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Adjustablesubwavelengthlocalizationinahybridplasmonicwaveguide S. Belan1,2, S. Vergeles1,2, P. Vorobev1,2 1Landau Institute for Theoretical Physics RAS, Kosygina 2, 119334, Moscow, Russia and 2MoscowInstituteofPhysicsandTechnology,Dolgoprudnyj,Institutskijper. 9,141700,MoscowRegion,Russia The hybrid plasmonic waveguide consists of a high-permittivity dielectric nanofiber embedded in a low- permittivity dielectric near a metal surface. This architecture is considered as one of the most perspective 3 candidatesforlong-rangesubwavelengthguiding. Wepresentqualitativeanalysisandnumericalresultswhich 1 revealadvantagesofthespecialwaveguidedesignwhendielectricconstantofthecylinderisgreaterthanthe 0 absolutevalueofthedielectricconstantofthemetal. Inthiscasethearbitrarysubwavelengthmodesizecan 2 beachievedbycontrollingthegapwidth. Ourqualitativeanalysisisbasedonconsiderationofsandwich-like n conductor-gap-dielectric system. The numerical solution is obtained by expansion of the hybrid plasmonic a modeoversinglecylindermodesandthesurfaceplasmon-polaritonmodesofthemetalscreenandmatching J theboundaryconditions. 6 1 PACSnumbers: ] s I. INTRODUCTION our approach we theoretically investigate propagation of the c HPP-mode. First we give qualitative analysis basing on the i pt The creation of the waveguides capable of guiding light considerationofplanesandwich-likeconductor-gap-dielectric waveguidestructure(CGD)[5].Wederiveexactanalyticalex- o with deep subwavelength confinement is of great interest for . practical applications. These devices may throw open the pression for effective index of the fundamental CGD-mode s and give criterion when the HPP-mode is CGD-like. Fi- c doorstonanoscaleopticalcommunications,quantumcomput- i ing,nanoscalelasersandbio-medicalsensing.Themainprob- nally, we present semi-analytical approach for describing of s theHPP-modepropagation. Similarapproachhasbeenprevi- y lemonthewaytopracticalrealizationisthediffractionlimit h oflightindielectricmedia. Electromagneticenergycannotbe ously applied for plane wave scattering by a cylinder placed p localized into nanoscale region much smaller than the wave- neartheplanesurface[6]-[7]. [ The scheme is based on expansion of the HPP-mode over length of light in the dielectric [1]. The possible solution to single cylinder modes and the surface plasmon polariton 4 thisproblemisusingofthematerialswithnegativedielectric v permittivity. For example, metals are known to exhibit this modes of the metal-dielectric interface and matching the 9 property below the plasma frequency. Metal structures pro- boundary conditions for electromagnetic field components. 8 Numericallyobtaineddispersionrelationsconfirmtheadvan- vide guiding of the surface plasmon-polaritons (SPP), which 9 tagesofourdesignofthehybridwaveguide. can be strongly localized near metal-dielectric interfaces[2]. 1 Howeverthepropagationlengthofthestronglyconfinedplas- . 1 monicmodesisnotlargeenoughduetothepresenceofOhmic 1 lossesinthedissipativemetalregions. 2 The new approach for this challenge integrates dielec- 1 : tric waveguide with plasmonic one. The hybrid plasmonic v waveguideconsistsofahigh-permittivitydielectricnanofiber i X separated from a metal screen by low-permittivity dielectric nanoscale gap [3]. Both the single fiber and the silver- r a dielectric interface cannot provide strong mode confinement at optical and near infrared frequencies, but such hy- bridconductor-gap-dielectricarchitecturehasexperimentally FIG.1: a)Geometryofthewaveguide; b)Plainwaveguidewith demonstrated deep subwavelength optical waveguiding [4]. thesamewidthofthegap. Relativelylargepropagationdistancehasbeenachieveddueto lowlosstangenttg = ε(cid:48)(cid:48)/ε(cid:48) attheoperatingfrequencyand