High-frequency scattering by aconic frustum

N.C. Albertsen

Indexing terms: Electromagnetic-wave scattering

Abstract: The radiation from a monopole mounted on conic frustum is calculated by the geometrical theoryof diffraction. The results are compared with experiments. Emphasis is placed on the shadow region wherecreeping waves play an important role.

1 Introduction

In Keller's geometrical theory of diffraction (g.t.d.),1 theconcept of diffracted rays is introduced. A diffracted raymay either be a space ray, i.e. with properties similar togeometrical-optics rays, but originating from a point ofdiffraction, or it may be a surface ray (creeping wave). Forcertain applications, e.g. scattering by a polyhedral body,only the first kind of rays are needed whereas scatteringproblems involving smooth bodies require the second kindof rays. The purpose of the present paper is to consider acase where the diffracting body has both smoothly curvedsurfaces and sharp edges. Both kinds of rays will thereforebe needed for a complete description of the scattered field.

2 Ray tracing

The geometry considered is shown in Fig. la. It consists ofa frustum of a right circular cone. The body is placed in aCartesian co-ordinate system with z as the axis of symmetryand the top coinciding with the xy-plane. The angles 6 and</> shown in Fig. la are ordinary polar co-ordinates. On thetop of the frustum atP0 a quarter-wave monopole is placed.The electromagnetic far field radiated by the monopole iscalculated in the shadow zone, i.e. for 6 > (TT/2). The zoneis divided into three regions. In region I the main contri-bution to the field comes from a diffracted ray emanatingfrom the upper edge of the body. The boundary betweenregions I and II on the far-field sphere consists of thosevalues of (0, <p) for which the edge-diffracted ray becomestangential to the curved surface of the frustum. Inside regionII the edge-diffracted ray has been replaced by ray 1 inFig. la. This ray consists of three parts: Between Po and Piit follows a straight line, between Pi andP3 it is a creepingwave following a geodesic and beyond P$ it is a straightline. The ray satisfies the extended Fermat principle,whence the tangents to the ray on both sides of P\ formthe same angle with the tangent to the rim at P^, and at P3

the tangents to the ray on both sides of this point coincide.The boundary between regions II and III on the far-fieldsphere consists of those values of (6, <f>) for which P3 lieson the lower rim of the frustum. In region III ray 1 nolonger exists. Ray 2 in Fig. la also consists of three parts:Between Po and P2 it is a straight line, between P2 andP4

it follows a geodesic and beyond Pa, it is again a straightline. Furthermore, the tangents to the ray before and after

diffraction form the same angle with the tangent to theedge both in P2 and P*. Ray 3 is constructed according tothe same principles as ray 2, only it follows a maximum-length path instead of a minimum-length path as does ray 2.Rays 2 and 3 contribute to the field in all the regionsmentioned.

Paper T388 M, received 23rd March 1979Dr. Albertsen is with the Laboratory of Applied MathematicalPhysics, Technical University of Denmark, DK-2 800 Lyngby,Denmark

MICROWA VES, OPTICS AND ACOUSTICS, JULY 1979, Vol. 3, No. 4

Fig. 1 Ray tracing on conic frustuma Ray paths for rays 1, 2, and 3b Development of curved surface

133

0308-6976/79/030133 + 04 $01-50/0

In Fig. \b the developed surface is shown and thegeodesies followed by rays 1 and 2 are indicated. Sincethe surface is developable the geodesies are straight linesin Fig. \b.

3 Field calculation

The far field is calculated by vector addition of the field onthe rays contributing in the region in question. Considerfirst ray 1. It undergoes diffractions at Px and P3. At Pt the

dB

-10

-20-

-30

-40140

-II

160

III

180

a

V

200 220

dB

-10

-20-

- 3 0 -

-40140

-II

160

III

180

b

200

\

220*

220-40

dB

200 220

0

10

•>n

30

t,r\

I

-

II

s

\

III

. K .

ii

f

i

-

140 160 180

f

200 220*

Fig. 2 Far field from monopole on conic frustum

(a) and (ft) 0 = 30° cut(c) and {d) 0 = 60° cut(e) and (/) 0 = 90° cut(a), (c) and (e) show 0 polarisation(b), (d) and (/) show 0 polarisation

experimenttheory

134 MICRO WA VES, OPTICS AND ACOUSTICS, JULY 1979, Vol. 3, No. 4

process is described by a hybrid diffraction coefficient2

whereas the process at P 3 is a normal surface diffraction.3

Between PQ and P\ the ray propagates as a geometricaloptics (g.o.) ray, between P\ and P2 it propagates as acreeping wave (c.w.) with an exponential decay determinedby the curvature of the geodesic, and beyond P3 it againpropagates as a g.o. ray. Ray 2 undergoes diffractions atP2 and P 4 . Both processes are described by hybrid dif-fraction coefficients. Between Po and P2, and beyond P 4 ,the ray propagates as a g.o. ray and between P2 and P4 itpropagates as a c.w. The surface is assumed to be perfectlyconducting and, since the monopole is perpendicular to thesurface, only c.w. modes with the electric-field vectororthogonal to the surface are excited.

