Journal of Microwaves, Optoelectronics and Electromagnetic Applications, Vol. 13, SI-1, November 2014

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1 Journal of Microwavs, Optolctronics and Elctromagntic Applications, Vol., SI-, Novmbr 04 SI-0 Rtriving th Effctiv Prmittivity of an Optical Mtamatrial Structurd with Mtallic Cylindrical Nanorods An Analytical Approach Basd on th Calculation of th Dpolarization Fild Andrson O. Silva and João C. W. A. Costa Fdral Univrsity of Para Faculty of Elctrical Enginring, Av. Augusto Corra, 0, CEP , Blm-Pa, Brazil, Abstract Th optical proprtis displayd by matrials nanostructurd with mtallic inclusions ar mainly drivn by th xcitation of surfac plasmon-polaritons. With this in mind, a valid homognization procdur must b abl to dscrib appropriatly th rlationship btwn th gomtry of th inclusions and th natur of local fild mods. In this papr, w driv a corrctd vrsion of th Maxwll-Garntt modl for th homognization of nonmagntic matrials structurd with cylindrical nanorod inclusions. Th analytical formulation combins th Mir-Wokaun approach with th Grn s function mthod for sourc rgions to rigorously dpict th impact of shap and siz of th inclusions on th ffctiv prmittivity of thos mdia. From th comparison to th rspctiv rsults obtaind by th Maxwll-Garntt formalism corrctd for sphrs and prolat sphroids, w show that our dduction provids furthr rliability in dscribing th spctral variation of th ffctiv prmittivity, particularly at th rsonanc rgions. Morovr, w prform a comparison to th rsults givn by th Bruggman modl in ordr to vidnc that th natur of local fild mods is also crucial in rgarding th Maxwll-Garntt approach as a suitabl choic to rtriv th ffctiv rsonanc charactristics. Indx Trms dpolarization filds, ffctiv mdium thory, optical mtamatrials, surfac plasmon-polaritons. I. INTRODUCTION Mtamatrials ar man-mad structurs nginrd to display lctromagntic proprtis that cannot b found in naturally-occurring mdia. Such proprtis rsult from th local fild mods xcitd in subwavlngth structurs that compos th unit cll, which is constitutd by diffrnt matrials arrangd in a rgular pattrn []. Svral optical proprtis associatd to ths mdia hav bn vrifid xprimntally and includ ngativ rfraction [], hyprlns imaging [], local fild nhancmnt [4] and hyprbolic disprsion [5]-[6], to nam a fw. Th ida implicit in th concpt of mtamatrials is that th intraction of light with subwavlngth structurs can b quivalntly dscribd in trms of macroscopic ffctiv paramtrs corrsponding Brazilian Microwav and Optolctronics Socity-SBMO rcivd 5 Mar 04; for rviw 8 Mar 04; accptd Oct 04 Brazilian Socity of Elctromagntism-SBMag 04 SBMO/SBMag ISSN

2 Journal of Microwavs, Optolctronics and Elctromagntic Applications, Vol., SI-, Novmbr 04 SI- to a homognous mdium. By its turn, an fficint homognization modl should b capabl to assign ffctiv constitutiv paramtrs that bring togthr a microscopically tailord comprhnsion of local fild ffcts [7]-[9]. Standard thortical approachs basd on fild avrag calculations at th long wavlngth limit ar oftn applid to stimat th ffctiv paramtrs of mtamatrials at th optical domain. Maxwll-Garntt (MG) and Bruggman thoris ar rmarkabl xampls whos applications rmain currntly vigorous [0]-[]. Both approachs dtrmin th macroscopic constitutiv paramtrs by considring th raction filds inducd in an inclusion mbddd in a host mdium. Thy ar xtnsions of th Clausius-Mossotti formulation [] and diffr in th mannr by which inclusions and host ar distinguishd. In th Bruggman approach, both matrix (or background) and particulat structurs ar considrd inclusions whil th host mdium is th homognizd matrial itslf. Unlik th Bruggman modl, th MG approach rquirs th clar idntification of which constitunts ar inclusions and host []-[4]. Undr a statistical point of viw, th MG approach can also b considrd a homognization modl in which th position of th inclusions in th structurd mdium ar uncorrlatd among thmslvs, as dmonstratd in [5]. Rgarding th stimation of th local filds, th original MG thory computs th macroscopic paramtrs basically from th avrag of th lctrostatic filds gnratd by th dipols inducd in sphrical inclusions and corrsponds to th first trm of th xpansion of scattrd filds obtaind in th Mi thory [6]. Thrfor, it nglcts inclusions whos gomtry dviats from th sphrical profil. To ovrcom this limitation, svral corrctions hav bn proposd to bttr stimat th filds inducd in th unit cll by th incidnt lctromagntic wav, which strongly dpnds on th gomtry of th particls and matrial composition. Ths corrctions can b xprssd in th ffctiv prmittivity as a dpolarization factor γ, which in gnral is a tnsor [7]. A qualitativ discussion on th corrction introducd by γ in th Bruggman and MG modls was brifly carrid out in [8] using th assumption of a quasi-static approximation. It addrsss th cas whr all th dimnsions of th inclusions ar in subwavlngth siz. To xtnd byond this, Mir and Wokaun [9] drivd a modl basd on th calculation of th dpolarization lctric fild (th fild that ariss as a raction to th applid lctric fild) to obtain a dynamical countrpart for γ and invstigat th fild nhancmnt causd by surfac plasmon-polariton xcitations on sphrical mtallic particls. This dynamical trm corrcts th polarization P for th spatial dphasing btwn ach infinitsimal dipol momnt inducd within th particl volum. From ths calculations aris an additional imaginary contribution, proportional to th total particl volum and attributd to radiativ losss. Foss t al. [0] substitutd th static trm of th Mir-Wokaun modl for sphrs by th corrsponding valu for thin longatd cylindrs to rtriv th ffctiv prmittivity of an arrangmnt of nanoscopic ndl-shapd gold particls. In [], th quivalnt circuit modl for th dipolar radiation is applid to th MG mixing formula as a way to includ damping du to scattring in th final rsult of th ffctiv prmittivity. Liu t al. [] usd th modifid xprssion dducd by Brazilian Microwav and Optolctronics Socity-SBMO rcivd 5 Mar 04; for rviw 8 Mar 04; accptd Oct 04 Brazilian Socity of Elctromagntism-SBMag 04 SBMO/SBMag ISSN

