side lobes. So in the subsequent discussion, P N (w) will be designated the "pattern polynomial". Finally, the aperture distribution A(u) corresponding to the pattern of Eq. 34 is, from (Eq. 28): A(u) = \ A k e * (35) JL 2 where u/s has the range of 1/2 to -1/2 over the aperture. Finally, since k takes on only integral values, the A k coefficients which determine the aperture distribution (Eq. 35) may be found from the values of g(w) where w takes on the integral values of k from -N/2 through N/2. Hence, A k = [P N (w) Q^w)] w = k , for k - -f , -1+1,..., |. (36) This completes Taylor's formulation of a polynomial synthesis method for rectangular apertures. In the next section, a synthesis formulation due to the author is described for obtaining broadside patterns having preassignable beamwidth and side-lobe level using the techniques above. •23 2.3.3 Synthesis Methods for Circular Apertures Very little work has been done in the synthesis of plane aperture antennas having a circular boundary and specified aperture symmetry, and such results that have been obtained are not applicable to arbitrary space factors. 12 In Section 4 is described a synthesis method for circular apertures developed by the author which provides a more general solution to this problem. J # 24- 3 TCHEBYSCHEFF POLYNOMIAL SYNTHESIS OF RECTANGULAR APERTURES A major application of aperture antennas in the transmission of electromagnetic waves occurs in systems which impose maximum design restrictions on beamwidth and side lobe level In particular, radar applications which demand high resolution of the target position with minimum interference from off-target objects are concerned with limiting these parameters to minimal values In aircraft radar, those designs that minimize noise pickup from predominant reflecting objects (such as the earth's surface) by means of inherent low side lobe pattern structure may be considered desirable This Section is concerned with a class of designs of rectangular aperture antennas which provide low side lobes without an excessive sacrifice of beamwidth and at the same time provide physically realizable aperture distributions. 3.1 Equal Side Lobe Formulation of a Pattern Polynomial In Section 2.3.2 was discussed a synthesis method for rectangular aperture antennas which expressed a pattern space factor g(w) as the product of a transcendental taper function Q[yj(w) by a polynomial Pj^(w), in terms of the real variable w s sin 9 The formulation of Eq 34 suggests two ways of synthesizing a prescribed pattern space factor g(w): (1) the first method involves finding a polynomial Pm(w) which will represent the quotient of a prescribed space factor g(w) divided by the appropriate taper function Qj^(w) (2) The second method consists of initially ignoring the taper function and considering the essential characteristics of the desired pattern to be embodied in the polynomial Pj\j(w), then, once the polynomial has been found which represents a desired pattern over an appropriate interval, the actual space factor Pj^Qjyj will possess the additional side lobe tapering effect of Qj^(w) which, for certain designs, may be considered desirable This latter approach is restricted to the synthesis of broadside rectangular aper- ture antennas having symmetrical space factors with respect to the principal aperture axis, and is the object of the following discussion The approach of method (1) is deferred until Section 3,4 in connection with the optimum side lobe problem The representation of even-function patterns for which beamwidth and side lobe level can be assigned beforehand has been conveniently carried out in terms of transformed Tchebyscheff polynomials in connec- 25 tion with designs of arrays of discrete sources 14 But whereas space factors for discrete arrays are expressible as finite cosine series, the formulation required by Eq 34 in this synthesis problem is a poly- nomial in w ■ s sin 9. The Tchebyscheff polynomials can be transformed into polynomials having the essential characteristics of desired pattern space factors in the following way. The Tchebyscheff polynomials are defined by T n (x) cos(n cos ' x) , for | x - 1 Because cos n9 is a polynomial in cos become cosh(n cosh 4 x) , for | x | - 1 (37) successive orders of Eq. 37 To(x) = 1 Ti(x) = x T 2 (x) - 2x 2 - 1 T n (x) - 2x T (x) ii n - 1 (x) (38) A graph of these Tchebyscheff polynomials for n = 1, 2, . '. , 6 is given in Fig. 6. These polynomials have roots which are all real and which occur in the interval (-1,1). They are found from Eq 37 by setting the expression for that interval to zero, whence the kth root, ordered from the one nearest x = 1, occurs at L ok cos 2k - 1 K , k - 1,2 (39) There is a point x between each adjacent pair of roots at which T (x) .mk n takes on a relative maximum or minimum of ±1. These positions are given by setting the expression of Eq 37 to ±1, whence in an ordered manner . x = cos mk krc 1,2, n- 1 (40) 26 1 1 i 1 I '\ \ \ 1 t> / ml / 17 / \ \ l.*r 1 9 V / / \ 1.2 \ / "\l A \ 1 / V \ ' I A \ / i 1 1 j - / 1 /"• •\1 » -t / i 7 1" 2 \ M • / V i V 6 1/ • 3 / 1 (. X- \ 1 /? N V / / vac / \ 1 \/ -A / / \ / ^ n= I n?2 A O ^6 / / 1 1 FIGURE 6. FUNCTION T N (X) PLOTTED FOR SIX ORDERS. 27 A table of positions of ordered zeros and relative maxima or minima is given in Appendix II. A pattern polynomial is now desired such that its zeros are spaced in a manner providing a high- amplitude, positive beam of preassignable level at w = 0, and n = N/2 zeros are to be symmetrically distributed to either side of w = in the interval (~s,s) such that alternating relative maxima and minima of ±1 are produced between adjacent zeros. A polynomial having this form is shown in Fig, 7. P|yj(w) must evidently *~W FIGURE 7. EVEN PATTERN POLYNOMIAL P N (w) be an even polynomial, of degree N Such a polynomial can be derived from a Tchebyscheff polynomial T n (x), where n = N/2, by means of a second-degree polynomial transformation x = f(w) given by the linear function of w : 2 a + b w (41) where a and b are constants chosen to provide the desired mapping. If one chooses the constants so as: (1) to map some preassigned x m > 1 of a given T n (x) function into Pj^(0) such that the central relative maximum of Fig. 7 becomes T n (x m ); and so as (2) to map the point (-1,1) or (-1,-1) of T n (x) into the points (s,±l) and (~s,±l) of the Pj^(w) func- tion; then the n roots of T_(x) are distributed symmetrically as 2n = N nulls for Pjyj(w) as in Fig. 7, and the side lobe level of unity is maintained within the visible range (-s ( s). A transformation which performs this desired mapping is x m - - Q TT- w (42) 28 Its inverse is evidently w ' ±s x m ~ x (43) x + 1 To illustrate the results of such a transformation, this pro- cedure is used to generate a Pi (w) polynomial from a T 5 (x) Tchebyscheff polynomial as shown graphically in Fig- 8. Hence the polynomial P 10 (w) - -16c 5 w 10 + 80x m c A w B - 20(8x m ' - l)c w D + 20(8x m 3 - 3x m ) c 2 w 4 - 20(4x m 4 - 3x m 2 ) cw + (16x m 5 - 20x m 3 + 5x m ), . (44) where c = (x + l)/s 2 , is the result of applying the transformation of Eq. 42 to the Tchebyscheff polynomial T 5 (x) ■ 16x 5 - 20x 3 + 5x. (45) It is of interest to note that another mapping function which gene- rates polynomials Pjyj(w) having the same properties as those described above is the elliptical transformation _*I +ll - 1.- (46) m This function will generate a desired P^(w) from a Tchebyscheff polyno- mial Tm(x) having the same degree as Pm(w) for this case, instead of half the degree of the pattern polynomial which is obtained when using the transformation of Eq. 42. For example, the function Pio(w) of Eq. 44 can be generated from T 10 ( x ) by means of Eq 46^ In either case however, the function Pm(w) having the desired properties is a unique solution. Some properties of the pattern polynomial Pm(w) just developed which are useful to know from the standpoint of the later development are readily found from the properties of the related Tchebyscheff polynomial and the appropriate transformation Following the notation given in Fig. 8, one obtains: (a) The principal beam maximum, or Pj\j(0), is found from the speci- fied ratio of the principal beam maximum to the side lobe level. Since 29 T 5 W Ffolw) y/t VJ < FIGURE 8 GENERATION OF A BROADSIDE PATTERN POLYNOMIAL P n (W) FROM T n (X)„ FOR THE CASE N - 2n - 10 30 the latter is unity, it is evident from Fig 8 that this ratio is P N (0) - T n (xj, (47) where N = 2n. So if P[\j(0) is given, Eq. 47 permits finding x m , using the definition of T n (x). (b) The beamwidth of Pj^(w) is defined as 2wj ;) , where w^ is the abscissa of Pj^(w) and the latter is equal to (l//2)Ppyj(0) . But Pj^(O) = T n (x ), so from Eq. 43, = 2s / x m " x b Vx + 1 where xj^ = cosh 1 u- 1 - cosh /~2 (48) (49) (c) The first-null position w 01 (adjacent to the principal beam) is found from the first-null position of the corresponding Tchebyscheff polynomial, given by Eq. 39. Hence, 'o 1 ±s Am X Q 1 V x + 1 where x oi cos (50) (51) 3.2 The Aperture Distribution The remaining characteristic to consider for the synthesis method of the preceding section is the aperture distribution A(u). This is provided by Eq. 35, for which the amplitudes A^ for that finite series are supplied by Eq. 36. But the discussion of the preceding section is restricted to even functions P]\j(w) > and since Qj^(w) is also even, then "from Eq. 36, A^ = A_j c . This means that the total aperture dis- tribution can be written, for an even Pj\j(w), as N/2 A(u) = A + 2 Yi A cos 2k(u/s), (52) k- l where the constants A^ are determined from the equispaced ordinates on 31 the synthesized pattern: A k - [Pn Oft] for k - 0,1,2,.., N/2. (53) In the practical designs of rectangular aperture antennas based on these concepts, it is usually desirable to limit the designs to aperture distributions which can be relatively easily realized from a physical viewpoint. Since the aperture distribution is, by means of Eq. 53, directly associated with space factor values g(w) at integral w-values, i t is apparent that a design which permits a relatively large amount of pattern field energy to occupy the invisible region |w| > s will also require large order (i.e., rapidly oscillating), high- amplitude cosine components to exist over the aperture. This condition is always brought about when the beamwidth is forced to be too small for the available aperture size. 3.3 Narrow-Beam, Tapered-Si de-Lobe Designs; Supergain Effects. A design example is given here to illustrate the effect of attempt- ing to extract an excessively narrow beam from a given aperture width. Assume a polynomial pattern P 2 o(w) having a side lobe level 40 db below the principal beam maximum is to be used in the design. The assumption of a 20th degree polynomial implies an associated Tchebyscheff polynomi- al Tio(x) and an associated taper function Q 2 o(w), values of which may be found in Appendix I. A 40 db side lobe level means from Eq. 47 that Tio(* m ) = 100, whence from Eq. 37, x m = 1.14372. It is now nec- essary to assume an aperture width s for which an aperture distri- bution is to be calculated using Eqs. 52 and 53, but examination of the latter has shown that the A^ coefficients of the finite cosine-series distribution are determined by the values of the pattern space factor g(w) itself for integral w-values, taken out to w = N/2 = 10 for this case. Two cases of assumed aperture widths are here submitted for comparison: (1) If an aperture width s - 10 is assumed for this design, then P 2 o(w) has the value of unity at w 10, whence the last cosine term of the aperture distribution has the negligible amplitude of the space factor g(10) - P 20 ( 10) Q 2Q ( 10) , or 1 * (5 389 * 10~ 6 )» (2) If aperture width s - 9 is assumed, then P 20 (9) is unity, making the next- to-last cosine term of the aperture distribution negligible when P 20 (9) 32 is multiplied by Q 2 o(9). But P 2 o(10), corresponding to the last cosine term, takes on in this instance the enormous value of 6.9715 x 10 , and this value, even when multiplied by Q 2 o(10) - 5.4 x 10" - , still leaves the last cosine term of the aperture distribution with an amplitude of 3765. This latter case, for which the given polynomial pattern P 2 o(w) is synthesized by means of too small an aperture to yield a practical aperture distribution, is termed a "supergain" case, in which consider- able energy must be stored in the aperture in order to make the assumed beamwidth fit the assumed aperture length. For either case, the aperture distribution is obtained from Eq. 52, or 10 ~~7 A(u) = A + 2 5 A cos 2 k(u/s), 4i i lit k = 1 and the coefficients of this series for the two cases are compared in the table: For s 9 For s = 10 A 100 100. 