GAUSSIAN VERTEX PLATE IMPROVESRETURN LOSS AND FAR-OUTSIDELOBES IN PRIME-FOCUSREFLECTOR ANTENNAS*Jian Yang1 and Per-Simon Kildal11 Department of ElectromagneticsChalmers University of TechnologyS-412 96 Gothenburg, Sweden

Recei ed 24 September 1998

ABSTRACT: Vertex plates are commonly used to impro¨e the return lossof primary-fed reflector antennas. We propose a new type of ¨ertex platewith a Gaussian thickness profile that gi es lower far-out sidelobes of theradiation pattern of the reflector antenna than the standard ¨ertex platewith constant thickness. We deri e the dimensions of this ¨ertex plateneeded to cancel the field at the focus of the reflector. We apply the

( )design rules to impro¨e the return loss RL of a primary-fed reflectorwith hat feed, and simulate it by using a computer code based on the

( )finite-difference time-domain FDTD technique. The results ¨erify theimpro¨ements of the radiation pattern obtained by using a Gaussian¨ertex plate rather than a standard one. Q 1999 John Wiley & Sons,Inc. Microwave Opt Technol Lett 21: 125]129, 1999.

Key words: ¨ertex plate; reflector antenna; Gaussian beam

1. INTRODUCTION

An important problem in the design of a reflector antenna isthe contribution from the reflector to the reflection coeffi-cient at the input flange of the feed. This problem was

w xstudied in 1 by analyzing reradiation from the inducedcurrent distribution on the reflector. By locating a flat vertexplate in the center of the reflector, the problem was reducedand the radius and thickness of this flat plate were deter-mined. Numerical optimization of the flat vertex plate for a

w x w xwide frequency band was reported in 2 and 3 . The vertexplate may also have a positive effect on the radiation patternof the reflector, such as an improvement in gain and suppres-sion of sidelobe levels. This is because the negative effects,which are caused by both the center blockage of the apertureand multiple reflections between the feed and the reflector,are strongly reduced due to the central dip in the aperturefield provided by the vertex plate. The emphasis of thepresent paper is to introduce a new vertex plate with aGaussian thickness profile, which has the potential of produc-ing even lower sidelobes than the standard flat plate with arectangular profile. We show how simple analytical expres-sions for the dimensions of both the rectangular and theGaussian vertex plates can be derived by using Gaussianbeam formulas. We have applied this approach to design areflector with hat feed, and we have simulated it with the

Ž .V2D code based on a finite-difference time-domain FDTDw xalgorithm 6 . The results verify the validity of the derived

formulas, and the comparison between results with a Gauss-ian vertex plate and with a traditional flat vertex plate showsthe superiority of the former.

* The vertex plate described in this paper is subject of the US patentapplication No US 60r056, 220 filed August 21, 1997, and the interna-tional application No PCTrSE98r01478 with priority August 21, 1997.

2. PERTURBATION THEORY FOR VERTEX PLATES

A parabolic reflector with a vertex plate can be described bythe coordinates

Ž .r s r q tz 1ˆ¨ p

where

12r s rr q z z; z s yF q rˆ ˆp p p 4F

describes the parabolic reflector without a vertex plate, andŽ .t s t r for r - r and t s 0 for r G r is the thickness¨ ¨

profile of the vertex plate with r the radius of it. The¨reflector with a vertex plate is shown in Figure 1, where thethickness of the vertex plate has been grossly exaggerated forthe sake of clarity.

