Analysis of scattering by plasmonic gratings of circular nanorods using lattice sums technique

senso s A icle Analysis o Sca e ing by Plasmonic G a ings o Ci cula Nano ods Using La ice Sums Technique Vakh ang Jandie i 1,*, Kiyo oshi Yasumo o 2, Ja omi Pis o a 3and Daniel E ni 1 1 Gene al and Theo e ical Elec ical Enginee ing (ATE), Facul y o Enginee ing, Uni e si y o Duisbu g-Essen, and CENIDE—Cen e o Nanoin eg a ion Duisbu g-Essen, 47048 Duisbu g, Ge many; [email p o ec ed] 2Facul y o In o ma ion Science and Elec ical Enginee ing, Kyushu Uni e si y, Fukuoka 819-0395, Japan; [email p o ec ed] 3Nano echnology Cen e, VSB-Technical Uni e si y o Os a a, 17. lis opadu 15, 708 33 Os a a—Po uba, Czech Republic; ja omi [email p o ec ed] *Co espondence: [email p o ec ed] Recei ed: 7 Augus 2019; Accep ed: 8 Sep embe 2019; Published: 11 Sep embe 2019   Abs ac : A sel -con ained o mula ion o analyzing elec omagne ic sca e ing by a signi ican class o plana g a ings composed o plasmonic nano ods, which we e in ini e leng h along hei axes, is p esen ed. The p ocedu e o he la ice sums echnique was implemen ed in a cylind ical ha monic expansion me hod based on he gene alized e lec ion ma ix app oach o ull-wa e sca e ing analysis o plasmonic g a ings. The me hod p o ided a high compu a ional e iciency and can be conside ed as one o he bes -sui ed nume ical ools o he op imiza ion o plasmonic senso s and plasmonic guiding de ices bo h ha ing a plana geome y. Al hough he p oposed o malism can be applied o analyze a wide class o plasmonic g a ings, h ee con igu a ions we e s udied in he manusc ip . Fi s ly, a mul ilaye ed g a ing o sil e nanocylinde s o med analogously o pho onic c ys als was conside ed. In he egion a om he esonances o a single plasmonic nanocylinde , he s uc u e showed simila p ope ies compa ed o con en ional pho onic c ys als. When one o a ew nano ods we e pe iodically emo ed om he o iginal c ys al, hus o ming a c ys al wi h de ec s, a new band was o med in he spec al esponses because o he esonan unneling h ough he de ec laye s. The igo ous o mula ion o plasmonic g a ings wi h de ec s was p oposed o he i s ime. Finally, a plasmonic plana g a ing o me al-coa ed dielec ic nano ods coupled o he dielec ic slab was in es iga ed om he iewpoin o design o a e ac i e index senso . Dual-abso p ion bands a ibu able o he exci a ion o he localized su ace plasmons we e s udied, and he nea ield dis ibu ions we e gi en in bo h abso p ion bands associa ed wi h he esonances on he uppe and inne su aces o a single me al-coa ed nanocylinde . Resonance in he second abso p ion band was sensi i e o he e ac i e index o he backg ound medium and could be use ul o he design o e ac i e index senso s. Also analyzed was a phase-ma ching condi ion be ween he e anescen space-ha monics o he plasmonic g a ing and he guided modes inside he slab, leading o a s ong coupling. Keywo ds: pe iodic s uc u es; plasmonics; e ac i e index senso s; sca e ing; bandgap s uc u es 1. In oduc ion Wi h he apid de elopmen o nanoscience and nano echnology, he in e ac ion o ligh wi h nanoscaled objec s emains as an impo an issue in pho onics because o hei no el applica ions o senso s, imaging, and densely in eg a ed de ices [ 1 – 7 ]. Unique op ical p ope ies o me al nanos uc u es ha e become easible because o he eme gence o su ace plasmons, which exis Senso s 2019,19, 3923; doi:10.3390/s19183923 www.mdpi.com/jou nal/senso s Senso s 2019,19, 3923 2 o 11 when he eal pa o