Plane waves with negative phase velocity in Farada

arXiv:physics/0402057v1[physics.optics]11Feb2004RevisedonOctober3,2003PlanewaveswithnegativephasevelocityinFaradaychiralmediumsTomG.Mackay∗SchoolofMathematics,UniversityofEdinburghJamesClerkMaxwellBuilding,TheKing’sBuildingsEdinburghEH93JZ,UnitedKingdomAkhleshLakhtakia†CATMAS—ComputationalandTheoreticalMaterialsSciencesGroupDepartmentofEngineeringScienceandMechanicsPennsylvaniaStateUniversity,UniversityPark,PA16802–6812,USAAbstract.ThepropagationofplanewavesinaFaradaychiralmediumisinvestigated.Condi-tionsforthephasevelocitytobedirectedoppositetothedirectionofpowerflowarederivedforpropagationinanarbitrarydirection;simplifiedconditionswhichapplytopropagationparalleltothedistinguishedaxisarealsoestablished.Thesenegativephase–velocityconditionsareexplorednumericallyusingarepresentativeFaradaychiralmedium,arisingfromthehomogenizationofanisotropicchiralmediumandamagneticallybiasedferrite.Itisdemonstratedthatthephasevelocitymaybedirectedoppositetopowerflow,providedthatthegyrotropicparameteroftheferritecom-ponentmediumissufficientlylargecomparedwiththecorrespondingnongyrotropicpermeabilityparameters.PACSnumber(s):41.20.Jb,42.25.Bs,83.80.Ab1IntroductionHomogeneousmediumswhichsupportthepropagationofwaveswithphasevelocitydirectedop-positetothedirectionofpowerflowhaveattractedmuchattentionlately[1,2,3].Thearchetypalexampleofsuchamediumistheisotropicdielectric–magneticmediumwithsimultaneouslynega-tiverealpermittivityandnegativerealpermeabilityscalars,asfirstdescribedVeselagointhelate1960s[4].Arangeofexoticandpotentiallyusefulelectromagneticphenomenons,suchasnegativerefraction,inverseDopplershift,andinverse˘Cerenkovradiation,werepredictedforthistypeofmedium[3,4].Recentexperimentalstudiesinvolvingthemicrowaveilluminationofcertaincom-positemetamaterials[5,6]—whichfollowedonfromearlierworksofPendryetal.[7,8]—aresupportiveofVeselago’spredictionsandhavepromptedanintensificationofinterestinthisarea.∗Fax:[phone];e–mail:[e-mail]†Fax:[phone];e–mail:[e-mail]1Inparticular,ageneralcondition—applicabletodissipativeisotropicdielectric–magneticmedi-ums—hasbeenderivedforthephasevelocitytobedirectedoppositetopowerflow[9];andthisconditionshowsthattherealpartsofboththepermittivityandthepermeabilityscalarsdonothavetobenegative.Aconsensushasyettobereachedonterminology.Forthepresentpurposes,amediumsupport-ingwavepropagationwithphasevelocitydirectedoppositetopowerflowismostaptlyreferredtoasanegativephase–velocitymedium.However,thereaderisalertedthatalternativeterms,suchasleft–handedmaterial[3],backwardmedium[10],double–negativemedium[1],andnegative–indexmedium[11],arealsoincirculation.Adiscussionofthisissueisavailableelsewhere[12].Thescopeforthephasevelocitytobedirectedoppositetopowerflowmaybegreatlyex-tendedbyconsideringnon–isotropicmediums,ashasbeenindicatedbyconsiderationsofuniaxialdielectric–magneticmediums[10,13,14].Thefocusofthepresentcommunicationisthepropa-gationofnegativephase–velocityplanewavesinFaradaychiralmediums(FCMs)[15,16].Thesemediumscombinenaturalopticalactivity—asexhibitedbyisotropicchiralmediums[17]—withFaradayrotation—asexhibitedbygyrotropicmediums[18,19,20].AFCMmaybetheoreticallyconceptualizedasahomogenizedcompositemedium(HCM)arisingfromtheblendingtogetherofanisotropicchiralmediumwitheitheramagneticallybiasedferrite[21]oramagneticallybiasedplasma[22].TheHCMcomponentmediumsareenvisionedasrandomparticulatedistributions.Thehomogenizationprocessisjustifiedprovidedthattheparticulatelengthscalesinthemixtureofcomponentsaresmallcomparedwithelectromagneticwavelengths.Avastliteratureontheesti-mationofconstitutiveparametersofHCMsexists;seeRefs.[23,24],forexample.TheconstitutiverelationsofFCMshavebeenrigorouslyestablishedforsometime[16],althoughinappropriateusestilloccurs[25].Inthefollowingsections,wavenumbersandcorrespondingelectricfieldphasorsaredelineatedfromeigenvalue/vectoranalysisforplanewavepropagationinanarbitrarydirection.Simpli-fiedexpressionsforthesequantitiesarederivedforpropagationparalleltothebiasing(quasi)–magnetostaticfield[19,p.71].Ageneralconditionforthephasevelocitytobedirectedoppositetopowerflowisestablished.Thetheoreticalanalysisisillustratedbymeansofarepresentativenumericalexample:theconstitutiveparametersofFCMsarisingfromaspecifichomogenizationscenarioareestimatedandthenusedtoexplorethewavepropagationcharacteristics.Asregardsnotation,vectorsareunderlinedwhereasdyadicsaredoubleunderlined.Allelectro-magneticfieldphasorsandconstitutiveparametersdependimplicitlyonthecircularfrequencyωoftheelectromagneticfield.Unitvectorsaredenotedbythesuperscriptˆsymbol,whileI=ˆxˆx+ˆyˆy+ˆzˆzistheidentitydyadic.Thecomplexconjugateofaquantityqiswrittenasq∗;therealpartofqiswrittenasRe{q}.Thefree–space(i.e.,vacuum)wavenumberisdenotedbyk0=ω√ǫ0μ0whereǫ0andμ0arethepermittivityandpermeabilityoffreespace,respectively;andη0=pμ0/ǫ0representstheintrinsicimpedanceoffreespace.22Analysis2.1PreliminariesThepropagationofplanewaveswithfieldphasorsE(r)=E0exp(ik0˜kˆu•r)H(r)=H0exp(ik0˜kˆu•r))(1)inaFCMisconsidered.Suchamediumischaracterizedbythefreq