Three-dimensional ray-tracing to model internal reflections in off-axis lens antennas
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Three-Dimensional Ray-Tracing to Model Internal Reflections in Off-Axis Lens AntennasAndrew P.Pavacic,Daniel Llorens del Río,Juan R.Mosig,Fellow,IEEE,andGeorge V.Eleftheriades,Senior Member,IEEEAbstract—A three-dimensional ray tracing technique for inter-nally reflected rays is developed and applied to compute the ra-diation pattern of an integrated lens antenna architecture with a twin arc-slot printed feed.The feed is represented in simulation by equivalent currents computed by the method of moments.The sim-ulated results are compared against measurements at30.2GHz,in-cluding the case of an offset feed,in which internal reflections lead to pattern contributions of up to20dB that cannot be predicted byfirst-order physical optics(PO)techniques described in prior work.The twin arc-slot design reduces undesired sensitivity in di-agonal lateral directions with respect to a comparable rectangular slot feed.Low-directivity feed model basis functions are shown to reduce discretization noise in the simulated patterns with respect to an array factor model.A frequency sweep analysis is introduced to test the sensitivity of individual sidelobes to the fabrication tol-erances of the dielectric lens.Index Terms—Integrated lens antennas,internal reflections,slot antennas,surface roughness.I.I NTRODUCTIONR ADIOMETRIC remote sensing applications have createda demand for millimeter-wave and submillimeter-wave re-ceivers operating at frequencies from30GHz to3THz[1],[2]. Dielectric lens antennas backed by printed feed networks,such as the one illustrated in Fig.1,are of interest due to their relative ease of fabrication,compatibility with planar Schottky mixer diodes[3],and mechanical robustness suitable for spaceborne payloads.A dielectric lens focuses the printed feed radiation pattern to provide quasioptical compatibility with Cassegrain re-flectors,which in turn provide directivity.Commercial full-wave simulation software is often inefficient for the design of such integrated lens antennas.To ensure suffi-cient directivity in the free space pattern,and to avoid undesired resonances[4],the lens should be several dielectric wavelengths in diameter,and thus constitutes an electrically large volume. Attempting to solve for thefields within the lens by brute force techniques such as the Finite Element Method(FEM)generates large mathematical matrices of unknown variables to be solved.Manuscript received June2,2005;revised October21,2005.This work was supported in part by the European Space Research and Technology Centre (ESTEC)Contract14062/00/NL/GD.A.P.Pavacic and G.V.Eleftheriades are with the Edward S.Rogers, Sr.Department of Electrical and Computer Engineering,University of Toronto,Toronto,ON M5G1G6,Canada(e-mail:pavacic@waves.utoronto.ca; gelefth@waves.utoronto.ca).D.L.del Río and J.R.Mosig are with the Laboratory of Electromagnetics and Acoustics,Swiss Federal Institute of Technology,CH-1015Lausanne,Switzer-land.Digital Object Identifier10.1109/TAP.2005.863143Fig.1.Extended hemispherical printed dielectric lens antenna dimensions,asfabricated.The printed feed consists of a slot antenna in the ground plane of amicrostrip matching network.This is also true for full-wave methods based on boundary el-ements,including the combinedfield integral equation(CFIE)for homogeneous dielectric bodies[5].Previous researchers have thus used a physical optics(PO)approach to reduce this complexity and more efficiently com-pute lens antenna radiation patterns[6],[7].The technique issimilar to the one applied to treat radome-covered antennas[8],[9].Under PO,only the tangential electric and magneticfieldsat the lens interface between dielectric and free space are cal-culated.The Schelkunoff equivalence principle is then appliedto substitute equivalent magnetic and electric currents for thesurface electric and magneticfields,respectively,and the radia-tion patterns are then computed from these equivalent currents.The PO method of calculatingfields only along the lens surfaceeffectively reduces the numerical complexity,by1)treating asurface instead of a volume,and2)avoiding the inversion of alarge linear system.Discrepancies observed between measured lens patterns andthose predicted by PO software have led some researchers toinclude the effects of internal reflections in the numerical model[6],[10].These are calculated via geometrical optics(GO)ray-tracing techniques that compute the