Method for Compute E-Plane Patterns of Horn Ant

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A Method for Computing .&Plane Patterns of Horn Antennas

Absfracf-This paper introduces a method for calculating thetotal antenna pattern in the E plane, including the backlobe region,of a horn by applying diffraction theory. Treatment of diffraction bya thick edge permits hornsf various edge thicknesses to bereated.The diffraction concepts developed by Sommerfeld and Pauli, whichtreated plane wave diffractioo by a wedge, are extended so thatdiffraction of plane and cylindrical waves by thn or thick edges maybe treated. The diffracted fields are obtained by applying the relationsdeveloped by Pauli in conjunction with reciprocity.

When the theory is applied to the horn, the si flcan t radiationmechanism s are direct radiation from the source at the horn apexand diffracted radiation due to the edges. The computed E-planepatterns of a horn antennaagree with measured patterns.

IKTRODUCTION

AERTUREmethods forevaluating he fields ofaperture antennas are restrict ed in the practicalsense that t he fields behind the aperture plane,or in the back direction of the antenna, are not easilyobtained,Such ormulations sSchelkunoffs quiv-alenceprinciplecould, n heory,be used to evaluatethe fields of anyantenna inallspace.However, theelectric and magnetic currents or fields must be knownover heentireantennasurface,and hen a difficultsurface integral must be evaluated. Usually the exteriorsurface currents on an tenna st ructures, xclusive of th eantenna aperture, do not contribute significantly to thefields in the region of the antenna main beam. The radi-ation in this region is due to a distribution of Huygenssources n the anten na aper ture . For example, by ex-tending the aperture for the horn antenna (as in Fig. 1)and implicitly assuming tha t the fields are identicallyzero on this plane, except over the aperture, the equiva-ence principle can be applied to obt ain the fields. I t iscommon practice to determine the aperture distributionby geometricalopticsor by experiment. Th e fieldsto th e left of this infinite plane cannot be obtained be-cause of the assumptions involved; but t he ields to theright of the aperture can be computedwithexcellentresults, provided the aperture distribution is accuratelymeasured. Thehorn ntenna epresents a caseorwhich this aperture method is used to obtain the for-ward fields, whereas the backlobe region must generallybe obtained experimentally.

This paper introduces a method for calculating thetotalE-planepattern, ncluding hebacklobe egion,

Manuscript received March 20, 1964; revised September 28,1964.The work reported here was supported in part by Contract AF30(602)-3269 between Rome Air Development Center, Griffiss AirForce Base, N. Y . , nd The Ohio State University Research Founda-tion, Columbus, Ohio.P. M. R u s o is with Systems Research Labs., Dayton, Ohio.Dept. of Electrical Engineering, The Ohio Stat e University, Colum-R. C. Rudduck and L. Peters, Jr. are with the Antenna Lab.,Ohio.

I u = o

I u . 01

Fig. 1. Application of equivalence principle to horn.

of a hornby pplyingdiffractionheory.nitially,Kellers1s2geometrical theory of diffraction was appliedto hisproblem; however, the diffractioncoefficientsused in his echnique are not genera lly adequate forthesecomputations,since heappropriatediffractioncoefficients used in the geometrical theory of diffractionare for a plane wave incident on a perfectly conductingwedge whereas cylindrical wave incidence is applicableto th e horn. Insofar as the diffraction in the E plane isconcerned, the horn may be considered to have a cylin-drical wave emanating from its throat.

Consequently, hediffraction or ylindricalwaveincidencewas determined hrough he use of Paulis8equations for diffraction of a plane wave b y a perfectlyconducting wedge, together with the principle of reci-procity. Thi s diffraction solution allows he E-p lane pat -terns to be calcu lated with good accuracy for horns ofvariousedge hicknesses.Previous reatment of an -tennasby half-planediffractionmethodshavebeenintroduced by LysheI-i for paraboloid reflector ant enn asand Ohba6 for corner reflector antennas .

REVIEW F EDGEDIFFRACTIONHEORYThe problem of straight-edgediffractionby a per-

fectly conducting wedge was first solved by Somrner-feld.6 H e considered a plane electromagnetic wave nor-mally incident on thewedge of an angle 2 - n ) ~shownin Fig. 2 ) . The azimuthal angle + and +O are associatedwith the angle of diffraction and the ang leof incidence,respectively.

28 , Apr 1957, pp 42644.2.~ 0 1 5 2 . eb 1962. DD 116-130.

