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1989, Ulversoy,Kildal 1628 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 31, NO. 12, DECEMBER 1989 Radiation from Slots in Artificially Soft and Hard Cylinders point TORE ULVERSBY AND PER-SIMON KILDAL, SENIOR MEMBER, IEEE Abstmct- Artificially soft and hard surfaces for electromagnetic waves 4 .lot have been defined in a recent paper. Here, the radiation fields of cir- cumferential slots in soft and hard cylinders are calculated, when the slots are excited by a TEII circular waveguide mode. The results are compared to those of a smooth conducting cylinder; showing that the soft surface strongly reduces the intensity of the field radiated along the cylinder whereas the hard surface enhances it. The results are applicable to the design of self-supported feeds for paraboloids, and to the analysis of the line feeds of the radiotelescope in Arecibo. The calculations are I!,!! confirmed by measurements. Fig. 1. Cylinder with circumferential slot. (a) Cross section near slot. (b) Coordinate system. I. INTRODUCTION II. CALCULATIONMODEL We consider an infinitely long cylinder that is located along the Prediction of the fields radiated from slots in cylinders is an impor- z-axis and has a circular cross section with diameter 2a (Fig. 1). At tant problem in several antenna designs, e.g., in slotted waveguide = 0, the cylinder is divided in two by a circumferencial slot with arrays like the line feeds of the radiotelescope in Arecibo [l], and z width 6. We assume that the slot is radiating, as it is excited by the in self-supported feeds for reflector antennas like the hat feed [2]. If fields of a radial parallel-plate waveguide. This is again excited by the cylinder is circular and conducting, the component of the E-field the TEI mode of an inner circular waveguide, so that the fields in that is normal to the cylinder may radiate strongly along the cylin- the radial waveguide will have the same first-order cos cp cp der, whereas the tangential component does not [2]. This is in some or sin azimuthal dependence as the TEll mode. Also, if we assume that cases disadvantageous as cross polarization may be created. It can 6 0.5 A, only the basic modes of the radial waveguide will be be avoided if the cylinder surface is made artificially soft or hard, as 5 shown in this communication. Artificially soft and hard surfaces for present over the slot as the higher order modes become strongly electromagnetic waves have been defined in [3], by using an analogy evanescent. From this, we can express the field over the cylindrical aperture of the slot in terms of two modes; one with z-directed with the soft and hard surfaces in acoustics. In [3], a soft surface is E- characterized by the radiated field along the surface being zero for fields which we refer to as a z-mode, and one with cp-directed E- either polarization, whereas a hard surface provides strong radiation fields which we refer to as a p-mode. If we consider excitation along the surface. Therefore, soft and hard cylinders can be used to from a TE, circular waveguide mode with linear polarization -in obtain low cross polarization of fields radiated around them, at the y-direction, the z- and p-components of the E- and H-fields in the same time as the field is either zero (soft case) or has a maximum aperture become [4, sec. 5-31. (hard case) along the cylinder. = V, sin p The calculation model in this communication is based on the cylin- E,, (la) drical transform method in [4, sec. 5-12]. This is extended to handle soft and hard surfaces. The results are evaluated for the case that the E,, = V, cos (7)cos cp slot is excited by a TEI I circular waveguide mode propagating inside the cylinder. This makes the results comparable to those obtained for H,, = V,KV2 cos cos cp /q a smooth conducting cylinder in [2]. The results are compared with (7) measurements that confirm the main characteristics of the soft and hard surfaces. It should be mentioned that the hard cylinder with a circumferential slot was already investigated and measured during Spring 1986 [5]. In fact, the concept of artificially soft and hard surfaces was a result of this work, done in connection with improve- for IzI < 6/2 and zero for all other z, where V, and V, are constants ments on the hat feed and the improved calculation model for the determining the excitations of the z- and cp-modes, respectively, and Arecibo line feed [6]. where K,, KV1,and