Description of bow-tie nanoantennas excited by. localized emitters using conformal transformation

Transcription

1 Dscription of bow-ti nanoantnnas xcit by localiz mittrs using conformal transformation Víctor Pachco-Pña,, Migul Brut,, Antonio I. Frnánz-Domínguz, Yu Luo, Migul Navarro-Cía * Ω, Antnnas Group TERALAB, Univrsia Pública Navarra, 006 Pamplona, Spain Optical an Smiconuctor Dvics Group, Imprial Collg Lonon, Lonon SW7 AZ, UK Institut of Smart Citis, Public Univrsity of Navarra, 006 Pamplona, Spain Dpartamnto Física Tórica la Matria Connsaa an Conns Mattr Physics Cntr IFIMAC, Univrsia Autónoma Mari, Mari 8049, Spain Th Photonics Institut an Cntr for OptoElctronics an Biophotonics, School of Elctrical & Elctronic Enginring, Nanyang Tchnological Univrsity, Singapor 69798, Singapor Ω School of Physics an Astronomy, Univrsity of Birmingham, Birmingham B5 TT, UK *m.navarro-cia@bham.ac.u; Phon: ; Fax: KEYWORDS conformal transformation, bow-ti, nanoantnna, plasmonic, transformation optics

2 ABSTRACT Th unprcnt avanc xprinc by nanofabrication tchniqus an plasmonics rsarch ovr th past fw yars has ma possibl th ralization of nanophotonic systms ntring into th so-call strong coupling rgim btwn localiz surfac plasmon LSP mos an quantum mittrs. Unfortunatly, from a thortical point of viw, th fil is hinr by th lac of analytical scriptions of th lctromagntic intraction btwn strongly hybriiz LSP mos an nanomittrs vn within th Marovian approximation. This gap is tacl hr by xploiting a conformal transformation whr a bow-ti nanoantnna xcit by a ipol is mapp into a prioic slab-ipol framwor whos analytical solution is availabl. Solving th problm in th transform spac not only provis a straightforwar analytical xplanation to th original problm valiat using full-wav simulations but also grants a p physical insight an simpl sign guilins to maximiz th coupling btwn localiz ipols an th bow-ti LSP mos. Th rsults prsnt hr thrfor pav th way for a full analytical scription of ralistic scnarios whr quantum ots or y molculs moll byon a two-lvl systm ar plac nar a mtallic bow-ti nanoantnna. Antnnas ar wll-nown nabling vics for fficint transuction btwn lctronic signals gui wavs an raio or microwav raiation non-gui wavs., Sinc thir incption at th n of th 9 th cntury, thy hav bn intimatly boun to wirlss communication systms. Howvr, this viw has tan a iffrnt prspctiv in rcnt yars within th fil of nanophotonics. Bnfit from th rcnt avancs in nanofabrication an optical charactrization tchniqus, as wll as th accuracy an prictiv valu that classical lctromagntics has monstrat own to th nanoscal, th antnna concpt has bn rvisit at optics. 4 7 Th so-call nano-antnnas ar vics that oprat in th visibl rang in a similar way as convntional low frquncy antnnas. Braing th iffraction limit of classical

3 optics, ths nanomtric vics nabl nar-to-far-fil coupling an vic-vrsa of optical signals with unprcnt fficincy. This nanoscal control ovr th propagation an confinmnt of visibl light has alray foun applications in aras compltly iffrnt from th traitional wirlss communications such as spctroscopy, 8 biosnsing, 9 photovoltaics, 0 optolctronics, phototction an nonlinar optics. Ali raio an microwav antnnas, th lctromagntic rspons of nano-antnnas is govrn by thir gomtris an by th matrial proprtis of thir componnts. Howvr, mtals hav a mor complx scription at visibl frquncis, maing th molling an optimization of ths nano-vics mor challnging from a thortical prspctiv. Hnc, th analytical scription of nano-antnna prformanc xists only for a fw simpl gomtris, such as sphrs, cylinrs or cubois. 4,5 Vry rcntly, a quasi-analytical tratmnt of mor complx nanostructurs has bn vlop using transformation optics, 6 0 a framwor similar to conformal mapping 4 but oprating xactly at th lvl of Maxwll Equations. 5 Bow-ti nano-antnnas ar compos by two triangular-shap mtal nanoparticls facing against ach othr, connct at thir apxs or sparat by a nanomtric gap. This is on of th most thoroughly invstigat structurs in th litratur. Exprimntal an numrical rports hav shown th suitability of this antnna an its variations for th implmntation of optical rcivrs an transmittrs. 6 4 Compar to th othr gomtris xamin unr transformation optics such as crscnts an cylinrical imrs, 0 bow-ti nano-antnnas promis strongr gr of fil localization an nhancmnt. This bnfits an it is in ssntial for a myria of plasmonic applications; for instanc, th strongr th local fil, th brightr th fluorscnc/harmonic-signal is or th largr th Rabi splitting of molcular rsonanc pas is in hybri mtal-molcul/nonlinar-matrial scnarios. In this wor, w xtn th st of nano-

