Soft & Hard Horn Antennas

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    IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 36, NO. 8, AUGUST 1988 1153by

    z , = o IZrl=oO, (soft)characterized by

    E, 1 - 0 Er I = Owall walland

    I Z z l = ~ Z,=O, (hard)characterized by

    aEr- = o - =o.ar wal l ar wall(3)

    (4)The cylindrical coordinate system (r , (p, z ) has been used forsimplicity. How ever, similar definitions could have been given forconical waveguides by using the spherical coo rdinate system (R, 8,(p), as well as for waveguide cross sections different from circular.The wall impedances in (1) and (3) are defined as the ratio betweenthe electric (E) and magnetic (H ) ield components tangential to th ewall, given by

    Soft and hard can be recognized as Dirichlet and Neumann types ofboundary conditions, respectively. Th e definitions in (1) and (3) ar eforced to meet the balanced hybrid condition given by [3]

    where TO is the free-space wave impedance. This means that the fielddistribution over the cross section of the waveguide is rotationallysymmetrical with straight field lines, giving zero cross polarization.In other words, the terms soft and hard boundaries assume thebalanced hybrid condition to be satisfied. The balanced hybridcondition as well as identity with zero in (2) and (4) can only beobtained ideally when the diameter of the waveguide tends to infinity.Approximate soft and hard conditions can be defined by 1Z , I % 1Z , Iand and lZ,l IZ,l, respectively, when (6) is satisfied.

    m. JELDDISTRBUTIONSND RADIATION PATTERN FORARBlTRARY WALL IMPEDANCESBy using a wall-imped ance model to descr ibe the field in a conical

    horn, the aperture field distributions and radiation patterns can becalculated for chosen values of the wall impedances Z , and Z R n thespherical coordin ate system (R , 8, (p). Let us assume walls withoutlosses and apply the reactance X , and susceptance BR at the wall,given by

    z = j x , Z R= 1 JBR (7)where

    to meet the balanced hybrid condition defined in the sphericalcoordinate system. The field distribution within the horn can becalculated numerically by assuming a spherical hybrid-mode. Thecharac teristic equation for determinatio n of the eigenvalue is given inAppendix I. The far-field radiation pattern is calculated by aKirchoff-Huygen integration over the spherical phase front.

    Fig. 1 illustrates the calculated field intensity ov er the hornaperture for given horn dimensions where the wall reactance X, s aparameter. As IX,l increas es, the field intensity at the wall increases.In the limit when X, = 0 (soft), he field intensity is zero at the wall,

    1.O98

    8 .7*I0 .63 .5a .444 3k

    21

    0.00.0 .I 2 .3 .4 S .6 7 .8 .9Normalized radius

    d lFig. 1. Calculated aperture field distributions for conical horn antenna with5 semiflare angle and aperture diameter 2X under balanced hybridcondition (rotationally symmetrical distribution) for different va lues ofX,,

    - 0 5.HARDI -5 .-.s4J ( X p X JE -10

    I -

    *In- I S

    1 J *U J U I-ta (W )

    Fig. 2. Calculated radiation patterns from a conical horn antenna with 5 semiflare angle and aperture diameter 2h under balanced hybrid condition(rotationally symmetrical distribution) for different values of X, .

    while when X , = 03 (hard), the field intensity is constant over theaperture.Fig. 2shows the calculated radiation patterns from the sam e hornaperture for different field distributions corresponding to differentvalues of X,. t can be seen that hard boundary co nditions give higherdirectivity and smaller beamwidth due to the increased apertureefficiency compared to soft conditions, which may be an advantagefor som e applications. Howev er, the disadvantage is that the sidelobelevel is much higher.

    The hard horn has significantly higher directivity than a soft hornonly if it has almost constant phase over the aperture, i.e., aperturecontrolled as defined in [18]. Such a horn is termed narrow bandin [19], as it has a frequency-dependent beamwidth. For a flare-angle controlled horn [18] or correspondingly wide-band horn[19], which is frequency-independent with respect to the beamwidth,the increased directivity cannot be obtained. The ideal aperture-controlled horn has a constant phase front over the aperture. This canbe obtained by using a ray-correcting lens in the aperture, byprofiling the horn, or approximately by using a very long horn.