3 Journal of Microwavs, Optolctronics and Elctromagntic Applications, Vol., SI-, Novmbr 04 SI- Mir and Wokaun as an xtnsion of th MG modl to comput th ffctiv prmittivity of a hyprbolic-disprsion mtamatrial composd by silvr nanowirs. Moroz [] xtndd that approach to prolat and oblat sphroidal particls in th cass whn th applid lctric fild is prpndicular or paralll to th principal axis. Th work carrid out by Körmöczi and Szabó [4], for instanc, applid th static trm of th dpolarization factor of sphroidal particls to corrct th MG formulation and rtriv th prmittivity of a mtamatrial layr usd as a building block of an invisibility cloak. In all ths works, th xprssion for γ is obtaind by modls in which th cylindrical gomtry is approximatd by sphrs or sphroidal particls with smi-major axis much gratr than th smi-minor axis. In this papr, w combin th Mir-Wokaun approach [9] with th Grn s function mthod for sourc rgions [7] to formulat rigorously th dynamical MG modl for cylindrical structurs. W apply it to calculat th ffctiv rlativ prmittivity of an optical mtamatrial composd by a priodic array of silvr nanorods mbddd in alumina background. Th charactristics of this formulation ar analyzd upon comparison to th rsults calculatd by th MG vrsions corrctd for prolat sphroids and sphrs. For th lattr gomtry, w substitut th static componnt of th corrction factor by that of cylindrical particls, analogously to what is mad in []. W also comput th ffctiv rlativ prmittivitis using th Bruggman modl in ordr to vidnc that th natur of local fild mods is bttr considrd in th MG formalism, which maks it an appropriat homognization tchniqu for optical mtamatrials, particularly for small volum fractions of th mtallic inclusions in th dilctric matrix. Comparisons btwn analytical prdictions and numrical rsults for rflctanc and transmittanc spctra ar also carrid out as an additional way to assss th accuracy of th formulation dducd in this work. II. THE CORRECTION FACTOR IN THE MAXWELL-GARNETT MODEL Th MG formalism for th homognization of mdia with mtallic inclusions is xprssd by: ff d m f d, () ff d m d whr f is th fill-ratio (or volum fraction) of th mtal in th structur, ζ is th corrction factor (also calld scrning paramtr) and ε m, ε d and ε ff ar, rspctivly, th rlativ prmittivitis of th mtal inclusions, th dilctric matrix and th homognizd mdium. This modl diffrs from th Bruggman approach in th sns that th lattr rgards mtallic particls and dilctric matrix as inclusions and th mbdding mdium is th homognizd structur. Th xprssion for th Bruggman modl can b drivd straightforwardly from th gnral formulation givn in th appndix. Th original MG modl is addrssd for sphrical particls and adopts ζ=. Othr gomtris as ndl-shapd particls hav ζ= or tnding to infinit, dpnding on th orintation of thir Brazilian Microwav and Optolctronics Socity-SBMO rcivd 5 Mar 04; for rviw 8 Mar 04; accptd Oct 04 Brazilian Socity of Elctromagntism-SBMag 04 SBMO/SBMag ISSN

4 Journal of Microwavs, Optolctronics and Elctromagntic Applications, Vol., SI-, Novmbr 04 SI- rvolution axs rlativly to th dirction of light incidnc. In fact, such valus corrspond to a first approximation for ζ sinc only th static countrpart in th mod xpansion of th Mi thory is considrd. By rgarding highr-ordr trms in this xpansion, w can achiv a mor complt valuation of ζ. In doing so, a compact xprssion for th corrction factor is obtaind [0]: whr γ is calld dpolarization factor., () Substituting () in (), th MG xprssion is modifid to xplicit th dpndnc on th dpolarization factor: f ff d d d m d f m d. () Using th procdur proposd by Mir and Wokaun [9] combind with th Grn s function mthod for sourc rgions [7], w calculat analytically in th nxt sction th dpolarization factor γ for nanorods with circular cross-sction (Fig. ). Th dvlopd xprssion is valid for an arbitrary rlation btwn th radius and th hight of th nanorod. III. THE ELECTRIC DEPOLARIZATION FIELD FOR CYLINDERS According to th Mir-Wokaun approach, w dfin an lmntal dpolarization lctric fild d gnratd by a tim-oscillating dipol momnt dp PdV xp jr t Ed in an infinitsimal volum dv ( is th wavnumbr, r is th sphrical radial coordinat and P is th magnitud of th polarization). Following th procdur outlind in [], w start with th xprssions for th radial and tangntial componnts of th lctric fild (in gaussian units): E r dp dp cos, (4a) r cr dp dp dp E sin, (4b) r cr c r whr th uppr dots indicat tmporal drivativs. Aftr prforming th drivativs, w substitut j r by: r jr r j r, (5) 6 and dcompos (4a-b) into th azimuthal and longitudinal countrparts, obtaining th following xprssions for th scalar componnts of d Ed : Brazilian Microwav and Optolctronics Socity-SBMO rcivd 5 Mar 04; for rviw 8 Mar 04; accptd Oct 04 Brazilian Socity of Elctromagntism-SBMag 04 SBMO/SBMag ISSN