internally reflectedray paths to their next points of intercept at the lens surface.The transmitted electric and magneticfields at these pointsare evaluated and substituted with Schelkunoff equivalentcurrents as before,to provide an auxiliary contribution to thecomputed lens antenna radiation pattern.This modeling ofinternal reflections has been extended in the current work toinclude the special case of an asymmetrical(i.e.,“off-axis”)lens feed,using software techniques similar to those applied toan off-axis radome[9].Earlier PO lens models have often simplified the internal re-flection ray-tracing by treating the printed feed as a point source 0018-926X/$20.00©2006IEEEcentrally located on the major axis of the lens[6],[7],[10],[11]. The internally reflected ray paths then exhibit a rotational sym-metry,such that each path lies entirely within a single plane, allowing all ray tracing calculations to be effectively two-di-mensional(2-D).The contribution of internal reflections for this on-axis case has not been found to significantly distort the com-puted radiation pattern,but rather to affect mainly the feed ad-mittance[11].Imaging array lens antenna applications employ multiple feeds mounted away from the main lens axis to achieve beam steering and multiple beam capability[2].In such off-axis cases,internally reflected power can be intuitively expected to affect the lens in directions opposite to that of the steered main beam.Simulating a displaced,off-axis feed is more com-plicated because it lacks rotational symmetry,thus requiring three-dimensional(3-D)ray tracing.This off-axis case is of particular interest when studying internal reflections,because a reflected pattern contribution that is too weak to measurably distort the main beam in the on-axis case may be detectable in the off-axis case.Custom3-D ray tracing software for mod-eling the lens is combined in this present work with method of moments(MoM)techniques for modeling the printed feed, in order to more fully investigate the causes of previously observed discrepancies between lens pattern simulation and measurement,and the contribution of internal reflections to off-axis lens antenna patterns.Section II of this paper proposes the use of a twin arc-slot printed feed antenna,including simulation and measurement of its return loss.The lens geometry under investigation is also de-scribed.Section III describes how MoM is applied to model the printed feed as a distribution of magnetic current elements.Sec-tion IV describes the3-D ray tracing algorithm used to simulate lens patterns with internal reflections,and Section V compares these results against measurement.II.D ESCRIPTION OF A NTENNAA.Lens DesignFor ease of fabrication,an extended hemispherical design was selected for the dielectric lens.The dimensions were chosen to approximate the desirable focusing properties of an elliptical lens with a feed located at one of itsfoci(1)(2)(3)where is the lens radius at the maximumwaist,is the majorsemiaxis of ellipticalcurvature,is the cylindrical extensionlength for an elliptical lensand is the total combined cylin-drical extension for the extended hemispherical approximation(see Fig.1).The lens radius of25.0mm was chosen for a30GHz scalemodel to test design methods applicable to smaller,submil-limeter-wavelength antennas.This size ensures that the lens sur-face is located in the farfield of the printed feed’s radiation pat-tern for both the“first-order”rays,which have a single point ofintercept with the lens surface as shown in Fig.2,and the inter-nally reflected rays,referred to in this paper as“second-order.”Fig.2.Dielectric lens modeling:2-D ray tracing for an elliptical lens,with2nd-order internally reflectedrays.Fig.3.30.2GHz twin arc slot feed architecture,as fabricated.Ground planeand microstrip layers are shown separately for clarity.(a)Twin arc slots in theground plane layer.