The radiation field of the feed at an arbitrary observationpoint r in the far field can be expressed as

1yjk rŽ . Ž . Ž .E r s e G u 2f r

Ž .where G u is the radiation field function. If we assume thatŽ .the phase reference point of the radiation field function G u

of the feed is at the origin of the reflector coordinate system,and that this origin coincides with the focus of the reflector,the incident field on the reflector becomes

Ž . Ž . Ž .E r s E r . 3i ¨ f ¨

The reflected field at the reflector surface r with and with-¨out a vertex plate is, respectively,

Ž . Ž . ? Ž . Ž .@ Ž . Ž .E r s yE r q 2 n r .? E r n r 4ˆ ˆr ¨ f ¨ ¨ f ¨ ¨

Ž . Ž . ? Ž . Ž .@ Ž . Ž .E r s yE r q 2 n r .? E r n r 5ˆ ˆr p f p p f p p

Ž .Ž Ž .. Ž .where n r n r is the normal to the reflector with withoutˆ ˆ¨ pa vertex plate. Let us consider the central region of thereflector where r is small. Assuming t small also, we can usethe approximations

1yjk r¨Ž . Ž . Ž . Ž .n r f n r f z, E r s y e G 0 ,ˆ ˆ ˆp ¨ r ¨ r¨

1yjk r¨Ž . Ž . Ž . Ž . Ž .E r s y e G 0 , and G u s G 0 y 6ˆr ¨ r¨

Figure 1 Geometry of vertex plate in reflector antenna

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 21, No. 2, April 20 1999 125

where we also have assumed a linear y-polarized antenna.Furthermore, we may use

Ž .r f r f F in the amplitude expression and 7a¨ p

Ž .r f r y t f F y t in the phase expression. 7b¨ p

Then,

1ykŽFyt .Ž . Ž .E r f y e G 0 yrFrr ¨ ¨ F

1yk FŽ . Ž . Ž .E r f y e G 0 y. 8ˆr p rFr ¨ F

Ž .We now transform the reflected fields E r andr ¨ r F r ¨Ž .E r along parallel rays to the plane defined by z sr p r F r 0¨

yF q t , where t is the maximum value of t, Figure 1.0 0Then,

Ž . Ž . yjk ŽyFqt0qr pyt .E z f E r erFrr ¨ 0 r ¨ ¨

1j2 k t y jk ŽFqt .0Ž .f y G 0 e e y, r F rˆ ¨F

Ž . Ž . yjk ŽyFqt0qr p.E z f E r erp 0 r p rFr ¨

1yjk ŽFqt .0Ž . Ž .f y G 0 e y, r F r . 9ˆ ¨F

We can now express the reflected field when the vertex plateis present as

Ž . Ž . Ž . Ž .E r s E r q DE z , 10r ¨ r p r 0

i.e., as the field due to the reflector without a vertex plateplus a correction term:

Ž . Ž . Ž .DE z s E z y E zr 0 r ¨ 0 r p 0

Ž .G 0j2 k t y jk ŽFqt .0Ž .f y e y 1 e y

FŽ .G 0

yjk ŽFqt yt .0 Ž .f y j2kte y, r F r . 11ˆ ¨F

Ž .The latter approximation in 11 is valid when kt < pr2 andj2 k t Ž . jkt jktis obtained from e y 1 s j2 sin kt e f j2kte . TheŽ .separation in terms in 10 makes it easy to find the aperture

field at the focus of the reflector, as this must be the sum ofthe same two contributions evaluated at the focal point. Theterm representing the undisturbed aperture field at the focuscan be found by assuming that the feed is of the BOR type1w x Ž . w x4 and by using geometrical optics GO , to be 5

1 u2 y j2 k FŽ . Ž . Ž .E r , w s y cos e G u 12ap ž /F 2

y1Ž Ž ..where u s 2 tan rr 2 F . The correction term needs to betransformed from its apparent location at the surface of thereflector to the aperture. This can be done by conventional

aperture integration formulas, which for our case becomes

kˆ ˆŽ . w Ž .E r s yj hJ y hJ ? R RHH s s4p S

eyjk R

ˆŽ . x Ž .qh J = z = R dS 13ˆs R

Ž .where J s n = DH f DE z and R s r y r with r sˆs r r 0 0Ž .r r, r s r r q yF q t z, and dS s r dr dw. A moreˆ ˆ ˆa 0 0 0 0 0

elegant and understandable alternative is to evaluate theŽ .correction term DE z in the aperture of the reflector atr 0

z s 0 by using Gaussian beam formulas. This will be done inthe next section.

