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APRIL 2005 VOLUME53 NUMBER 4 IETMAB (ISSN0018-9480) PART II OF TWO PARTS SPECIALISSUEONMETAMATERIALSTRUCTURES,PHENOMENA,ANDAPPLICATIONS GuestEditorial .. ... ... ... ... ... ... ... ... .... ..... ... .... ... ... ... ... ... ... ..T.ItohandA.A.Oliner 1413 PAPERS Analysis andModeling Homogenizationof3-D-ConnectedandNonconnectedWireMetamaterials... .....M.G.SilveirinhaandC.A.Fernandes 1418 FundamentalModalPropertiesofSurfaceWavesonMetamaterialGroundedSlabs .. ... ... ... ... ... ... ... .... .. ... ... .... ... ... .. P.Baccarelli,P.Burghignoli,F.Frezza,A.Galli,P.Lampariello,G.Lovat,andS.Paulotto 1431 RefractionLawsforAnisotropicMediaandTheirApplicationtoLeft-HandedMetamaterials. ... ... ... ... ... .... .. ... ... .... ... ... ... ... ... ... ... ... ... .T.M.Grzegorczyk,M.Nikku,X.Chen,B.-I.Wu,andJ.A.Kong 1443 Equivalent-Circuit Models for Split-Ring Resonators and Complementary Split-Ring Resonators Coupled to Planar TransmissionLines ... ... ... ... ... ... ... ...... ... ... .... ... ... ... .. J.D.Baena,J.Bonache,F.Martín, R.MarquésSillero,F.Falcone,T.Lopetegi,M.A.G.Laso,J.García–García,I.Gil,M.FloresPortillo,andM.Sorolla 1451 EfficientModelingofNovelUniplanarLeft-HandedMetamaterials.. ... ... ... ... ... ... ... ... ... ... ... .... .. ... ... .... ... ... ... ... ... ... ... ... ... ... .Y.Guo,G.Goussetis,A.P.Feresidis,andJ.C.Vardaxoglou 1462 Numerical Analysis Propagation Property Analysis of Metamaterial Constructed by Conductive SRRs and Wires Using the MGS-Based Algorithm .... ... ... ... ... ... ..... .... ... ... ... ... .... .. H.-Y.Yao,W.Xu,L.-W.Li,Q.Wu,andT.-S.Yeo 1469 FDTDStudyofResonanceProcessesinMetamaterials. ... ... .... ... ... ... ... ... ... ... ... ... ... ... .... .. ... ... .... ... ... ... ... ... ... ... ..E.A.Semouchkina,G.B.Semouchkin,M.Lanagan,andC.A.Randall 1477 PeriodicFinite-DifferenceTime-DomainAnalysisofLoadedTransmission-LineNegative-Refractive-IndexMetamaterials .. ... ... .... ... ... ... ... ...... ... ... ... ... ... ... .... . T.Kokkinos,C.D.Sarris,andG.V.Eleftheriades 1488 ModelingofMetamaterialsWithNegativeRefractiveIndexUsing2-DShuntand3-DSCNTLMNetworks .. ... .... .. ... ... .... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ...P.P.M.So,H.Du,andW.J.R.Hoefer 1496 (ContentsContinuedonBackCover) (ContentsContinuedfromFrontCover) Experimental SimulationofNegativePermittivityandNegativePermeabilitybyMeansofEvanescentWaveguideModes—Theoryand Experiment ... ...... ...J.Esteban,C.Camacho-Peñalosa,J.E.Page,T.M.Martín-Guerrero,andE.Márquez-Segura 1506 EffectiveElectromagneticParametersofNovelDistributedLeft-HandedMicrostripLines ... ... ... ... ... ... .... .. ... ... .... ... ... ... ... ... ... ... ... ... ... ... .... ... ... . S.-G.Mao,S.-L.Chen,andC.-W.Huang 1515 ExperimentalRealizationofaOne-DimensionalLHM–RHMResonator. ... ... ... ... ... ... ... ... ... ... .... .. ... ... .... ... ... ...Y.Li,L.Ran,H.Chen,J.Huangfu,X.Zhang,K.Chen,T.M.Grzegorczyk,andJ.A.Kong 1522 Super-CompactMultilayeredLeft-HandedTransmissionLineandDiplexerApplication .... ..Y.Horii,C.Caloz,andT.Itoh 1527 APositiveFutureforDouble-NegativeMetamaterials(InvitedPaper) ... .... ..... ... . N.EnghetaandR.W.Ziolkowski 1535 InformationforAuthors.. ... ... ... ... ... ... ... ... ... .... ...... ... ... ... ... ... ... ... ... ... ... .... 1557 IEEEMICROWAVETHEORYANDTECHNIQUESSOCIETY TheMicrowaveTheoryandTechniquesSocietyisanorganization,withintheframeworkoftheIEEE,ofmemberswithprincipalprofessionalinterestsinthefieldofmicrowavetheoryandtechniques.Allmembers oftheIEEEareeligibleformembershipintheSocietyandwillreceivethisTRANSACTIONSuponpaymentoftheannualSocietymembershipfeeof$14.00plusanannualsubscriptionfeeof$24.00.Forinformation onjoining,writetotheIEEEattheaddressbelow.MembercopiesofTransactions/Journalsareforpersonaluseonly. ADMINISTRATIVE COMMITTEE K.C.GUPTA,President K.VARIAN,VicePresident A.MORTAZAWI,Secretary M.HARRIS,Treasurer M. P. DE LISO D. HARVEY S.KAWASAKI T. LEE V. J. NAIR W. SHIROMA K.VARIAN S. M.EL-GHAZALY J. HAUSNER J.S.KENNEY D. LOVELACE B. PERLMAN R. 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Univ.ofColoradoatBoulder MitsubishiElectricCorp. MidasConsulting Raleigh, NC Germany Korea USA Japan U.K. 27695-7911USA email: email: email: email: email: Phone:+19195155191 [email protected] [email protected] [email protected] [email protected] [email protected] Fax:+19195131979 ANDREASCANGELLARIS AMIR MORTAZAWI DYLANF.WILLIAMS RUEY-BEEI WU MANH ANH DO email: Univ.ofIllinois,UrbanaChampaign Univ.ofMichiganatAnnArbor NIST NationalTaiwanUniv. NanyangTechnologicalUniv. [email protected] USA USA USA Taiwan, R.O.C. Singapore email: email: email: email: email: [email protected] [email protected] [email protected] [email protected] [email protected] ANTTI RÄISÄNEN YOSHIO NIKAWA PETER RUSSER ALESSANDROCIDRONALI VITTORIORIZZOLI HelsinkiUniv.ofTechnol. Kokushikan Univ. 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DigitalObjectIdentifier10.1109/TMTT.2005.847875 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.53,NO.4,APRIL2005 1413 Guest Editorial I. INTRODUCTION II. TERMINOLOGY METAMATERIALS, the topic of this TRANSACTIONS’ The class of materials or structures that are referred to as SpecialIssue,isavagueterm.Broadly,itmeanssome- metamaterials are characterized by various terms such as: thingbeyondnaturalmaterials,andimpliessomethingartificial, 1) left-handed (LH); 2) BW; 3) negative-index (NI) or nega- notordinarilyfoundinnature. tive-refractiveindex (NRI); or 4) double-negative (DNG), and Originally,thewordmetamaterialsincludedawiderrangeof perhapsothers.Theterm“left-handed”wasusedintheoriginal artificialmaterials,butinthisTRANSACTIONS’SpecialIssue,the paper by Veselago, and many people use it for that reason. It wordappliestothosematerialsorstructuresinwhichthephase signifies a contrast with the well-known “right-hand rule” for velocity and the group (power transmission) velocity point in the direction of the Poynting vector relative to the directions opposite directions. This property leads to a series of fasci- of the electric and magnetic fields. An objection to its use nating performance features, such as a backward electromag- is that “LH” is used in classifying chiral media and types of neticwaveinthematerial,anegativeindexofrefraction,are- circularly polarized radiation. The term BW is not as widely versedDopplereffect,etc.Theseperformancefeaturesinthem- usedbecausebackwardwavescanbeproducedinseveralother selves arestimulating tocontemplate,butwe arelearning that ways.TheNRIpropertywasoriginallythoughttobeobtainable novelandpracticalmicrowavecomponentscanalsobedevised. only from a metamaterial medium, but it is now known that Among these are forward or backward directional couplers it is possible to obtain this property from a special periodic with improved features like arbitrary coupling ratios, shorter arrangementofordinarymaterials.Nevertheless,theNRIterm couplinglengths,broaderbandwidths,etc.Othercouplingcom- is a very appropriate description when dealing with two- or ponents with improved features include branch-line couplers three-dimensionalstructures. andhybridrings.Adifferenttypeofnovelcomponentthatfol- Thetermsaboverepresentineachcaseapropertythatresults lowed directly from the backward-wave property is a leaky- fromawavethatpropagateswithinthemetamaterialstructure. wave antenna that can be frequency scanned or electronically The remaining term, double negative,follows from the funda- scanned from backward endfire through broadside to forward mental properties of the metamaterial itself. Double-negative endfire. Previously, such scanning could be obtained by em- signifies that the permittivity and permeability of the ma- ployingthefirsthigherspaceharmonic,buthere,theradiation terial (or the effective values of and of the structure com- issuppliedbythedominantspaceharmonic.Yetanothernovel prisingthemetamaterial)arebothnegative.Forordinarymate- deviceisaperfectlens,consistingofaplanarslabofmetama- rials,thesevaluesarebothpositive.ThelettersDNGhavebeen terial,whichhasthestartlingpropertythatthesharpnessofthe foundtobeeasilygeneralizabletodouble-positive(DPS)orto image isnotlimitedbythewavelengthoflight.Thistrulyun- single-negative materials, as only-epsilon-negative (ENG) and usualfeaturefollowsfromitsnegative-refractive-indexproperty only-mu-negative(MNG). andthefactthatevanescentwavesareamplifiedwhentraversing Unfortunately, there is no commonly agreed-upon term, so themetamaterialslab. we have permitted in this TRANSACTIONS’ Special Issue each Allofthesestructureshavebeenbuilt,andtheirperformances papertoemployitsownpreference. havebeenverifiedbymeasurementsatmicrowavefrequencies. Thistopicisrathernew,aboutahalf-dozenyearsold,butit hasexpandedveryrapidly,particularlyinthephysicscommu- III. BRIEFHISTORICALBACKGROUND nity.Inthemicrowavecommunity,thecontributionsbeganonly