2Ai 110.08 121.62 2A 2 • 10.63 -20.23 2A 3 -.272 -.839 2A 4 .078 ,415 2A 5 .052 -.163 2A e .018 . 041 2A 7 -.0004 -.0074 2A 8 -.0018 .0013 2A 9 0002 -.0002 2A 10 7530. .000005 These aperture distributions, together with their pattern space factors g(u), are illustrated in Fig. 9 Evidently the effect of using too small an aperture for the assumed order of pattern polynomial is one of requiring an exorbitant amount of energy in the invisible region, as can be seen from Fig. 9e. This energy, not utilized in establishing the radiation field which must occur in the visible region (~s,s), is mani- fest in the aperture field distribution of Fig. 9f where it exhibits 33 (a) Pattern polynomial and taper function for s = 10. 9 10 11 (d) Pattern polynomial and taper function for s = 9. 4g(w) to 3765 Visible^ limit \^\S**sj£ * ^^ B (b) Space factor; s = 10. -d — o 10 II -Z- 1 A(u) I0£-- i i 1 L_ 1 u 5 (c) Aperture distribution; s = 10. (a) Pattern Polynomial Pu(w), ( b) Resulting Space factor S(w), FIGURE 12. FUNCTIONS RELATED TO OPTIMUM PROBLEM FOR APERTURES. l/Q^Cw) curves occurs at the unknown points w^ along the positive w-axis, where k = 1,2,.,, N/2 - 1, these conditions are embodied in the equations PN(0) = Mo» Pfl(w k ) Qn (*k) " *!> PN (w k ) On ( w k> = ±l > (55) (56) (57) where the signs of Eqs. 56 and 57 depend on whether k is odd or even as interpreted from Fig. 12a„ If to these N -1 equations is added -40 = an end condition amounting to the assignment that the most remote tangency point is given as w = s, then sufficient conditions are -£-- i 2 provided for finding the N/2 - 1 tangency points and the N/2 + 1 co- efficients of the even pattern polynomial Pj^(w). Since these are non- linear relations, approximation methods must be used to find the un- knowns. Methods for evaluating this optimizing polynomial Pm(w) are treated in an article in connection with a different problem of opti- mizing patterns for discrete arrays of anisotropic elements, 4 and are not discussed here. A completely different approach to determining the optimum pattern 46 md some re- S(w) directly is given in an excellent paper by Taylor, suits of his analysis are provided in a yet unpublished paper. This method is based on a result of G.J. van der Maas 47 that the space factor of a Dolph-Tchebyschef f array, whose number of elements is indefinitely increased, approaches the function S(w,A) = cos n K IV/W (58) where A = (1/n:) cosh - M, if M is the value of S(w) at w = 0. The function of Eq. 58 is a unity side lobe pattern, or "ideal" pattern, of the type illustrated by Fig. 13, whose zeros are given by setting that expression to zero, whence 7a 2 + (k - y 2 y (59) »~w figure 13. "ideal" unity side lobe pattern. These zeros are, for large k, asymptotic to the half-integers ±(k For an admissible space factor S(w), the integral -00 I r> 1 \ I 2 lw - Vz) |S(w)| 2 dv -/. no 41 must exist, and this is not true for the S(w) of Eq . 57 because of the constant side lobe amplitude As pointed out by Taylor, the nearest approach to this ideal pattern which is yet an admissible function is provided by an entire function whose asymptotic behavior on the real w-axis is proportional to (sin Kvt) /kv/ . The zeros of the latter are of course at the integral values of w„ Taylor constructed a function which has approximately the constant side lobe property of the ideal space factor of Eq . 57 while at the same time possessing the asymptotic behavior of the (sin Ti;w)/rtw function, by means of a canonical product on the zeros of the desired function. Hence, the scale of the ideal function of Eq . 57 can be expanded by a desired amount a > 1 on the w-axis so as to provide the function Si(w,A) :: cos Tt a/(w/g) 2 - A 2 the zeros of which are w -a on /A 2 + (n -Y 2 f (60) (61) It is then proposed to construct a function having, on the w-axis, a set of zeros as follows: w on + (n - Y 2 ) 2 , for 1 i n ifi r - < tor n - n (62) The value of a is chosen such that the nth zero of this set occurs, in fact, at a w-value which is an arbitrary integer n, and which pro- vides the relation for finding a once n is picked. Hence w on which yields the desired A 2 + (H - Y 2 ) 2 (63) a = 7 a 2 + (5 - y 2 ) 2 (64) The value of n selected would probably be of the order of the aperture width in wavelengths or s ; i.e., roughly the limit of the visible range of the space factor The canonical product on the zeros of Eq . 62 is 42 then S(w,A,n) - c T. (w/a) 2 A 2 + (n - %) R - 1 00 _£- (w/n) 2 ] (65) c Q (w) 2(5 - 1) n U j 1 - (w/a) A 2 + (n - Yi) (66) where the infinite product of Eq - 65 has been represented by the sym- and c is an arbitrary constant. Note that from the bol Q 2(5 1) infinite product form of (sin rcw)/rcw, Q Q (w) 2(n - 1) 2(n - 1) (sin rcw)/Ttw IT [l - (w/n)^] may also be written n r 1 (67) (n - D! (n - 1 + w)! (n - 1 - w) (68) which is identical with the Qj^(w) taper function defined by Eq . 33 in connection with Woodward's synthesis method, provided (n - 1) is replaced by N/2- If the arbitrary constant c is given the value M which defines the main beam to side lobe ratio as in connection with Eq, 57, then Eq. 66 becomes the trancendental-polynomial product having the desired zeros w expressed by Eq. 62: on n - 1 S(w,A,n) =MQ (w) 1(1 - 1) n M x 1 - (w/a) A 2 + (n - Vi) (69) or the desired space factor which provides a physically realizable approach to the ideal space factor of Fig. 13. Values of the taper function Q are given in Appendix I. The finite product of Eq . 2(n- 1) 69 is obviously an approximating solution for the pattern polynomial Pj^(w) of Fig. 12a. The aperture distribution required to produce the approximation space factor S(w,A,n) of Taylor above, by virtue of the correspondence of Eq. 69 to Taylor's transcendental-polynomial formulation of Eq.34 ■43- is given by an expression equivalent to Eq. 35 or Eq. 52: ii A(u) = A + 2 LL Au cos 2rck(u/s), (70) k = i K where the constants A^ are given by the values of the space factor S(w,A,h) at integer w-values: Ai, = [S(w,A,n] , for k = 0, 1, .., n. (71) K w=k 3.6 Comparison of Tapered Side Lobe and Optimum Side Lobe Designs It is of interest to make a comparison of a typical optimum side lobe design obtained from the discussion of the last section with a tapered side lobe design given by the method of Section 3.2. Let the side lobe level be 40 db down for both designs, which for the ideal optimum pattern of Eq . 57 makes M = 100, or A = 2.844. Then the beamwidth 2wj ;) , where w b is the value of w where S(w,A) = M/v2, becomes from Eq. 57: !w bo = lv c osh -1 M - cosh _1 (M/^5"). (72) For this 40 db case, the "ideal beamwidth" is 2wl = 1.202. The actual beamwidth corresponding to the pattern of Eq. 68, which behaves near the origin essentially the way the "expanded ideal" function of Eq. 61 does, is practically the ideal value of Eq. 72 multiplied by a, or 2w b = a(2w bo )^ (73) But typical values of a calculated from Eq. 65 for moderately large n-values, and for the A of interest, are found to be of the order of perhaps 1.03 or 1.04. Hence, for this 40 db case, the "practical beamwidth" is of the order of 2w b = 1.25- (74) This result may be compared with beamwidths obtainable for a tapered side lobe design as given by the curves of Fig. 10. For aperture widths of the order of 10 or 12 wavelengths, the beamwidth obtainable without supergaining is found to be about 1.4. Hence, this example shows that better than 10% improvement in beamwidth is obtained from -44 the optimum design This improvement is observed to become less with larger aperture size and with lower first side lobe level. The exact solution to the optimum problem discussed in this section is not provided by the solution given by Taylor as just described, because of the effect of side lobe tapering within the interval -n <_ w £ »i produced by the particular choice of zeros given by Eq . 62. A better choice of these zeros is provided by the attack suggested in connection with Fig. 12, which requires finding a polynomial whose envelope of side lobes is given by the curves ±1/Qiv(w)„ This requires that Eqs. 55, 56, and 57 be simultaneously satisfied. This exact problem, as has already been pointed out, is however quite difficult, and instead of possessing a degree of universality as does Taylor's method, would probably require a separate solution for each new set of parameters assigned. 45 4. CIRCULAR APERTURE SYNTHESIS While a certain amount of work has been done in the synthesis of pattern space factors of rectangular apertures having particular symme- tries of aperture field distributions as described in Section 2, it was pointed out in Section 2.3.3 that very little consideration has been given in the past to the synthesis problem for circular apertures. Just as for the rectangular aperture problem, consideration must be given to the kind of symmetry to which the aperture distribution is to be restricted. The most interesting of these is circular symmetry which produces, for a plane-wave aperture excitation condition, a pattern space factor of revolution. It is well known that many rectangular or circular aperture designs have pattern space factors in the 45° tilted planes which deviate considerably from the patterns in the principal planes for which the antennas were designed, especially with regard to side lobe conditions. A truly circularly symmetric aperture design obviates this difficulty. The analysis of space factors attributable to circularly symmetric aperture distributions was considered in Section 2.2.2.2, and the para- meters of the problem are shown in Fig. 3. In this section, the prob- lem of synthesizing prescribed pattern space factors for such aperture antennas is considered, using the Hankel transform pair of Eqs. 12 and 13 as a basis for the development. In the first part of this Sec- tion, a synthesis approach analogous to the "mutilated function" in- terpretation of Ramsay, mentioned in Section 2.2.2, is considered. In the remaining sections, a more general method of synthesis in- volving an orthogonal set of pattern functions related to the Hankel transform pair is developed and applied to several examples. 4.1 Circular Aperture Synthesis by Integral Approximation This method of synthesis for circularly symmetric aperture antennas is based on the approximation properties of the Hankel transform pair. Consider that a space factor g(u) is to be produced by a distribution f(r) over a circularly symmetric aperture of finite size. Let us retain the notation used in connection with the aperture shown in Fig. 3. The space factor g(u) is given by Eq . 