3. VERTEX PLATE WITH RECTANGULAR PROFILE

We consider first a vertex plate with a rectangular profile, i.e.,t s t for r - r . Then, the correction term of the field at0 ¨z s yF q t becomes0 0

2kt0 y jk FŽ . Ž . Ž .DE z f yj G 0 e y, r F r . 14ˆr 0 ¨F

We approximate this field distribution by a Gaussian distribu-tion according to

2kt 2 2a y jk F yr r raŽ . Ž . Ž .DE z s yj G 0 e e , 0 F r F `. 15r 0 F

We determine the unknowns t and r by requiring that thea apowers and directivities of the radiation patterns resulting

Ž .from the two versions of DE z be equal, i.e.,r 0

r `¨ 2 22 2yr r ra< < < <t r dr s t e r drH H0 a0 0

r `¨ 2 2yr r ra Ž .t r dr s t e r dr . 16H H0 a0 0

These expressions are satisfied when

1Ž .t s 2 t and r s r . 17a 0 a ¨'2

w xFrom the Appendix or 5 , we know that a Gaussian aperturedistribution radiates a beam which, for all distances zX fromthe reference plane z s yF q t , is described by the for-0 0mula

2kt ra aX y jk FŽ . Ž .DE r , z s yj G 0 e Xa Ž .F r z

= yr 2 r r 2Ž zX . y jŽ1r2.kC Ž zX .r 2 jf Ž zX . y jk zX Ž .e e e e y 18ˆwhere

2X2 zX 2Ž .r z s q ra(ž /kra

zX

XŽ .C z s 2X 2 2Ž .z q kr r2a

p kr 2aXŽ . Ž .f z s y arctan . 19Xž /2 2 z

By using the vertex plate, we want to create a null in theaperture field at the focus of the reflector where the feed is

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 21, No. 2, April 20 1999126

located, i.e., for zX s F y t corresponding to z s 0. This0gives the condition

Ž . Ž . Ž .E 0 q DE 0, F y t s 0 20a a 0

Ž . Ž .where, by using 12 and 19 ,

Ž .G 0yj2 k FŽ .E 0 s y e ya F

Ž .DE 0, F y ta 0

Ž .2kt G 0 ra a y j2 k F y jŽk t qf ŽFyt ..0 0s yj e e yŽ .F r F y t0

p kr 2aŽ . Ž .f F y t s y arctan . 210 Ž .ž /2 2 F y t0

Ž . Ž .From 21 , the condition in 20 for the cancellation of thefield at the focus becomes, by also using F 4 t ,0

2kt 2kta af s 12 22 2w Ž . Ž .x w Ž .x' '1 q 2 F y t r kr 1 q 2 Fr kr0 a a

kr 2 kr 2a a Ž .kt sarctan f arctan . 220 ž /Ž .ž /2 F y t 2 F0

The solutions to these equations can be found numerically to' 'Ž .be t s 2 t s 0.18l and r s 1r 2 r s 0.42 lF , i.e.,a 0 a ¨

' Ž .t s 0.09l and r s 0.6 lF . 230 ¨

4. VERTEX PLATE WITH GAUSSIAN PROFILE

The rectangular vertex plate profile will have an actual radia-tion pattern of the same form as a uniformly distributedaperture. We have treated this uniformly distributed aperture

Ž . Ž .as if it is Gaussian in order to derive 23 . Therefore, 23 isonly valid as a guideline for the vertex plate with the rectan-gular profile. Uniform aperture distributions are known togive high sidelobes. Therefore, we may expect that there existother vertex plate profiles which could give lower sidelobesthan the rectangular one, and in particular, we would expecta Gaussian profile to be beneficial. Let us therefore intro-duce a vertex plate with a Gaussian profile, i.e.,

yŽ r r r 0 .2 Ž .t s t e 240

where t is the thickness of the vertex plate at r s 0 and r0 0defines the 1re width of the plate. In order to find ananalytic form for the aperture field, we need to find aconvenient expression for eyjk t. We choose to use

yjk t y jk t0Ž1yr 2 rŽ2 r 02 .. Ž .e f e 25

Ž . Ž 2 Ž 2.. Ž .by approximating 24 as t f t 1 y r r 2 r . Thus, 110 0becomes

Ž .2kt G 0 20 y jk F yŽ r r r .0Ž .DE z f yj e er 0 F

yjk t0 r 2 rŽ2 r 02 . Ž .= e y, r F r 26ˆ ¨

Ž .which is the same distribution as that in 15 , except for theyjk t0 r 2 rŽ2 r 0