In1968,Veselago[1]authoredapaperthataskedwhatelectro- aboutthreeyearsago,butprogresshasalsobeenrapid,although dynamicbehaviorcouldbefoundifthe and ofamaterialare thegoalsandstresshaveusuallybeensomewhatdifferentfrom simultaneouslynegativeratherthanpositive.Hefoundthatthe those in the physics community. The nature of these contribu- resultswouldbeveryinterestingandunusual,andhepointedout tionsandtheirpotentialfornovelapplications,however,arenot that,amongtheseneweffects,onewouldfindbackwardwaves, yetwidelyunderstoodorappreciatedbymostmicrowaveengi- anegativerefractiveindex,areversedDopplereffect,areversed neers. It is hoped that this TRANSACTIONS’ Special Issue will Cˇerenkoveffect,aplanarslablens,etc.Hethenaskedifanyma- helptoclarifythestate-of-the-artforourcommunityandillus- terialsinnaturecouldbefoundwithsimultaneouslynegative tratesomeoftheopportunitiesforfuturenovelapplicationsin and values,andheconcludedthattherearenone. themicrowavefield. Asaresultofthisconclusion,nofurtherprogressalongthese Sincewebelievethatmanypeopleinthemicrowavefieldare lineswasmadeuntiltherecentturnofthecentury.Ithadbeen notveryfamiliarwiththistopic,wearedevotingmuchofthis known for some time that a medium consisting of an array of GuestEditorialtosomebackgroundinformationforthosewho conducting rods can serve as a medium with a negative value mayfitintothiscategory. of ,butwhatwasmissingwashowtoachieveanegativevalue of .Thebreakthroughwasprovidedin1999byPendryetal. in[2],whichpresentedseveralexamplesofhowonecanobtain DigitalObjectIdentifier10.1109/TMTT.2005.845126 negative values of from conductors. The most interesting of 0018-9480/$20.00©2005IEEE 1414 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.53,NO.4,APRIL2005 them was a split-ring resonator, an array of which provided a themeantime.ThatwasthesituationforVeselago’spaperuntil negative value of over a portion of its resonance frequency recently.Nowthatweknowhowtocreateanartificialmedium range. The combination of the rods and split-ring resonators withsuchproperties,alloftheinterestingperformancepredic- yielded an artificial medium possessing both a negative and tions are being verified experimentally. “What-if” papers can, overanarrowfrequencyrange. therefore,beveryvaluableundertherightcircumstances. Encouragedbytheimplicationsoftheseresults,ateamatthe In this TRANSACTIONS’ Special Issue, Engheta and Zi- University of California at San Diego, La Jolla, built a com- olkowskihavesubmittedanexcellentinvitedreviewpaperwith positemediumalongtheselinesatmicrowavefrequencies,and the intriguing title “A Positive Future for Double Negative mademeasurementsthatclearlydemonstratedthatthismedium Metamaterials.” Many portions of their paper inquire into exhibited a negative index of refraction. These measurements structuresthatmayleadtopotentialapplications.Inthatsense, [3] provedthatthe physical effect is real, and they triggered a these inquiries are “what-if” papers that may be tested exper- floodofpapers,especiallyinthephysicscommunity. imentally in the near future. One example that is discussed in The microwave community was also excited by the poten- theirpaperwasinitiallyapurely“what-if”paper(in2002),but tialfornoveleffectsandnovelcomponents,butitrealizedthat ithasnowbeenexperimentallyverifiedinoneofthepapersin theguidingstructureandconstituentelementsthatwereavail- this TRANSACTIONS’ Special Issue. The physical structure is a ableatthattimewerenotsuitableforapplicationtomicrowave bilayer in which one layer is DNG and the other layer is DPS integrated circuits. Since the structure on which the measure- (an ordinary medium), forming a two-layer resonant cavity mentsweremadeincludedasplit-ringresonatorfortheeffective that can have a sub-wavelength arbitrarily small thickness. negative , a widespread perception developed that the array This experimental paper by Li et al. is entitled “Experimental ofelementsmustcontainaresonantconstituent,whichisobvi- RealizationofaOne-DimensionalLHM–RHMResonator.” ouslyundesirablebecausetheresonantelementisnarrow-band We were hoping for numerous papers with novel applica- andalsolossyinpreciselythemetamaterialrangeofoperation. tions,butthemicrowavecommunityisnotyetreadyformany Another key problem is that the physics community preferred suchpapers.Instead,thehardware-orientedcreativityisdirected theirstructurestobethree-dimensional(3-D),oratleasttwo-di- more toward novel ways to obtain negative values of and . mensional(2-D),whereasformicrowaveintegratedcircuits,the OnesuchpaperinthisTRANSACTIONS’SpecialIssueisentitled guidingstructuremustbeone-dimensional(1-D). “SimulationofNegativePermittivityandNegativePermeability DuringJune2002,attheIEEEMicrowaveTheoryandTech- by Means of Evanescent Waveguide Modes—Theory and Ex- niques(MTT)andAntennasandPropagation(AP)/International periment,”byEstebanetal.Thekeyconceptinvolvestherecog- Scientific Radio Union (URSI) Symposia, solutions were pre- nitionthatanemptymetallicwaveguidebelowcutoffsupportsa sentedbythreedifferentgroupsthatsuccessfullyaddressedthe decayingwavecomparabletoawaveinastopbandorto“prop- above-mentionedproblems.Allthreeusedexistingtransmission agation”throughanover-denseplasma.Theauthorsextendthe linesasthesupportingguidingstructure;Oliner[4]chosestrip earlierpaperbyMarquésetal.,whichrelatesamediumwitha line,whileIyerandEleftheriades[5]andCalozetal.[6]selected negative to evanescent TE modes, to a new equivalence, be- microstripline.Allthreealsochosebasicallysimilardisconti- tweenevanescentTMmodesandanegative . nuity elements (shunt inductances and series capacitances) to A secondnovelstructure involves split-ring resonators. The providetherequirednegative and values.Thebigdifference firststepsaretominiaturizethesizeofthesplitringsandplace between[4]andboth[5]and[6]isthat[4]employedanalytical them in an array along the guide direction. As is well known expressionsforthediscontinuitiesvalidoverawidefrequency from earlierstudies,such arrayswill producea negativevalue range,whereas[5]and[6]usedLandCtermsandstressedthe ofpermeability.Anewercontributionisadualofsuchsplit-ring low-frequencyfeaturesandthedesignsimplicityassociatedwith resonators,calledacomplementarysplit-ringresonator,which theL,Ctransmission-lineapproach.Thesepapersprovidedthe isfabricatedbyetchingthenegativeimageofthesplitringinthe foundation for both the approach and methodology regarding groundplaneofamicrostriplineunderneaththeconductorstrip. howtoproceedwhendesigningfor1-Dtransmissionlines.This The complementary resonator produces a negative value of , isespeciallytruefortheapproachesthatuseLandC.Theauthors ratherthan .Thepropertiesoftheoriginalsplit-ringstructure of [5] and [6], together with various additional later contrib- anditscomplementaryversionarediscussed,andtheirequiva- utors, then proceeded to develop a series of novel microwave lentcircuitsaredeveloped,togetherwithhowtheygiveriseto componentsforusewithmicrostripline. bandpass and bandstop filters. The narrow-band and lossy na- tureoftheseresonatorsarestillconcerns. It is not our intention to cover all the novel features in this IV. ADDITIONALOBSERVATIONS TRANSACTIONS’ Table of Contents, but we have listed these Veselago’s original paper [1] is a classic “what-if” paper. It few examples as illustrative of their flavor. The field has been asks,“Whatifamediumweretohavenegativevaluesof and moving ahead nicely, and we hope that this TRANSACTIONS’ instead of positive values? In what ways would the electro- Special Issue will serve to stimulate further creativity, partic- dynamicbehaviorsbedifferent?”Ifthebehaviorsarenotmuch ularlyinthedirectionofnovelmicrowavecomponents. different,ortheresultsotherwiseuninteresting,thestudywould Intheseintroductoryremarks,wehavepresentedmuchinthe servetotellusthatthestructureisnotworthinvestigatingfur- way of background information in an attempt to be of help to ther.Or,iftheresultsareinteresting,butthereisnowayinwhich thosewhoarelesswellinformedonthistopicandareinneed measurements can be made, the structure must be set aside in ofsomeguidance. IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.53,NO.4,APRIL2005 1415 We wish to express our sincere thanks to our many re- the review process, and diligently made the modifications re- viewers who conscientiously contributed their time and effort quested by the reviewers. We are particularly indebted to this in order to help us produce a quality Special Issue. We re- TRANSACTIONS’ Editor-in-Chief, Prof. Michael B. Steer, for ceived almost 40 submitted manuscripts and, in many cases, providing the necessary guidance in both matters of principle we needed to call on the same reviewer to evaluate more andinthemanyproceduraldetails.Wealsogreatlyappreciate thanonemanuscript.Wealsowishtoexpressourappreciation the many ways in which he graciously offered help and kept to those authors who patiently cooperated