pe mi i i y o he me al is nega i e o he wa eleng h o he assumed op ical exci a ion. Su ace plasmon esonances c ea e sha p spec al abso p ion and sca e ing peaks, as well as a s ong elec omagne ic nea - ield enhancemen . S udies on he in e ac ion o ligh wi h he plasmonic s uc u es a e o ganized in many di e en ways, being dependen on he dimensionali y o he unde lying objec s and exci ing sou ces. Re iew wo ks abou su ace plasmon esonance senso s a e p esen ed in [8,9]. Sca e ing o an inciden plane wa e by plasmonic g a ings composed o a pe iodic a angemen o me al and me al-coa ed dielec ic nanocylinde s was igo ously in es iga ed u ilizing ou o iginal o malism based on he la ice sums echnique [ 10 – 12 ]. The me hod is b ie ly discussed in Sec ion 2. Compu a ional e iciency de ines he main ad an age o he de eloped o malism. Hence, i could be conside ed as a use ul ool o designing and op imizing compac plasmonic de ices ailo ed o ad anced applica ions, such as in bio-analy ics and/o medical diagnos ics. The goal o he manusc ip is o apply he p oposed o malism o a ious con igu a ions o plasmonic s uc u es and demons a e i s use ulness om he iewpoin o lexible design, no only o plasmonic senso s, bu also il e s, e lec o s, and guiding plasmonic de ices. We should poin ou one impo an ac . The p oposed me hod is sel -con ained and can easily be modi ied o analyze he modal ields and adia ed ields in a wide class o plana plasmonic s uc u es, hus ep esen ing a sma app oach o unc ional plasmonic de ice design. He e, he de eloped p ocedu e was implemen ed in FORTRAN and applied o he analysis o elec omagne ic sca e ing encompassing h ee con igu a ions o plasmonic s uc u es; howe e , he class o plasmonic s uc u es ha can be analyzed using he p oposed o malism is much b oade han his. The i s con igu a ion was a mul ilaye ed pe iodic s uc u e composed o cylind ical me al (sil e ) nanocylinde s. Analogous o a con en ional pho onic c ys al (PhC), i can inhibi elec omagne ic wa e p opaga ion wi hin a pa icula equency ange (i.e., a pho onic bandgap). An unde s anding o he equency bands, in which he elec omagne ic wa e p opaga ion is supp essed, is an impo an issue in lexibly designing plasmonic guiding de ices wi h ela i e low losses. The second con igu a ion was o med by in oducing pe iodic de ec s in o he o iginal PhC. The de ec s we e in oduced by pe iodically emo ing one o a ew nanocylinde s om a PhC laye . This app oach gi es us addi ional eedom o lexibly design he e lec ion and ansmission bands as desi ed unde he p esc ibed s uc u al pa ame e s, which is no possible o con en ional con igu a ions (s uc u es wi hou de ec s) . Ou in es iga ions showed ha he pe iodic de ec s led o he appea ance o a new band in he spec al esponses due o he esonan unneling h ough he de ec laye s. In o de o analyze he g a ings wi h pe iodic de ec s, a modi ica ion o he o iginal o mula ion was needed and is b ie ly p esen ed in he manusc ip . The hi d con igu a ion was a single a ay o me al-coa ed dielec ic nanocylinde s coupled o an iso opic dielec ic slab and loca ed in a backg ound medium, whose e ac i e index was no equal o one. The s uc u e was conside ed om he iewpoin o design o e ac i e index plasmonic senso s. 2. Fo mula ion o he P oblem In case o mul ilaye ed plana a ays o sca e e s, as shown as inse s o Figu es 1–4, he sca e ed space ha monics impinged on he neighbo ing a ays as new inciden wa es and we e sca e ed in o ano he se o space ha