(b)Microstrip matching network.Dimensions in m.B.Printed Feed ArchitectureAs shown in Fig.2,the lens’collimating property is only ef-fective over its rounded surface above the plane of its maximumwaist.Geometric optic analysis reveals that feed radiation inter-cepting the lens surface below the maximum waist,at the surfaceof a cylindrical extension,is not collimated,but rather propa-gates laterally in undesired directions.For this reason,the mostefficient feed architectures for use with lens antennas should bedesigned to minimize radiation in lateral directions along theground plane.Such lateral radiation was simulated and com-pared for two feed architectures-the conventional twin rectan-gular slot feed employed by other researchers(not illustrated;used here only for comparison)[7],and the twin arc-slot design[12].The dimensions of the fabricated twin arc-slot feed are shownin Fig.3.Whereas two rectangular slots are generally sepa-rated by0.5dielectric wavelengths to provide E-plane cancel-lation of the lateral wave,the arc-slots form a ring of diam-parison of simulated S return loss of twin arc-slot feed into in finite dielectric half-space (permittivity "=12:0)versus measured S return loss into fabricated dielectric lens of equal permittivity.eter 0.6wavelengths,to minimize radiation in all lateral direc-tions.The width of the slot apertures is 25%of the arc length,which increases their feed point impedance,ensuring a wide coupling bandwidth of about 10%to the50microstrip line.Designing the arc-slots to radiate into the dielectric lens re-duces their radius of curvature with respect to arc-slots that are designed to radiate into free space,leaving reduced physical space for the microstrip layout.The lines are thus curved and arranged to provide equal and maximal spacing between the edges of the microstrip and apertures,in order to minimize un-desired coupling.A stepped-impedance transformer consisting of three quarter-wavelength sections is also used to help opti-mize matching,bandwidth,and microstrip line width.The sim-ulated return loss of the printed feed is compared against mea-surement in Fig.4.The simulation approximates the lens as an in finite dielectric half-space ofpermittivity ,so the comparison is used only to con firm the feed ’s resonant fre-quency,since previous work has shown that including the ef-fects of internal re flections on return loss (periodic resonances in measurement)makes negligible contribution to the radiation patterns [11],[13].The simulated radiation patterns into the interior of the di-electric lens (here treated as an in finite dielectric half-space)as calculated by a commercial MoM software package are illus-trated in Fig.5.This 3-D representation illustrates the manner by which the rectangular slot array and element factors cancel the undesired lateral wave radiation in the E-and H-planes,re-spectively,without doing so in diagonal directions [Fig.5(a)].The twin arc-slot architecture minimizes lateral wave radiation in all directions [Fig.5(b)],reducing the lateral radiation in the diagonal plane by 20dB with respect to the rectangular slot feed,and thus ensuring that a larger fraction of radiated power reaches the rounded surface of the lens above its maximum waist,as de-sired.III.F EED M ODELLING BY M ETHOD OF M OMENTSA.Integral Equation for Microstrip-Fed SlotsFor the purpose of simulating the printed feed,it is assumed that the lens surface is in its far field and does not affectitsFig. 5.Printed feed radiation pattern into in finite dielectric.(a)Twin rectangular slot architecture.(b)Twin arc-slot architecture.behavior.The validity of this assumption was tested by Van der V orst et al.[14].Therefore the feed radiates into an in finite,homogeneous dielectric half-space.A standard MoM technique for layered media has been adopted [15].The ground plane (assumed to be in finite)is extended to cover the slots.These are accounted for withequivalent magneticcurrentson either side.Equivalent electriccurrentsare used to represent the microstrip transmission ing the Green ’s function of a homogeneous dielectric half-space on the side of the lens and that of a grounded dielectric slab on the other side of the ground plane,the following system of integral equations may becast:(4)(5)where on the substrate side of the groundplane,is the Green ’s function for electric fields and electric currentsources,is the Green ’s function for electric fields and magnetic currentsources,is the Green ’s function for magnetic fieldsand electric current sources,andis the Green ’s function for magnetic fields and magnetic current sources;on the dielec-tricside,is the Green ’s function for magnetic fields and magnetic current sources,andis the electric field of the excitation.After expansionofand as a linear combina-tion of triangular and rectangular rooftops,a linear system in the expansion coef ficients is obtained.Therefore the equivalent currents are the direct product of the MoM analysis.Only theFig.6.Longitudinal equivalent magnetic current s M dR along the arc slots.magnetic currents,which radiate into the lens,are considered in the next stage of the analysis,as illustrated in Fig.6.A delta-gap at the microstrip input line is used