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 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. Both f,,(w) and gn(w) become functions of the two excitation of the slot, so that (13) can be separated into

G(e, p) = v,[A,(e) sin pi +c,(e)COS p+l

+V,[A,(B) sin pi + C ,(e) cos p91 (14) nw - k,H~”’(k,a)f,,(w) - -H!?(k,a)g,(w) = HT, +Hrv. where A,(@and C ,(e) are the radiation patterns in E- and H-plane, JwPa of the z-mode, and A,(@ and C ,(e) are the radiation patterns in E- and H-plane of the p-mode. 1

1630 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 31, NO. 12, DECEMBER 1989

III. SMOOTHCONDUCTING CYLINDER RADIATION PATTERN. SOFT CILlroER This case was treated in [l], but is summarized here to make this communication more complete. For a smooth conducting cylinder

7, =O and 7, =O (15) which from (10) means that EL = 0 and ET, = 0. Therefore, f,(w) c. can be solved from (9a) and thereafter g,, (w)from (9b). The resulting radiation patterns of the two modes in the slot become, by using (13) and (14),

(ka)wo(-kcos e) -40. A,(@ = n(ka sin O)Hf’(kasin e) - (ka cos O)Wo(-k cos e) C,(@ = n(ka sin e)ZHY)’(kasin e) -20. A,(e) = o w1(-k cos e) C ,(e) = j (16) n~f”(kasin e) ’ .. -30.1 v..... T..,.. I .I....,. , .....* These radiation patterns are plotted in [l, fig. 21 for 6 << X, i.e., 0. U). 60. 90. WO(-k cos 0) = 6. They are characterized by A, (e) and C ,(e) being TNTA equal and high-valued for 8 = 0. A,@)decreases slowly and C,(O) Fig. 2. Radiation pattern, soft cylinder. 6 = 0.47 A, U = A. rapidly for 0 increasing up to 90°. C,(@ is zero for 0 = 0 and equal E- and H-plane patterns and hence zero cross polarization increases according to (ka sin e)’ when 0 increases. Therefore, the z- for a proper excitation of the slot. If 6 is not small we must in mode radiates mainly in the E-plane and strongly along the cylinder, addition require that WO(W)= WI which means that the aperture and the cp-mode radiates only in the H-plane and has a zero along (w), distribution of the E- and H-plane modes in z-direction must be the cylinder. equal. This can be obtained by using soft or hard walls of the radial IV. SOFTCYLINDER waveguide between the inner and outer surface of the cylinder. i The surface impedances of a soft cylinder are defined by [3] The radiation patterns in (18) are plotted in Fig. 2 for 6 = 0.47 X and a = X. We see that the E-plane pattern of the z-mode and the 7, = 00 and 7, =O. (17) H-plane patterns of the 9-mode dominates. They are not identical because 6 is too large. This means that ET, and HT, are zero, so that f,(w) and gn(w)can be found by solving (9b) and (9d). The resulting radiation patterns V. HARDCYLINDER are The set of equations (9a)-(9d) also applies to a hard cylinder, for A,(@ = jK,(ka sin O)’Hf’’(kasin B)Wo(-k cos O)/D(e) which the surface impedances are given by [3]

c,(e) = -jK, COS e(ka sin O)Hf’(kasin O)Wo(-k cos O)/D(e) 7, =O and H, =CO. (20) A,(e) = 1-j cos e(ka sin e)Hf)(ka sin e)wl(-k cos e) Then, E;, = 0 and H;, = 0, so that the solutions off,, and g, are +jK,,(ka sin Q2Hf)’(kasin O)W2(-k cos O)l/D(e) particularly easy to find from (9a) and (Sc), respectively. Thereafter we get by using (13), (14) the radiation patterns c ,(e) = ~(kasin e)ZHf)‘(ka sin e)wI(-k cos e) kaWo(-k cos e) . A,(e) = -jKVl cos O(ka sin O)Hf’(ka sin O)W2(-k cos O)]/D(O) (18) A,@)= n(ka sin O)Hf’(kasin e) ’ where the denominator is - KVzkaWI(-kcos e) , we)= c,(e)= 0. (21) D(B) = ~{[kasin OHf”(ka sin e)]’ - [cos OHy’(ka sin O)]’}. a(ka sin e)Hf)(ka sin e) ’

If here we also let 6 << A, we get Wo(-kcosO) = WO@)= 6, In this case we see that the E- and H-plane patterns have the same W,(-kcosO) = W1(0) = 26/n, and Wz(-kcos8) = Wz(0)= 0. shape except for the minor difference between W&) and Wl(w). Also, the z-mode radiates entirely in E-plane and the cp-mode entirely Then, we see that A,@)= K,C ,(O)n/S and C,(@ = K,A,(B)n/G, and also easily derive that A,(O) = C,(O) = A,(O) = C ,(O) = 0 in H-plane. The correct excitation of the two modes to obtain low by expansion of the Hankel functions for small arguments. The total cross polarization is E- and H-plane patterns become for 6 << X V, = -KpZVp.