monics, which hen impinged back on he o iginal a ay. This desc ibes he mul iple sca e ing p ocess be ween he plana a ays. The sca e ing p ocess om each laye o he s uc u e was cha ac e ized by he e lec ion and ansmission ma ices, which ela e a se o he inciden space ha monics o a se o e lec ed and ansmi ed ones. The inciden and sca e ed ields we e exp essed as se s o cylind ical wa es, and he e lec ion Ri and ansmission Fi ma ices o he a ays loca ed on he i- h laye we e gi en as ollows [10–12]: Ri=U+(kx0)[I−T(k0)L(kx0h,k0h)]−1T(k0)P(kx0), (1) Senso s 2019,19, 3923 3 o 11 Fi=I+U−(kx0) [I−T(kx0)L(kx0h,k0h)]−1T(k0)P(kx0), (2) wi h P(kx0)=hPsq(kx0)i=h(−j)se−js cos−1(kxq/k0)i, (3) U±(kx0)=hU± ps (kx0)i="2js kyphe∓js cos−1(kxp/k0)#, (4) L(kx0h,k0h)=hLq−s(kx0h,k0h)i, (5) whe e k 0 is he ee-space wa enumbe , he p- h space ha monic has an x-dependence as e−jkxpx , wi h a wa enumbe k xp =k x0 +2 π p/h(o hogonal sys em and hpa ame e a e speci ied in Figu e 1). He e, P (k x0 ) labels he ma ix ha ans o ms he down-going q- h inciden space ha monic wa e o he s- h cylind ical ha monic wa e. U+ and U− a e he ma ices ans o ming he s- h cylind ical ha monic wa e back o up-going and down-going p- h space-ha monic wa es (wi h p,s, q = − M, − M+1, . . . , 0, . . . ,M−1, M). The p- h ow and q- h column elemen s o he e lec ion Ri= [Rpq i]and ansmission Fi= [Fpq i] ma ices ep esen he e lec ion and ansmission coe icien s o he p- h o de e lec ed and ansmi ed space ha monics o he q- h o de space ha monic o he inciden wa e, espec i ely. No e ha I deno es he uni ma ix, T (k 0 ) is he T-ma ix o he isola ed cylind ical od [ 10 ] desc ibing he na u e o he sca e ed ield pe uni cell, and L (k x0 h,k 0 h) is called he la ice sums. The la ice sums L (k x0 h,k 0 h) in Equa ion (5) cha ac e ize he pe iodic a angemen s o he sca e e s and a e independen o he pola iza ion, o hei obse a ion poin s, and geome ical pa ame e s. The con e gen and e y e icien calcula ion o he la ice sums o complex alues o k x0 (in he modal analysis) we e demons a ed in ou ecen wo k [ 12 ]. The si ua ion was sligh ly di e en when we had a single pe iodic laye . In his case, we we e no dealing wi h he e lec ion and ansmission ma ices, bu wi h he e lec ion and ansmission ec o s. This was because we did no ha e any mul iple in e ac ion be ween he laye s, and P in Equa ion (3) is no a ma ix, bu a he a ec o . Finally, since he e lec ion and ansmission ma ices o single a ays we e de ined, he gene alized e lec ion ma ix iewed om he egion o plane wa e incidence could be easily ob ained based on a ecu si e algo i hm [ 10 , 13 ]. The calcula ion p ocess was e y as , since only simple ma ix mul iplica ions we e in ol ed. Senso s 2019, 19, 3923 3 o 10 () () () 10 00 cos / () − −  ==−    Pxq s xsqx js k k kPk je , (3) () () () 10 00 cos / 2− ±±   ==    Uxp s xx yp js k k ps j kUk e kh , (4) ()() 00 00 ,, − =  Lxxqs khkh L khkh , (5) whe e k0 is he ee-space wa enumbe , he p- h space ha monic has an x-dependence as − x p j kx e, wi h a wa enumbe kxp = kx0 + 2πp/h (o hogonal sys em and h pa ame e a e speci ied in Figu e 1). He e, P(kx0) labels he ma ix ha ans o ms he down-going q- h inciden space ha monic wa e o he s- h cylind ical ha monic wa e. U+ and U- a e he ma ices ans o ming he s- h cylind ical ha monic wa e back o up-going and down-going p- h space-ha monic wa es (wi h p, s, q = - M, - M +1, …, 0, …, M -1, M). The p- h ow and q- h column elemen s o he e lec ion []=R p q ii R and ansmission []=F p