m m the specific spatial structure of the guiding mode with field confinementwithinnondissipativegapregion. II. QUALITATIVEDESCRIPTION In the present paper we show that the hybrid plasmon polariton(HPP)modeconfinementcanbeconsiderablyrisenby The geometry of the hybrid waveguide is the following: a aspecificchoiceofthematerials,whenthedielectricconstant circular dielectric cylinder of diameter d and permittivity ε d of the cylinder is greater than the absolute value of the di- isplacedaboveametalscreenofpermittivityε . Thewidth m electricconstantofthemetalscreen. Themainadvantageof of the gap between the cylinder and the metal screen is h, the choice is the hyperbolic-like dependence of the effective see Fig. 1(a). Let us choose the Cartesian reference system index on the gap width. This feature allows to achieve ar- as it is shown in Fig. 1: z-axis is directed along the waveg- bitrary subwavelength mode size at any frequency by tuning uide,whereasx-axisisdirectednormallytothemetalscreen. the distance between the cylinder and the metal. To justify We consider a plasmon-polariton mode of frequency ω and 2 the propagation constant β propagating along z-axis. Thus, all electromagnetic field components depend on time and z- coordinate as exp[iβz − iωt]. We assume, that responses of both dielectric and metal on electromagnetic field are described by dielectric constants, which are ε and ε respec- m d tively. Generally, the outermedium maybenot vacuum, but somedielectricmediumhavingdielectricconstantbeingequal to ε . All the materials are assumed to be nonmagnetic. To g describe the mode confinement, it is convenient to introduce effectiverefractiveindexn ,whichisdefinedasn =β/k, eff eff wherek = ω/cinthewavenumberinvacuum. Theeffective indexdeterminesthefieldpenetrationdepthintothematerial (cid:112) withpermittivityεas1/k n2 −ε.Thepenetrationdepthof eff the bound mode should be real in the unbounded waveguide constituents (metal and outer dielectric space), and may be imaginaryforboundedconstituents(fiber).Thegreatern is eff thestrongerdegreeofconfinement. FIG.2: (Coloronline)Effectiveindexn oftheCGD-modever- CGD Optimization for transversal field confinement imple- susgapwidthhfordifferentmetalpermittivityandcriticalgapwidth mentedinpaper[3]forhybridwaveguideshowsthatthethin- (εm;hc).Thedielectricconstantsofthedielectricandgapregionare ner gaps provide higher localization. The fiber diameter is εd =5.76andεg =1respectivelyatwavelengthλ=490nm. muchgreaterthanthethegapwidthinthecase,andthemode issufficientlylocalizedintheregionwherethegapcanbecon- plain part of the gap is comparable with the mode penetra- sideredasapproximatelyplain. Intheregion, thewaveguide (cid:112) tiondepthintothedielectric1/k n2 −ε . Thesizeofthe shapeisclosetoplainsandwichlikeconductor-gap-dielectric e√ff d (CGD)structure,seeFig.1(b).ThelimitofplainCGD-model plain√partoftheg(cid:112)apisevaluatedas2 hd,thusthecondition is 2 hd∗ ≈ 1/k n2 −ε . For d greater enough than d∗ wasconsideredin[5],wherethepropertiesoftheboundfun- eff d theguidingmodecanapproachthestronglyconfinedmodeof damental mode were investigated. One of the main advan- thesandwichlikesystemevenifthediameterofthecylinder tage of the CGD-mode is that effective index of the mode √ ismuchlessthanfreespacewavelength. n isgreaterthantherefractiveindexofthedielectric ε , CGD √ d In order to give general physical argumentation of our re- n > ε . This implies, that the electromagnetic field CGD d sultsletusconsiderplanarsandwich-likeCGD-waveguidein ofthemodedecaysexponentiallyintothedielectriccladding. detail. ThewavevectoroffundamentalCGD-mode(whichis Nevertheless,theanalysisproposedin[5]isnotapplicableto TM-mode)canbecalculatedfromequation[5] HPP-modeofhybridwaveguidewithoptimaldiameterfound in [3]. The