Cloaking a sensor for threedimensional Maxwell s equat - PDF document

Cloakingasensorforthree-dimensionalMaxwell'sequations:transformationopticsapproachXudongChen1;andGuntherUhlmann2;31DepartmentofElectricalandComputerEngineering,NationalUniversityofSingapore,117576Singapore2DepartmentofMathematics,UniversityofCaliforniaatIrvine,Irvine,CA92697,USA3DepartmentofMathematics,UniversityofWashington,Seattle,WA98195,USA elechenx@nus.edu.sg Abstract:Theidealtransformationopticscloakingisaccompaniedbyshielding:externalobservationsdonotprovideanyindicationofthepresenceofacloakedobject,norisanyinformationabouttheeldsoutsidedetectableinsidethecloakedregion.Inthispaper,atransformationisproposedtocloakthree-dimensionalobjectsforelectromagneticwavesinsensormode,i.e.,cloakingaccompaniedbydegradedshielding.Theproposedtransformationtacklesthedifcultycausedbythefactthatthelowestmultipoleinthree-dimensionalelectromagneticradiationisdipoleratherthanmonopole.Thelossofthesurfaceimpedanceofthesensorplaysanimportantroleindeterminingthecloakingmodes:idealcloaking,sensorcloakingandresonance.©2011OpticalSocietyofAmericaOCIScodes:(260.2710)Inhomogeneousopticalmedia;(260.2110)Electromagnetictheory;(160.1190)Anisotropicopticalmaterials;(290.3200)Inversescattering. 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1.IntroductionTransformationopticshasbeenapopularresearchtopicinthepastfewyearsduetoitsexcitingpropertyofdesigningcloakswhichcancompletelyhideobjectsfromelectromagneticdetec-tion.ThefundamentalideaistheinvarianceofMaxwell'sequationsunderaspace-deformingtransformationifthematerialpropertiesarealteredaccordingly[1–9].However,theidealcloakingisaccompaniedbyshielding:Thereisadecouplingoftheeldsinsideandoutsideofthecloakedregion,sothatexternalobservationsdonotprovideanyindicationofthepresenceofacloakedobject,norisanyinformationabouttheeldsoutsidedetectableinsidethecloakedregion.Inmanyreal-worldapplications,however,thereareneedsforeffectivelycloakingsen-sorsanddetectorssothattheirpresencemaybelessdisturbingtothesurroundingenvironment.Forexample,inareceivingantennaarray,weaimatreducingthecouplingamongantennael-ementssothatanelementisabletoreceiveelectromagneticsignalwithoutdisturbingtheeldreceivedbyotherelements.Whenthenear-eldscanningopticalmicroscope(NSOM)operatesincollectionmode,thetipshouldbeveryclosetothescatteringobjects,whichinevitablyyieldsundesiredmultiplescatteringbetweentipandscattererandthusdegradestheaccuracyofthemeasuredeld.Inthepastfewyears,therehavebeenattemptstodesignsensorsthatareabletoperceiveelectromagneticradiationandatthesametimehavenegligibledisturbancetothesurroundingenvironment,whichisreferredtoas`cloakingsensors'or`cloakinsensormode'.Therstpaperinthisdirectionis[10],whereaplasmoniccoatingallowselectromagneticwavetoreachthesensorandatthesametimethescatteringduetothesensoriscanceledoutbythatofthecloak.TheideahasbeenappliedtocloakaNSOMtipforthepurposeofneareldimaging[11],whichisofgreatindustrialandscienticsignicance.However,therearealsosomelimitationsofthemethod.Forexample,itcanonlycloaksensorssmallerorcomparabletowavelengthandtheshapeandmaterialofthecloakdependonthegeometricalandphysicalpropertiesofthesensor.Anotherapproachtocloakasensoristransformationoptics.In[12],asensorthatiscladdedwithaspherewithsurfaceimpedanceisplacedinsidethecloak.Thismethodworksforsensorsofarbitrarysizeandshapeaslongasthesensorsareabletomeasuretheeldattheboundaryofthecladdingsphere.Asmentionedin[12],thistransformationopticsapproachisabletoachieveincloakingasensorforacousticwavesandtwo-dimensional(2D)electromagneticwaves.However,afurthercalculationshowsthatthetransformationopticsapproachproposedin[12]cannotcloakasensorforthree-dimensional(3D)electromagneticwaves.Thisismainlyduetothefactthatthelowestmultipolein3Delectromagneticradiationisdipoleratherthanmonopole.Theasymptoticbehavior,forsmallarguments,ofRiccati-Besselfunctionsoforderoneandaboveleadstovanishingelectromagneticeldsinsidethecloak,i.e.,theshieldingeffectisstillhandinhandwiththecloakingeffect.Anothertransformationopticsapproachispresentedin[13,14],wherebothacloaklayerandananti-cloaklayerplaytheroletoachievecloakingsensoreffect.Whereasthisapproachcancloaksensorsofarbitrarysizeandworksforboth2Dand3Delectromagneticwaves,thesensorismodeledasadielectricsphere andthepapersdonotdiscusswhetherthemethodworksforagenericsensor,i.e.,whetherboththecloaklayerandanti-cloaklayerdependontheshapeandmaterialofthesensor.Inaddition,thecloakingdeviceconsistsofvelayers,eightunknownsforeachorderofmultipole,andtwosmallparameterstobeadjusted,whichmakesittedioustoderiveanalyticalsolutions.Inthispaper,weproposeanoveltransformationopticsmethodtocloakasensor,whichcanbeofarbitrarysize,shape,andmaterial,for3Delectromagneticwaves.Thecloakingdeviceconsistsinthreelayers,veunknownsforeachorderofmultipole,andonesmallparametertobeadjusted.Analyticalresultscanbeobtainedforeachofveunknownsevenwithoutresourcetosmallparameterapproximation.Comparedwith[12],theproposedtransformation,fromthephysicalspacetothevirtualspace,circumventsthedifcultycausedbythefactthatthelowestmultipolein3Delectromagneticradiationisdipoleratherthanmonopole.Thelossofthesurfaceimpedanceofthesensorplaysanimportantroleindeterminingthecloakingmodes:idealcloaking,sensorcloakingandresonance.Numericalsimulationsvalidatetheproposedtransformationmodel.2.CongurationofthecloakandthesensorAsshowninFig.1.ThesensorisplacedinsideasphereofradiusR0,withsurfaceimpedanceboundaryconditionEq=HfEf=Hqa0(see[15]andreferencestherein).Thesensorisabletomeasurethetangentialelectriceldsatthesurfaceofthesphere.ThecloaklayeriswithinanannuluswithinnerandouterradiiR1andR2,respectively.Allthreespheresmentionedaboveareconcentric,withthecenterbeingtheoriginofthecoordinatesysteminthephysicalspace.ThespacebetweenspheresofradiiR0andR1,aswellasthespaceoutsideofthesphereofradiusR2,arefreespace.Thepermittivityandpermeabilityofthecloaklayerareobtainedusing (a)(b) Fig.1.Thegeometryofthesensorandthetransformationfunction.