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Beamforming by Left-Handed Extraordinary Transmission Metamaterial Bi- and Plano-Concave Lens at Millimeter-Waves

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Beamforming by Left-Handed Extraordinary Transmission Metamaterial Bi- and Plano-Concave Lens at Millimeter-Waves
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  IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 6, JUNE 2011 2141 Beamforming by Left-Handed ExtraordinaryTransmission Metamaterial Bi- and Plano-ConcaveLens at Millimeter-Waves Miguel Navarro-Cía  , Member, IEEE  , Miguel Beruete, Igor Campillo, and Mario Sorolla Ayza  , Senior Member, IEEE   Abstract— A deep analysis of bi- and plano-concave lens made of stacked subwavelength hole arrays in terms of focus and angularpower distribution is presented. The key difference between theseleft-handed extraordinary transmission lenses (LHET-lenses) andthe classical metallic lenses is based on the fact that contrary tothe latter ones, LHET-lenses work in the cut-off region of the cir-cular waveguide formed by consecutive stacked holes. This leadsto a negative index of refraction, whereas metallic lenses exhibit apositive but less than one index of refraction.  Index Terms— Eggcrate lens, extraordinary transmission,metallic lens, metamaterials. I. I NTRODUCTION I Tiswell-knownthatartificialdielectricmaterialswereusedto alleviate experimental difficulties of performing mea-surements of electromagnetic waves propagating in plasmas inthe 1960s [1]. However, artificial dielectric materials trace backto the 1940s, when they were initially used as beamformers forantenna applications. Pioneering work in metallic lens foundedon parallel metallic plates was done by Kock [2], [3]. The com-bination of line lengths—in the case of TE-lenses as a resultof a different phase velocity relative to free space, whereas inthe case of TEM-lenses due to an increment of the physicallength—and array geometries acts as a phase transformer, con-verting the spherical wave emitted by the feed into a plane waveor, the other way around, focusing the incident plane wave intoa spot. Outstanding in the TE-lens is that concave and convexchange their roles compared with common dielectric lenses be-cause of the positive but less than one index of refraction,,ofthefirstpropagationmodeofthisparallelplateswave-guide.Similar to common dielectric lenses, the artificial metallicTE-lens has a tradeoff between matching and thickness. Whenthe index of refraction is closer to 1, the reflection is lower butthe lens needs to be thicker to achieve focusing. On the other ManuscriptreceivedApril27,2009;revisedApril18,2010;acceptedJanuary15, 2011. Date of publication April 21, 2011; date of current version June 02,2011. This work was sponsored by the Air Force Office of Scientific Research,Air Force Material Command, USAF, under Grant FA8655-10-1-3078.M. Navarro-Cía, M. Beruete, and M. Sorolla are with the Millimeter and Ter-ahertz Waves Laboratory, Universidad Pública de Navarra, Campus Arrosadía,31006 Pamplona, Spain (e-mail: mario@unavarra.es).I. Campillo is with CIC nanoGUNE Consolider, Tolosa Hiribidea 76, 20018Donostia, Spain (e-mail: i.campillo@nanogune.eu).Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TAP.2011.2143651 hand, the closer the is to 0 the thinner the lens will be, butalso the greater the mismatch. These properties arise from thenon-variabilityofthemagneticpermeability , whichisconsid-ered .TE-lenses as reported by Kock were limited to one polariza-tion. Therefore, a further improvement was to combine rows of vertical parallel plates with stacked horizontal plates, yieldingto an eggcrate structure [4], which also provides a more robustassembly.A new surge of interest in artificial effective media recentlyfollowed the so-called split-ring resonator (SRR) [5]. By over-lapping the negative magnetic behavior of this particle with anegative electric permittivity media such as a wire mesh array, arevival of the left-handed media (LHM) proposed by Veselago[6]inthelate1960swasattainable.Amongothers,anintriguingproperty of LHM is negative refraction which comes as a resultof the negative index of refraction (NIR) of these media.The major consequence of the negative refraction regardinglens issue is that a slab of thickness , made of a LH metama-terial with can focus in a focal point the radiation of anotherpointsourcelocatedatadistance