4 antnna configurations with analytical tratmnt incluing a two-imnsional bow-ti gomtry prsnting translational symmtry along on irction, as shown in Figur. W xploit transformation optics concpts to xplain th pnnc of th non-raiativ cay spctra i., th powr absorb, P abs, by th bow-ti nanoantnna unr ipol illumination 0,5 on th bowti gomtrical paramtrs an to giv physical insights on th coupling btwn oscillating classical lin ipols an th localiz surfac plasmon LSP mos support by th bow-ti gomtry. RESULTS AND DISCUSSION Figur a shows th gnral problm unr consiration: th coupling btwn a lin ipol nanomittr with arbitrary orintation an a bow-ti nanoantnna ma of silvr Ag. Notic that th tip of th bow-ti nanoantnna stui hr is concav to facilitat th conformal mapping. Th ipol is locat on th x -axis nm away from th cntr of th bow-ti. This is in a mor ralistic situation than placing th ipol insi th gap, sinc nanomtr-siz gaps ar in gnral inaccssibl for nano- an micromtr-siz mittrs. Th bow-ti is fin by th arm lngth, L +L, th arm angl,, an th gap btwn arms. Th arm lngth along with th gap givs th total lngth of th bow-ti l. W rstrict th stuy to bow-ti gomtris much smallr than th illumination wavlngth to b within th ralm of nar-fil quasi-static approximation. In this scnario, magntic an lctric fils ar coupl, an th lattr can b fully scrib by an lctrostatic potntial satisfying Laplac s quation. For simplicity, th bow-ti gomtris ar mb in vacuum, an th ilctric function of Ag is tan from Pali s xprimntal ata s th Mthos sction for mor tails of th numrical stuy. 6 4

5 Figur. a Schmatic rprsntation of a mtallic bow-ti nanoantnna with a gap on its cntr illuminat with a ipol plac at x, y = nm, 0 grn arrow. b Transform gomtry aftr th conformal mapping is appli to th bow-ti nanoantnna Th systm can b qualitativly xplain with a simpl huristic analysis. Th raiation from th localiz oscillating ipol an atom or a quantum ot in an xcit stat, for instanc is coupl to th iffrnt LSP mos support by th bow-ti nanoantnnas. This pump lctromagntic nrgy is vntually issipat u to mtal absorption, i.., non-raiativ amping. Givn th sub-wavlngth siz of th bow-ti, raiation loss, i.., raiativ amping, is ngligibl. Th strngth of th coupling, an thus, th non-raiativ amping, pns on th position of th ipol within th fil istribution of th LSP mos. In gnral, th problm of fining th optimum st of paramtrs for a spcific xprimnt is arss by prforming brut-forc computations. An altrnativ to ruc th computational rquirmnts is vising analytical solutions. In th nxt sction w riv a conformal mapping solution for th bow-ti nanoantnna xcit by a ipol. W transform th problm into a gomtry that can b asily solv analytically, simplifying th calculation an analysis of th original problm. Thortical analysis: Conformal mapping Th bow-ti can b transform into th multi-slab gomtry shown in Figur b by applying th following conformal transformation: 5

6 z lnz' whr z = x+iy an z = x +iy ar th spatial coorinats in th transform an original fram, rspctivly. Through this conformal transformation, circular raial lins in th original gomtry ar mapp into vrtical horizontal lins in th transform fram. 6,0 This transformation rsults in a multi-slab gomtry with th imnsions of all mtal slabs as L +L an = along th x-an y- axis, rspctivly. Th original ipol is manwhil convrt into an array of ipols with th sam strngth plac along th y-axis with a prioicity π, i.., at x = 0, y = m, whr m is an intgr. It is worth pointing out that a scnario involving a nanoantnna with thr arms woul b convrt into a multi-slab gomtry with an aitional slab pr prio s Supporting Information. By looing at th multi-slab gomtry, a qualitativ an quantitativ tail nxt unrstaning of th LSP mos support by th bow-ti nanoantnna can b achiv. As it is shown in Figur b th ipol array mission triggrs surfac plasmons propagating along both positiv an ngativ irctions of x in th multi-slab gomtry; which ar mapp into th plasmonic mos xcit by th singl mittr along both arms of th bow-ti nanoantnna. Bcaus of th finit lngth of th slab/bow-ti-arms, ths surfac plasmons ar rflct bac an forth btwn th two ns of th structur, forming a staning wav pattrn that givs ris to th LSP mos. Hnc, th continuous surfac plasmon polariton spctrum of an infinit slab or bow-ti is convrt into a finit st of iscrt LSPs, charactriz by th mo orr n 7,8 s th Mthos sction an Supporting Information. Th D conformal transformation nsurs that th matrial proprtis rmain unchang, unli th D countrpart. 7 0 In aition, it prsrvs th potntial in ach coorinat systm: 7 x, y ' x', y' 6