(a)AsensorofarbitraryshapeisplacedinsideaspherewithradiusR0andsurfaceimpedancea0,anditisabletomeasurethetangentialelectriceldonthesphericalsurface.ThecloaklayerisinbetweenspheresofradiiR1+dandR2,wheredisansmallpositivenumberandisnotshowninthegureduetoitssmallmagnitude.Otherregionsarefreespace.(b)Thetransformationr0=(r)fromthephysicalspacetothevirtualspace. theinvarianceofMaxwell'sequationsundertransformationofthespatialcoordinatesystems.Consideracoordinatetransformationbetweenthevirtualspace(curvedfreespace)withthephysicalspace.Theformerhasspatialcoordinatesr0;q0;f0,permittivity0,andpermeabilitym0,whereasthelatterhasspatialcoordinatesr;q;f,andparameters ; m.Thetransformationisonlyintheradialdirection,i.e.,r0f(r);q0q;f0f:(1)Theassociatedpermittivityandpermeabilitytensorsaregivenby r(r)ˆrˆrt(r)ˆqˆqt(r)ˆfˆf(2) mmr(r)ˆrˆrmt(r)ˆqˆqmt(r)ˆfˆf;(3)wherer0f2(r) r2f0(r);t0f0(r)(4)mrm0f2(r) r2f0(r);mtm0f0(r):(5)Itiswellknownthatthetransformthatsatisesbothf(R1)=0andf(R2)=R2yieldsperfectinvisibility.Toavoidsingularities,welettheboundaryoftheinnercloakingmaterialbeatR1d,wheredisasmallpositivenumber.Theincidentwaveisgeneratedbyatime-harmonic[exp(iwt)]sourcethatislocatedoutsideofthesphereofradiusR2.Toachievecloakingasensor,weaimatobtaininganegligiblescatteredeldoutsideofthesphereofradiusR2andatthesametimeperceivableelectromagneticeldatthesurfaceofthecladdingsphere.3.AnalyticalresultsforelectromagneticeldsElectromagneticeldscanbedecomposedintotwoindependentmodes,TEandTMmodes(transversetoradialdirectioninsphericalcoordinatesystem),whicharedualtoeachother.FortheTMmode,theBeldcanbeexpressedasBшrf0FM
;(6)wherethescalarpotentialFMsatises1 f2sinq¶ ¶qsinq¶FM ¶q1 sinq¶2FM ¶f2¶2FM ¶f2k20FM0;(7)wherek20w20m0.SeparationofvariablesyieldsFMNån=1nåm=nAnmˆBn(k0f)Ymn(q;f);(8)whereNisthehighestordermultipoleusedinthenumericalsimulations,ˆBn(z)=zbn(z)istheRiccati-Besselfunction,andYmnaresphericalharmonics.NowthemagneticeldHandtheelectriceldEcanbeexpressedasHm101 rsinq¶FM ¶fˆqm101 r¶FM ¶qˆf(9)Em1010 iwf0¶2FM ¶f2k20FMˆr1 r¶2FM ¶f¶qˆq1 rsinq¶2FM ¶f¶fˆf:(10) ThepotentialFMinthethreeregionsiswrittenasFextM(r;q;f)=ånåmhKnmˆJn(k0r)+AnmˆH(1)n(k0r)iYmn(q;f);forrR2(11)FcloM(r;q;f)=ånåmhBnmˆJn(k0f)+CnmˆH(1)n(k0f)iYmn(q;f);forR1drR2(12)FintM(r;q;f)=ånåmhDnmˆJn(k0r)+EnmˆH(1)n(k0r)iYmn(q;f);forR0rR1d:(13)Sincethesourceisgiven,Knmisuniquelydetermined.Foreachmultipoleofordern,wewillsolveforveunknowsAnm,Bnm,Cnm,Dnm,andEnm.Eqs.(9)and(10)indicatethatthecontinu-itiesoftangentialcomponentsofHandEacrosstheboundariesrR1dandrR2amounttothecontinuitiesofFMand¶M ¶(k0f),respectively.Theimpedanceboundaryconditionreducesto¶M ¶(k0f)aFM0,whereaisproportionaltoa0.Thus,wehavethefollowingboundaryconditionsBnmˆJn(k0R2)+CnmˆH(1)n(k0R2)=KnmˆJn(k0R2)+AnmˆH(1)n(k0R2)(14)BnmˆJ0n(k0R2)+CnmˆH(1)0n(k0R2)=KnmˆJ0n(k0R2)+AnmˆH(1)0n(k0R2)(15)BnmˆJn(k0(R1+d))+CnmˆH(1)n(k0(R1+d))=DnmˆJn(k0(R1+d))+EnmˆH(1)n(k0(R1+d))(16)BnmˆJ0n(k0(R1+d))+CnmˆH(1)0n(k0(R1+d))=DnmˆJ0n(k0(R1+d))+EnmˆH(1)0n(k0(R1+d))(17)DnmˆJ0n(k0R0)+EnmˆH(1)0n(k0R0)=ahDnmˆJn(k0R0)+EnmˆH(1)n(k0R0)i:(18)ItiseasytoseethatEqs.(14)and(15)yieldBnmKnmandAnmCnm.TherearetwocasesforEq.(18).3.1.Case1WhenˆJ0n(k0R0)aˆJn(k0R0)0,wehaveDnmˆH(1)0n(k0R0)aˆH(1)n(k0R0) ˆJ0n(k0R0)aˆJn(k0R0)Enmg(a)Enm(19)NowwesolvefortwounknownsAnmandEnmfromtwolinearequations,KnmˆJn(k0(R1+d))+AnmˆH(1)n(k0(R1+d))=Enmhg(a)ˆJn(k0(R1+d))+ˆH(1)n(k0(R1+d))i(20)KnmˆJ0n(k0(R1+d))+AnmˆH(1)0n(k0(R1+d))=Enmhg(a)ˆJ0n(k0(R1+d))+ˆH(1)0n(k0(R1+d))i(21)Weobtain:Enm=Knmi=Fn;(22)Anm=KnmGn=Fn;(23)whereFn=hgˆJn(k0(R1+d))+ˆH(1)n(k0(R1+d))iˆH(1)0n(k0(R1+d))hgˆJ0n(k0(R1+d))+ˆH(1)0n(k0(R1+d))iˆH(1)n(k0(R1+d))(24)Gn=hgˆJn(k0(R1+d))+ˆH(1)n(k0(R1+d))iˆJ0n(k0(R1+d))+hgˆJ0n(k0(R1+d))+ˆH(1)0n(k0(R1+d))iˆJn(k0(R1+d))(25) andtheWronskianˆJnˆH(1)0nˆJ0nˆH(1)niisused.ItisworthhighlightingthatwehaveobtainedanalyticalresultforEnmwithoutusinganyapproximation.Fromhereonwards,wewillusethefactthatdisasmallparametertosimplifytheobtainedanalyticalresult.Foraninnitesimalparameterz,wehavetheasymptoticsˆJn(z)pnzn+1andˆH(1)n(z)qnzn[16].UsingTaylor'sexpansionandthefactthatf(R1)=0,weobtainFngˆJn(k0R1)+ˆH(1)n(k0R1)+k0dhgˆJ0n(k0R1)+ˆH(1)0n(k0R1)i+(k0d)2 2hgˆJ00n(k0R1)+ˆH(1)00n(k0R1)i(nqn)[k0(R1+d)](n+1)ngˆJ0n(k0R1)+ˆH(1)0n(k0R1)+k0dhgˆJ00n(k0R1)+ˆH(1)00n(k0R1)i+O(d2)oqn[k0(R1+d)]n=qn[k0(R1+d)](n+1)(A1+A2+A3+A4+A5+A6);(26)whereA1nhgˆJn(k0R1)+ˆH(1)n(k0R1)i;(27)A2nk0dhgˆJ0n(k0R1)+ˆH(1)0n(k0R1)i;(28)A3k0f(R1d)hgˆJ0n(k0R1)+ˆH(1)0n(k0R1)i;(29)A4k0f(R1d)k0dhgˆJ00n(k0R1)+ˆH(1)00n(k0R1)i;(30)A5n(k0d)2 2hgˆJ00n(k0R1)+ˆH(1)00n(k0R1)i;(31)A6O(d2)f(R1d);(32)wheretheLandaubig-onotationO(
)denotestermsofthesameorder,i.e.,neglectingconstantmultipliersandhigher-orderterms.DependingonwhetherornottheleadingterminFniszero,wediscussthetwofollowingtwocases.3.1.1.Case1.1WhengˆJn(k0R1)+ˆH1n(k0R1)0,Enm Knmiq1n[k0f(R1d)]n+1A11:(33)TofurtherquantifytheasymptoticbehaviorofEnm,wecanexpressthetransformfunctionf(R1d)as,consideringthefactthatf(R1)=0anddissmall,f(R1d)=bdso(ds);(34)forsomebands,wheres0.NotethattheLandaulittle-onotationo(