with the rigors of us on track. List of Reviewers AlkimAkyurtlu DavidJackson HiroshiShigesawa IkuoAwai DieterJaeger DanSievenpiper MarioSorollaAyza RolfJakoby AriSihvola MaurizioBozzi VikramJandhyala RichardSnyder PaoloBurghignoli MikkoKärkkäinen AlexsanderSochava ChristopheCaloz ShigeoKawasaki RobertoSorrentino SteveCammer EdwardKuester CostasSoukoulis FilippoCapolino YasuoKuga SergeiTretyakov KaiChang PaoloLampariello MikioTsuji WengChoChew RalphLevy Ching-KuangC.Tzuang PeterdeMaagt Le-WeiLi TetsuyaUeda MichaelDydyk DidierLippens YuanxunWang GeorgeEleftheriades Wen-taoLu DylanWilliams NaderEngheta RicardoMarqués Bae-IanWu FabrizioFrezza WolfgangMenzel KeWu SpartakGevorgian FranciscoMesa FanYang ThomaszGregorczyk EricMichielssen JunhoYeo WojciechGwarek HosseinMosallaei TsukasaYoneyama WolfgangHoefer MichalMrozowski JanZehentner YasushiHorii MichalOkoniewski RichardZiolkowski AkiraIshimaru AtsushiSanada TATSUOITOH,GuestEditor UniversityofCaliforniaatLosAngeles(UCLA) DepartmentofElectricalEngineering LosAngeles, CA 90095-1594 USA ARTHURA.OLINER,GuestEditor PolytechnicUniversity DepartmentofElectricalEngineering Lexington, MA 02421 USA REFERENCES [4] A.A.Oliner,“A periodic-structurenegative-refractive-indexmedium withoutresonantelements,”inIEEEAP-S/URSIInt.Symp.Dig.,San [1] V. G. Veselago, “The electrodynamics of substances with simulta- Antonio,TX,Jun.16–21,2002,p.41. neously negative values of " and (cid:22),” Sov. Phys.—Usp., vol. 10, pp. [5] A.K.IyerandG.V.Eleftheriades,“Negativerefractiveindexmetamate- 509–514,Jan.–Feb.1968. [2] J.B.Pendry,A.J.Holden,D.J.Robbins,andW.J.Stewart,“Magnetism rialssupporting2Dwaves,”inIEEEMTT-SInt.MicrowaveSymp.Dig., fromconductorsandenhancednonlinearphenomena,”IEEETrans.Mi- Seattle,WA,Jun.2–7,2002,pp.1067–1070. crow.TheoryTech.,vol.47,no.11,pp.2075–2084,Nov.1999. [6] C.Caloz,H.Okabe,T.Iwai,andT.Itoh,“Transmissionlineapproach [3] R.A.Shelby,D.R.Smith,andS.Schultz,“Experimentalverificationof ofleft-handedmaterials,”inIEEEAP-S/URSIInt.Symp.Dig.,SanAn- anegativeindexofrefraction,”Science,vol.292,pp.77–79,Apr.2001. tonio,TX,Jun.16–21,2002,p.39. 1416 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.53,NO.4,APRIL2005 TatsuoItoh(S’69–M’69–SM’74–F’82)receivedthePh.D.degreeinelectricalengineeringfrom theUniversityofIllinoisatUrbana-Champaign,in1969. FromSeptember1966toApril1976,hewaswiththeElectricalEngineeringDepartment,Uni- versityofIllinoisatUrbana-Champaign.FromApril1976toAugust1977,hewasaSeniorRe- search Engineer with the Radio Physics Laboratory, SRI International, Menlo Park, CA. From August1977toJune1978,hewasanAssociateProfessorwiththeUniversityofKentucky,Lex- ington.InJuly1978,hejoinedthefacultyatTheUniversityofTexasatAustin,wherehebecamea ProfessorofElectricalEngineeringin1981andDirectoroftheElectricalEngineeringResearch Laboratory in 1984. During the summer of 1979, he was a Guest Researcher with AEG-Tele- funken,Ulm,Germany.InSeptember1983,hewasselectedtoholdtheHaydenHeadCentennial ProfessorshipofEngineeringatTheUniversityofTexasatAustin.InSeptember1984,hewas appointedAssociateChairmanforResearchandPlanningoftheElectricalandComputerEngi- neeringDepartment,TheUniversityofTexasatAustin.InJanuary1991,hejoinedtheUniversity ofCaliforniaatLosAngeles(UCLA)asProfessorofElectricalEngineeringandHolderoftheTRWEndowedChairinMicrowave and Millimeter Wave Electronics. He was an Honorary Visiting Professor with the Nanjing Institute of Technology, Nanjing, China,andattheJapanDefenseAcademy.InApril1994,hewasappointedanAdjunctResearchOfficerwiththeCommunica- tionsResearchLaboratory,MinistryofPostandTelecommunication,Japan.HecurrentlyholdsaVisitingProfessorshipwithThe UniversityofLeeds,Leeds,U.K.Hehasauthoredorcoauthored350journalpublications,650refereedconferencepresentations, andhaswritten30books/bookchaptersintheareaofmicrowaves,millimeterwaves,antennas,andnumericalelectromagnetics. Hehasgenerated64Ph.D.students. Dr. Itoh is a member of the Institute of Electronics and Communication Engineers of Japan, and Commissions B and D of USNC/URSI.HeservedastheeditoroftheIEEETRANSACTIONSONMICROWAVETHEORY ANDTECHNIQUES(1983–1985).He serves on the Administrative Committee of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S). He was vice presidentoftheIEEEMTT-Sin1989andpresidentin1990.Hewastheeditor-in-chiefofIEEEMICROWAVEANDGUIDEDWAVE LETTERS(1991–1994).HewaselectedanHonoraryLifeMemberoftheIEEEMTT-Sin1994.Hewaselectedamemberofthe NationalAcademyofEngineeringin2003.HewasthechairmanoftheUSNC/URSICommissionD(1988–1990)andchairman ofCommissionDoftheInternationalURSI(1993–1996).HeischairoftheLongRangePlanningCommitteeoftheURSI.He servesonadvisoryboardsandcommitteesfornumerousorganizations.Hehasbeentherecipientofnumerousawardsincluding the1998ShidaAwardpresentedbytheJapaneseMinistryofPostandTelecommunications,the1998JapanMicrowavePrize,the 2000IEEEThirdMillenniumMedal,andthe2000IEEEMTT-SDistinguishedEducatorAward. IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.53,NO.4,APRIL2005 1417 ArthurA.Oliner(M’47–SM’52–F’61–LF’87)wasbornonMarch5,1921,inShanghai,China. HereceivedtheB.A.degreefromBrooklynCollege,Brooklyn,NY,in1941,andthePh.D.degree fromCornellUniversity,Ithaca,NY,in1946,bothinphysics. He joined the Polytechnic Institute of Brooklyn (now Polytechnic University) in 1946, and became Professor in 1957. He then served as Department Head from 1966 to 1974, and was Director of its Microwave Research Institute from 1967 to 1982. He was a Walker-Ames Vis- itingProfessorwiththeUniversityofWashington,in1964.HehasalsobeenaVisitingProfessor withthe Catholic University,Rio de Janeiro,Brazil,the TokyoInstituteofTechnology, Tokyo, Japan,theCentralChinaInstituteofScienceandTechnology,Wuhan,China,andtheUniversity of Rome, Rome, Italy. In 2003, the University of Rome (“La Sapienza”) granted him an Hon- oraryDoctorate,andorganizedanassociatedspecialsymposiuminhishonor.Heisamemberof the BoardofDirectorsofMerrimac Industries.Hehasauthored over300papers, variousbook chapters, and has coauthored or coedited three books. His research has covered a wide variety of topics in the microwave field including network representations of microwave structures, guided-wave theory with stress on surfacewavesandleakywaves,wavesinplasmas,periodicstructuretheory,andphased-arrayantennas.Hehasmadepioneering andfundamentalcontributionsinseveraloftheseareas.Hisinterestshavealsoincludedwaveguidesforsurfaceacousticwavesand integratedoptics,novelleaky-waveantennasformillimeterwaves,andleakageeffectsinmicrowaveintegratedcircuits.Lately,he hascontributedtothetopicsofmetamaterials,andtoenhancedpropagationthroughsubwavelengthholes. Dr.OlinerisaFellowoftheAmericanAssociationfortheAdvancementofScience(AAAS)andtheInstitutionofElectrical Engineers(IEE),U.K.HewasaGuggenheimFellow.HewaselectedamemberoftheNationalAcademyofEngineeringin1991. Hehasbeentherecipientofprizesfortwoofhispapers:theIEEEMicrowavePrizein1967forhisworkonstrip-linediscontinu- ities,andtheInstitutionPremiumoftheIEEin1964forhiscomprehensivestudiesofcomplexwavetypesguidedbyinterfacesand layers.HewasPresidentoftheIEEEMicrowaveTheoryandTechniquesSociety(IEEEMTT-S),itsfirstDistinguishedLecturer, andamemberoftheIEEEPublicationBoard.HeisanHonoraryLifeMemberofIEEEMTT-S(oneofonlysixsuchpersons). In 1982, he was the recipient of its highest recognition, the MicrowaveCareer Award. A specialretrospective session was held in his honor at the IEEE MTT-S International Microwave Symposium (reported in detail in the December 1988 issue of this TRANSACTIONS,pp.1578-1581).In1993,hewasthefirstrecipientoftheDistinguishedEducatorAwardoftheIEEEMTT-S.He wasalsoarecipientoftheIEEECentennialandMillenniumMedals.HeisalsoapastU.S.chairmanofCommissionsAandDof theInternationalUnionofRadioScience(URSI),along-timememberofandactivecontributortoCommissionB,andaformer memberoftheU.S.NationalCommitteeofURSI.In1990,hewastherecipientoftheURSIvanderPolGoldMedal,whichis giventriennially,forhiscontributionstoleakywaves.In2000,theIEEEawardedhimasecondgoldmedal,theHeinrichHertz Medal,whichisitshighestawardintheareaofelectromagneticwaves. 