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2572IEEETRANSACTIONSONANTENNASANDPROPAGATION,VOL.51,NO.10,OCTOBER2003ComparativeAnalysisofEdge-andBroadside-CoupledSplitRingResonatorsforMetamaterialDesign—TheoryandExperimentsRicardoMarqués,Member,IEEE,FranciscoMesa,Member,IEEE,JesúsMartel,andFranciscoMedina,SeniorMember,IEEEThispaperdevelopsaquasi-analyticalandself-con-sistentmodeltocomputethepolarizabilitiesofsplitringresonators(SRRs).Anexperimentalsetupisalsoproposedformeasuringthemagneticpolarizabilityofthesestructures.Experimentaldata MARQUÉSetal.:EDGE-ANDBROADSIDE-COUPLEDSRRthemodel.Further,theadvantagesanddisadvantagesofbothtypesofparticleswillbediscussed.Thepaperisorganizedasfollows:SectionIIthoroughlydevelopstheproposedmodelfortheEC-SRRandtheBC-SRR.Theanalysisinthissectionfollowsandcompletesthepreviousanalysisin[20]and[9],whichresultsinaself-consistentandquasi-analyticalmethodforcomputingthefrequencyofresonanceandthepolariz-abilitiesoftheanalyzedparticles.Next,SectionIIIpresentsanewmethodtoobtainexperimentallytheaboveparameters.Specifically,theproposedmethodprovidesthefrequencyofresonanceandthemagneticpolarizabilityoftheSRRs,whicharethemostrelevantparametersformetamaterialdesign.InSectionIV,thetheoreticalresultsarecomparedwithourexper-imentaldataandpreviouslypublishedresults.Oncethemodelhasbeensuccessfullyvalidatedbytheabovecomparisons,itisusedtoevaluatethepotentialofbothEC-SRRsandBC-SRRsformetamaterialdesign.II.THEORYThissectionwilldescribethemodelsforthepolarizabilitiesofbothtypesofSRRsandforthecompositeNMPMandLHMmetamaterialsbasedonthoseparticles.TheEC-SRRandBC-SRRtobeanalyzedareshowninFig.1(a)and(b).Inbothcasestwosimilarsplitringsarecoupledbymeansofastrongdistributedcapacitanceintheregionbetweentherings(theslitsaremeaningfullywiderthanthedistancebetweenthe and ,respectively).Whenatime-harmonicexternal)magneticfieldofangularfrequency isappliedalongthe axisofthesestructures,anelectromotiveforcewillappeararoundtheSRRs.ProvidedthattheelectricalsizeoftheSRRcanbeconsideredsmall,aquasistaticbehaviorisexpected.Withthisquasistaticmodelinmind,itisnotdifficulttoseehowtheinducedcurrentlineswillpassfromoneringtotheotherthroughthecapacitivegapsbetweenthemintheformoffielddisplacementcurrentlines(thecurrentlinescanbeviewedastracingalmostcirculartrips).Therefore,thetotalcurrentintensityflowingonbothringsremainsthesameforanycrosssectionofthestructure(i.e.,itisindependentoftheangularpolarcoordinate).Thewholedevicethenbehavesasan circuitdrivenbyanexternalelectromotiveforce.Thetotalcapacitanceofthis circuitwillbetheseriescapacitanceoftheupperandthelowerhalves(withrespectthelinecontainingtheringgaps)oftheSRRandtheresonancefrequency givenby (1)where istheperunitlength(p.u.l.)capacitancebetweentherings, isthetotalinductanceoftheSRR,and istheaverageradiusoftheconsideredSRR.Theaboveresultfortheresonancefrequency(1)isconfirmedbyamoredetailedelectromagneticanalysis[20]thatalsopro-videsthepolarizabilitiesasafunctionof , andothergeo-metricalandconstitutiveparametersoftheEC-SRR.Thisanal-ysiscanbeextendedtotheBC-SRRbysimplyneglectingthe (a) Fig.1.TwotypesofSRR.(a)EdgecoupledSRR(EC-SRR).(b)BroadsidecoupledSRR(BC-SRR).cross-polarizationeffectsthatwerepresentintheEC-SRRpar-ticle[20].Theresultingequationscanbesummarizedasfol-lows:•FortheEC-SRR: (2) (3) •FortheBCSRR: (5) (6) (7) 2574IEEETRANSACTIONSONANTENNASANDPROPAGATION,VOL.51,NO.10,OCTOBER2003 and arethemagneticandelectricinduced and theexternalfieldsand thepo-larizabilities,whicharefoundtobe[20] (8) (9) (10) (11)with and beingthep.u.l.capacitancebetweentheringswhenthedielectricslabisremoved.In(5)–(11)subscriptsstandforcartesiancomponentsandsuperscriptsformagnetic/magnetic( ),electric/elec-tric( )orelectric/magnetic( )interactionbetweentheparticleandtheexternalfield.Ohmiclossescanbeapproximatelyincorporatedtothemodelbymeansofaneffectiveresistanceoftherings