2 . Ž .phase e . Thus, the field DE z due to a vertexr 0

plate with a Gaussian profile has approximately a GaussianŽ .distribution within r F r . We further assume that DE z¨ r 0

is a complete Gaussian aperture distribution over the rangeof 0 F r F `, which can be done without much error. The

X ŽGaussian beam is now for all z described by see the.Appendix

2kt r X X2 2 20 0X y jk F yr r r Ž z . y jŽ1r2.kC Ž z .rŽ .DE r , z s yj e e eXa Ž .F r z

= jf Ž zX . y jk zX Ž .e e y 27ˆ

where

22X X2 z t z0X 2Ž .r z s q 1 q r0( 2ž / ž /kr r0 0

X 2 Ž 2 X .4 z q k t r q z t0 0 0XŽ .C z sX2 X2 2Ž .4 z q k r q z t0 0

p kr 2 kt0 0XŽ . Ž .f z s y arctan q . 28Xž /2 2 z 2

By using the vertex plate, we also want now to create a null inthe field at the focus of the reflector where the feed islocated. This gives, in the same way as in the previoussection,

Ž . Ž . Ž .E 0 q DE 0, F y t s 0 29a a 0

where, for F 4 t ,0

Ž .G 0yj2 k FŽ .E 0 s y e ya F

Ž .DE 0, F y ta 0

Ž .2kt G 0 r0 0 y j2 k F jŽk t qf ŽFyt ..0 0s yj e e yŽ .F r F y t0

22Ž . Ž .2 F y t F y t t0 0 0 2Ž .r F y t s q 1 q r)0 02ž / ž /kr r0 0

222 F Ft0 2f q 1 q r0( 2ž / ž /kr r0 0

p kr 2 kt0 0Ž .f F y t s y arctan q0 Ž .ž /2 2 F y t 20

p kr 2 kt0 0 Ž .f y arctan q . 30ž /2 2 F 2

Ž .The condition for cancellation of the field at the focus in 29Ž .becomes now, by using 30 ,

222 F Ft0 22kt r s q 1 q r0 0 0( 2ž / ž /kr r0 0

kr 2 kt0 0 Ž .kt s arctan q . 310 ž /2 F 2

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 21, No. 2, April 20 1999 127

The solution to these two equations can be found numerically2 Ž .to be kt s 0.92 and kr r 2 F s 0.85, which corresponds to0 0

t s 0.15l0

' Ž .r s 0.5 Fl . 320

These are the dimensions of the Gaussian vertex plate.

5. APPLICATION

We have used the results in the previous sections to dimen-sion vertex plates with both rectangular and Gaussian heightprofiles for a reflector antenna which is fed by a hat feed.The depth of the reflector is 1008 and the focal length is

Ž .4.78l. By using Eq. 23 , the radius and thickness of the flatŽ .vertex plate are r s 1.3l and t s 0.09l, and by using 32 ,¨ 0

the Gaussian vertex plate should have r s 1.09l and t s0 00.15l. The reflection coefficients and the radiation patternsof the antenna were calculated for both cases, and for the

w xcase of no vertex plate, by the V2D code 6 based on theFDTD method, and are presented in Figures 2 and 3. Thethree antennas are exactly the same except for the vertexplates. We see that the Gaussian vertex plate gives a reflec-tion coefficient which is below y20 dB over a 15% frequencyband, and the radiation pattern is improved compared withthe flat vertex plate case, especially in the far-out sidelobes.

