A UTD-OM technique to design slot arrays on a perfectly conducting paraboloid

A UTD-OM technique to design slot arrays on a perfectly conducting paraboloid
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  1688 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 5, MAY 2005 A UTD-OM Technique to Design Slot Arrays on aPerfectly Conducting Paraboloid Theodoros N. Kaifas, Theodoros Samaras  , Member, IEEE  , Katherine Siakavara  , Member, IEEE  , andJohn N. Sahalos  , Senior Member, IEEE   Abstract— A procedure that can be employed to design slot ar-rays on smooth convex conducting surfaces is presented. The uni-form theory of diffraction (UTD) for the analysis and the orthog-onal method (OM) for the synthesis have been used. Detailed de-scription of the method is given and it is accompanied by a designstudy of a slot array on a perfectly conducting paraboloid. UTDhas been applied after a proper re-construction and re-evaluationof the diffraction integral. Thus, a method for treatment of caus-tics formed on convex surfaces comes out as a by-product of ourstudy. Coupling phenomena between two slots on convex surfacesare also addressed.  Index Terms— Caustics, conformal arrays, diffraction theory,mutual admittance, orthogonal method (OM), ray-tracing, slotarrays. I. I NTRODUCTION A RRAYS conformed to curved platforms are in many casesuseful in the air- and space-born vehicles. They have thecharacteristic of maintaining the aerodynamic integrity of theairface. Such arrays are dictated by the geometry of the sup-porting structure and have to meet the EM requirements. It isnoticed that much work has been devoted in the past but a gen-eral and widely accepted method for the analysis and synthesisproblems is still lacking.In the present paper the uniform theory of diffraction (UTD),[1],[2],fortheanalysisofslotarraysandtheorthogonalmethod(OM), [3], for the synthesis are employed. Our effort has to dowith the study of slot arrays on a curved conducting surface.Especially slot arrays positioned on a perfectly conducting pa-raboloid will be presented in the examples.The paper is organized as follows. First of all the UTD is re-viewed and the ray tracing implementation is presented. Recon-struction and re-evaluation of the diffraction integral are usedto treat caustics, so that the far field of slots can be accuratelyderived. Mutual coupling between two slots positioned on theparaboloid is studied and the results are compared with the cor-responding ones of the literature. Then the functional descrip-tion of the synthesis procedure (OM) is given and finally the Manuscript received February 26, 2004; revised July 28, 2004.T. N. Kaifas is with the Radiacommunications Laboratory, Department of Physics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece(e-mail: Samaras is with the Department of Physics, Aristotle University of Thes-saloniki, Thessaloniki 54124, Greece (e-mail: Siakavara and J. N. Sahalos are with the Radiocommunications Group,School of Science, University of Thessaloniki, Thessaloniki 54124, Greece(e-mail:; Object Identifier 10.1109/TAP.2005.846791Fig. 1. Ray paths on a paraboloid surface. UTD-OM analysis-synthesis technique is introduced for the de-signofanarray.Arraysareexaminedwithandwithoutcouplingbetween the slots. The coupling results of slots on curved sur-faces are compared with those of planar surfaces.II. A NALYSIS  P ROCEDURE It is well-knownthattheUTD is an asymptotic technique thatcanbeeasilyimplementedwithrelativelysmallrequirementsoncomputational recourses. However,UTD is problematic when itis treated in caustics and paraxial regions, [1], [2]. With the ex-ception of some simple geometries there is not an explicit for-mulationthatderivesthegeodesicpaths.Inmostofthecasesnu-merical techniques have to be applied. Even for simple configu-rations, thousands of geodesic detachment points must be takeninto account in order to derive a radiation pattern. In the caseof a paraboloid (Fig. 1) three detachment points, connecting apoint source to the far field direction, must be included in thecalculation.  A. Ray Tracing Thegeometryunderstudyisaconductingparaboloidofrevo-lution, which is a coordinate surface of the parabolic coordinatesystem. The parametric equations of the above system are [4](1)where is the sharpness/flatness parameter, and are the in-dependent variables of the surface.The receiver is considered to be at the direction towardthe infinity on the plane. Its position is defined by the angle. 