thatcanbeintroducedasanimaginarypartofthetotalinductance givingthefollowingcomplexinductance: whichsubstitutesto intheaboveexpressionsforthepolariz-ThecomputationoftheSRRpolarizabilitiesusingtheaboveexpressionsrequirestheevaluationof , and Assumingthattheringcurvaturehasnegligibleeffect,alargenumberofmethodsforcomputingthep.u.l.capacitancebetweentheringsofboththeEC-SRRandtheBC-SRRcanbefoundintheliterature(see,forinstance,[21]andreferencestherein).Forthepresentpurposesithasbeenfoundthattheclosedexpressionsgivenin[21,Tables2.6and2.7]giveenoughaccuracy.Thesetablesprovidedesignformulasforthephaseconstant andtheimpedance ofamicrostriptransmissionlineandapairofcoupledmetallicstripsonadielectricsubstrate,respectively.Thep.u.l.capacitance, ,ofthesestructuresareobtainedfromthewell-knownexpression: [21].Oncetheyareobtained,thep.u.l.capacitanceofthepairofcoupledstripsdirectlygives oftheEC-SRR.FortheBC-SRRitcanbeobservedthatthep.u.l.capacitanceofapairofcoupledmetallicstripsseparatedbyadielectricsheetofthickness [seeFig.1(b)]ishalfthep.u.l.capacitanceofasinglemicrostriptransmissionlineonasheetmadeofthesamedielectricandthickness ,whichcanbeobtainedfromtheformulasin[21,Table2.6].Theproposedmethodofcomputationisfastand,aswasaforementioned,givesenoughaccuracyintheframeofthepresentmodel.However,betterresultscouldbeobtainedbyusingmorespecificmethodswhichcantakeintoaccounttheeffectsofcurvature(see[22]foradeeperdiscussiononthistopic).Nevertheless,thisimprovementonlywouldhavesenseifotheraspectsofthemodel,suchasthecomputationoftheringinductance,areimprovedtoo.Thecomputationofthetotalinductance oftheSRRsisnotsostraightforward,althoughanappropriateapproximationcanprovideaconsiderablesimplificationwhilekeepingreason-ableaccuracy.Accordingtothepreviousassumptionsonthebe-haviorofthelinecurrentsalongtheSRRs,itcanbeassumedthatthetotalinductanceofbothSRRscanbeapproximatedbytheinductanceofaequivalentringwhoseaverageradiusistheaverageradiusoftheconsideredSRRandwidthequaltothewidth ofeachoriginalring(seeFig.2).Theinductancecanbethencomputedmakinguseofthevariationalexpression ,where isthemagnetostaticenergyforthetotalcurrentintensity supportedbythering.SolvingforthemagnetostaticpotentialintheFourier-Besseldomain,andaftersomealgebraicmanipulations,itisfinallyobtainedthat(seeAp-pendix) (13)where istheFourier-Besseltransformofthecurrentfunc-onthering, definedby (14)with beingtheazimuthalsurfacecurrentdensityonthering.Forpracticalcomputationsithasbeenassumedaconstantvaluefor onthering,thatis, for Thisapproximation,takingintoaccountthevariationalnatureof(13),givesareasonableapproximationfor .Inthiscase,theFourier-Besseltransform, ,isanalyticallyobtainedintermsoftheStruveandBesselfunctions(seeAppendix)andtheintegrationin(13)iscarriedoutnumerically.Abetterap-proximationcouldbeobtainedifamoreaccuratedescription (multiplebasisfunctions)hadbeenemployed.Never-theless,thisnumericalimprovementisnotexpectedtoenhancesubstantiallythequalityoftheapproachsinceotherapproxima-tionsarealreadyinvolvedinthetheory.Finally,asaforementioned,ohmiclossesareintroducedinthemodelbymeansoftheeffectiveresistance, ,oftheSRR.ThiseffectiveresistanceisobtainedbyusingtheequivalentringmodelforthecurrentdistributionontheSRR.Ifaconstant isassumedontheringofFig.2,theresistancecanbeapproxi-matedas if otherwise(16)where istheskindepthand and thethicknessandconduc-tivityofthemetallization,respectively.OncetheEC-SRRandtheBC-SRRpolarizabilitieshavebeenobtainedinaself-consistentway,theycanbeusedinalocalfieldtheoryinordertodeterminethemacroscopicconstitutiveparametersofmediaconsistingofaregulararrayofSRRs.Thislocalfieldtheorymakesuseofthewell-knownLorentztheory[13],[12]anddirectlyappliestoanySRR-basedNMPM.ItcanbealsoappliedtotheanalysisofdiscreteLHMmadebythesuperpositionofanartificialplasmaandaSRR-basedNMPM.Inthislatterapplication,itwillbeimplicitlyassumedthattheconstitutiveparametersoftheLHMmediaarethesuperposi-tionofthoseoftheartificialplasmaandtheNMPM.Although