Figure 2 Reflection coefficients of antennas with different vertexplates

Figure 3 Radiation patterns of antennas with different vertex plates

Beyond 808, the improvement is about 5 dB. The pattern forthe reflector without a vertex plate has the highest sidelobesdue to the blockage and multiple reflections between thefeed and the reflector. It has to be mentioned that thereflection coefficient with the Gaussian vertex plate may beadditionally improved by adjusting the radius and thicknessnumerically. In addition, the improvement in the far-outsidelobes when using the Gaussian vertex plate is moresignificant in reflector antennas with large diameters. Theresults obtained by the V2D code agreed well with measure-

w x w xments in 7 and 8 , both for the radiation pattern and thereflection coefficient.

6. CONCLUSION

We have derived formulas which can be used to dimensionvertex plates with both rectangular and Gaussian heightprofiles. The Gaussian vertex plate is shown to reduce theinput reflection coefficient well, and it gives a radiationpattern with lower sidelobes than a rectangular profile.

APPENDIX

We consider a Gaussian aperture distribution of the form

Ž . yr 2 r ra2 yjk ŽCa r2.r 2 Ž .E x , y s E e e y A1ˆa 0

where E is a constant, r is the 8.7 dB radius in the0 aaperture, and C is the wavefront curvature. The Gaussianabeam radiated by this aperture, for all z from the aperture,can be expressed as

r 2 2 2a yr r r Ž z . y jk ŽC Ž z .r2.r jf Ž z . y jk zŽ . Ž .E r s E e e e e y A2ˆ0 Ž .r z

where r s rr q zz andˆ ˆ

2 2Ž . Ž Ž .. Ž Ž .. Ž .'r z s r z q r z A3diff GO

is the 8.7 dB radius of the beam at r s zz,

2Ž .C 1 r za diffŽ . Ž .C z s 1 q A4ž /Ž .1 q zC zC r za a GO

is the wavefront curvature at r s zz, andˆ

Ž .p r zGOŽ . Ž .f z s y arctan A5ž /Ž .2 r zdiff

is the phase slippage. In these equations,

Ž . Ž . Ž .r z s 2 zr kr A6diff a

is the diffraction cone radius, and

Ž . Ž . Ž .r z s 1 q C z r A7GO a a

is the GO cone radius.

REFERENCES

1. S. Silver, Microwave antenna theory and design, McGraw-Hill,New York, 1949.

2. H. Copy and Y. Leviatan, Reflection coefficient optimization atfeed of parabolic antenna fitted with vertex plate, Electron Lett 16Ž .1980 , 945]947.

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 21, No. 2, April 20 1999128

3. G.T. Poulton and T. Almoyyed, Optimum vertex plate design forreflector antennas, Proc 1973 European Microwave Conf, Brus-sels, Belgium, Sept. 1973, c. 4.4.

4. P.-S. Kildal and Z. Sipus, Classification of rotationally symmetricantennas in BOR and BOR types, IEEE Antennas Propagat0 1

Ž .Mag 37 1995 , 114]117.5. P.-S. Kildal, Modern antenna theory, Compendium, Microwave

Antennas, CTH, 1997.6. W.K. Gwarek, V2D-solver version 1.9: A software package for

electromagnetic modeling of microwave circuits of vector 2-DŽclass, Warsaw University of Technology e-mail: gwarek@ire.pw.

.edu.pl .7. P.-S. Kildal and J. Yang, FDTD optimizations of the bandwidth of

the hat feed for mm-wave reflector antennas, Proc 1997 IEEEAP-S Int Symp, Montreal, Canada, July 1997, pp. 1638]1641.

8. J. Yang and P.-S. Kildal, Design of hat fed reflector antennasusing FDTD, Proc EMB 98}Electromagnetic Computations forAnalysis and Design of Complex Syst, Linkoping, Sweden, Nov.1998.