On the far-field sphere (r0 = constant) the electric fieldson rays 1 and 2 (except for a factor e'fer°/r0) are denotedEx and E2, respectively. Here k is the propagation constantand the time variation e~iu>t is suppressed. The asymptoticexpressions for Ex and£ 2 are:

v /—L vsc, irx 2 sin fa

x dsr s (0 , p i , n) exp {iksx ~ TX n(s)ds} ds r (0 , p3 , AZ)Jo '

(1)

eikT* dr< r(7T - V, 0 , 7T - V)

r2 2 sin j32i

*2

, « 2

x dst.iS(0, P 2 , n) exp {iks2 -J r2>n(s)ds} dsr<s(0, p 4 , n)

2 sin j34(2)

The symbols used in eqns. 1 and 2 are as follows: Thedistances rx, r2, sx, and s2 are shown in Fig. 1 togetherwith the angles v, fa, /32, and j34. The parameter r3(/*4)denotes the distance from P 3 (P4) to a fixed reference planethrough the origin and orthogonal to the rays beyond^3(^4), while P i , p 2 , p 3 , and p4 are the radii of curvatureof the geodesies at Px, P2, P 3 , and P 4 , respectively. Theangle \p0 is the dihedral angle between two planes, onespanned by the tangent to the rim at P4 and a rectilineargenerator of the cone and the other spanned by the tangentto the rim at P4 and the ray from P4 to the far field. Theparameters sc> x, sC) 2 and sCt 4 are caustic distances whichdetermine the divergence of a pencil of rays in the plane ofthe edge after an edge diffraction. They may be found from

1 1 _ 1 — cos v

rx Rxsinfa

1r2

J_«C,2

_L = _Sc> 4 S2

1 — cos v

sin

Sc< 2

cos (2; + i//0)~cos v

/?2 sin]34

(3)

(4)

(5)

where Ri and R2 are the radii of the top and bottom plateof the frustum, respectively. The exponential decay of thenth mode of the creeping waves is determined by TX „ and

r2 > n, where

ri.B(s) = ^ (6)

Here x'n is the absolute value of the nth zero of the deriv-ative of the Airy function, Ai(x), and K I (s) is the curvatureof the geodesic between Px and P 2 , s being the arclengthmeasured from P i . For r2i „ a similar expression holds withKX(S) replaced by K 2 (S) , the curvature of the geodesicbetween P2 andP 4 .

The diffraction coefficients used in eqns. 1 and 2 are:a 2-dimensional source factor,2 dfr\ a uniform wedgediffraction coefficient,4 drr(a, </>«, ^o)» where a is thewedge angle and i//,-(i//o) is the dihedral angle between oneface of the wedge and a plane spanned by the edge and theincoming (outgoing) ray (sin^0Dh in Reference 4 equalsd^drr), a surface diffraction coefficient,2 dsrs(Z,p,n),where Z is the surface impedance, p the radius of curvatureof the geodesic and n the mode number, and a diffractioncoefficient, dsr(Z,p,n), describing the transition from acreeping wave to a space ray. In the present case Z = 0 and

(7)

The polarisation vector of the field on ray 1 is parallel tothe outward-pointing normal to the body at P 3 and thefield on ray 2 is polarised along a unit vector which isorthogonal to the tangent to the rim at P4 and pointedaway from the body.

The field on ray 3 is constructed similarly to the field onray 2.

4 Experiment

For the experiment an aluminum conical frustum wasmachined. The dimensions were: height 200mm, Rx =100 mm and R2 = 174 mm. The monopole was placed50 mm from the centre of the top plate and excited by abattery-powered oscillator inside the body at 10 GHz.

In Fig. 2 comparisons between experimental and theo-retical data are shown. The patterns are far-field cuts forfixed values of0, and only 9 values close to 180° are retainedin order to emphasise the creeping wave contributions. Inall cases 5 creeping wave modes were included to ensureconvergence. The cut 0 = 0° is not shown since for this casethe geodesies become straight lines (rectilinear generators)and the theory does not apply. This explains why thetheory erroneously predicts zero field strength for 0 = 180°in all cases shown in Fig. 2.

The agreement between experimental and theoreticaldata is good in regions I and III, away from 9 = 180°,whereas the behaviour of the theoretical curve in region IIis anomalous. This is ascribed to the fact that the contri-bution from ray 1 is calculated in a penumbra region forwhich no correction was introduced.