reason is that the mode of plain CGD-model in- (ε κ −ε κ )(ε κ −ε κ ) exp[2hκ ]= d g g d m g g m , (2) deeddescribestheHPP-modeonlyforlargeenoughfiberdi- g (ε κ +ε κ )(ε κ +ε κ ) d g g d m g g m ameter d otherwise HPP-mode should be considered as a re- (cid:112) sultofhybridizationofsurfaceplasmonpolaritonmodesand whereκi = k n2CGD−εi foreachmaterial,i = m,g,dand themodesofthesingledielectriccylinder. nCGD is the effective index of the mode. In particular, 1/κd √ √ and1/κmarethepenetrationdepthsintothedielectricandthe h = √λ logεm√εd−εg−εg√εd−εm. (1) metalcorrespondingly. Itisknownthatsuchplanethree-layer c 4π ε −ε ε ε −ε +ε ε −ε waveguidesupportsthepropagationoftheboundeigenmode d g m d g g d m only if the width of the intermediate layer is less than some The main goal of the present work is to give the theoretical cut-off value h which is determined by the permittivities at c description of the hybrid waveguide and to find approaches givenfrequency todeeperlocalizationoftheHPP-mode. Comparativeanaly- When the thickness exceeds this critical value the funda- sisof[3,5]suggests,thatinordertogetstrongertransversal mental CGD-mode becomes radiative with energy leaking miniaturizationofthehybridwaveguidetheCGD-likeregime √ intoupperdielectrichalfspace. ofpropagation(withn > ε )shouldbeachievedforthe eff d Thereisasignificantdifferencebetweenthedependenceof diameterwhichismuchlessthanfreespacewavelength. Our theeffectiveindexn onthegapthicknesshforthecases analysis of the plain CGD-structure shows that the localiza- CGD oflowandhighindexdielectric,seeFig.2. Forrelativelylow tion of the fundamental mode can be significantly risen for refractive index of dielectric, ε < |ε |, there exists surface d m specialsetofmaterials, whenabsolutevalueofthemetaldi- plasmon-polariton mode when the gap is absent, h = 0. It electricconstantislessthanthedielectricconstantofthedi- (cid:112) has effective index n = ε ε /(ε +ε ). Then the electriccladding,|ε |<ε .Forthecase,theeffectiverefrac- md m d d m m d effective index of the fundamental CGD-mode is bounded, tive index n is proportional to inverse width of the gap, √ CGD ε < n < n . Just the case was considered in the n ∝ 1/kh, when the width h is small enough. To use d CGD md CGD papers[5]and[3]. Otherwisewhenpermittivityofdielectric the same effect for the hybrid waveguide, the cylinder diam- isrelativelyhigh eter should sufficiently exceed some critical value d∗, which isdeterminedbytheconditionthatthetransversalsizeofthe ε >|ε |>ε , (3) d m g 3 theeffectiveindexn unlimitedlydivergesasthegapthick- CGD √ nesstendstozero,h(cid:28)λ/ ε : d 1 (ε −ε )(ε −ε ) n ≈ ln d g m g . (4) CGD 2kh (ε +ε )(ε +ε ) d g m g This leads to extremely strong light confinement in a trans- parentdielectricgaplayerlocatedbetweenthehigh-indexdi- electricandtheconductor.Theactualdegreeoflocalizationis restrictedonlybyadditionalfactors,suchasincreasingOhmic lossesinthemetal,spatialdispersionandatomicstructureof thematerials. Inthisrespectthepropertiesoftheconductor- gap-dielectricplasmonicmodesimilartothatofthegapplas- mon polaritons in a conductor-gap-conductor structure [8]- [9]. This feature is the principle behind our idea: in prac- tice one should choose the metal of the absolute permittivity less than the permittivity of the cylinder and place cylinder FIG.3: (Coloronline)Effectiverefractiveindexofthefundamental at distance h < hc from the metal plane. When such metal hybridmode versuscylinderdiameter d(colouredlines) compared is involved the effective index of the HPP mode can be sig- withthoseofsinglefiber(blacksolidline)andSPPmode(lowerblack nificantlygreaterthaneffectiveindexofelectromagneticfield brokenline). Thedielectricconstantsofthecylinder,dielectricand inbulkmaterialofthecylinderevenforverysmalldiameters