0018-926X/$20.00 © 2005 IEEE  KAIFAS  et al. : UTD-OM TECHNIQUE TO DESIGN SLOT ARRAYS ON A PARABOLOID 1689 One of the main steps in the process of our study is the de-termination of the geodesic paths on the parabolic surface. Theclassical method to find a geodesic path is to use the nonlinearsecond order differential equations of the independent variablesof the surface [4]. The equations are treated as an initial valueproblem and are solved by the help of a class of computationalroutines called ordinary differential equation (ODE) routines.Another approach is to use a special coordinate system calledgeodesic coordinate system (e.g., the geodesic constant method[5]).The basic problem with the first approach has to do with theneed of computational resources. The second one becomes in-accurateforgeodesicsthatlieonplanesnormaltothesymmetryaxis on body-of-revolution surfaces. For example this happenswhen the system includes ring arrays.In either case, the key step in computing the geodesics is thedetermination of the shadow and light separatrix. Actually theseparatrixisthelocusoftheshadowboundary.Ifthepositionsof the source and the receiver are given, the separatrix contains allthepossibledetachmentpoints(Fig.1).Arayfollowsageodesicpath, on the surface, connecting the source with the detachmentpoint. Then it is transmitted to the receiver along a straight linethat coincides with the direction of the tangent to the surface atthe detachment point.For the geometry of the present problem it is proved thatmore than one detachment point fulfil the above conditionmeaning that more than one ray reach the receiver. Apart fromthe geodesics themselves, the paraboloid separatrix equationas well as the ray tube factors and must be defined. Theseparatrix (see Appendix I) obeys the following:(2)The factor is specified as the distance between adjacentrays and the factor assigns the angle formed by the tangentsof adjacent rays.The differential equations relating the above factors with thelength along the geodesic path are(3)(4)where is the Gaussian curvature of the surface [4].A rigorous study of the ray tracing led to the conclusion that,in general, more than one ray scattered by the paraboloid reachthe receiver. This means that there are cases, where more thanone diffraction point exist on the separatrix.An example that clarifies this ray tracing mechanism follows.In Fig. 2 the ray paths leaving the source positioned atarepresented.Thesharpness/flatnessparameterissetequal to unity.In the first case these paths end at the separatrix defined fora receiver angle at which from (2) gives .In the second case the paths end at the separatrix defined for. From the rays reaching the separatrix,in general, only one ray, and in some cases up to three, will bedirected to the receiver. In fact, one ray reaches the receiver in Fig. 2. Geodesics and separatricies on a paraboloid—cases 1 and 2. the first case (receiver at ) and three in the second case(receiver at ). The ray that travels on the height path(central ray) always reaches the receiver, while in the secondcase two identical side rays also contribute to the field. It isobvious that there must be an angle between 20 and 50 degrees,for the geometry under study, at which one ray would break intothree and a caustic is generated. A detailed calculation of thespreading parameters and , as a function of the angle ,proves that this caustic break occurs when .The next step is to find the dependence of the angle on theparameters of the system. This is an easy task, which comes outfrom the combination of (1) and (2). The caustic angle dependson the position of the source, the position of the receiver and thesharpness/flatness parameter of the paraboloid.  B. The Paraboloid Caustics It is well-known that the coalescence of diffraction pointsleads to the formation of caustics. Near the caustic regions thefieldisnotpresentedbytheraytechnique.Thus,transitionfunc-tions need to be included in the solution. It must be pointed outthat the evaluation of the fields in regions of caustics is not new,[6]. Remarkable efforts can be found in the literature; amongthem we mention here [7] and [8]. In [7] fold caustics (two coa-lescencepoints,Airyfunctions)appearedforthescatteringcase.In[8]cuspcaustics(threecoalescencepoints,paraboliccylinderfunctions) for the radiation from a source on a flat plate with acurved edge were presented. In our study we have the case of radiation from a convex surface and we are dealing with a cuspcaustic.The treatment of the caustic requires the construction of thediffraction integral without using the simplifying assumptionof the single, isolated, diffraction point. Several theories havebeen introduced for the diffraction integral. Catastrophe theoryentails functions that could be used for this purpose, [9]. Theincrementaltheoryofdiffraction(ITD)[10],thephysicaltheoryof diffraction (PTD) [11], or the concept of equivalent currents[12] have appeared as useful tools for this process. 