THE UNIVERSITY OF MICHIGAN 5780-5-T The Pattern Synthesis Problem for a Slotted Infinite Cylinder O. Einarsson, F. B. Sleator and P. L. E. Uslenghi Report 5780-5-T on NASA Grant NsG-444 August 1965 National Aeronautics and Space Administration Langley Research Center Langley Station Hampton, Virginia 23365

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THE UNIVERSITY OF MICHIGAN 5780-5-T TABLE OF CONTENTS INTRODUCTION 1 II THE SYNTHESIS PROBLEM WITH CONTINUOUS SOURCE DISTRIBUTIONS 5 2. 1 Plane Aperture 5 2.2 Aperture on an Infinite Circular Cylinder 11 2. 3 The Quality Factor 14 2. 4 Comparison Between the Plane and Cylindrical Cases 19 2. 5 Formulation of the Synthesis Problem 24 2. 6 Numerical Results 33 III ARRAYS OF AXIAL AND CIRCUMFERENTIAL HALFWAVE LENGTH SLOTS ON AN INFINITE CIRCULAR METAL CYLINDER 52 3.1 Introduction 52 3.2 Array of Axial Slots 53 3. 3 Array of Circumferential Slots 58 3. 4 Numerical Results 66 3.5 Array of Axial and Circumferential Slots 73 3. 6 Final Considerations 75 APPENDIX 78 BIBLIOGRAPHY 82 l'St

THE UNIVERSITY OF MICHIGAN 5780-5-T INTRODUCTION Any attempt to classify or organize the tremendous body of existing theory of antennas is certain to lead almost immediately to the distinction between the analysis problem, i. e. the determination of field patterns or radiation characteristics of given antenna forms, and the synthesis problem, i. e. that of determining the form and excitation of an antenna which will produce a prescribed field pattern. The two problems are of roughly equal practical importance, but the latter, being inherently less well defined and straightforward and therefore more difficult, has received a rather meager share of the attention. A number of significant contributions have appeared recently, however, some of which are noted below, and the present report is an attempt to extend and clarify some of the results and conclusions reached. The choice of the particular questions considered here was motivated initially by a problem in satellite communication. Simply stated, the requirement is for a flush-mounted antenna on an essentially cylindrical body which will produce a farfield pattern of sufficient uniformity so that radio contact can be maintained at any orientation within a certain angular region. Disregarding the questions of implemen tation, which are by no means trivial, we are left with considerable leeway in the formulation of an appropriate analytical problem. Further restrictions, however, are afforded by considerations of simplicity and feasibility of solution, and we accordingly limit ourselves here to the problem of determining the excitations required in slots of various types in the surface of a conducting infinite cylinder in order to produce the best approximation in a certain sense, to certain prescribed far-field radiation patterns. To the best of our knowledge the treatment presented here is essentially new. However, there are numerous recent papers whose substance is related in some degree to that of the present report and which deserve some mention as antecedents

THE UNIVERSITY OF MICHIGAN 5780-5-T and sources of inspiration. Results obtained for the radiated fields of slotted cylinders with specified excitation are legion. A comprehensive bibliography of these is contained in the book by Wait (1959), and among the more recent contributions are papers by Knudsen (1959), Nishida (1960), Logan, Mason and Yee (1962) and Hasserjian and Ishimaru (1962a, b). Solutions of synthesis problems for such structures are, however, few. Notable among these are the results of Wait and Householder (1959), who developed a procedure for synthesizing a given radiation pattern by means of a cylinder excited by a circumferential array of axial slots, and DuHamel (1952), who considered antenna arrays on circular, elliptical, and spherical surfaces, and proved for the circular case that the radiation pattern obtained with a certain minimum number of antennas differs by only a few percent from that produced when the antennas are replaced by a continuous current distribution. The literature on the synthesis problem for a single aperture on a conducting cylinder is even more sparse. The authors are currently aware of no other investigations which treat such a problem explicitly. There are a number of papers, however, which deal with single apertures of various shapes in infinite conducting plane screens, employing formulations and techniques similar to some of those used here for the cylindrical case. Among these we note the following. Various extremal problems, with the common stipulation of a fixed number of spherical wave function in the field representations, have been considered by Chu (1948) and Harrington (1957), the variational quantities being the gain, quality factor, the ratio of these, and the side lobe level for given main beam width. The problem of finding a pattern function which takes given values at a certain number of specified points and which minimizes the square integral of its corresponding aperture function was treated by Woodward and Lawson (1948) and by Yen (1957). Determination of the aperture function specifiable with a given number of harmonics which minimizes the sidelobe level for given width of the main beam was carried out by Taylor (1955), 2 -

THE UNIVERSITY OF MICHIGAN 5780-5-T Mittra (1959) and Fel'd and Bakhrakh (1963). The latter paper, along with one by Kova[cs and Solyma'r (1956), deals also with the question of the best mean-square approximation over the whole space (visible and invisible) to a function which equals some given function in the visible region and vanishes outside it. Solymaer (1958) and Collin and Rothschild (1963) consider the maximization of the directivity with a given number of harmonics in the aperture function and a specified value of the supergain ratio or quality factor. In a paper by Ling, Lefferts, Lee and Potenza (1964) the normalized second moment of the far-field power pattern is minimized for various plane aperture shapes, including the rectangle, circle, annulus and ellipse. Finally, in a mathematically elegant analysis, Rhodes (1963) has exhibited the optimum mean-square approximation to a given pattern function with a fixed number of terms in the aperture field representation and a given value of the supergain ratio, as defined by Taylor (1955). It might be observed here that all of the above analyses concern themselves with field strength rather than power patterns. The only valid example of power pattern synthesis known to the authors at present is a paper by Caprioli, Scheggi and Toraldo di Francia (1961) which makes use of a technique of interpolation between sampling points. As remarked above, some of the formulations and techniques developed in the treatment of plane apertures have a direct bearing on the cylindrical problem. However, there are several important differences here which limit their applicability and necessitate certain modifications. If one attempts, for example, to follow the procedure of Rhodes (1963), it develops immediately that the kernel of the integral equation relating the aperture and far fields in the cylindrical case is not symmetric, and thus possesses no orthogonal set of eigenfunctions. One of the principal features of Rhodes method, namely the orthogonality of the set of pattern functions corres - onding to an orthogonal set of aperture functions, is therefore not available here,

THE UNIVERSITY OF MICHIGAN 5780-5-T and the solution of a system of linear equations apparently cannot be avoided. Certain extremal properties of the eigenfunctions for the plane case are also lacking in the cylindrical case. Even more important, perhaps, is the fact that whereas in the plane case all the known pattern synthesis procedures yield an approximating pattern which is real if the prescribed pattern is, this does not hold for the cylindrical case. As a consequence, it turns out that here the best admissible mean-square approximation to a given real pattern may be an extremely poor approximation in amplitude. These are among the principal considerations which governed the formulation of the problems treated in the present report. The choice of the mean-square deviation as the measure of the degree of approximation is more or less mandatory from an algebraic standpoint. Since it appears that there is little to be gained in the cylin drical case through the use of special basis functions, we have employed only exponential or trigonometric functions, with the inclusion of a weight factor in some cases which satisfies an edge condition at the slot boundaries. Because the amplitude of the radiation pattern far outweighs the phase in practical importance, an iteration scheme was developed for the case of a single slot, in which the phase of the prescribed pattern is sacrificed for the sake of substantially improving the amplitude approximation. This scheme has not been used for numerical computations in cases with multiple slots such as those considered in Section III, though there seems to be no reason why it could not be. The authors wish to acknowledge the considerable and sustained efforts of certain colleagues, in particular D.R. Hodgins, T. L. Boynton, J.A. Rodnite, J.A. Ducmanis and Miss Austra Maldups, who programmed the numerical work reported here. Credit is also due to the University of Michigan Computing Facility, which actually produced the numbers. __________________ _ 4

THE UNIVERSITY OF MICHIGAN 5780-5-T II THE SYNTHESIS PROBLEM WITH CONTINUOUS SOURCE DISTRIBUTIONS 2. 1 Plane Aperture Consider a finite aperture S in an infinite conducting screen lying in the yz plane of a rectangular coordinate system and introduce polar coordinates x=rsinOcos, y=rsinOsin A, z=rcos.| It is well known that the far field can be expressed in terms of the distribution over the aperture of the tangential components of the electric field strength (Silver, 1949). Thus it is found that ik eikr _E 2 cosIJ \ E (O, y, z)exp ik(y sine sin O+z cos e dydz (2.1) -ik eisin j(oyz)+ cossin Eeo ) E - Csino 0 ( (OIYJZ)+ sin Ez(0,y,z r —w oo 2 7r y sin 0 exp Eik(y sin 0 sin O+z cos ] dydz (2.2) -iwt. where the time dependence e is everywhere suppressed. By virtue of Babinet's principle the substitution E -H, H -H- -E makes (2. 1) and (2. 2) valid also for the complementary problems where the fields are caused by surface currents H, H on a conducting plane disk of the same shape as the aperture. If we assume the aperture to be rectangular and aperture field linearly polarized (say E-=O) and separable, that is E (O,y,z)=el (y) e2(z) the radiation pattern in y -f 5

THE UNIVERSITY OF MICHIGAN 5780-5-T the xy and xz planes depends solely on el(y) and e2(z) respectively. Thus the problem of synthesizing the radiation patterns in the main planes in this case simplifies to finding two independent one-dimensional aperture distributions. The synthesis problem is usually formulated as such a one-dimensional problem. For a general rectangular aperture the assumptions of a separable and linearly polarized aperture field seem quite questionable (c.f. Collin, 1964) but for a narrow slot (width <X/10) they are certainly accurate enough. One other case in which a one-dimensional aperture distribution can be used is when the aperture is infinite in one direction and the derivatives of all field components with respect to that direction vanish. We can now write the relation between the radiation pattern and the aperture field as kL g(0) = e f(rl)dr, t[l 1, (2.3) where the physical significance of the functions f, g in the cases of an infinite slot and a line current is given in Table II-1. The notation corresponds to Fig. 2-1. It is natural to restrict the functions g and f in eq. (2. 3) to be complex valued functions, square integrable over the interval (-1, 1) (notations: f, g, e L2). It was early recognized that the synthesis problem as expressed by (2. 3) has the following properties (see Bouwkamp and DeBruijn, 1946). a) For an arbitrary function h(Q)e L2, there is in general no aperture function f(rl) L2 such that the corresponding pattern function g(~)= h(g). b) We can obtain, however, for every positive quantity E an aperture function f(r) E L2 whose corresponding g(g) satisfies i h()- 2d < V~6 _1_

C PLANE APERTURE LINE SOURCE TM FIELD TE FIELD _ I _ i(kp- -) i(kp- ) ~ ~ kL e kL e kL e E o E - cos E g(sin ) EH - sin) g(cos) o zp -aP- OD 0 2 4 F7 r-k- -p op — >00 0 2 qr2-,,- -k p-"' Vr oo 0 2 4ikr! 2 2y 0Ho 2Z E z(0, y)=Eof( ) E (0, y)=E f(L I(z) I (z) f z OL 0 L TABLE II-1: RELATION OF PHYSICAL FIELD QUANTITIES TO THE APERTURE AND - PATTERN FUNCTIONS OF Eq. (2. 3).