as excitation.A full account of the method can be found in [16].IV .3-D R AY T RACINGA.Constant Power Ray Tubes for Off-Axis FeedsThe ray tracing PO method most often used to compute lens radiation patterns treats the feed as a point source of radiation located at one elliptical focus,as shown in Fig.2.The radia-tion of this feed into the lens is expressed as a function of the elevation and azimuth angles(and ,respectively),and used to compute the field intensity and phase along the ray paths ex-tending outwards from the feed.These values are used to com-pute equivalent electric and magneticcurrentsand at the lens surface according to Schelkunoff equivalence[7](6)(7)whereand are the transmitted electric and magnetic fields,and is the local unit normal to the lens surface.These equiva-lent surface currents are then used to compute the lens radiation pattern.If the printed feed is centered at one focus of an elliptical lens,simulation of internally re flected rays may bene fit from the ro-tational symmetry of the lens curvature,and the fact that rays re-flected from the elliptical lens surface all pass through the ellip-tical focus.This rotational symmetry for on-axis feeds reduces the complexity of the model by ensuring that each internally re-flected ray path is contained within in a single plane,allowing simpli fied 2-D ray tracing to be used for internal re flections,as in [6].The transmitted electric and magnetic fields are computed at the lens surface for the internally re flected 2nd-order raysjustFig.7.3-D PO ray tracing,oblique view.as they are for the 1st-order rays,and the Schelkunoff principle is applied to replace these with equivalent currents according to (6)and (7).The 2nd-order pattern contribution of internally re-flected rays can then be added to the 1st-order lens patterns by superposition.The full 3-D ray tracing method described in this work goes further by accommodating asymmetrical off-axis feeds,for which the internal re flection contribution can be experimentally observed and used to validate the simulation.Rays subtended from such off-axis feeds are not con fined to a single plane.The 3-D off-axis ray tracing software must therefore dynamically calculate 3-D re flection paths throughout the interior of the lens,as illustrated in Fig.7.The law of power conservation is observed by giving the individual rays a finite cross-section (i.e.,“ray tubes ”as in [17])of constant power (for lossless media),converging or diverging according to re flections from the curved interior lens surfaces between one point of re flection and the next.B.“Filling-in-the-Nulls”Effect of Surface Discretization The 3-D ray tracing method allows an arbitrary feed architec-ture to be modeled as a discretized source current distribution,and the radiation patterns inside and outside the lens calculated by superposition of the patterns of all the current elements.It was found that this superposition method can enhance the fi-delity of null modeling in the simulated patterns.This advan-tage is demonstrated by comparing the patterns computed with superposition applied “inside ”and “outside ”the lens,as in the example described below,and illustrated in Fig.8.The far-field transverse electric fields are given by the Schelkunoff equiva-lence(8)(9)where is thewave-number,is the intrinsic free space waveimpedance,is the distance from the lens surface to the point in the far field,and the radiationintegralsand are de finedas(10)(11)Fig.8.Two-element array patterns computed by the PO method.The higher directivity of the array factor formula-based feed model (effectively,combining the element patterns “inside ”the lens,before computing Schelkunoff equivalent currents at the lens surface)may cause a “filling-in-the-nulls ”effect due to discretization puting the less directive,individual dipole element patterns separately,and superposing the resultant patterns external to the lens,reduces this artifact.where is the surface of thelens,is the distance from the feed current source to the lens surface,and is the anglebetweenand .In the numerical simulation,these radiation integrals are dis-cretized,and thus differ from (10)and (11)by discretization noisetermsand(12)(13)Theterms and appear as a result of the assumption that the transmitted fields remain locally constant over a small subdivi-sion of the lens surface.The greater the directivity of the feed element radiation pattern within the dielectric lens,the stronger will be the discretization noisecontributions and .These noise terms can be effectively reduced