1) The Diffraction Integral:  In our work the concept of equivalent currents is adopted [12]. Correction of caustics  1690 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 5, MAY 2005 is made in order to study radiation from slots positioned onconvex surfaces. In the shadow region of the source the far fieldis due to the creeping waves contribution. The expression of this contribution is given in the form of a line integral over theseparatrix. In other words the evaluation of the far field (in theshadow region) produced by a slot on a convex surface can beconsidered as a two-step process. Initially, the source produces,through the mechanism of the creeping waves, equivalent mag-netic currents on the separatrix. Then the separatrix, as a curvedline magnetic source, radiates the far field at the receiver.The electric field due to equivalent magnetic currentsover the separatrix can be given by [12](5)where is the unit vector toward the receiver, is the abscissa(the arc length) and is the infinitesimal length on the sepa-ratrix. is the distance between a point on the separatrix andthe receiver, is the wave number of the free space. It is no-ticed that in [12] (5) is used to express the diffracted field due toequivalent currents along an edge discontinuity. Here the sameequation is applied on a separatrix rather than on an edge. Thisis a new approach and the results presented below justify ourchoice.The key step in the evaluation process is the determination of the equivalent magnetic currents, which are(6)is the field that, having been produced by the source andhaving travelled in the form of a creeping wave, leaves the pa-raboloid surface detached from a point of the correspondingseparatrix. To attach a mathematical form to this field we usea well-known formula from [1](7)where is the transmission dyadic given in [1], is the mag-neticcurrentdensity of asource positionedatpoint andis a point along the separatrix having an abscissa equal to .For convenience, an infinitesimal source is used. If an extendedsource is to be used, superposition must be employed by inte-grating overthesource’sarea.Substitutingthemagnetic currentdensity into (5) we arrive at(8)By explicitly stating the dominant phase variation, cantake the form(9)where is the length of the geodesic connecting the sourceto a point on the separatrix having an abscissa equal to . Sub-stituting (9) into (8) we have(10)where is the position vector that starts at the phase centerand ends on the separatrix at the point having an abscissa equalto . In the above equation the far-zone approximationhasbeenintroduced,where isthedistancebetweenthe receiver and the structure’s phase center. Also, in (10) thenotation has been introduced. 2) Evaluation of the Diffraction Integral:  The creepingwave contribution in the shadow region has been expressed bythe diffraction integral, which is(11)Inordertotreatthecaustics,onehas,first,tochangethevariableto properly express the phase function and to takethe Chester expansion, [13], [14], for the amplitude . Forthe case at hand, the treatment of the cusp with three equidistantdiffraction points gives [14](12)The roots of the first derivative of the above function give thepositions of the diffraction points(13)The amplitude expansion is given as(14)where should be equal to four.The substitution of the expansion into the diffraction integralgives(15)The other terms of the amplitude expansion (14) donot contribute because the integral is evaluated at , wheretheybecomeequal tozero.Forthecaseunderstudy,thePearceyintegrals [15] via parabolic cylinder functions are used.  KAIFAS  et al. : UTD-OM TECHNIQUE TO DESIGN SLOT ARRAYS ON A PARABOLOID 1691 The phase parameters and and the amplitude expansionparameters and can be found by matching them, awayfrom the caustic, with the respective UTD ones. After the deter-mination of those parameters, the total field is given by(16)where and are the fields predicted by the UTD for thecentral and the noncentral rays correspondingly, whileand are given by(17)where and are Parabolic cylinder functions of order 1/2 and 3/2, respectively. The argument is definedas(18)where is the position, on the separatrix, of the noncentraldiffraction point and of the central one.Expressions(16)–(18)canbeeasilyevaluatedwhenthreerealrays exist. When there is only one real ray (and two complex),one must perform complex ray tracing to extract the values of various parameters appearing in those equations. In the caseat hand the length of the geodesic and the parameter , areknown in closed form and the above analysis can be imple-mented without resorting to complex ray tracing. Moreover, inthis case, the Taylor rather than the Chester expansion of thephase and the amplitude functions can be employed. In Ap-pendixIItheclosedformofthevariouscausticparameters,bothfor the shadow and the lit region of the caustic, is given. C. Implementation Results In this section implementationresults ofthe previouslystatedtheory are presented. The study provides an analysis for an-tennas consisting of slot radiators positioned on a paraboloid of revolution. First, the E-plane radiation field of a circumferentialslot and the H-plane radiation field of an axial slot, both posi-tioned on the paraboloid, are given. Then the mutual couplingbetween two slots having various configurations is presented. 