1418 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.53,NO.4,APRIL2005 Homogenization of 3-D-Connected and Nonconnected Wire Metamaterials MárioG.Silveirinha,Member,IEEE,andCarlosA.Fernandes,Member,IEEE Abstract—Thehomogenizationofcompositestructuresmadeof long thin metallic wires is an important problem in electromag- neticsbecausetheyareoneofthebasiccomponentsofthedouble- negativemedium.Inthispaper,weproposeanewanalyticalmodel tocharacterizetheeffectivepermittivityofthethree-dimensional- wiremediuminthelongwavelengthlimit.Westudytwodifferent topologies for the wire medium. The first structure consists of a latticeofconnectedwires,whereasthesecondoneconsistsofalat- ticeinwhichthewiresarenotconnected.Ourresultsshowthatthe propagationofelectromagneticwavesinthetwometamaterialsis verydifferent.Whileoneofthestructuresexhibitsstrongspatial dispersion,theotheroneseemstobeagoodcandidateforimpor- tantmetamaterialapplications.Wealsofoundthat,forextremely Fig.1. (a)Fragmentofametamaterialformedbyalatticeofconnectedwires. lowfrequencies,oneofthestructuressupportsmodeswithhyper- (b) Fragment of a metamaterial formed by a lattice of nonconnected wires bolicwavenormalcontours,originatingnegativerefractionatan (adjacentorthogonalwiresarespacedofhalf-latticeconstant). interfacewithair.Wevalidatedourtheoreticalresultswithnumer- icalsimulations. Theconfigurationoriginallyproposedin[1]ischaracterized IndexTerms—Double-negative(DNG)medium,homogenization byDNGparametersonlyforauniquepolarization,andforprop- theory,metamaterials,negativerefraction,wiremedium. agation along a specific direction of space. The fabrication of anisotropicDNGmediumclearlycompelsforabasiccellwith extrasymmetry.Apparently,apartfrompracticaldifficultiesre- I. INTRODUCTION latedtotechnologicallimitations,thegeneralizationseemstobe IN [1],Smith et al. proposed an original structure thatcon- straightforward. sists of a lattice of long metallic wires and split-ring res- However,thesituationmaynotbesoplain.Forexample,one onators(SRRs).Theoreticalandexperimentalstudiesshowthat might expect that a medium formed by wires that are parallel the composite structure behaves as a double negative (DNG) to the coordinate axes would interact with the radiation as an medium (also known as left-handed medium) with a negative isotropicmediumwithnegativepermittivity.However,numer- index of refraction [1]–[3]. This remarkable result motivated ical results reported by Silveirinha and Fernandes [8] support muchresearchonmetamaterialsandpotentialapplications.For that,atleastforthenonconnectedtopologystudiedin[8],that example,numerousstudiestrytoexplorethefocusingproperty is not the case. Indeed, we found that near the “plasma fre- ofaDNGslabsurroundedbyconventionalmedia[4]–[6].Itwas quency” the referred metamaterial is not isotropic. Moreover, evensuggested,withconsiderablecontroversy,thatmetamate- weverifiedthatthenumericalresultsareconsistentwiththehy- rials could be used to fabricate a super lens with no limit of pothesis that, analogous with the one-dimensional (1-D)-wire resolution[6].Otherstudiesshowthatmetamaterialsmayfavor mediumformedbyanarrayofparallelwires[9]andotherre- theminiaturizationofsomedevices,andtherealizationofsub- latedstructures[10],theconsideredgeometryforthethree-di- wavelengthcavityresonators[7]. mensional (3-D)-wire medium has strong spatial dispersion in ThenegativerefractionoftheDNGmetamaterialisattributed thelongwavelengthlimit,i.e.,thepermittivitydependsnotonly to the combined effect of the arrays of wires and SRRs. As is onthefrequency,butalsoonthewavevector. well known, the lattice of wires is modeled as a medium with The results reported in [8] raise the obvious question: “Is negativepermittivity,whereasthelatticeofSRRsischaracter- itpossibletofabricateanisotropicmetamaterialwithnegative izedbyanegativepermeability.Toafirst-orderapproximation, permittivity?”Thisisafundamentalsubjectnotonlytounder- thereisnocouplingbetweenthetwobasicinclusions. stand the possible limitations of DNG metamaterials, but also becausemetamaterialswithnegativepermittivity,whenpaired withmetamaterialswithnegativepermeability,mayhaveinter- ManuscriptreceivedMay20,2004;revisedSeptember20,2004.Thiswork wassupportedbytheFundaçãoparaCiênciaeaTecnologiaunderProjectPOSI estingpropertiesandapplications,asdescribedin[11]. 34860/99. Inthispaper,westudytwodifferenttopologiesforthe3-D- M. G. Silveirinha is with the Electrical Engineering Department–Instituto wiremedium.Thefirststructureconsistsofasimplecubiclat- de Telecomunicações, Polo II da Universidade de Coimbra, 3030 Coimbra, Portugal(e-mail:[email protected]). ticeofconnectedwires,andthesecondoneconsistsofasimple C. A. Fernandes is with the Instituto Superior Técnico–Instituto de cubiclatticeinwhichthewiresarenotconnected(i.e.,thege- Telecomunicações,TechnicalUniversityofLisbon,1049-001Lisbon,Portugal ometry studied in [8] using numerical methods). In Fig. 1, we (e-mail:[email protected]). DigitalObjectIdentifier10.1109/TMTT.2005.845128 depictfragmentsofbothmetamaterials. 0018-9480/$20.00©2005IEEE SILVEIRINHAANDFERNANDES:HOMOGENIZATIONOF3-D-CONNECTEDANDNONCONNECTEDWIREMETAMATERIALS 1419 A. ConnectedTopology In this case, the metallic region in the unit cell consists of threecylindricalwireswithradius andlength .Wehave (1) Intheabove, representsthesurfaceofthewiresection oriented along the -direction. Note that the wires intersect Fig. 2. Geometry of the unit cell. (a) Connected wire geometry. mutually,formingajunctionneartheorigin. (b)Nonconnectedwiregeometry. Ourobjectiveistoproposeananalyticaltreatmentfortheho- mogenizationproblem.Thus,itisdesirabletohaveamodelas We propose a new analytical procedure to homogenize the simpleaspossibleforthecurrentthatflowsalongthewires.We will assume that , where is the wavelength of periodic metamaterials. We will prove that the two topologies radiation in the dielectric region. Therefore, the thin-wire ap- forthe3-D-wiremediumarenotequivalent,andthattheinter- proximationcanbeused. actionofelectromagneticwaveswiththetwostructuresisvery Within the thin-wire approximation, the density of current different.Wewillderiveanapproximateanalyticalformulafor over each wire flows along the direction of the axis, and theeffectivepermittivityofthecompositestructures.Toourbest has approximately circular symmetry. We will also admit that knowledge, no results were reported in the literature that con- is a traveling wave characterized by the wave vector cern the homogenization of related structures with the excep- (thejustificationforthisassumptionwillbegiven tionofthenumericalanalysisdescribedin[8],andtheexperi- inSectionIII).Therefore,weadmitthat mentalresultsreportedin[12](thegeometryconsideredin[12] isslightlymorecomplexthantheonesconsideredhere). This paper is organized as follows.1 Sections II–IV concern (2) the characterization of the Floquet modes that propagate in the periodic material. In Section II, we describe the geometry where is a periodic function that stands for the unknown of the wire medium. In Section III, we formulate the modal current. In order to take into account the coupling at the junc- problem, and we obtain the characteristic system using an tion, we admit the possibility of the current induced over the integral-equation-based approach. In Section IV, we propose th wire being discontinuous at . In general, we have a simplified characteristic system that can be solved using .Theconservationofcurrentre- analytical methods. Sections V and VI concern the homoge- quiresthat(seealso[13]) nizationofthemetamaterials.InSectionV,weexplainhowthe effectivepermittivitydyadic canbe obtained directlyfrom the (3) characteristic system. In Section VI, we discuss the physical phenomena implied by our formulas, and we compare the developed theory with full-wave numerical results. Finally, in Notethat,withinthedescribedmodel,wecompletelyneglect the shape of the wire junctions. Essentially, in our model, the SectionVII,wepresentconclusions. junctions are replaced by infinitesimal gaps located at (i.e.,thepointswhereweallowthecurrenttobedis- II. GEOMETRY continuous)withacirculartransversesection.Nevertheless,our We denote a generic point of space by , simplified model is expected to yield good results since each andtheunitvectordirectedalongthe -directionby junctionoccupiesonlyaverysmallfractionoftheunitcell. .Thewiremediumisobtainedbytheperiodicrepetition oftheunitcell showninFig.2.Theunitcelliscenteredatthe B. NonconnectedTopology origin. The wires are arranged into a simple cubic lattice with The nonconnected geometry is depicted in Fig. 2(b). As in lattice constant . The boundary of the metallic region in the the previous case, the metallic region in the unit cell consists unitcellisdenotedby ,andtheoutwardunitnormalvector ofthreecylindricalwireswithradius andlength .Nowthe isdenotedby .Forsimplicity,weadmitthatthewiresareem- wiresdonotintersect,andthewireaxesarespacedby ,i.e., beddedinair.Weconsidertwodifferenttopologiesforthewire ahalf-latticeconstant.Thewireorientedalongthe -direction medium. The geometry shown in Fig. 2(a) corresponds to the iscenteredat connected case, and the geometry of Fig. 2(b) corresponds to the nonconnected case. In Sections II-A and –B, we describe (4) eachconfigurationindetail. Asbefore,weadmitthatthedensityofcurrentoverthe th 1Whilethispaperwasbeingprepared,SimovskiandBelovstudiedtheho- wireisgivenby(2).Now isacontinuousperiodicfunction mogenizationofthenonconnectedwiremediumusingalocalfieldapproach [23]. becausetherearenojunctions.