5 Journal of Microwavs, Optolctronics and Elctromagntic Applications, Vol., SI-, Novmbr 04 SI-4 cos sin cos sin de PdV, (6a) r r de z cos r cos j PdV r, (6b) whr only th trms up to and longitudinal dirction, rspctivly. wr considrd. Th subscripts ρ and z rfr to th azimuthal plan Fig.. Schmatic rprsntation of a nanorod with circular cross-sction. Dashd lins dlimitat th prolat sphroid inscribd in th nanorod. Th point 0 is th cntroid of th cylindr. In (6a-b), w can obsrv thr contributions: th trms rlatd to /r corrspond to th lctrostatic countrpart of lctric fild gnratd by an infinitsimal tim-oscillating dipol; th trms dpnding on r ar calld dynamic componnts and ar rlatd to th dphasing among th radiatd filds of infinitsimal dipols; th trm rgards damping du to radiation. Th lctric dpolarization fild is obtaind from th intgration of (6a-b) ovr th particl volum. As discussd in th following, this calculation rquirs som important rmarks. A. Th static componnt of th lctric dpolarization fild Du to th singularity associatd to /r, th intgration of th static countrpart of th dpolarization fild is divrgnt. To ovrcom this obstacl, w rsort to th Grn s function mthod for sourc rgions, whos cntral formula is xprssd hr in trms of th polarization vctor: E d / r lim G ~ P dv L ~ 4 P, V 0 (7) In (7), G ~ is th dyadic Grn s function and L ~ is th static dpolarization tnsor. Th trm V δ is namd principal volum, a small rgion that ncloss th sourc and xcluds it from th volum V. Brazilian Microwav and Optolctronics Socity-SBMO rcivd 5 Mar 04; for rviw 8 Mar 04; accptd Oct 04 Brazilian Socity of Elctromagntism-SBMag 04 SBMO/SBMag ISSN

6 Journal of Microwavs, Optolctronics and Elctromagntic Applications, Vol., SI-, Novmbr 04 SI-5 With th assumption that P is uniform and choosing V δ of th sam gomtrical shap as V but in rducd siz, w can infr that th intgral opration on th right-hand sid of (7) is null if w attmpt to th idntity: P d S G ~ P dv ds 0, (8) V V SV V r providd that S(V-V δ ) is th closd surfac that bounds (V-V δ ) and th vctor Thus, th final rsult for th static componnt of th lctric dpolarization fild is: Ed 4L ~ P. / r d S is outward to S. (9) For th cylindr shown in Fig., th gnral xprssion for th static dpolarization tnsor is givn by [7]: / cos L ~ / cos 0. (0) 0 0 cos B. Th dynamic and radiation componnts of th lctric dpolarization fild In contrast to th static componnt, th dynamic and radiation countrparts of th lctric dpolarization fild can b calculatd dirctly from th intgration: cos sin cos E ˆ ẑ j d ẑ / r, rad dv. () r r Th valuation of () rsults in: Ed V C ~ P V j I ~ / r, rad pr pr P, () whr V pr is th volum of th prolat sphroid inscribd in th cylindr, I ~ is th idntity dyadic and C ~ is th dynamic tnsor: Brazilian Microwav and Optolctronics Socity-SBMO rcivd 5 Mar 04; for rviw 8 Mar 04; accptd Oct 04 Brazilian Socity of Elctromagntism-SBMag 04 SBMO/SBMag ISSN

7 Journal of Microwavs, Optolctronics and Elctromagntic Applications, Vol., SI-, Novmbr 04 SI C ~ 0 0 0, E E 0 0 n 4h / E a. h / () C. Th final xprssion for th lctric dpolarization fild With (9) and () w dtrmin th vctorial xprssion for th total lctric dpolarization fild: Ed V L ~ pr 4 C ~ 4 Vpr j I ~ P 4 (4) Th major diffrncs btwn th lctric dpolarization fild for cylindrical inclusions and th corrsponding on for prolat sphroids li on th xprssions of th static dpolarization factor L ~ and th dynamic tnsor C ~. Particularly, th lattr gomtry prsnts an azimuthal componnt for C ~ whras th formr possss solly a longitudinal countrpart. Th xprssions for L ~ and C ~ approachs th corrsponding formulations for prolat sphroids whn h/a tnds to infinity. In practic, this mans that th lctric dpolarization fild for prolat sphroids is a valid approximation only for rods rsmbling ndl-shapd particls. As mntiond prviously, lctric fild E d is th raction fild to th incidnt lctric fild E 0. Hnc, th total E total is E d addd to E 0. Sinc: with ~ bing th polarizability tnsor. W can rlat E V ~ 0 P, (5) Etotal to ~ to achiv th complt xprssion of th total dpolarization factor γ. Such calculation is outlind in th following sction. Th dvlopd xprssion is valid for an arbitrary rlation btwn th radius and th hight of th nanorod. IV. THE DEPOLARIZATION FACTOR FOR CYLINDRICAL NANORODS Rgarding that: 4 P m E total. (6) W can assign th following quation to rlat th scalar componnts of ~ and P : Brazilian Microwav and Optolctronics Socity-SBMO rcivd 5 Mar 04; for rviw 8 Mar 04; accptd Oct 04 Brazilian Socity of Elctromagntism-SBMag 04 SBMO/SBMag ISSN