1 Keller, J. B., Diffraction by an aperture, J. A p p l . Pltys., volKeller, J. B. , Geometrical theory of diffraction, J. Opt. SOC.m .,

diffraction of light, Phys. Rev. , ~ 0 1 5 4 , ec 1938, pp 924-931.* Pauli, I$., &asymptotic series for functions in the theory ofantenna, kI.Sc., thesis, The Ohio State University, Columbus, Ohio,4 Lysher, L., J., ,4 study of the near-field behind a parabolic1961.

5 Ohba, Y . ,On the radiation pattern of a corner reflector finitein width, IEEE Trans. on Antennas and Propagation, vol AP-11,Mar 1963. DD 127-132.pp 245-265.6 Sommeifeld, A.,Optics, New York: Academic Press, Inc., 1954,

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220 IE E ER A N S A C T I O S S ON ANTEN:Y',4S ANDROPAGATION MaTch

Fig. 2 . Diffraction of plane wave by wedge.

Th e field u t point P is a solution to the scalar waveequationsubject o heappropriateboundary condi-tions; t he solution may be formulated as

u(r,+)= v ( r . + + + a ) * v(r,+-+o) (1)Polarizationdetermines he choice of sign such hat,with the electric vector perpendicular (parallel) to theedge, the positive(negative)sign schosen. It is con-venient to represent the incident oreflected field in theform

z'(r.6) = * ( r , + f + o ) (2)such that the( - ) sign yields the incident fields and the(+) sign yields the reflected fields. The com ponent ieldis given by

V(r.6) = v* + v 5 (3)where Y* is the geometrical optics field given b yij p ms ( 4 + 2a~zN) - < ++ 2 n n X < rv* = 3 7 = 0, f 1, 52 . . . (4)

0 otherwiseand vg is the diffracted field given by

1 r

wherep = kr (6)

Here C is the appropria te path in the plane of the com-plex variable.

An asymptotic expression for ( 5 ) , obtained by Som-merfeld, is of the form

This formyields nfinite fields in hevicinity of theshadow boundary and is valid only if

iT

However,for the specialcase of n= 2 , the diffractedfield expression was obtained by Sommerfeld in termsof the Fresnel Integ ral as

wherea = l + C O S + (10)

I t should be noted th at th e positive square root of (ap)shouldalwaysbe aken. The sum v*+vB can also berepresented more compactly by

e j ( r / 4 )c(,,+) = ejp cos Q e-jr'd, (11)

d r

It is seen that no discontinuities exist for 4 = T n (11).By making a transformation of variables and choos-ing an appropria te contour, Pauli3 obtained an expres-sion for Sommerfeld's solution (5 ) which is expressed inseries form as

85(,,.$)= n& T 4cos- os-n 12. iky COS Q SW e-jr2d7 + . . . (12)(=p)'/2

where the higher order terms may be eglected for largekr . In the case of the thin half-plane (n=2), he higherorder terms are identically zero and (12) reduces to (9).If the higher order terms may be neglected, the valuefZ'B converges to one-half the incidentfield at the shadowboundary or at =IT.

If theassumption hat ( u p ) is large is valid, heasymptotic form of (12) which s given in (7 ) may beused to advantage as

where1 . T- in-

cos- cos-nand higher order terms are neglected. t should be notedthat (13) is valid provided the condi tion of (8) is satis-fied.

The geometrical heory of diffraction developed byKeller1.2 makes use of the form given in (13) to extendgeometrical optics to include diffraction phenomena a tedges.


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222 IE E E TRAATSACTIOhTSO N AATTERTNASA N D PROPAGATION MarchI

Fig. 4.Model of thick edge.

..L/I--11

sourceFig. 5. Cylindrical wave diffraction by a thick edge.

culty of using plane wave diffractioncoefficients for thedouble diffraction across the edge is overconle by asymp-totic expansion of th e Green's Theorem Integrals. Effec-tively, the total field on the edge surfaces is integratedto obta in the diffracted field inregions near multiplediffraction shadow boundaries. However, RIorse's equa-tionsapply o hick edgediffraction orplanewaveincidence bu t no t for cylindrical wave ncidence.