12, or g(u) = \ r f(r) Jo(ur) dr (75) 46- for an aperture distribution f(r) over a circular aperture of any size Conversely, the aperture distribution is then f(r) oo u g(u) Jo(ur) du, (76) Now there exist certain functions g(u) which one might prescribe as space factors to which there correspond aperture distributions f(r) which must range over an r-interval (0, °°) if the g(u) are to be pre- cisely represented by Eq. 75. That is, space factors exist which re- quire an aperture of indefinitely large size to produce them within an arbitrary degree of accuracy. One such space factor is the discontin- uous function g(u) £ u < u |u| > u . (77) Then the aperture distribution required to produce this g(u), from term- by-term integration of Eq. 76 over the interval (0, u ) becomes f(r) = u Ji (u r) for '■ r < °°. (78) Thus to produce precisely the step- function space factor of Eq . 77, illustrated in Fig. 14a, the infinite-aperture distribution of Eq . 78, shown in Fig. 14b, is required. So as to avoid this physical g(u) " u f(r)'i (a) Prescribed space factor. (b) Required aperture distribution. FIGURE 14. A DISCONTINUOUS SPACE FACTOR AND ITS APERTURE FUNCTION. *The inherent difficulties of defining the space factor of an aperture of infinite size are dis- cussed in reference 4 47- impossibility, let one consider an approximation to the space factor g(u) by means of a finite aperture size, i.e., by considering what the nature of an approximation to g(u) which we shall call g a (u) will be when the aperture distribution of Eq. 78 is defined over an aperture of finite size b. Call this truncated or '"mutilated" function *a (cf. Eq. 15), it is apparent that insofar as choosing a countable number of these pattern functions is concerned, one is re- quired only to designate the manner in which successive orders |i are to be chosen. The circularly symmetric aperture distribution function f(r) which corresponds to the space factor of Eq. 87 is f(r) (1 - r 2 )' 0, < r < 1 r > 1 (88) Although it is quite permissible to choose a sequence of complex |i- values for the purpose of building up an orthogonal set of space factor functions from Eq. 87, it is well known that an imaginary component of (i, which has the effect of assigning a phase shift to the aperture dis- tribution (Eq. 88,) will generally produce a broadening of the princi- pal beam in the pattern space factors given by Eq. 87, as well as render the nulls indistinct along the real u-axis of interest. These effects are such as to diminish the available resolving power of any given pattern element of Eq. 87. Also, in a specific design, variable phase shift over an aperture may not be as readily controlled as variable amplitude. For these reasons, the values of |i will be restricted to real numbers, necessarily greater than zero. Having restricted the values of |i to real numbers, it is now desifed to pick a countable set of discrete p,-values from the con- tinuum |i > 0. Since a choice of \x such that < \i < 1 produces a singu- larity at the aperture edge, these values are omitted from the set. This limits the range to p, > 1. It is here submitted that aperture distributions which can be represented by a sum of aperture functions of the form of Eq. 88, for which u. is assigned the positive integers, will define an aperture distribution of sufficient generality for the purposes of this synthesis. Such an aperture distribution will be represented by r co ^ V A k (l-r 2 )l<-\ 1 (89) 52* where the Au are real constants. The corresponding space factor is the sum of elements of the type given by Eq . 87 and becomes, from the Hankel transform of Eq. 89, oo (90) g(u) = Y A k 2 k - * (k - 1)! J k< u > k-i » k These latter results are not directly applicable to the synthesis problem at hand, since for a preassigned g(u) to be represented by a series of Bessel functions as Eq. 90, there is no ready method for de- termining the coefficients Ak, except perhaps from a large number of simultaneous equations obtained from the power series expansion of g(u) and the corresponding series of Bessel functions of the right-hand side of Eq. 90. Such a procedure is out of the question from the gener- alized viewpoint of interest because of the slow convergence of the Bessel functions for even moderately large u-values. The coefficients of a series representation such as Eq. 90 may be determined, however, if the functions involved are orthogonal over an assigned u-interval. Although the set of functions Jk(u)/u of Eq. 90, for k = 1, 2, ... are not orthogonal over any assignable u- interval, it is possible to transform them linearly into a new set whose elements are orthogonal, A procedure for doing this is discussed in the next section. 1.2.1 An Orthogonal Set of Pattern Functions Let the set of space factor functions of Eq. 87, for |i equal to the positive integers, be represented by the symbol [h] , and order the elements according to [h] = hi (u), h 2 (u), ... , h n (u), ... , (91) wh ere h n (u) = 2 n n! iaJOtl. (92) u n 53 > This makes the corresponding aperture distribution function f n (r) = f 2n(l r ) n " \ £ r < 1 r > 1. (93) The interval of orthogonalization is chosen to be (0, °°) in order to make the synthesis method sufficiently general so as to apply to all finite aperture sizes. In order to construct an orthogonal set [H] from [h] , it is possible to form n linear combinations of the elements of [h] to define n elements of the orthogonal set [H] . This is done after a method devised by E. Schmidt. 49 Assume that the orthogonal set [H] has elements defined by the ordered set H t (u) = hju) H 2 (u) = C 21 Hi(u) + h 2 (u) H 3 (u) = C 31 Ht(u) + C 32 H 2 (u) + h 3 (u) Hi. H n (u) - h n (u) + ^ C nk H k (u) n - 1 X, (94) The orthogonality condition existing between any two elements is defined by poo \ ^ ^ du = 6 mn N nn , m, n = 1, 2, ... , (95) where 6 mn is the Kronecker delta, and N nn is a normalizing factor for preserving the value of unity for 6 mn whenever m = n. The functions h n (u) of Eq- 94 are given by Eq. 92 and it remains only to find co- efficients Cpk i n order to define the I^(u). These are found by system- atically applying the orthogonality condition of Eq..95 to that par- ticular Hk(u) of Eq . 94 whose coefficients are to be found, by means of an integration of its product with each preceding H-element. *On an infinite interval of orthogonalization for orthogonal functions in general, cf. refer- ence 23. = 54- In particular, the coefficient C 2 i of the H 2 (u) element of Eq. 94 is found from the orthogonality of H 2 with Hi by Eq. 95: \ H 2 H x du = . (96) The coefficient C 21 is found by substituting the H 2 (u) expression from Eq. 94 into Eq. 96, yielding poo \ [C^Mii) + h 2 (u)] HJu) du = . (97) Solving for C 21 : H* du pOC Jo which is the desired result. Again, the coefficients C 3 i and C 32 of the H 3 (u) element of Eq. 94 are found from the two orthogonality relations poo \ H 3 Hi du = (99) S. oo H 3 H 2 du = (100) o respectively. The coefficients in question are determined upon substi- tuting the expression for H 3 from Eq. 94 into each integral and then applying the orthogonality condition of Eq. 95 again. Hence, Eqs. 99 and 100 become pco \ [C 31 Hi + C 32 H 2 + h 3 ] Hi du = (101) Ooo [C 31 Hi + C 32 H 2 + h 3 ] H 2 du = (102) 'O respectively, and applying orthogonality property (Eq. 95) reduces each -55- *> equation to two terms, whence 00 C. ■ " J > h3 "' d " (103) \ H* du 00 h 3 H 2 du O "3 "2 -3 2 ^00 Hs du o (104) In general, the kth coefficient C^ °f tne ntn element H n of the orthogonal set [H] is given by C^-Ak, (105) where for convenience the N's denote the integrals poo N n k = \ *n H k du (106) pco poo N kk = \ H k du = \ h k H k du - ( 107 > Thus the orthogonal set [H] of Eq. 94 is specified once the integrals of Eqs . 106 and 107 have been evaluated. The numerical evaluation of such integrals as they occur in this problem is postponed until Section 4.2.3. For the present purposes, let it be supposed that the C^ have been found. The next task is to express the H n (u) of Eq. 94 as a linear sum of h k (u) only. Such an expression, which can be written in the form "n " £_ Bn k h k n = 1, 2, ... , (108) is desirable if the H n (u) functions are to be tabulated or applied to a computational process. For such purposes, the B nk coefficients of 56- Eq. 108 can be expressed as functions of C n ^ coefficients. This re- lationship can be evolved upon comparing Eq. 108 with Eq. 94. Its deri- vation is also deferred until Section 4.2.3. Once the coefficients of the expressions for the orthogonal ele- ments of [H] have been evaluated for Eq . 108 or 94, these equations may be applied to the synthesis of a prescribed pattern space factor g(u). To this end, and for the purpose of simplifying the Fourier processes eventually employed, the orthogonal elements may first be normalized by finding a constant a R such that for any 1^, fl£ d* - l. (109) Hence the normalization factor a n becomes Hn du n nn' (110) where N nn is obviously the same integral as the denominator of the CLk expression of Eq . 105, as well as the N nn of Eq . 95. So an ortho- normal set [G] is derived from orthogonal set [H] if the elements H_(u) of the latter are replaced by the elements G^u) - N$ H» (111) Then the identity G* du = 1 (112) is certainly satisfied. Furthermore, an equation similar to Eq. 108 can then be written for the C^ elements, or Gn n nn Bnk h k 1 (113) k h k k = 1 (114) -57- > where the constants b n j^ are obviously b nk = <£ %k ■ C115) The elements G n (u) of Eq . 114 comprise the orthonormal set [G] upon which is based the development of the following sections. 4.2.2 Approximation of the Integral Equation Solution Provided now with orthonormal set [G] , let us consider the ques- tion of representing a prescribed pattern function g (u) by means of a series of the orthonormal functions G n (u) furnished by Eq. 114, i.e., by means of the series oo g(u) = Y A k Gjju) , (116) k = 1 where the Ak are arbitrary constants, and the symbol "=" is used to designate that the mean-square error between the series of Eq. 116 and the g(u) function which it is supposed to represent may not necessarily be made arbitrarily small unless the orthonormal set on which the expansion is based is a complete set. It is shown later that the ortho- normal set [G] is, in fact, incomplete, which is synonymous with saying that not every preassigned g(u) can be represented by series (Eq. 116) within an arbitrary number e for all u in the interval of orthogonali- zation. Whether the set [G] is complete or not, however, does not disturb the essential approximation properties of the series of Eq. 116 if the coefficients A^ of that series are Fourier coefficients. For if this is so, i.e., if the Ai are given by ^oo A k = \ g( u ) Gk(u) du , (117) then the Bessel inequality is true, or oo K 1 } g (u) du, (118) k = 1 -58- implying that the sum of the squares of the expansion coefficients of g(u) always converges, assuming that the improper integral of Eq. 115, pertaining to the infinite interval of orthogonalization , exists. The equality of Eq . 