THE UNIVERSITY OF MICHIGAN 5780-5-T L I FIG. 2-1: APERTURE IN AN INFINITE SCREEN AND LINE SOURCE There is, thus, in general no'exact solution' of the synthesis problem but we can approximate any prescribed pattern arbitrarily closely in the mean-square sense. It is consequently possible,for every aperture however small, to find an aperture function f e L2 which delivers a radiation pattern with arbitrarily high directivity. Any attempt to obtain'supergain' from practical antennas will, however, result in an unrealizable aperture function with high amplitude and rapidly varying phase. As a measure of the'realizability' of the aperture function, Taylor (1955) introduced the supergain ratio, which in our notation is defined as 47kL'1( 2 (2.4) 1 8g(c _2d_ L ~~~~~~~~~~~8,

THE UNIVERSITY OF MICHIGAN 5780-5-T Other'quality factors' have been introduced by Chu (1948) and Collin and Rothschild (1963, 1964). We will discuss these different factors in some detail in Section 2. 3. In order to achieve the pattern synthesis we write the aperture function f(r) as f(r7) a f (r) (2. 5) 0 where {fnlis given as a set of functions defined on the interval (-1, 1). We denote the pattern functions corresponding to f (ri) as gn (). There is a large degree of freedom in the choice of {fn; the only necessary property is that this set be complete in the subspace of L2 where the solution of our problem is to be found. We will consider two examples of functions which have been used in different synthesis procedures. Example 1: f (r/)= eiunr 11 < 1 NkL 2 *Xa t z g-u[ |(2. 6) 2 sin( - -un) gn() kL I 1 -2 -u If we choose the parameters u =nir and allow n to take negative as well as positive values the expression (2. 5) is an ordinary Fourier series. In this case the radiation pattern in certain directions is related to the coefficients a by the expression 2n7r kL L gn( (kL)=2an n kL L X and an approximating pattern which coincides with the prescribed pattern at these points is readily obtained. _ _ _ _ _ __ 9 _

THE UNIVERSITY OF MICHIGAN 5780-5-T Example 2: n(77)= S n( 2 kL) 11711 n On 2' (2. 7) 2 (1) kL kL gno= inR ( 2'1)SO'k2,) (21t14 gn~ ~' n On On 1 R0n is a radial and SOn an angular prolate spheroidal function in the notations of Flammer (1957). The interesting properties of these functions are that they are the eigenfunctions of Eq. (2. 3) and that they bear a close relationship to the supergain ratio. They have been studied by Slepian and Pollak (1961) and Landau and Pollak (1961, 1962) and from their work we take the following results. i. ~ S 0is orthogonal and complete in the interval (-1, 1). It is also orthogonal in the interval (-oo, wo) (but not, as Rhodes (1963) claims, complete there). ii. The smallest possible value of the supergain ratio y is IT kin 00d~~ji~~ 13"L~oo( 21 9(2. 8) 7mmn (1\ kL 2 and is obtained for the aperture function f(rl) = S00(, rk ) iii. Consider the class of aperture functions which corresponds to a given value of the supergain ratio yO (notation: f E E(-yo)) and which are normalized such t apro) ds = 1 f. I ~fn1* then |SOn} are the best possible choice in the following sense: They _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 1 0

THE UNIVERSITY OF MICHIGAN 5780-5-T are the functions which achieve 1 N-1 2 min max min f(r)- a f(r) dr. (2. 9) nfN-1 f6E(%) E aN 1 nObserve that iii ensures the best mean-square approximation of the aperture function, which does not necessarily imply the best fit of the pattern function. The set of pattern functions {gn} is here orthogonal and is consequently suited for a meansquare approximation of the prescribed pattern. 2. 2 Aperture on an Infinite Circular Cylinder Consider an infinite conducting circular cylinder of radius a with a finite aperture S. We introduce a cylindrical coordinate system (p, 0, z) such that the generating surface is given by p = a,and also polar coordinates (r, 0, 0) (p=r sin 0, z=r cos 0). In analogy with the plane case the far field can be expressed as i eikr _ ein_ E0 27r2 r Ez(an r -e>o n=-co sin 0 iH (ka sin 0) exp [-i(no'+kz cos iO do' dz (2.10) 1 er eino ncos 0 rr - n=- (ka sin e) exp [-i(nO'+kzcos )] d'dz, (2. 11) where H(1) and H(1) are the Hankel function of the first kind and its derivative with n n i____________......11..

THE UNIVERSITY OF MICHIGAN 5780-5-T respect to the argument (Silver and Saunders, 1950). Just as in the planecase the infinite axial slot and the narrow circumferential slot can be treated as one-dimensional problems. We write the relation between the radiation field and the aperture field as P(M)= K(0 -r < < _r. (2.12) The physical significance of P and A is given in Table 11-2 where the notations correspond to Fig. 2-2. X J+=-~ tZ=- Z =8 2a 2 FIG. 2-2: INFINITE AXIAL AND NARROW CIRCUMFERENTIAL SLOT ON INFINITE CYLINDER It turns out that the fundamental properties of the plane synthesis problem as expressed by a) and b) on page 6 are still valid in the cylindrical case. Thus; a) For an arbitrary function F(O)C& L7, there is in general no aperture function A(o') EL2 such that the corresponding pattern function P(o)=F( 0). b) We can obtain, however, for every positive quantity c, an A(') E6 L whose corresponding P(0) satisfies j F(o)-P()jI d<12._ __ __ __ __ __,__ 12....

NARROW CIRCUMFERENTIAL INFINITE AXIAL SLOT SLOT TM FIELD TE FIELD i(kr'+ ) i(kr.ikr Or (E) E7 E E E 0 e P() Ep E2 P() ZOr Z 0 kr ZEr -o E 6r;:ooEr O IF -6/2 V(O)= E (a,, z)dz=-A(O) E (a, )=E A(O) | (a 0)=EOA(O)O (6 = width of slot) H(ka) 2 n 27r2i n=0 inH (ka) 2 2 = in (ka) (e0=1, e2= e2.... = 2) TABLE II-2: RELATION OF PHYSICAL FIELD QUANTITIES TO THE APERTURE AND PATTERN FUNCTIONS OF EQ (2.12)

THE UNIVERSITY OF MICHIGAN 5780-5-T A proof of these statements is given in the appendix. To obtain a meaningful synthesis problem, we must apparently, as in the plane case, put some constraint on the permissible aperture functions. The question of the most significant or appropriate constraint is discussed in the next section. 2.3 The Quality Factor Integration of the complex Poynting vector over the aperture S in the infinite plane or on the cylinder yields 1 fEAH* ~ inds = P - 2i(W -W ) (2.13) 2''- r m e A A where n is a unit vector normal to the aperture a or p respectively). P is the radiated power and W -W is the difference between the time-averages of e m 1 the electric and magnetic energy densities I ((E EE* - _H HI*) integrated over 4 0the half space or the region outside the cylinder respectively. If we try to calculate the stored electric and magnetic energies separately as the integrals of 1 1 T O' E 0 E* and H - H H- H respectively, the results will be infinite because of the slow decrease of the far fields. To overcome this, Collin and Rothschild (1963, 1964) defined the electric and magnetic energy densities as w = - 0 E.E* - U e 4.- - e (2.14) 1,,,,,,, ~~1 4............... _ _ _ _ _ _ _ _ _

THE UNIVERSITY OF MICHIGAN 5780-5-T Here Ue and Um are quantities equal to the energy densities in the far field when the distance from the source tends to infinity. In the cylindrical case we take U =U R (E A H') (2.15) e m 4 e- -p The quanities given by Eq. (2.14) are finite if integrated over all space and can be considered as the remaining energies when a part corresponding to the power flow in the radial directions is subtracted. A quantity which in the cylindrical case corresponds to the supergain ratio is apparently 1 -Ia 2 1JA(o)j do ~r 1 |I~a(0 (2.16) It is often assumed that the supergain ratio is a measure of the reactive power, 2w(W -W ), orthe stored energy. This opinion has been critized by Collin and m e Rothschild (1963). The result of their investigation for the plane case is that a large value of the supergain ratio indicates a high amount of reactive power and consequently also a high amount of the stored energy, but the converse is not always true. Consider for example the TM-field in the infinite aperture (cf. Table II-1 ). The aperture field which pertains to the smallest possible value of the supergain ratio (Eq. 2. 8 ) has a value different from zero at the end points of the aperture. This clearly violates the edge condition (cf. Meixner, 1949) and implies an energy density around the edges which tends to infinity in such a manner that it is not integrable. In spite of the fact that the supergain ratio takes its lowest possible value in this case both the reactive power and the stored energy are infinite. Observe that in the corresponding TE-case there is no violation of the edge condition. For a narrow 15

THE UNIVERSITY OF MICHIGAN 5780-5-T slot the stored energy in the vicinity of the aperture tends to infinity as the width of the slot tends to zero. In the limit (i. e. a line source) the reactive power and stored energy are infinite for every aperture function. quite independently of the value of the supergain ratio. Collin and Rothschild (1963 and 1964) have proposed a quality factor for radiation problems defined as 2t0W Q-= (2.17) r where W is the larger of the time-averaged magnetic or electric energies stored in the "evanescent" field, as defined by (Eq. 2.14). P is the radiated power. The definition (2. 17) is in accordance with the usual definition of quality factor for a network or microwave cavity and can be considered as characteristic of a radiating system which is tuned for resonance by the addition of a lossless reactive element. If we express the fields as sums of cylindrical modes and thus expand the pattern function in a Fourier series Pp ) = Ze in (2.18) n=-00 there is no interaction energy between different modes and we can calculate the energy for each mode separately. The total quality factor is then obtained as O 2 Q n=-o (2.19) nr=-oo For the infinite axial slot the factors Qn can be expressed explicitly. They are equal for the TM- and TE- cases and are calculated by Collin and Rothschild (1964) as 16

THE UNIVERSITY OF MICHIGAN 5780-5-T 4ka+ n 2+1-(ka) 2] (2+Y) - (n+l 1)J-ka Ji+ - (n+ 1)Y -ka Y+] Qn='~' " n n n n Jn n (2.20) The argument of the cylinder functions is everywhere ka. The narrow circumferential slot delivers an expression for Qn which depends on the distribution of the electrical field strength across the slot. It can be expressed as an infinite integral containing a combination of cylinder functions similar to Eq. (2.20). Chu (1948) obtained a quality factor for the spherical case by using the recurrence relation for the spherical Bessel functions to define an equivalent RLC net work for each mode. The quality factor was then defined as the ordinary Q related to this circuit. This procedure has been shown by Collin and Rothschild (1963) to be equivalent to the definition in Eq. (2. 17). The method leads to tedious calculations for higher modes and Chu therefore introduced a simplified equivalent circuit and a slightly different quality factor which is not restricted to spherical modes and can be expressed as W a W(We- W ) + w(W e m)l -' &) eme(2.21) p r where the quantities involved are defined by Eq. (2.13). The derivative with re-,spect to w shall be taken with the tangential component of the electric field 3trength over the aperture kept constant. If the aperture is small compared to the wavelength this is equivalent to keeping the feeding voltage constant and we can vrite 2 s~ E^A Hi I i dS = Ve (G + iB) (2.22) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _,1 7