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APRIL 2005 VOLUME53 NUMBER 4 IETMAB (ISSN0018-9480) PART II OF TWO PARTS SPECIALISSUEONMETAMATERIALSTRUCTURES,PHENOMENA,ANDAPPLICATIONS GuestEditorial .. ... ... ... ... ... ... ... ... .... ..... ... .... ... ... ... ... ... ... ..T.ItohandA.A.Oliner 1413 PAPERS Analysis andModeling Homogenizationof3-D-ConnectedandNonconnectedWireMetamaterials... .....M.G.SilveirinhaandC.A.Fernandes 1418 FundamentalModalPropertiesofSurfaceWavesonMetamaterialGroundedSlabs .. ... ... ... ... ... ... ... .... .. ... ... .... ... ... .. P.Baccarelli,P.Burghignoli,F.Frezza,A.Galli,P.Lampariello,G.Lovat,andS.Paulotto 1431 RefractionLawsforAnisotropicMediaandTheirApplicationtoLeft-HandedMetamaterials. ... ... ... ... ... .... .. ... ... .... ... ... ... ... ... ... ... ... ... .T.M.Grzegorczyk,M.Nikku,X.Chen,B.-I.Wu,andJ.A.Kong 1443 Equivalent-Circuit Models for Split-Ring Resonators and Complementary Split-Ring Resonators Coupled to Planar TransmissionLines ... ... ... ... ... ... ... ...... ... ... .... ... ... ... .. J.D.Baena,J.Bonache,F.Martín, R.MarquésSillero,F.Falcone,T.Lopetegi,M.A.G.Laso,J.García–García,I.Gil,M.FloresPortillo,andM.Sorolla 1451 EfficientModelingofNovelUniplanarLeft-HandedMetamaterials.. ... ... ... ... ... ... ... ... ... ... ... .... .. ... ... .... ... ... ... ... ... ... ... ... ... ... .Y.Guo,G.Goussetis,A.P.Feresidis,andJ.C.Vardaxoglou 1462 Numerical Analysis Propagation Property Analysis of Metamaterial Constructed by Conductive SRRs and Wires Using the MGS-Based Algorithm .... ... ... ... ... ... ..... .... ... ... ... ... .... .. H.-Y.Yao,W.Xu,L.-W.Li,Q.Wu,andT.-S.Yeo 1469 FDTDStudyofResonanceProcessesinMetamaterials. ... ... .... ... ... ... ... ... ... ... ... ... ... ... .... .. ... ... .... ... ... ... ... ... ... ... ..E.A.Semouchkina,G.B.Semouchkin,M.Lanagan,andC.A.Randall 1477 PeriodicFinite-DifferenceTime-DomainAnalysisofLoadedTransmission-LineNegative-Refractive-IndexMetamaterials .. ... ... .... ... ... ... ... ...... ... ... ... ... ... ... .... . T.Kokkinos,C.D.Sarris,andG.V.Eleftheriades 1488 ModelingofMetamaterialsWithNegativeRefractiveIndexUsing2-DShuntand3-DSCNTLMNetworks .. ... .... .. ... ... .... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ...P.P.M.So,H.Du,andW.J.R.Hoefer 1496 (ContentsContinuedonBackCover) (ContentsContinuedfromFrontCover) Experimental SimulationofNegativePermittivityandNegativePermeabilitybyMeansofEvanescentWaveguideModes—Theoryand Experiment ... ...... ...J.Esteban,C.Camacho-Peñalosa,J.E.Page,T.M.Martín-Guerrero,andE.Márquez-Segura 1506 EffectiveElectromagneticParametersofNovelDistributedLeft-HandedMicrostripLines ... ... ... ... ... ... .... .. ... ... .... ... ... ... ... ... ... ... ... ... ... ... .... ... ... . S.-G.Mao,S.-L.Chen,andC.-W.Huang 1515 ExperimentalRealizationofaOne-DimensionalLHM–RHMResonator. ... ... ... ... ... ... ... ... ... ... .... .. ... ... .... ... ... ...Y.Li,L.Ran,H.Chen,J.Huangfu,X.Zhang,K.Chen,T.M.Grzegorczyk,andJ.A.Kong 1522 Super-CompactMultilayeredLeft-HandedTransmissionLineandDiplexerApplication .... ..Y.Horii,C.Caloz,andT.Itoh 1527 APositiveFutureforDouble-NegativeMetamaterials(InvitedPaper) ... .... ..... ... . N.EnghetaandR.W.Ziolkowski 1535 InformationforAuthors.. ... ... ... ... ... ... ... ... ... .... ...... ... ... ... ... ... ... ... ... ... ... .... 1557 IEEEMICROWAVETHEORYANDTECHNIQUESSOCIETY TheMicrowaveTheoryandTechniquesSocietyisanorganization,withintheframeworkoftheIEEE,ofmemberswithprincipalprofessionalinterestsinthefieldofmicrowavetheoryandtechniques.Allmembers oftheIEEEareeligibleformembershipintheSocietyandwillreceivethisTRANSACTIONSuponpaymentoftheannualSocietymembershipfeeof$14.00plusanannualsubscriptionfeeof$24.00.Forinformation onjoining,writetotheIEEEattheaddressbelow.MembercopiesofTransactions/Journalsareforpersonaluseonly. ADMINISTRATIVE COMMITTEE K.C.GUPTA,President K.VARIAN,VicePresident A.MORTAZAWI,Secretary M.HARRIS,Treasurer M. P. DE LISO D. HARVEY S.KAWASAKI T. LEE V. J. NAIR W. SHIROMA K.VARIAN S. M.EL-GHAZALY J. HAUSNER J.S.KENNEY D. LOVELACE B. PERLMAN R. SNYDER R.WEIGEL M.HARRIS L.KATEHI N.KOLIAS J.MODELSKI D.RUTLEDGE R.SORRENTINO S.WETENKAMP HonoraryLifeMembers DistinguishedLecturers PastPresidents T. ITOH A.A.OLINER K. TOMIYASU L. E. DAVIS T. ITOH D. RYTTING R.J.TREW(2004) T.S.SAAD L.YOUNG W. GWAREK B. KIM M. SHUR F.SCHINDLER(2003) W. HEINRICH J. LASKAR P. 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Univ.ofColoradoatBoulder MitsubishiElectricCorp. MidasConsulting Raleigh, NC Germany Korea USA Japan U.K. 27695-7911USA email: email: email: email: email: Phone:+19195155191 [email protected] [email protected] [email protected] [email protected] [email protected] Fax:+19195131979 ANDREASCANGELLARIS AMIR MORTAZAWI DYLANF.WILLIAMS RUEY-BEEI WU MANH ANH DO email: Univ.ofIllinois,UrbanaChampaign Univ.ofMichiganatAnnArbor NIST NationalTaiwanUniv. NanyangTechnologicalUniv. [email protected] USA USA USA Taiwan, R.O.C. Singapore email: email: email: email: email: [email protected] [email protected] [email protected] [email protected] [email protected] ANTTI RÄISÄNEN YOSHIO NIKAWA PETER RUSSER ALESSANDROCIDRONALI VITTORIORIZZOLI HelsinkiUniv.ofTechnol. Kokushikan Univ. 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DigitalObjectIdentifier10.1109/TMTT.2005.847875 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.53,NO.4,APRIL2005 1413 Guest Editorial I. INTRODUCTION II. TERMINOLOGY METAMATERIALS, the topic of this TRANSACTIONS’ The class of materials or structures that are referred to as SpecialIssue,isavagueterm.Broadly,itmeanssome- metamaterials are characterized by various terms such as: thingbeyondnaturalmaterials,andimpliessomethingartificial, 1) left-handed (LH); 2) BW; 3) negative-index (NI) or nega- notordinarilyfoundinnature. tive-refractiveindex (NRI); or 4) double-negative (DNG), and Originally,thewordmetamaterialsincludedawiderrangeof perhapsothers.Theterm“left-handed”wasusedintheoriginal artificialmaterials,butinthisTRANSACTIONS’SpecialIssue,the paper by Veselago, and many people use it for that reason. It wordappliestothosematerialsorstructuresinwhichthephase signifies a contrast with the well-known “right-hand rule” for velocity and the group (power transmission) velocity point in the direction of the Poynting vector relative to the directions opposite directions. This property leads to a series of fasci- of the electric and magnetic fields. An objection to its use nating performance features, such as a backward electromag- is that “LH” is used in classifying chiral media and types of neticwaveinthematerial,anegativeindexofrefraction,are- circularly polarized radiation. The term BW is not as widely versedDopplereffect,etc.Theseperformancefeaturesinthem- usedbecausebackwardwavescanbeproducedinseveralother selves arestimulating tocontemplate,butwe arelearning that ways.TheNRIpropertywasoriginallythoughttobeobtainable novelandpracticalmicrowavecomponentscanalsobedevised. only from a metamaterial medium, but it is now known that Among these are forward or backward directional couplers it is possible to obtain this property from a special periodic with improved features like arbitrary coupling ratios, shorter arrangementofordinarymaterials.Nevertheless,theNRIterm couplinglengths,broaderbandwidths,etc.Othercouplingcom- is a very appropriate description when dealing with two- or ponents with improved features include branch-line couplers three-dimensionalstructures. andhybridrings.Adifferenttypeofnovelcomponentthatfol- Thetermsaboverepresentineachcaseapropertythatresults lowed directly from the backward-wave property is a leaky- fromawavethatpropagateswithinthemetamaterialstructure. wave antenna that can be frequency scanned or electronically The remaining term, double negative,follows from the funda- scanned from backward endfire through broadside to forward mental properties of the metamaterial itself. Double-negative endfire. Previously, such scanning could be obtained by em- signifies that the permittivity and permeability of the ma- ployingthefirsthigherspaceharmonic,buthere,theradiation terial (or the effective values of and of the structure com- issuppliedbythedominantspaceharmonic.Yetanothernovel prisingthemetamaterial)arebothnegative.Forordinarymate- deviceisaperfectlens,consistingofaplanarslabofmetama- rials,thesevaluesarebothpositive.ThelettersDNGhavebeen terial,whichhasthestartlingpropertythatthesharpnessofthe foundtobeeasilygeneralizabletodouble-positive(DPS)orto image isnotlimitedbythewavelengthoflight.Thistrulyun- single-negative materials, as only-epsilon-negative (ENG) and usualfeaturefollowsfromitsnegative-refractive-indexproperty only-mu-negative(MNG). andthefactthatevanescentwavesareamplifiedwhentraversing Unfortunately, there is no commonly agreed-upon term, so themetamaterialslab. we have permitted in this TRANSACTIONS’ Special Issue each Allofthesestructureshavebeenbuilt,andtheirperformances papertoemployitsownpreference. havebeenverifiedbymeasurementsatmicrowavefrequencies. Thistopicisrathernew,aboutahalf-dozenyearsold,butit