118 holds for every admissible function g(u) only if the set [G] is complete, and in such an instance is called Parseval's formula. Also, the process of minimizing the familiar mean- square error E between g(u) and the "nth-order approximation to g(u)", given by N terms of a series of type 116 but having arbitrary co- efficients a k , or poo n E = J (g - Y a k Gj,) 2 du > , (119) k = 1 will lead to the Fourier coefficients of Eq. 114, or a k = A k . (120) The identity of coefficients a k to the Fourier coefficients A k is applicable to an expansion of g(u) in terms of an orthonormal set G (u) whether or not the set is complete. In the event of an expansion (116) in terms of an incomplete set, the mean-square error defined by Eq. 119 merely does not have the limit zero, no matter how many terms of the series (116) are used. Thus in the limit, the series representation will still exhibit an oscillation about the function g(u). The convergence in the mean of the series for g(u) to an oscil- lating approximation of the function g(u) is precisely the result desired in the problem of synthesizing a prescribed pattern function. It has already been shown in Section 4.1 that there exist pattern space factors g(u) such that no aperture function f(r) over an r-range of (0, 1) can succeed in arbitrarily closely approximating that g(u) over its entire u-range of (0, °°) . The step- function space factor of Fig. 14 is an example of such a g(u). The oscillating approximation to the step function shown in Fig. 15b is indicative of the kind of solu- tion, though not necessarily the same solution, which the series of Eq. 116 provides. Finally, in view of integral transform (Fig. 15) which determines the aperture distribution f(r) over (0, 1) once an admissible repre- 59- sentation of g(u) has been found, one may write for f(r) f(r) u g(u) Jo(ur) du /o 00 ) ^ \ u G n (u) Jo(ur) du. (121) n = 1 The Hankel transform of G n (u) appearing in this equation is evaluated in terms of Eq. 114. Hence, CO yi /-sen (r) = ^ A n f_ b nk \ u M u > Jo(ur) du n = 1 k ■= 1 k b nk (1 - r 2 ) k - 1 n = 1 k = 1 < r < 1 r > 1, (122) upon substituting for h^(u) from Eq. 92 and using the Hankel trans- form of Eq. 93. Thus Eq . 122 represents the required aperture dis- tribution which will produce the series approximation to g(u) given by Eq. 116. The aperture distribution corresponding to the orthonormal space factor element GLdi) of Eq. 114 is therefore evidently k = 1 which permits writing Eq. 122 as n £ k b nk (l - r 2 ) k - 1 - F n (r) f(r) r \ n oo ^ *n v: , (r) , £ r < 1 r > 1 (123) (124) •60- 1.2.3 Determination of the Orthonormal System Coefficients It remains yet to find the b n j^ coefficients of Eq. 122 to permit the evaluation of an unknown f(r) of a given synthesis problem. The numerical values of the b n ^ coefficients can ultimately be related to the orthogonalization integrals as indicated in the following para- graphs. Equation 115 shows that the b n j^. are related to the B^ coefficients through the normalization integral N^ of Eq . 110. Let us concen- trate on the evaluation of the B^ first. If expressions for H^ are inserted into Eq. 94 over the indicated k-range from lower order H- expressions, then one may write Eq. 94 as n n -- k H n = Y h k / ^(p + k - 1) B (p + k - l)k> < 12 5) k - 1 p - 1 where the C-coef ficients are given by the ratio of integrals of Eq. 105. Upon comparing Eq. 125 and Eq. 108, obviously n - k B nk = ) ^(p + k - 1) B (p + k - l)k' (126) P ' 1 This relation shows that any B n j. coefficient can be determined as a linear combination of lower-order B coefficients. All that remains toward evaluating these B n ^ is expressing the integrals N_i. appearing in Eq . 105 as functions of a definite integral fundamental to this development, namely poo J o K h n du - B™ . (127) It can readily be shown that if the expressions for H_ given by the linear equation (125) are substituted into either integral N ^ of Eq . 105, the resulting expansion permits writing a sequence of N-u, each written as a linear combination of R^ and lower-order N-values, or k - 1 N nk = Rnk + Y_ •"kp n np ■61- » This expression, when used together with the C n ^ of Eq. 105, permits finding the C-coef ficients in the ordered fashion: C 2 1 ; C 31 , C 32 ; C 41 , ^42. C* 3 ; etc., whereupon the B-coef ficients, using Eq. 126, can also be evaluated in order. Then the b n ^ of Eq. 115 may be determined. The numerical evaluation of the foregoing quantities which lead to finding the desired b n j^ depends upon finding the value of the integral E mn of Eq. 127. This can be expressed, upon using the definition of h n (u) given by Eq. 92, as /-NX) R^ = 2 m + n m! n! \ J m (u) J n (u) iT" du, m,n > 0. (129) The integral of this expression is Struve's integral and is evaluated by substitution of the integral representation of J (z)/zM- (where z is a complex variable whose real part is u) and contour integration as 48 r/2 o-m-n \ J m (u) J n (u) u« du = L p , V { \\ l ] -< U7, (* + n) > 0. Jo m n (m + n + l A) r (m + Y 2 ) T (n + %) (130) But for k a positive integer, the half-integer gamma function is r k > 1 (135) Nnk «nk (n = k _ 1)!(n _ k) , (4k _ 3)I[(k _ fijj7> n _k _ 1. (135) C, - ~2 A(n " k) n n! \(n - 1) ! 1 3 (4k - l)!(n + k - D! n > k (136) ^ k (n - k)!k k! [(k - l)!] 3 (2n - 1)! (2n + 2k - 1)! ' r . = (-1)" ~ k 2 4n ~ 4 k - 2 n s kf(n _ n!l 4 (2k )!( 2n + 2k - 2)! > , ^ k (2n - l)(4n - 3)!(n - k)!(n + k - l)!(k!) 4 ' " ' (137) where note in particular that B nn = 1. Finally the coefficients b k , needed for the orthonormal set [G] specified by Eq. 115 and for the corresponding aperture distribution expansions given by Eq . 123, are found from Eq . 115, in which the normalizing coefficient N_"^ 2 is found from Eq. 135 for n = k. A table of the b k corresponding to the first nine G n (u) functions is given in Appendix V. Note that the coefficients listed there omit a common nIT factor which is contained in the expression for the normalizing factor Nljf of Eq. 115. nn 4.2.1 Properties of the Orthonormal Set Now that the b nk coefficients have been obtained, the orthonormal set elements G n (u) can be written down as the linear combination of Bessel functions h k (u) given by Eq. 114. Hence the expressions for the first three elements of the orthonormal set are: Gi(u) - 4tc[.-433 013 hj G 2 (u) = >nr [-1.620 185 h x + 1.518 924 h 2 ] G 3 (u) = >nr [3.211 310 h x - 8.429 688 h 2 + 5.268 555 h 3 ] . (138) The functions h k (u) have been tabulated to four places, 20 but can be obtained to better accuracy from tables of Bessel functions of the first kind by applying definition (94). Since the absolute values of the ■63- orthonormal Gj,(u) functions as well as of the hj c (u) functions are bound- ed within a value of unity, it is apparent that the decimal place accuracy required of the b n j < coefficients must be approximately the same as the place accuracy desired for the G n (u) functions being calculated. This rather limits the available accuracy for the higher-order G n (u) functions, since the coefficients acquire increasingly larger values with increase in order. This difficulty is removed in part by deriving a series expression for each G n (u) function, which permits applying such a series to that part of the u-range where more accuracy may be required. The series expressions for G n (u) may be found by combining the coefficients of like powers of u for the finite sum of Bessel series which define each h(u). The results of these manipulations yields series for G^u) of the form: CO G n (u) IT* 1 nn (~D k A n (k) k! u 2k 2 (140) where the A^k) are functions of polynomials in k as follows Ai 1 (k + 1)! -2(1 ± 8kl_ l!3°5(k + 2)! 2 2 -3 2 (3 - 8k + 32k 2 ) 3!3°5°7°9(k + 3)! (141) The development of the series for the G (u) of Eq. 140, together with additional expressions for A^(k), are given in Appendix VI. Comparison of the first few terms of the series for the G n (u) functions (cf. Eqs . 188 through 193) with the power series expansion of the cosine function. cos u = 1 - 2(u/2) + .6667U/2) 5333(u/2) + .. (142) shows that the higher order G n (u) functions appear to tend toward a cosine-function behavior near the origin, for G e (u) this is the case to 64^ essentially a thi rd- order approximation. The lack of a recursion formula for A_(k) in series (Eq. 140) prevents a conclusive statement in this regard. A tabulation of the G n (u) functions for the first nine orders and for u- values through 25 is given in Appendix VII. Values were computed from Bessel function tables and the finite series of Eq. 138 using the definition of h^Cu) given by Eq . 92. In such calculations for which the decimal accuracy was impaired using this method, as for the higher order functions near the origin, the series of Eq. 137 was found use- ful. Figure 16 shows a plot of the G n (u) functions corresponding to the tabular values obtained. The forms of the G^u) curves are of interest from a pattern view- point. They represent space factors whose main beams are directed more and more away from the aperture principal axis (u = 0) as the order n is increased. 1.2.5 Properties of the Aperture Distributions Corresponding to G n (u). It has been shown that to each orthonormal pattern function G n (u) there corresponds a field distribution over the circular aperture given by Eq. 123. These aperture distributions F n (r) have been calculated from Eq. 120, and are tabulated in Appendix VIII. A graph of these re- sults is shown in Fig. 17. It is of interest to note that each of the aperture functions F n (r) is an even polynomial of degree 2(n - 1), all of whose roots are real and half of which lie within the positive aperture interval (0, 1). 1.2.6 Synthesis Examples for Circular Apertures The method for synthesizing a given space factor has been described in Section 4.2.2. It was found that by using Eq. 113, the Fourier co- efficients A^ will determine the aperture distribution f(r) from Eq. 124 as well as the approximation to the desired pattern g(u). Synthesis problems for circularly symmetric apertures may involve the design of aperture distributions for producing approximations to such pattern types as sector patterns, triangular patterns, Gaussian- taper patterns, secant patterns, and others. Let us consider such designs in the fol- lowing discussion. (1) Sector Patterns This synthesis problem concerns the design of apertures which pro- duce essentially a constant pattern space factor over a specified sector ■65- ♦) * \ - 3 1 J x^ 3 id/ la ©/ 3 \. \ ^ 3 / to \ y 4 3 ■w/ /, 3 c5^ 1 1 1 i — . (M m u> m co en ao m to CO CO o o 3 o o X h- q: o CO UJ CD = 66- o in o m O in o m «• * ro IO CO CM m o in o m o O O — — CJ " r 1 r 1 / - ♦ F n (0 1 6 5 4 X 6 ^\ 3 2 Cn _F\ F2 1 F, ( ) .2 >\ \ .; 5 \ A ^/v- 5 > / < i 7\ ) lr i 3 \ 1 1.0 "f -I -2 -4 -9 -10 I- FIGURE 17. APERTURE DISTRIBUTIONS F N (R) CORRESPONDING TO G N (U) -67 4 of the visible range, and zero space factor outside that sector. This situation is represented in Fig. 18. Let us consider the problem of g(u) u (a) In rectangular coordinates. (b) In polar coordinates. FIGURE 18. SECTOR PATTERN. synthesizing this pattern by the method outlined in the foregoing sections. If the desired pattern is given by the function • g(u) = J 1 , £ u £ u tO, u > u , (143) the Fourier coefficients applying to this problem are found from Eq. 117, or for this case, A k G^(u) du (144) These coefficients may be found by substituting into this expression the series for G^(u) from Eq. 140. Since the orthonormal Gj^(u) functions are continuous and uniformly convergent, the resulting series for the coefficients A^ may be integrated term by term as follows: A, =N^ < T U ° f | -'^ |ll f.' A k 1N nn \ 2- , 9 Jo m = o m! 2 2m du = k¥ 00 (-l) m A, (m) \ u ° /12m " du m = nn „Tn (2m + 1) m! 2 SSL (~l) m A k (m) /u |2m Bl = ' (145) 68 ) This series converges reasonably quickly for moderate values of sector half-width u - Hence, for u = 4, evaluation of the A^' s from Eq . 145 produces the approximation to g(u) given by Eq. 116 or oo g(u) * Y A k G^u) (146) k = 1 = .9446 d + .5149 G 2 + .0261 G 3 + .0295 G 4 - .0325 G 5 + . . . , (147) The corresponding aperture distribution is given by these same A^- coefficients applied to Eq. 122, or, for £ r £ 1, f(r) = 2 [.9446 F x + .5149 F 2 + .0261 F 3 + .0295 F 4 - .0325 F 5 + ...]