THE UNIVERSITY OF MICHIGAN 5780-5-T and Q, w. (2.23) 2G If B f O, we can tune the system for resonance by adding (connecting in parallel) a positive susceptance wC or a negative one -1/oL. If we denote the resulting susceptance after tuning as Bo, we see that to -a = w a- + iB. Thus Q' can be considered as a measure of the frequency sensitivity of the input susceptance if we tune the system to resonance. In the plane case Q' = Q (Collin and Rothschild, 1963) but for cylindrical modes they are slightly different. A straightforward calculation yields, for example, Q' for a single cylindrical TM-mode as 2 1 n1 n Q 221 - ka (2.24) n n Jr j2(ka)+Y2(ka) n n The quantities used in the definition of Q' in Eq. (2.21) can all be obtained as surface integrals over the aperture. It is therefore possible to use this definition for a quite arbitrary conductive body with an aperture on its surface. In a general case, however, this Q' will have no connection with the energy stored in the vincinity of the body and it is, for example, possible for Q' to take negative values. We can define the "supergain ratio" for a single cylindrical TM-mode as r= 2.[J2(ka)+Y2(ka n =0, 1, 2,... (2.25) The factor r defined by Eq. (2. 16) is then obtained as z ln n i~ r~~~~= n —~ ~~(2.26) n_1=- 0 18

THE UNIVERSITY OF MICHIGAN 5780-5-T The definition of n in Eq. (2.25) is such that r1- Q1 as ka- -0. Qn, Q and r have been calculated from Eqs. (2.20), (2. 24) and (2. 25) for n 10 and ka, 15 n and are shown in Fig. 2-3 for some values of n. The question of how high values of the quality factors are admissible in practical design of slot antennas is outside the scope of this report. However, it seems clear that there can be no close connection such that two different aperture distributions with equal quality factor are always equally easy (or difficult) to realize practically. All the quality factors considered here have the property that a high value implies impractical design. It is seen from Fig. 2-3 that Q is the most restrictive one and if we prescribe Q during the synthesis procedure the corresponding values of Q' and r will also be under control. 2.4 Comparison Between the Plane and Cylindrical Cases As we have seen in Sections 2.1 and 2.2, the plane and cylindrical synthesis problems have the following main features in common: a) there is no "exact solution" and b) there is a theoretical possibility of obtaining supergain. Thus the approach for the cylindrical problem should be in general the same as in the plane case and in conformity with Eq. (2. 5) we write the aperture distribution as A(p) <= m1'm( -a< a (2.27) m=0 ~ where ~{J is a given set of linearly independent functions, square integrable over the interval -a< (<a (notationm e L2a) We denote the pattern function that via Eq. (2.12) corresponds to m (p) as TTm(0). As in the plane case, we have considerable freedom in the choice of { as long as the set is complete in the subset of L2 where the solution of our synthesis problem is to be found. The simplest type of meaningful restriction on the aperture function is to use a finite set of functions {~ o in the expression (2.27). We can then calculate the N+1 coefficients o, ~1' "'~N either in such a way that the corresponding pat19

THE UNIVERSITY OF MICHIGAN 5780-4-T 4 10 1 I i 10 I I i I 22 I~~~~~~~~~k FI. -: IFEEN UA~T ACOS ORCLIDICL MME %~~~2

THE UNIVERSITY OF MICHIGAN 5780-5-T tern function is equal to the prescribed pattern for N+ 1 given values of 0 in the interval -ir ~ 0 <?r, or so the approximating pattern is the best mean-square approximation of the prescribed pattern. The unknown coefficients are obtained in either case as the solution of a system of N+ 1 linear equations. In contrast to the plane case, there is no set of simple functions such as those of Eq. (2.6) which yield a direct relationship between the coefficients and the radiation pattern in certain directions. The mean-square approximation, on the other hand, would be simplified if one could find a set {I such that the corresponding pattern functions were orthogonal, but there seem to be no well-known functions with this property either. A linearly independent set could be orthogonalized by the usual Schmidt process, but this is of course equivalent to solving the original system of linear equations. A more satisfactory constraint on the aperture function than merely limiting the number of terms in the expansion (2.27) is to keep some quality factor constant during the synthesis procedure. If we use the mean-square approximation, such a scheme can be treated analytically by introducing a Lagrange multiplier. A process which for the plane case delivers the best mean-square approximation to a prescribed pattern for a given value of the supergain ratio has been proposed by Rhodes (1963). The functions used by Rhodes for expansion of the aperture function are the spheroidal functions of Eq. (2. 7). Due to the orthogonality of the corresponding set of pattern functions and the special choice of constraint, the optimum pattern and aperture functions can be determined directly without solving any set of linear equations, i. e. the matrix of the system degenerates to a diagonal form. Rhodes uses a finite number of terms in the expansions (2. 5) and thus the spheroidal functions are also the best set to use in approximating the aperture function in the sense of iii on p. 10. This, of course, does not mean that they are the best choice for a specific prescribed pattern, but only that they are the best for the worst possible pattern

THE UNIVERSITY OF MICHIGAN 5780-5-T with the prescribed value of the supergain ratio. It may also be remarked that for the cases of a TM-field in an infinite aperture and a line source, the pattern functio of Eq. (2.3) differs from the radiated field in the far zone by a cosine factor (cf. Table II-1). This means that in the procedure of Rhodes, deviation from the prescribed radiated field will be somewhat overemphasized in directions away from the normal to the aperture, to the detriment of the fit in directions close to the normal. If we should define instead a pattern function h(~) directly proportional to the radiated far field in these cases, the integral relation corresponding to Eq. (2.3) would be i( ~ SkL h() = ef(r) dr, (2.28) -1 - 14 AS 1. The eigenfunctions of this equation are, as pointed out by Fel'd and Bakhrakh (1963), odd periodic Mathieu functions of argument are cos. It may seem desirable to find a set of functions in the cylindrical case which have properties similar to i, ii and iii on p. 10. If we write the integral relation in Eq. (2. 12) so that the aperture function and the pattern function are defined in the same interval we get P(0) = - K(0- 0)A( 0)dO, -7r < ( 7r. (2.29) One important difference between this and the corresponding formula for the plane case, Eq. (2.3), is that the kernel is no longer symmetric. The eigenfunctions of Eq. (2.29) (if there are any) will consequently certainly not be orthogonal. The property that the spheroidal function S00 minimizes the supergain ratio is also a direct consequence of the symmetry of the kernel in the plane case. The aperture function in the cylindrical case for which the corresponding quantity r defined by Eq. (2. 16) takes its lowest possible value is the eigenfunction which corresponds to the lowest eigenvalue of the kernel 22

THE UNIVERSITY OF MICHIGAN 5780-5-T KL(O, 0) K'0'- - )K(0'- a O)d'. (2.30) - 7r This kernel is symmetric and consequently has orthogonal eigenfunctions. However, if we use them to express the aperture function, the corresponding set of pattern functions will not be orthogonal and not much is gained in the synthesizing of an arbi trary prescribed pattern. The simplifications in the mean-square optimization procedure that can be achieved by a more sophisticated choice of OM and rTTM must be weighed against the fact that we must deal numerically with more complicated functions. In the plane case these "best" functions turned out to be functions which were already tabulated and had suitable expansions available. However, even in this case it is no obvious that the expansion of the aperture function in a Fourier series, for example, instead of a series of spheroidal functions, would involve a significantly greater total amount of numerical calculations. In the cylindrical case where, due to the non-symmetry of the kernel, there appears to be no orthogonal set with V}mn also orthogonal, it seems reasonable in the first instance to choose a set of aperture functions such that the corresponding set of pattern functions is easy to calculate. Only if it then turns out that this special choice delivers a numerically intractable optimization process, should there be any reason to look for a "better" set of functions. Such an approach gives no precedence to the "supergain ratio" over other possible quality factors in the formulation of the constraint on the aperture functions, and as indicated in Section 2.3, it may be appropriate to use a more restrictive quality factor. Generally, only the amplitude of the prescribed pattern (i. e. the power pattern) is of interest in the synthesis process. If the phase of the prescribed pattern has any influence on the result of the synthesis, we have a possibility of improving the approximation by choice of a suitable phase function. A simple application of such a scheme for a plane aperture is given by Caprioli et al (1961)......_ _..........__ 23.............

THE UNIVERSITY OF MICHIGAN 5780-5-T There is one more (perhaps more basic) difference between the synthesis problems in the plane and cylindrical cases. Within the authors' knowledge, all proposed synthesis procedures in the plane e5 deliver a real approximating pattern if the prescribed pattern itself is a real function. In the cylindrical case, however, the approximating pattern related to a real prescribed pattern will, in general, be a complex-valued function. As we shall see in Section 2.6 this means that the best mean-square approximation to a real pattern can be a very poor approximation to the power pattern. 2.5 Formulation of the Synthesis Problem The aim of our pattern synthesis is to achieve such an aperture function that the corresponding power pattern is the best possible approximation in some sense to a prescribed pattern. The realizability of the aperture function should be controlled during the process by limiting some factor connected with the energy stored in the near field around the aperture. This type of constraint is relatively tractable if we use a mean-square approximation, and a suitable definition of the optimum achievable pattern corresponding to a prescribed pattern Pg(0) would be the function P(0) which pertains to the quantity min 1 p() 2min (IP (0)I - IP(0, )2 do (2.31) under an appropriate subsidiary condition. If the aperture function A(0) is expressed as a linear combination of given functions as in Eq. (2. 27) we can consider A as a function of the appropriate coefficients yo, Y.... Use of calculus to determine the minimum results in an infinite nonlinear system of equations for {Yv} and since this is extremely intractable we will modify the formulation somewhat. It is easy to see that A is also obtained as 24

THE UNIVERSITY OF MICHIGAN 5780-5-T A= {} Pg(0)] - P()f2d (2.32) where the minimum of the right hand side shall be taken simultaneously with respect to the set of numbers {TYv} and the function 0(0). A necessary condition for this minimum is of course that we have a minimum of {y} alone if we keep 0(0) constant and vice versa. At least one such {yv} and 0(0) together with the corresponding value of the integral in Eq. (2.32) can be constructed in the following way. Define A mm (97f P51 (02 )' -r 1I g()1 - P s(0) do (2.33) where P (0) is a given function. The minimization is here an ordinary meansquare approximation of P (0) with respect to the given function P IPg(O) Is_ ( Thus the integral on the right hand side can be expressed as a positive definite quadratic form in TyvJwhich has a single minimum obtained by solving a system of linear equations in'Yo.Y1... For a specified P (0) the numbers,A 2,.. form a positive, monotonic, decreasing, and accordingly convergent sequence. The monotonicity is shown by the following reasoning: if we substitute Ps() P-1() L5(0)I| for Ps_1(0)1l in Eq. (2.33), the value of the integral will clearly diminish. The succeeding minimization with respect to Ps+1(0) to obtain As+ can then only result in a still smaller value. Now s Z <:otsis apparently the desired stationary value, but as 25