8 Journal of Microwavs, Optolctronics and Elctromagntic Applications, Vol., SI-, Novmbr 04 SI-7 4 P ˆ P ẑ L ~ V V cyl cyl P ˆ z Vcyl C ~ 4 Pz ẑ Vcyl I ~ 4 4 j P ˆ P ẑ. m z m z (7) In (7), w dcomposd th polarization and th polarizability into thir azimuthal and longitudinal componnts. Furthrmor, th rlation V pr =(/)V cyl, whr V cyl is th cylindrical volum, is applid. Thn th componnts of ~ ar calculatd and th azimuthal and longitudinal formulations for γ ar idntifid: Vcyl m,z, (8) 4,z m whr: cos j Vcyl, (9a) 4 z cos Vcyl Cz j Vcyl, (9b) 4 4 whr C z is th nonzro trm in C ~. Th diffrncs btwn th lctric dpolarization filds for cylindrs, sphrs and prolat sphroids influnc dcisivly th final xprssions for γ. For th cylindrical gomtry, th azimuthal componnt γ ρ possss only th static and radiation countrparts whras th longitudinal componnt γ z includs th dynamic trm. On th othr hand, th total dpolarization factors for prolat sphroids xhibit dynamic trms in th azimuthal and longitudinal componnts as wll as th corrsponding formulations for sphrs. Tabl I summarizs γ ρ and γ z for all ths gomtris. As mntiond prviously, th static componnt of γ for sphrical particls is substitutd for that of cylindrical inclusions. Brazilian Microwav and Optolctronics Socity-SBMO rcivd 5 Mar 04; for rviw 8 Mar 04; accptd Oct 04 Brazilian Socity of Elctromagntism-SBMag 04 SBMO/SBMag ISSN

9 Journal of Microwavs, Optolctronics and Elctromagntic Applications, Vol., SI-, Novmbr 04 Brazilian Microwav and Optolctronics Socity-SBMO rcivd 5 Mar 04; for rviw 8 Mar 04; accptd Oct 04 Brazilian Socity of Elctromagntism-SBMag 04 SBMO/SBMag ISSN SI-8 TABLE I. DEPOLARIZATION FACTORS FOR CYLINDERS, SPHERES AND PROLATE SPHEROIDS cylindrs γ ρ γ z cyl V j cos 4 cyl z cyl V j C V cos 4 4 sphrs γ ρ γ z 9 r j r cos 9 r j r cos prolat sphroids γ ρ γ z pr pr pr V j V D Z 4 4 pr pr z pr V j V D Z 4 4 In tabl I, th paramtrs for prolat sphroids ar []: n Z pr, (0a) z Z pr h D, (0b) h D n D z, (0c) with: 4 4 h a h, () As γ is rsponsibl for corrcting th MG modl to th homognization of mdia with cylindrical inclusions, th aftrmath accuracy is dirct rsult of th application of (9a-b) in dpicting () ovr th spctral rang undr analysis. This aspct is illustratd in Fig., whr th magnituds of th diffrncs btwn th polarization factors for cylindrs, prolat sphroids and sphrs (dnotd as

10 Journal of Microwavs, Optolctronics and Elctromagntic Applications, Vol., SI-, Novmbr 04 SI-9 Δγ ρ and Δγ z ) ar plottd as function of optical wavlngths. Two cass ar considrd: h/a=500 and h/a=55. With h fixd at 5 μm, thos ratios ar narly th ons obtaind by choosing a qual to 0 nm and 4 nm, rspctivly. In th cas of cylindrical nanorods, for instanc, such valus can b achivd by slf-assmbling fabrication mthods [6]. For th azimuthal dpolarization factors of cylindrs and prolat sphroids, a rduction of 0% in th rlation h/a is accompanid by a growing of 50% in Δγ ρ at th wavlngth λ= 500 nm and is furthr incrasd for shortr wavlngths. A similar bhavior is obsrvd for Δγ z. Whn sphrs ar takn into account, th valus of Δγ ρ and Δγ z ar mor pronouncabl, which indicats a significant impact in applying th MG modl to th homognization of optical mtamatrials. Nglcting thos diffrncs may lads to an imprcis spctral dscription of th ffctiv prmittivity of ths mdia at th optical domain. Fig.. Magnitud of th diffrnc btwn th dpolarization factors for cylindrs, prolat sphroids and sphrs. Two valus of h/a ar considrd: 500 and 55. As xampl of application, w applid () with th dynamical dpolarization factors (9a-b) to calculat th ffctiv rlativ prmittivitis of an optical mtamatrial composd by silvr cylindrical nanorods mbddd in alumina. Th obtaind rsults ar compard to th ons providd by th MG modl corrctd for prolat sphroids and sphrs. W join to this comparison th valus prdictd by th Bruggman modl in ordr to show that an accurat homognization approach should also considr th natur of th xcitd local fild mods. Such calculation is outlind in th following sction. Brazilian Microwav and Optolctronics Socity-SBMO rcivd 5 Mar 04; for rviw 8 Mar 04; accptd Oct 04 Brazilian Socity of Elctromagntism-SBMag 04 SBMO/SBMag ISSN