MARQUÉSetal.:EDGE-ANDBROADSIDE-COUPLEDSRR Fig.2.EquivalentsingleringmodelforcomputingofbothSRRs.Inbothisthewidthoftheoriginalringsand theaverageradiusofthewholestructure(seealsoFig.1).thisassumptionisnotalwaysjustified(thegeneralconditionsforthevalidityofthisassumptionarediscussedin[23]),ithasbeenfoundtobeingoodagreementwithexperimentalresults[6]–[9]andnumericalsimulations[6],[10],[11].Subjectedtothisrestriction,theapplicationoftheproposedtheorytodis-creteLHMwouldaccountfortheartificialplasmabysimplyintroducinganadditionaleffectivedielectricsusceptibility, (whichmaybetensorialforanisotropicartificialplasmas).Fora2-Dartificialplasmamadeofaregulararrayofparallelmetallicplatesseparatedadistance ,andforelectricfieldpolarizationandwavepropagationbothparalleltotheplates, isgivenby[1] (17)where playstheroleofaneffectiveplasmafrequency,whichcoincideswiththecutofffrequencyoftheparallel-platewaveg- .For2-and3-Darraysofwires,theexpressionsfor and maybecomemorecomplicated[1],[3],[4].III.EXPERIMENTALAtthefrequenciesofinterestformetamaterialdesign(i.e.,nearthefirstresonance)thedominanteffectinboththeEC-SRRandBC-SRRparticlesturnsouttobethemagneticpolariz-abilityaccountedforbythe parameter.Thispolarizabilitygivesrisetoastrongdiamagneticbehaviornearandaboveres-onance[4],[20],thusplayinganessentialroleinthedesignofNMPMandLHM.AnotherrelevanteffectintheEC-SRRpar-ticleiscross-polarization,whichresultsinabianisotropicbe-haviorofthoseNMPMandLHMdesignedusingthisparticle.Sincetheexperimentalevidenceofthislattereffecthasbeenal-readydiscussed[20],thissectionwillfocusspecificallyontheexperimentaldeterminationofthemagneticpolarizabilityoftheEC-SRRandBC-SRR(thislatterparticledoesnotexhibitmag-netoelectriccoupling).Accordingto(8),themainparameterstobedeterminedaretheresonancefrequency, ,andthenonres-onantfactor Asimplewayofdeterminingtheresonancefrequencycon-sistsinplacingtheresonatorinahollowwaveguideataloca- Fig.3.SchemeoftheexperimentalsetupformeasuringtheSRRsmagneticpolarizabilities.TheSRRisplacedinsideacircularaperturemadeinametallicscreenplacedintherectangularwaveguide.Forthereportedexperimentsthediameteroftheaperturewas6mm. Fig.4.Atypicalplotofthe measuredusingtheexperimentalsetupofFig.3.AnEC-SRRwasusedinthisparticularexperiment.EC-SRRdimensions 2.6mm,0.6mm,0.2mm.Substrateofthicknessmmandpermittivity ,withmetallizationsmadeofcopperwith=35mandconductivity tionwheretheelectromagneticfieldcouldexciteitandtomea-suretheinthetransmissioncoefficient(or,alternatively,theinthereflectioncoefficient)[24].Infact,thismethodonlyprovidestheresonancefrequencyofaninfinitearrayofperi-odicallyrepeatedparticles,whichisshiftedovertheresonancefrequencyofasingleparticle.However,ifthedimensionsofthewaveguidecrosssectionareseveraltimesthoseoftheSRR,thisshiftisnotveryimportant[24].Thistechnique,however,doesnotprovideanyinformationabout .Inordertoobtainamea-sureof ,theexperimentalsetuphasbeenmodifiedinthewayshowninFig.3.TheSRRisplacedinsideasmallcircularaper-turepracticedinametallicscreenlocatedinthemiddleoftherectangularwaveguide.Theinputandtheoutputaretwocom-mercialcoaxialtorectangularwaveguidetransitions.Whenthe forthisstructureismeasured,typicalplotsasthatshowninFig.4areobtained.Thecorrespondstotheresonancefrequency(1),forwhichthemagneticdipoleoftheparticlebe-comesamaximum.Theinthe canbeexplainedbyusingBethe’stheoryofdiffractionthroughsmallapertures[25],[26].Followingthistheory,theaperturewithouttheSRRwillradiatetowardtheoutputasanequivalentmagneticdipoleof (18)where istheradiusoftheapertureand themagneticfieldinthewaveguideiftheaperturewasnotpresent.SincetheSRRisparamagneticbelowresonance[see(8)],theremustbecertain 