. (148) These results are plotted, using values for the Gj < (u) and the F^(r) functions tabulated in Appendices VII and VIII, in Fig. 19, together with similar results obtained for several other sector half-widths u . Note that the approach to the desired pattern becomes better as u is made larger, and that the larger sector widths demand more and more phase reversals of the aperture distribution. These curves are of course universal in the sense that they apply to any aperture size for a given wavelength; a larger aperture radius "a" will provide a greater visible range (3a along the u-axis, as discussed in Section 2.2.2.2. It is useful to note that the A^- coefficients for the sector pattern problem, determined by the integration of Eq . 145, could also have been obtained by graphical integration under the G n (u) curves of Fig. 16 if desired, for the integral (144) for A^ is merely the area under the G^(u) curve up to the value u = u . A planimeter may con- veniently be used to perform such integrations and will give sufficient accuracy for problems utilizing this synthesis method, while avoiding the need for evaluating series which may converge very slowly. (2) Triangular Patterns Also of interest are approximations to the triangular space factor of Fig. 20 which this synthesis method provides. Let the ideal space ■69- ♦ g(u) Ideal sector pattern u =IO Actual pattern u = IO 8 - 6 - 4 - fir) 2 - Jt><^$0 figure 19, approximations to sector patterns 70 ♦ factor be given by g(u) 1 - -H- Uo' , 1 u < u u > u • (149) The expansion coefficients for the orthonormal series which approximates this pattern are found by integration to be oo A k = Nn"n ;s £ (-l) m A k (m) u \2m m!(2m + l)(2m + 2) 2 (150) m = The approximation to the desired pattern of Fig. 20 is given by Eq. 146 in terms of the A^ of Eq . 150. For values of u = 3, 4, 5, and # U FIGURE 20, TRIANGULAR PATTERN. 6, the designs resulting from this expansion are shown in Fig. 21. It is of interest to observe the effect of parameter u G on the first side lobe and on the beamwidth in these designs. These properties are summa- rized in the following table, and a comparison with the (sin u)/u func- tion and with three integer-order J n (u)/u n patterns taken from the well- known class of Eq. 21 are given: Pattern Type 3 First Side Lobe Level Beamwidth, 3.0 2 u b Triangular, u = 19 db " 4 25 3.4 " 5 36 3.95 " ii = 6 36 4.7 (sin u)/u 13.2 2.78 Ji(u)/u 2 17.6 3.24 J 2 (u)/u 24 6 4.0 J 3 (u)/u 30 6 4.64 ■71 deal Triangular Pattern, u = 3 * o For u =5 J r For u = 6 I r 2 4 6 B 10 .2 .4 .6 FIGURE 21. APPROXIMATIONS TO TRIANGULAR PATTERNS •72- ♦ I From this comparison it is obvious that it is readily possible to find aperture distributions from certain synthesis examples which will pro- duce more favorable low side lobe conditions for a given beamwidth than the conventional (1 - r ) n smooth-taper distributions frequently mentioned in the literature. Hence, the triangular pattern case for u = 5 provides essentially the same beamwidth as the J 2 (u)/u pattern, while having a side lobe level better than 10 db further down. (3) Approximations to the (Sin u)/u Pattern Given that the space factor g(u) sin u u (151) is to be synthesized for a circularly symmetric aperture, the Fourier coefficients become, from Eq. 117, oo sin u u G^(u) du . (152) For this problem, instead of using the power series (140) for G^ (u), use Eq. 114 in terms of the Bessel functions h n (u): A k = n = 1 k /~oo bkn \ ""J"* h n (u) du . (153) Substitution of the spherical Bessel function for (sin u)/u, and for h n (u) with its equivalent from Eq. 92, yields for the integral of Eq. 153: rpoo | — — j | — —i Sia-a h n (u) du = ^ || u^ Jy 2 (u) 2 n n! u- n J n (u) = 2 n n! Ji 2~ n ~^ 2 >Trc r (n + l A) r(n + l)r(n + ^)T(l) du (154) ■73- This yields for the A^ coefficients of Eq. 153, k « I J kn n = 1 If <*(0). (155) The approximation to the (sin u)/u function of Eq. 153 then becomes the series oo g (u) -= i y_ G k ( ° ) G k (u) (156) k = 1 That this series converges precisely to the desired g(u) of Eq. 153 may be shown on the basis of the coefficients A^ of Eq. 155 satisfying Parseval's identity 00 I Ak = g (u) du (157) The integral of the right-hand side of this equation converges to n/2 for the function (sin u)/u of Eq. 151 in question. But substitution of coefficients Ak determined from Eq. 155 above shows that the sum of the left-hand side of Eq. 157 tends to n/2 as well; for example, seven terms of that summation have the value 1.567. The graphical results of approximations to the (sin u)/u pattern and its corresponding aperture distribution provided by series (156) and the form of Eq. 124 applying to this problem or f(r) V (yO) F k (r) , : r < 1 , r > 1, are shown in Fig. 22. The closeness of the approximation provided by only two terms of each series is indicated, and it is evident that six ■74- g(u) 2A K G K (u) 10 - 8 - f(r) =22A K F K (r) 6 2 - - + --Two terms Eight terms L « r FIGURE 22 APPROXIMATIONS TO (SIN U)/U PATTERN -75- * \ or eight terms of the series provide a very good approach to the pat- tern. For each term added, the aperture edge value at r = 1 becomes correspondingly larger until in the limit, the singularity of the Hankel transform of Eq. 151 is developed there. The series representations for the (sin u)/u pattern and its aper- ture distribution just discussed are of interest for at least two rea- sons. First, a function which can be arbitrarily closely approximated by means of a series of Gk(u) functions (other than the trivial case of a finite linear combination of G^(u) functions) is non-rigorously de- monstrated to exist. This is mentioned by way of contrast with the previous discussions on sector and triangular space factor syntheses, which were observed to yield oscillating approximations to given space factors. Second, it is observed that the approximation to the limiting forms of the sector and triangular space factor becomes precisely the result (156) for the (sin u)/u function as the design width u of those space factors is allowed to approach zero. This is proved by noting that the expansion coefficients Ak for these two cases become, as u - 0, proportional to G^(0), as for Eq. 155. It becomes apparent that attempts to synthesize excessively narrow-beam space factors will tend to produce high amplitude fields at the aperture edge. (4) Graphical Integration Methods in Circular Aperture Synthesis If a given pattern g(u) to be synthesized is of such analytical form as to cause the series of Eq. 117 for finding the Ak to converge quite slowly, or alternatively, if the given g(u) is merely an arbitrary curve for which no analytical expression is assigned, then the integra- tion of Eq. 117 can be effectively performed by graphical or mechanical means. To do this, one need merely modify the Gk(u) functions by multi- plying each curve of Fig. 16 by the g(u) to be synthesized, and plot the resulting product functions. The areas under the appropriate u- interval of these g(u) Gk(u) curves, which may be determined by means of a planimeter, are the desired expansion coefficients Ak of Eq. 117. As an example of a graphical solution, consider the synthesis of the given space factor g(u) = exp(-.179 u 2 ) , (158) a function of the Gaussian type, for which the 20% amplitude point occurs at u 1 = 3. The expansion coefficients Aj^ of Eq . 117 for the ■76- # series (116) representing this function are readily found by graphical integration of Eq. 117. Hence, multiplying each (^(u) function of Fig. 16 by the given g(u) of Eq . 158 above will yield a set of curves as shown in Fig. 23. The areas under these curves, obtainable by means g(u)G,(u) TI79U' -I-*- u g(u)Gjju) FIGURE 23. CURVES OF G(U) ^(U) PRODUCTS, FOR GAUSS i AN SPACE FACTOR. of a planimeter, are by Eq . 117 proportional to the expansion co- efficients A^. Normalizing the first coefficient so obtained yields for these coefficients: A x = 1, A 2 = .116, A 3 ; .0423, etc., and so the expansion for the approximation to the Gaussian pattern becomes g(u) ; G x + .116 G 2 + .0423 G 3 - .0141 G 6 + .004 G 7 0211 G 4 + .0106 G £ (159) This curve is shown in Fig. 24 compared with the idealized Gaussian function of Eq. 158 which it is observed to approach in the mean. Also shown are two more design examples for 20% amplitude points at u 4 = 4 and 5. Corresponding aperture distributions, calculated from Eq. 124, are also shown. Note that the highest aperture edge values are again obtained for designs for the narrowest beam, i.e., for u x = 3 in this set of examples. The first side lobe levels and beamwidths are tabulated here for these cases for the purpose of comparison with other results in this text: Uj (20% Position) First Side Lobe Level Beamwidth 2 ui 3 23 db 3.9 4 36 4.2 5 42 4.5 ■77 Gaussian pattern ir ffr) Aperture distribution for u,= 3 2 .4 .6 .8 for Uis4 i I u 8 10 12 14 16 .2 .4 .6 .8 I. FIGURE 24 APPROXi MAT SONS TO GAUSSIAN PATTERNS •78 g(u) ,' — Ideal Pattern Actual Pattern (a) Displaced sector pattern and aperture function. * g(u) Ideal Pattern Actual Pattern f(r) > (b) Displaced Gaussian pattern and aperture function. FIGURE 25. APPROXIMATIONS TO DISPLACED SECTOR AND GAUSSIAN PATTERNS •79 ' It is of interest to determine the effect of displacing the sector pattern of Fig. 18 away from the normal to the circular aperture. An example of this is represented by the dotted line of Fig. 25a. The coefficients of the expansion representing this pattern were found once more by means of a planimeter. The resulting expansion is g(u) = -.24 d - .45 G 2 + 6.96 G 3 + 10.13 G 4 + 6.17 G 5 + 2.05 G 6 + .42 G 7 + . 02 G 8 + (160) This result is represented by the solid curve of Fig. 25a. Note that the high amplitude of G* in series (Eq. 160) is brought about by the location of the lobe center of this sector pattern in the region of the principal beam of the G 4 function. The aperture distribution corre- sponding to this sector pattern is shown in Fig. 25a. A pattern having lower side lobes than the sector approximation of Fig. 25b can be obtained from the displaced Gaussian pattern defined by g(u) = exp[-.101(u - 7) 2 ] , (161) which has its 20% amplitude points located four units to either side of the main beam maximum, which in turn is assigned at u = 7. This space factor is represented by the dotted line in Fig. 25b and has the same shape as the Gaussian curve of Fig. 24, for the case u x = 4. (5) Secant Patterns The graphical method of synthesis, just described in connection with Gaussian-pattern approximations, is useful in the design of a "secant-pattern" space factor of the type shown in Fig. 26. This *-U FIGURE 26 SECANT PATTERN SPACE FACTOR space factor shape might be useful for applications requiring a circu- larly symmetric space factor possessing a gradual increase of radiated 80- $ power with increase in deviation from the principal aperture axis, and is analogous to the cosecant designs related to aircraft search antennas discussed in Sec. 2.3.1. A value u is selected to be at or near the limit of the visible range of a given design, and since the angular deviation 9 from the aperture principal axis is related to the aperture size through u = (3a sin 9, letting Pa = u provides the equation for the curve of Fig. 26: g(u) = sec [sin -1 £-] - (162) This curve may be truncated at some arbitrary u-value to avoid the singularity at u . Two secant-pattern designs are illustrated in Fig. 27, for u = 9.5 and 14.5, and with the secant curves truncated at u - %• The series coefficients were obtained by planimeter measurement as discussed above. Note that the approximations to the desired function become better with a larger u value. The aperture diameters required by these two designs are given by 2u /k, or about 6X