A(e)= wue)+ V,A,(W Computed radiation patterns are shown in Fig. 3 for 6 = 0.47 X and 1 a = X. We observe that the axial radiation is strong. c(e)= v,c,(e)+ v,c,(e) = K,v,A,(e)n/6 + -v,A,(o)~/~ Kz VI. MEASUREMENTS (19) We have done measurements at 12 GHz to check the theoretical from which we see that the E- and H-plane patterns are equal if V, = radiation patterns. Such measurements introduce three problems, V,(K,s/6)?This means that for a soft cylinder it is possible to obtain which are discussed below. IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 37, NO. 12, DECEMBER 1989 163 1

- IO. m- a $ 0.

Y- + -I %-IO.

-20.

-30.

THETA

A Az(6)

0. 50. MTA Fig. 3. Radiation pattern, hard cylinder. 6 = 0.47 A, a = A.

TILTA (b) Fig. 5. Measured and calculated radiation patterns for soft cylinder with reflecting plate. 6 = 0.47 A, a = A. (a) z-mode. (b) cp-mode.

the radiation patterns of the two modes separately, by blocking the excitation of the opposite mode. The z-mode was blocked by using a grid of thin z-directed wires across the slot. These short-circuit the Fig. 4. Photo of hard cylinder with slot and soft reflecting plate. z-component of the field and have very little effect on the fields of the cp-mode. The cp-mode was blocked by dividing the radial wave- 1) The radiation pattern of the slot is omnidirectional, so that the guide into two radial waveguides of half the height by using a thin supports of the cylinder and the cable connecting it to the receiver conducting disk with a hole in the center, as the cp-mode is strongly will cause serious measurement errors. This was solved by locating evanescent with a short height. The radial waveguide was filled with a reflecting disk at one edge of the slot (Fig. 4) (i.e., z = -6/2), dielectric material to keep together all mechanical parts of the mea- so that the radiated field behind the plate is strongly reduced. The surement model. disk was provided with a soft surface by using circular corrugations as in the brim of the hat feed [2]. Thereby, the reflection coefficient A. Smooth Cylinder becomes polarization-independent, and the sum of the direct and The measurements of this case have been done before, see [2, reflected fields can be obtained by multiplying A,(@,A,(@, C,(O) fig. 51, and are therefore not repeated here. They show extremely and C ,(e) by the same interference function good agreement with the calculations, as the calculations then also = ejk cos 9612 - e -jk cos 9612 - 2 . included edge-diffracted fields. z(e) - J sin (ksin 86/2). (22) This is valid only if the disk diameter is infinitely large, whereas we B. Soft Cylinder have used d = 6.0 X in practice. The finite size could easily have The soft surface of the cylinder was realized by using transverse been included in the calculations by using the uniform geometrical (i.e., circular) corrugations according to [3] and [7]. The depth of the theory of diffraction, as in [2], but this is not necessary for the testing corrugations were chosen to be 0.24 X according to 17, fig. 41 for the of the analytical models of this communication. chosen cylinder diameter 2a = 2.0 X. There were three corrugations 2) The length of the cylinder will have to be finite in practice, per wavelength and the width of each were 0.08 A. The results of thereby introducing measurement errors. The actual measured cylin- the measurements are shown in Fig. 5. We see that the agreement ders have lengths of 14 X measured from the reflecting disk. The with the theory is good for both the principal patterns, i.e., the E- measurement errors are discussed later. The radiation field is mea- plane pattern of the z-mode and the H-plane pattern of the p-mode. sured at a distance of 24 m, i.e., about 96 X at 12 GHz. Discrepancies are present at 0 = f90' due to diffraction from the 3) It is extremely difficult in practice to control the relative ex- edge of the disk, and at 0 = 0 due to diffraction from the end of the citation between the z- and cp-modes of the slot, which is necessary cylinder. However, we clearly see the low radiated level for 0 = 0, in order to know what we measure. This was solved by measuring a characteristic of the soft cylinder. 1632 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 37, NO. 12, DECEMBER 1989

gations, there will be strong radiation along the cylinder for both the z- and p-modes of the excitation. The radiation pattern is fairly symmetrical for both the soft and hard cylinders, in the sense that the E-plane pattern of the z-mode, i.e., A,@), has nearly the same shape as the H-plane pattern of the p-mode, i.e., C v(0). This means that an antenna consisting of a slot in a long soft or hard cylinder can be designed to have low cross polarization if the slot is correctly excited.