7 whr an ar th lctrostatic potntials in th transform an original frams, rspctivly. Thrfor, th x an y componnts of th lctric fil istribution E x an E y, rspctivly in th original gomtry can b irctly uc from q as: 8,9 E' x ' ' z ' z' * z' x' z' * x' ' ' z' z' * E' y ' ' z ' z' * z' y' z' * y' ' ' i i z' z' * 4 Hnc, by solving th problm in th multi-slab fram, th bow-ti scnario is solv straightforwarly. Notic that, in th multi-slab gomtry, th fil istribution along th y irction E y actually rprsnts th azimuthal componnt of th lctric fil E in th bowti scnario, which can b calculat from th x an y componnts qs -4 as E = E x sin +E y cos, with = tan - y /x. On th othr han, th fil istribution along th x irction E x is irctly transform into th raial componnt of th lctric fil in th original gomtry E, which can b obtain as E = E x cos +E y sin. From hr on, th azimuthal an raial componnts will b us to rprsnt th lctric fil istribution in th bow-ti nanoantnnas hr stui. Th quantitativ tails of th analytical formulation to calculat th plasmonic rspons of th bow-ti nanoantnna is riv in th Mthos sction, whr th problm is solv for th multi-slab gomtry. Non-raiativ cay in th gap bow-ti nanoantnna Sinc th nrgy is consrv in th transformation, th powr issipation is th sam in both frams. Hnc, th non-raiativ cay of th nanomittr can b uc by calculating th powr issipat in th multi-slab gomtry. This can b obtain by valuating th lctric fil at th ipol position in th original fram, as follows: s s Pnr Pabs Im pxex x, y 0 pye y x, y 0 5 7

8 whr P nr is th non-raiativ powr mission, =πc/ 0 is th angular frquncy at th woring wavlngth 0 an c is th vlocity of light in vacuum, p x an p y ar th componnts of th ipol momnt along th x an y irctions, an s E an E ar th componnts of th lctric fil s x y along x an y irctions in th rgion whr th ipol is plac < y <. Importantly, in our calculations, an intrinsic quantum yil qual to is assum for th nanomittr, which allows intifying th non-raiativ cay xprinc by th mittr an th powr absorb by th bow-ti nanoantnna. 40 Morovr, not that as q 5 is riv in th quasi-static approximation, th xprssion for th xtinct powr by a point ipol can b us to scrib th nanomittr non-raiativ cay. Plasmonic rspons of gap bow-ti nanoantnnas Changing th bow-ti arm angl Lt Γ 0 ω an Γ nr ω b th isolat ipol raiativ rat an th non-raiativ cay rat for th full systm. Figur rnrs th volution of th non-raiativ Purcll nhancmnt rat spctra Γ nr ω/γ 0 ω calculat as th ratio of th powr absorb by th nanoparticl P nr an th total powr raiat by th isolat localiz mittr P 0, i.., Γ nr ω/γ 0 ω = P nr /P 0,0,5 as a function of for a bow-ti nanoantnna with total lngth l =0 nm an a normaliz gap btwn both arms of 0.05l. Th analytical rsults ar valuat using q 5 along with th powr raiat by th ipol P 0 =/6 µ 0 p with µ 0 th prmability of vacuum an p th magnitu of th ipol momnt, rspctivly. Th analytical rsults ar compar with numrical calculations on with th commrcial softwar Comsol Multiphysics s th Mthos sction. Th analytical rsults for th vrtically orint ipol cas Figur a show that th maximum of Γ nr ω/γ 0 ω shifts from 698 nm to 94 nm whn th angl varis from 5º to 45º. This pa originats from th first n = LSP mo support by th bow-ti 8