Q 1999 John Wiley & Sons, Inc.CCC 0895-2477r99

TRANSIENT SCATTERING BYCONDUCTING CYLINDERS } IMPLICITSOLUTION FOR THE TRANSVERSEELECTRIC CASESadasiva M. Rao,1 Douglas A. Vechinski,1 and Tapan K. Sarkar21 Department of Electrical EngineeringAuburn UniversityAuburn, Alabama 368492 Department of Electrical Engineering and Computer ScienceSyracuse UniversitySyracuse, New York 13244-1240

Recei ed 26 September 1998

ABSTRACT: In this work, we present an implicit solution of thetime-domain integral equation to calculate the induced current profile on

( )infinite conducting cylinders illuminated by a trans erse electric TEincident pulse. A detailed description of the numerical procedure alongwith representati e results are presented for two types of integral equa-

) ( ) )tions, ¨iz. 1 the electric field integral equation EFIE , and 2 the( )magnetic field integral equation HFIE . Q 1999 John Wiley & Sons,

Inc. Microwave Opt Technol Lett 21: 129]134, 1999.

Key words: transient scattering; conducting cylinders; integral equations

1. INTRODUCTION

Recently, a numerically efficient, implicit solution schemewas proposed to solve the time-domain integral equationarising in transient electromagnetic scattering problems. Thisapproach was successfully applied to conducting cylindersŽ . w x w xtransverse magnetic case 1 , arbitrary wires 2 , and arbi-

w xtrarily shaped conducting bodies 3 . Note that the implicitsolution scheme provides a stable solution even at late times,which is one of the major advantages.

In this work, we extend the implicit solution scheme to thetransverse electric case of two-dimensional conducting cylin-ders. For the numerical solution, we consider both the elec-

Ž .tric field integral equation EFIE and the magnetic fieldŽ .integral equation HFIE . The EFIE case is considered for

the sake of completeness, whereas the HFIE solution isrequired to extend this technique to dielectric scatterers.Further, we note that the EFIE is applicable to both open

and closed contours, whereas the HFIE is applicable toclosed cylinders only.

2. INTEGRAL EQUATION FORMULATION

The scattering geometry under consideration is shown inFigure 1. Let C denote the cross section of an open or closed

Ž .perfectly electric conducting PEC cylinder parallel to thez-axis. At each point on C, let a represent an outward-di-nrected unit vector normal to the contour. The circumferentialvector a is then obtained by a s a = a .t t z n

The incident field is a plane wave with its magnetic fieldŽ .polarized in the z-direction TE incidence . The incident

electromagnetic field, defined in the absence of the scatterer,Ž .induces a surface current J r, t on the scatterer. The bound-

ary conditions require that the total tangential electric fieldon the conducting surface be zero or

w sw x inc x Ž .E J q E s 0 on C 1tan

sw xwhere E J is the scattered electric field due to the inducedcurrent J. The scattered field radiated by the current J maybe written in terms of the magnetic vector and electric scalarpotentials as

­Asw x Ž .E J s y y =F 2

­ t

where

RXJ r , t yž /m ` c X XŽ . Ž .A r , t s dz dC 3H H

X4p RC z sy`

RXq r , t ys ž /1 ` c X XŽ . Ž .F r , t s dz dC 4H H

X4pe RC z sy`

X 2 X 2< <'and R s r y r q z , the distance from the field pointŽ X X. Ž . Ž .r to the source point r , z . In 3 and 4 , m and e are the

permeability and permittivity of the surrounding medium,and c is the velocity of propagation of the electromagneticwave. The electric surface charge density q is related to theselectric surface current density by the continuity equation

­ qs Ž .= ? J s y . 5s ­ t

Ž . Ž .Combining 1 and 2 gives

­Aincw x Ž .q =F s E 6tan

­ t tan

and represents the electric field integral equation formula-tion.

We may also develop an integral equation that uses theboundary condition on the magnetic field. From boundaryconditions, we obtain

w sŽ . inc x Ž .J s a = H J q H 7n

where H inc is the incident magnetic field, and H s is thescattered magnetic field due to the induced currents J. Asbefore, the scattered field can be written in terms of thepotential functions, and is given by

1sŽ . Ž .H J s = = A 8

m

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 21, No. 2, April 20 1999 129