    It should also be mentioned that a cylindrical waveguide radiator isideally aperture controlled. The horn calculated in Figs. 1an d 2 isalmost aperture controlled.When the horn has an arbitrary flare angle, the method shown in

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    1154 IEEE TRANSACTIONS ON ANTENNAS AN D PROPAGATION, VOL. 36, NO. 8, AUGUST 1988Appendix I should be used to calcu late the field distribution interior tothe horn. For a very long horn with a small flare angle, or for acylindrical waveguide radiator, cylindrical waves can be assumedwithin the horn. Thus for waveguides with circular or rectangularcross sections, simple analytical express ions can be derived both forthe interior field distribution and for the radiation pattern (seeAppendices 11and III).

    Fig. 3 illustrates the relative directivity and relative sidelobe levelagainst the edge taper A (relative field intensity at the wall defined inAppendix II) for a waveguide with circular cross section. The term(1 + co s 0)/2 is neglected in the calculations of the sidelobes (seeAppendix 11),which is valid only when the radius a is not too sm all.When A increases, the directivity increases. The relation between Aand the wall reactance X, s shown in Fig. 4 where the relativediameter D / h of the waveguide is a parameter (A is the free-spacewavelength). The theoretical achievable increase from an ideally softto an ideally hard boundary horn with circular cross section is 1.60dB,which is equivalent to a reduction of the beamwidth of about 22percent (see Table I). This corresponds to a 69-percent apertureefficiency for a soft horn. Alternatively, a hard boundary horn hasabout 17-percent smaller aperture diameter than a soft boundary hornwith the same directivity and beamwidth. At the same time, thesidelobe level increases by about 10 dB to 17.6 dB below the mainlobe level when comparing hard with soft horns. In the sam e table itcan be seen that the improved performance fo r rectangular horns iseven larger than for circular horns. H oweve r, the sidelobe level is ashigh as 13.3 dB in a plane parallel to either of the waveguide walls.Most often there will be a trade-off between increased directivity,decreased beamwidth, and increased sidelobe level which should bekept below a certain value. Note that the sidelobe level for the softhorn in Table I varies with the horn diameter in general, and a typicalvalue is used.

    Note that for small horn apertures the sidelobe level andbeamwidth will be reduced compared to the values given in Table Iaccording to the factor (1 + co s 0)/2. For small horns the relativepower in the wall region may be large and reduce the gain.w. EALIZATION OF HARD HORNSAND HORNSWITH ARBITRARYWALL MPEDANCES

    In [20] a horn having longitudinally oriented corrugations wasanalyzed. It was concluded that the horn could support a modifiedTEll-mode with straighter field lines than for the nominal TEII-mode, which gives low cross polarization. In [16] a similar horn wasanalyzed where the longitudinal corrugations were filled with adielectric material. How ever, none of these articles mentioned that analmost uniform aperture illumination was achieved, resulting in anincreased directivity.In fact, it can be shown that the horn antennas with longitudinalcorrugations approximate the hard boundary conditions when thedepth of the corrugations is [16], [14]

    and the dominant hybrid mode is assumed. We see that when thecorrugations are air filled (relative permittivity of dielectric materialer = l) , the hard boundary conditions can only be met exactly in thelimit when the corrugations are infinitely deep.

    The strip-loaded horn with longitudinally oriented strips presentedin [15] has approximately the same electrical performance as thepreviously described corrugated horn [16]. It can be shown that thehorn meets the hard boundary conditions at a given frequency whenthe approximate thickness of the dielectric wall is given by (9), whichis asymptotically correct for large waveguide dimensions [2 11. Atlower frequencies the dominant mode exhibits TEII-properties. At

    1 I I I I I I I I I 1 -1 . 60. 0 0.2 0 .4 0 . 6 0.8 1 .of SO F l E d g e t a p e r A HARD t

    Fig. 3 . Calculated relative directivity and typical sidelobe level versus edgetaper for horn antenna with constant phase over aperture under balancedhybrid condition. (Directivity is normalized to directivity for constantamplitude and phase distribution.)