9 nanoantnna, as w show blow through th fil istribution inspction. Similarly, for a horizontal ipol, th first non-raiativ pa u to th first LSP mo is blu-shift from 650 nm to 7 nm, s Figur. Although similar trns ar obsrv in th full-wav simulations, thr is an vint blu-shift btwn simulation rsults Figur c,g an analytical calculations for both ipol orintations Figur a,. Th blu-shift ariss from th assumption that th LSP mos acquir a phas shift of π upon rflction at ach n of th mtal slabs, i.., at th opn bounaris at L an -L. To account for a iffrnt rflction phas shift, a corrction is introuc in th form of an xtra phas. Th calculation of is on by fitting th analytically-comput wavlngth of th funamntal mo n = LSP mo to th simulations. Sinc highr orr LSP mos may xprinc iffrnt rflction conitions than th funamntal on, this corrction may not apply for highr orr mos. Th valus of for a vrtical an horizontal ipol as a function of th angl ar shown in Figur,h, rspctivly. A linar slop is obtain for angls from 5º to 5º whil this tnncy varis for largr angls. Th corrct Γ nr ω/γ 0 ω is shown in Figur b,f for a vrtical an horizontal ipol, rspctivly. Now, a goo agrmnt btwn analytical an numrical rsults is obtain. As xplain bfor, u to th finit siz of th bow-ti nanoantnnas an th quivalnt transform problm th LSPs ar istribut as a st of iscrt mos in th spctra. Th rsonant conition of ths iscrt LSP mos an thir spctral istribution ar provi in th Supporting Information for svral bow-ti angls xcit by both vrtical an horizontal ipols. From now on, = Γ nr nr ω/γ 0 ω will rfr to th corrct rsults. 9

10 Figur. Non-raiativ Purcll nhancmnt spctra as a function of th bow-ti angl for a ipol with vrtical a- an horizontal orintation -h: analytical rsults without a, an with corrction b,f to fit simulation rsults c,g. Phas corrction appli in th analytical calculations for a ipol with vrtical an horizontal h polarization. Nxt, w analyz in tail th analytical an simulation rsults of th non-raiativ Purcll nhancmnt spctra for two bow-ti nanoantnnas with = 5º an 0º xcit by a vrtical Figur a an horizontal ipol Figur b. Ltting Γ r ω b th raiativ cay rat for th full systm, th simulation rsults of th raiativ Purcll nhancmnt = Γ r r ω/γ 0 ω calculat as th ratio of th powr raiat by th systm nclos by th nanoparticl-ipol P r an P 0 ; Γ r ω/γ 0 ω = P r /P 0,0,5 ar also shown in th sam figur for compltnss. Notic that it is consistntly at last two orrs of magnitu smallr than nr, an thus ngligibl, as w assum initially. A goo quantitativ agrmnt btwn analytical an numrical rsults is shown in Figur a,b for th first non-raiativ pa whil th othr pas ar slightly blushift, as xpct from th abov iscussion on. An avrag blu-shift of 0.9% an % is 0

11 obsrv btwn th simulation an analytical rsults for th pa associat to th n = LSP mo for th bow-ti nanoantnna with = 5º for a vrtical an horizontal ipol, rspctivly. Th simulation rsults of an nr along with th absorption cross sctions of th bow-ti r nanoantnnas unr plan-wav illumination ar shown in th Supporting Information. Figur. Analytical ots an simulation soli lins rsults of th non-raiativ Purcll nhancmnt spctra along with th simulation rsults of th raiativ Purcll nhancmnt spctra ash lins for two bow-tis with angl = 5 light lins an = 0 ar lins whn a vrtical a an horizontal b ipol is us as a raiativ sourc. Th lttrs nxt to th pas rfr to th iffrnt panls in Figur 4. Figur 4 shows th spatial absorption profils across th bow-ti nanoantnna with = 5 an iffrnt ipol orintation calculat at th wavlngths highlight in Figur. Th sam rsults for = 0 can b foun in th Supporting Information. A goo agrmnt btwn analytical an simulation rsults is notic. As xpct, whn svral absorption maxima xist, th absolut on is obtain closr to th apxs for all cass. This is a consqunc of th largr

12 fil concntration clos to th gap which happns u to th spatial comprssion of th surfac plasmon mos. 0 Th spatial absorption istribution for th funamntal mo unr a vrtical ipol illumination Figur 4a,b has an absorption minimum point out by whit horizontal arrows at y =. nm y =.6 nm in th analytical numrical calculation. This absorption minimum rprsnts th no of th n = LSP mo. For th pas associat to th n = LSP mo, Figur 4c,, on can howvr notic a local maximum at y =.06 nm y =.8 nm in th analytical numrical rsults locat at ach arm of th bow-ti nanoantnna, point out by horizontal grn arrows. This occurs bcaus this position corrspons to th anti-no of th n = LSP mo. Unr horizontal ipol illumination, th positions of th maxima an minima chang accoring to th anti-nos of th corrsponing LSP mos, as it is monstrat nxt through lctric fil istribution pattrns. Thrfor, th bow-ti nanoantnnas invstigat hr hav a multi-ban absorption rspons that ariss from th fficint coupling of th localiz mittr to th multipl LSP mos support within th rang from 00 nm to 900 nm. A snapshot of th fil istribution for a bow-ti nanoantnna with = 5º is shown in Figur 5 for th first an scon pa of of ach ipol orintation. For convninc, hr w plot nr E an E for th vrtical an horizontal ipol xcitation, rspctivly. From ths color plots w can clarly intify th mo orr of th various LSPs. Unr a vrtical ipol illumination, th azimuthal fil istribution at th first pa has a null btwn th fil maxima at th nr gs of ach bow-ti arm Figur 5a. For th scon pa Figur 5b, w hav thr antinos an two nulls along th raial irction in ach arm, which corrspons to th n = LSP mo. On th othr han, for th cas of a horizontally orint mittr, two minima appar at both ns of ach arm with an anti-no btwn thm at th lowst pa Figur 5c, which nr corrspons to th n = LSP mo. At th scon pa Figur 5 th fil istribution can b