hasexpandedveryrapidly,particularlyinthephysicscommu- III. BRIEFHISTORICALBACKGROUND nity.Inthemicrowavecommunity,thecontributionsbeganonly In1968,Veselago[1]authoredapaperthataskedwhatelectro- aboutthreeyearsago,butprogresshasalsobeenrapid,although dynamicbehaviorcouldbefoundifthe and ofamaterialare thegoalsandstresshaveusuallybeensomewhatdifferentfrom simultaneouslynegativeratherthanpositive.Hefoundthatthe those in the physics community. The nature of these contribu- resultswouldbeveryinterestingandunusual,andhepointedout tionsandtheirpotentialfornovelapplications,however,arenot that,amongtheseneweffects,onewouldfindbackwardwaves, yetwidelyunderstoodorappreciatedbymostmicrowaveengi- anegativerefractiveindex,areversedDopplereffect,areversed neers. It is hoped that this TRANSACTIONS’ Special Issue will Cˇerenkoveffect,aplanarslablens,etc.Hethenaskedifanyma- helptoclarifythestate-of-the-artforourcommunityandillus- terialsinnaturecouldbefoundwithsimultaneouslynegative tratesomeoftheopportunitiesforfuturenovelapplicationsin and values,andheconcludedthattherearenone. themicrowavefield. Asaresultofthisconclusion,nofurtherprogressalongthese Sincewebelievethatmanypeopleinthemicrowavefieldare lineswasmadeuntiltherecentturnofthecentury.Ithadbeen notveryfamiliarwiththistopic,wearedevotingmuchofthis known for some time that a medium consisting of an array of GuestEditorialtosomebackgroundinformationforthosewho conducting rods can serve as a medium with a negative value mayfitintothiscategory. of ,butwhatwasmissingwashowtoachieveanegativevalue of .Thebreakthroughwasprovidedin1999byPendryetal. in[2],whichpresentedseveralexamplesofhowonecanobtain DigitalObjectIdentifier10.1109/TMTT.2005.845126 negative values of from conductors. The most interesting of 0018-9480/$20.00©2005IEEE 1414 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.53,NO.4,APRIL2005 them was a split-ring resonator, an array of which provided a themeantime.ThatwasthesituationforVeselago’spaperuntil negative value of over a portion of its resonance frequency recently.Nowthatweknowhowtocreateanartificialmedium range. The combination of the rods and split-ring resonators withsuchproperties,alloftheinterestingperformancepredic- yielded an artificial medium possessing both a negative and tions are being verified experimentally. “What-if” papers can, overanarrowfrequencyrange. therefore,beveryvaluableundertherightcircumstances. Encouragedbytheimplicationsoftheseresults,ateamatthe In this TRANSACTIONS’ Special Issue, Engheta and Zi- University of California at San Diego, La Jolla, built a com- olkowskihavesubmittedanexcellentinvitedreviewpaperwith positemediumalongtheselinesatmicrowavefrequencies,and the intriguing title “A Positive Future for Double Negative mademeasurementsthatclearlydemonstratedthatthismedium Metamaterials.” Many portions of their paper inquire into exhibited a negative index of refraction. These measurements structuresthatmayleadtopotentialapplications.Inthatsense, [3] provedthatthe physical effect is real, and they triggered a these inquiries are “what-if” papers that may be tested exper- floodofpapers,especiallyinthephysicscommunity. imentally in the near future. One example that is discussed in The microwave community was also excited by the poten- theirpaperwasinitiallyapurely“what-if”paper(in2002),but tialfornoveleffectsandnovelcomponents,butitrealizedthat ithasnowbeenexperimentallyverifiedinoneofthepapersin theguidingstructureandconstituentelementsthatwereavail- this TRANSACTIONS’ Special Issue. The physical structure is a ableatthattimewerenotsuitableforapplicationtomicrowave bilayer in which one layer is DNG and the other layer is DPS integrated circuits. Since the structure on which the measure- (an ordinary medium), forming a two-layer resonant cavity mentsweremadeincludedasplit-ringresonatorfortheeffective that can have a sub-wavelength arbitrarily small thickness. negative , a widespread perception developed that the array This experimental paper by Li et al. is entitled “Experimental ofelementsmustcontainaresonantconstituent,whichisobvi- RealizationofaOne-DimensionalLHM–RHMResonator.” ouslyundesirablebecausetheresonantelementisnarrow-band We were hoping for numerous papers with novel applica- andalsolossyinpreciselythemetamaterialrangeofoperation. tions,butthemicrowavecommunityisnotyetreadyformany Another key problem is that the physics community preferred suchpapers.Instead,thehardware-orientedcreativityisdirected theirstructurestobethree-dimensional(3-D),oratleasttwo-di- more toward novel ways to obtain negative values of and . mensional(2-D),whereasformicrowaveintegratedcircuits,the OnesuchpaperinthisTRANSACTIONS’SpecialIssueisentitled guidingstructuremustbeone-dimensional(1-D). “SimulationofNegativePermittivityandNegativePermeability DuringJune2002,attheIEEEMicrowaveTheoryandTech- by Means of Evanescent Waveguide Modes—Theory and Ex- niques(MTT)andAntennasandPropagation(AP)/International periment,”byEstebanetal.Thekeyconceptinvolvestherecog- Scientific Radio Union (URSI) Symposia, solutions were pre- nitionthatanemptymetallicwaveguidebelowcutoffsupportsa sentedbythreedifferentgroupsthatsuccessfullyaddressedthe decayingwavecomparabletoawaveinastopbandorto“prop- above-mentionedproblems.Allthreeusedexistingtransmission agation”throughanover-denseplasma.Theauthorsextendthe linesasthesupportingguidingstructure;Oliner[4]chosestrip earlierpaperbyMarquésetal.,whichrelatesamediumwitha line,whileIyerandEleftheriades[5]andCalozetal.[6]selected negative to evanescent TE modes, to a new equivalence, be- microstripline.Allthreealsochosebasicallysimilardisconti- tweenevanescentTMmodesandanegative . nuity elements (shunt inductances and series capacitances) to A secondnovelstructure involves split-ring resonators. The providetherequirednegative and values.Thebigdifference firststepsaretominiaturizethesizeofthesplitringsandplace between[4]andboth[5]and[6]isthat[4]employedanalytical them in an array along the guide direction. As is well known expressionsforthediscontinuitiesvalidoverawidefrequency from earlierstudies,such arrayswill producea negativevalue range,whereas[5]and[6]usedLandCtermsandstressedthe ofpermeability.Anewercontributionisadualofsuchsplit-ring low-frequencyfeaturesandthedesignsimplicityassociatedwith resonators,calledacomplementarysplit-ringresonator,which theL,Ctransmission-lineapproach.Thesepapersprovidedthe isfabricatedbyetchingthenegativeimageofthesplitringinthe foundation for both the approach and methodology regarding groundplaneofamicrostriplineunderneaththeconductorstrip. howtoproceedwhendesigningfor1-Dtransmissionlines.This The complementary resonator produces a negative value of , isespeciallytruefortheapproachesthatuseLandC.Theauthors ratherthan .Thepropertiesoftheoriginalsplit-ringstructure of [5] and [6], together with various additional later contrib- anditscomplementaryversionarediscussed,andtheirequiva- utors, then proceeded to develop a series of novel microwave lentcircuitsaredeveloped,togetherwithhowtheygiveriseto componentsforusewithmicrostripline. bandpass and bandstop filters. The narrow-band and lossy na- tureoftheseresonatorsarestillconcerns. It is not our intention to cover all the novel features in this IV. ADDITIONALOBSERVATIONS TRANSACTIONS’ Table of Contents, but we have listed these Veselago’s original paper [1] is a classic “what-if” paper. It few examples as illustrative of their flavor. The field has been asks,“Whatifamediumweretohavenegativevaluesof and moving ahead nicely, and we hope that this TRANSACTIONS’ instead of positive values? In what ways would the electro- Special Issue will serve to stimulate further creativity, partic- dynamicbehaviorsbedifferent?”Ifthebehaviorsarenotmuch ularlyinthedirectionofnovelmicrowavecomponents. different,ortheresultsotherwiseuninteresting,thestudywould Intheseintroductoryremarks,wehavepresentedmuchinthe servetotellusthatthestructureisnotworthinvestigatingfur- way of background information in an attempt to be of help to ther.Or,iftheresultsareinteresting,butthereisnowayinwhich thosewhoarelesswellinformedonthistopicandareinneed measurements can be made, the structure must be set aside in ofsomeguidance. IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.53,NO.4,APRIL2005 1415 We wish to express our sincere thanks to our many re- the review process, and diligently made the modifications re- viewers who conscientiously contributed their time and effort quested by the reviewers. We are particularly indebted to this in order to help us produce a quality Special Issue. We re- TRANSACTIONS’ Editor-in-Chief, Prof. Michael B. Steer, for ceived almost 40 submitted manuscripts and, in many cases, providing the necessary guidance in both matters of principle we needed to call on the same reviewer to evaluate more andinthemanyproceduraldetails.Wealsogreatlyappreciate