2576IEEETRANSACTIONSONANTENNASANDPROPAGATION,VOL.51,NO.10,OCTOBER2003frequency atwhichtheinduceddipoleintheSRRcancelsouttheequivalentdipoleoftheaperturewithouttheSRR(18).Atthisfrequency,theradiationfromtheSRR-loadedaperturetowardtheoutputwillbesubstantiallyreduced,thusoriginatingtheobservedinthe plot.Byequatingthemagnitudesofthemagneticdipolesin(2),(5),and(18), in(8)canbedetermined.Itshouldbenotedthattheexternalmagneticfluxdensityatthecenteroftheapertureishalftheexternalmagneticfluxdensitythatwouldbeinthewaveguideiftheaperturewasnotpresent[26].Takingthisfactintoaccount,thefinalresultfor isgivenby thus,providingamethodtoobtaintheexperimentalvalueof fromthemeasurementoftheresonancefrequency andthefrequency Finallyitshouldbementionedthatanaccuratemeasurement and bymeansoftheaboveprocedurerequirestheaperturediametertobechoseninsuchawaythattheresonancefrequencyisnotsubstantiallyaffectedbytheSRR-aperturecoupling.Measurementsoftheresonancefrequency,madebyplacingtheSRRinthewaveguidewithoutthemetallicscreen,haveshownthatbothmeasurementsagreewithanerrorlessthan1%forthechosendiameterofthescreen(6mm)andfortheparticlesizesconsideredintheexperiments(seeSectionIV).IV.NUMERICALANDXPERIMENTALESULTSInordertoverifytheaccuracyoftheproposedtheoreticalmodel,asetofEC-andBC-SRRswereprintedonacommer-cialmetallizedsubstratewithdielectricconstant ,thickness 0.49mmandcoppermetallizationswiththickness mandconductivity S/m.Theresonancefrequencyand weremeasuredusingthetechniquereportedinSectionIIIforeachSRR.Thetheoreticalresultswerecom-putedusingthemodelreportedinSectionII.Thecomputedthe-oreticalresultsandthemeasuredresultsareshowninFig.5(a)to(d).Differentvaluesoftheexternalradius andstripwidth oftheSRRswereconsidered.Anacceptableagreementcanbeobservedbetweenthetheoreticalandexperimentalresults.Thisagreementshowsthattheproposedmodel,despiteofitssimplicity,canaccountforthedescriptionofthemainEC-andBC-SRRscharacteristics.Disagreementsmainlyoccurforthehighestvaluesof .Thisfactissomewhatexpectedsincethedescriptionoftheringresponsetoanexternalexcitationintermsoftheringinductanceislessaccurateforthesecases.Thebehavioroftheresonancefrequencyand withrespecttotheexternalradiusissimilarforboththeEC-andtheBC-SRR.However,thedependencewiththestripwidth isquitedif-ferent.TheBC-SRRseemstobealmostinsensitiveto theresonancefrequencyoftheEC-SRRincreaseswith .Thisfactcanbeexplainedconsideringthatthep.u.l.capacitanceandtheinductancehaveoppositevariationswith fortheBC-SRR.However,thep.u.l.capacitanceisratherinsensitiveto forthe (a) (b) Fig.5.Measured(dotsanddiamonds)andcomputed(solidlines)valuesfortheresonancefrequency, =! ,and forsomeEC-SRRandBC-SRRprintedonadielectricsubstratewiththesamespecificationsasinFig.4.:0.49mm,0.2mm(forEC-SRR), ,withmetallizationsmadeofcopperwithathickness=35mandconductivity TheremainingEC-SRRandBC-SRRdimensionsaregivenineachplot.Fromthestandpointofmetamaterialsdesign,themostim-portantparametercharacterizingtheSRRsispossiblythereso-nancefrequency.Thisparametergivestheelectricalsizeoftheparticlewithintheregionofinterest,whichisthemainlimita-tionfortheaccuracyofthecontinuous-mediumdescriptionofthemetamaterial.TheSRRsanalyzedinFig.5(a)to(d)havean MARQUÉSetal.:EDGE-ANDBROADSIDE-COUPLEDSRR Fig.5.