13 lin to th n = LSP mo as it has thr nulls on at th cntr an two at th xtrms of ach arms an maxima in btwn conscutiv nulls. Notic that th lctric fil is strongr at th anti-nos narby th apx of th bow-tis, as xpct from th spatial absorption profils. Altrnativly, th fil istribution can b asily associat to staning wav pattrns in th transform fram, as it is laborat in th Supporting Information. Figur 4. Analytical a,c,,g, an simulation-comput b,,f,h absorption for th bow-ti with angl = 5 whn th illuminating ipol is vrtical top an horizontal bottom: funamntal a,b,,f an scon non-raiativ cay rat pa c,,g,h within th spctral winow of intrst. Th scal color bar is saturat to facilitat th intlligibility. Horizontal grn an whit arrows inicat rspctivly th location of th maxima an minima on th top arm of ach bowti.

14 Figur 5. Snapshot of E -fil top row an E -fil bottom row for a bow-ti nanoantnna with angl = 5 an illuminat with a vrtical a,b an a horizontal c, ipol at th pas in Figur : a 869 nm, b 556 nm, c 698 nm an 545 nm. Not that th scal bar has bn saturat from -0.0 to 0.0 an from to 0.05 to bttr apprciat th fil istribution across th whol spac. Changing th gap of th bow-ti nanoantnna All th rsults iscuss in th prvious sctions hav bn obtain consiring bow-ti nanoantnnas with varying. W iscuss nxt th influnc of th gap siz in th non-raiativ 4

15 spctra of two bow-ti nanoantnnas with = 5º, for a fix antnna lngth l = 0 nm. Th Supporting Information contains th rsults for = 0º. Th analytical rsults for as a nr function of th gap istanc btwn both arms ar shown in th first row of Figur 6 whn a vrtical Figur 6a an a horizontal Figur 6b ipol is plac at x = nm, y = 0 nm. It can b obsrv that all pas rlat to a spcific LSP mo for both polarizations an angls ar blu-shift whn th gap is incras. This blu-shift can b asily xplain in trms of th transform multi-slab gomtry: an incrmnt of th gap btwn both arms of th bow-ti nanoantnna is quivalnt to rucing th total lngth of th slabs in th transform fram i.., L = L + L is ruc. Hnc, th rsonant conition of th staning wav pattrn happns for shortr wavlngths. To facilitat th scription an comparison, th corrsponing numrical spctra ar shown in th scon row of Figur 6. Ths panls monstrat a vry goo agrmnt with th analytical rsults. For = 5º an a vrtical ipol Figur 6a,c th pa rlat to th n = LSP mo nr is blu-shift from nm not shown to 845 nm whn th normaliz gap gos from 0.0l to 0.06l. Intrstingly, this pa is not always th absolut maximum, in contrast to nr what happns for th absorption cross sction for a bow-ti unr plan-wav illumination shown in th bottom row of Figur 6. For instanc, is largr for th n = LSP mo for a nr normaliz gap of 0.0l. This shows that thr ar prfrr positions to incras th transfr of nrgy from th raiativ ipol sourc to th iffrnt LSP mos. In particular, for th cas 0.0l, th vrtical ipol is locat at a fil istribution null of th n = LSP ign-mo not shown hr. Hnc th pa associat to this mo vanishs. For th cas of a horizontal ipol Figur 6b, th pa u to th n = LSP mo is blu-shift from 769 nm to 697 nm whn th normaliz gap is incras from 0.0l to 0.06l. Th two othr non-raiativ pas 5