    , ."0.0 0 . 2 0. 4 0.6 0 .8 1 .oEd g e t a p e rA

    Fig. 4. Wall reactance versus edge taper for circular cylindrical waveguideunder balanced hybrid condition where normalized diameter is parameter.

    TABLE ICOMPARISON BETWEEN ARD AND SOFT HORNSIncreased Reduced Reduced Sidelobe Level

    Cross Directivity Beamwidth Size Hard Horn Soft HornSection (dB) (Percent) (Percent) (dB) (dB)-17 .6 -27 .5ircular 1.60 22 11Rectangular 1.82 26 19 -13 .3 -23 .0

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    IEEE TRANSACTIONS ON ANTENNAS AN D PROPAGATION, VOL. 36, NO . 8, AUGUST 1988 1155zq5r =z z+r =zIhI777j jil

    (a) (b)Structurefor arbitrary wall impedance. (a) Combination of grid andig. 5.

    dielectric. (b)Transmission line equivalent.

    Fig. 6. Dielectric core horn with inverse-graded ndex.higher frequencies the dominant mode turns into a surface wave,characterized by an imaginary radial wavenumber internal to thehorn, but still having a real propagation constant.In a horn with a hard boundary, surface waves (in the case oflongitudinal strips) and higher order modes may exist, which willlimit the gain increase caused by the assumed uniform apertureillumination. This effect has to be studied and quantified further.Hard horns with longitudinal strips and longitudinal corrugationshave been measured [15]. Sidelobes levels from - 15 to - 0 dBhave been obtained, with peak cross polarization below -2 5 dB.This indicates that an approximately uniform aperture illuminationhas been obtained as assumed in this article. The relatively highcross polarization is probably caused by surface waves as well ashigher order m odes.A horn which is required to have maximum possible gain, wherethe sidelobes are to be kept below a given level, is ideally achieved byproviding finite wall impedances (see Figs. 3 and 4). Such a horn maybe realized by a combination of dielectric lining and a conducting gridstructure on the wall, as shown in Fig. 5.By a proper choice of theparameters, the balanced hybrid conditions can be met, where X , isdifferent from zero and infinity. By choosing a complex gridstructure with two or more dielectric layers with grids in between, amore broad-band wall impedance can in principle be achieved.A chosen field tapex at the wall with finite values of X , canalso in principlebe achieved by a dielectric core horn with an inversegraded index behavior (Fig. 6), which spreads the field away fromthe symmetry axis. The phase error can be corrected by shaping theaperture surface properly.From the literature it is known that increased directivity due toincreased aperture efficiency can be achieved by using a dielectriclining on the horn wall [23], [24]. In [24] an almost uniform apertureillumination has been reported for a circular cylindrical horn havingapproximately 1-wavelength diameter. However, the peak crosspolarization is not better than -2 9 dB.

    V. CONCLUSIONIn this communication horn antennas having soft and hardboundary conditions, denoted by soft and hard horns, respectively,are discussed. The corresponding soft and hard boundaries aredefined in terms of an anisotropic surface impedances. The defini-tions of soft and hard horns are based on circular cylindrical crosssections, but may easily be extended to horns with arbitrary crosssections, e.g., rectangular or elliptical.