THE UNIVERSITY OF MICHIGAN 5780-5-T usual when we use necessary but not sufficient conditions for an extremum we have to check separately whether we have obtained the absolute minimum. There may exist several limit points of As, depending on the choice of Po(), and there seems to be no simple rule which tells how to choose P (0) so that the absolute minimum corresponding to Eqs. (2.31) and (2.32) is obtained as the limit. For a prescribed real valued pattern function P (?)E L7 and a given initial function PO( L2 we now calculate the set of functions consisting of those functions e Lr which minimize the corresponding quantities -1 1 ~Ps() 2 kZ pg |(p (0) | do + 2 2wW (2.34) s 27r 9 IP -1 (A s 27 E2 s' 3 s-i 2ir E S = 1, 2,... where W is the larger of the time-averaged magnetic or electric energies stored s in the evanescent field as defined by Eq.(2.14), connected with Ps(p). The quantity It is a given parameter which can be interpreted as the "weight" assigned to the stored energy compared to the deviation from the prescribed pattern. Since /s can also be considered as a Lagrange multiplier the functions P1(0), P2(p),'. are also the ones which pertain to A in Eq. (2.33) under the constraint 2wW <G, where S S G is a constant. If A is reasonably small the radiated power of the approximating s pattern is close to that of the given pattern. In that case the above constraint is nearly equivalent to keeping Q as defined by Eq. (2.17) constant. It may seem more natural to prescribe a value of the stored energy or the quality factor than of /A during the optimization procedure. The reason for not doing so is that the problem then would contain an unknown Lagrange multiplier which would have to be determined by a "cut and try" procedure, i.e. we would have to guess a value of the multiplier and then solve the problem and check whether the solution satisfied the subsidiary condition. Even if we used the information from earlier trials to improv the subsequent guesses as much as possible the procedure would involve the solution,__ __, 26

THE UNIVERSITY OF MICHIGAN 5780-5-T of several times as many minimum problems as in our formulation. For a fixed prescribed pattern the parameter p is a monotonic decreasing function of the stored energy and it may be assumed that it is as good (or bad) as this quantity as a measure of the realizability of an aperture function. We will henceforth consider only a TM-field in an infinite axial slot. The aperture and pattern functions we obtain are thus valid also for the narrow circumferential slot. The corresponding procedure for a TE-field in an infinite slot is completely analogous. It follows from Eq. (2.34) that the approximating pattern is an even or odd function of 0 if the prescribed pattern has the same property, that is, if P (0) is even, all Ps(0) will be so and vice versa. As the kernel K(0) in Eq. (2.12) is an even function, this means that the corresponding aperture function is also either even or odd, in accordance with parity of P (0). Thus, if we divide a general prescribed pattern function into an even and an odd part the solution of the synthesis problem is the sum of the solutions for the even and odd parts separately. It is numerically advantageous to make this separation, and we express the aperture function as N AN( C1()2 ~ cos A (0 )=1 - (0/)Ym sin a'(2.35) where the superscript N indicates the number of terms used in the expansion. In general, we will give the expressions for the even and odd case in the same formula, with the upper alternative pertaining to the even and the lower to the odd functions. The factor l - (p/a) makes the aperture function fulfill the edge condition and thus ensures a finite amount of stored energy. The kernel K(0) in Eq. (2.12) is given as a trigonometric series with period -|r <0 ( r and we want to express AN (0) in the same way, 27

THE UNIVERSITY OF MICHIGAN 5780-5-T 00 N Ncos A N() = a in -7r < 0 7r. (2.36) n=O We write the coefficients a as n N aN = y d (2.37) n m= m nm m=O where according to Eq. (2.35), nm 27r sin ac sin d (2.38) nc' (.J(m4 r - nA) J(m7r +) 4 mmir - rra mr + nu o 1=2=...=2 Since a Fourier series always can be integrated term by term we substitute A (0) from Eq. (2.36) into Eq. (2.12) and obtain the following series for the corresponding pattern function 00 N ~~N cos PN() = p sin n -7r < 0 r (2.39) Pn sin n=0 where N N 1 n =.....(2.40) Pn 7r i nH(l)(ka) n For a TM-field in an infinite axial slot the stored magnetic energy for each mode is always greater than the electric energy, and using Eqs. (2.17) and (2.19) we get..... 28

THE UNIVERSITY OF MICHIGAN 5780-5-T N=1 OD00 r 2E2 2wWN rrN O f~ E1 IsN rn 1 Z 2 Pn Qn Z k E JPn Qn e 0n o(2.41) nn where Qn is given by Eq. (2.20). Employing Parseval's relation we can now write Eq. (2.34) as A n=0 - 2 Re(P In Nn 1 (2.42) where c n Cos nr PN () nd (2. 43) is the nth Fourier coefficient of the prescribed pattern after s-1 steps of the iterative scheme. In the numerical computations a good approximation of pg can be obtained without integration by constructing a finite trigonometric sum M 5 cos k0 Lkk sin k k=O whose value coincides with that of PN (0) s-1 at a sufficient number of equidistant values of 0. If we use Eqs. (2.37) to express the coefficients pN in Eq. (2.40) in terms |of {^m we see that i\ is a (positive definite) quadratic form in these quantities and we obtain the minimum of AS by putting

THE UNIVERSITY OF MICHIGAN 5780-5-T A2N aoN s O,_ =0. a(Re m) a(0mm) This leads to a system of linear equations for the N+ 1 (even case) or N (odd case) unknowns ym, N E mjZA Q C=C = 0, 1,2,...N (2.44) where ~2 001+PQ (m7r - na) A =A _2 2n0 1 Al A c m m- 16 en=0 n J2(ka)+ y2(ka) m7r - na n n.l(m7r + naI) (1-r - Dm) + J(7r - na) + 1 (1 + 1 - m7r+ na 1 1r-na 7r+na (2.45) n n The advantage of separating(ka) the prescribed pattern in an even and an odd part is that we have only to solve two independent systems of equations in N and N+ 1 unknowns instead of one system with 2N+ 1 unknowns. As the set of functions we used in the expansion of A (0) in Eq. (2.35) apparently is complete in the subset of L2 which consists of aperture functions with a finite amount of stored energy, we obtain the aperture function related to the stationary value of the integral in Eq. (2.32) as lirm lirm AN(. (2.47) A(I) =N-+o s0 (2. 47) |

THE UNIVERSITY OF MICHIGAN 5780-5-T Thus the computational scheme is to iterate the phase of the prescribed pattern until there is nosfurther improvement, for a fixed value of N so large that the resulting A(0) is sufficiently close to the limit. It may be noted that the system of equations (2. 44) is intrinsically ill suited for numerical solution if N is large. If we compute the elements Ar of the coefficient matrix to a fixed number of significant figures we can replace the infinite sum in Eq. (2.45) by a summation up to n = M if M is large enough. But for n > M the column vectors of {AmJ are no longer linearly independent and the solutions of the system (2.44) do not minimize A. Thus the numerical stability of the problem is dependent on how rapidly the series in Eq. (2. 45) converges. A simple calculation shows that due to the factor tuQ the terms behave as 1/n2 when n is large and consequently, as could be expected, the stability increases with increasi value of /u. When {mo is calculated (after a sufficient number of iterations of the phase of the prescribed pattern) the aperture function is obtained from Eq. (2.35) and the corresponding pattern function from Eqs. (2.37) - (2.4). We introduce the real quantities XN= 2 1,C (2.48) 2=0 j 7~Tr =m U=0= (2.49) Nr =0 ~l m 2 Re(Y'Ym) (2.50) | 7r =0 m=1 (m2_ 2)2 31....

THE UNIVERSITY OF MICHIGAN 5780-5-T Also we normalize the prescribed pattern as 2 IP (p)]2do = 1 (2.51) and obtain AN = 1-XN (2.52) mm N (P() IPN(0)|d = 1+Y N 2X. (2.53) The quality factor is then expressible as N 2wWN XN yN Q 2WN XY- (2.54) Q N -N PN MYN r and the "supergain ratio" as - aAN(C) d2do 2 2r ir Y Sj |PN(02d0 2 N(2.55)..........._ _ _ -32..

THE UNIVERSITY OF MICHIGAN 5780-5-T 2. 6 Numerical Results The accompanying Figs. 2-4 through 2-8 have been taken from a large body of numerical data computed for the case of a single axial slot in an infinite cylinder under the sole constraint that the number of harmonics in the aperture field shall be fixed. The forms actually used in these computations were somewhat different from those described in the preceding sections, and a brief listing is perhaps desirable here. The aperture function is first expanded in the two sets of exponentials corresponding to the two angular intervals (-a, a) and -w, r), with the expansion pertaining to the former limited to a fixed number of terms, thus, N i (2. 56) = N in e a e'M (2 56) n=This yields at once the relation N a sin (mr -na) a (2. 57) n wr m —N m (mwr-na) (2.57) If the corresponding pattern function is )PN PP Nein9 (2.58) the fundamental integral relation (2. 12) provides that N N a n n wi *n (1) inH1(ka) and the application of the minimizing conditions results in the linear system of equations in the unknowns Tml......_~~~~ ~33

THE UNIVERSITY OF MICHIGAN 5780-5-T N jy A n=Bn n=-N....., N (2. 59 m mn n where co = g sin(mr -rr)sin(n7r -ro) r=0-(0 H (lka)H()(ka)(m7r -rro)(nvr -ra) O * (irsin (nr-ra) Pg B= (2) H (ka) (n7r-ra) Pg(p) being the prescribed pattern function. If the inverse of the matrix Amn} is denoted by -ln A, then the actual pattern function PN(d) can be written A B Em(m) (2. 60) PN(~) Jr m=;N =- mn n m where sin (mT -noeino Em(0) = i nH((( ka)(mn r -n! and the mean-square error between the actual and prescribed patterns is finally CN~ ~' _CON mn mn (2.61) The infinite sums in the above forms were of course truncated at some point where 34

THE UNIVERSITY OF MICHIGAN 5780-5-T the accuracy was found to be sufficient. The iteration procedure in this formulation was essentially the same as that described in Section 2. 5. The various parameters involved are defined as follows: ka = 27r (cylinder radius/wavelength) a = 1/2 angular width of slot f = 1/2 angular width of sectoral prescribed pattern N = highest order harmonic in aperture function. The range of values of ka used in the computations was 12 - 21. In general the depen dence of the phenomena of interest here on this parameter is not striking, and consequently only two values are treated in the results presented. Values of a ranged from. 25 up to 2. 6 radians, and those of f from. 5 up to r radians. The maximum order N ranged from 1 to 5. The iteration procedure in general was continued until two successive values of EN were obtained which differed by less than 10 percent. In most cases this required only from two to four iterations. Figure 2-4 shows values of eN plotted against a for a given value of ka and various values of N. The final iterated values shown for N = 3, 5 are not necessarily the minimum values obtainable by this process, but are very near these values, and the linearity of the behavior for N = 3 is perhaps noteworthy. Figure 2-5 shows the effect of the iteration procedure for a relatively narro slot and a uniform (omnidirectional) prescribed pattern. The values of eN obtained here ranged from. 880 down to. 325. The aperture field for the final iteration in the same case is shown in Fig. 2-6. It was found in general, as expected, that the combination of narrow slot and omnidirectional prescribed pattern resulted in the most widely fluctuating aperture fields. Figure 2-7 shows the final iterated pattern functions for a prescribed pattern of approximately sectoral form, i. e. essentially a step function, of width 3 radians, with slot width 2 radians and various values of N. The corresponding aperture fields are shown in Fig. 2-8. 35