metalareε = 12.25ε = 2.25andε = −129+3.3irespec- d g m ofthecylinder. Note, thattocalculategroupvelocityv and tivelyatwavelengthλ = 1.55µm. Theseparametersarechosenin g chromaticdispersionforthemodeusingtheformula(4),one accordancewiththepaper[3]. Thecriticalgapwidthhc = 5nm. shouldknowthedispersionlawsforpermittivitiesε andε . TheHPP-to-CGDcrossoverpoint:d∗ ≈17µmforh=2nm. m d Forthingaph(cid:28)h ,thegroupvelocityscalesasv /c∝h/λ. c g Thus divergence of the CGD-mode effective index with gap results. It follows from Maxwell’s equations that the elec- widthdecreasingleadstostrongreductionofthegroupveloc- tromagnetic field of guiding mode can be fully described by ity. z-componentsoftheelectricandthemagneticfields,E and There is reverse side of the strong localization which is z B [13]. These fields satisfy the following two-dimensional small propagation distance. It was shown in paper [3] that z Helmholtzdifferentialequationinsidethehomogeneousareas thestrongestlocalizationoftheHPP-modecorrespondstothe wherepermittivityisconstant: lowestpropagationlength. Itiscommonplaceofwaveguides whichusemetalasaconstructivecomponent. Letusconsider (cid:26) (cid:27) (cid:26) (cid:27) limitwhenthegapindexislow,soεg (cid:28)|εm|. Forthecase ∆⊥ Ez −(cid:0)β2−εk2(cid:1) Ez =0, (6) H H z z 1 (cid:18) |ε | ε(cid:48)(cid:48) (cid:19) n ≈ 1− m +i m , (5) CGD kh|εm| εd |εm| where ∆⊥ = ∂x2 + ∂y2 and k = ω/c is the free space wavenumber. Theboundaryconditionsonthebothinterfaces where ε(cid:48)(cid:48) is the imaginary part of the metal permittivity. It arecontinuityofcomponentsE ,H ,εE andH ,whereξ- m z z ξ ξ followsfromEq. (5),thatthelocalizationradiusisoftheor- componentofavectorisitsnormalcomponent. der of h|εm| in the limit h (cid:28) hc. Note, that our approach Oursemi-analyticalmethodisbasedontherepresentation allows to squeeze the mode at arbitrary frequency into any of the hybrid waveguide as an integration of the dielectric subwavelengthscalesimplybytuningthegapwidthinaccor- fiber and plane plasmonic waveguide. We express the elec- dancewith(4). Hence,ourwaveguidedesignbreaksconnec- tromagnetic field of the HPP-mode as a linear combination tionbetweenmodelocalizationandthecarryingfrequencyof of cylindrical modes around the fiber and evanescent plane the mode. In particular, the approach may be interesting for waves above the metal screen. Boundary conditions provide design waveguides at THz frequencies [10, 11]. The prop- the system of linear equations on the expansion coefficients. agation length (cid:96) ∼ h|εm|/|tg| i.e. reduces with the mode Suchanapproachleadstohighlyefficientmethodofnumer- sizereduction. Tokeepthepropagationlengthacceptablefor icalsolvingadifficultboundary-valueproblemsthatdescribe practical implementation at fixed degree of localization one thepropagationofwavesinacomplexsystems[14]-[15].The shouldminimizelosstangenttg.Thus,aprospectinglike[12] schemeisdevelopedindetailinAppendixA. is needed to propose the optimal choice of materials for our Toverifyoursemi-analyticalmethod,inFig.3wepresent approach(3). thedependenceoftheeffectiveindexofthefundamentalhy- brid mode on the cylinder diameter d for a range of the gap widthshinthecaseoftelecommunicationwavelengthwhen III. SEMI-ANALYTICALDESCRIPTION ε < ε < |ε |. These dispersion curves are obtained from g d m ournumericalprocedureandshowagoodagreementwiththe In the section, we present the semi-analytical approach to resultsobtainedin[3]byusingfinite-elementpackageFEM- the propagation of the HPP-mode and discuss the numerical LabfromCOMSOL. 