K,2 are constants given by Manuscript received November 18, 1988. This work was supported by the - j .HF”(ka) .IrK(ffa) . Royal Norwegian Council for Scientific and Industrial Research. This com- K, = ;K,1= munication is based on a thesis submitted by T. Ulversdy to the Norwegian Hf’(ka) -J a6kaK ‘(aa)’ Institute of Technology in partial fulfillment of the requirements for the M.S. degree. T. UlversBy is with SINTEF, Norwegian Institute of Technology, N-7034 K,~=J-. aK(aa) withcx = {(i)2-k2. (2) Trondheim, Norway. kK I (aa) P-S. Kilda1 was with ELAB-RUNIT, Norwegian Institute of Technology, N-7034 Trondheim, Norway. He is now with the Division of Network Theory, Chalmers University of Technology, S-41296 Gothenburg, Sweden. Here k is the wavenumber, q is the wave impedance in free s ace IEEE Log Number 8929316. Hf’ is the Hankel function of first order and second kind, He’’ i6 0018-926X/89/1200-1628$01.00 0 1989 IEEE IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 37, NO. 12, DECEMBER 1989 1629 the derivative of Hy’ , K is the modified Bessel function of first order Equation (9) contains four linear equations of the six unknowns and K is the derivative of K. fn(w), gn(w), EL, E&, HL and Hf,. Two extra linear equations We will now derive the field solution outside the cylinder by using are obtained by using (3), that clearly must be valid also for the an extension of the method in [4, sec. 5-12] for the case that the cylindrical transforms of the induced fields as the transformation of surface impedance of the cylinder is anisotropic. The anisotropic all field components are the same, i.e., surface impedance is in a cylindrical coordinate system defined in terms of its two components ETI, = -%HT, and ET9 = 7,HE (10) The actual expressions for the source field components in (9) are 7, = -&/HIP and vv = E,,/H,, (3) obtained by using (1) and (5). The result is where E,, and E,, (HI2 and H,,) are the z- and p-components, respectively, of the E-field (H-field) at the cylinder surface. The ET, = V, WO@) total E-field (and H-field) anywhere along the cylinder including the 2j slot can be written as a sum of this induced E-field (H-field) at the cylinder surface and the source fields in (1) which is bounded by the slot, i.e., Ez = Esz + Eiz (4) and corresponding for the other field components. We introduce the cylindrical transform E: of E,, i.e., 2% +m w) = & 1 E,(a, p, z)e-J“pe-JwZdzdp for n = fl and zero for the other values of n with WO@), 1m Wl(w),Wz(w) being the Fourier transforms of a function that is (5) constant, equal to cos (?rz/6)and equal to sin (xz/6),respectively, and correspondingly for the other field components. The inverse for IzJ5 6/2, and zero elsewhere, i.e., transforms have the form sin (w6/2) Wo(w)= 6 ~ E,@, cp. Z) =F-’[ET(n, w)] wS/2 = -1 +m elnqLrE:(n, w)ejwzdw. (6) cos (w6/2) cos (w6/2) W,(w)=6 _____ 2s *-W6 r+w6 n=-m ( +--I It is clear that each field component at e = a may be expressed as an cos (w6/2) cos (wS/2) inverse transform of its own cylindrical transform along the cylinder. ~ - ~ Also, the cylindrical transformation is linear, so that from (4) x+w6 ?r-w6 E:(n, W)= ET,@, w)+EL@, W) (7) Finally, the desired field solution outside the cylinder is found from the solutions of fn(w) and g,(w) by an inverse transformation of the where ET, is the Fourier transform of E,, in (l), and EL is the vector potentials in (8) and then by using eqs. (5-19)]. In unknown transform of the z-component Ei, , and corresponding for [4, (5-18), the far field of the slot, the resulting expression become particularly the other field components. simple, see eq. (5-151)]. Using this, and fl(w) = fP1(w) and Let us now introduce the field solution outside the cylinder. In the [4, gl(w) = -gp1(w), which we know must be satisfied from the form space of the cylindrical transform this can be written as the (w,n) of the excitation in (1), we get sum of TE, and TM, solutions with electrical and magnetic vector potentials of the form where where g,(w) and fn(w) are unknown functions. The E- and H- fields of (8) are readily found by using [4,eqs. (5-18), (5-19)]. For e = a, these components must be equal to the corresponding field 2jk +-sin eg,(-k COS 0) COS cp+ components in (7). This gives the following equations for each of the x field components: with 8 and + unit vectors in the directions of increasing polar angle 0 and azimuth angle cp, respectively.
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