(.)Measured(dotsanddiamonds)andcomputed(solidlines)valuesfortheresonancefrequency, =! ,and forsomeEC-SRRandBC-SRRprintedonadielectricsubstratewiththesamespecificationsasinFig.4.:0.49mm,0.2mm(forEC-SRR), ,withmetallizationsmadeofcopperwithathickness=35mandconductivity S/m.TheremainingEC-SRRandBC-SRRdimensionsaregivenineachplot.electricalsizeofapproximatelyatenthofthefree-spacewave-lengthatresonance.Thesevaluesareofthesameorderthanthosereportedinpreviousdesigns[6]–[9],allofthembeingclosetothelimitofapplicabilityofthecontinuous-mediumap-proach.Therefore,itisveryconvenienttoexplorethefeasibilityofdesigningSRRswithsmallerelectricalsizesatresonance.AssumingthatthetotalsizeoftheSRRremainsapproximatelyconstant( ),(1)saysthat isgovernedbytwopa-rameters:thetotalinductance andthep.u.l.capacitance Ofthesetwoquantities,thep.u.l.capacitanceismoreeasilytunable.Changesin fortheBC-SRRcanbeachievedbyvaryingthestripwidth thesubstratethickness andthedi-electricconstantofthesubstrate, .However,Fig.5(d)showsthatvariationsof with arecancelledoutbytheassoci-atedvariationsof .Therefore,thereremaintwomainparam-etersfortuning: and .FortheEC-SRR, ismainlygov-ernedbytheringsspacing, ,andthepermittivity, ;twopa-rametersthatdonotaffect .Fig.6(a)and(b)showsthevari-ationoftheresonancefrequencyandthenormalizedelectricalsize(definedas ,with beingthefreespacewave-lengthatresonance)fortheEC-SRRandtheBC-SRRwithre-specttotheaboveparameters.InFig.6(a)thetuningparameteristhespacingbetweentherings(withthesubstratethicknessbeingconstant, 0.4mm)whereasthetuningparameterinFig.6(b)isthesubstratethickness.ItcanbeseenthatmuchsmallerelectricalsizescanbeachievedusingBC-SRRsinsteadofEC-SRRs:aparticlediameterofabout canbeob-tainedemployingBC-SRRsprintedonacommercialsubstrate ofapproximately mat10GHz.Evensmallerelectricalsizescouldbeachievedwiththinnersubstrates(suchasoxidelayers)and/orhigherdielectricpermittivitysubstrates(asferroelectricsubstrates,forinstance).Conversely,theelec-tricalsizeoftheEC-SRRsremainsalmostconstantforsmallvaluesofthespacingbetweentherings.Moreover,veryintenseelectricfieldsshouldappearattheringedgesinEC-SRRswithverysmallspacing,whichmaycausehighlossesand/ordielec-tricbreakdown.SincetheelectricfielddistributionissmootherinBC-SRRs,theseeffectsareexpectedtobelessimportantforthisparticle.Inthefollowing,theproposedlocalfieldtheoryisappliedtothecomputationofthedispersioncurvesofsomeNMPMsandLHMs.Thus,theNMPMandtheLHManalyzedin[6]arenewlyanalyzedusingourmodelinFig.7.Forcomparisonpur-poses,theresultsobtainedusingthecommercialelectromag-neticsolverMAFIAreportedin[6]arealsoshown.NoticethatthereisamismatchbetweenthefrequencypassbandsfortheLHMandtheNMPMthatappearsbothinnumericalcalcula-tionsandexperiments.TheexperimentalvaluesfortheLHM andtheabovemismatch extractedfrom[6,Fig.3],areequallyshowninthefigure.(Asisexplainedin[20],the mismatchiscloselyrelatedtothecrosspo-larizationeffectsappearingintheEC-SRR.)Thisplotillus-tratestheaccuracyoftheproposedsimpletheory:ourresultsfrtheleft-handedpassbandappearclosertotheexperimentsthanthoseobtainedin[6]byusingacommercialfull-wavesolver.Aswasmentioned,theuseofBC-SRRsforbuildingupmeta-materialsallowsforasignificanreductionintheelectricalsizeoftheunitcell.Thisfactwillbeillustratedinthefollowing.Thedispersioncharacteristicofanisotropic2-DLHM(seetheinsetinthefigure)madeusingsmall-sizeBC-SRRsisshowninFig.8.ThereportedLHMisdesignedfollowingthetheoryreportedin[9].Inthisdesign,theartificialplasmaissimu-latedbyanarrayofparallelmetallicplates[1]althoughforpracticalsimulationsonlyapairofplatesisneeded.Thisarraysimulatesalosslessplasmaforwavepropagationandelectricfieldpolarizationbothparalleltotheplates[1].Forthiswavepolarization,theeffectivedielectricsusceptibilityoftheartifi-cialplasmaisgivenby(17).ThediscreteNMPMismadebyplacingBC-SRRssymmetricallybetweentheplates.SincetheBC-SRRsresponseisisotropicforexternalmagnetizationper-pendiculartotheBC-SRRplane,theelectromagneticresponseofthedeviceisthenisotropicinthisplaneandforthisspecificpolarizationoftheincidentwave[9].Asitwastheoreticallyandexperimentallyshownin[9],theproposeddevicebehavesasanLHM(fortheappropriatewavepolarization)incertainfrequencybandthatliesabovetheresonancefrequencyoftheBC-SRR.Thisfacthasbeenadvantageouslyused,alongwiththedataprovidedinFig.6(b),todesignaLHMwhoseunitcellhasaverysmallelectricalsize.Thus,theresultsshowninFig.8havebeencomputedforBC-SRRswithexternalradius 