fromtheslab,see[6]. Moreover, shaped metamaterial lenses behave as the men-tioned metallic TE-lens does: when the refractive index is ei-ther negative or positive but less than 1, plane wave focusingcan be achieved using plano-concave lenses, in contrast to theconvex profile needed for ordinary dielectrics. Contin-uing with the comparison, it is worth mentioning that for thesame focal length a larger radius of curvature is allowed withnegative index of refraction metamaterials, and thus, metamate-rials lenses are thinner and aberrations may be minimized [7].Furthermore, since the double-resonance scheme for achievingLHM allows, in principle, independent control of the electricpermittivity, , and the magnetic permeability, , it is possibleto obtain maintaining the normalized characteristicimpedance equal to 1, , a feature that hap-pens when both permittivity and permeability are .Therefore, the compromise between low reflection and thick-ness can be overcome by this approach.However, there is more to the story. In a first approximation,an ideal (i.e., lossless) slab of LHM withis actually a perfect lens [8] (reformulated as near-perfect lensbecause of the singularity arisen when and are exactly equalto [9]). And this fact has become one of the main drivingforces in metamaterials along with the quest of a low-loss meta-material in the visible range. The fabrication of metamaterialsusing a pair of subwavelength hole arrays drilled in very thin 0018-926X/$26.00 © 2011 IEEE  2142 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 6, JUNE 2011 metallic plates [10], [11], termed fishnet structure, has becomethe most promising scheme at optical wavelengths. In the mil-limeter wave range, similar results have been observed [12]. In-deed,losseshavebeenfurtherminimizedduetotheuseofprop-erly engineered stacked subwavelength hole arrays under extra-ordinary transmission (ET) [13], [14]. This structure presents negative refraction as it has been shown in a straightforwardpure geometrical and experimental work [15].The next step in ET-based metamaterials has been thecharacterization of the foci of a plano-concave parabolicNIR metamaterial lens operating at millimeter wavelengthscomposed of stacked subwavelength hole arrays patterned inaluminum plates [16]. Moreover, a bi-concave lens has beenreported following the same route [17]. This LH extraordinarytransmission lens, LHET-lens, resembles the eggcrate lens,but the physics is completely different since the former worksin the cut-off region of the circular waveguide determinedby the stacked subwavelength holes, whereas the latter dealswith modes in propagation. Finally, it is worth mentioning thatnegative refraction lenses with constant or gradient index havebeen constructed by employing other metamaterial topologiesor photonic crystals [18]–[20].In this work we provide a further analysis of LHET-lenses,providing a comparison with metallic lenses. The frequencydependence of the focus of both bi- and plano-concaveLHET-lenses is measured. Furthermore, we show the angulardependenceoftheantennaformedbyanopen-endedwaveguidesource and a plano-concave lens. All these results are recordedalong E- and H-plane, - and - planes, respectively.Thus, this work complements previous publications by in-cluding: a comparison with old metallic lenses, highlightingsimilarities and differences; a full band characterization of thefocal chromatic aberration of both plano-concave and bi-con-cave lens, instead of particularizing to a single frequency asin [16], [17]; experimental focus detection employing horn an-tennas and open-ended waveguides for the bi-concave lenses;of importance for theantennas community is that we present forthefirsttimetheangularpowerdistribution(co-andcross-polar)of the plano-concave lens along with a characterization of itsmatching.The outline of this paper is the following: next section iswritten primarily to give basic knowledge about metallic lensesso as to compare their performance with that of LHM lenses; inthe third section the design of a LHET-lens is presented; and,finally, the last three sections are devoted to results for bi- andplano-concave LHET-lenses.II. B ACKGROUND  M ETALLIC  L ENSES Althoughnotalwaysstrictlyaccurate,theeasiestwaytograspthe profile of generic lens is to look at Fig. 1 and apply raytracing analysis. By enforcing equality between electrical pathlengths of a general ray and a ray that crosses the lens throughits principal axis, we