    Compared to the traditional soft boundary horn with zero fieldintensity at the wall, 1.60-dB increased directivity and 22-percentreduced beamwidth can theoretically be obtained with a hardboundary horn with a circular cross section, having constantamplitude and phase distribution ove r the aperture. At the sam e time,the relative sidelobe level increases to 17.6 dB. For a horn withrectangular cross section the corresponding directivity can beincreased by 1.82 dB and the beamwidth reduced by 26 percent,while the sidelobe level increases to 13.3 dB. The advantage of usinga hard horn is to obtain maximum directivity, or minimumbeamwidth, for given aperture dimensions. The maximum theoreticalreduction of the aperture diameter for the same directivity andbeamwidth is about 1 7 and 19 percent when com paring hard with softhorns with circular and rectangular cross sections, respectively. Thecondition for the increased directivity is that the phase be almostconstant over the aperture of the horn and that the relative power inthe wall region is negligible.Ideally hard boundary conditions may be obtained by a horn withlongitudinally oriented corrugations filled with dielectric, or alterna-tively, by longitudinally oriented conducting strips on a dielectricallylined wall. Arbitrary w all impedances, which may be desired to keepthe sidelobes below a given level, can, in principle, be realized by acombination of conducting grids and dielectric on the wall or by aninverse graded index horn. The characteristic equation for thespherical hybrid mode in a conical horn with arbitrary wallimpedances is derived here as well.Hard boundary hor ns may be applied as elements in cluster feeds toreduce the size of the array or in high-resolution systems where thehorns should be placed close together. Possible problems might becaused by surface waves and coupling between the horns when usedin an array configuration, as well as by radiation from the w allregion. These effects will be studied in more detail.

    APPENDIXCHARACTERISTIC EQUATIONOR TH E EIGENVALUEN A CONICAL

    HORNWITH ARBITRARYALL IMPEDANCESThe characteristic equation for the eigenvalue v of a corrugatedhorn is given in [22] with the wall impedance 2, as a parameter and

    2, = 0. The general characteristic equation where 2, # 0 will nowbe constructed based on the expressions in [22]. The expressions forthe wall impedance when the field interior to the horn is described in

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    1156 IEEE TRANSACTiONS ON ANTENNAS AND PROPAGATION, VOL. 36, NO. 8, AUGUST 1988terms of a single spherical hybrid-mode become For ideal soft and hard conditions we obtain

    .ikR/s [ A (soft)- Z - = H ,R W ~ I=-- V ( v + l ) P $4 - ] (11) p;d= a J l ( k u s i n 8 ) ] [ + ? e ] 21 [47ria s i n 8 ,where the incident field is the TEll-mo de, kR 9 1 where R denotes

    the distance from the horn apex and k is the free-space propagationconstant, CY is the semiflare angle of the horn and should be not closeto zero, and

    where Pt is an associated Legen dre function of order 1 and deg ree v.By eliminating A from (10) and (l l) , the characteristic equationbecomes

    z, + - P t ( a )( v + l );.-&PbCY] [ - JkR 1= [ kR ] . ( I , )v( v + 1) sin CY

    I hard) (18)When the radius a is large, the term (1 + co s 0)/2 has negligibleeffect on the main lobe and first sidelobes as 8 s small.A more convenient param eter than pa is the edge taper defined by

    A = Jo(Pu). (19)When A = 0 and 1 , the boundary is soft and hard, respectively. Therelation between pa and the wall reactance X, = qtBR can bederived from (19) and the field expressions in [19],

    For a given value of the edg e taper A or pa X , increases with a/A .The eigenvalue v can be solved from the last equation by an iterativeprocess, where the first estimate may be expressed by

    APPENDIXIIRADIATION FROM A RECTANGULAR WAVEGUIDE ITH SOFTAN DHARDBOUNDARIESThe field distribution interior to a rectangular waveguide with2 (14) dimensions a x b can be ex pressed byp i x )os ( Z Y ) , (soft)

    APPENDIX II It is assumed that the dimensions are large compared to thewavelength A. The directive gain can than be e xpressed asRADIATION FROM A CIRCULAR CYLINDRICAL WAVEGUIDE UND ERBALANCEDYBRID ONDITlON AN D WITH ARBITRARY WALL

    IMPEDANCEThe y-polarized field distribution in a circular cylindrical wave-

    guide can be expressed by2.405Ey= qoHx=Jo(pr) p = - (1 -6), (05651) (15)

    E: dx dy4 -a/ 3 -b/ZU (23)where J is a Bessel function of the first kind. It is assumed that the

    field is propagating under the balanced hybrid condition and that theradius a is large compared to the wavelength A (straight field linesand rotationally symmetrical distribution). When 6 = 0, th eboundary is soft, while when 6 = 1, the boundary is hard.