16 rlat to th n = an n = LSP mos, rspctivly ar also blu-shift as th gap is incras. Hr, th analytical pas u to scon an thir highr orr mo ar also nr blu-shift from simulation rsults by an avrag prcntag of.% an.7%, rspctivly, for a vrtical ipol, an.67% an.% for an horizontal ipol. Th blu-shift is smallr for a horizontal ipol bcaus of th comparativly shortr phas corrction appli to this configuration. As it has bn scrib bfor, pning of th angl, gap, an orintation of th ipol, th localiz mittr cannot transfr nrgy fficintly to th LSP mos isplay as minima in th non-raiativ Purcll nhancmnt. This phnomnon can b asily xplain by looing at th multi-slab gomtry. Lt us thn analyz th cas of th bow-ti with = 5º illuminat with a horizontal ipol Figur 6b,, analytical an simulation rsults, rspctivly. It can b obsrv that thr is a rang of gaps btwn 0.08l to 0.08l whr th pa lin to th n = LSP mo vanishs. To invstigat in tail this fatur, th for this bow-ti nr nanoantnna using a horizontal ipol is shown in Figur 7a, for a normaliz gap of 0.057l an 0.05l, rspctivly ths panls hav bn xtract from th blac ash lins of Figur 6b. For th cas of a gap of 0.057l, Figur 7a shows that thr pas ar prsnt at 69 nm, 5 nm an 454 nm which ar thos rlat to th LSP mos with n =, an, rspctivly. On th othr han, whn th gap is 0.05l Figur 7 all pas ar r-shift, as xpct, to 7 nm, 556 nm an 4 nm. Nvrthlss, th LSP mo with n = at 4 nm almost isappars. 6

17 Figur 6. Analytical first row an simulat scon row non-raiativ Purcll nhancmnt spctra along with th simulation rsults of th raiativ Purcll nhancmnt spctra thir row an absorption cross sction fourth row as a function of th gap btwn th arms for th bowti with angl = 5 whn a vrtically first column an horizontally scon column polariz ipol top thr rows or plan-wav is us as a sourc bottom row. 7

18 Figur 7. Analytical non-raiativ Purcll nhancmnt spctra for a bow-ti with = 5 an gap = 0.057l a an 0.05l. Analytical normaliz magnitu of th lctric fil for th paralll-plat gomtry at th rlvant spctral position shown in a an. Th position of th illuminating horizontal ipol is shown as a grn arrow for ai. Not that th scal in an h has bn saturat from to bttr obsrv th fil istribution This phnomnon can b xplain by analyzing th fils in th transform gomtry, as follows: first, th analytical rsults of th normaliz magnitu of th lctric fil istribution in th multi-slab gomtry for th cas of a gap of 0.057l at th first, scon an thir pas is shown in Figur 7b-, rspctivly. Th fil istribution at ths pas corrspons to th fil istribution of th LSP mos with n =, an, rspctivly, as it has bn xplain bfor. For th cas of th first LSP mo n = th horizontal ipol schmatically shown as an 8

19 horizontal grn arrow is plac clos but not just at th no at L i.., th no in th staning wav pattrn of th fil istribution; thrfor, th ipol can coupl to this LSP mo. Howvr, for n = Figur 7c th ipol is closr to th first no, whr poorr transfr of nrgy btwn th ipol to th LSP is xpct. Hnc, a ruction of tas plac for nr this mo compar to th first on. Similar prformanc can b obsrv for n =. In this cas, th ipol is vn closr to th no compar with th first an scon mos; thrfor, th amplitu of th pa is ruc although it still appars in th spctrum. Now, lt us analyz th cas whn th gap is 0.05l. For this gomtry, th normaliz magnitu of th lctric fil at 7, 556 an 4 nm is shown in Figur 7f-h, rspctivly. As it can b obsrv in Figur 7h, th fil istribution corrspons to th LSP mo with n =, as xplain bfor. Morovr, it is shown that th horizontal ipol is xactly at th position whr th istribution of th lctric fil has a no. Thrfor, th lctromagntic nrgy rlas by th ipol os not coupl fficintly to this LSP mo giving ris to a null in. On th contrary, for th cas of th first nr an scon pas s Figur 7f,g, rspctivly th horizontal ipol is locat at a mor favorabl position for nrgy transfr to th LSP mos than for th n = LSP mo an th n = an LSP mos for 0.057l gap; Hnc, th non-raiativ cay rat is highr for thm. CONCLUSIONS In conclusion, an analytical solution for bow-ti nanoantnnas bas on conformal transformation in th quasi-static approximation has bn rigorously riv. For situations byon th quasi-static limit, on coul xplor th implmntation of a raiativ corrction bas on a fictitious absorbing ipol in th transform spac. 0,8 Th conformal transformation prmits to convrt th original problm of a bow-ti nanoantnna xcit by a local ipol into a multi-slab gomtry with an array of ipols whos solution can b foun 9