get the following relation:(1) Fig.1. Schematicofalensthatfocusesaplanewaveortransformsacylindricalwave-front into a plane wave. where is the distance between the focus and the lens, andand theindexofrefractionoftheair andthelens,respectively. By rearranging terms(2)which is the expression of the locus of an ellipse. The specificprofile depends on the index of refraction. In general, the pro-files obtained are ellipses or hyperbolas, regardless the sign of . A case of particular interest is when (its positivecounterpart is obviously trivial), in which the equation simpli-fies to , and the profile is parabolic. Curiously,in geometrical optics an index of refraction has beenused as a mathematical tool to describe the reflection at the in-terface between air and a metal [21], which particularized toparabolic-shaped mirrors leads to the well-known property of focusing parallel rays into a point called focus of the parabola.The advent of LH metamaterials made it possible the fabri-cation of a true media with index of refraction . There-fore,suchparabolic-profiledmediarefract(i.e.,transmit)therayrather than reflecting it, and thus, allow the possibility to workin transmission. In the particular case when there is impedancematching between the two media , waves can gothrough the surface without any reflection [6], [22]. Finally, for other values of , the general shapes deduced from (1) areellipses and hyperbolas.On the other hand, previous metallic lenses designs donot consider negative parameters and the profiles employedare ellipses or hyperbolas, but never parabolas. For instance,Kock’s lenses can be sorted into two main categories: TEM-and TE-lens. When an electromagnetic wave is polarized per-pendicular to the plates, the waveguide formed by the platessupports a TEM mode. Conversely, when the electric field isparallel to plates, the first propagation mode is .In the case of the TEM-lens [3], the phase velocity in bothmedia, air and parallel plates waveguide, is identical. So, inorder to change the electric path length compared with freespace, plates are bent as meanders or tilted to force an anglewiththedirection ofpropagation, and increasethephysical pathlength.TE lenses [2] provide more degrees of freedom because thephasevelocitybetweenthewaveguideandfreespaceisdifferentand then the electric path is also different. As shown in [2] the  NAVARRO-CÍA  et al. : BEAMFORMING BY LHET METAMATERIAL BI- AND PLANO-CONCAVE LENS AT MILLIMETER-WAVES 2143 index of refraction is between 0 and 1 unlike dielectric mate-rials, whose index of refraction is always positive and greaterthan 1. Again, concave and convex lenses change their roles asa consequence of having .III. D ESIGN OF  S TACKED  S UBWAVELENGTH  H OLE  A RRAYS L ENS Once we know the profile required for a lens whose indexof refraction is , we concentrate our effort in findingthe frequency at which our LHET metamaterial exhibits thisindex, although this does not necessarily involve matching tofree space as stressed in the introduction. To this end, the dis-persion diagram of the infinite structure is firstly calculated byusing the eigenmode solver of the finite-integration time do-main commercial software CST Microwave Studio. Let us takethe unit cell and apply periodic boundary conditions with spe-cific phase shift across the cell in the longitudinal dimension,i.e., the stacking direction, whereas in the cross-sectional di-mensions electric and magnetic walls are employed. The elec-tromagnetic wave propagating in this artificial waveguide, thus,resembles the one of a TEM plane wave [23]. The unit cell pa-rameters are: hole diameter mm, transversal latticeconstants mm and mm, longitudinal latticeconstant mm , and metal thicknessmm. We use doubly periodic subwavelength hole arrays to re-duce the size of the lens [24] assuming the penalty of polariza-tion dependence. We have modeled the metal as a perfect elec-tricalconductor(PEC),whichisareasonableapproximationformetalsatmillimeter-waves.Forthisparticularsetofparameters,ET emerges, for a single plate, around 57 GHz despite the in-dividual hole starts to propagate at 70 GHz, that is, the cut-off frequency of the circular waveguide defined by the hole. Fromthe dispersion diagram, the effective index of refraction is di-rectly calculated through the relation:(3)Both,firstpropagationmodeofthedispersiondiagramanditscorresponding index of refraction are plotted in Fig. 2. Clearly,the mode of the stacked