by using basis functions of low directivity (e.g.,elemental dipoles)to model the printed feed architecture within the lens,and then applying the superpo-sition of patterns computed from the equivalent currents “out-side,”at the lens surface.The difference is shown in Fig.8,where the PO method is used to compute a simple example two-element array pattern with no focusing (i.e.,the lens is “em-bedded ”in an in finite dielectric of its own permittivity).Mod-eling this array feed as a spatial functionofand according to the array factor (effectively,superposing the dipole element patterns “inside ”the lens),can cause a “filling-in-the-nulls ”ef-fect (since the relatively weak discretization noise is only no-ticeable in the vicinity of pattern nulls),which has occasionally been observed in comparisons between simulation and measure-ment of previous work [14].Computing the radiation patterns for each low-directivity dipole element individually,however,and superposing the lens patterns computed from the resulting equivalent currents at the lens surface (i.e.,superposition “out-side ”the lens),yields sharper nulls in the computed paring Lens Patterns With 3-D Ray TracingThe lens radiation patterns into free space were simulated with the 3-D ray tracing technique described in Section IV-A for the twin rectangular-and arc-slot feed architectures.TheseFig.9.Simulated 30GHz lens radiation patterns from 3-D ray tracing,with comparison of rectangular and arc-slot printed feeds (solid line-co-polarization;dashed curve-cross-polarization).are compared in Fig.9for the case of the extended hemispher-ical lens with radius 25.0mm,cylindrical extension height 8.5mm,andpermittivity.Since the receiver is intended to couple to a quasioptical Gaussian beam,the co-and cross-polar-ization direction vectors are de fined in a flat plane,according to Ludwig ’s cross-polarization de finition #1of [18].The dominant cross-polarization sensitivity appears near the first sidelobes in the diagonal plane,and is some 5dB lower for the arc-slots than for the rectangular slots.The twin arc-slot feed exhibits a 0.25dB improvement in gain with respect to the twin rectangular slots,due to the improved cancellation of the lateral wave in di-agonal directions.The cancellation of the lateral wave also pro-vides a safe place to mount mechanical fixtures around the lens,without affecting its radiation pattern.V .S IMULATION V ERSUS M EASUREMENTparison at 30.2GHz With Centered FeedFig.10compares the simulated patterns to measurement.Both the E-plane and H-plane measured patterns have a 3dB beamwidth of 13,exhibiting the desired rotational sym-metry.Con firming the findings of previous researchers [6]and [12],Fig.10shows that the agreement between simulation and measurement does not exhibit signi ficant changes when 2nd-order internal re flections are included in the patterns for a centered,on-axis feed.Even for a high-permittivity lenswith,the pattern contribution of the internal re flections isoverpowered by the basically co-directional main beam.parison of 3-D ray tracing simulation versus measurement for 30.2GHz twin arc slot lens patterns.(a)E-plane;(b)H-plane.Higher order sidelobes in Fig.10exhibit apparent multi-pathing effects in the measured patterns,and in some directions do not consist of symmetrical or easily identi fiable lobes.These pattern features exhibit levels generally below those ofthe first sidelobe levelsat dB.The measured cross-po-larization sensitivity ofabout dB also seems affected by multipathing,but corresponds with the maximum expected levels in the diagonal plane (shown separately in Fig.11).“Multipathing ”pattern spreading may be related to the finite manufacturing tolerances of the lens curvature (explored in more detail in Section V-C).parison at 30.2GHz With Off-Axis FeedWhen the feed position was offset from the main axis of the lens by 4.5mm,the difference between pattern levels with and without the internal re flection contribution reached 20dB at an elevation angle of 40—a difference likely to be easily distin-guishable in measured patterns,since it is located in a direction of low 1st-order pattern levels,well away from the mainbeamparison of lens pattern D-plane cross-polarization levels.and first sidelobes.This simulated case is compared against measurement in Fig.12,in which the measured pattern at 40agrees with the simulation including 2nd-order rays.At eleva-tion angles of about 35–50,the 1st-order simulated pattern alone does not accurately predict the measured pattern levels.The 20–25dB difference