THE UNIVERSITY )F MICHIGAN 5780-5-T In Table II-3 are listed the values of EN at each iteration for the majority of the cases which have'been computed with AN( )given by Eq. (2. 56) under the sole constraint that N shall be fixed. TABLE 11-3: VALUES OF eN ka a f3 N s=0 s=1 s=2 s=3 s=4 15.25 7r 3.850.417.289.266 1.0.5 5.0059.0047.0044 1.0 1.027.024 3.021.012.010.010 5.0175.0085.0068.0063 2.0 1.300.171.168 3.297.100.079.073 5.290.070.037.030.027 r 1.670.507.503 3.668.412.378 5. 662.312.228.217 1.4 7r 3.542.327.295 1.7.447.262.242 2.0 1.0 3.012.012 5.0066.0066 7r 3.352.195.186 5.351.151.119.111 2.3 7r 3.257.132.130 2. 6.161.076.075 21.25.5 1.053.016.014.014 3.035.022.017.014.013 1. 5 1.384.130.072.063.061 3.356.084.044.034.029 r 1.910.620.555.547 3.880.465.344.325 1.0 1.5 1.146.076.076 3.144.047.031.029 5.138.055.029.022.019 2.0 1.0 3.013.013 5.0078.0078,r 3.355.220.216 5.355.183.160.151 36

No Iteration: N = 1 2 3 —-.8 4 Final Iterated Values. N: 3 0 - 5A.6 C;' EN -.2a 4 n g.2FC) 0 30 60 90 120 150 180 C) a in degrees FIG. 2-4: MEAN SQUARE ERROR VS SLOT WIDTH, ka = 15

THE U N IVERSY5 TOF MICHIGAN 1.8 0 q P () ( — - -bed) — 1.6 I/ 2 I 1.4 ~k/\ { vi. \ 2 3II.8' I.20 20 40 60 80 I00 12.0 140 160 180 in degrees 38

THE UNIVERSITY OF MICHIGAN 5780-5-T.80 I |A(n, 160 I~~~~~~~~ arg A(q) 140.-_.___. j 120.7R ~i -, oo 80.6 1 I - 60 |A(O)1 I \ | / I - 40 arg A(O).5 I 20 360.4 I I 340 9I 320.3 300 280' -2 I - 260 240 -I - 220 200 180 0.2.4.6.8 1.0 39

THE UNIVERSITY OF MICHIGAN 5780-5-T 1.5 Pg (<)(prescribed). /\N= I (EN =.076) PN.(k3 (E)N.029) igN=5 es(EN =.019) 1.04 PNNo,.5 30 60 90 130 150 180 Cf in degrees FIG. 2-7: LIMITING FAR-FIELD AMPLITUDE VS ANGLE; ka=21, a = 57. 30, 3 85. 90

THE UNIVERSITY OF MICHIGAN 5780-5-T 4.3 IA(O)I 0.2.4.6.8 1.0 (a) q-/a I00 80 0OO s/ argA(~)! 44 N=3 20 N =5 0.2.4.6 8 1.0 (b) FIG. 2-8: LIMITING APERTURE FIELD DISTRIBUTION; ka = 21, a = 57. 3~, / = 85. 9~.

THE UNIVERSITY OF MICHIGAN 5780-5-T The pattern functions and the corresponding aperture fields given in Figs. 2-9 through 2-16 are computed according to the formulation of the synthesis problem in Section 2.5. The value of ka used in the computation was 15 and a (1/2 angular width of slot) was 1 or 2 radians. The reason for the choice of those relatively large values of a is that the series in Eq. (2. 45), as mentioned before, only converges as 1 /n2 and a simple method for calculating the limit is easiest to obtain for large values of a. The maximum number of terms (N) in the expansion of A(A) in Eq. (2. 35) was 10, which in most cases gave a satisfactory approximation to the limit N-4 co. The number of terms in the series of Eqs. (2. 39) and (2. 46) was restricted to 39, which due to the factors H() (ka) and H(2)(ka), respectively, in the denominator of the inn n dividual terms was quite sufficient. In accordance with this, the prescribed sectorial pattern with 1/2 angular width 3=2 radians was defined as given by the first 39 terms in the Fourier expansion, normalized in such a way that the mean square valu was equal to one. The iteration procedure was continued until the last value of N Amin differed by less than 1 percent from the preceeding one. This required from three to twelve iterations. The different values of the Lagrange multiplier, p used in the calculations are listed in Table II-4 together with the obtained values of the minimized quantity A and the mean-square difference E between the actual and prescribed patterns. 9 10 Values for N equal to 9 and 10 are given; the difference between A and A is a measure of how close the result is to the limit N --- o. Also listed are the quality factors defined by Eqs. (2. 54) and (2. 55). 42

THE UNIVERSITY OF MICHIGAN 5780-5-T TABLE II-4 ka = 15 9 10 9 10 10 10 p AA E Q P 0. 1 0. 379 0. 379 0. 282 0. 282 1. 85 0. 176 0. 01 0.213 0.213 0.157 0. 157 7.74 0. 437 0.1 0.133 0.127 0. 075 0. 064 0. 78 0.140 0. 01 0. 051 0. 041 0. 036 0. 025 1.76 0.174 0.1 0.163 0. 163 0. 081 0. 081 1.09 0.138 0. 01 0. 055 0. 055 0. 035 0. 035 2.22 0.178 =2 1 0. 098 0. 098 0. 040 0. 040 0. 07 0. 077 ta2 0.1 0. 031 0. 030 0. 019 0. 017 0.14 0. 083 43

THE UNIVERSITY OF MICHIGAN 5780-5-T I I "', -.I',Z'~4000 0 400 A(o)I ~ =0. 01 0~000 argA() 10 0 ~*Q =7.74 ~10' -3000 = 0.437 300 ~~~5 -~ 6~~~~~~~~~ 0 ~ ~- I ~~I \~~~ arg A(O) l 00 * 9 0 sO 2~~~~~~~~000 "10~~~~~~' 0e0 ~ 30 1. 0 2__ _ _ __ _ _ _ __ _ _ _ __ _ _ _ __ _ _ _ - 00 i~~~~~~~~~~~ *: P (~)(prescribed) IP 0(o~o ) arg A o(o) 1~~~~~~~ ~ 1000 0. 5 - ~' \~ ~~~-~ -- 200 05 00~~~~~~ 0 00 0 0 0 0~~~~~~ I I I'I I'''''I O0 0 0.10 0 0 00 500 1000 150 FIG. 2-9: FAR-FIELD AMPLITUDE AND APERTURE IELD DISTRIBUTION, ka=15, a =57*30, 13=180, N=10. 44

THE UNIVERSITY OF MICHIGAN 5780-5 -T 300~ _ |A(>p) |- =0. 1 5* - arg A(MO) Q =1.85. -' 10 0,*eee.e.e****e*,-".. = 0.176. - 200~ | A(0) | l ~~-@@ 0 argA(o) - 100 0 i, I I'' I 0 0.5 1.0 T) IIEw'ciii,,, ~1.5T1 — P1(PIo0) * 0 0 0 300 o o o * arg P 0(). 1.0<4 *. -- * * 11 200. IP,,ce~l argP10(B) ~. 50~ 1 00~ 1 50 -— ka=15 -=57.3,,180a, N=100 45 0 ~ 0. 0 FIG. 2-1 F F 00

THE UNIVERSITY OF MICHIGAN 5780-5-T A-) f =0. 01 10 3000 argA(M) Q 1.76 2..O yo =0. 174 2 000 11, ~~. O~10 100 0 0 00~~~~~~~~~~~~~~~~ 1.5 Ip~~~ce~~1 3000,.~~se.0..- ".I1 ~~~~~ o I "0! (prescribed) 9 1~ ~~~~~~~~.0' ~~ 1. 0 001 \j'.o'.' \..' ~00 I I I I I I I I I I I I I 1 j~ ~~~~~~~~~~/ I,.'.' 10. F - 0~ 0 ~~ ~ ~ ~~~0. 1.0'a'10( )110 300000 q argP10(P) P( )prescribed), 1.0 e 0~~~~~~~~~~~~~~~~.,5 5~~~~~ S 0 o 00 ~~ e ~ 0~~~5 10 FIG. 2-11: FAR-FIELD AMPLITUDE AND APERTURE FIELD DISTRIBUTION, ka=15, a~=114.6~, /3=180~, N=10. 46

THE UNIVERSITY OF MICHIGAN 5780-5-T. | 10 I 300' argA(p) Q =0.78 2.0 F- r. 140.05e..'._...'-C -200'" 1.0 o 100~ o 0 0.5 1.0,10~~~, 1.5 _ [P10([o)l' 0 300 arg P10( ) 3 ~ P (0)(prescribed) g 1.0'' 200~ 100 -0. 5.. 00 ~' 50 1 00 150 FIG. 2-12: FAR-FIELD AMPLTeUDE AND APERTURE FIELD DISTRIBUTION, ka-15, Ta=114.60, 131800, NA10. 47

THE UNIVERSITY OF MICHIGAN 5780-5-T 400~ T I I I I I l I I I i I A(O) / * =0. 01 10.... arg A(O) Q =2. 22 pr10 =0.178 300 5 _. C.._ o ~ T 1 I*_00 ( 0. 5 1.' *0 0 01, a** 200 ~ **0~ 1 00.50g''|'''' I.' |. | lX I. 1. 5' I do 1~0*''300 00 0 1.FIG 2-3: FAR-FIELD AMPLITUDE AND APERTURE FIELD DSTR IBUTION, ka=15, (=57.30, =114.6, N=10. 0.5 ~. 00 els' 0 0 0 50 100(j5 48~~~~~~~~ -

THE UNIVERSITY OF MICHIGAN 5780-5-T 4000 IA()l u=o0. 1 10 arg A({) Q 1, 09 F10= 0. 138 - 300O 5000000000 ~-t 200 0~ *0) *l 0*.. @ --. P arg A(p) 1000 ~0 ~ ~ 0 1 1.5, 300 1.00 arg AP': 1o(0 1 (prescribed). \1. * 100~ [p.Or~~~~~~~ _(~~ ~ 0. 0 500o 1000 1500 FIG. 2-14; FAR-FIELD AMPLITUDE AND APERTURE FIELD DLISTRIBUTION, ka=15, a=57.30, 3=114.6~, N=10. 49

THE UNIVERSITY OF MICHIGAN 5780-5-T 1.4I_[ A(P3) I 0.8 1.64 - 10 400~' argA() Q =0.14 1.2 - P0 20.083 1.0 _3000 0. 8 I A(I O. 6 - 2000 0.4 - arg A(O) 0.2 - 1000 I I I I " 0 0. 4 0. 8 1.2 1. 6 2. 0 1. 5 3000.J: 300~ 1.0 Po()l \ 200~ 1 01~~~~~ a lP10(f~l 2 — (0) ~ 0. 5 -g (prescribed) 100 0., 5 -ooo~~ 00 *~0 50

THE UNIVERSITY OF MICHIGAN 5780-5-T 400~ 1.4 IA(0),- =1.0 argA(P) Q =0.07 1.2 _ = 0. 077 300 1.0 0 0,8 200 )A(p 0.6 1 \ arg A(O) 0.4 \ 100~ 0.2 o I' I I'O 0 0.4 0.8 1.2 1.6 2..0 * 001 15>< 2 3000 1.. Ipl0(1 \. 200~ IPioc~,l arg Pi0,\I argPl0(b) g (prescribed) *,.' t 1000 0. 5 P01: 0 ~ 1 )~ 5~ 9'0 135 1800 FIG. 2-16: FAR-FIELDAMPLITUDEANDAPERTURE F0ELDD;TRBUTION, ka=15, a=114.60, 3=114.6;, N=10. 51