20 analytically, an is also solution of th original bow-ti nanoantnna scnario. Our conformal mapping approach also nabls us to scrib in tail all th spctral faturs in th nonraiativ Purcll nhancmnt of a nanomittr plac in th vicinity of iffrnt bow-ti nanoantnnas. Ths rsults shoul as th sign of bow-ti nanoantnnas for multipl applications. In particular, it may hol promis to mol analytically th ynamics of ralistic strong coupling scnarios whr localiz surfac plasmon mos intract with stats lin to fw-lvl mittrs such as quantum ots or y molculs. METHODS Multi-slab gomtry mimicing th gap bow-ti nanoantnna Hr, th multi-slab gomtry shown in Figur b is solv. Taing into account that th imnsions of th bow-ti nanoantnna ar sufficintly smallr than th oprational wavlngth l <<, th nar-fil approximation can b us an thus, th lctric fil can b fully scrib by an lctrostatic potntial satisfying Laplac s quation. As it is nown, in th multislab gomtry shown in Figur b, it is possibl to xcit surfac plasmon mos in both transvrsal an longituinal irctions, with thir propagation along th x- an y- axis, rspctivly. Howvr, th intrst hr is focus in th rivation of th surfac plasmon mos xcit in th multi-slab gomtry whn L + L >> ; thrby, th contribution of th longituinal LSP mos i.., thos with phas variation along y can b nglct an assum that th xcit LSP mos ar mainly u to th transvrsal mos i.., thos with phas variation along x. Bas on this, th lctrostatic potntials outsi an insi th mtal strips in Figur b can b calculat as a sum of all iscrt transvrs mos, as follows: i L L ix ixil i A y B y B y,0 y i 6 0

21 0, y B B A y y y i il ix ix i L L i 7, y E E y y i il ix ix i L L i 8, y C C y y i il ix ix i L L i 9, y D D y y i il ix ix i L L i 0 Whr is th wav vctor of th transvrs LSP mos calculat as =nπ-/l +L with n =,,, rprsnting th iscrt transvrs SP mo, is th corrction of phas appli to th bow-ti nanoantnna to ta into account th complx rflction xprinc by th surfac plasmon wavs at th xtrms of th nanoparticl, A + an A - ar th xpansion cofficints of th incint potntial, B + an B - ar th cofficints rlat to th scattring potntial in th rgion whr th ipol is plac < y <, E + an E - ar th cofficints of associat to th scattring potntial in th rgion whr a ipol is absnt + an C +, C -, D + an D - ar thos corrsponing to th potntial insi th mtal strips. Th cofficints associat with th incint potntial can b obtain by xpaning th ipol potntial along th x irction using a Fourir transform: 0 sgn ip p A x y whr p y an p x ar th componnts of th ipol momnt along th x an y irctions, rspctivly, an ε 0 is th prmittivity in vacuum.

22 Th othr ight unnown cofficints B +, B -, C +, C -, D +, D -, E + an E - can b solv by using th bounary conitions at ach intrfac of Figur b. First, th conition of consrvation of th paralll componnt of th lctric fil at th bounaris, +, an + + is appli, as follows: D D B B A D D E E C C B B A 4 C C E E 5 Also, th conition of consrvation of th normal componnt of th isplacmnt fil at th sam bounaris as qs -5 is appli, as follow: D D B B A 6 D D E E 7 C C B B A 8 C C E E 9 whr ε is th prmittivity of th mtal us in th structur Silvr in this cas. Th solutions of th potntials in th ral spac for th rgion whr thr is < y < an thr is not a ipol +, s an s, rspctivly, can b thn obtain by applying an invrs Fourir transform to th inuc potntials:

23 y y p xsin x sinx L pycos x cosx L B B 0 s L L 0 n y y p xsin x sinx L py cos x cosx L E E s L L 0 n Similarly, th potntials insi both mtallic slabs m an m ar: y y p xsin xsinx L pycos x cosx L C C m 0 L L n y y p xsin xsinx L pycos x cosx L D D m 0 L L n whr is fin as: cos L L sin L L 4 Finally, th x an y componnts of th lctric fil can b calculat by simply iffrntiating th potntials: y y p cos x cosx L p sin x sinx L B B s E x x y L L n 0 5 y y p cos x cosx L p sin x sinx L E E s Ex x y L L n 0 6 y y p cos x cosx L p sin x sinx L C C m E x x y L L n 0 7 y y p cos x cosx L p sin x sinx L D D m Ex x y L L n 0 8 y y p sin xsinx L p cos x cosx L B B s E y x y L L n 0 9