)denoteshigherorderterms,i.e.,thoseapproachingtozeroatfasterrates.Thus,Enm KnmO(d(n+1)s):(35)FromEq.(23),astraightforwardcalculationgivesthatAnmO(d(2n+1)s).Sincen1,weseebothEnmandAnmapproachzeroasdapproacheszero,indicatingthatcloakingeffectiscoupledwithshieldingeffect,i.e.,thereisnosensoreffect. 3.1.2.Case1.2WhengˆJn(k0R1)+ˆH(1)n(k0R1)=0,theWronskianˆJnˆH(1)0nˆJ0nˆH(1)niimpliesthatgˆJ0n(k0R1)+ˆH(1)0n(k0R1)0.ItisalsoimportanttostressthatthedifferentiationoftheWron-skianwithrespecttok0R1yieldsgˆJ00n(k0R1)+ˆH(1)00n(k0R1)=0.Subsequently,weseefromEq.(28)toEq.(31)that,A20,A30,andA4A50.DuetothepresenceofanonzeroA3,FncontainsatermoforderO(dns),whichisinniteandeventuallyleadstozerovalueofEnmasdgoestozero.Inthiscase,thereisnosensingeffect.Toachieveanon-vanishingEnm,wehavetoeliminateA3andtheonlywayistouseA2tocancelit.TheconditionA2A3o(ds)canbesatisedwhenndbds0,i.e.,s1;bn:(36)TheconditionthatgˆJn(k0R1)+ˆH(1)n(k0R1)=0andEq.(36)canbesatisedbydifferentordermultipoles.Itisimportanttonotethatoncetheaforementionedtwoconditionsaresatisedforaparticularorder,sayn0,theycannotbesimultaneouslysatisedbyothersorders.Fornn0,weeasilyobtainthatEnmO(dn02)andAnmO(d2n01);Fornn0,weobtainfromCase1.1thatEnmO(dn+1)andAnmO(d2n+1).Thebehaviorsofcloakingandshieldingeffectaredifferentforvariousorderofmultipoles.•Caseofn01:E1mO(d1)approachesinnityandA1mO(d1)approacheszero,whichmeanstheresonancemode.Itisworthmentioningthattheinteriorresonancedoesnotdestroythecloakingeffect,whichisdifferentfromtheconclusionof[12].Thisdifferenceispossiblyduetothefactthat[12]usestheCauchydata,i.e.,themappingfromthetotaleldtoitsnormalderivative,whereasthispaperstudiesthemappingfromtheincidenceeldtothescatteredone.•Caseofn02:E2mO(d0)isinthesameorderastheincidencewaveandA2mO(d3)approacheszero,whichmeanssensormode.•Caseofn03:BothEnmandAnmapproachzero,whichmeansidealcloakingmode,i.e.,cloakingeffectandshieldingeffectarehandinhand.FromtheconditionofgˆJn(k0R1)+ˆH(1)n(k0R1)=0andthedenitionofg(a)inEq.(19),weeasilyobtainthevalueofathatleadstotheresonancemode(forn01)andthesensormode(forn02),aˆJ0n0(k0R0)ˆY0n0(k0R0)ˆJn0(k0R1)=ˆYn0(k0R1) ˆJn0(k0R0)ˆYn0(k0R0)ˆJn0(k0R1)=ˆYn0(k0R1):(37)3.2.Case2WhenˆJ0n(k0R0)aˆJn(k0R0)=0,wendthatEnm0andaˆJ0n(k0R0)=ˆJn(k0R0).WecarryoutananalysissimilartothatofSection3.1,simplyreplacinghgˆJn(k0R1)+ˆH(1)n(k0R1)iEnmbyˆJn(k0R1)Dnm.Thus,thenecessaryconditionsforanelectromagneticwavetopenetratetheinnerboundaryofthecloakareˆJn(k0R1)=0andf(R1d)=ndo(d)foraparticularn0.Theresonancemode(forn01)andthesensormode(forn02)arethesameasthosediscussedinSection3.1.3.3.RemovalofsingularityAsshowninSection3.1andSection3.2,oneoftheconditionsforachievingthesensormodeisf(R1d)=ndo(d).Wenotethattheconditionofzeroscatteringoutsideofthecloak requiresthatf(R2)=R2.Thus,ifr0f(r)isacontinuousfunction,inevitablythereisaparticularvalueofr,sayRb,forwhichr00.Consequently,fromEq.(12)weconcludethatasingularityappearsinthecloaklayersincetheargumentforH(1)n(
)iszero.Toremovesuchasingularity,weletthefunctionftobediscontinuousatrRb,butatthesametime,wekeepthecontinuityofthepermittivityandpermeability.Withthispurpose,weseefromEqs.(4)and(5)thatafunctionf(r)thatsimultaneouslysatisesf(R+b)=f(Rb)0andf0(R+b)=f0(Rb)canachieveboththecontinuityofpermittivityandpermeabilityandtheremovalofsingularity.OneexampleofsuchfisdepictedinFig.1.Itisinterestingtodiscussthemeaningofthefunctionr0f(r)forthecaser00.Inde-rivingthepermittivityandpermeabilityinthephysicalspace(Eqs.(4)and(5)),thekeystepistousex0=xy0=yz0=zr0=r[3,17].Whenr00,apointinthephysicalspaceandthecorrespondingpointinthevirtualspaceareontheoppositesideoftheorigin.Theinvisibilityconditionsf(R1)=0andf(R2)=R2indicatethataswemovefromtheinnerboundaryofthecloaktotheouteroneinthephysicalspace,thecorrespondingpointinthevirtualspacemovesfromtheoriginaltor0R2.However,sincethevirtualspaceisfreespace,thereisnoscatteringatall.Consequently,itdoesnotmatterinwhichwaywemovefromtheoriginaltor0R2inthevirtualspace.Thelineartransformthatispresentedin[1,17]indicatesauniformspeeddis-placement.Theconcentratortransformthatispresentedin[3]indicatesadisplacementallthewayoutsideoftheouterboundaryofthecloak,followedbyabackwarddisplacementtocomebacktotheouterboundary.Here,inthispaper,ourtransformfunctionshowsadisplacementto-wardtheoppositedirection,followedadirectionalchangeandthenaall-the-waydisplacementtowardstheouterboundaryofthecloak.4.NumericalsimulationsFollowingthecriteriathatwerepresentedinSection3,weproposedaparticulartransformation,asdepictedinFig.1(b),totesttheperformanceofthecloakingdevice.Wechoosethefollowingparameters:R00:5l,R11:5l,R23:0l,Ra2:0l,Rb2:33l,Rc2:66l,andh0:15l.ThetransformfunctionintherangeofR1drR2isgivenbyf(r)=n0rn0R1forR1drRahn0(Ra+R1) RbRa(rRa)+n0(RaR1)forRarRbhn0(Ra+R1) RbRa(rRb)+hforRbrRcR2n0(RaR1) R2Rc(rRc)+n0(RaR1)forRcrR2:(38)Weobservethatthevaluesof and marebothnegativeandniteintheregionR1drRa,andtheradialcomponentsapproachzeroattheboundaryrR1d.Inotherregions,both and marepositiveandnite.ItisimportanttostressthatalthoughfourtransformationsareusedintheregionofR1drR2,thepotential,andconsequentlytheelectromagneticeld,isexpressedinasingleformula,Eq.(12).ThesurfaceimpedanceaiscalculatedthroughEq.(37).Weconsideranx-polarizedplanewavewithaunitamplitudeEincˆxeik0zisincidentuponthecloakingdevicealongthezdirection.Sincethegivenincidencewavecontainsonlythejmj1term,fromhereonwards,wedropofftheminthesubscriptforsimplicity.Forexample,thecoefcientKnmiswrittenasKn.ItiswellknownthatKnisproportionaltoin(2n1)=[n(n1)]forbothTEandTMcomponentsoftheincidentplanewave[3].Weaimatexaminingthecloakingeffectandpenetratingeffectofthecloakingdeviceastheparameterdapproacheszero.ThecloakingeffectisquantiedbyjAn=Knj.ThelowerthevalueofjAn=Knj,thebetterthecloakingeffect.SinceEq.(19)showsthatEn=Dnisindependentofd,wecanuseonlyjDn=Knj 105 104 103 102 101 10 10 10 105 10 10 (|D/K n=1 n=2 n=3 105 104 103 102 101 10 10 10 10 10 10 105 10 (|A/K n=1 n=2 n=3 (a)(b) Fig.2.Forn0=1,thecloakingandpenetratingeffectsfordifferentordersofmultiplesinthelimitofdapproachingzero.(a)Thecloakingeffect.(b)Penetratingeffect. 10 108 106 104 102 10 10 10 10 105 10 Loss tangent|A/K n=1 n=2 n=3 1010 108 106 104 102 10 1012 1010 108 106 104 102 10 10 10 Loss tangent|D/K n=1 n=2 n=3 (a)(b) Fig.3.Forn0=1andd=103,theeffectofsurfaceimpedancelossonthecloakingandpenetratingeffects.(a)Thecloakingeffect.