0.6mmandringwidth 0.2mmprintedonadi-electricsubstrateof andthickness 0.01mm.Forthechosenlatticeparameter 1.5mmthedeviceshowsanLHMpassbandaround3.3GHz.Noticethatwithinthispass-bandthenormalizedelectricalsizeoftheunitcell( , thefree-spacewavelength)isapproximately0.015.Thiselec-tricalsizeisoneorderofmagnitudesmallerthanthepreviouslyreportedones[6],[7],[9].ThisfactprovestheusefulnessoftheproposedBC-SRRsfordesigningdiscreteLHMwithsmallsizeunitcells;namely,discreteLHMthatcanbeaccuratelyde-scribedbyacontinuousmediumapproach.V.CAself-consistentquasi-analyticalmodelforthepolarizabili-tiesofEC-SRRandBC-SRRhasbeenpresented.Theaccuracy 2578IEEETRANSACTIONSONANTENNASANDPROPAGATION,VOL.51,NO.10,OCTOBER2003 (a) Fig.6.Resonancefrequencyandnormalizedelectricalsizeforseveral(a)EC-SRRand(b)BC-SRRwiththesameexternalradius 0.6mmandringwidth0.2mm,printedonseveralsubstrates.ofthismodelhasbeenvalidatedbyexperiments.Inparticular,anexperimentalsetupformeasuringtheresonancefrequencyandthemagneticpolarizabilitiesofbothtypesofSRRshasbeenproposedandrealized.SincetheSRRpolarizabilitieshavenotbeenpreviouslymeasured,theproposedexperimentalmethodforthecharacterizationofSRRsisoneofthemostrelevantcon-tributionsofthiswork.Theobtainedexperimentalresultshavebeensatisfactorilycomparedwiththeoreticalresultscomputedfollowingtheproposedmodel.Startingfromtheproposedmodel,andmakinguseofthewell-knownLorentztheory,alocalfieldmodelforSRR-basedNMPMhasbeendeveloped.Thismodelisalsousefulforcom-putingthedispersioncharacteristicsofLHMmadebyasuper-positionoftheaforementionedSRR-basedNMPMandwire-orparallelplate-madeartificialplasmas.ThislocalfieldmodelhasbeentestedbycomputingthedispersioncharacteristicsofsomepreviouslyreportedLHM,whichshowanacceptableagreementwithpreviouslyreportedtheoreticalandexperimentaldata.Thesimplicityandsmallnumericaleffortinvolvedintheproposedlocalfieldmodelmakesthisapproachtobeausefulandeffi-cientalternativefortheanalysisanddesignofdiscreteNMPMandLHM.Asanadditionaladvantageofthisapproach,itmay MARQUÉSetal.:EDGE-ANDBROADSIDE-COUPLEDSRR Fig.7.DispersioncurvesfortheLHMandNMPMmediareportedin[6].Theresultsreportedin[6,Fig.2(a)and(c)]arealsoshownforcomparisonpurposeTheexperimentalvaluesfortheLHMpassband, ,andforthemismatchbetweentheLHMandtheNMPMpassbands, ,obtainedfrom[6,Fig.3]areindicatedintherightaxisofthefigure. Fig.8.Dispersioncurvesforthe2-DisotropicLHMandNMPMshownintheinset.Thelatticeparameteris1.5mmandtheBC-SRRdimensionsare 0.6mmand0.2mm.TheBC-SRRareprintedonasubstrateofthickness0.01mmand=10 providefurtherphysicalinsightonsomecharacteristicsofthesemetamaterials(forinstance,intheanalysisofbianisotropicef-fectsinEC-SRRmadeLHM).Oncetheaccuracyoftheproposedmodelhasbeenchecked,acomparativeanalysisofbothEC-andBC-SRRhasbeencarriedout.Ithasbeenfoundthatthislatterparticlehastwomainad-vantagesovertheEC-SRR:isotropyintheplaneofthestructure(fornormallyincidentmagneticfield)andapotentiallymuchsmallerelectricalsize.Thefirstpropertycanbeusefulinde-signingisotropicLHMandNMPMforexperimentationinneg-ativerefractiveindexandrelatedphenomena.Thesecondprop-ertyisusefulfordesigningdiscretemetamaterialswithelectri-callysmallunitcells.Sincethesuitabilityofthecontinuous-mediumapproachcriticallydependsontheelectricalsizeoftheunitcell,thispropertymayresultinsubstantialimprovementsinthedesignofnewcontinuous-likemetamaterials. 