and 9A. respectively. Sever- al phase reversals of the field across the aperture are necessarily re- quired by such designs in keeping with the condition of deflecting the principal beam away from the aperture central axis. The approximations to the desired secant space factors shown in Fig. 27 are obviously not very good ones. The reason, for this is that this synthesis method will not give good approximations to assigned patterns which undergo rapid changes in slope for only a small variation of u. A much better approximation is obtained if the assigned pattern has properties which this synthesis method is capable of reproducing rather well, e.g., has a principal beam or beams of sufficient breadth, has no sharp discontinuities, and has no sudden changes in slope. An example of an assigned pattern having such qualities is illustrated by the dotted curve of Fig. 28- This curve was arbitrarily sketched by hand, and has a beam maximum assigned at u = 11. The solid curve shows the approximation to this curve which is provided by the graphical synthesis approach used in previous examples, and is given by the series g(u) = 2.6 Gi + 3.86 G 2 + 0.86 G 3 + 12.7 G 4 + 25. G 5 + 20.8 G 6 + 10.5 G 7 + 3.7 G 8 + 0.72 G 9 + ••• . (163) 81 Ideal secant pattern g(u) 2 Actua pattern Ideal secant pattern 0^ I I I I I I I I LU I 2 4 6 8 10 *^*« f(r) 2.4 2.0 1.6 Aperture distribution Y^— for u = l4.5 1.2 \ .8 .4 \\>-^ for u o=9.5 ( _L l\ l\ 1 /l 1 > ""1 I ) .1 .2 \ .3 \4 / .5 .6^ ■Xj .8 > /.9 \l.O FIGURE 27, APPROXIMATIONS TO SECANT PATTERNS, 82- g (u) 10 — 8 — 6 — ^^ Arbitrary Curve - • • -\ / \ — '* / \ \ // \ w \ — Approximation Pattern w > 1 1 // — // // 1 // s / 1 I 1 _i_ _l _l i V i k^ 4 — 2 — 2 4 6 6 10 12 14 16 18 20 U f(0 i FIGURE 28, APPROXIMATION TO ARBITRARY CURVE 83 Comparison of this synthesis result with the secant pattern results of Fig. 26 gives an indication as to how the ideal secant patterns of the latter might be altered in order that a better fit may be obtained over the important part of the ideal curves. To this end, instead of termi- nating the ideal secant patterns abruptly in a discontinuity at u , the termination should be made rather gradual - for example, by means of a sloping straight line, or a curve of more nearly Gaussian form. Also, the truncation at the top of the ideal curves should be made broader so as to permit a higher beam-maximum value. The designs discussed in this section are suggestive of others which can be attacked by means of the computational or graphical methods described. 1.2.7 The Low Side Lobe Problem for Circular Apertures It is apparent from the discussion of the examples of the previous section that the synthesis method developed for circular apertures hav- ing circular field symmetry does not provide a direct way of obtaining an accurate control of the side lobe level. For one thing, the approxi- mation to a pulse pattern of diminishing width was pointed out in ex- ample (3) of Section 4.2.6 to approach a (sin u)/u pattern. This func- tion having a first side lobe level 13.2 db below the principal beam am- plitude, may be considered to be the limiting result of all broadside- beam, low-side-lobe designs based on this design method and being de- signed for excessively narrow beamwidth. Therefore a side lobe level considerably lower than this 13.2 db figure is attainable only at the expense of a broader beamwidth, and the precise side lobe level obtained will be determined by the nature of the given g(u) function upon which the design is based. It is furthermore evident that the so-called "optimum" pattern, an equal-side-lobe formulation based on properties of the Tchebyscheff polynomials equivalent to that discussed in Sections 2 and 3, cannot be found for the circularly symmetric aperture using this synthesis method. In spite of these inherent limitations of this synthesis method, low-side- lobe designs can still be achieved by trial-and-error methods based on observed properties of specific design results. For example, the triangular pattern designs of Fig. 21 were observed to produce low side lobe results. Also, two separate designs might be combined lin- early such that the resulting side lobe level is lower than for either 84- component. The pattern results of Fig. 21 also exhibit possibilities of this kind. Again, it is sometimes possible to combine some higher- order G n ( u ) function with a given pattern result so as to produce cancellation of lobe amplitudes in the visible region, while the main beam of that GLCu) function is assigned to the invisible region. This method possesses the disadvantage of requiring rapid amplitude changes of the aperture distribution which result from the addition of the particular high-order F n (r) aperture function corresponding to that CL(u). Such designs are also inclined to be quite frequency-sensitive in practice. 1.2.8 Summary of Design Method The design examples of the previous section serve to demonstrate that once the orthonormal set of space factor functions G n (u) and the corresponding aperture distribution functions F n (r) were developed, the procedure for designing an aperture distribution f(r) to produce an approximation to a given space factor g(u) reduces to the relatively simple Fourier process for determining the unknown coefficients of both functions . For convenience, the design procedure is summarized here as follows: (1) Given: a real space factor g(u), expressed either analyti- cally or graphically. (2) Coefficients A^ are to be found from Eq . 117. If g(u) is expressed analytically, the integral for successive A^, A k =\ g( u ) G k (u) du , k = 1, 2, ••■ (117) may be evaluated by substituting for successive G k (u) with their infi- nite series expressions from Appendix VI, and then integrating term by term. If these results converge too slowly, or if g(u) is expressed graphically, curves (or tables) of the successive product functions g(u) Gk(u), for k = ■ 1 , 2, ... , may be formed. Integration under these curves in accordance with the interpretation of Eq. 117 above may then be carried out by any convenient means (such as a planimeter) to yield the A^-values of interest (3) The approximation space factor is now given by the series of G^(u) functions to which there correspond significant values of Au ; ■85- hence, oo g(u) - \ A k (^(u) , k = 1 (116) where the G^(u) functions may be taken from Fig. 16 or Appendix VII. (4) The aperture distribution which produces the approximation space factor (116) above is r f(r) Y A k F k (r) , £ r < 1 k = I , r > 1 , (124) where the F k (r) functions may be taken from Fig. 17 or Appendix VIII. % 86 = 5. CONCLUSION The purpose of this report has been to develop new approaches and techniques of synthesizing rectangular and circular aperture antennas having certain symmetries. Methods of aperture design and synthesis which have been achieved in recent years in the field of microwaves have also been reviewed in an effort to provide the reader with a reasonably comprehensive background for the synthesis methods proposed. The synthesis method for tapered side lobe designs derived from the Tchebyscheff polynomial as proposed in Section 3 is seen to provide a method for designing rectangular aperture distributions which produce a given first side lobe level and a beamwidth compatible with dis- tributing the zeros of the space factor inside the visible region such that a supergain solution is not developed. The corresponding aperture distributions for such designs are observed to taper to essen- tially zero at the aperture edge, and so they correspond to aperture con- ditions which are more nearly physically realizable than distributions having a substantial edge discontinuity or pedestal. Comparison of these designs with low side lobe designs provided by smooth taper aperture functions such as power-cosine distributions shows that a first side lobe level of the order of 10 db lower than that provided by power-cosine designs is obtainable for a comparable beamwidth. Also, the differences between the aperture distributions for these Tchebyscheff- polynomial -derived designs and power-cosine designs providing equal beamwidth is quite small in spite of the differences in side lobe levels noted, which points out the inherent difficulty of solving the low side lobe problem from a practical standpoint. Comparison of the tapered-side -lobe, Tchebyscheff-polynomial-derived designs developed by the author with equal-side - lobe approximations discussed by Taylor shows that slightly smaller beamwidths are avail- able from the latter. This condition is due to some of the total radi- ated energy being distributed among the more remote side lobes within the visible region, those lobes being considerably larger for Taylor's results than the corresponding side lobes obtained from the tapered side- lobe designs. Taylor's results are quite important from an academic point of view in providing an approximation to the optimum synthesis problem for rectangular apertures. His calculations show that a more •87- or less considerable edge discontinuity can be expected from typical optimum designs The tapered side lobe designs provide more nearly physically realizable aperture edge results, although at the expense of beamwidth. The synthesis method discussed in Section 4 provides a new class of solutions for circular aperture antennas having fields of circular symmetry. Besides providing a mean-square approximation to patterns of arbitrary shape, it also offers low-side lobe, narrow-beam solutions which represent an improvement over existing designs The labor in- volved in obtaining the orthonormal functions essential to this method was observed to be rather extensive, but once the functions were ob- tained, the additional work needed to synthesize a prescribed pattern is relatively small. Furthermore, the functions involved are inde- pendent of aperture size, which permits using the same orthogonal set for all designs. To accomodate a given aperture size, one need only assign the appropriate limit of the visible region to the u-axis for the space factor. Since the method is designed for the case of plane- wave aperture excitation, it is evident that it might also find applica- tion in sonic or optical field problems involving circular symmetry. It seems reasonable to suspect that this synthesis method for circular field symmetry, utilizing an incomplete orthonormal set of space factors as a basis for the synthesis method, might be extensible to variations of this problem involving other aperture symmetries, or perhaps aperture excitations which are not necessarily plane-polarized. To establish such a synthesis method, it would be necessary to find an appropriate class of integral representations of far zone space factors in terms of their corresponding aperture distributions, and provided the appropriate orthogonalization integrals could be evaluated, an orthogonal system of space factors might be constructed from that class using the Schmidt method. The synthesis procedure would then follow, in terms of the familiar Fourier processes Complex field variables might be used if desired, since orthogonality conditions apply equally well in such a case. In the light of the theoretical results which have been supplied by the synthesis methods described, there yet remains the question of finding practical antenna designs which will produce the circularly symmetric amplitude variations over the aperture as required by the de~ 88 signs. Also, it is recalled that one of the requirements of the aper- ture fields for these designs is that such fields be transverse over the aperture. This condition is generally not the case for aperture antennas currently used in practice, e.g., metal-plate lenses or parabo- loid reflectors fed by means of small horn primary sources. Recent work has shown that plane-wave fields are obtainable from special primary sources which use a combination of an electric dipole feed coupled with a complementary source amounting to a magnetic dipole. 