    The directive gain from the aperture can be expressed by

    wherekx =k sin 8 co s cpky= k sin 8 sin 9 (24)

    k = 2a/A an d (r ,8, c p ) denote the spherical coordinates for the far

    p;d =A[ 1JO(pr)eJkrsinecos~rr dcp

    [J;(pr)r dr dcp47r 0 0a ka sin BJo(pa)Jl(kain e)-puJl(pa)Jo(ku in e)

    = [4. h [(ka in e ) z - (pa )2 ] [J~(pu )+J~(pa ) ]1 2where (r, 8, c p ) denote the spherical coordinates for the far field.

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    IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 6, NO . 8, AUGUST 1988 1157field. For ideal soft and hard conditions we obtain

    Py=

    cos2 (k x 4) cos2 (ky:)256 a b7r3 A A [ - ( x - 1 2 [ - ( y-2 1 2

    a b4 r - -A A2

    [y] (hard).2

    ACKNOWLEDGMENTThe authors want to thank Tom Cwik for commenting on the

    article, and Allan Love for a detailed review.REFERENCES

    P. D. Po tter, A new horn antenna with suppressed sidelob esand equalbeamwidth, Microwave J., vol. VI, pp. 71-78, June 1963.A. J. Simmons and A. F. Kay, The sca lar feed-A high performanc efeed for large paraboloidal reflectors, in Inst. Elec. Eng. Conf.Publ. 21, June 1966, pp. 213-217.H. C. Minnett and B. MacA. Thomas, A method of synthesizingradiation patterns with circular symmetry, ZEEE Trans. AntennasPropagat.. vol. AP-14, pp. 654-656, Sept. 1966.V. H. Rumsey, Horn antennas with uniform power patterns aroundtheir axis, ZEEE Trans. Antenna s Prop aga t., vol. AP-14, pp. 656-658, Sept. 1966.H. E. Bartlett and R. E. Mosely, Dielguides-Highly efficien t lownoise antenna feeds, Microwave J., vol. 9, pp. 53-58, Dec. 1966.C. M. b o p , Y.-B. Cheng, and E. L. Ostertag, On the fields in aconical horn having an arbitrary wall impedance, ZEEE Trans.Antennas Propagat., vol. AP-34, pp. 1092-1098. Sept. 1986.US Patent 4 410 892, Oct. 1983; European Patent pending.P. J. B. Clarricoats, A. D. Olver, and M. S . A. S . Rizk, A dielectricloaded conical feed with low crosspolar radiation, in Proc. URSZSymp. EM Theory, Santiago de Compostela, Spain, Aug. 23-26,R. Lier and J . A. Aas, Simple hybrid mode horn feed loaded with adielectric cone, Electron. Lett., vol. 21, no. 13, pp. 563-654, June20, 1985.E. Lier, A dielectric hybrid mode antenna feed: A simple alternativeto the corrugated horn, ZEEE Tran s. Antenna s Propa gat., vol. AP-E. Lier, T. Schaug-Pettersen, and J. A. Aas, New classes of hybridmode antennas-Alternatives to corrugated horn feeds, in Proc.ZEEEAJRSZSymp., Vancouver, BC, June 1985, sec. B-18-6, p. 241.E. Lier and T. Schaug-Petterson, The strip-loaded hybrid-mode feedhorn, ZEEE Trans. Antennas Propagat., vol. AP-35, pp. 1086-1089, Sept. 1987.S . F. Mahmoud and M. S . My , A new version of dielectric linedwaveguide with low cross-polar radiation, ZEEE Trans . Ante nna sPropagat.. vol. AP-35, pp. 210-212, Feb. 1987.P-S. Kildal, Definition of artificially soft and hard surfaces forelectromagn etic waves, Electron. Lett., vol. 24, no. 3, pp. 168-170,Feb. 4, 1988.E. Lier and P-S. Kildal, A novel type of high-gain horn antennas, inProc. ZCAP Conf., York, England, March 30-April 2, 1987, pp.43 1-433.M. S . Aly and S. F. Mahmoud, Prop agation and radiation behavior of

    1983, pp. 351-354.