The steps employing the diffraction of a cylindricalwave by a 90" wedge to ob ta in he diffraction by athick edge are illustrated in Fig. 6. Here a line sourcelocated a t ( x a , $0) illuminates corner A of a 90' wedge,and t he diffracted fields are obtained as previously dis-cussed. I t is desired to locate an equivalent source SAat corner A of such a magnitude that both the originaland equ ivalen t sources have identical far fields in thedirection of the incident ray of Fig. 6(b). Kow when sAis ocated a t corner A inFig.6, either the equivalentsource or the original source illuminates corner B withthe s am e field. Diffraction from corner B is readily ob-tained rom Fig. 6(b)and (16). To obtain henexthigher,multiple-diffracted ield, the ame process isapplied to Fig. 6(c); that is, an equivalent source SB islocated at corner B such that the far diffracted field ofSA and the farield of the equivalent source are identicalin t he direction of the incident ray of Fig. 6(c). Koweithersource, SB or the diffracted field from SA , willilluminate corner A with the same intensity. Diffractionfrom corner A due to the source S B shown in Fig. 6(c)is readily obtained in the same manner as before. Thisprocess is then repeated to obta in higher order, multiplediffracted fields, butusually heyare negligible. Bysumming heapprop ria te fields, the otaldiffractionfrom corners A or B can be obtained in losed form. Forinstance, for the situation illustrated in Fig. 7, he totaldiffracted fields are

(c)Fig. 6. Construction of diffracted fields by a thick edge.

Fig. i . Coordinate system for cylindrical wave diffractedby a thick edge.

exp ( - j k d cos 8) corner B (174and

Z'B = (~B(zo .?70- )o ) t - ~B(zO.Zio+$,,))e'B(d ,O)Z'E(d ,B,)1- i ( d . 0 )+ ~ B ( ~ , . s - + , ) ~ J B ( ~ , , , + + + ~ ) )orner A (llib)

where the exponential factor in (17a) refers the phaseto corner A .In practical situations, especially for square edges ofsmall thickness, i.e., d


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1965 RUSSO,t ai.: E-Plane Patterns of Horn Antennas 2283~lg(,,d). This becomes tediouswork, if necessa ry;andto compound the difficulty, Pauli3 gives only two higherorder terms. Therefore, P B ( ~ , Q ) as not been calculatedfor very small values of r , i.e., v g ( d . p ) , v g ( d , e ),and w g ( d , o )cannot be evaluated for d


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24 I EEE TRANSACTIOKS O N Ah TEXNAS AND PROPAGATION

Fig. 12. Pyramidal horn adapted for experiment.

es of varying hicknesses could be securedon topf the b rass shee ts, making it possible to var y the edge

as shown in Fig. 12.Figures 13 and 14 compare the measured and calcu-

patte rns for the horn with the thin edge and withn edge thickness of 0.5 inch, respectively. Remarkableis obtained.

13 hows the esul tsobtained whenKellersl Theory of Diffraction is employed for the

is very good agreement, except inimmediate regions of the shadow boundary.By this

the diffraction at th e edges is tak en as planediffraction at each edge is given

y (13) if the incident wave is put in the forme-fk (r--zg)

dF0 -4 # (19)

r is measuredfrom the horn hroat. However,n the method is applied to the thi ck dge, the region

f the shadow boundary dominates the pat tern, shown14, endering this otherwise valuable techniques for this application.I t should be note d tha tMorses2 treatment of shadowis not valid for the shadow boundariesf the

cylindrical wave from the horn throat. Alongboundarydirec tions , which coincide withhe

orses technique, s well as Kellers,a plane wave field as th e diffracted field, i.e., arm field with amplitude independent of dis tance ,

technique of this paper gives the diffractedwavedependence.However, i t appears hat Morses techniqueshouldvalid for treat ing the horn if i t is viewed as havingtwo walls formed by two rectangular cylinders. Theas receiving a plane wave from

directions, and the field at the point of contactthe two cylinders should be representat ive of

e horn antenna pattern. The same viewpoint may beo th e thin-walledhorn in order to obtain itsttern by use of Sommerfeldssolution 9), bu t he

of a cylindrical wave is necessary

I

F i w 13. E-plane pa ttern of thin-walled horn. (a) Measured pattern,-i b) calculated using plane wax-e diffraction (13), (c ) eithermethod of calculation, and (d) calculated using cylindrical wavediffraction (18).

Angle

Fig. 14. E-plane patte rn of a thick walled horn. (a) Measured pat-tern, (b) calculated using cylindrical wave diffraction (17), and( c ) calculated using plane wave diffraction (13).

I

Degrees

Fig. 15. Comparison of measured horn patterns for variouswall thicknesses.

paper. Morse

Method for Compute E-Plane Patterns of Horn Ant

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