4 FIG.4:Wavevectorofthefundamentalhybridmodeversuscylinder FIG.5:Wavevectorofthefundamentalhybridmodeversuscylinder diameterd(colouredlines)comparedwiththoseofsinglefiber(black diameterd(colouredlines)comparedwiththoseofsinglefiber(black solid line) and SPP mode(lower black broken line). The dielectric solid line) and SPP mode(lower black broken line). The dielectric constantsofthecylinder,dielectricandmetalareε =5.76,ε =1 constantsofthecylinder,dielectricandmetalareε =5.76,ε =1 d g d g and ε = −9.2 respectively at wavelength λ = 0.49µm. These andε =−4respectivelyatwavelengthλ=0.49µm. Thecritical m m parametersarechoseninaccordancewiththepaper[16].Thecritical gapwidthh =13,4nm. TheHPP-to-CGDcrossoverpoints(black c gapwidthh =7nm.TheHPP-to-CGDcrossoverpointsare:d∗ ≈ arrows)are:d∗ ≈40nmforh=2nm,d∗ ≈65nmforh=5nm, c 310nmforh=2nm,d∗ ≈875nmforh=5nm. d∗ ≈220nmforh=10nm. In accordance with general argumentation given in Sec- diameter and large-diameter asymptotics is defined by the tionIIwenextpresenttwosetsofplots.Fig.4(a)corresponds equation nSF(d0) = nmg, where nSF(d) is the diameter de- to the case of fiber with comparatively low refractive index, pendenceoftheeffectiveindexofthesinglefiberfundamen- √ εd <|εm|,theparametersofthewaveguidearetakenaccord- tal mode. If the condition εdkd (cid:28) 1 is valid one can de- inglytoexperimentalwork[16]. Fig.4(b)correspondstoop- rivethatthelocalizationofthismodeisexponentiallysmall, √ √ posite limit, when εd > |εm|. Parameters of these two plots nSF = εg+κ2g/(2 εgk2),where differ only for metal permittivity ε , the value ε = −4 is m m 16e−2γ+1 (cid:26) 8(ε +ε ) (cid:27) chosenforFig.4(b). Here,wedonotconcretizethematerial κ2/k2 ≈ exp − d g (cid:28)1, (8) of the metal screen, our goal is just to demonstrate the qual- g (kd)2 εg(εd−εg)(kd)2 itativedifferenceoftheguidingmodepropertiesforthecase andγ =0.5772...isEuler-Mascheroniconstant. (3). Letussupposetheeffectiveindexn oftheCGD-mode Presentedresultsindicatethatwhenfiberdiameterdisde- CGD not to be significantly above that of bulk plane wave in the creased,theHPP-modelosesconfinementalongthemetaland √ fiber medium, ε . For example, this assumption is true at eventually (at d = 0) becomes a surface plasmon-polariton d theconventionalplasmonicconditionwhentheabsolutevalue mode of the flat metal-vacuum interface. Herewith the ef- of metal permittivity is sufficiently greater than the dielec- fective index of the HPP-mode monotonically decreases to tric permittivity, |ε |/ε −1 (cid:38) 1. Just the case is realised m d that of this SPP-mode. Thus all dispersion curves have the (cid:112) at Fig.3 and Fig.4(a). Then the field penetration depth into sameasymptoticn → n = ε ε /(ε +ε )atsmall eff mg m g m g theupperdielectricisquitelargeaswellastheHPP-to-CGD d. Two different behavior are possible at the opposite limit crossoverdiameterd∗ (cid:38) λ, sotheCGD-modedoesnotpro- of large diameter. As the diameter d → ∞, the HPP-mode vide the strong confinement. Therefore there are no advan- canasymptoticallytendeitherfundamentalsinglefibermode tages of CGD-like limit in the case from the view of HPP- or the fundamental mode of the planar three-layer system, modeconfinement. Foragivenfrequencyandgapwidththe the choice depends on the gap width h. If the gap thick- choice with the strongest coupling of the fiber mode and the ness h is below than h (Eq.(1)) the HPP-mode approaches c surface plasmon polariton mode, corresponding to d = d , 0 theCGD-modewiththediameterincreasing. Inthecasethe providesthestrongestlocalizationofthefieldwithinnanogap crossoverbetweentheasymptoticsoccursatd∗(blackarrows due to the great contrast of permittivities [3]. At the same atFig.4(b))whichisdeterminedas time significant part of energy is transferred inside the fiber, 1 thusthewaveguidemodeconfinementisachievedlargelydue d∗ ≈ . (7) to the boundedness of the high-permittivity dielectric part of 4(n2 −ε )hk2 CGD d the waveguide. Once the diameter of the fiber is optimum For h > h the HPP-mode becomes the cylinder-like and