    34, pp. 21-29, Jan. 198 6.

    a longitudinally slotted horn with dielectric-filled slots, Proc. Inst.Elec. Eng., vol. 132, pt. H ., no. 7, Dec. 1985.[I71 E. Lier and P-S. Kildal, Dielectrically lined horn antennas,presented at the Informal Workshop on Primary Feeds and RF-SensingSystems, ESTEC, The Netherlands, June 10-11, 1987.P-S. Kildal, A Gaussian beam model for aperture-controlled andflareangle-controlled corrugated horn antennas, Proc. Inst. Elec.Eng., pt. H, to be published.[I91 B. MacA. Thomas, Design of corrugated conical horns, ZEEETrans. Antennas Propagat., vol. AP-26, pp. 367-372, Mar. 1978.[20] T. Scharte n,J. Nellen , and F. van den Bogaart, Longitudinally slottedconical horn antenna with small flare angle, hoc. Inst. Elec. Eng.,vol. 1 28, pt. H., no. 3, pp. 117-123, June 1981.E. Lier, Theoretical analysis of the strip-loaded hybrid-mode hornantenna with soft and hard boundaries based on a circular cylindricalmodel, ZEEE Trans. Antennas Propagat., submitted.[22] P. J. B. Clarricoats and A. D. Olver, Corrugated horns formicrowave antennas, in Inst. Elec. Eng. Electromagnetic WavesSeries 18.[23] G. N. Tsandoulas and W. D. Fitzgerald, Aperture efficiencyenhancement in dielectric ally loaded horns, ZEEE Trans . Ante nna sPropagat. , vol. AP-20, pp. 69-74, Jan. 1972.[24] A. Kumar, Die lectric-lin ed waveguide feed, ZEEE Tran s. Ant en-

    nas Propagat., vol. AP-27, pp. 279-282, Mar. 1979.

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    [21]

    London: Peregrinus, 1984, ch. 3.

    The Input Impedance of a Probe-Fed RectangularCavity which Transmits Through aPlasma-C overed Cylindrical BodyJOHN M. JAREM, MEMBER, IEEE, AND CHAO-MING MA

    Abstract-A method of moments solution for the input impedance of aprobe-fed cavity which radiates through a plasma-covered cylindricalbody is presented. A general description of the meth od of momen tssolution and cylindrical aperture analysis is given. M athematical expres-sions are given for the cylindrical aperture admittances. Numericalexamples of the input impedance which correspond to cylinders of severaldifferent radii have been generated when free space is assumed to coverthe antenna aperture and when an in homog eneou s plasma is assumed tocover the anten na aperture.

    I. INTRODUCTIONAn important problem in the area of missile antenna theory is the

    problem of determining the input impedance of a probe-fed cavityaperture antenna which radiates from a cylindrical missile bodywhich is covered by an inhom ogeneous plasma sheath. The solutionof this problem has been determined approximately by the use of asingle mode aperture admittance analysis [l], [2] and has beendetermined more fully by the method of mom ents [3]. In the apertureadmittance analysis the region exterior to the plasma aperture wasapproximated by both an infinite ground plane [l ] and an infinitely

    i

    Manuscript received March 29, 1987; revised November 5, 1987. Thiswork was supported in part by the Sandia National Laboratories underAntenna Development Contract DOC 95-1400. This work was presented atthe 1987 IEEE Antennas and Propa gation Society International Symposiumand URSI Radio Science Meeting, Blacksburg, VA. (See [4].)

    J . M. Jarem is with the Electrical and Computer Engineering Department,University of Alabama, Huntsville, AL 35899.C. M. Ma is with the Electrical Engineering Department, University ofTexas at El Paso, El Paso, TX 79912.IEEE Log Number 8821189.oO18-926X/88/08oO-1157$01oO 0 988 IEEE