THE UNIVERSITY OF MICHIGAN 5780-5-T III ARRAYS OF AXIAL AND CIRCUMFERENTIAL HALF-WAVELENGTH SLOTS ON AN INFINITE CIRCULAR METAL CYLINDER 3. 1 Introduction In this section, the case of an array of either axial or circumferential slots uniformly spaced around the circumference of an infinite metal cylinder is considered. All slots are assumed to be half a wavelength long and very narrow, so that the voltage distribution along them is sinusoidal. A good fore-and-aft coverage is then achieved in the case of circumferential slots and, to a lesser degree, also for axial slots. The realization of a nearly omnidirectional pattern in a plane perpendicular to the axis of the cylinder (azimuthal plane) is difficult to obtain whenever the cylinder radius is large compared to the wavelength; the considerations which follow are therefore directed to the synthesis of this azimuthal pattern. If all the feeding voltages across the centers of the slots have the same amplitude and phase, then the best mean-square approximation to an omnidirectional azimuthal field pattern is achieved. In the following sections, formulas are derived which give the minimum mean squared error between the preassigned and the actual patterns, as well as the feeding voltage necessary to produce a far field of prescribed intensity. Computations were carried out for both the mean squared error and the feedin voltage. The numerical results are tabulated and plotted below for a number of slots, N, varying from 2 to 6, and for values of ka varying from 9. 00 to 21. 75 (k = 2ir /X is the free space wave number, and a is the radius of the cylinder). In general, a smaller mean squared error is obtained when the number of slots is increased, for a given value of ka. However, the computed results for the case of axial slots show that this rule is not always valid. Finally, it is shown how to obtain an omnidirectional equatorial pattern having a preassigned elliptical polarization by alternating axial and circumferential slots 52

THE UNIVERSITY OF MICHIGAN 5780-5 -T around the cylinder and by properly choosing the amplitudes and the relative phase of the two feeding voltages. In particular, a circularly polarized equatorial pattern can be obtained in this way. 3. 2 Array of Axial Slots Let us consider an array of N half-wavelength axial slots equally spaced around the circumference of an infinite circular metal cylinder of radius a surroundedby free space. Let us introduce a system of spherical polar coordinates (r, 0, 0) connected to the orthogonal Cartesian coordinates (x, y, z) of Figure 3-1 by the usual relations x = r sin 0 cos 0, y=r sin 0 sin 0, z=r cos O. The slots are symmetrically located with respect to the plane z = 0. If we indicate by 2 a the angular width of each slot as seen from the cylinder axis, and assume that the first slot is centered at 0 = 0, then the electric field produced by the ith slot has, at a large distance from the cylinder, only a 0-component which in the equatorial plane 0 = 7r /2 is given by the well known formula (see, for example Wait, 1955): -imr 0, =~ 2k f 6m C1o(ka)' mcosm ( (3.1) ir ka n=o H (ka) m <(A=1, 2, -, N), where /m=sin(ma)/(ma), (3.2) k=27r /AX is the free space wave number, VI is the voltage across the center of the Ith slot (that is, the product of the 0-component of the electric field at the center of the ith slot times the width of the slot), 6=1, 6>1 1=2, and theprime indicates the derivative of the Hankel function with respect to its argument ka. 53

THE UNIVERSITY OF MICHIGAN 5780-5-T FIG. 3-1: ARRAY OF AXIAL SLOTS 54

THE UNIVERSITY OF MICHIGAN 5780-5-T The far field E0 due to the array of slots is obtained by adding together the fields produced by each slot: N ikr E ekr P(o)k (3 3) Ep= ~ E =E ea krp0 where E is a normalizing constant with the dimensions of an electric field intena sity, and the field pattern P(o) is given by -imr 0 6 e P() m(1)' F (, (3.4) m=o HH (ka) ivith N Fr(0)- ~A~ecos m ~ —(-1) (3. 5) and V >A = * (3. 6) = 2 r aE a The coefficients Al are to be chosen in such a way as to approximate the preassigned far field ikr (Eo)give= E -- (3. 7) ~given a kr as closely as possible. The mean squared error between preassigned and actual patterns is defined by the relation 2121 lp~2d r(3.8) r2v55 55

THE UNIVERSITY OF MICHIGAN 5780-5-T We want to choose Ax so that c be minimum; a simple calculation shows that we must take H(1)'(ka) 0 A1=A 2...=AN N(1+B) where 2,o, B=2H1 (ka)H(2(ka) (3. 10) m~l HmN(ka)H mN(ka) If the coefficients Ae are chosen according to (3. 9), then the mean squared error assumes its minimum value: EA=min e = B/(I+B) (3.11) From (3. 6) and (3. 9) it follows that a7r ka H(1) (ka) VA (3. 12) XE 2N(1+B) a where V 2=. =VN is the voltage across the center of each slot. Formula (3. 12) gives the feeding voltage as a function of frequency, cylinder radius and number of radiating slots for every preassigned value of the far field intensity. Computations of both EA and |VAt were performed with the aid of the IBM 7090 computer at The University of Michigan, for the parameter values ka=9. 00(0. 25) 21. 75 and N=2(1)6. In these calculations, the slot was assumed to be of infinitesimal width, that is, the quantity imN was taken as equal to unity for all values of m and N. Since Wait (1955) has shown that the difference between the radiation pat - tern of a slot whose width is less than about one-tenth of a wavelength and the 56

20 ka 10 io A I I I I I I I 0 10 0.15- Z~r.now-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~n O~~~~~~~~~~~~i5 ~~~~~~ ~ ~ ~ IA IVAIr I~~~~~~~~~~~~~~~~~~~IP O~io-1~ 0.105 -ka 20 FIG. 3-2: FEEDING VOLTAGE FOR AXIAL SLOTS. Bottom and left coordinate scales are valid for N=2, 3; the right and top scales are valid for N=4, 5 and 6.

THE UNIVERSITY OF MICHIGAN 5780-5-T radiation pattern of a slot of infinitesimal width is negligible, we may conclude that the numerical results obtained under the hypothesis a = O remain valid for all ac 7r /(lO0ka) radians. The numerical results are tabulated in Section 3. 4; only three or four figures of the seven that were obtained for each number are given. The same results are plotted in Figs. 3-2 and 3-3. It is seen that for a given N, the dimensionless parameter IVAltends to decrease as ka increases, whereas EA increases with ka; also, bothlVAI and EA present an oscillatory behaviour which becomes more and more pronounced as N increases. For a given ka, the mean squared error EA generally decreases when the number of slots is increased; however, it is easily seen from Fig. 3-3 that this is not always the case: for example, the mean squared error for five slots is less than that for six slots in the range 11. 5 <ka<13. The radiation pattern corresponding to the minimum mean squared error is given by =( O p + = () (3.13) where -imN b(o)=2H) (ka) cos (mNO). (3. 14) o (1)' H0(ka) mN The pattern (3. 13) is symmetrical with respect to 0 = O and periodic with period 27r /N; it is therefore sufficient to calculate it in the range O, 0,< I/N. 3. 3 Array of Circumferential Slots Let us now consider an array of N half-wavelength circumferential slots equally spaced around the circumference of the infinite metal cylinder of Fig. 3-4. If the first slot is centered at 0 = 0, then the electric field produced by the th slot has, at a large distance from the cylinder, only a z-component which in the 58

1.0-t N-2 3 4 5 6 EA 0.9 C.0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i 0.8 - I I'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0.7 10 I''''' I' ('' 15 ka 20 FIG. 3-3: MINIMUM MEAN SQUARED ERROR FOR AXIAL SLOTS z

THE UNIVERSITY OF MICHIGAN 5780-5-T i/4 27r/ N x 2 a-I FIG. 3-4: ARRAY OF CIRCUMFERENTIAL SLOTS........... 60

THE UNIVERSITY OF MICHIGAN 5780-5-T equatorial plane 0 = r /2'is given by the formula (Papas, 1950) co 7 ikr kaV i l Kz -We=7 e- [2Xmml), (3.15) where Cos( n m m 2 2 (3.16) (ka) -m and V't is the voltage across the center of the Ith slot (that is, the product of the z-component of the electric field at the center of the slot times the width of the slot). The far field due to the array of N slots is then given by N ikr e Ez= t Ez = Ec - P, (3.17) where E is the normalization constant with the dimensions of an electric field inc tensity, and -i(m+l) - oo 2 e P()= (1) m Fm(0) (3. 18) with F()= Alcos m - -l) (3. 19) The mean squared error between the preassigned far field ikr (E,)given =EHk (3.21) 61

ivc0.25Cl - N=2_ o. 20- - I,,, 1,, 1, I I 10 15 kao 20 FIG 3-5: FEEDING VOLTAGE FOR CIRCUMFERENTIAL SLOTS

THE UNIVERSITY OF MICHIGAN 5780-5-T and the actual far field (3. 17) is minimized by choosing 0 (ka) (2H )l)ka li A1=A2 N N(1+C) (3.22) where 2 C=2(ka) 4H() (ka)H (k)) a) mN (3.23) 0 o0(m H (ka)H (2)'ka) mN mN If the coefficients A, are chosen according to (3. 22), then it follows that EC = min E = C/(l+C), (3.24) and that (1) V wkaH( (ka)e VC =~~ = (3.25) VC XE 2N(1+C) where Vc=Vl=V2=-* =VN is the voltage across the center of each slot. Formula (3. 25) gives the feeding voltage as a function of frequency, cylinder radius and number of radiating slots for every preassigned value of the far field intensity. Computations of EC and IvCI were carried out using the same values of ka and N that were previously adopted in the case of axial slots. The numerical results are tabulated in Section 3. 4 and plotted in Figs. 3-5 and 3-6. It is seen that IV C I decreases rapidly as ka increases, for a given N. If N is not large, then VC is almost independent of N: the curve of IVCI as a function of ka for a given N exhibits small oscillations about the curve N=2, whose amplitudes appear to increase as N becomes larger. The minimum mean squared error EC increases with ka for a given N, and decreases as N increases for a given ka (Fig. 3-6). 63

1.0O EC N 2 EC 0.9 - 3 0.9 z 0.7 0.6 0 10 15 ka 20o FIG. 3-6: MINIMUM MEAN SQUARED ERROR FOR CIRCUMFERENTIAL SLOTS

THE UNIVERSITY OF MICHIGAN 5780-5-T The radiation pattern corresponding to the minimum mean squared error is given by [P] -0p= 1+C)'(3.26) -where -imN 2 c(o)=2(ka) 2H()(ka) os mN (3.27) 005 (mN~). (3. 27) HmN (ka) As in the case of axial slots, the pattern (3. 26) is symmetrical with respect to 0=O and periodic with period 2w /N, and it is therefore sufficient to calcllate it in the range O:0 < 7/N. 65

THE UNIVERSITY OF MICHIGAN 5780-5-T 3.4 Numerical Results ka N VA, 104 VC 104 EA- 103 EC103 2 1366 2774 927 852 3 1313 2772 895 779 9. 00 4 1391 2770 852 705 5 1315 2788 826 629 6 1693 2778 731 556 2 1337 2741 930 856 3 1321 2740 896 784 9.25 4 1418 2740 851 712 5 1215 2748 841 639 6 1703 2757 733 566 2 1315 2709 932 860 3 1333 2709 897 789 9.50 4 1436 2711 852 719 5 1149 2710 852 649 6 1692 2734 738 575 2 1300 2678 934 863 3 1343 2679 897 795 9. 75 4 1440 2682 853 726 5 1115 2673 858 658 6 1654 2708 747 584 2 1291 2648 935 866 3 1343 2650 899 799 10. 00 4 1425 2653 856 732 5 1106 2640 861 666 6 1586 2681 760 594 2 1282 2619 936 869 3 1329 2620 901 804 10.25 4 1388 2624 862 738 5 1115 2608 861 675 6 1488 2652 778 603 2 1269 2592 938 872 3 1301 2592 904 808 10.50 4 1333 2595 869 744 5 1135 2579 860 682 6 1370 2620 798 613 2 1250 2565 939 875 3 1262 2565 908 813 10.75 4 1266 2567 877 750 5 1162 2552 859 689 6 1250 2588 818 622 66