11 Journal of Microwavs, Optolctronics and Elctromagntic Applications, Vol., SI-, Novmbr 04 SI-0 V. THE EFFECTIVE PERMITTIVITY OF AN OPTICAL METAMATERIAL STRUCTURED WITH CYLINDRICAL SILVER NANORODS Nonmagntic mtamatrials composd by mtallic nanorods mbddd in a dilctric matrix ar uniaxial anisotropic mdia []. Th silvr nanorods of radius a and hight h ar arrangd in a hxagonal lattic, as schmatically shown in Fig.. For this arrangmnt, th fill-ratio f can b dducd by f=(a/d) /[π/() / ], whr a is th diamtr of th nanorod of circular cross-sction and D is th distanc btwn th cntroids of adjacnt inclusions [6]. By using (9a-b) in (), w assign to th ordinary and th xtraordinary rlativ prmittivitis th dpndnc on th azimuthal and longitudinal dpolarization factors, as xprssd by: f d ff ( o ) d d f ff( ) d d z m d f m d d m d f m d, (a), (b) th subscripts o and rfr to th ordinary and xtraordinary prmittivitis, rspctivly. Ths rlativ prmittivitis corrspond to th diagonal componnts of th tnsor: xx 0 0 ~ 0 yy 0, () 0 0 zz whr ε ff(o) =ε xx =ε yy and ε ff() =ε zz. Fig.. Schm of th mtamatrial composd by silvr cylindrical nanorods (black rgions) mbddd in alumina matrix (gry rgion). Fig. 4 shows th valus for ε ff(o) and ε ff() calculatd by th MG modl corrctd by th total dpolarization factors for sphrs, prolat sphroids and cylindrical inclusions. In ths calculations, w usd th rlativ prmittivity of silvr dpictd in [5] for th optical spctrum. Th rlativ prmittivity of alumina is rgardd as.0 [6]. Furthrmor, th corrsponding spctral curvs obtaind from th Bruggman modl corrctd for cylindrical rods ar shown in ordr to compar to th MG formulation. Th thicknss of th mtamatrial layr is h=5 μm and th distanc D btwn Brazilian Microwav and Optolctronics Socity-SBMO rcivd 5 Mar 04; for rviw 8 Mar 04; accptd Oct 04 Brazilian Socity of Elctromagntism-SBMag 04 SBMO/SBMag ISSN

12 Journal of Microwavs, Optolctronics and Elctromagntic Applications, Vol., SI-, Novmbr 04 SI- th cntrs of adjacnt nanorods is 60 nm. Calculations wr prformd for a=0 nm and a=4 nm. Such valus rsult in h/a=500 and h/a=55, approximatly. a=0 nm a=4 nm Fig. 4. Ral and imaginary parts of th ffctiv ordinary and xtraordinary rlativ prmittivitis for th lattic of silvr cylindrical nanorods immrsd in alumina matrix calculatd from: MG modl corrctd for cylindrs (blu lin), prolat sphroids (grn lin) and sphrs (rd lin) and Bruggman modl (black lin). According to Fig. 4, th rsonanc wavlngths in th Bruggman modl ar significantly rdshiftd with rspct to th ons stimatd by th MG modl for ε ff(o). This aspct can b analyzd taking into account th contrast of th prmittivitis of th matrials that compos th mdium. As dscribd in th appndix, th main diffrnc btwn th Bruggman and th MG modls lis on th dfinition of inclusions and th mbdding mdium. In th Bruggman modl, th ffctiv prmittivity ε ff corrsponds to th mbdding mdium itslf and both particls and background ar rgardd as inclusions immrsd in this homognizd nvironmnt. This conjctur dos not dpnd on th idntification of particls and background, i.., th valu of ε ff is not snsibl to intrchangs btwn th fill-ratios of ach constitunt in a structurd mdium. As rsult, th natur of th local fild mods cast by th Bruggman modl ar limitd to bulk wavs xcitd in compounds involving matrials with a slight contrast btwn prmittivitis. This is not th cas of optical mtamatrials with mtallic inclusions, in which th high contrast displayd by th mtal-dilctric composition in th unit cll allows th xcitation of surfac plasmon-polaritons mods (SPP). Th lctromagntic filds of SPP mods ar highly confind to th mtal-dilctric intrfac, bing Brazilian Microwav and Optolctronics Socity-SBMO rcivd 5 Mar 04; for rviw 8 Mar 04; accptd Oct 04 Brazilian Socity of Elctromagntism-SBMag 04 SBMO/SBMag ISSN

13 Journal of Microwavs, Optolctronics and Elctromagntic Applications, Vol., SI-, Novmbr 04 SI- rsponsibl for rsonanc faturs at smallr wavlngths than ons of bulk wavs. On th contrary to Bruggman modl, th MG formalism rquirs idntification of inclusions and dilctric matrix. Consquntly, rsonancs associatd to surfac wavs ar rgardd by this approach. This fatur togthr with an appropriat gomtrical corrction factor maks th MG modl a mor suitabl analytical tchniqu to th homognization of optical mtamatrials with low volum fraction of mtallic inclusions. Fig. 5 shows th spatial distribution of th lctric fild componnt E y computd by simulations in th FEM (Finit Elmnt Mthod) basd commrcial softwar COMSOL 4.b [7] for a nanorod inclusion of radius a=0 nm in th mtamatrial mdium. Th sctional viw is paralll to th z axis and th lctric fild distribution is considrd for thr wavlngths along th spctrum of th mtamatrial: 00 nm, 40 nm and 500 nm. Th fild distribution is highly confind to th mtal-dilctric intrfac, which is associatd to a local SPP mod whos rsonanc wavlngth bing th sam as th on prdictd by th MG modl corrctd for cylindrical inclusions, as it is indicatd in Fig. 4. Fig. 5. Spatial distribution of th amplitud of th E y fild componnt computd for thr wavlngths: 00 nm, 40 nm and 560 nm. Th radius of th cylindrical nanorods (dlimitd by thick black lins) is a=0 nm and th distanc btwn adjacnt inclusions is 60 nm. Th sctional viw is paralll to th z axis. An intrsting aspct in Fig. 4 compriss th influnc of particl siz on th spctral curvs for ε ff(o) calculatd by th MG modl. Th largr is th volum of th mtallic inclusion, th lowr is th fild nhancmnt associatd to th local SPP mods. It implis a rd-shiftd and broadr rsonanc bandwidth, as dpictd by th comparison of th ε ff(o) plots for sphrs, prolat sphroids and cylindrical nanorods. For th xtraordinary ffctiv rlativ prmittivity ε ff(), it is found that th MG modl corrctd for sphrical particls undrstimats th losss. This fact is also consqunc of th particl siz. Th xistnc of a gratr latral ara in th nanorods lads to major valus of th imaginary part of ε ff() Brazilian Microwav and Optolctronics Socity-SBMO rcivd 5 Mar 04; for rviw 8 Mar 04; accptd Oct 04 Brazilian Socity of Elctromagntism-SBMag 04 SBMO/SBMag ISSN