q ii Fma ices ep esen he e lec ion and ansmission coe icien s o he p- h o de e lec ed and ansmi ed space ha monics o he q- h o de space ha monic o he inciden wa e, espec i ely. No e ha I deno es he uni ma ix, T(k0) is he T-ma ix o he isola ed cylind ical od [10] desc ibing he na u e o he sca e ed ield pe uni cell, and L(kx0h, k0h) is called he la ice sums. The la ice sums L(kx0h, k0h) in Equa ion (5) cha ac e ize he pe iodic a angemen s o he sca e e s and a e independen o he pola iza ion, o hei obse a ion poin s, and geome ical pa ame e s. The con e gen and e y e icien calcula ion o he la ice sums o complex alues o kx0 (in he modal analysis) we e demons a ed in ou ecen wo k [12]. The si ua ion was sligh ly di e en when we had a single pe iodic laye . In his case, we we e no dealing wi h he e lec ion and ansmission ma ices, bu wi h he e lec ion and ansmission ec o s. This was because we did no ha e any mul iple in e ac ion be ween he laye s, and P in Equa ion (3) is no a ma ix, bu a he a ec o . Finally, since he e lec ion and ansmission ma ices o single a ays we e de ined, he gene alized e lec ion ma ix iewed om he egion o plane wa e incidence could be easily ob ained based on a ecu si e algo i hm [10,13]. The calcula ion p ocess was e y as , since only simple ma ix mul iplica ions we e in ol ed. Figu e 1. Powe e lec ion (blue line) and powe ansmission ( ed line) o a 31-laye ed plasmonic c ys al o med by he co esponding pe iodical a angemen o cylind ical Ag nanocylinde s. Pe iod o he g a ing was equal o h = 300 nm and adius o he nanocylinde amoun s o = 40 nm. Exci a ion was p o ided by an impinging H-pola ized plane wa e (Hz, Ex, Ey) wi h no mal incidence (along he posi i e y-axis). Fu he mo e, he me hod can be ex ended o sca e e s wi h non-ci cula c oss sec ion, by nume ically de i ing he ele an T-ma ix exp ession [14]. In case o a ci cula cylinde , he T-ma ix Figu e 1. Powe e lec ion (blue line) and powe ansmission ( ed line) o a 31-laye ed plasmonic c ys al o med by he co esponding pe iodical a angemen o cylind ical Ag nanocylinde s. Pe iod o he g a ing was equal o h=300 nm and adius o he nanocylinde amoun s o =40 nm. Exci a ion was p o ided by an impinging H-pola ized plane wa e (H z ,E x ,E y ) wi h no mal incidence (along he posi i e y-axis). Senso s 2019,19, 3923 4 o 11 Fu he mo e, he me hod can be ex ended o sca e e s wi h non-ci cula c oss sec ion, by nume ically de i ing he ele an T-ma ix exp ession [ 14 ]. In case o a ci cula cylinde , he T-ma ix has a qui e simple o m—i is a diagonal ma ix— ha can be e y easily calcula ed in closed o m. Howe e , o some o he non-ci cula c oss sec ions, he T-ma ix is no anymo e diagonal and should be calcula ed nume ically. 3. Nume ical Resul s and Discussions In he nume ical expe imen s, we assumed Ag (sil e ) o he me al [ 15 ], which is commonly used in plasmonics. I is wo h men ioning ha an impo an ad an age o he p oposed app oach is i s compu a ion e iciency. The desk op CPU un ime on he 3.6 GHz In el Co e i7 wi h 8 GB RAM pe one equency poin was app oxima ely 0.03–0.05 s o all h ee con igu a ions shown below. 