thanonemanuscript.Wealsowishtoexpressourappreciation the many ways in which he graciously offered help and kept to those authors who patiently cooperated with the rigors of us on track. List of Reviewers AlkimAkyurtlu DavidJackson HiroshiShigesawa IkuoAwai DieterJaeger DanSievenpiper MarioSorollaAyza RolfJakoby AriSihvola MaurizioBozzi VikramJandhyala RichardSnyder PaoloBurghignoli MikkoKärkkäinen AlexsanderSochava ChristopheCaloz ShigeoKawasaki RobertoSorrentino SteveCammer EdwardKuester CostasSoukoulis FilippoCapolino YasuoKuga SergeiTretyakov KaiChang PaoloLampariello MikioTsuji WengChoChew RalphLevy Ching-KuangC.Tzuang PeterdeMaagt Le-WeiLi TetsuyaUeda MichaelDydyk DidierLippens YuanxunWang GeorgeEleftheriades Wen-taoLu DylanWilliams NaderEngheta RicardoMarqués Bae-IanWu FabrizioFrezza WolfgangMenzel KeWu SpartakGevorgian FranciscoMesa FanYang ThomaszGregorczyk EricMichielssen JunhoYeo WojciechGwarek HosseinMosallaei TsukasaYoneyama WolfgangHoefer MichalMrozowski JanZehentner YasushiHorii MichalOkoniewski RichardZiolkowski AkiraIshimaru AtsushiSanada TATSUOITOH,GuestEditor UniversityofCaliforniaatLosAngeles(UCLA) DepartmentofElectricalEngineering LosAngeles, CA 90095-1594 USA ARTHURA.OLINER,GuestEditor PolytechnicUniversity DepartmentofElectricalEngineering Lexington, MA 02421 USA REFERENCES [4] A.A.Oliner,“A periodic-structurenegative-refractive-indexmedium withoutresonantelements,”inIEEEAP-S/URSIInt.Symp.Dig.,San [1] V. G. Veselago, “The electrodynamics of substances with simulta- Antonio,TX,Jun.16–21,2002,p.41. neously negative values of " and (cid:22),” Sov. Phys.—Usp., vol. 10, pp. [5] A.K.IyerandG.V.Eleftheriades,“Negativerefractiveindexmetamate- 509–514,Jan.–Feb.1968. [2] J.B.Pendry,A.J.Holden,D.J.Robbins,andW.J.Stewart,“Magnetism rialssupporting2Dwaves,”inIEEEMTT-SInt.MicrowaveSymp.Dig., fromconductorsandenhancednonlinearphenomena,”IEEETrans.Mi- Seattle,WA,Jun.2–7,2002,pp.1067–1070. crow.TheoryTech.,vol.47,no.11,pp.2075–2084,Nov.1999. [6] C.Caloz,H.Okabe,T.Iwai,andT.Itoh,“Transmissionlineapproach [3] R.A.Shelby,D.R.Smith,andS.Schultz,“Experimentalverificationof ofleft-handedmaterials,”inIEEEAP-S/URSIInt.Symp.Dig.,SanAn- anegativeindexofrefraction,”Science,vol.292,pp.77–79,Apr.2001. tonio,TX,Jun.16–21,2002,p.39. 1416 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.53,NO.4,APRIL2005 TatsuoItoh(S’69–M’69–SM’74–F’82)receivedthePh.D.degreeinelectricalengineeringfrom theUniversityofIllinoisatUrbana-Champaign,in1969. FromSeptember1966toApril1976,hewaswiththeElectricalEngineeringDepartment,Uni- versityofIllinoisatUrbana-Champaign.FromApril1976toAugust1977,hewasaSeniorRe- search Engineer with the Radio Physics Laboratory, SRI International, Menlo Park, CA. From August1977toJune1978,hewasanAssociateProfessorwiththeUniversityofKentucky,Lex- ington.InJuly1978,hejoinedthefacultyatTheUniversityofTexasatAustin,wherehebecamea ProfessorofElectricalEngineeringin1981andDirectoroftheElectricalEngineeringResearch Laboratory in 1984. During the summer of 1979, he was a Guest Researcher with AEG-Tele- funken,Ulm,Germany.InSeptember1983,hewasselectedtoholdtheHaydenHeadCentennial ProfessorshipofEngineeringatTheUniversityofTexasatAustin.InSeptember1984,hewas appointedAssociateChairmanforResearchandPlanningoftheElectricalandComputerEngi- neeringDepartment,TheUniversityofTexasatAustin.InJanuary1991,hejoinedtheUniversity ofCaliforniaatLosAngeles(UCLA)asProfessorofElectricalEngineeringandHolderoftheTRWEndowedChairinMicrowave and Millimeter Wave Electronics. He was an Honorary Visiting Professor with the Nanjing Institute of Technology, Nanjing, China,andattheJapanDefenseAcademy.InApril1994,hewasappointedanAdjunctResearchOfficerwiththeCommunica- tionsResearchLaboratory,MinistryofPostandTelecommunication,Japan.HecurrentlyholdsaVisitingProfessorshipwithThe UniversityofLeeds,Leeds,U.K.Hehasauthoredorcoauthored350journalpublications,650refereedconferencepresentations, andhaswritten30books/bookchaptersintheareaofmicrowaves,millimeterwaves,antennas,andnumericalelectromagnetics. Hehasgenerated64Ph.D.students. Dr. Itoh is a member of the Institute of Electronics and Communication Engineers of Japan, and Commissions B and D of USNC/URSI.HeservedastheeditoroftheIEEETRANSACTIONSONMICROWAVETHEORY ANDTECHNIQUES(1983–1985).He serves on the Administrative Committee of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S). He was vice presidentoftheIEEEMTT-Sin1989andpresidentin1990.Hewastheeditor-in-chiefofIEEEMICROWAVEANDGUIDEDWAVE LETTERS(1991–1994).HewaselectedanHonoraryLifeMemberoftheIEEEMTT-Sin1994.Hewaselectedamemberofthe NationalAcademyofEngineeringin2003.HewasthechairmanoftheUSNC/URSICommissionD(1988–1990)andchairman ofCommissionDoftheInternationalURSI(1993–1996).HeischairoftheLongRangePlanningCommitteeoftheURSI.He servesonadvisoryboardsandcommitteesfornumerousorganizations.Hehasbeentherecipientofnumerousawardsincluding the1998ShidaAwardpresentedbytheJapaneseMinistryofPostandTelecommunications,the1998JapanMicrowavePrize,the 2000IEEEThirdMillenniumMedal,andthe2000IEEEMTT-SDistinguishedEducatorAward. IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.53,NO.4,APRIL2005 1417 ArthurA.Oliner(M’47–SM’52–F’61–LF’87)wasbornonMarch5,1921,inShanghai,China. HereceivedtheB.A.degreefromBrooklynCollege,Brooklyn,NY,in1941,andthePh.D.degree fromCornellUniversity,Ithaca,NY,in1946,bothinphysics. He joined the Polytechnic Institute of Brooklyn (now Polytechnic University) in 1946, and became Professor in 1957. He then served as Department Head from 1966 to 1974, and was Director of its Microwave Research Institute from 1967 to 1982. He was a Walker-Ames Vis- itingProfessorwiththeUniversityofWashington,in1964.HehasalsobeenaVisitingProfessor withthe Catholic University,Rio de Janeiro,Brazil,the TokyoInstituteofTechnology, Tokyo, Japan,theCentralChinaInstituteofScienceandTechnology,Wuhan,China,andtheUniversity of Rome, Rome, Italy. In 2003, the University of Rome (“La Sapienza”) granted him an Hon- oraryDoctorate,andorganizedanassociatedspecialsymposiuminhishonor.Heisamemberof the BoardofDirectorsofMerrimac Industries.Hehasauthored over300papers, variousbook chapters, and has coauthored or coedited three books. His research has covered a wide variety of topics in the microwave field including network representations of microwave structures, guided-wave theory with stress on surfacewavesandleakywaves,wavesinplasmas,periodicstructuretheory,andphased-arrayantennas.Hehasmadepioneering andfundamentalcontributionsinseveraloftheseareas.Hisinterestshavealsoincludedwaveguidesforsurfaceacousticwavesand integratedoptics,novelleaky-waveantennasformillimeterwaves,andleakageeffectsinmicrowaveintegratedcircuits.Lately,he hascontributedtothetopicsofmetamaterials,andtoenhancedpropagationthroughsubwavelengthholes. Dr.OlinerisaFellowoftheAmericanAssociationfortheAdvancementofScience(AAAS)andtheInstitutionofElectrical Engineers(IEE),U.K.HewasaGuggenheimFellow.HewaselectedamemberoftheNationalAcademyofEngineeringin1991. Hehasbeentherecipientofprizesfortwoofhispapers:theIEEEMicrowavePrizein1967forhisworkonstrip-linediscontinu- ities,andtheInstitutionPremiumoftheIEEin1964forhiscomprehensivestudiesofcomplexwavetypesguidedbyinterfacesand layers.HewasPresidentoftheIEEEMicrowaveTheoryandTechniquesSociety(IEEEMTT-S),itsfirstDistinguishedLecturer, andamemberoftheIEEEPublicationBoard.HeisanHonoraryLifeMemberofIEEEMTT-S(oneofonlysixsuchpersons). In 1982, he was the recipient of its highest recognition, the MicrowaveCareer Award. A specialretrospective session was held in his honor at the IEEE MTT-S International Microwave Symposium (reported in detail in the December 1988 issue of this TRANSACTIONS,pp.1578-1581).In1993,hewasthefirstrecipientoftheDistinguishedEducatorAwardoftheIEEEMTT-S.He wasalsoarecipientoftheIEEECentennialandMillenniumMedals.HeisalsoapastU.S.chairmanofCommissionsAandDof theInternationalUnionofRadioScience(URSI),along-timememberofandactivecontributortoCommissionB,andaformer memberoftheU.S.NationalCommitteeofURSI.In1990,hewastherecipientoftheURSIvanderPolGoldMedal,whichis giventriennially,forhiscontributionstoleakywaves.In2000,theIEEEawardedhimasecondgoldmedal,theHeinrichHertz Medal,whichisitshighestawardintheareaofelectromagneticwaves. 