5 Conclusion

An experimental verification of hybrid diffraction coef-ficients for creeping waves has been carried out. The fieldfrom a quarter-wave monopole on a conic frustum wascalculated in the shadow region and good agreement withexperimental data was found, except in penumbra regionsand regions where the theory is inapplicable.

MICROWA VES, OPTICS AND ACOUSTICS, JULY 1979, Vol. 3, No. 4 135

6 Acknowledgment

The author wishes to thank Jesper Hansen of the Electro-magnetics Institute, Technical University of Denmark, forhelpful encouragement and suggestions.

7 References

1 KELLER, J.B.: 'Geometrical theory of diffraction', J. Opt. Soc.Am., 1962,52, pp. 116-130

2 ALBERTSEN, N.C., and CHRISTIANSEN, P.L.: 'Hybrid dif-fraction coefficients for first and second order discontinuitiesof two-dimensional scatterers', SIAM J. Appl. Math., 1978, 34,pp. 398-414

3 KELLER, J.B.: 'Diffraction by a convex cylinder', IRE Trans.,1956, AP-4, pp. 312-321

4 KOUYOUMJIAN, R.G., and PATHAK, P.H.: 'A uniform geo-metrical theory of diffraction for an edge in a perfectly con-ducting surface', Proc. IEEE, 1974, 62, pp. 1448-1461

Book reviewAntennas and propagation: IEE Conference Publication169. Pts. 1 and 2IEE, 1978, Pt. 1 452 pp and Pt. 2 186 pp, £2600 (UK) or£3050 (overseas), ISBN 0 85296 195 2'Antennas and propagation' is the published proceedings ofthe international conference on these subjects which washeld at the IEE, Savoy Place, London, England on 28-30thNovember 1978. The conference, which is widely regardedas having been highly successful, was the first internationalantennas and propagation event to be held in Europe andattracted contributions from many countries.

The work contains the collected papers presented at theconference, including 5 invited review papers and 131specialist papers covering a wide variety of antenna andpropagation topics. The work, which is produced in softcovers, has 638 pages and, sensibly in view of its size, isdivided into two volumes dealing separately with antennasand propagation. Despite the obvious difficulties in pre-paring conference publications, a uniform, high standard ofpresentation has been achieved.

Volume 1, containing 75% of the contributions, coverspapers on antennas, with interest ranging from h.f. to milli-meter-wave frequencies and applications in both com-munications and radar. Three invited review papers areincluded, on advances in adaptive and scanned arrays (R.C.Hansen), new developments in vJi.f./uJi.f. antennas (F.M.Landstorfer), and antenna and propagation work at theEuropean Space Agency (J. Aasted). In the specialist paperssections, array antennas feature prominently with severalsections being devoted to scanned and multibeam arrays,particularly for radar applications, adaptive arrays for nullsteering in communications systems, and planar andconformal arrays including those using microstrip radiatingelements. Reflector antennas are, predictably, also wellcovered, with a number of papers being devoted to reflectorand feed design and the generation of shaped beams.

Another topic receiving much attention is antenna measure-ments, and several papers describe novel approaches to thissubject, including the use of near-field and holographictechniques. Other papers in volume 1 cover the subjects ofradomes, h.f. and vJi.f. antennas, scattering and diffraction,numerical techniques and basic antenna theory.

Volume 2 covers contributions on propagation subjects.The two invited review papers in this volume describe theionosphere as a propagation medium (E.D.R. Shearman),and propagation and applications of millimeter waves (D.C.Hogg). Predictably, fading and rainfall effects on terrestrialcommunication links are well covered, and the results ofinvestigations in both European and tropical environmentsare described. Similarly, several papers describe the proper-ties of ground-to-satellite communications links, includingthe effects of rain on the polarisation of these signals. Othersubjects covered are surface-wave propagation at mediumfrequencies in built-up areas, and propagation and remotesensing over sea surfaces.

In general, the publication provides a useful review ofsome of the work currently being undertaken in theseactive subjects, and, in particular, the review papers will beseen as valuable summaries of present work in the fieldsdescribed. Conference publications such as this serve asuseful media presenting research results, including thosewhich, although of considerable interest, may not be in aform directly suitable for publication as a formal researchpaper in a learned society journal. Although the restrictionon the length of the individual papers clearly limits theirdepth of coverage, they can provide a valuable introductionto a line of research work, more details of which may, inmany cases, be obtained subsequently. 'Antennas andpropagation' will be found a useful source of informationby other research workers in these fields.

J. AUSTIN

136 MICROWA VES, OPTICS AND A COUSTICS, JULY 19 79, Vol. 3, No. 4