24 y y p sin xsinx L p cos x cosx L E E s E y x y L L n 0 0 y y p sin xsinx L p cos x cosx L C C m E y x y L L n 0 y y p sin xsinx L p cos x cosx L D D m E y x y L L n 0 Th complt solution for ach constant is not shown hr u to thir complxity; thrfor th cofficints ar us in orr to ruc th quations of potntials an lctric fil. Howvr, ths solutions can b irctly obtain ithr manually or using a mathmatic softwar. A similar mathmatical rivation can b appli for a bow-ti nanoantnna compos of thr arms. Th corrsponing rsults can b foun in Supporting Information. Numrical simulations Th numrical rsults ar calculat using th commrcial Finit Elmnt Analysis softwar Comsol Multiphysics. Th mol of mtal us in this wor is Silvr Ag mol as a Dru- Lorntz function with th form ε r = ε - p /-i+ε l l / l - +i l ; with ε =.74, Dru plasma frquncy p = ra/s, Lorntz plasma frquncy l = ra/s, ε l =.69, Dru amping constant = ra/s an Lorntz amping constant l = 890 ra/s. This function fits Pali s xprimntal ata. 6 Th bow ti antnnas, with a total lngth of l = 0 nm ar immrs in a vacuum mol as a two-imnsional squar of 600 nm 600 nm. In orr to ruc unsirabl rflctions from th systm, scattring bounary conitions i.., prfctly match layrs hav bn appli to th bounaris of th squar of vacuum. Th D TM point ipol us to illuminat th nanoantnna is mol using two anti- 4

25 paralll in-plan magntic currnts with a sparation of 5 pm. Also, an xtrmly rfin msh has bn us with a maximum an minimum msh siz of nm an pm, rspctivly for th box of vacuum. For th bow-ti nanoantnnas, a rfin msh twic smallr than th msh us for th box of vacuum is appli to nsur accurat rsults. For th systmatic stuy shown in Figur c,g, th non-raiativ powr was valuat by simulating in a frquncy rang from 00 THz to 000 THz with a stp of 0 THz for th following rang of angls of aprtur of th antnnas: from 5º to 5º with a stp of 0.5º an from 5º to 5º with a stp of 0.5º an from 5º to 45º with a stp of º. This paramtrical stuy was appli for both, vrtically an horizontally polariz ipol. With ths paramtrs, th stimat tim to solv ach simulation i.. for on angl of aprtur of th antnnas with on polarization was in man 90 minuts ach. ASSOCIATED CONTENT Supporting Information. Rsonant conition an iscrt istribution of th LSP mos as a function of th angl for both polarizations of th localiz mittr is shown in Figur Sa. Th cut-off wavlngth of th first 5 LSP mos for th bow-ti nanoantnnas with an angl of 5º, 5º an 45º whn a vrtical an a horizontal ipol illuminats th bow-ti nanoantnnas is shown in Figur Sb,c, rspctivly. Numrical rsults of th non-raiativ an raiativ nhancmnts along with th absorption cross sction for two bow-ti nanoantnnas with = 5º an = 0º for both polarizations ar shown in Figur S. Figur S illustrats th spatial absorption istribution for th bow-ti with = 0º. Dscription of th staning wav pattrn in th transform fram to support th rasoning rvolving aroun Figur 5. Th nonraiativ an raiativ Purcll nhancmnt spctra along with th absorption cross sction spctra for th bow-ti with = 0º an iffrnt polarization illumination ar pict in 5

26 Figur S4. Th scnario comprising a lin ipol nanomittr with arbitrary orintation an a tripo nanoantnna ma of silvr along with th associat rsults for = 0º ar rport in Figur S5. This matrial is availabl fr of charg via th Intrnt at AUTHOR INFORMATION Corrsponing Author * School of Physics an Astronomy, Univrsity of Birmingham, Birmingham B5 TT, UK. m.navarro-cia@bham.ac.u; Phon: ; Fax: ACKNOWLEDGMENT Th authors woul li to than Prof. Sir J. Pnry for fruitful iscussions. This wor was support in part by th Spanish Govrnmnt unr Contract TEC C--R. V.P.-P. is sponsor by Spanish Ministrio Eucación, Cultura y Dport unr grant FPU AP M.B. is sponsor by th Spanish Govrnmnt via RYC Y. L. woul li to acnowlg th funing support from NTU-A*STAR Silicon Tchnologis Cntr of Excllnc unr th program grant No M.N.-C. was support by an Imprial Collg Junior Rsarch Fllowship an is now support by a Birmingham Fllowship. ABBREVIATIONS LSP, Localiz Surfac Plasmon REFERENCES Balanis, C. A. Antnna Thory: Analysis an Dsign; Thir Eit.; John Wily & Sons: Hobon, Nw Jrsy, 005. Pozar, D. M. Microwav Enginring; John Wily & Sons: Nw Yor, 998. Novotny, L.; Hcht, B. Principls of Nano-Optics; Scon Ei.; Cambrig Univrsity Prss: U.K., 0. 6

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