2580IEEETRANSACTIONSONANTENNASANDPROPAGATION,VOL.51,NO.10,OCTOBER2003OMPUTATIONOFTHEOTALNDUCTANCE Letusassumeacurrent ontheringofFig.2.Sincethereisnofieldand/orcurrentdependenceontheazimuthalcoordinate themagnetostaticenergycanbecomputedas (20)where istheazimuthalcomponentofthevectormagneticpotential.Integratingbypartsandusingthat ,where isthe componentofthemagnetostaticfieldand thepartialderivativewithrespect ,itisfoundthat (21)where isdefinedby(14)and(15).Sincethecurrentsarerestrictedtothe plane, canbederivedfromascalarmagneticpotential ThisscalarmagneticpotentialmustsatisfyLaplace’s ,subjectedtothefol-lowingboundaryconditions: ; and TakingtheFourier-Besseltransform,whichisdefinedas theaboveproblemisanalyticallysolvedfor ,whichisfoundtobe wheretheupper(lower)signstandsfor ( ).Equa-tion(13)isobtainedafterintroducing into(21),using(22)and(23)andmakinguseoftheParsevaltheoremandtherelation Finally,if isassumedtobegivenby(14)–(15),aftersomealgebraicmanipulations,theringinductancecanbeob-tainedasthefollowingintegral: (24)where , ,andfunction isdefined (25)with and beingthe thorderStruveandBesselfunctions,respectively.[1]W.Rotman,“Plasmasimulationbyartificialdielectricsandparallel-platemedia,”IRETrans.AntennasPropagat.,vol.AP-10,pp.82–95,Jan.1962.[2]J.B.Pendry,A.J.Holden,W.J.Stewart,andI.Youngs,“Extremelylowfrequencyplasmonsinmetallicmesostructures,”Phys.Rev.Lett.,vol.76,pp.4773–4776,June1996.[3]J.M.Pitarke,F.J.GarcíaVidal,andJ.B.Pendry,“Effectiveelectronicresponseofasystemofmetalliccilinders,”Phys.Rev.B,vol.57,pp.15261–15266,June1998.[4]J.B.Pendry,A.J.Holden,D.J.Ribbins,andW.J.Stewart,“Magnetismfromconductorsandenhancednonlinearphenomena,”IEEETrans.Mi-crowaveTheoryTech.,vol.47,pp.2075–2084,Nov.1999.[5]V.G.Veselago,“Electrodynamicsofsubstanceswithsimultaneouslynegativeelectricalandmagneticproperties,”Sov.Phyis.USPEKHI,vol.10,pp.509–517,1968.[6]D.R.Smith,W.J.Padilla,D.C.Vier,S.C.Nemat-Nasser,andS.Schultz,“Compositemediumwithsimultaneouslynegativeperme-abilityandpermittivity,”Phys.Rev.Lett.,vol.84,pp.4184–4187,May[7]R.A.Shelby,D.R.Smith,S.C.Nemat-Nasser,andS.Schultz,“Mi-crowavetransmissionthroughatwo-dimensional,isotropic,left-handedAppl.Phys.Lett.,vol.78,pp.489–491,Jan.2001.[8]R.Marqués,J.Martel,F.Mesa,andF.Medina,“Left-handedmediasimulationandtrnsmissionofEMwavesinsubwavelengthsplit-ring-resonator-loadedmetallicwaveguides,”Phys.Rev.Lett.,vol.89,pp.13901(1)–13901(4),Oct.2002. ,“Anew2-Disotropicleft-handedmetamaterialdesign:Theoryandexperiment,”MicrowaveOpt.Tech.Lett.,vol.36,pp.405–408,Dec.[10]T.Weiland,R.Schumann,R.B.Greegor,C.G.Parazzoli,A.M.Vetter,D.R.Smith,D.C.Vier,andS.Schultz,“Abinitionumericalsimula-tionsofleft-handedmetamaterials:Comparisonofcalculationsandex-J.Appl.Phys.,vol.90,pp.5419–5424,Nov.2001.[11]P.MarkosandC.M.Soukoulis,“Numericalstudiesofleft-handedma-terialsandarraysofsplitringresonators,”Phys.Rev.B,vol.65,pp.036622(1)–036622(8),2002.[12]A.Sihvola,ElectromagneticMixingFormulasandApplica-.London,U.K.:IEE,1999.[13]R.E.Collin,FieldTheoryofGuidedWaves.NewYork:IEEE,1991.[14]Y.N.Kazantsev,M.V.Kostin,G.A.Kraftmakher,V.I.Pomonarenko,andV.V.Shevshenko,“Artificialparamagnetic,”J.Commun.Tech.Electron.,vol.39,pp.78–81,1994.[15]M.V.KostinandV.V.Shevshenko,“Artificialmagneticsbasedondoublecircularelements,”inProc.Chiral’94,3rdInt.WorkshoponChiral,Bi-IsotropicandBi-AnisotropicMedia,F.MariotteandJ.-P.Parneix,Eds.,Perigueux,France,May1994,pp.50–56.[16]A.J.BahrandK.R.Clausing,“Anapproximatemodelforartificialchiralmaterial,”IEEETrans.MicrowaveTheoryTech.,vol.42,pp.1592–1599,Dec.1994.[17]F.Mariotte,S.A.Tretyakov,andB.Sauviac,“Isotropicchiralcompositemodeling:Comparisonbetweenanalytical,numerical,andexperimentalMicrowaveOpt.Tech.Lett.,vol.7,pp.861–864,Dec.1994.[18]S.A.Tretyakov,F.Mariotte,C.R.Simovski,T.G.Kharina,andJ.-P.He-AnalyticalAntennaModelforChiralScatterers:CamparisonwithNumericalandExperimantalData,vol.44,pp.1006–1014,July1996.[19]M.M.I.SaadounandN.Engheta,“AreciprocalphaseshifterusinganovelpseudochiralorMicrowaveOpt.Tech.Lett.,vol.5,pp.184–188,Apr.1992.[20]R.Marqués,F.Medina,andR.Rafii-El-Idrissi,“Roleofbianisotropyinnegativepermeabilityandleft-handedmetamaterials,”Phys.Rev.B,vol.65,pp.144440(1)–144440(6),2002.[21]I.BahlandP.Bhartia,MicrowaveSolidStateCircuitDesign.NewYork:Wiley,1988.