between pattern levels in this direction demonstrates the potential importance of including the contri-bution of internally re flected rays in the simulated patterns,and the utility of 3-D ray tracing when modeling lens antennas with off-axis feeds.C.Lens Fabrication Errors Studied by Frequency Sweep Despite the improved agreement between simulation and measurement at elevation angles between 35–50as a result of including 2nd-order internal re flections in the simulated pat-terns of Fig.12,the same comparison exhibits sidelobe levels in other directions where the bene fits of including internal re flec-tions are less clear.For example,at elevation angles between 20–30in the same Fig.12,the measured pattern corresponds more closely with the 1st-order simulated pattern,in which internal re flections are not included.It would be reasonable to wonder which pattern features are actually caused by 1st-and 2nd-order rays,and which are the less predictable results of multipathing effects and finite fabrication tolerances of the surface of the lens.A method of analysis devised to study this question uses a frequency sweep to approximate phase errors at the lens sur-face.The analysis is based on the principle that the phase of a propagating wave varies more rapidly over a given distance in-side the dielectric of a high-permittivity lens than it does in the free space outside the lens.A local imperfection in the surface of the lens can thus be approximated as a simple phase error in the ray intercepting that point on the lens surface.An easy way to introduce phase errors to the simulated patterns is to slightly vary the frequency of the 3-D ray tracing simulation,without changing the MoM feed currents that were calculated at a fixed,single frequency.This introduces a variable perturbation to the phase of the Schelkunoff equivalent currents calculated at the surface of the lens,providing a series of simulated patterns eachparison of 3-D ray tracing simulation versus measurement for 30.2GHz twin arc slot lens patterns.H-Plane with 4.5mm offset.with its own amount of surface phase error.Since the amount of phase error is proportional to both the frequency shift and the length of each ray,this models a proportional change of lo-cation of the surface of the lens (e.g.,a 1%reduced frequency approximates a lens surface that 1%closer to the printed feed).Such a frequency-swept series of simulated patterns is shown in Fig.13(a).The frequency is swept over a 3%bandwidth about the center frequency of 30.2GHz,in 100MHz steps.While the 1st-order patterns in Fig.13(a)exhibit sidelobes and nulls that are fairly consistent and insensitive to frequency sweep in all directions,the patterns including internal re flections exhibit some elevation angles where the pattern is insensitive to the sweep,and other directions where the sweep causes large variations in the pattern levels.Of particular note-the simulated pattern levels between 35–50(where the measured pattern of Fig.12showed better agreement with simulations including in-ternal re flections)appear stable and insensitive to the sweep,whereas pattern levels between 20–30(where there was less agreement in Fig.12)are sensitive to phase errors at the lens surface and exhibit variations of up to 13dB.This suggests that when internal re flections are included in the 3-D ray tracing sim-ulation,remaining discrepancies between simulation and mea-surement may be due to finite fabrication tolerances of the di-electric lens affecting the pattern measurements.Fig.13(b)illustrates the patterns obtained when the frequency sweep is applied to measured patterns.In this case the goal is again to help identify which sidelobes are sensitive to the sweep and which are not,and to compare these results to the simulated frequency sweep of Fig.13(a).Fig.13(b),exhibits only one sidelobe that is clearly observable and insensitive to the frequency sweep –the first sidelobe to the left of the steeredmain beam,at the elevation angleof.This sidelobe is also shown to be insensitive to frequency sweep in the simulated pat-terns of Fig.13(a).Other sidelobes are more dif ficult to discern clearly in the measured patterns,but fairly insensitive nulls doappear at elevation anglesof,and ,showing good agreement with the sweptsimulations.Fig.13.Investigation of internal re flection modeling with offset feed and sweep analysis.(a)Simulation,29.8–30.7GHz with 0.1GHz step.(b)Measurement,29.8–30.7GHz with 0.1GHz step.(c)30.2GHz measurement superposed on simulated 29.8–30.7GHz sweep with 0.1GHz step.。