subwavelength hole array structure re-mains below the cut-off frequency of the hole and presents neg-ative slope, i.e., phase velocity opposite to the group velocity.On the other hand, this frequency response predicts an index of refraction at 53.5 GHz. The inherent frequency varia-tioncausesdispersioninthestructurewhichlimitsitsbandwidthand causes chromatic aberration as TE-lenses do. The index of refraction of a TE-lens with cut-off frequency GHz,that is, mm, is also depicted in Fig. 2 for comparisonpurposes. In terms of bandwidth and chromatic aberration usualmetallic lenses may have better performance. A more thoroughanalysis would include the dependence of on the angle of in-cidence; however, this is beyond the scope of this paper, since,as shown below, the conclusions drawn from Fig. 2 match wellwith the measured results. This is due to the fact that refractionis governed by the lens profile and axial backward propagation, Fig.2. (a)Sketchofaplano-concavelens,anddetailsoftheunitcell(inset).(b)Analytical dispersion diagram (green) of an infinite structure made of stackedsub-wavelength hole arrays, whose unit cell parameters are: transversal lattices     mm,      mm, longitudinal lattice constant         mm, holediameter         mm, and metal thickness         mm. Corresponding indexof refraction associated to this LH mode (red), and index of refraction of thefirst propagation TE mode supported by parallel metallic plates with separationbetween plates         mm (cut-off frequency      GHz (diamonds). as happened in our previous wedge experiment [15]. For a morecomplete two-dimensional analysis, see [25] and [26]. Following previous discussion, a bi- and plano-concaveLHET-lens working at 53.5 GHz is constructed with focallength mm , i.e., the focus is centered at a dis-tance of 27 mm from the nearest transversal plane of the lens.The 3D lens is generated by revolving the parabola defined by. We approximate the smooth parabolic shape witha staircase profile with step equal to the size of the cross-sectiondimensions of the unit cell of the subwavelength hole array.This fact limits the sharpness of the foci when we illuminatethe plano-concave lens with a plane wave, as it will be apparentin the experimental results shown below. The bi-concave lensis constructed by simple placing back to back two identicalplano-concave lenses.The whole structure, including the frame for the assembly,has maximum dimensions of 125 mm 115 mm 23 mm (46mm) for the plano-concave (bi-concave) lens, see Fig. 3. These  2144 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 6, JUNE 2011 Fig. 3. Pictures of the prototypes: (left) bi-concave LHET-lens; (right) plano-concave LHET-lens. Unit cell parameters are: transversal lattices      mm,     mm,longitudinal latticeconstant         mm,holediameter        mm, and metal thickness         mm. dimensions along with the design focal length ensure a negli-gible spill over.Finally, experimental results are recorded with an ABmmQuasiopticalVectorNetworkAnalyzerwithintheV-bandofthemillimeter-wave range (45 to 70 GHz). This instrument is basedon a solid state multiplier that generates the millimeter-submil-limeter wavefrequencieswhich aredetected byharmonic mixerheterodynedownconversion.Theparticularexperimentalset-upforeachLHET-lens,namelybi-andplano-concave,isdescribedin each section, since the illumination is changed in each case.IV. R ESULTS OF  B I -C ONCAVE  L ENS Initially, we explorethequestion of point source.Toillustratetheeffectoftheillumination,wecompareresultsbetweencorru-gated horn antenna and open-ended rectangular waveguide illu-mination. To this end, we place firstly the prototype in the pathbetween a transmitting corrugated horn antenna and an iden-tical one acting as receiver. The transmitting antenna launchesa linearly polarized Gaussian beam [27], whose E-field is alongthe large transversal periodicity , of the structure. The dis-tance between the sample and the transmitting antenna is 45mm (experimental position of the focal length [17]), whereasthe receiving antenna scans the transmission coefficient for fourdifferent distances along the principal axis of the lens,mm. Thus, these values along with the dimen-sions of the lens given at the end of the preceding section, implythat we are working in the radiating near-field, sincemm, where is the largest dimension of theparaboloid [28]. The calibration is done by recording the trans-mission coefficient at each