(b)Penetratingeffect,wherethehorizontaldottedlinedenotesthesensormode.toquantifythepenetratingeffect.ThelowerthevalueofjDn=Knj,thepoorerthepenetratingability.ThecaseofjDn=Knj1correspondstotheresonanceeffect.First,weletn01.ThequantitiesofjAn=KnjandjDn=KnjfordifferentvaluesofthesmallparameterdandordernumbernareshowninFig.2.WeseefromFig.2(a)thatthecloakingeffectappliestoallorders,includingnn01.Inaddition,theslopesofthecurves,whichareinthelogarithmicscale,agreewiththetheorypresentedinSection3.Thatistosay,wheneverdisdecreasedbyafactor10,thevaluesofjAn=Knjdecreasesbyafactorof10,105,and107forn1,n2,andn3,respectively.WeseefromFig.2(b)thattheresonanceeffectappliestotheordernn01andtheshieldingeffectapplieston2andn3.Inaddition,theslopesofthecurvesagreewiththetheory,i.e.,wheneverdisdecreasedbyafactor10,thevaluesofjDn=Knjincreasebyafactorof10fornn01anddecreasesbyafactorof103and104forn2andn3,respectively.Tosummarize,forn01,thewavecomponentcorrespondingtonn0yieldsresonanceeffectwithoutexternalscattering,whereasthewavecomponentcorrespondingtonn0simultaneouslyyieldscloakingandshieldingeffect. x(z( !3 !2 !1 0 1 2 3 !3 !2 !1 0 1 2 3 !6 !4 !2 0 2 4 6 x(z( !3 !2 !1 0 1 2 3 !3 !2 !1 0 1 2 3 !6 !4 !2 0 2 4 6 x(z( !3 !2 !1 0 1 2 3 !3 !2 !1 0 1 2 3 !60 !40 !20 0 20 40 60 80 (a)(b)(c) Fig.4.Forn0=1,thexcomponentoftheelectriceldinthexzplaneforthelosstangentof(a)102,(b)104,and(c)107,correspondingtotheidealcloaking,sensor,andresonancemode,respectively.Sincethesensorsurfacewillresultinsomeenergyloss,weareinterestedinanalyzinghowthislossaffectsthecloakingandpenetrating.Tothispurpose,wecalculatejAn=KnjandjDn=Knjinthecasewhentherealsurfaceimpedanceaisreplacedbyacomplexonea(1iLt),whereLtdenoteslosstangent.Fig.3depictsthevaluesofjAn=KnjandjDn=Knjford103fortherangeoflosstangentfrom1010to101.Fig.3(a)and(b)showthatthelossenhancesthecloakingeffectandatthesametimedecreasesthepenetratingeffectfortheordernn01,butitbarelyaffectsthecloakingorpenetratingeffectforotherorders.WeseefromFig.3(b)thatfornn01,asthelossdecreases,thepenetratingabilitychangesfromtheshieldingmode(i.e.,idealcloakingmode)tothesensormodeandthentotheresonancemode.Itisinterestingtoseethatamoderateloss,suchaslosstangentof104,isabletoenhancethecloakingeffectandatthesametimetoshifttheresonancemodetothesensormode.InFig.4,weplotthexcomponentoftheelectriceldinthexzplaneforthelosstangentof102,104,and107,correspondingtotheshielding(i.e.,idealcloaking),sensor,andresonancemode,respectively. x(l)z(l) -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -1 -0.5 0 0.5 1 Fig.5.Thexcomponentoftheelectriceldinthexzplaneforthecasewhenonlythesensorexists,withoutthepresenceoftheoutercloakinglayer. 105 104 103 102 101 10 10 10 10 10 10 105 10 10 (|A/K n=1 n=2 n=3 105 104 103 102 101 1020 1015 1010 105 10 10 (|D/K n=1 n=2 n=3 (a)(b) Fig.6.Forn0=2,thecloakingandpenetratingeffectsfordifferentordersofmultiplesinthelimitofdapproachingzero.(a)Thecloakingeffect.(b)Penetratingeffect.Incomparison,whenonlythesensorexists,withoutthepresenceoftheoutercloakinglayer,thedistributionoftheelectriceldisplottedinFig.5,whereweseethatthedistortedeldpatternindicatesthepresenceofascatterer.Next,forn02,wealsoanalyzethecloakingandpenetratingeffects.WeseefromFig.6(a)thatthecloakingeffectappliestoallorders,includingnn02andtheslopesofthecurvesagreewiththetheories,i.e.,wheneverdisdecreasedbyafactor10,thevaluesofjAn=Knjdecreasesbyafactorof103,103,and107forn1,n2,andn3,respectively.WeseefromFig.6(b)thatthesensormodeappliestotheordernn02andtheshieldingeffectapplieston1andn3.Theslopesofthecurvesagreewiththetheory,i.e.,wheneverdisdecreasedbyafactor10,thevaluesofjDn=Knjkeepsthesameorderfornn02anddecreasesbyafactorof102and104forn1andn3,respectively.Tosummarize,forn02,thewavecomponentcorrespondingtonn0yieldssensoreffectwithoutexternalscattering,whereasthewavecomponentcorrespondingtonn0simultaneouslyyieldscloakingandshieldingeffects.Fig.7depictsthevaluesofjAn=KnjandjDn=Knjford103forpresenceofloss 10 108 106 104 102 10 10 10 10 10 10 108 106 104 Loss tangent|A/K n=1 n=2 n=3 1010 108 106 104 102 10 1010 108 106 104 102 10 10 Loss tangent|D/K n=1 n=2 n=3 (a)(b) Fig.7.Forn0=2andd=103,theeffectofsurfaceimpedancelossonthecloakingandpenetratingeffects.(a)Thecloakingeffect.(b)Penetratingeffect,wherethehorizontaldottedlinedenotesthesensormode. x(z( !3 !2 !1 0 1 2 3 !3 !2 !1 0 1 2 3 !4 !2 0 2 4 6 x(z( !3 !2 !1 0 1 2 3 !3 !2 !1 0 1 2 3 !4 !2 0 2 4 6 (a)(b) Fig.8.Forn0=2,thexcomponentoftheelectriceldinthexzplaneforthelosstangentof(a)104and(b)107,correspondingtotheidealcloakingandsensormode,respectively.inthesurfaceimpedance.Weobservethatthelossenhancesthecloakingeffectandinthemeanwhiledecreasethepenetratingeffectfortheordernn02,butitbarelyaffectsthecloakingorpenetratingeffectforotherorders.WeseefromFig.7(b)thatfornn02,asthelossdecreases,thepenetratingabilitychangesfromtheshieldingmodetothesensormode.InFig.8,weplotthexcomponentoftheelectriceldinthexzplaneforthelosstangentof104and107,correspondingtotheshieldingandthesensormode,respectively.Inpractice,fornn02,thesensormodecaneasilybedestroyedbytheloss.Forexample,inFig.8alosstangentof104isabletoshiftthesensormodetotheshieldingmode.5.ConclusionTheidealtransformationopticscloakingisaccompaniedbyshielding:externalobservationsdonotprovideanyindicationofthepresenceofacloakedobject,norisanyinformationaboutthe eldsoutsidedetectableinsidethecloakedregion.Inthispaper,weproposedatransformationthatcloaksthree-dimensionalobjectsforelectromagneticwavesinsensormode,i.e.,cloakingaccompaniedbydegradedshielding.Theproposedtransformationtacklesthedifcultycausedbythefactthatthelowestmultipoleinthree-dimensionalelectromagneticradiationisdipoleratherthanmonopole.Itisworthemphasizingthatwehaveobtainedtheanalyticalsolutiontotheelectromagneticeldsineachregionofthewholespace.Wendthatforthedipoleterm(n=1),weareabletoachieveresonancemodewithoutexternalscattering.Forthequadrupoleterm(n=2),weareabletoachievesensormodewithoutexternalscattering.However,thelossthatispresentedonthesurfaceofthesensordegradesthepenetratingeffectsothatamoderatelossisabletoshifttheresonancemodetothesensormodeforn1andtoshiftthesensormodetotheidealcloakingmodeforn2.Thus,inrealworldapplications,wherelossispresentonthesurfaceofsensor,itismoredesirabletoachievethesensormodeusingthedipoleterm.AcknowledgementTheauthorChenacknowledgesthenancialsupportfromtheSingaporeTemasekDefenceSystemsInstituteundergrantTDSI/09001/1A. eldsoutsidedetectableinsidethecloakedregion.Inthispaper,weproposedatransformationcloaksthree-dimensionalobjectsforelectromagneticwavesinsensormode,i.e.,cloakingaccompaniedbydegradedshielding.Theproposedtransformationtacklesthedifcultycausedbythefactthatthelowestmultipoleinthree-dimensionalelectromagneticradiationisdipoleratherthanmonopole.Itisworthemphasizingthatwehaveobtainedtheanalyticalsolutiontotheelectromagneticeldsineachregionofthewholespace.Wendthatforthedipoleterm(n=1),weareabletoachieveresonancemodewithoutexternalscattering.Forthequadrupoleterm(n=2),weareabletoachievesensormodewithoutexternalscattering.However,thelossthatispresentedonthesurfaceofthesensordegradesthepenetratingeffectsothatamoderatelossisabletoshifttheresonancemodetothesensormodefor1andtoshiftthesensormodetotheidealcloakingmodefor2.Thus,inrealworldapplications,wherelossispresentonthesurfaceofsensor,itismoredesirabletoachievethesensormodeusingthedipoleterm.AcknowledgementTheauthorChenacknowledgesthenancialsupportfromtheSingaporeTemasekDefenceSystemsInstituteundergrantTDSI/09001/1A. Received 27 Jul 2011; revised 7 Sep 2011; accepted 9 Sep 2011; published 3 Oct 2011 (C) 2011 OSA 10 October 2011 / Vol. 19, No. 21 / OPTICS EXPRESS 20530 1010 108 106 104 102 100 1018 1016 1014 1012 1010 108 106 104 Loss tangent|An/Kn| n=1 n=2 n=3 1010 108 106 104 102 100 1010 108 106 104 102 100 102 Loss tangent|Dn/Kn| n=1 n=2 n=3 (a)(b) Fig.7.For,theeffectofsurfaceimpedancelossonthecloakingandpenetratingeffects.(a)Thecloakingeffect.(b)Penetratingeffect,wherethehorizontaldottedlinedenotesthesensormode. x(l)z(l) !3 !2 !1 0 1 2 3 !3 !2 !1 0 1 2 3 !4 !2 0 2 4 6 x(l)z(l) !3 !2 !1 0 1 2 3 !3 !2 !1 0 1 2 3 !4 !2 0 2 4 6 (a)(b) Fig.8.Forcomponentoftheelectriceldinthexzplaneforthelosstangentof(a)10and(b)10,correspondingtotheidealcloakingandsensormode,respectively.inthesurfaceimpedance.Weobservethatthelossenhancesthecloakingeffectandinthemeanwhiledecreasethepenetratingeffectfortheorder2,butitbarelyaffectsthecloakingorpenetratingeffectforotherorders.WeseefromFig.7(b)thatfor2,asthelossdecreases,thepenetratingabilitychangesfromtheshieldingmodetothesensormode.InFig.8,weplotthecomponentoftheelectriceldinthexzplaneforthelosstangentofand10,correspondingtotheshieldingandthesensormode,respectively.Inpractice,2,thesensormodecaneasilybedestroyedbytheloss.Forexample,inFig.8alosstangentof10isabletoshiftthesensormodetotheshieldingmode.5.ConclusionTheidealtransformationopticscloakingisaccompaniedbyshielding:externalobservationsdonotprovideanyindicationofthepresenceofacloakedobject,norisanyinformationaboutthe Received 27 Jul 2011; revised 7 Sep 2011; accepted 9 Sep 2011; published 3 Oct 2011 (C) 2011 OSA 10 October 2011 / Vol. 19, No. 21 / OPTICS EXPRESS 20529 x(l)z(l) -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -1 -0.5 0 0.5 1 Fig.5.Theoftheelectriceldinthexzplaneforthecasewhenonlythesensorexists,withoutthepresenceoftheoutercloakinglayer. 105 104 103 102 101 1035 1030 1025 1020 1015 1010 105 100 105 d (l)|An/Kn| n=1 n=2 n=3 105 104 103 102 101 1020 1015 1010 105 100 105 d (l)|Dn/Kn| n=1 n=2 n=3 (a)(b) Fig.6.Forthecloakingandpenetratingeffectsfordifferentordersofmultiplesinthelimitofapproachingzero.(a)Thecloakingeffect.(b)Penetratingeffect.Incomparison,whenonlythesensorexists,withoutthepresenceoftheoutercloakinglayer,thedistributionoftheelectriceldisplottedinFig.5,whereweseethatthedistortedeldpatternindicatesthepresenceofascatterer.Next,for2,wealsoanalyzethecloakingandpenetratingeffects.WeseefromFig.6(a)thatthecloakingeffectappliestoallorders,including2andtheslopesofthecurvesagreewiththetheories,i.e.,wheneverisdecreasedbyafactor10,thevaluesofdecreasesbyafactorof10,10,and102,and3,respectively.WeseefromFig.6(b)thatthesensormodeappliestotheorder2andtheshieldingeffectappliesto1and3.Theslopesofthecurvesagreewiththetheory,i.e.,wheneverisdecreasedbyafactor10,thevaluesofkeepsthesameorderfor2anddecreasesbyafactorof10and101and3,respectively.Tosummarize,for2,thewavecomponentcorrespondingtoyieldssensoreffectwithoutexternalscattering,whereasthewavecomponentcorrespondingtosimultaneouslyyieldscloakingandshieldingeffects.Fig.7depictsthevaluesofforpresenceofloss Received 27 Jul 2011; revised 7 Sep 2011; accepted 9 Sep 2011; published 3 Oct 2011 (C) 2011 OSA 10 October 2011 / Vol. 19, No. 21 / OPTICS EXPRESS 20528 x(l)z(l) !3 !2 !1 0 1 2 3 !3 !2 !1 0 1 2 3 !6 !4 !2 0 2 4 6 x(l)z(l) !3 !2 !1 0 1 2 3 !3 !2 !1 0 1 2 3 !6 !4 !2 0 2 4 6 x(l)z(l) !3 !2 !1 0 1 2 3 !3 !2 !1 0 1 2 3 !60 !40 !20 0 20 40 60 80 (a)(b)(c) Fig.4.Forcomponentoftheelectriceldinthexzplaneforthelosstangentof(a)10,(b)10,and(c)10,correspondingtotheidealcloaking,sensor,andresonancemode,respectively.Sincethesensorsurfacewillresultinsomeenergyloss,weareinterestedinanalyzinghowthislossaffectsthecloakingandpenetrating.Tothispurpose,wecalculateinthecasewhentherealsurfaceimpedanceisreplacedbyacomplexone,wheredenoteslosstangent.Fig.3depictsthevaluesoffortherangeoflosstangentfrom1010to10.Fig.3(a)and(b)showthatthelossenhancesthecloakingeffectandatthesametimedecreasesthepenetratingeffectfortheorderbutitbarelyaffectsthecloakingorpenetratingeffectforotherorders.WeseefromFig.3(b)that1,asthelossdecreases,thepenetratingabilitychangesfromtheshieldingmode(i.e.,idealcloakingmode)tothesensormodeandthentotheresonancemode.Itisinterestingtoseethatamoderateloss,suchaslosstangentof10,isabletoenhancethecloakingeffectandatthesametimetoshifttheresonancemodetothesensormode.InFig.4,weplotthecomponentoftheelectriceldinthexzplaneforthelosstangentof10,10,and10correspondingtotheshielding(i.e.,idealcloaking),sensor,andresonancemode,respectively. Received 27 Jul 2011; revised 7 Sep 2011; accepted 9 Sep 2011; published 3 Oct 2011 (C) 2011 OSA 10 October 2011 / Vol. 19, No. 21 / OPTICS EXPRESS 20527 105 104 103 102 101 1020 1015 1010 105 100 105 d (l)|Dn/Kn| n=1 n=2 n=3 105 104 103 102 101 1035 1030 1025 1020 1015 1010 105 100 d (l)|An/Kn| n=1 n=2 n=3 (a)(b) Fig.2.Forthecloakingandpenetratingeffectsfordifferentordersofmultiplesinthelimitofapproachingzero.