3.1. Ligh Sca e ing by Mul laye ed S uc u es o Plasmonic Nanocylinde s wi h and wi hou De ec s Pe iodic de ec s we e in oduced by pe iodically emo ing one o se e al ods om he laye . The Equa ions (1)–(5) canno be di ec ly used o he a ays ha ing di e en pe iods. A modi ica ion o Equa ions (1)–(5) was needed. Wi hou loss o gene ali y, le us assume ha he s uc u e is composed o a ays ha ing pe iods hand 2h( he laye wi h a pe iod 2his o med by pe iodically emo ing one nanocylinde ). The p oblem can be sol ed by decomposing he o iginal se o space ha monics wi h he pe iod 2hin o subse s o space ha monics wi h a pe iod h. In o he wo ds, o he a ays wi h he pe iod h, we in oduced wo subse s o space ha monics as ollows [16]: nejkxpx;kxp =kx0+2πp/2ho=1 ∪ ν=0ej(ξν+2πp h)x;ξν=kx0+νπ/h. (6) Taking in o accoun he o hogonal na u e o he space ha monics, he e lec ion and ansmission ma ices o he a ays wi h he pe iod hcan be calcula ed using he simila exp essions o Equa ions (1)–(5), bu unde he incidence o wo di e en subse s o space ha monics. The esul s we e hen ea anged on he basis o he o iginal se o space ha monics wi h he pe iod 2h. Fo he a ays wi h he pe iod h, we used wo subse s o space ha monics wi h he wa enumbe s ξ0 and ξ1 , as shown in Equa ion (6). The gene aliza ion o he p oblem o a wide class o s uc u es wi h pe iodic de ec s is s aigh o wa d. In Figu es 1–3, mul ilaye ed s uc u es ha ing a pe iod h=300 nm and composed o Ag nanocylinde s wi h a adius =40 nm a e nume ically in es iga ed. Wi hou loss o gene ali y, 31 laye s we e conside ed and he backg ound medium was a ee space. Usually, a “plana plasmonic s uc u e” means ha i s hickness is no la ge han se e al hund ed nanome e s. Ou s uc u e p obably was no he bes example o he plana geome y; howe e , he wide plasmonic gap could be obse ed o he mul ilaye ed s uc u e and hus i is a eason o conside his con igu a ion. None heless, ou o malism can be easily applied o a ious plana con igu a ions such as he second example in he manusc ip . We discuss he equency esponse o he powe e lec ion and powe ansmission unde no mal incidence o an exci ing plane wa e. The no mal incidence is he usual con igu a ion o he plasmonic c ys al o he use as a equency and pola iza ion selec i e de ice. All nume ical esul s we e ca ied ou wi h an e o measu e in he ene gy conse a ion o less han 10 −7 [ 17 ]. Fo be e unde s anding he impac o pe iodic de ec s on he spec al esponses, we conside ed h ee con igu a ions, as illus a ed as inse s in Figu es 1–3. Figu e 1displays he o iginal con igu a ion wi h no de ec included. I consis ed o a squa e la ice o med by cylind ical Ag nanocylinde s. In Figu e 2, only he in e media e laye ( he 16 h laye ) o plasmonic c ys al was eplaced by a de ec laye o he la ice, in which one cylinde was pe iodically emo ed, and hus a laye wi h a pe iod 2hwas o med. In Figu e 3, he plasmonic c ys al was gene a ed by al e na ely s acking a comple e laye and a de ec laye whe e one nanocylinde was acco dingly emo ed. The de ec s we e bo h pe iodic in he x- and y-di ec ion wi h a pe iod 2h. The spec al esponse o bo h he powe e lec ion (blue line) and he powe ansmission ( ed line) o he incidence o (H z ,E x ,E y ) Senso s 2019,19, 3923 5 o 11 wa e a e plo ed in Figu e 1. F om he spec a, i can be seen ha an o iginal plasmonic c ys al had a wide e lec ion band (plasmonic gap) wi hin he isible wa eleng h egion 600 nm < λ <680 nm, which was a om he plasmon esonance o he indi idual plasmonic nanocylinde a 355 nm. No e ha a sepa a ion dis ance be ween he nanocylinde s along he ans e se y-di ec ion was qui e la ge h>2 and, he e o e, no coupling occu ed be ween localized plasmons on he me al nanocylinde s. I could be seen ha he ansmission was supp essed wi hin a sho wa eleng h band ( 250 nm <λ<380 nm ). When he 16 h laye o he la ice was eplaced by a de