distance when the LHET-lens is notin the way.The results are plotted in Fig. 4(a). Frequency filtering be-havior (transmissionpassband centered attheLHET frequency)and Wood’s anomaly (at 60 GHz) aside, it should be notedthat lens effect is not observed since the transmission coeffi-cient does not exceed calibration, although, at least, a relativehighertransmissionisscannedfordistancesaround mm.Quite a different scenario arises if we replace corrugated hornantennas by open-ended rectangular waveguides. This feed pro-vides much less directive illumination, i.e., a broader pattern al-though slightly asymmetric both in E- and H-plane. From nowon, we will use open-ended rectangular waveguides as feeder Fig. 4. Transmitted power recorded when the lens is illuminated by (a) corru-gatedhornantennaand(b)openrectangularwaveguide.Besides,(b)transmittedpower through a doubly periodic subwavelength hole array plate (pink curve).Insets of each figure sketched the experimental set-up used. anddetectorinthissection.Withthisset-up,atransmittedpowerenhancement of 11 dB at 54.2 GHz appears for the conjugatedpoint of the source at the image zone, that is, mm, seeFig. 4(b). Moreover, this enhancement is not only caused bythe collimation associated with the subwavelength hole array[29] but mainly due to the lens profile, since one perforated alu-miniumplatepresentsonly3dBenhancementat54.2GHz,pinkcurve in Fig. 4(b).Subsequently we generate maps of power lateral distribu-tion as a function of frequency and transversal position. Thetransmitting antenna remains fixed at 45 mm, whereas the re-ceiving antenna is moved transversally at three different dis-tances and 75 mm in order to scan the power in bothplanes (H-plane) and (E-plane) planes. In Fig. 5,clear filtering is observed, which is caused by the LHET-lens inthe three cases. Besides, H-plane presents higher sidelobes thanthe E-plane for the whole ET band. This is in accordance withthe simulation results of [17].From panels Fig. 5(b) and (e), the spatial resolution at 54.2GHz determined by the distance between the peak and the loca-tion of the first null can be measured. In the E-plane, the re-solving power is , whereas in the H-plane the value is.RecallingthattheRayleighcriterionforacircularlensof diameter definesthespatialresolutionas ,it is worth noting that the experimental resolution of our pro-posed lens in the E-plane is close to the theoretical value for a  NAVARRO-CÍA  et al. : BEAMFORMING BY LHET METAMATERIAL BI- AND PLANO-CONCAVE LENS AT MILLIMETER-WAVES 2145 Fig. 5. Measured lateral power distribution (dB) as a function of frequencyfor E-plane (a)–(c) and H-plane (d)–(f) at three different distance of the imageplane,      mm (a) and (d);      mm (b) and (e); and      mm (c) and(f). Excitation and reception was done using open waveguides. dielectric lens with a focal length-to-diameterratio similarto our bi-concave LHET-lens, despite the employed non pointsource. On the other hand, the longer transversal length of thesource along maybe thereason ofthe deteriorationof thespa-tial resolution along this axis.V. R ESULTS OF  P LANO -C ONCAVE  L ENS In [16] we showed the asymmetrical focus at 53.5 GHz of our plano-concave LHET-lens and it was tentatively attributedto the larger step in the staircase approximation of the parabolaalong the -direction. Now, we investigate in more detail thisasymmetrical focus vs. frequency, as well as the cross-polar be-havior of the lens.A schematic of the experimental set-up can be found on topof Fig. 6. The LHET-lens is illuminated from its flat back faceby a Gaussian beam [27] generated by the corrugated horn an-tenna. For the dimensions of the experimental set-up, when theGaussian beam reaches the LHET-lens, it is expanded enoughso as to consider the illumination as a good approximation toa plane wave. The normalized lateral power distribution rela-tive to the absolute maximum of each panel is shown in Fig. 6for the two principal cutting planes, that is, E- and H-cutting Fig. 6. Co-polar measurements of lateral normalized power distribution as afunction of frequency for E-plane (left column) and H-plane (right column) atfour different distance of the image plane,            and 75 mm. Excita-tion and reception was done using corrugated horn antennas. planes, on the left and right column of the figure respectively.Thementionedasymmetrybetweenplanesisstillpresentforthe
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