THE UNIVERSITY OF MICHIGAN 5780-5-T ka N VA 104 VC-104 EA. 103 EC 103 2 1228 2538 941 878 3 1220 2538 912 817 11.00 4 1201 2539 885 756 5 1193 2528 857 696 6 1143 2555 835 631 2 1208 2513 943 880 3 1184 2513 916 821 11.25 4 1148 2512 891 761 5 1223 2505 855 702 6 1061 2523 849 640 2 1193 2489 944 883 3 1161 2488 918 824 11.50 4 1112 2487 895 766 5 1253 2484 853 708 6 1008 2492 858 648 2 1183 2465 945 885 3 1149 2464 920 828 11.75 4 1093 2462 898 771 5 1278 2463 851 713 6 0980 2462 863 656 2 1176 2442 946 888 3 1149 2441 921 831 12.00 4 1094 2439 899 776 5 1298 2443 851 719 6 0973 2434 866 664 2 1167 2419 947 890 3 1154 2419 921 835 12.25 4 1103 2416 900 780 5 1309 2423 851 724 6 0980 2407 866 671 2 1156 2397 948 892 3 1160 2397 922 838 12.50 4 1119 2395 899 784 5 1308 2403 853 729 6 0997 2382 865 677 2 1142 2376 949 894 3 1163 2376 922 841 12.75 4 11.37 2375 898 788 5 1292 2383 856 734 6 1019 2360 864 684........ 67

THE UNIVERSITY OF MICHIGAN 5780-5-T ka N VA. 104 VC- 104 EA103 EC. 103 2 1126 2355 950 896 3 1160 2355 923 844 13.00 4 1154 2355 898 791 5 1259 2362 861 739 6 1045 2339 861 689 2 111i 2334 951 898 3 1147 2335 925 846 13.25 4 1168 2335 898 795 5 1210 2341 868 743 6 1073 2320 859 695 2 1100 2315 952 899 3 1126 2315 927 849 13.50 4 1174 2316 898 799 5 1151 2320 875 748 6 1101 2302 857 700 2 1092 2295 953 901 3 1100 2295 929 852 13.75 4 1171 2297 899 802 5 1089 2299 883 753 6 1128 2286 855 705 2 1086 2276 954 902 3 1074 2277 931 854 14.00 4 1156 2278 901 806 5 1032 2278 890 757 6 1153 2270 853 709 2 1079 2258 954 904 3 1053 2258 933 857 14.25 4 1130 2260 904 809 5 0989 2258 896 761 6 1176 2255 851 714 2 1069 2240 955 906 3 1039 2240 935 859 14.50 4 1095 2241 908 812 5 0961 2238 899 765 6 1195 2241 8j0 718 2 1057 2223 956 908 3 1033 2223 936 861 14. 75 4 1056 2223 912 815 5 0947 2220 902 769 6 1208 2227 849 722 68

THE UNIVERSITY OF MICHIGAN 5780-5-T ka N VA 104 VC 104 EA 103 EC 103 2 1045 2206 957 909 3 1033 2206 936 864 15.00 4 1019 2206 916 818 5 0946 2202 903 773 6 1215 2212 850 726 2 1034 2189 958 911 3 1036 2189 937 866 15.25 4 0990 2189 919 821 5 0954 2185 903 777 6 1212 2197 852 731 2 1026 2173 958 912 3 1039 2173 937 868 15.50 4 0972 2172 921 824 5 0967 2169 902 780 6 1196 2182 855 735 2 1020 2157 959 913 3 1039 2157 937 870 15.75 4 0965 2156 922 827 5 0985 2153 901 783 6 1168 2167 859 739 2 1014 2141 960 914 3 1035 2142 938 872 16. 00 4 0966 2140 923 830 5 1003 2139 900 787 6 1126 2151 865 743 2 1007 2126 960 916 3 1024 2126 939 874 16.25 4 0973 2125 923 832 5 1022 2125 899 790 6 1074 2134 873 746 2 0999 2111 961 917 3 1008 2111 941 876 16.50 4 0983 2111 923 834 5 1039 2111 898 793 6 1016 2118 880 750 2 0990 2097 961 918 3 0990 2097 942 877 16. 75 4 0995 2097 922 836 5 1053 2097 897 796 6 0960 2101 888 754 69

THE UNIVERSITY OF MICHIGAN 5780-5-T ka N VA- 104 VC 104 EA 103 EC-103 2 0980 2082 962 919 3 0972 2082 944 879 17. 00 4 1005 2083 922 839 5 1062 2084 897 798 6 0911 2085 894 758 2 0971 2068 963 921 3 0958 2068 945 881 17.25 4 1012 2069 922 841 5 1064 2071 898 801 6 0875 2068 899 762 2 0965 2054 963 922 3 0949 2054 946 882 17.50 4 1014 2055 923 843 5 1059 2057 899 804 6 0851 2053 903 765 2 0959 2041 964 923 3 0945 2041 946 884 17. 75 4 1009 2042 924 845 5 1044 2044 901 806 6 0840 2038 905 768 2- 0954 2028 964 924 3 0945 2028 947 885 18. 00 4 0996 2028 925 847 5 1020 2030 904 809 6 0839 2023 905 772 2 0949 2015 965 925 3 0947 2015 947 887 18.25 4 0976 2015 927 849 5 0988 2017 908 812 6 0845 2009 905 775 2 0942 2002 965 926 3 0949 2002 947 889 18.50 4 0952 2002 929 851 5 0952 2003 912 814 6 0856 1996 905 778 2 0934 1990 966 927 3 0947 1990 948 890 18.75 4 0926 1989 932 853 5 0916 1990 916 817 6 0871 1984 904 781 70

THE UNIVERSITY OF MICHIGAN 5780-5-T ka N VA 104 VC 104 EA-103 EC 103 2 0926 1977 966 928 3 0942 1977 948 891 19. 00 4 0904 1977 934 855 5 0884 1977 919 819 6 0888 1972 903 783 2 0919 1965 967 929 3 0933 1965 949 893 19.25 4 0886 1965 936 857 5 0861 1964 922 821 6 0905 1961 901 786 2 0913 1953 967 929 3 0921 1953 950 894 19.50 4 0876 1953 937 859 5 0846 1952 924 824 6 0923 1950 900 789 2 0909 1942 967 930 3 0907 1942 951 895 19.75 4 0872 1941 937 861 5 0839 1940 925 826 6 0940 1940 899 791 2 0904 1930 968 931 3 0895 1930 952 897 20.00 4 0874 1930 938 863 5 0840 1929 925 828 6 0955 1930 898 793 2 0899 1919 968 932 3 0885 1919 953 898 20.25 4 0879 1919 938 864 5 0847 1917 925 830 6 0968 1920 897 796 2 0893 1908 969 933 3 0879 1908 954 899 20.50 4 0887 1908 938 865 5 0857 1907 925 832 6 0978 1910 897 798 2 0886 1897 969 934 3 0876 1897 954 900 20.75 4 0894 1897 937 867 5 0869 1896 924 834 6 0984 1900 897 800 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ ~ 7 1.

THE UNIVERSITY OF MICHIGAN 5780-5-T ka N VA 104 VC 104 EA-103 EC'103 2 0880 1887 969 934 3 0876 1887 954 901 21. 00 4 0901 1887 937 869 5 0882 1886 923 836 6 0984 1890 897 803 2 0874 1876 970 935 3 0877 1876 954 903 21.25 4 0904 1877 937 870 5 0894 1876 923 838 6 0978 1880 898 805 2 0869 1866 970 936 3 0878 1866 955 904 21.50 4 0904 1866 938 872 5 0904 1866 922 839 6 0964 1870 900 807 2 0865 1856 970 936 3 0876 1856 955 905 21.75 4 0898 1856 939 873 5 0912 1856 922 841 6 0943 1860 903 809 -__ _ _ _ _ __ ~ 72

THE UNIVERSITY OF MICHIGAN 5780-5-T 3. 5 Array of Axial and Circumferential Slots An elliptically polarized far field pattern in the azimuthal plane may easily be obtained by alternating N uniformly spaced axial slots and N uniformly spaced circumferential slots around the cylinder. It is not necessary to assume that the two angles between the center of a circumferential slot and the centers of the two adjacent axial slots, as seen from the cylinder axis, be equal; we shall only assume that the angle between two adjacent axial slots, or between two adjacent circumferential slots, is 2wr/N radians (Fig. 3-7). From formulas (3. 11),(3. 12),(3. 24) and (3. 25) it follows that V H(l)(ka).?r E a 1-EA o -17 a (3.28) circumferential slots are sensibly omnidirectional, then relation (3. 28) gives the ratio of the two feeding voltages as a function of the far field polarization, for prescribed values of N and ka. For a linearly polarized far field, the quantity E /E is real. For a circularly polarized far field, one must choose, Ea/E = e and therefore [+ ]-EA a E _ a) H (ka). (3.29) Vc _J1- 1-EC a(ka) o cir. pol. In the case of a large cylinder, further simplification is achieved by observing _______________________________________ 7 3 l

THE UNIVERSITY OF MICHIGAN 5780-5-T z 2a1 F~~2 a FIG 3-7: ARRAY OF AXIAL AND CIRCUMFERENTIAL SLOTS 74

THE UNIVERSITY OF MICHIGAN 5780-5-T that (1)' 7 Ho (ka) -i 71 o 2 H) e J1k I(ka >> 1); 0 (ka) in this case: V E a 1-EA a V EC (ka >> 1). (3. 30) V 1-EC E c c 3. 6 Final Considerations The main results of Section III may be summarized as follows. a). If all the feeding voltages have the same amplitude and phase, then the best mean-square approximation to an omnidirectional azimuthal field pattern is achieved; the actual optimum pattern is given by formulas (3. 13) and (3. 26). b). The feeding voltage as a function of frequency, cylinder radius, number of slots and far field intensity is given by formulas (3. 12) and (3. 25), and is tabulated in Section 3. 4 and plotted in Figs. 3-2 and 3-5. c). A far azimuthal field with a prescribed polarization may be obtained through formula (3. 28), provided that the actual patterns are sensibly omnidirectional. The optimization process that was used in this section is based on the minimization of the mean squared error (3. 8). In order to have some information on the features of the patterns thus obtained, computations of P(0) as given by formula (3. 26) were carried out for the case of ka = 10 and five circumferential slots. The results are plotted in Fig. 3-8; it is seen that the amplitude of the field pattern is quite far from unity. 75

THE UNIVERSITY OF MICHIGAN 5780-5-T 1.0 +800 0.8- >namplitude +40~ phase 0.6~I oo~ 00 0.4 I ~, IC-~ -400 0.2 -80~ 0- I 10 I 1 O I 00~ 120 24 36 FIG. 3-8: AMPLITUDE AND PHASE OF [P()]opt FOR ka=1O AND FIVE CIRCUMFERENTIAL SLOTS 76