ec laye , as shown in Figu e 2, he wide e lec ion band was ba ely a ec ed by he de ec s and e ained i s o iginal bandwid h. When he de ec laye s we e pe iodically assembled in he y-di ec ion, as shown in Figu e 3, he equency esponse o he e lec ion was signi ican ly modi ied compa ed o hose o Figu es 1and 2. As shown in Figu e 3, an addi ional na ow band was o med in he sho e wa eleng h egion a ound 425 nm because o he esonan unneling h ough he de ec laye s. Since he pe iod o he la ice along he x-di ec ion was 2h, he i s space ha monic began o p opaga e wi hin he indica ed wa eleng h ange, wi h he spec al esponse o i s ampli ude also being ma ked by a ed line. Nex , o compa ison, we s udy he se en-laye ed s uc u e composed o Ag nanocylinde s, whils he o he geome ical pa ame e s we e he same as hose al eady used in Figu e 1. The co esponding spec al esponses o he powe e lec ion and powe ansmission coe icien s a e plo ed in Figu e 4 by blue and ed lines, espec i ely. The nea ield dis ibu ions wi hin he plasmonic la ice’s supe cell we e calcula ed a h ee di e en wa eleng hs, namely a 298 nm (ma ked by “1”), a 355 nm (ma ked by “2”), and a 630 nm (ma ked by “3”). A s ong e lec ed ield was obse ed a 298 nm, associa ed wi h he Rayleigh wa eleng h (λ≈h/pa he no mal incidence, whe e pdeno es he space-ha monic), whe eas he ield was enhanced a he su ace o he uppe mos nanocylinde laye a λ =355 nm. The la e was ela ed o he in e ac ion o he inciden wa e and he uppe mos indi idual nanocylinde . Unlike Figu e 1, he s ong plasmonic gap was no o med in case o se en laye s, and he e o e he nea ield dis ibu ion a λ =630 nm showed only a pa ially e lec ed ield om he mul ilaye ed pe iodic s uc u e. Ou analysis showed ha a leas 13 laye s we e needed o ealize a plasmonic gap ha ully e ol ed o he one shown in Figu e 1(only when o he s pa ame e s we e ixed). Senso s 2019, 19, 3923 5 o 10 h > 2 and, he e o e, no coupling occu ed be ween localized plasmons on he me al nanocylinde s. I could be seen ha he ansmission was supp essed wi hin a sho wa eleng h band (250 nm < λ < 380 nm). When he 16 h laye o he la ice was eplaced by a de ec laye , as shown in Figu e 2, he wide e lec ion band was ba ely a ec ed by he de ec s and e ained i s o iginal bandwid h. When he de ec laye s we e pe iodically assembled in he y-di ec ion, as shown in Figu e 3, he equency esponse o he e lec ion was signi ican ly modi ied compa ed o hose o Figu es 1 and 2. As shown in Figu e 3, an addi ional na ow band was o med in he sho e wa eleng h egion a ound 425 nm because o he esonan unneling h ough he de ec laye s. Since he pe iod o he la ice along he x-di ec ion was 2h, he i s space ha monic began o p opaga e wi hin he indica ed wa eleng h ange, wi h he spec al esponse o i s ampli ude also being ma ked by a ed line. Figu e 2. The same se ings as in Figu e 1, excep ha he in e media e 16 h laye o he o iginal c ys al was eplaced by a de ec laye whe e one nanocylinde was pe iodically emo ed wi h a pe iod o 2h. Figu e 3. The same se ings as in Figu e 1, bu he c ys al was o med by al e na ely s acking he comple e laye wi h a pe iod h and he de ec laye , whe e one nanocylinde was pe iodically emo ed acco ding o he pe iod 2h. Figu e 2. The same se ings as in Figu e 1, excep ha he in e media e 16 h laye o he o iginal c ys al was eplaced by a de ec laye whe e one nanocylinde was pe iodically emo ed wi h a pe iod o 2h. Senso s 2019,19, 3923 6 o 11 Senso s 2019, 19, 3923 5 o 10 h > 2 