8 Such sources, used in conjunction with suitable lens designs, may provide a practical solution to many of the pattern design results provided by the synthesis method discussed. 89 APPENDIX I THE Q N (w) FUNCTION AND TABULATION The general expression for Qm(w), w integral or not, is the un- bracketed factor of Eqs. 32 or 33. But for w an integer, say w = w 1} the expression contains the limit, evaluated using L'Hopital's rule: lim sin Ttw = cosjrjvi w-*w 1 rc(w 2 -w 1 2 ) 2wi (164) Hence the desired expression, for w x a non-negative integer, is JL (-1) 2 (N/2)! cos TCwi Qm(W!) = — — 2w 1 2 {(w 1 2 - l 2 )( Wl 2 -2 2 )»-[ Wl 2 -(w t - 1) 2 ]}{[ W1 2 -(w! + l) 2 ]-'[w 1 2 -(N/2) 2 ]} ... ... (165) This is alternatively written, for w 1 a positive integer: Q N (wJ (*)(| - 1)-"(| - w t + 1) (§■ + !)(«■ + 2)---(| + wj (166) Noting also that 0^(0)^ 1, from Eq. 165. A tabulation of Qj^(w) for w an integer in the useful range is given in Table I. Table I. The Qyj(w) Function ±w N = 8 N = 12 N - 16 N = 20 N = 24 N = 30 1 1 1 1 1 1 1 .8 . 85714 . 88888 . 90909 . 92307 .93750 2 .4 . 53571 ■ . 62222 .68182 . 72527 .77206 3 . 114284 . 23809 . 33939 . 41958 .48351 . 55760 4 .014285 .07143 . 14141 .20979 .27198 . 35217 5 .01299 . 04315 .083916 . 12799 . 19369 6 • .00108 .00932 . 026223 . 04977 . 08895 7 .001243 . 006170 .015718 .03773 8 7.77 x HT 5 .001028 .003929 .01312 9 .000108 .000748 .00383 10 5.41 x HT 6 .000102 .000919 11 8.87 x 10 -6 .000177 12 3.65 x HT 7 2.62 x 10 13 2.80 x 10- 14 1.93 x 10- 15 6.45 x 10' 16 —9 90 APPENDIX II TABLE II. ORDERED ZEROS AND RELATIVE MAXIMA OF T n (x) T 3 (x) T e (x) T 8 (x) T, o(x) k x ok x mk x ok "ink x ok "ink x ok 'Snk 1 .8660 .5 .9659 .8660 .9808 .9239 .9877 .9511 2 -.5 .7071 .5 .8315 .7071 .8910 .8090 3 .8660 . 2504 .5556. .3827 .7071 .5878 4 -.2504 -.5 . 1951 .4540 .3090 5 -.7071 -.8660 -.1951 - .3827 . 1564 6 -.9659 -.5556 - .7071 -.1564 -.3090 7 - . 9659 -.8315 - .9239 - . 4540 -.5878 8 =.9808 -.7071 -.8090 9 -.8910 -.9511 10 -.9877 91 APPENDIX III CIRCULARLY SYMMETRIC APERTURE DISTRIBUTIONS AND THEIR CORRESPONDING SPACE FACTORS f(r), - r 1 1 g(u) 1. 2. 1 3. 2 r 4. 4 r 5. 2 U (1 - r r 6. 2 -H (1 - r r* 7. 2 Jf (1 - r ) 8. cos mrtr 10. - 1 e ar cos a v/l - r : r yr eo z / l^ k 2k ( -1) u k^O 2 2k (k!) 2 (2k + m + 2) J t (u) u 2 J 2 (u) _ J 3 (u) u 2 u 8 J 3 (u) 8 J 4 (u) u V - 1 u - J 5 (u) ■ rUjiii^, Relixl > - 1 u sin u in vu 2 2 + a / 2 2 v u + a k + Z. E 2k k =0 P = l 2 k! (mrc) k - 2 P + 1 (2k - 2p + 2)! -1 a A 2k+l , ( 2k + l )!u 2k fi o L la ( ^ L) 2k., ,,2 2k a^ k=0 p=0 2 (k!) a 2k + 1 - p 2k + 1 _a_ (-1) (2k + 1)! 2k + 2 (2k + 1 - p)! K (u 2 ♦ a 2 ) % + a (u 2 ♦ a 2 / 72 - a 2 J ° j Jo 2 -92- APPENDIX IV EVALUATION OF COEFFICIENTS RELATED TO DEVELOPMENT OF ORTHONORMAL SET [ G] . Evaluation of N mn From Eq. 106, and using the expression for ^ from Eq. 94, one may write f N mn KPn du n-l N N i - s ^^ k=lN kk R mn (167) It is convenient to evaluate N mn in the order N m ^, N m 2» ... so as to obtain a recursion relation. First, N m ^ = R^. Then, from Eq. (167) above, whe re Hence, N m2 = R m2 "l - N 21 N n,l' N n R m2 N 21 = R 2 i , 2 N»l - 5»1 _ 3 (2m + 3) N n Rli 3 7 5' R m9 "' R m9 2 9 (m + 1) 2 (2m + 3) N m2 = R m2 1 - 5(m + 1) * Next, where and = R 1 m2 5(m + 1) N R 3 1 ml N LI K: N32 N m2 N 2 2 R m3 N 31 N ml _ R 31 R ml _ 2-3 (2m + 5) (2m + 3) 5-7 (m + 2) (m + 1) N ll R m3 R ll R m 3 N 32 N m2 - N 22 R m3 R 3 2 j* R 2 2 2 R m2 (m - 1) 2(m - l)(2m + 5) R m ,5(m + 1) 3-5 (m + 1) (m + 2) 93 Hence, N , = R_ a. i - 2-3(2m + 5)(2m . + 3) 2(2m + 5)(m - 1) 5 7(m + 2 Km + 1) (m - l)(m - 2) 3-5(m + D(m + 2) 3'7(m + l)(m + 2) In a similar fashion one obtains (169) N m4 = R m4 5(m - l)(m - 2 Km - 3) 3-ll-13(m + D(m + 2)(m + 3) (170) Finally, by induction, l-3-5 — (2n - l)-m(m - l)(m - 2)-"(m - n + 1) N = R mn mn (2n - l)(2n + l)----(4n - 3)°m(m + l)(m + 2)----(m + n - 1) (2n - l)![(2n - 2)!] 2 m!(m - 1)! ■ R mn (4n - 3)![(n - l)!] 2 (m + n - l)!(m - n) ! (171) The following result is also useful for finding C and for normal- izing the B-coef ficients : 2 (172) N nn " R nn n[(2n - 2)!] (4n - 3)! 2 Evaluation of C mn From Eq. 105 and using the results above, -N mn /N nn -2 mUn- l )![(m - l)(m - 2)— n1 3 -m(m - l)-°-(n + 1) n(2m - l)i(m - n) ! (2m * 2n -• 1) (2m + 2n - 2)---(m + n) 4 ( m - n ) -2 mm! f(m - PiT (4n • l)!(m j m -1) ! (m - n)! nn! [(n 1) !] 3 (2m 1) ! (2m + 2n - 1)1 ' m -» n. (173) -94- 3 Evaluation of B, mn These coefficients are readily evaluated on the diagonals of the triangular matrix corresponding to the sequence of equations (108). The B values are given by Eq. 126. Hence, on the principal diagonal, (174) B mm 1, for all integers m - 1. The next diagonal, for which m - 2, yields, using Eq. 173 above: B m (m- 1 ) m(m- 1 ) = -2V(m - 1) (4m - 3) (2m - 1) Again, for m - 3, and using the previous result: B " C m(m-2) m{m-2) Y + m(m- 1) g C ,»(m-2) (■-!>(■-!: where it has already been determined that C and that 2* m 2 (m - l) 3 (m - 2) mU-2) 2! (4m - 5) (4m - 7) (2m - 1) (2m - 3) (175) (176) (177) = 2! (4m - 5) (4m - 6) (4m - 7) C m(m-2) 23 < 4m - 3) (m - I) 2 (m - 2) Thus Eq. 176 becomes I _ 2 (4 m 5) B = C m(m- 2) m(m- 2) (4m 3) 2 m (m 1) (m = 2) 2! (4m 3) (4m - 5) (2m - 1) (2. In a similar manner, one obtains if m (178) 3) (179) B o e 2 I - z m (m 1) (m 2) a (m - 3) (m-3) 3! (4m - 3) (4m - 5) (4m - 7) (2m - 1) (2m - 3) (2m • 5) (180) 95 and finally by induction, for the (m k)th diagonal, where m > k: B (-l) k 2 2k m 2 r(m - l)(m 2) — "(m - k + 1)T (m - k) m(m-k) k![(4m 3) (4m 5)- (4m 2k ■ 1)] [(2m l)(2m • 3) • • • -(2m -2k+ 1)] k al-2 4, = (-1) 2m (m - kU(m - DM (2m - 2k)!(4m 2k - 2)! (2m - l)k! [(m - k)!] 4 (4m - 3)!(2m k 1)! ' n , , (181) This result, the one desired may also be written in terms of row m and column n of the matrix: , ,1-n 2u-2n r ...3 B = LJJ I n [m(m ■ l)(m - 2) — (n + 1)1 mn m(m - n)! [(4m - 3)(4m - 5)— (2m 2n - 1)] [(2m - l)(2m - 3)*°°°(2n + 1)] = (-l) m " n 2 4m " 4n " 2 m 2 n r(m - 1)!1 4 (2n)!(2m + 2n - 2)! ; (182) (2m - l)(4m - 3) ! (m n)!(m + n - l)!(nl) 4 ' if the integer m > n 96 APPENDIX V TABLE III. COEFFICIENTS b nk OF THE ORTHONORMAL FUNCTIONS G n (u) bii - .433 012 702 b 2 i = -1.620 185 175 b 3 i = 3.211 308 144 b 2 2 = 1.518 923 601 b 32 b 33 = -8.429 683 878 = 5.268 552 424 b 41 = -5.123 475 38 b 5 i = 7.310 095 716 b 6 i = -9.740 380 b 42 = 25.937 594 11 b 6 2 = -60-308 289 85 b 82 ■ 118.710 879 7 b4-3 = -39.626 879 90 b 5 3 = 163.334 951 7 b 6 3 = -494.628 665 3 b 44 = 18.781 489 95 b B4 = -178.647 603 4 b 8 4 - 919.700 174 4 t>55 = 68.332 708 29 b 6 5 bee - 786.343 649 - 252.285 254 1 b 71 - 12.392 033 7 b 8 i = -15.247 950 54 b 9 i = 17.443 089 3 b 72 = -209.115 569 b 82 ■ 340.219 899 2 b 82 ■ -497.128 044 6 t>7 3 = 1234.362 73 b 8 3 = -2693.407 538 b 83 = 5047.848 789 b 74 ■ -3420.213 41 t>84 ■ 10310.700 73 b 9 4 ■ -25532.832 97 i>75 = 4848.152 74 t»86 = -21343.150 52 b 96 = 71811.092 73 b 7 e = -3407.173 85 b 8 e ■ 24455.693 30 b 8 e = -118488.303 b 77 = 941.608 418 b 8 7 - -14598.551 6 b 8 7 = 113954 311 8 b 8 e ■ 3543.733 23 b 98 b 99 ■ -59139.235 48 ■ 12799.811 38 Note that the coefficients b n j { listed here omit a common v^T factor. Since this factor is common to every G^(u) expression, it may be ignored in any synthesis problem, or the final space factor may be multiplied by vK if correct scaling is desired. 97 APPENDIX VI ORTHONORMAL G n (u) EXPRESSED AS SERIES Where the series expression for h k (u) defined by Eq. 92 for in tegral k is h. (u) = 2 k k! u" k J.(u) oo . . m 2m k . C ( ~ u u "-° 2 2m m! (k + m) (183) then combining this series into Eq. 113 for G n (u), one obtains, upon interchanging the order of summations, G „< u) - N * 1 ^ nn jf=\) k! 1! B nl 2! B n2 (1 + k)! (2 + k)! + . o .+ n! B. (n + k)! V 2k ( 184) Eq. 184 may also be written G„(u) N nn L k-0 (-1) A (k) ,„ 2k n nn ifrft u\ v 2~' ' k! '*' (185) /here A^(k) represents the bracketed rational expression of Eq.184, or A n (k) - Z J m! B„ i= l (m + k)! (186) These A n (k) may be evaluated by inserting appropriate values of B nm to yield, for the first six expressions: 1 A ft (k + 1)! -2(1 ± 8k) l!3°5(k + 2)! 2«2 a = 2*3 (3 - 8k t 32k^) 2!3-5-7-9(k + 3) ! A 4 -~ 2 3 (75 + 784k - 576k' + 512k°) 3!3°5-7 13(k + 4)! e„3 _2 A = 2°3 5 (735 - 2992k + 14464k - 5120k + 2048k ) 5 4! 3 -5 7» -17(k + 5)! A = -2 3 5 (19845 ± 245976k - 269120k + 343040k - 71680k + 16384k ) 5!3»5-7»--21(k + 6)! (187) 98 No recursion relation has been obtained for successive /^(k). Series expressions for G n (u) are then as follows, for the first few terms: d(u) = .433 013vn [1 - 5(u/2) 2 + .08333(u/2) 4 - . 006 9444(u/2) 6 + .000 017 36111U/2) 8 - •••]. (188) G 2 (u) =-.101 262v^k [1 - 3(u/2) 2 + .708 333(u/2) 4 - .069 444(u/2) e + .003 81944(u/2) 8 - 1.355 82 x i(T 5 (u/2) 10 + •••]. (189) G 8 (u) = .050 177^ [1 - 2.25(u/2) 2 + .958 333(u/2) 4 - .123 611(u/2) e + .008 15145U/2) 8 - .000 393849(u/2) 10 + ■■•■•}. (190) G 4 (u) =-.031 271v^ [1 - 2.12(u/2) 2 + .763 333U/2) 4 - .117 lll(u/2) 6 + .008 85020(u/2) 8 - .000 393849(u/2) 10 + 1.159 06 x 10" 5 (u/2) 12 - 7.602 87 x 1(T 7 (u/2) 14 + ■ • •] . (191) G 5 (u) = .022 6247v^ [1 - 2.07142(u/2) 2 + .719 388(u/2) 4 - .120 629U/2) 6 + .007 81351(u/2) 8 - .000 370196(u/2) 10 + .000 116556(u/2) 12 - 2.60342 x 10~ 7 (u/2) 14 + ■ • •] . (192) G e (u) - -.016 386^ [1 - 2„04762(u/2) 2 + .700 539U/2)* - .096 199(u/2) e + .007 11513(u/2) 8 - .000 331771(u/2) 10 + 1.05629 * l(T 6 (u/2) 12 - 2.42166 x io- 7 (u/2) 14 + 4.16657 x UT 8 ^) 1 - 5.56362 x 10 _11 (u/2) 18 + •■•••]. (193) 99 • CD ^ r~~ t- r— no no CO •* M U\ vo o o oo o lo o no i— i co r— no cm t— LO LO NO ON NO <— I NO ON CO © "— I ' — I rH ■— I 00 00 CO rH ON CO NO LO r- 1 T? co lo co on r~- CO LO CO on <~ • r— I O O O r— ' I I I OD ^ no o t— lo Tf r— tJ NO Tf CO NO CN O LO Tf O NO CO 1—1 o o <— I o o O o CO o o o i— I CM CO NO CO h m in -* o © r— CN O i— I |— I CO O i— I i— I O O O O CO NO CM CN ■— I Tf CM NO r~ OO t— C— O CM LO On co rH co r- CM O O O O i— I OO i— I CM CM O Tf O 00 CM >—i NO CO CM ON t— t— ON CO CM Tf r— I rH rH rrA q co oo co Tf i— i LO CO r— I NO CO i— I NO © Tf 00 Tf o cm r~ © O i — I i — ioo C3 t c O U* o C/3 3 W 3 in -J a ^ ^ ^ I— I ON ON O C~- NO t-~ LO NO LO oo co lo t— o ■— i CM NO LO CM CO Tf ■— I O O ■— I o o O O O o o o I I no ■* m in h CO ON LO LO f— I CO CO NO i— I CM t— I CM no oo r- no o lo r— I O O I— I r— I O O O O o o o CO O rH 00 iH CO NO ON Tf © ON © CO Tf t— NO T* i— I i— I rH ON "— I CM t— I CM r- 1 O CN] r— I r— I o o o o o o t- m ^ in o CO O ON Tf r~ LO NO NO CO oo CM ON O LO t— r- 1 O O O O o o o o o CO LO NO i— I CO CO ON t— O ON NO CM Tf CM CO Tf NO On Tf iH f— I O ■— I LO ON o o o O o ON CO Tf NO ON i— I O t"- ON NO o i— i r- on o "if h^o in CO LO t— CM CO O O O •— I i— I CM Tf -*f sO CM CO NO rH O CM t— CO Tf CM ON r— I CO r— NO Tf ■<* 0O CO I s - On O O i— I r— I rH CO Tf CM CO O LO O CO O rH O »* NO NO CM -H CM CM NO Tf CO t~- O ON CO i— I rH CM I— I >H CM NO LO t— CO O CO CM rH LO © Tf f- CO CO CM NO © O CO CO rH CM CM CO CM CM rH CO rH CM ON NO CM NO O r— ( O ON ON Tf O CO 00 rf CM O r— I NO Tf 0O CO Tf LO CM rH tH O O H i— I CM CM CO rH NO f- t— CO ON co co co m Tf CO ON r~ NO NO (N CO "* H H O © rH r—\ O I I I I t— CO CM LO o O SO rH Tf O LO NO ON NO r— Tf NO I—l f— >H LO t~ Tf rH O rH O O rH iH CO t— O CO CM r— I 0O ON f- CO *f t— CO O Tf LO NO iH ON rH CM LO Tf © © I— I O O r- < r— I I 1 CO O 00 CO CM rH O rH LO O CO -H O O O LO CM rH NO NO CO CM