14 Journal of Microwavs, Optolctronics and Elctromagntic Applications, Vol., SI-, Novmbr 04 SI- than what is calculatd for sphrs or prolat sphroids. Th Bruggmman modl also dos not achiv accuratly th imaginary part of ε ff() ovr th optical domain du to nglct th xcitation of SPP mods. W also carrid out computational simulations in COMSOL Multiphysics 4.b to valuat th rflctanc R and transmittanc T of lightwav incidnc upon a unit cll of th mtamatrial schmatizd in Fig.. In this calculation, w considrd that light incids onto th plan of th circular cross-sction of th nanorods. Th layr thicknss of th matmtarial is h=5 μm. Fig. 6 shows th spctra of R and T for normal incidnc. Th numrical rsults computd for a=0 nm and a=4 nm ar plottd togthr with th analytical valus obtaind from th Bruggman approach and th MG modls corrctd for cylindrs, prolat sphroids and sphrs. Analytical rsults for R and T wr calculatd by th transfr matrix tchniqu [8]. It is shown that th curvs corrsponding to th MG modl corrctd for cylindrs ar narly th rflctanc spctra computd numrically for th two valus of nanorod radius. Morovr, for both cass, th quantity T+R is smallr than th unit. This is du to th highly absorptiv charactr prsntd by th mtamatrial layr with thicknss of 5 μm. Indd, th analytical and numrical modls indicat null transmittanc ovr th analyzd spctrum, with xcption of th MG modl for sphrs, which prdicts a nonzro transmittanc for wavlngths largr than 400 nm. a=0 nm a=4 nm Fig. 6. Rflctanc and transmittanc spctra upon normal incidnc calculatd from: MG modl corrctd for cylindrs (blu lin), prolat sphroids (grn lin), sphrs (rd lin), Bruggman modl (black lin) and numrical rsults (pink dashpoint lin). A good agrmnt btwn analytical prdictions providd by th MG modl corrctd for cylindrical inclusions and numrical rsults is also vrifid for obliqu incidnc, as it is shown in Fig. 7 and Fig. 8. Calculations wr prformd for two angls of incidnc (0º and 45º) rgarding Brazilian Microwav and Optolctronics Socity-SBMO rcivd 5 Mar 04; for rviw 8 Mar 04; accptd Oct 04 Brazilian Socity of Elctromagntism-SBMag 04 SBMO/SBMag ISSN

15 Journal of Microwavs, Optolctronics and Elctromagntic Applications, Vol., SI-, Novmbr 04 SI-4 both polarizations TE (lctric fild paralll to th plan of th nanorod cross-sction) and TM (magntic fild paralll to th plan of th nanorod cross-sction). As in th cas of normal incidnc, th MG vrsion for sphrs prdicts a spctral rang of nonzro transmittanc for φ=45º and TM polarization as opposd to th othr analytical approachs and computational simulations. a=0 nm TE polarization φ=0º φ =45º TM polarization φ =0º φ =45º Fig. 7. Rflctanc and transmittanc spctra upon obliqu incidnc calculatd from: MG modl corrctd for cylindrs (blu lin), prolat sphroids (grn lin), sphrs (rd lin), Bruggman modl (black lin) and numrical rsults (pink dashpoint lin). Th radius of th nanorods is a=0 nm and two angls of incidnc ar considrd: φ= 0º and φ=45º. Brazilian Microwav and Optolctronics Socity-SBMO rcivd 5 Mar 04; for rviw 8 Mar 04; accptd Oct 04 Brazilian Socity of Elctromagntism-SBMag 04 SBMO/SBMag ISSN

16 Journal of Microwavs, Optolctronics and Elctromagntic Applications, Vol., SI-, Novmbr 04 SI-5 a=4 nm TE polarization φ =0º φ =45º TM polarization φ =0º φ =45º Fig. 8. Rflctanc and transmittanc spctra upon obliqu incidnc calculatd from: MG modl corrctd for cylindrs (blu lin), prolat sphroids (grn lin), sphrs (rd lin), Bruggman modl (black lin) and numrical rsults (pink dashpoint lin). Th radius of th nanorods is a=4 nm and two angls of incidnc ar considrd: φ= 0º and φ=45º. In Fig. 8, it is obsrvd that th rflctanc curvs calculatd numrically ar slightly rd-shiftd in comparison to th prdictions providd by th MG modl corrctd for cylindrs. It is consqunc of an inhomognous polarization of th inclusions. In fact, th inhomognous polarization profil bcoms mor pronouncabl with th rduction of th aspct ratio h/a, analogously to which is discussd in [] for sphroidal particls. A limit of application of th MG formalism corrctd for cylindrs compriss th siz of th structural unit of th mtamatrial. As discussd in [], dscribing th ovrall lctromagntic Brazilian Microwav and Optolctronics Socity-SBMO rcivd 5 Mar 04; for rviw 8 Mar 04; accptd Oct 04 Brazilian Socity of Elctromagntism-SBMag 04 SBMO/SBMag ISSN