the gap width is small enough the advantages of the hy- c in the limit of the large diameter. In the case the critical brid architecture are used completely: cross section size of diameter d corresponding to the transition between small- the system can be much less than the wavelength and mode 0 5 confinementismuchstrongerthanforuncoupledsinglefiber or flat metal-dielectric interface. To achieve further increase oftheHPP-modeconfinementthefiberwithhigherdielectric constantshouldbeused. Next let us assume that effective index of the CGD-mode is significantly larger than the refractive index of the fiber medium. Thiscanbeachievedbydiminishingthegapthick- ness in the case |ε | < ε that corresponds to the Fig.4(b). m d ThentheCGD-modehasstrongconfinementsothecrossover diametercanbedecreasedtodeepsubwavelengthscale,d∗ (cid:28) λ,bytuningthegapwidth. ThereforetheattractiveCGD-like asymptotic can be achieved by HPP-mode with very small diameter of cylinder providing the wished structure of the FIG.6:Referencesystem. mode with the strong transversal localization in two dimen- sionswithinthegapregionandexponentialdecayingintothe cylinder. Note that in the case the top part of the fiber cross AppendixA:Numericalmethod sectionisatdistancesmuchlargerthan1/κ fromthegapand d itsparticularshapedoesnotplayroleanymore. The theoretical description of the hybrid waveguide is in- hibitedbyitscomplexgeometry. Ingeneralweshouldchose suchsystemofcoordinateswherethesurfacesofthewaveg- uidearetheisolinesandHelmholtzequationcanbesolvedby separation ofvariables. Thehybrid geometry correspondsto thesocalledbipolarcoordinatesbasedontwosetsoforthog- IV. CONCLUSION onalcircles.InthiscoordinatesystemtheHelmholtzequation has quite complicated form and accordingly the set of eigen In the paper we have proposed the novel approach for hy- functionscannotbefoundanalytically.Howevertheunknown brid plasmonic waveguide design providing wide opportuni- hybrid eigen functions can be expressed in terms of known ties for HPP-mode property controlling. When the absolute solutions of the Helmholtz equation in other coordinate sys- permittivityofthemetalislessthanthatofthedielectricthe tems. It is convenient to represent the total electromagnetic hybrid effective index is unlimitedly diverges (Eq.(4)) with fieldofHPP-modeasthesuperpositionoftheallmodesofsin- gap width decreasing. High effective index provides strong glefiber(cylindricalfunctions)andallSPPmodes(evanescent confinement of the electromagnetic field in two dimensions planewaves)withsomeunknowncoefficientsofexpansion. within the nanometer-scale gap region. Thus the mode size We chose the Cartesian system of coordinates with origin canbesimplycontrolledbytuningthewaveguide’geometry at axis of the cylinder which is z-axis (Fig. A). Supposing at fixed frequency and materials constituting the waveguide. thestructureofthefundamentalhybridmodetobesymmetric Theadvantagesofthecase|εm| < εd areconfirmedbyboth withrespecttox-axiswecandescribethelongitudinalcom- qualitativeanalysiswithinplanarthree-layermodelandrigor- ponentoftheelectricfieldas oussemi-analyticalmethoddescribingtheHPP-modepropagation in general. The propagation distance of hybrid mode rperdoupcaegsatwiointhatthfiexmedoddeesgirzeeeroefdulocctiaolniz.aTtioonacohnieevsehloounlgd-rmaningie- EEz((dg))==(cid:80)(cid:80)∞∞n=0baEEnKJn((χκdrr))ccoossnnϕϕ+ mnoitzeedltohsasttsainmguelntatntegou=ss|aεt(cid:48)mi(cid:48)s/fyεi(cid:48)mng| oofftbhoethmceotanld.itIiotnssho|εumld|b<e +Ez((cid:82)m0∞) =cEq(cid:82)e∞xn=pd(0EQneκxgp(nx(−−QgDκ))(cxo−sqDκg)y)cdoqsqκ ydq ε and tg (cid:28) 1 at optical and near infrared frequencies is a z 0 q m m d (A1) challengingtask. Thusimplementationofthewaveguideloss andz-componentsofthemagneticfieldas compensation techniques would be required