[22]F.TefikuandE.Yamashita,“Capacitancecharacterizationmethodforthick-conductormultipleplanarringstructuresonmultiplesubstrateIEEETrans.MicrowaveTheoryTech.,vol.40,pp.1894–1902,Oct.1992.[23]D.R.Smith,W.J.Padilla,D.Vier,R.Shelby,S.Nemat-Nasser,N.Kroll,andS.Schultz,“Left-handedmetamaterials,”inCrystalsandLightLocalizationinthe21stCentury,C.M.Soukoulis,Ed.Dordrecht,TheNetherlands:Kluwer,2001.[24]P.Gay-BalmazandO.J.F.Martin,“Electromagneticresonancesinin-dividualandcoupledsplit-ringresonators,”J.Appl.Phys.,vol.92,pp.2929–2936,Sept.2002.[25]H.A.Bethe,“Theoryofdiffractionbysmallholes,”Phys.Rev.,vol.66,pp.163–182,Oct.1944.[26]D.M.Pozar,MicrowaveEngineering.NewYork:Wiley,1998. MARQUÉSetal.:EDGE-ANDBROADSIDE-COUPLEDSRRRicardoMarqués(M’95)wasborninSanFernando,Cádiz,Spain.HereceivedthePh.D.degreefromtheUniversidaddeSevilla,Spain,in1987.HeiscurrentlyanAssociateProfessorwiththeDepartamentodeElectrónicayElectromagnetismo,UniversidaddeSevilla.Hismainscientificactivityhasbeenformanyyearsinthecomputer-aideddesignofplanartransmissionlinesandcircuitsatmicrowavefrequencies,withemphasisintheinfluenceandappli-cationsofcomplexmedia,suchasanisotropicdielectrics,magnetizedferritesandplasmas,aswellasbi(iso/aniso)tropicmaterials.Heisalsointerestedintheelectromagneticanalysisandcharacterizationofdiscretemetamaterials,in-cludingbianisotropicandLHM.Prof.Marqueshasbeenand/orisareviewerfortheIEEETRANSACTIONSICROWAVEHEORYANDECHNIQUES,theIEEETRANSACTIONSONNTENNASANDROPAGATION,andotherscientificandtechnicaljournalsandFranciscoMesa(M’02)wasborninCádiz,Spain,onApril1965.HereceivedtheLicenciadodegreeinJune1989andtheDoctordegreeinDecember1991,bothinphysics,fromtheUniversityofSeville,Spain.HeiscurrentlyanAssociateProfessorwiththeDepartmentofAppliedPhysic1,UniversityofSeville.Hisresearchinterestfocusonelectromagneticpropagation/radiationinplanarstructureswithgeneralanisotropicmaterialsandmetamaterials.JesúsMartelwasborninSeville,Spain,in1966.HereceivedtheLicenciadoandDoctordegreesinphysicsfromtheUniversityofSeville,in1989and1996,respectively.Since1992,hehasbeenwiththeDepartmentofAppliedPhysicsII,Univer-sityofSeville,where,in2000,hebecameanAssociateProfessor.Hiscurrentresearchinterestisfocusedonthenumericalanalysisofplanartransmissionlines,themodelingofplanarmicrostripdiscontinuities,thedesignofpassivemicrowavecircuits,microwavemeasurements,andartificialmedia.FranciscoMedina(M’90–SM’01)wasborninPuertoReal,Cádiz,Spain,inNovember1960.HereceivedtheLicenciado(withhonors)andDoctordegreesfromtheUniversityofSeville,Seville,Spain,in1983and1987,respectively,bothinphysics.From1986to1987,hespenttheacademicyearattheLaboratoiredeMi-croondesdel’ENSEEIHT,Toulouse,France,thankstoascholarshipoftheMinisteriodeEducaciónyCiencia,Spain,andMinistèredelaRechercheetlaTechnologie,France.From1985to1989,hewasanAssistantProfessorwiththeDepartmentofElectronicsandElectromagnetism,UniversityofSeville,andsince1990,hehasbeenanAssociateProfessorofelectromagnetism.HeisalsocurrentlyHeadoftheMicrowavesGroup,UniversityofSeville.Hisresearchinterestincludesanalyticalandnumericalmethodsforguidance,resonantandradiatingstructures,passiveplanarcircuits,periodicstructures,andtheinflu-enceofanisotropicmaterialsonsuchsystems.Heisalsointerestedinartificialmediamodelinganddesign.Dr.MedinawasamemberoftheTechnicalProgrammeCommittees(TPC)ofthe23rdEuropeanMicrowaveConference,Madrid,Spain,1993,theIS-RAMT’99,Málaga,Spain,1999,andmemberoftheTPCoftheMicrowavesSymposium2000,Tetouan,Morocco,2000.Hewasco-organizerofthework-NewTrendsonComputationalElectromagneticsforOpenandBoxedMi-crowaveStructures,Madrid,Spain,1993.HeisontheeditorialboardoftheIEEETRANSACTIONSONICROWAVEHEORYANDECHNIQUESandactsasreviewerforotherIEEE,InstitutionofElectricalEngineers(IEE),U.K.,andAmericanPhysicsAssociationjournals.Hewastherecipientoftwoscholar-