17 Journal of Microwavs, Optolctronics and Elctromagntic Applications, Vol., SI-, Novmbr 04 SI-6 proprtis of a structurd mdium by homognizd constitutiv paramtrs is only valid for a unit cll considrably smallr than th wavlngth of th incidnt radiation. Whn th siz of th lattic constant is comparabl to th wavlngth, th lctromagntic proprtis of th mdium xcitd by th impinging wav ar no longr attaind by homognizd constitutiv paramtrs and diffractiv ffcts bcom dominant. In our work, th distanc btwn adjacnt nanorods is in th ordr of 60 nm, which satisfis th condition of a scal siz much smallr than th incidnt wavlngth ovr th analyzd spctral rang. Othr limiting conditions concrn multipolar intractions and an inhomognous polarization. As th radius of th mtallic inclusions is incrasd and th nanorods ar closr to ach othr, quadrupolar filds bcom dominant on th final xprssion of th dpolarization factor. Rgarding only dipolar intractions into th calculation of th ffctiv prmittivity is a valid approach for lowr fill-ratios (typically blow 0%, as is discussd in []). Th nt ffct of quadrupolar xcitations is a fild distribution that rsmbls th on associatd to bulk wavs and, as consqunc, th application of th Bruggman thory can also lad to accurat rsults. Th importanc of quadrupol plasmon rsonancs is analyzd in [9], which dmonstrats that quadrupolar filds can play significant rol in th optical proprtis of mtallic particls whos shap dviats from sphrs or sphroids. Furthrmor, th polarization of th mtallic nanorods is in fact inhomognous, what is mor pronouncabl for largr radii of th inclusions and it is also a sourc of additional corrctions in th MG formulation. VI. CONCLUDING REMARKS Th rsults prsntd in this papr show us that th gomtry of th inclusions and th natur of th local fild mods ar of crucial importanc in rtriving th ffctiv proprtis of optical mtamatrials. Basd on this framwork, w argu that th Maxwll-Garntt modl is a mor appropriat choic than th Bruggman approach, sinc th lattr has its accuracy limitd to matrials structurd by constitunts of slight contrast btwn prmittivitis. This allows rgarding only ffcts rlatd to bulk local mods in th homognization procdur. On th othr hand, th MG formalism is abl to compris surfac lctromagntic mods that can b xcitd in a unitary cll of th mdium. This aspct is of particular intrst in mtamatrials involving mtallic particls of small radius for which th ffctiv optical rspons is dtrmind by th local filds associatd to SPP rsonancs. Th suitability of th MG modl is also strongly dpndnt on th adquat dpiction of siz and shap of th inclusions. To dmonstrat this, w applid th MG modl to rtriv th ffctiv prmittivity of a mtamatrial composd by silvr cylindrical nanorods mbddd in alumina. Whn compard to th calculations providd by th MG modl corrctd for sphrs and prolat sphroids, our formulation showd a rd-shiftd rsonanc with a widr bandwidth, which bcoms mor pronouncabl as th ratio btwn th hight and radius of th nanorod is lowrd. This aspct rvals th influnc of th nanorod gomtry on th nhancmnt of th local SPP mods. Th largr Brazilian Microwav and Optolctronics Socity-SBMO rcivd 5 Mar 04; for rviw 8 Mar 04; accptd Oct 04 Brazilian Socity of Elctromagntism-SBMag 04 SBMO/SBMag ISSN

18 Journal of Microwavs, Optolctronics and Elctromagntic Applications, Vol., SI-, Novmbr 04 SI-7 volum of th cylindrical mtallic particls rducs th fild nhancmnt, shifting th ffctiv rsonanc spctrum toward largr wavlngths and broadning th bandwidth. All ths charactristics indicats that a rliabl accuracy of th MG modl rquirs a propr dfinition of th rlationship btwn th particl gomtry and local fild mods. APPENDIX Analytical thoris dvlopd to stimat th ffctiv paramtrs of inhomognous mdia start with th qustion of avraging local filds. Although it ncompasss th conjctur of long wavlngth limit, th aggrgation of othr assumptions givs a mor profound acquaintanc with thir application constraints. Th mainfram of ths thoris is formd by Clausius-Mossotti, Maxwll- Garntt and Bruggman modls. Considring a mdium structurd by spcimns α and β, thos rspctiv formulations can b agglutinatd in th gnral xprssion: ff h ff h f h h h f h, (A) In (A), ε ff is th ffctiv dilctric constant. Th trm ε h rfrs to th dilctric constant of th host mdium and f is th fill-ratio of ach constitunt. If th host is st up to hav unitary valu, Clausius-Mossotti rlation is obtaind. This approach is limitd by th ambiguity in dfining ζ whn nithr constitunt is th fr spac. If a distinction btwn th composits α and β is mad in ordr to dfin uniquly which ar th inclusions, (A) fits th Maxwll-Garntt modl. In this cas, th host mdium is idntifid as on of th constitunts. It is thn pointd out that ε ff is not prsrvd upon an intrchang btwn α and β trms. Mor spcifically, th MG modl yilds diffrnt valus of th ffctiv prmittivity for ach dfinition of th host (α or β). If thr is no rquirmnt to diffrntiat inclusions and background and th host is itslf th homognous mdium, th Bruggman modl is achivd. By this formalism, w can prmut ε α with ε β and f α with f β and th rsult for ε ff rmains th sam, howvr, th prvious dtrmination of which gomtry ζ dnots is still ncssary for a dfind ffctiv prmittivity. ACKNOWLEDGMENTS This work is supportd by th Brazilian agncis CNPq (National Council for Scintific and Tchnological Dvlopmnt) and CAPES (Coordination of Advancd Studis of Ministry of Education). REFERENCES [] W. Cai and V. M. Shalav, Optical Mtamatrials Fundamntals and Applications. Nw York: Springr, 00. [] A. Fang, T. Koschny and C. M. Soukoulis, Optical anisotropic mtamatrials: ngativ rfraction and focusing, Phys. Rv. B, vol. 79, 457, 009. [] B. D. Cass, W. T. Lu, Y. J. Huang and E. Gultp, Supr-rsolution imaging using a thr-dimnsional mtamatrials nanolns, Appl. Phys. Ltt.. vol. 66, 04, 00. Brazilian Microwav and Optolctronics Socity-SBMO rcivd 5 Mar 04; for rviw 8 Mar 04; accptd Oct 04 Brazilian Socity of Elctromagntism-SBMag 04 SBMO/SBMag ISSN