REFERENCES [l] P.4. Kildal, “Study of element patterns of excitations of the line feeds of the spherical reflector antenna in Arecibo,” IEEE Trans. Antennas Propagat., vol. AP-34, no. 2, pp. 197-207, Feb. 1986. [2] -, “The hat feed: A dual mode rear-radiating waveguide antenna having low cross-polarization,” IEEE Trans. Antennas Propagat ., vol. AP-35, no. 9, pp. 1010-1016, Sept. 1987. [3] -, “Definition of artificially soft and hard surfaces for electromag- netic waves,” Electro. Lett., vol. 24, no. 3, pp. 168-170, Feb. 1988. [4] R. F. Hamngton, Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961. [5] T. UlversQy, “Radiation patterns of a circumferential slot in a cylinder with different kinds of surface impedances,” (in Norwegian), Norwe- gian Inst. Technol., Trondheim, Student Rep., Apr. 1986. [6] T. UlversQy and P.-S. Kildal, “Improved element pattern for the line feeds of the spherical reflector antenna in Arecibo,” IEEE Trans. Antennas Propagat., pp. 1624-1627, this issue. [7] P.3. Kildal, “Artificially soft and hard surfaces in electromagnetics,” IEEE Trans. Antennas Propagat., submitted.

(b) TIETA Fig. 6. Measured and calculated radiation patterns for hard cylinder with Equivalent Circuits for Electrically Small Antennas reflecting plate. 6 = 0.47 A, a = A. (a) z-mode. (b) p-mode. Using LS-Decomposition with the Method of Moments C. Hard Cylinder T. L. SIMPSON, SENIOR MEMBER, IEEE, JAMES C. LOGAN, MEMBER, The hard cylinder was realized by using dielectric-filled longitu- IEEE, AND JOHN W. ROCKWAY, MEMBER, IEEE dinal corrugations (Fig. 4), see [3] and [7]. The corrugations were rectangular, so the depth was chosen to be 0.25 A/- = 0.2 h Abstract-As part of an investigation into methods for accelerating the process of filling the method of moments impedance matrix [Z], it for E = 2.54. The diameter of the cylinder was 2u = 1.9 h, and there were 32 corrugations around it, each with a width of 0.12 A. was found that [Z] could be decomposed into three parts: a real matrix [L] from the magnetostatic vector potential, a real The measured patterns (Fig. 6) show much stronger ripples than for (inverse ) matrix [SI from the electrostatic static the soft cylinder. The reason is that the diffraction from the end of scalar potential, and a complex impedance matrix [z(w)] of residual the cylinder is stronger, as the field propagating along the cylinder frequency-dependent contributions. By neglecting [ z(o)] at sufficiently is strong. Also, this diffraction effect causes radiation in H-plane for low frequencies, static and quasi-static charge and current distributions the z-mode and in E-plane for the 0-mode that is not present when were obtained. For electrically small antennas, a complete RLC-curcuit the cylinder is infinitely long. was obtained directly from a single quasi-static solution rather than as an approximate characterization of the impedance as a function of fre- quency. This gives precise definition of the circuit parameters limiting the VII. CONCLUSION performance of electrically small antennas. The radiation pattern from a slot in an infinitely long cylinder has INTRODUCTION been examined, when the cylinder is made of both artificially hard and soft surfaces. Simple closed-form expressions have been derived. In the analysis of wire antennas it is often required to compute the Also, patterns have been presented for specific parameters. Compar- frequency response over a wide band of frequencies. The work ison with measurements show good agreement, although there are differences near the axial direction, caused by the finite length of Manuscript received December 10, 1987; revised July 13, 1988. This work the measurement model. In spite of this we are able to draw the was supported in part by the Naval Ocean Systems Center under the Navy/ following conclusions. ASEE 1987 Summer Faculty Research Program. If the surface of the cylinder is made soft by circular corrugations, T. L. Simpson is with the Department of Electrical and Computer the radiation along the cylinder will vanish for both the z- and p- Engineering, University of South Carolina, Columbia, SC 29208. J. C. Logan and J. W. Rockway are with the Naval Ocean Systems Center, modes of the excitation in the slot. 271 Catalina Boulevard, San Diego, CA 92152. If the surface of the cylinder is made hard by longitudinal corru- IEEE Log Number 8929317.

0018-926X/89/1200-1632$01.00 0 1989 IEEE