and, he e o e, no coupling occu ed be ween localized plasmons on he me al nanocylinde s. I could be seen ha he ansmission was supp essed wi hin a sho wa eleng h band (250 nm < λ < 380 nm). When he 16 h laye o he la ice was eplaced by a de ec laye , as shown in Figu e 2, he wide e lec ion band was ba ely a ec ed by he de ec s and e ained i s o iginal bandwid h. When he de ec laye s we e pe iodically assembled in he y-di ec ion, as shown in Figu e 3, he equency esponse o he e lec ion was signi ican ly modi ied compa ed o hose o Figu es 1 and 2. As shown in Figu e 3, an addi ional na ow band was o med in he sho e wa eleng h egion a ound 425 nm because o he esonan unneling h ough he de ec laye s. Since he pe iod o he la ice along he x-di ec ion was 2h, he i s space ha monic began o p opaga e wi hin he indica ed wa eleng h ange, wi h he spec al esponse o i s ampli ude also being ma ked by a ed line. Figu e 2. The same se ings as in Figu e 1, excep ha he in e media e 16 h laye o he o iginal c ys al was eplaced by a de ec laye whe e one nanocylinde was pe iodically emo ed wi h a pe iod o 2h. Figu e 3. The same se ings as in Figu e 1, bu he c ys al was o med by al e na ely s acking he comple e laye wi h a pe iod h and he de ec laye , whe e one nanocylinde was pe iodically emo ed acco ding o he pe iod 2h. Figu e 3. The same se ings as in Figu e 1, bu he c ys al was o med by al e na ely s acking he comple e laye wi h a pe iod hand he de ec laye , whe e one nanocylinde was pe iodically emo ed acco ding o he pe iod 2h. Senso s 2019, 19, 3923 6 o 10 Nex , o compa ison, we s udy he se en-laye ed s uc u e composed o Ag nanocylinde s, whils he o he geome ical pa ame e s we e he same as hose al eady used in Figu e 1. The co esponding spec al esponses o he powe e lec ion and powe ansmission coe icien s a e plo ed in Figu e 4 by blue and ed lines, espec i ely. The nea ield dis ibu ions wi hin he plasmonic la ice's supe cell we e calcula ed a h ee di e en wa eleng hs, namely a 298 nm (ma ked by “1”), a 355 nm (ma ked by “2”), and a 630 nm (ma ked by “3”). A s ong e lec ed ield was obse ed a 298 nm, associa ed wi h he Rayleigh wa eleng h (λ ≈ h/p a he no mal incidence, whe e p deno es he space-ha monic), whe eas he ield was enhanced a he su ace o he uppe mos nanocylinde laye a λ = 355 nm. The la e was ela ed o he in e ac ion o he inciden wa e and he uppe mos indi idual nanocylinde . Unlike Figu e 1, he s ong plasmonic gap was no o med in case o se en laye s, and he e o e he nea ield dis ibu ion a λ = 630 nm showed only a pa ially e lec ed ield om he mul ilaye ed pe iodic s uc u e. Ou analysis showed ha a leas 13 laye s we e needed o ealize a plasmonic gap ha ully e ol ed o he one shown in Figu e 1 (only when o he s pa ame e s we e ixed). Figu e 4. (le ) Spec al esponse o he powe e lec ion/ ansmission o he same con igu a ion in Figu e 1, bu o se en laye s along he y-axis. ( igh ) Nea ield |Hz| dis ibu ions a h ee di e en peak wa eleng hs, i.e. 298 nm (associa ed wi h he Rayleigh wa eleng h), 355 nm (associa ed wi h a esonance wa eleng h o a single sil e nanocylinde ), and 630 nm (associa ed wi h plasmonic gap in Figu e 1). 