(a)Thecloakingeffect.(b)Penetratingeffect. 1010 108 106 104 102 100 1020 1015 1010 105 100 Loss tangent|An/Kn| n=1 n=2 n=3 1010 108 106 104 102 100 1012 1010 108 106 104 102 100 102 104 Loss tangent|Dn/Kn| n=1 n=2 n=3 (a)(b) Fig.3.For,theeffectofsurfaceimpedancelossonthecloakingandpenetratingeffects.(a)Thecloakingeffect.(b)Penetratingeffect,wherethehorizontaldottedlinedenotesthesensormode.toquantifythepenetratingeffect.Thelowerthevalueof,thepoorerthepenetratingability.Thecaseofj1correspondstotheresonanceeffect.First,welet1.ThequantitiesoffordifferentvaluesofthesmallandordernumberareshowninFig.2.WeseefromFig.2(a)thatthecloakingeffectappliestoallorders,including1.Inaddition,theslopesofthecurves,whichareinthelogarithmicscale,agreewiththetheorypresentedinSection3.Thatistosay,wheneverisdecreasedbyafactor10,thevaluesofdecreasesbyafactorof10,10,and102,and3,respectively.WeseefromFig.2(b)thattheresonanceeffectappliestotheorder1andtheshieldingeffectappliesto2and3.Inaddition,theslopesofthecurvesagreewiththetheory,i.e.,wheneverisdecreasedbyafactor10,thevaluesofincreasebyafactorof10for1anddecreasesbyafactorof10and102and3,respectively.Tosummarize,for1,thewavecomponentcorrespondingyieldsresonanceeffectwithoutexternalscattering,whereasthewavecomponentcorrespondingtosimultaneouslyyieldscloakingandshieldingeffect. Received 27 Jul 2011; revised 7 Sep 2011; accepted 9 Sep 2011; published 3 Oct 2011 (C) 2011 OSA 10 October 2011 / Vol. 19, No. 21 / OPTICS EXPRESS 20526 requiresthat.Thus,ifisacontinuousfunction,inevitablythereisaparticularvalueof,say,forwhich0.Consequently,fromEq.(12)weconcludethatasingularityappearsinthecloaklayersincetheargumentforiszero.Toremovesuchasingularity,weletthefunctiontobediscontinuousat,butatthesametime,wekeepthecontinuityofthepermittivityandpermeability.Withthispurpose,weseefromEqs.(4)and(5)thatafunctionthatsimultaneouslysatises)=0and)=canachieveboththecontinuityofpermittivityandpermeabilityandtheremovalofsingularity.OneexampleofsuchisdepictedinFig.1.Itisinterestingtodiscussthemeaningofthefunctionforthecase0.Inde-rivingthepermittivityandpermeabilityinthephysicalspace(Eqs.(4)and(5)),thekeystepistouse[3,17].When0,apointinthephysicalspaceandthecorrespondingpointinthevirtualspaceareontheoppositesideoftheorigin.Theinvisibility)=0and)=indicatethataswemovefromtheinnerboundaryofthecloaktotheouteroneinthephysicalspace,thecorrespondingpointinthevirtualspacemovesfromtheoriginalto.However,sincethevirtualspaceisfreespace,thereisnoscatteringatall.Consequently,itdoesnotmatterinwhichwaywemovefromtheoriginaltothevirtualspace.Thelineartransformthatispresentedin[1,17]indicatesauniformspeeddis-placement.Theconcentratortransformthatispresentedin[3]indicatesadisplacementallthewayoutsideoftheouterboundaryofthecloak,followedbyabackwarddisplacementtocomebacktotheouterboundary.Here,inthispaper,ourtransformfunctionshowsadisplacementto-wardtheoppositedirection,followedadirectionalchangeandthenaall-the-waydisplacementtowardstheouterboundaryofthecloak.4.NumericalsimulationsFollowingthecriteriathatwerepresentedinSection3,weproposedaparticulartransformation,asdepictedinFig.1(b),totesttheperformanceofthecloakingdevice.Wechoosethefollowing,and.Thetransformfunctionintherangeofisgivenby)= )+ )+ )+Weobservethatthevaluesof eand arebothnegativeandniteintheregion,andtheradialcomponentsapproachzeroattheboundary.Inotherregions, eand arepositiveandnite.Itisimportanttostressthatalthoughfourtransformationsareusedintheregionof,thepotential,andconsequentlytheelectromagneticeld,isexpressedinasingleformula,Eq.(12).ThesurfaceimpedanceiscalculatedthroughEq.(37).Weconsideran-polarizedplanewavewithaunitamplitudeincxeisincidentuponthecloakingdevicealongthedirection.Sincethegivenincidencewavecontainsonlytheterm,fromhereonwards,wedropofftheinthesubscriptforsimplicity.Forexample,thecoefcientnmiswrittenas.Itiswellknownthatisproportionaltoton(n+1)]forbothTEandTMcomponentsoftheincidentplanewave[3].Weaimatexaminingthecloakingeffectandpenetratingeffectofthecloakingdeviceastheparameterzero.Thecloakingeffectisquantiedby.Thelowerthevalueof,thebetterthecloakingeffect.SinceEq.(19)showsthatisindependentof,wecanuseonly Received 27 Jul 2011; revised 7 Sep 2011; accepted 9 Sep 2011; published 3 Oct 2011 (C) 2011 OSA 10 October 2011 / Vol. 19, No. 21 / OPTICS EXPRESS 20525 3.1.2.Case1.2)=0,theWronskianimpliesthat)+0.ItisalsoimportanttostressthatthedifferentiationoftheWron-skianwithrespectto)+)=0.Subsequently,weseefromEq.(28)toEq.(31)that,0,and0.Duetothepresenceofanonzerocontainsatermoforderns,whichisinniteandeventuallyleadstozerovalueofnmgoestozero.Inthiscase,thereisnosensingeffect.Toachieveanon-vanishingnmwehavetoeliminateandtheonlywayistousetocancelit.Theconditioncanbesatisedwhenbd0,i.e.,Theconditionthat)+)=0andEq.(36)canbesatisedbydifferentordermultipoles.Itisimportanttonotethatoncetheaforementionedtwoconditionsaresatisedforaparticularorder,say,theycannotbesimultaneouslysatisedbyothersorders.Forweeasilyobtainthatnmnm;For,weobtainfromCase1.1thatnmnm.Thebehaviorsofcloakingandshieldingeffectaredifferentforvariousorderofmultipoles.•Caseofapproachesinnityandapproacheszero,whichmeanstheresonancemode.Itisworthmentioningthattheinteriorresonancedoesnotdestroythecloakingeffect,whichisdifferentfromtheconclusionof[12].Thisdifferenceispossiblyduetothefactthat[12]usestheCauchydata,i.e.,themappingfromthetotaleldtoitsnormalderivative,whereasthispaperstudiesthemappingfromtheincidenceeldtothescatteredone.•Caseofisinthesameorderastheincidencewaveandapproacheszero,whichmeanssensormode.