to use such hy- bcirricduwitsa.veAgnuoidtheearspaotceonmtipalonaepnptliocafttihoenmofinoiuatruwrizaevdegpuhidoetodneic- HHz((dg))==(cid:80)(cid:80)∞∞n=0baHHnKJn((χκdrr))ssiinnnnϕϕ+ sign lies in study field of the resonant plasmons. The reso- +z(cid:82)∞cHexnp=(0Qnκ (xn−gD))sinqκ ydq nmaentcael-dcioenledcittiroicnifnoterrafacsuerifsactehepfilansem-tounn-ipnoglaorfittohneaptearmpiltatinvair- H(m0) =q(cid:82)∞dHexgp(−Qκ (x−Dg))sinqκ ydq z 0 q m m ties, −(ε +ε ) (cid:28) ε [17]. Thus for particular metal the (A2) m d d (cid:112) resonance can be achieved only in a narrow spectral range. whereQ= 1+q2andD =d/2+h. While resonant increasing of the CGD-mode effective index Towritethecorrespondingequations,itisconvenienttoex- requires only the geometrical condition h (cid:28) λ. Thus for pressthefieldinsidedielectricintermsofonlyplaneevanes- the conductor-gap-dielectric structure the resonance of plas- centwaves,whenweimposethecontinuityconditionsonthe monic mode can be attained by gap width decreasing at any boundary of the metal, and in terms of angular harmonics, frequencyaslongasthecondition|ε |<ε isvalid. for the cylindrical surface. We solve it by using the evanes- m d 6 cent plane wave expansion of modified cylindrical functions andangularharmonicspectrumoftheevanescentplanewaves [18] (cid:90) ∞ ∞ (cid:88) (cid:90) ∞ Hz(g) = bHnFnH(q)e−Qκgxsinqκgydq + Kn(κgr)cosnϕ= FnE(q)e−Qκgxcosqκgydq, (A3) 0 n=0 0 (cid:90) ∞ + cHeQκg(x−D)sinqκ ydq. q g (cid:90) ∞ 0 K (κ r)sinnϕ= FH(q)e−Qκgxsinqκ ydq, (A4) n g n g 0 ∞ ItcanbeeasilyderivedfromtheMaxwell’sequationsthat (cid:88) eQκgxcosqκ y = GEcosnϕ, (A5) forthenormalcomponents g n n=0 ∞ (cid:88) eQκgxsinqκ y = GHsinnϕ, (A6) g n iβ ∂E ik ∂H n=0 Eξ = −β2−εk2 ∂ξz + β2−εk2 ∂ηz, (A11) where ik ∂E iβ ∂H H = −ε z − z, (A12) ξ β2−εk2 ∂η β2−εk2 ∂ξ (Q+q)n+(Q−q)n FE = , (A7) n 2Q (Q+q)n−(Q−q)n whereηistangenttotheinterfacecoordinateinthetransversal FnH = 2Q , (A8) plane. ThecontinuityconditionsonthemetalsurfaceforE ,B , z z εE and εB lead to the first system of linear homogeneous 2−δ x x GEn = 2 0n((Q+q)n+(Q−q)n)In(κgr), (A9) equations(SLE)oncoefficientsbEn, bHn, cEq, cHq, dEp, dHp. The correspondingcontinuityconditionsforE ,B ,εE andεB GH = ((Q+q)n−(Q−q)n)I (κ r). (A10) z z r r n n g onthecylindricalsurfaceproducethesecondSLEonampli- tudes aE, aH, bE, bH, cE, cH, which is now integral with re- Thus the electromagnetic fields in surrounding medium n n n n q q specttocE,cH. Inordertoavoidintegrationoftheunknown closetothedielectricwaveguidecanbewrittenas q q functionsweexpressthecoefficientscE,cHinthetermsofbE, q q n (cid:88)∞ bHfromthefirstSLEandsubstitutethemintothesecondSLE. E(g) = bEK (κ r)cosnϕ + n z n n g The procedureleads to theinfinite system oflinear homoge- n=0 neousalgebraicequationsforcoefficientsaE, aH, bE, bH. In n n n n ∞ (cid:90) ∞ order to solve the system numerically one should truncate it (cid:88) + cosnϕ cEGE(r,q)e−QκgD dq, to a finite size. Then the propagation constant of the funda- q n n=0 0 mentalhybridmodecanbedeterminedfromtheconditionof vanishingofthecharacteristicdeterminant. ∞ (cid:88) H(g) = bHK (κ r)sinnϕ + z n n g n=0 ∞ (cid:90) ∞ (cid:88) + sinnϕ cHGH(r,q)e−QκgD dq. q n n=0 0 Acknowledgments Thecorrespondingexpressionsforthefieldsclosetothesur- faceofthemetalare (cid:90) ∞ ∞ WethankI.R.GabitovforvaluableadvicesandV.V.Lebe- (cid:88) Ez(g) = bEnFnE(q)e−Qκgxcosqκgydq + devforfruitfuldiscussions. Theworkwaspartiallysupported 0 n=0 bytheRussianFederationGovernmentFRBR12-02-01365-a, (cid:90) ∞ CouncilofthePresidentoftheRussianFederationgrantNo.- + cEeQκg(x−D)cosqκ ydq, 324.2012.5,theRussianFTPKadry,andfoundationDynasty. q g 0 7 [1] BornM.,WolfE.,PrinciplesofOptics(CambridgeUniv.Press, multilayers,Phys.Rev.B75,241402(2007) 1999.) 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