t— CM ON CO rH o O O 1 ! co Tf o co co co no o co o r- 1 ON on Tf i — I Tf CO NO CO CO rH r- ON Tf CO o o o o o > x a Mf rH CO rH NO ON O t— CO LO CO LO NO cn h m to r- co rH NO Tf CM NO O CO r— I rH CO CM CM O O O o o o NO Tf CM CO iH •— I O t— CO On CM NO i— I O CO CM CO ON CM Tf LO CO CM t~- i— I O i— I CM CM CM R rH r— no r— o cm co o CO ON "* NO Tf O CN ON rH ON f- 00 LO CM O O O >H I— I O rH Tf CO NO Tf ON r— rH f— i—< © c— Cxi r— © rH O NO CO CO Q\ rH Tf Tf CO O rH O O O Tf NO ON CO CM O LO rH Tf Tf CM LO CO rH CO co co in on in LO rH NO LO O o o o o o O Z < a * t— CM CM NO o c— C— NO LO Tf o o rH oo Tf no o r— O Tf O rH ON rH LO CM CM CM LO ON O O O O O I— ( CO ON LO O CM NO ON CO O CO rH ON NO CO O CO NO CO CO ON ON CO NO O CM CM CM ■— I O •— 1 O rH o f- CM CM \C CO ON CM LO CM LO i— • 00 t— rH CM f- o CO LO LO ON NO rH O O O O O 00 CM CO CO r— o co oo no Tf ON Tf LO CO LO NO t~- CO LO rH NO LO O Tf o o o o o CO t— NO CM O LO OS O LO CO NO NO ON CO ON o co on co co in f— I CM Tf rH o o o o o 3 CM o H. cm o in co co on no on lo r— Tf cm CM NO Tf o O ON ■— i on r— o O Tf O CM CO ON CO CO H O r—t CM CO CM r- o cm co Tf Tf CO ON ON LO Tf i, in Tf lo co r— cm no no no NO t— "—I NO rH O O rH O O r-i <— I rH LO CM Tf t~- r~- on no O •—* 00 CM O NO CO rH O CM no in o Tf tt o o o o o ON Tf O f— ON no r— NO H ON Tf cm in r— no ON LO CO CM LO O CM CO rH rH o o o o o co co on co i— i NO NO rH rH CO ON ON CM t— rH nO co on i— i Tf CM i— I O CM i— I O O O o o I I c * CO LO On 00 ON 00 i— i on cm r— on co o o r— co cm r-- CO rH ON t~- Tf NO co oo Tf ctn iH in Tf CO CM o o o NO CT\ O LO LO co r- o co no on lo Tf no r— on o in co co CO O CM CM O o o o o o CO NO Tf o CM r— I CM 00 LO Tf O rH NO CM CO OO NO Tf 00 •— I rH rH O O i— I O O o O O CO LO LO CO Tf ON t~- Tf rH ON co on o co co Tf Tf ON Tf CM O CO O CO CO O O O o o t— CO CO CO CM LO rH CO LO Tf O NO Tf LO CO t^- Tf rH LO ^f o o o o o o o o o o I o h n ro ^ in NO f- CO ON o rH CM CO Tf LO no r— oo on o rH rH rH rH CM rH CM CO Tf LO CM CM CM CM CM 100 <7\ o On CM CM LO t— LO LO LO o f— CO CM ON r— 1 LO CO no LO CO in r— i— CM OO LO CM LO CM CM oo co rt" i—i CO CM i— i LO ON CO On ■«* o CO LO i— 1 o r- I— 1 NO 00 r~ i—i r~ LO LO NO ON CM CM oo oo r~ O o ON CM r~ NO CC oo I— oo o i—i m r- r~- cm CO CM r~ co ON co CM CM co co O 3 LO t~- r-H r~ LO r- 1 t— s LO co ON x* i— i co o CM 00 lo t— o NO o o NO On LO «* i—i co CM r~ CM NO o o oo o o co LO CM NO LO CO CM CM CM CM rj" r}< rj< tj" LO t— r- oo o 00 ON LO CM o# a CO ON NO r- NO co LO NO o ON -* LO CO 1— t ON o o ■^f ON CM co ON r— © 00 I-H ■>* o LO r- co r- LO ON O CM co o co r- CM 00 co T* LO 1— 1 ON o 1— 1 NO o x* oo I-H 0Q CO 3 ■^ r— co 5 t^- co NO CM O co 3 NO 1—1 o Tf ON oo r— co 1— i-H •*r LO 1—1 CM t— CO f— 1 c^ r— 1 1—1 r~ co ON NO CM r- co CO I-H 1—1 NO LO CO 00 CM \o ■^ r~ co •<* CO On co r— CM 00 o CM CO O CO OO co © ■^ ON t— co o CO CM o rA CO CM NO CO co ON CO ■>* co CM CM CM CM CM CO CO CO LO CM NO rt OO NO r— LO LO co LO ■<* co CO LO 1—1 1—1 CM CO t- NO CM o CO CM 00 CM i—l 1—1 NO CM •*P oo u co CO 00 LO CM o NO ON OO ON 1— t— o NO CM CM ON X* LO CM t— 1—1 CO t~- 00 co "<* I—I LO ON co On NO 00 t— o o rf a co oo -+ lO NO 1—1 ON 1—1 NO NO CM LO NO On LO o CM ON 1— 1 On NO ■* o ^■^ CO CM 1— 1 00 LO LO CM co ON CM o 00 ■<* r— I—I CM ■«* c— co LO CO CO & o NO 00 r^ "* r- »* LO CM CM t— CM o 00 ON CO CM oo NO NO 0\ co NO oo •<* t— co 1— 1 CM CM On co co i—i -** LO t~- CD rt ■«# co rH 1—1 1 1—1 i 1 i—i 1 1— 1 1— 1 1—1 1 I— 1 i I—I 1 [ ! r— 1 CM CM CM CM 1—1 i CM ] LO i ON o u o LO I—I r- t— CM CO NO LO i-H r— 00 ON CO LO co i—i i—i LO co r~ o 00 CM O ON oo LO co co oo I—l LO NO ON NO ON i-H CM CO "5f NO ON r- 00 00 co LO ON <«* o 1—1 LO t— 1—1 CO o LO ON ON 58 LO NO © o LO 00 o CM LO ■«* o CO i-H o ON co 1— 1 r~ NO CO 00 CO 1—1 ON CO ON 1— 1 oo co ON ON CO t— co NO CO LO LO o CO 00 LO 1—1 NO CO CO CM LO 1—1 ■<* CO LO 1—1 CO ON t- LO LO CO LO CM ON CM 00 ON o o i-H o I—l CO fcu co CO CM CM i—i i-H 1 i-H 1 1—1 1 I 3 1— 1 I—l I—l • 1— 1 1 1—1 i 1—1 1 CM I CM 1 CM 3 1— 1 I I 1—1 co LO r- 1 u CO o On ON ON CO ON CO ON NO CM i—l NO CO LO i-H NO i-H LO On CM CO LO NO CM ON CO NO o LO CM NO o 00 co o co o ON co CO NO LO CM 00 NO CO CO o NO co 00 CM LO o co CM ON CO i-H o CO r~ co ON 00 o I—l NO 00 NO co LO LO On 1—1 NO CM c— CM CM NO CM o LO NO On NO 1—1 ON CO 1—1 LO NO LO ON LO o CM 1—1 CO co t— LO CM i-H NO CM p— 1 ON o LO t— CO LO 5p oo ON o LO CM NO CO 1—1 co co CM 1—1 I—I CM CM CM CM 1—1 t— 1 i-H 1—1 I—l i-H I—l 1— 1 I— 1 i-H r— 1 1— 1 rH CM CO ■<* LO a z a Q NO NO CO f— 1 LO LO r— 1— 1 ■*f CO LO ON ON CM r~ LO co Oh CO CM r— i—l i—i CO I— ON CO r- t— i—l NO r- CO NO co co ON NO On LO LO LO NO NO On co NO LO 1—1 o i—l CM NO NO o r— NO 1—1 "* o co r~ CO CM co CO r— t— r~- i—I i—l CM 00 -t+ co LO ON s LO CM o t— CM oo -tf co t— NO o s CM i-H LO CM i—l co ON a LO 00 O CO -* LO CO LO t— co p— 1 00 LO o i—l r- ON NO r— 1 CO t- CO i— 1 i— i i—l o CO LO ON LO CM 1— 1 LO 00 I—l CM CM I—l CO NO co CM r-i CM LO ON "* t— ON CM LO 00 CM CM CM CM r— 1 I—l 1— 1 1—1 I—l i—l i-H 1— 1 i-H 1—1 CM CM CM CO NO NO CO 1—1 LO On NO o CM CO 1— 1 t— t— o r~ ?£ NO r- r—l 1—1 00 •«* On CM «* ■* CO co CM CM LO CM 3 CM i-H o o 1—1 •cf r— oo CM ^ co •<* 00 CO X* ON r- 00 ON i-H ** CO co CO LO NO LO CM r— o i-H co rH LO ON oo i-H i— i co ON LO LO ON CO NO CO t~- i-H lO i— 1 CM OO 00 -rf -* CO i-H NO "<* LO 1— ON Tf CM co NO S3 CM o ON On o -tf CO CO LO CTN CM co r-- ON CM CM CM h- CM co <* LO NO 00 3 ON LO CM o ON CO NO x* CO i— i o CM LO NO t— r— 00 ON o r— 1 CM CO co -* LO NO o CO co o co co • LO o o LO i-H CM LO LO CM CO CO LO LO LO LO CM •>* LO NO oo CM Tf LO ^D r- co o "* LO LO NO NO t»- r- 00 OO 00 oo oo 00 ON ON ON ON ON On ON On -101 REFERENCES 1. Allen, C. C. Radiation Patterns for Aperture Antennas with Non- Linear Phase Distributions. Convention Record of the I.R.E. , Part 2, 1953. 2. Ashmead, J. and Pippard, A. B. The Use of Spherical Reflectors as Microwave Scanning Aerials, J.I.E.E. 93, Part IIIA, 1946. 3. Barrow, W. L. and Greene, F. M. Rectangular Hollow-Pipe Radiators . Proc. I.R.E. , 26, Dec. 1938. p. 1498. 4. Booker, H. G. and Clemmow, P. C. The Concept of an Angular Spec- trum of Plane Waves and its Relation to that of Polar Diagram, and Aperture Distribution. Proc. I.E.E. 97, Part III, Jan., 1950. p. 11. 5. Bouwkamp, C. J. and De Bruijn, N. G. The Problem of Optimum Antenna Current Distr ibut ion. Phillips Research Report, 1, 1945. p. 135. 6. Brown, J. The Effect of a Periodic Var iation in the Field Intensity Across a Radiating Aperture. J.I.E.E. 97, Part III, Nov. 1950. p. 992. 7. Chait, H. N. A Microwave Schmidt System, N.R.L. Report 3989, May 14, 1952. 8. Chlavin, A. A New Antenna Feed Having Equal E- and H-Plane Pat- terns. Trans. I.R.E., AP-2, No. 3, July, 1954. p. 113. 9. Chu, L. J. Microwave Beam Shaping Antennas . M.I.T. Research Labora- tory of Electronics, Report 40, June, 1947. 10. Chu, L. J. Physical Limitations of Omnidirectional Antennas , J. Appl. Physics, Dec. , 1948. p. 1163. 11. Courant, R. and Hilbert, D. Methods of Mathemat ical Phys ics . New York: Interscience Publishers, 1953. p. 52. 12. Cox, D. V. Solution of Aper ture Dis tr ibut ion for a Given Field Pattern. Bumblebee Series Report 122, Johns Hopkins University Applied Physics Laboratory, Feb., 1950. 13. Debye, P. Polar Molecules . New York Dover Publications (reprint). 14. Dolph , C. L. A Current Distribution for Broadside Arrays which Optimizes the Relationship Between Beamwidth and Side Lobe Level. Proc. I.R.E., 34, June, 1946. p. 335. 15. Dunbar, A. S. Calculation of Doubly Curved Re f lee tor s for Shaped Beams. Proc. I.R.E., 36, 1948. p. 1289. 16. Eaton, J. E. An Extens ion of the Luneberg Lens. N.R.L. Report 4110 Feb. 16, 1953. 17. Friedlander, F. G. A Die le c tr ic -Lens Aerial for Wide -Angle Beam Scanning. J.I.E.E., 93, Part IIIA, 1946. p. 658. 102- 18. Fry, D. W. and Goward, F. K. Aerials for Centimeter Wavelengths, Cambridge Press, 1950. p. 63. 19. Gunter, R. C. The Mangm Mirror, paper presented at URSI Spring Meeting, Washington, D. C. , 1951. 20. Jahnke, E. and Emde , F. Tables of Func t ions New York: Dover Publications, 1943 (reprint), pp. 180 189. 21. Jones, E. M. T. Paraboloid and Hyperboloid Lens Antennas .. Trans. I.R.E., AP-2 No. 3, July, 1954. p. 119. 22. Jordan, E. C. Electromagnetic Waves and Radiating Systems , New York Prentice-Hall, 1950; Chap. 15. 23. Kaczmarz, S. and Steinhaus, H. Theorie der Orthogonalreihen Mono- grafje Matematyczne Vol. 6 Warsaw: Garasinski, 1935. pp. 100-102. 24. Kelleher, K. S. Antenna Wave fr ont Problems . N.R.L. Report 3530, Sept. 19, 1949. 25. Kock, W. E. Metal Lens Antennas Proc. I.R.E. , 34, 1946. p. 828. 26. Kock, W. E. Metallic Delay Lenses, B.S.T.J., 27, 1948. p. 58. 27. Luneberg, R. K. Mathematical Theory of Optics Brown University Press, 1944. 28. Magnus, W. and Oberhettinger , F. Special Func t ions of Mathematical Physics, New York: Chelsea, 1949. p. 37. 29. Marston, A. E. Double Curvature Reflectors for Beam Shaping with Quasi-Point-Source Feed N.R.L. Report 3981 May 29, 1952. 30. Page, L. Introduction to Theore t ical Physics New York: D. Van Nostrand, 1935. chap. XIV. 31. Peeler, G. D. M. and Archer, D. H. A Two -Dimens lonal Microwave Lune- berg Lens. N.R.L. Report 4115, March 2, 1953. 32. Peeler, G. D. M. ; Kelleher, K. S. and Coleman, H. P. Virtual Source Luneberg Lenses, Trans. I.R.E. , AP-2, No. 3, July, 1954. p. 94. 33. Ramsay, J. F. Fourier Trans forms in Aerial Theory, Marconi Review 9, 10, 11, 1946-48. 34. Rhodes, D. R. An Exper imental Invest igat ion of the Radiat ion Pat- terns of Electromagnetic Horn Antennas , Proc. I.R.E., 36, 1948 p. 1101. 35. Rinehart, R. F. A Solut ion of the Rapid Scanning Problem, J. Appl. Physics, 19, 1948. p. 860. 36. Rust, N. M. The Phase Correction of Horn Radiators J.I.E.E., 93 Part IIIA, 1946. p. 50. 103 % 37. Schelkunoff, S. A. Electromagnetic Waves, New York: D. Van Nostrand, 1943. p. 355. 38. Si lberstein , L. Simp I if ied Methods of Tracing Rays Through Any Optical System. Longmans, Green and Co. p. 1. 39. Silver, S. Microwave Antenna Theory and Design.. New York: McGraw- Hill Book Company, Inc. 1949. p. 164 and 192. 40. Sinclair, G. and Cairns, F. V. Optimum Patterns for Arrays of Non-Isotropic Sources, Trans. I.R.E. , AP 1, Feb., 1950. p. 50. 41. Sneddon, I. N. Fourier Trans for ms ■ New York: McGraw Hill Book Company, Inc. 1951. pp. 48-52. 42. Spencer, R. C. Paraboloid Diffraction Patterns from the Stand-point of Physical Optics. M.I.T. Radiation Laboratory Report T-7, Oct. 2 1, 1942. 43. Stratton, J. A. and Chu, L. J. Diffraction Theory of Electro- magnetic Waves Phys. Rev., 56, 1939. p. 99. 44. Stuetzer, 0. M. Development of Artificial Microwave Optics in Germany. Proc. I.R.E. , 38, 1950. p. 1053. 45. Taylor, T. T. An Antenna Pattern Synthe s is Method N.R.L. Report 4043 (Side Lobe Conference), April, 1952. p. 38. 46. Taylor, T. T. Design of Line Sources for Narrow Beamwidth and Low Side Lobes. Tech. Memo. No. 316, Hughes Aircraft Co., July 31, 1953. 47. Van der Maas, G. J. Simpl if ied Calculation for Dolph-Tchebyscheff Arrays. J. App. Physics, 25, Jan., 1954. p. 121. 48. Watson, G. N. Theory of Bessel Functions Cambridge Press, 1948. pp. 396 and 454. 49. Whittaker, E. T. and Watson, G. N. Modern Analys is New York: Mac millan, 1948. p. 224. 50. Woodward, P. M. A Method of Calculating the Field over a Plane Aperture Required to Produce a Given Polar Diagram. J.I.E.E., 93, Part IIIA, 1946. p. 1554. 51. Woodward, P. M. and Lawson, J. D. The Theoretical Prec is ion with which an Arbitrary Radiat ion Pattern May be Obtained from a Source of Finite Size. J.I.E.E. , 95, Part III, 1948. p. 363. * 104 I J ACCO USA #25973 5 05 05 N 2 5 973 11 DK.HLUE/RL.BLEU/AZUL OBSC