•Caseof3:Bothnmnmapproachzero,whichmeansidealcloakingmode,i.e.,cloakingeffectandshieldingeffectarehandinhand.Fromtheconditionof)+)=0andthedenitionofinEq.(19),weeasilyobtainthevalueofthatleadstotheresonancemode(for1)andthesensormode Case2)=0,wendthatnm0and.WecarryoutananalysissimilartothatofSection3.1,simplyreplacing)+nmnm.Thus,thenecessaryconditionsforanelectromagneticwavetopenetratetheinnerboundaryofthecloakare)=0and)=foraparticular.Theresonancemode(for1)andthesensormode(for2)arethesameasthosediscussedinSection3.1.3.3.RemovalofsingularityAsshowninSection3.1andSection3.2,oneoftheconditionsforachievingthesensormode)=.Wenotethattheconditionofzeroscatteringoutsideofthecloak Received 27 Jul 2011; revised 7 Sep 2011; accepted 9 Sep 2011; published 3 Oct 2011 (C) 2011 OSA 10 October 2011 / Vol. 19, No. 21 / OPTICS EXPRESS 20524 andtheWronskianisused.Itisworthhighlightingthatwehaveobtainedanalyticalresultfornmwithoutusinganyapproximation.Fromhereonwards,wewillusethefactthatisasmallparametertosimplifytheobtainedanalyticalresult.Foraninnitesimalparameter,wehavetheasymptotics[16].UsingTaylor'sexpansionandthefactthat)=0,we)+)+)+ )[)+)+)++k0f(R1+d)]�n=qn[k0f(R1+d)]�(n+1)(A1+A2+A3+A4+A5+A6);(26)whereA1=�nhgˆJn(k0R1)+)+)+)+ wheretheLandaubig-onotationdenotestermsofthesameorder,i.e.,neglectingconstantmultipliersandhigher-orderterms.Dependingonwhetherornottheleadingterminiszero,wediscussthetwofollowingtwocases.3.1.1.Case1.1)+nm k0f(R1+d)]n+1A�11:(33)Tofurtherquantifytheasymptoticbehaviorofnm,wecanexpressthetransformfunctionas,consideringthefactthat)=0andissmall,)=bdforsome,where0.NotethattheLandaulittle-onotationdenoteshigherorderterms,i.e.,thoseapproachingtozeroatfasterrates.Thus,nm FromEq.(23),astraightforwardcalculationgivesthatnm.Since1,weseenmnmapproachzeroasapproacheszero,indicatingthatcloakingeffectiscoupledwithshieldingeffect,i.e.,thereisnosensoreffect. Received 27 Jul 2011; revised 7 Sep 2011; accepted 9 Sep 2011; published 3 Oct 2011 (C) 2011 OSA 10 October 2011 / Vol. 19, No. 21 / OPTICS EXPRESS 20523 Thepotentialthethreeregionsiswrittenasext)=nm)+nm)=nm)+nmint)=nm)+nmSincethesourceisgiven,nmisuniquelydetermined.Foreachmultipoleoforder,wewillsolveforveunknowsnmnmnmnm,andnm.Eqs.(9)and(10)indicatethatthecontinu-itiesoftangentialcomponentsofacrosstheboundariestothecontinuitiesof ,respectively.Theimpedanceboundaryconditionreduces 0,whereisproportionalto.Thus,wehavethefollowingboundarynm)+nm)=nm)+nmnm)+nm)=nm)+nmnm))+nm))=nm))+nmnm))+nm))=nm))+nmnm)+nm)=nm)+nmItiseasytoseethatEqs.(14)and(15)yieldnmnmnmnm.TherearetwocasesforEq.(18).3.1.Case10,wehavenm nmnmNowwesolvefortwounknownsnmnmfromtwolinearequations,nm))+nm))=nm))+nm))+nm))=nm))+Weobtain:nmnmnmnm))+))+))+))+ Received 27 Jul 2011; revised 7 Sep 2011; accepted 9 Sep 2011; published 3 Oct 2011 (C) 2011 OSA 10 October 2011 / Vol. 19, No. 21 / OPTICS EXPRESS 20522 theinvarianceofMaxwell'sequationsundertransformationofthespatialcoordinatesystems.acoordinatetransformationbetweenthevirtualspace(curvedfreespace)withthephysicalspace.Theformerhasspatialcoordinates,permittivity,andpermeability,whereasthelatterhasspatialcoordinates,andparameters e; .Thetransformationisonlyintheradialdirection,i.e.,Theassociatedpermittivityandpermeabilitytensorsaregivenby e=er(r)ˆrˆr+et(r)ˆqˆq+et(r)ˆfˆf(2) m=mr(r)ˆrˆr+mt(r)ˆqˆq+mt(r)ˆfˆf;(3)whereer=e0f2(r) r2f0(r);et=e0f0(r)(4)mr=m0f2(r) Itiswellknownthatthetransformthatsatisesboth)=0and)=yieldsperfectinvisibility.Toavoidsingularities,welettheboundaryoftheinnercloakingmaterialbeat,whereisasmallpositivenumber.Theincidentwaveisgeneratedbyatime-harmonictime-harmonicexp)]sourcethatislocatedoutsideofthesphereofradius.Toachievecloakingasensor,weaimatobtaininganegligiblescatteredeldoutsideofthesphereofradiusandatthesametimeperceivableelectromagneticeldatthesurfaceofthecladdingsphere.3.AnalyticalresultsforelectromagneticeldsElectromagneticeldscanbedecomposedintotwoindependentmodes,TEandTMmodes(transversetoradialdirectioninsphericalcoordinatesystem),whicharedualtoeachother.FortheTMmode,theeldcanbeexpressedasrfwherethescalarpotential f2sinq¶ ¶q ¶q sinq¶2FM ¶f .Separationofvariablesyieldsnmisthehighestordermultipoleusedinthenumericalsimulations,)=istheRiccati-Besselfunction,andaresphericalharmonics.Nowthemagneticeldandtheelectriceldcanbeexpressedas rsinq¶FM ¶f r¶FM ¶q �iwf0¶2FM ¶f2+k20FMˆr+1 r¶2FM ¶q rsinq¶2FM ¶f Received 27 Jul 2011; revised 7 Sep 2011; accepted 9 Sep 2011; published 3 Oct 2011 (C) 2011 OSA 10 October 2011 / Vol. 19, No. 21 / OPTICS EXPRESS 20521 andthepapersdonotdiscusswhetherthemethodworksforagenericsensor,i.e.,whetherbothcloaklayerandanti-cloaklayerdependontheshapeandmaterialofthesensor.Inaddition,thecloakingdeviceconsistsofvelayers,eightunknownsforeachorderofmultipole,andtwosmallparameterstobeadjusted,whichmakesittedioustoderiveanalyticalsolutions.Inthispaper,weproposeanoveltransformationopticsmethodtocloakasensor,whichcanbeofarbitrarysize,shape,andmaterial,for3Delectromagneticwaves.Thecloakingdeviceconsistsinthreelayers,veunknownsforeachorderofmultipole,andonesmallparametertobeadjusted.Analyticalresultscanbeobtainedforeachofveunknownsevenwithoutresourcetosmallparameterapproximation.Comparedwith[12],theproposedtransformation,fromthephysicalspacetothevirtualspace,circumventsthedifcultycausedbythefactthatthelowestmultipolein3Delectromagneticradiationisdipoleratherthanmonopole.Thelossofthesurfaceimpedanceofthesensorplaysanimportantroleindeterminingthecloakingmodes:idealcloaking,sensorcloakingandresonance.Numericalsimulationsvalidatetheproposedtransformationmodel.2.CongurationofthecloakandthesensorAsshowninFig.1.Thesensorisplacedinsideasphereofradius,withsurfaceimpedanceboundarycondition(see[15]andreferencestherein).Thesensorisabletomeasurethetangentialelectriceldsatthesurfaceofthesphere.Thecloaklayeriswithinanannuluswithinnerandouterradii,respectively.Allthreespheresmentionedaboveareconcentric,withthecenterbeingtheoriginofthecoordinatesysteminthephysicalspace.Thespacebetweenspheresofradii,aswellasthespaceoutsideofthesphereof,arefreespace.Thepermittivityandpermeabilityofthecloaklayerareobtainedusing (a)(b) Fig.1.Thegeometryofthesensorandthetransformationfunction.