1) Radiation From a Single Slot:  The surface of the parab-oloid is given by (1). The parameter is set equal to unity forthe radiation pattern calculations, hereafter.InFig.3(a)thegeometryandin(b)the polarizedfar-fieldpat-tern on the plane (E-plane) of a circumferential slot are illus-trated.InFig.3(c)the polarizedfar-fieldpatternonthe plane(H-plane) of an axial slotis shown. In bothcases the slotdimen-sionsare andthecenterofeachoneislocatedat(19)The dotted line shows the UTD result, while the continuous lineillustrates the result of the present work. Away from the causticthe two results coincide. It is obviousthat our equivalent currentapproach does not show any divergence in the caustic region.This is important for the diffraction problem in curved surfaces. (a)(b)(c)Fig. 3. (a) Single circumferential slot radiator on a paraboloid. (b) The   polarized far-field pattern of an axial slot on the conducting paraboloid(E-plane). (c) The    polarized far-field pattern of an axial slot on theconducting paraboloid (H-plane). Theentirepatterncanbepartitionedintofiveregions,namelyA. Shadow region—one ray;B. Lit region—one ray;C. Shadow region—one ray—shadow region of thecaustic;D. Shadow region—three rays—lit region of the caustic;E. Shadow region—three rays.  1692 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 5, MAY 2005 Lack of published results, relevant to the paraboloid geom-etry, prevents us from comparing our predictions with those of other authors (even for a limited portion of the angular space).Instead we draw attention to the fact that the field has the ex-pected variation in the lit and shadow regions. In the lit re-gionthefar-fieldofthecircumferentialslotresemblesthecorre-sponding E-plane field of a slot residing on a plane conductingsurface. For the axial slot the field in the lit region resemblesthe H-plane field of a slot on a planar surface. In the shadowregion smaller values of the radiated pattern are observed. A re-markable singularity at the UTD field (dotted line) between theshadow and lit region of the caustic is exhibited. The presentwork, taking into account equivalent currents and the additionalrays produced on the caustic, achieves to suppress this singu-larity (continuous line). 2) Coupling Between Slots:  In the following, results will bepresented on the mutual coupling between two slots positionedon a perfectly conducting paraboloid. The slot dimensions areand the parameter is set equal to 5, in order tocompare our results with those in [16]. We consider the positionof the first slot to be fixed at and thesecondslottobemovingonthepath and[see Fig. 4(a)].We use the azimuth angle, , as a parameter. The computedresults are presented in Fig. 4(b) and are compared with thecorrespondingonesfortheplanarconfiguration.Thecontinuouslines illustrate the mutual admittance between two slots on theparaboloid,whilethedottedonespresentthecalculationsfortheplanar case. It is worth noticing that for the results areconsistent with those given in [16]. It is assumed that each slotis fed by a rectangular waveguide in the dominant vector mode.The details of the derivation of the planar calculations, as wellas farther comments on the results are given in Appendix III.For a double–humped behavior is observed. This be-havior was first observed in [16]. It mainly comes from the rela-tive inclination of the slots with respect to the geodesic tangent,as it is indicated in the Appendix III and [17].Due to the nonintegrable singularity, the pointis calculated using the spectral domainapproach for the respective planar slot [18]. The result agreeswith the one in [19]. The case where is equal to unity is givenin Fig. 4(c). It can be seen that the coupling level foris higher than that for , because the slots are closer for. Fig. 4(b) and (c) can be used for the design of E-planeslot arrays [Fig. 5(a)].III. S YNTHESIS  P ROCEDURE Uptothispointwehavepresentedtheanalysisprocedureandvariousresultsobtainedwithit.Inordertogiveacomprehensiveanalysis-synthesis scheme, a functional description of the OMas the adopted synthesis procedure is given.The far electric field of an array of slots can be expressed bythe formula(20) (a)(b)(c)Fig.4. (a)Receivermovementonaverticalplane.(b)Mutualcouplingofslotsof Fig. 4(a)      . (c) Mutual coupling of slots shown in Fig. 4(a)      . where is the far electric field and is the complexamplitude of the th current mode [3]. The magnetic currentmodes are used to model the slot radiators. The previous equa-tionshowsthatthefarfieldisexpressedonthebasis .This basis is not an orthonormal one. The dot products of its el-ements(21)
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