19 Journal of Microwavs, Optolctronics and Elctromagntic Applications, Vol., SI-, Novmbr 04 SI-8 [4] K.-S. L, J. M. Son, D. Y. Jong, T. S. L and W. M. Kim, Rsolution nhancmnt in surfac plasmon rsonanc snsor basd on wavguid coupld mod by combining a bimtallic approach, Snsors, vol. 0, 00, pp , 00. [5] C. L. Corts, W. Numan, S. Molsky and Z. Jacob, Quantum nanophotonics using hyprbolic mtamatrials, J. Opt., vol. 4, no. 6, 0600, 0. [6] L. M. Custodio, C. T. Sousa, J. Vntura, J. M. Tixira, P. V. S. Marqus and J. P. Araujo, Birfringnc swap at th transition to hyprbolic disprsion in mtamatrials, Phys. Rv. B, vol. 85, 65408, 0. [7] D. R. Smith and J. B. Pndry, Homognization of mtamatrials by fild avraging, J. Opt. Soc. Am. B vol., no., pp. 9-40, 006. [8] C. R. Simoviski, On lctromagntic charactrization and homognization of nanostructurd mtamatrials, J. Opt. vol., 000, 0. [9] C. Fitz and G. Shvts, Currnt-drivn mtamatrial homognization, Phys. B: Cond. Matt., vol. 405, no. 4, pp , 00. [0] W. T. Lu and S. Sridhar, Suprlns imaging thory for anisotropic nanostructurd mtamatrials with broadband allangl ngativ rfraction, Phys. Rv. B, vol. 77, 0, 008. [] B. Vasic, G. Isic, R. Gajic and K. Hingrl, Controlling lctromagntic filds with gradd photonic crystals in mtamatrial rgim, Opt. Exprss, vol. 8, no. 9, pp. 0-0, 00. [] D. E. Aspns, Optical proprtis of thin films, Thin Solid Films, vol. 89, no., pp. 49-6, 98. [] J. Elsr, R. Wangbrg, V. A. Podolskiy and E. E. Narimanov, Nanowir mtamatrials with xtrm optical anisotropy, Appl. Phys. Ltt., vol. 89, 60, 006. [4] J. Kanungo and J. Schilling, Exprimntal dtrmination of th principal dilctric functions in silvr nanowir mtamatrials, Appl. Phys. Ltt., vol. 97, 090, 00. [5] P. Mallt, C. A. Guérin and A. Sntnac, Maxwll-Garntt mixing rul in th prsnc of multipl scattring: drivation and accuracy, Phys. Rv. B, vol.7, 0405, 005. [6] C. F. Bohrn and D. F. Huffman, Absorption and scattring by a sphr, in Absorption and Scattring of Light by Small Particls, Nw York: John Wily, 98, pp [7] A. D. Yaghjian, Elctric dyadic Grn s functions in th sourc rgion, Proc. IEEE, vol. 68, no., 48-6, 980. [8] D. E. Aspns, Local-fild ffcts and ffctiv mdium thory: a microscopic prspctiv, Am. J. Phys. Vol. 50, no. 8, , 98. [9] M. Mir and A. Wokaun, Enhancd filds on larg mtal particls, Opt. Ltt., vol. 8, no., 58-58, 98. [0] C. A. Foss, G. L. Hornyak, J. A. Stockrt and C. R. Martin, Tmplat-Synthsizd nanoscopic gold particls: optical spctra and th ffcts of particl siz and shap, J. Phys. Chm., vol. 98, no., pp , 994. [] M. Y. Koldintsva, R. E. DuBroff and R. W. Schwartz, A Maxwll Garntt modl for dilctric mixturs containing conducting particls at optical frquncis, Prog. In Elctromag. Rs. (PIER), vol. 6, pp. -4, 006. [] Y. Liu, G. Bartal and X. Zhang, All-angl ngativ rfraction and imaging in a bulk mdium mad of mtallic nanowirs in th visibl rgion, Opt. Exprss vol. 6, no. 0, pp , 008. [] A. Moroz, Dpolarization filds of sphroidal particls, J. Opt. Soc. Am. B, vol. 6, no., pp , 009. [4] K. Körmöczi and Z. Szabó, Nar-infrard invisibility cloak nginrd with two-phas mtal-dilctric composits, IEEE Trans. on Mag., vol. 50, no, , 04. [5] P. B. Johnson and R. W. Christy, Optical constants of th nobl mtals, Phys. Rv. B, vol. 6, no. 470, pp , 97. [6] W. T. Lu and S. Sridhar, Suprlns imaging thory for anisotropic nanostructurd mtamatrials with broadband allangl ngativ rfraction, Phys. Rv. B, vol. 77, 0, 008. [7] [8] M. Born and E. Wolf, Principls of Optics, Cambridg Univrsity Prss, 999, pp [9] K. L. Klly, E. Coronado, L. L. Zhao and G. C. Schatz, Th optical proprtis of mtal nanoparticls: th influnc of siz, shap and dilctric nvironmnt, J. Phys. Chm. B, vol. 07, pp , 00. Brazilian Microwav and Optolctronics Socity-SBMO rcivd 5 Mar 04; for rviw 8 Mar 04; accptd Oct 04 Brazilian Socity of Elctromagntism-SBMag 04 SBMO/SBMag ISSN