(a)Asensorofarbitraryisplacedinsideaspherewithradiusandsurfaceimpedance,anditisabletomeasurethetangentialelectriceldonthesphericalsurface.Thecloaklayerisinbetweenspheresofradii,whereisansmallpositivenumberandisnotshowninthegureduetoitssmallmagnitude.Otherregionsarefreespace.(b)Thetransformationfromthephysicalspacetothevirtualspace. Received 27 Jul 2011; revised 7 Sep 2011; accepted 9 Sep 2011; published 3 Oct 2011 (C) 2011 OSA 10 October 2011 / Vol. 19, No. 21 / OPTICS EXPRESS 20520 13.G.Castaldi,I.Gallina,V.Galdi,A.AlandN.Engheta,“Cloak/anti-cloakinteractions,”Opt.Express106343(2009).14.G.Castaldi,I.Gallina,V.Galdi,A.Alu,andN.Engheta,“Analyticalstudyofsphericalcloak/anti-cloakinterac-tions,”WaveMotion,455–467(2011).15.Y.-L.Geng,C.-W.Qiu,andN.Yuan,“Exactsolutiontoelectromagneticscatteringbyanimpedancespherecoatedwithauniaxialanisotropiclayer,”IEEETrans.AntennasPropagat.,572–576(2009).16.M.AbramowitzandI.A.Stegun,HandbookofMathematicalFunctionswithFormulas,Graphs,andMathemat-icalTables(Dover,NewYork,1972).17.D.Schurig,J.B.Pendry,andD.R.Smith,“Calculationofmaterialpropertiesandraytracingintransformationmedia,”Opt.Express,9794–9804(2006). 1.Introductionransformationopticshasbeenapopularresearchtopicinthepastfewyearsduetoitsexcitingpropertyofdesigningcloakswhichcancompletelyhideobjectsfromelectromagneticdetec-tion.ThefundamentalideaistheinvarianceofMaxwell'sequationsunderaspace-deformingtransformationifthematerialpropertiesarealteredaccordingly[1–9].However,theidealcloakingisaccompaniedbyshielding:Thereisadecouplingoftheeldsinsideandoutsideofthecloakedregion,sothatexternalobservationsdonotprovideanyindicationofthepresenceofacloakedobject,norisanyinformationabouttheeldsoutsidedetectableinsidethecloakedregion.Inmanyreal-worldapplications,however,thereareneedsforeffectivelycloakingsen-sorsanddetectorssothattheirpresencemaybelessdisturbingtothesurroundingenvironment.Forexample,inareceivingantennaarray,weaimatreducingthecouplingamongantennael-ementssothatanelementisabletoreceiveelectromagneticsignalwithoutdisturbingtheeldreceivedbyotherelements.Whenthenear-eldscanningopticalmicroscope(NSOM)operatesincollectionmode,thetipshouldbeveryclosetothescatteringobjects,whichinevitablyyieldsundesiredmultiplescatteringbetweentipandscattererandthusdegradestheaccuracyofthemeasuredeld.Inthepastfewyears,therehavebeenattemptstodesignsensorsthatareabletoperceiveelectromagneticradiationandatthesametimehavenegligibledisturbancetothesurroundingenvironment,whichisreferredtoas`cloakingsensors'or`cloakinsensormode'.Therstpaperinthisdirectionis[10],whereaplasmoniccoatingallowselectromagneticwavetoreachthesensorandatthesametimethescatteringduetothesensoriscanceledoutbythatofthecloak.TheideahasbeenappliedtocloakaNSOMtipforthepurposeofneareldimaging[11],whichisofgreatindustrialandscienticsignicance.However,therearealsosomelimitationsofthemethod.Forexample,itcanonlycloaksensorssmallerorcomparabletowavelengthandtheshapeandmaterialofthecloakdependonthegeometricalandphysicalpropertiesofthesensor.Anotherapproachtocloakasensoristransformationoptics.In[12],asensorthatiscladdedwithaspherewithsurfaceimpedanceisplacedinsidethecloak.Thismethodworksforsensorsofarbitrarysizeandshapeaslongasthesensorsareabletomeasuretheeldattheboundaryofthecladdingsphere.Asmentionedin[12],thistransformationopticsapproachisabletoachieveincloakingasensorforacousticwavesandtwo-dimensional(2D)electromagneticwaves.However,afurthercalculationshowsthatthetransformationopticsapproachproposedin[12]cannotcloakasensorforthree-dimensional(3D)electromagneticwaves.Thisismainlyduetothefactthatthelowestmultipolein3Delectromagneticradiationisdipoleratherthanmonopole.Theasymptoticbehavior,forsmallarguments,ofRiccati-Besselfunctionsoforderoneandaboveleadstovanishingelectromagneticeldsinsidethecloak,i.e.,theshieldingeffectisstillhandinhandwiththecloakingeffect.Anothertransformationopticsapproachispresentedin[13,14],wherebothacloaklayerandananti-cloaklayerplaytheroletoachievecloakingsensoreffect.Whereasthisapproachcancloaksensorsofarbitrarysizeandworksforboth2Dand3Delectromagneticwaves,thesensorismodeledasadielectricsphere Received 27 Jul 2011; revised 7 Sep 2011; accepted 9 Sep 2011; published 3 Oct 2011 (C) 2011 OSA 10 October 2011 / Vol. 19, No. 21 / OPTICS EXPRESS 20519 Cloakingasensorforthree-dimensionalsequations:transformationopticsapproachXudongChenGuntherUhlmannDepartmentofElectricalandComputerEngineering,NationalUniversityofSingapore,117576SingaporeDepartmentofMathematics,UniversityofCaliforniaatIrvine,Irvine,CA92697,USADepartmentofMathematics,UniversityofWashington,Seattle,WA98195,USAhenx@nus.edu.sg idealtransformationopticscloakingisaccompaniedbyshielding:externalobservationsdonotprovideanyindicationofthepresenceofacloakedobject,norisanyinformationabouttheeldsoutsidedetectableinsidethecloakedregion.Inthispaper,atransformationisproposedtocloakthree-dimensionalobjectsforelectromagneticwavesinsensormode,i.e.,cloakingaccompaniedbydegradedshielding.Theproposedtransformationtacklesthedifcultycausedbythefactthatthelowestmultipoleinthree-dimensionalelectromagneticradiationisdipoleratherthanmonopole.Thelossofthesurfaceimpedanceofthesensorplaysanimportantroleindeterminingthecloakingmodes:idealcloaking,sensorcloakingandresonance.©2011OpticalSocietyofAmericaOCIScodes:(260.2710)Inhomogeneousopticalmedia;(260.2110)Electromagnetictheory;(160.1190)Anisotropicopticalmaterials;(290.3200)Inversescattering. 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Received 27 Jul 2011; revised 7 Sep 2011; accepted 9 Sep 2011; published 3 Oct 2011 (C) 2011 OSA 10 October 2011 / Vol. 19, No. 21 / OPTICS EXPRESS 20518