1418 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.53,NO.4,APRIL2005 Homogenization of 3-D-Connected and Nonconnected Wire Metamaterials MárioG.Silveirinha,Member,IEEE,andCarlosA.Fernandes,Member,IEEE Abstract—Thehomogenizationofcompositestructuresmadeof long thin metallic wires is an important problem in electromag- neticsbecausetheyareoneofthebasiccomponentsofthedouble- negativemedium.Inthispaper,weproposeanewanalyticalmodel tocharacterizetheeffectivepermittivityofthethree-dimensional- wiremediuminthelongwavelengthlimit.Westudytwodifferent topologies for the wire medium. The first structure consists of a latticeofconnectedwires,whereasthesecondoneconsistsofalat- ticeinwhichthewiresarenotconnected.Ourresultsshowthatthe propagationofelectromagneticwavesinthetwometamaterialsis verydifferent.Whileoneofthestructuresexhibitsstrongspatial dispersion,theotheroneseemstobeagoodcandidateforimpor- tantmetamaterialapplications.Wealsofoundthat,forextremely Fig.1. (a)Fragmentofametamaterialformedbyalatticeofconnectedwires. lowfrequencies,oneofthestructuressupportsmodeswithhyper- (b) Fragment of a metamaterial formed by a lattice of nonconnected wires bolicwavenormalcontours,originatingnegativerefractionatan (adjacentorthogonalwiresarespacedofhalf-latticeconstant). interfacewithair.Wevalidatedourtheoreticalresultswithnumer- icalsimulations. Theconfigurationoriginallyproposedin[1]ischaracterized IndexTerms—Double-negative(DNG)medium,homogenization byDNGparametersonlyforauniquepolarization,andforprop- theory,metamaterials,negativerefraction,wiremedium. agation along a specific direction of space. The fabrication of anisotropicDNGmediumclearlycompelsforabasiccellwith extrasymmetry.Apparently,apartfrompracticaldifficultiesre- I. INTRODUCTION latedtotechnologicallimitations,thegeneralizationseemstobe IN [1],Smith et al. proposed an original structure thatcon- straightforward. sists of a lattice of long metallic wires and split-ring res- However,thesituationmaynotbesoplain.Forexample,one onators(SRRs).Theoreticalandexperimentalstudiesshowthat might expect that a medium formed by wires that are parallel the composite structure behaves as a double negative (DNG) to the coordinate axes would interact with the radiation as an medium (also known as left-handed medium) with a negative isotropicmediumwithnegativepermittivity.However,numer- index of refraction [1]–[3]. This remarkable result motivated ical results reported by Silveirinha and Fernandes [8] support muchresearchonmetamaterialsandpotentialapplications.For that,atleastforthenonconnectedtopologystudiedin[8],that example,numerousstudiestrytoexplorethefocusingproperty is not the case. Indeed, we found that near the “plasma fre- ofaDNGslabsurroundedbyconventionalmedia[4]–[6].Itwas quency” the referred metamaterial is not isotropic. Moreover, evensuggested,withconsiderablecontroversy,thatmetamate- weverifiedthatthenumericalresultsareconsistentwiththehy- rials could be used to fabricate a super lens with no limit of pothesis that, analogous with the one-dimensional (1-D)-wire resolution[6].Otherstudiesshowthatmetamaterialsmayfavor mediumformedbyanarrayofparallelwires[9]andotherre- theminiaturizationofsomedevices,andtherealizationofsub- latedstructures[10],theconsideredgeometryforthethree-di- wavelengthcavityresonators[7]. mensional (3-D)-wire medium has strong spatial dispersion in ThenegativerefractionoftheDNGmetamaterialisattributed thelongwavelengthlimit,i.e.,thepermittivitydependsnotonly to the combined effect of the arrays of wires and SRRs. As is onthefrequency,butalsoonthewavevector. well known, the lattice of wires is modeled as a medium with The results reported in [8] raise the obvious question: “Is negativepermittivity,whereasthelatticeofSRRsischaracter- itpossibletofabricateanisotropicmetamaterialwithnegative izedbyanegativepermeability.Toafirst-orderapproximation, permittivity?”Thisisafundamentalsubjectnotonlytounder- thereisnocouplingbetweenthetwobasicinclusions. stand the possible limitations of DNG metamaterials, but also becausemetamaterialswithnegativepermittivity,whenpaired withmetamaterialswithnegativepermeability,mayhaveinter- ManuscriptreceivedMay20,2004;revisedSeptember20,2004.Thiswork wassupportedbytheFundaçãoparaCiênciaeaTecnologiaunderProjectPOSI estingpropertiesandapplications,asdescribedin[11]. 34860/99. Inthispaper,westudytwodifferenttopologiesforthe3-D- M. G. Silveirinha is with the Electrical Engineering Department–Instituto wiremedium.Thefirststructureconsistsofasimplecubiclat- de Telecomunicações, Polo II da Universidade de Coimbra, 3030 Coimbra, Portugal(e-mail:[email protected]). ticeofconnectedwires,andthesecondoneconsistsofasimple C. A. Fernandes is with the Instituto Superior Técnico–Instituto de cubiclatticeinwhichthewiresarenotconnected(i.e.,thege- Telecomunicações,TechnicalUniversityofLisbon,1049-001Lisbon,Portugal ometry studied in [8] using numerical methods). In Fig. 1, we (e-mail:[email protected]). DigitalObjectIdentifier10.1109/TMTT.2005.845128 depictfragmentsofbothmetamaterials. 0018-9480/$20.00©2005IEEE SILVEIRINHAANDFERNANDES:HOMOGENIZATIONOF3-D-CONNECTEDANDNONCONNECTEDWIREMETAMATERIALS 1419 A. ConnectedTopology In this case, the metallic region in the unit cell consists of threecylindricalwireswithradius andlength .Wehave (1) Intheabove, representsthesurfaceofthewiresection oriented along the -direction. Note that the wires intersect Fig. 2. Geometry of the unit cell. (a) Connected wire geometry. mutually,formingajunctionneartheorigin. (b)Nonconnectedwiregeometry. Ourobjectiveistoproposeananalyticaltreatmentfortheho- mogenizationproblem.Thus,itisdesirabletohaveamodelas We propose a new analytical procedure to homogenize the simpleaspossibleforthecurrentthatflowsalongthewires.We will assume that , where is the wavelength of periodic metamaterials. We will prove that the two topologies radiation in the dielectric region. Therefore, the thin-wire ap- forthe3-D-wiremediumarenotequivalent,andthattheinter- proximationcanbeused. actionofelectromagneticwaveswiththetwostructuresisvery Within the thin-wire approximation, the density of current different.Wewillderiveanapproximateanalyticalformulafor over each wire flows along the direction of the axis, and theeffectivepermittivityofthecompositestructures.Toourbest has approximately circular symmetry. We will also admit that knowledge, no results were reported in the literature that con- is a traveling wave characterized by the wave vector cern the homogenization of related structures with the excep- (thejustificationforthisassumptionwillbegiven tionofthenumericalanalysisdescribedin[8],andtheexperi- inSectionIII).Therefore,weadmitthat mentalresultsreportedin[12](thegeometryconsideredin[12] isslightlymorecomplexthantheonesconsideredhere). This paper is organized as follows.1 Sections II–IV concern (2) the characterization of the Floquet modes that propagate in the periodic material. In Section II, we describe the geometry where is a periodic function that stands for the unknown of the wire medium. In Section III, we formulate the modal current. In order to take into account the coupling at the junc- problem, and we obtain the characteristic system using an tion, we admit the possibility of the current induced over the integral-equation-based approach. In Section IV, we propose th wire being discontinuous at . In general, we have a simplified characteristic system that can be solved using .Theconservationofcurrentre- analytical methods. Sections V and VI concern the homoge- quiresthat(seealso[13]) nizationofthemetamaterials.InSectionV,weexplainhowthe effectivepermittivitydyadic canbe obtained directlyfrom the (3) characteristic system. In Section VI, we discuss the physical phenomena implied by our formulas, and we compare the developed theory with full-wave numerical results. Finally, in Notethat,withinthedescribedmodel,wecompletelyneglect the shape of the wire junctions. Essentially, in our model, the SectionVII,wepresentconclusions. junctions are replaced by infinitesimal gaps located at (i.e.,thepointswhereweallowthecurrenttobedis- II. GEOMETRY continuous)withacirculartransversesection.Nevertheless,our We denote a generic point of space by , simplified model is expected to yield good results since each andtheunitvectordirectedalongthe -directionby junctionoccupiesonlyaverysmallfractionoftheunitcell. .Thewiremediumisobtainedbytheperiodicrepetition oftheunitcell showninFig.2.Theunitcelliscenteredatthe B. NonconnectedTopology origin. The wires are arranged into a simple cubic lattice with The nonconnected geometry is depicted in Fig. 2(b). As in lattice constant . The boundary of the metallic region in the the previous case, the metallic region in the unit cell consists unitcellisdenotedby ,andtheoutwardunitnormalvector ofthreecylindricalwireswithradius andlength .Nowthe isdenotedby .Forsimplicity,weadmitthatthewiresareem- wiresdonotintersect,andthewireaxesarespacedby ,i.e., beddedinair.Weconsidertwodifferenttopologiesforthewire ahalf-latticeconstant.Thewireorientedalongthe -direction medium. The geometry shown in Fig. 2(a) corresponds to the iscenteredat connected case, and the geometry of Fig. 2(b) corresponds to the nonconnected case. In Sections II-A and –B, we describe (4) eachconfigurationindetail. Asbefore,weadmitthatthedensityofcurrentoverthe th 1Whilethispaperwasbeingprepared,SimovskiandBelovstudiedtheho- wireisgivenby(2).Now isacontinuousperiodicfunction mogenizationofthenonconnectedwiremediumusingalocalfieldapproach [23]. becausetherearenojunctions.

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