THE UNIVERSITY OF MICHIGAN 5780-5-T If one is interested only in approximating the power radiation pattern, then one must conclude that the minimization of the error (3. 8) does not constitute a good criterion of optimization. In this case, one should try to minimize either 27r E j ei@(0) P(O) 2do (3. 31) 0 where @(O) is a continuous real function of 0 to be chosen so that the mean squared error between I P(0)1 and unity be minimum, or the mean squared error between the actual and the preassigned power patterns: Epower- }. (3. 32) It appears that the mathematical difficulties encountered in minimizing e power cannot be overcome easily, so that it seems preferable to minimize the right-hand side of (3. 31), e. g. by successive approximations. One may choose: 27rP ( 2 1mEm 1 2,( |P'(0)- IPm-1( l do, (m=l, 2, 2. ) (3. 33) with PO () = 1, and determine the unknown feeding voltages which appear in the expression of P (0) so as to minimize em. The iteration procedure (3. 33) can easily be handled by a computer; however, it remains to be proven that em converges to a minimum value of e as m increases. Numerical results based on the approximation procedure (3. 33) have been obtained in Section It for the problem treated there. 77

THE UNIVERSITY OF MICHIGAN 5780-5-T APPENDIX Lemma For every function A(0), square integrable in the interval -a < a -a (notation: A(0) EL2 ) the corresponding function P(p) defined by P(p) = K(0- p') A(0')d0', -Tr < p 7r, a..,r (A. 1) where r c 0 cosn n n=0 i H( (ka) n K(0) = (A. 2) Z 1n n( nO i H (ka) e =1, e =... = 2 n o'1 r is an analytic function. We first note that by introduction of the variable C = e, the Fourier series in Eq. (A. 2) can be considered as Laurent series which are convergent in every region in the complex c-plane defined by 0 < a < ( I < b < o0, where a, b are positive constants. This is a consequence of the asymptotic behavior of H (1)(ka) and (1)' n H (ka) for ka fixed and n - o through real positive values: H (ka) i2 n r nn \eka n kae H(1) (ka) i-(n (A.+ 11 - 00 78

THE UNIVERSITY OF MICHIGAN 5780-5-T The Laurent series in r represents an analytic function in its region of convergence and, since ~ is an analytic function of p, K(0) is a function analytic in the entire complex 0-plane (cf. Whittaker and Watson, 1927, p. 160 ff.). From this it follows that P(0) is also an analytic function because it has the unique derivative o a K(o- p))A(01) do? Theorem If we take an arbitrary set of functions ( complete in L2 and construct a new set of functions {TT (), where TTn(0) is the P(0) in Eq. (A. 1) which corresponds to A(0) = n (p0), then the set TT,(I will be closed in L2r. That is, there is no function belonging to L7 which is orthogonal to all F- (0). 2 n To prove the theorem, we assume that the contrary is true, i.e. that there is a function F(W)E Lg such that for every n F;0)T-[()d =- F *(0)d0 K(0-0')0 (0')d' = 0 (A.4) -or -7r -a where F is the complex conjugate of F. It is obvious, by virtue of Fubini's theorem, that we can change the order of integration in Eq. (A. 4), and using the fact that according to Eq. (A. 2), K(p) is an even function we obtain aP n(')d'\ F*(0)K(I' - )do = S n(')G(')d' = O (A. 5) where G(0) is thus defined as -79

THE UNIVERSITY OF MICHIGAN 5780-5 -T 7r G(0) i K( -')F:(p')d', -7r < 0 r (A. 6) -7T As ~&(p) is complete in L2 a Eq. (A. 5) can hold only if G(0) 0 almost everywhere in the interval -a < 0 < a. But due to Eq. (A. 6), G(o) satisfies the conditions in the Lemma and is consequently analytic in the interval -ir < 0 7r and thus G(0) 0 — in this whole interval. Eq. (A. 6) also expresses G(0) as the convolution of the functions K(0) and F'(0). Thus G(P) 0 0 implies that F*(0) vanishes almost everywhere, which proves the theorem. In L2 completeness and closure are equivalent and we have the following corollary. Corollary To every pair of set of functions {fn(0 ) and TTn(0)} as defined in the theorem there is a finite set of functions ai such that 0 N 5? IF(P),- anTTn(0) 2d0 < e (A.7) n=0 -r for any given arbitrary function F(0) L2 and e> 0. This means that if we consider Eq. (A. 1) as an integral equation with 2 P(P) = F(0) there is no solution A({)EL2 except when F({) belongs to a certain class of analytic functions. However, we can always find an N A(p) = Jan n() EL2 n= 0............80

THE UNIVERSITY OF MICHIGAN 5780-5-T such that the corresponding N P(0) = ZI a 170) n=O approximates F(0) arbitrary close in the mean square sense.

THE UNIVERSITY OF MICHIGAN 5780-5-T BIBLIOGRAPHY Aigrain, P. (1952), "Les antennes super-directives, " L'Onde Elect., 32, No. 299, pp. 51-54. Bouwkamp, C. J. and N. G. DeBruijn (1946), "The Problem of Optimum Antenna Current Distribution, " Philips Res. Reports, 1, No. 2, pp. 135-158. Caprioli, L., A. M. Scheggi and G. Toraldo di Francia (1961), "Approximate Synthesis of a Prescribed Diffraction Pattern by Means of Different Aperture Distributions," Optica Acta, 8 pp. 175-179. Chu, L. J. (1948), "Physical Limitations of Omni-directional Antennas, " J. Appl. Phys. 19, pp. 1163-1175. Collin, R. E. (1964), "Pattern Synthesis with Non-separable Aperture Fields, " IEEE Trans. G-AP, AP-12, pp. 502-503. Collin, R. E. and S. Rothschild (1963), "Reactive Energy in Aperture Fields and Aperture Q, " Can. J. Phys., 41, No. 12, pp. 1967-1979. Collin, R. E. and S. Rothschild (1963), "Reactive Energy in Aperture Fields and Aperture Q, " Electromagnetic Theory and Antennas (Ed. E. C. Jordan) Pt. 2, Pergamon Press, pp. 1075-1077. Collin, R. E. and S. Rothschild (1964), "Evaluation of Antenna Q, " IEEE Trans. G-AP, AP-12, No. 1, pp. 23-27. DuHamel, R. H. (1952), "Pattern Synthesis for Antenna Arra: s on Circular Elliptical and Spherical Surfaces," University of Illinois Engineering Experiment Station, Technical Report No. 16. Fel'd Ya. N and L. D. Bakhrakh (1963), "Present State of Antenna Synthesis Theory," Radiotek. i. Elek., 8, No. 2, pp. 163-179. (Translated edition) Flammer, C. (1957), Spheroidal Wave Functions, Stanford University Press. Harrington, R. F. (1958), " On the Gain and Beamwidth of Directional Antennas, " IRE Trans. G-AP, AP-6 pp. 219-225. Hasserjian, G. and A. Ishimaru, (1962), "Excitation of a Conducting Cylindrical Surface of Large Radius of Curvature, " IRE Trans. G-AP, AP-10, No. 3, pp. 264273. Hasserjian, G. and A. Ishimaru (1962), "Currents Induced on the Surface of a Conducting Circular Cylinder by a Slot, " J. Res., NBS, 66D No. 3, pp. 335-365. Knudsen, H. L. (1959), "Antennas on Circular Cylinders:' IRE Trans. G-AP, AP-77 Special Supplement, pp. 361-370. 82.

THE UNIVERSITY OF MICHIGAN 5780-5-T Kovacs, R. and L. Solymar (1956), "Theory of Aperture Aerials Based on the Properties of Entire Functions of the Exponential Type, " Acta Phys. Budapest, 6 No. 2, pp. 161-183. Landau, H. J. and H. O. Pollak (1961), "Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty-II, " Bell Syst. Tech. J., 40, No. 1, pp. 65-84. Landau, H. J. and H. O. Pollak (1962), "Prolate Spheroidal Wave Function, Fourier Analysis and Uncertainty-mII, " Bell Syst. Tech. J., 41 No.4, pp. 1295-1336. Ling, C., E. Lefferts, D. Lee and J. Potenza (1964), "Radiation Pattern of Planar Antennas with Optimum and Arbitrary Illumination, " IEEE International Convention Record, Part 2, pp. 111-124. Logan, N. A., R. L. Mason and K. S. Yee, (1962), "Equatorial Plane Radiation Fields Produced by a Circumferential Slot on a Large Circular Cylinder, " IRE Trans. G-AP, AP-10, No. 3, pp. 345-347. Meixner, J. (1949), "Die Kantenbedingung in der Theorie der Beugung Elektromagnetischer Wellen an Volkommen Leitenden Ebenen Schirmen, " Ann. Phys., Vol. 6:2-9. Mittra, R. (1959), "On the Synthesis of Strip Sources, " University of Illinois Technical Report No. 44. Nishida, S. (1960), "Coupled Leaky Waveguides II: Two Parallel Slits in a Cylinder," IRE Trans. G-AP AP-8, No. 4, pp. 354, 360. Papas, C. H. (1950), "Radiation From a Transverse Slot in an Infinite Cylinder, " J. Math and Phys., 28, pp. 227-236. Rhodes, D. R. (1963), "The Optimum Line Source for the Best Mean-square Approximation for a Given Radiation Pattern," IEEE Trans. G-AP, AP-11, pp. 440446. Silver, S. (ed.) (1949), Microwave Antenna Theory and Design, McGraw-Hill Book Company, New York. Silver, S. and W. K. Saunders (1950), "The External Field Produced by a Slot in an Infinite Circular Cylinder, " J. Appl. Phys., 21, pp. 153-158. Slepian, D. and H. O. Pollak (1961), "Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty-I," Bell Syst. Tech. J., 40 pp. 43-64. (cf. Landau and Pollak)......1~~ ~83

THE UNIVERSITY OF MICHIGAN 5780-5-T Solymar, L. (1958), "Maximum Gain of a Line Source Antenna if the Distribution Function is a Finite Fourier Series, " IRE Trans. G-AP AP-6, pp. 215-219. Taylor, T. T. (1955), "Design of Line-source Antennas for Narrow Beamwidth and Low Sidelobes,"IRE Trans. G-AP, AP-3, No. 1, pp. 16-28. Wait, J. R. (1955), "Radiation Characteristics of Axial Slots on a Conducting Cylinder',' Wireless Eng., pp. 316-323 Wait, J. R. (1959), Electromagnetic Radiation from Cylindrical Structures, Pergamo Press, New York. Wait, J. R. and J. Householder (1959), "Pattern Synthesis for Slotted-cylinder Antennas, " J. Res., NBS., 63D No. 3, pp. 303-313. Whittaker, E. T. and G. N. Watson (1927), A Course of Modern Analysis, Fourth Edition, Cambridge at The University Press. Woodward, P. M. and J. D. Lawson, (1948), "The Theoretical Precision With Which an Arbitrary Radiation-Pattern May be Obtained from a Source of Finite Size," J. Inst. Elec. Eng. 95, No. 37, Pt.IHI, pp. 363-371. Yen, J. L. (1957), "On the Synthesis of Line-sources and Infinite Strip-sources, " IRE Trans. G-AP, AP-5, pp. 40-46. 84

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