3.2. Coupling be ween Plasmonic G a ing and a Dielec ic Slab o Applica ion as a Re ac i e Index Senso We s udied Ag-coa ed dielec ic nanocylinde s wi h he ollowing s uc u al pa ame e s: 1 40= nm, 220= nm, h = 300 nm, 0 /6.5= εε , d = 200 nm, and 0 /2.5= s εε (glass). No e ha in he i s pa , he pu e Ag nanocylinde s we e conside ed, and in he second pa , he Ag-coa ed elemen s we e conside ed. Fo a me al-coa ed nanocylinde , special a en ion was paid o he appea ance o “ he second abso p ion band” (Figu e 6), which was ne e obse ed o a single pu e me al od and was ound o be e y sensi i e o he e ac i e index o he backg ound medium. The geome y o he p oblem is illus a ed in Figu e 5. The e ac i e index o he backg ound medium was n. The s uc u e can be used o he design o e ac i e index plasmonic senso s. Figu e 6a shows he con ou plo s o he powe ansmission o he 0- h di ac ion o de and he abso p ion coe icien s as a unc ion o he wa eleng h wi hin he ange o 250 nm < λ < 1000 nm and he e ac i e index 11.7≤≤na no mal incidence (90)=° i ϕ . In his ange o e ac i e indices, he backg ound medium can, o example, be conside ed as a luid, such as wa e , chlo o o m, benzene, o ca bon disul ide. Figu e 6b demons a es wo abso p ion bands a ibu able o he exci a ion o localized su ace plasmons. The i s abso p ion band, 300 nm < λ < 380 nm, was a ibu able o he su ace plasmons suppo ed by he ou e in e ace o he me al coa ing esona ing wi h he inciden wa e, whe eas he Figu e 4. ( le ) Spec al esponse o he powe e lec ion/ ansmission o he same con igu a ion in Figu e 1, bu o se en laye s along he y-axis. ( igh ) Nea ield |H z |dis ibu ions a h ee di e en peak wa eleng hs, i.e. 298 nm (associa ed wi h he Rayleigh wa eleng h), 355 nm (associa ed wi h a esonance wa eleng h o a single sil e nanocylinde ), and 630 nm (associa ed wi h plasmonic gap in Figu e 1). 3.2. Coupling be ween Plasmonic G a ing and a Dielec ic Slab o Applica ion as a Re ac i e Index Senso We s udied Ag-coa ed dielec ic nanocylinde s wi h he ollowing s uc u al pa ame e s: 1=40 nm , 2= 20 nm, h=300 nm, ε/ε0= 6.5, d=200 nm, and εs/ε0= 2.5 (glass). No e ha in he i s pa , he pu e Ag nanocylinde s we e conside ed, and in he second pa , he Ag-coa ed elemen s we e conside ed. Fo a me al-coa ed nanocylinde , special a en ion was paid o he appea ance o “ he second abso p ion band” (Figu e 6), which was ne e obse ed o a single pu e me al od and was ound o be e y sensi i e o he e ac i e index o he backg ound medium. The geome y o he p oblem is illus a ed in Figu e 5. The e ac i e index o he backg ound medium was n. The s uc u e can be used o he design o e ac i e index plasmonic senso s. Figu e 6a Senso s 2019,19, 3923 7 o 11 shows he con ou plo s o he powe ansmission o he 0- h di ac ion o de and he abso p ion coe icien s as a unc ion o he wa eleng h wi hin he ange o 250 nm < λ <1000 nm and he e ac i e index 1 ≤n≤ 1.7 a no mal incidence (ϕi= 90 ◦) . In his ange o e ac i e indices, he backg ound medium can, o example, be conside ed as a luid, such as wa e , chlo o o m, benzene, o ca bon disul ide. Figu e 6b demons a es wo abso p ion bands a ibu able o he exci a ion o localized su ace plasmons. The i s abso p ion band, 300 nm < λ <380 nm, was a ibu able o he su ace plasmons suppo ed by he ou e in e ace o he me al coa ing esona ing wi h he inciden wa e, whe eas he second abso p ion band, 550 nm <λ<650 nm , was o med because o he exci a ion o he su ace plasmons along he bounda y laye be ween he me al and he inne dielec ic co e. Fo be e unde s anding o he p oblem, we also nume ically s udied he nea ield dis ibu ions o he magne ic ield Hz wi hin hese abso p ion bands, and he enla ged igu es o he supe cell a e shown as inse in Figu e 6b. F om hese igu es, i can be no ed ha in he i s abso p ion band egion he esul ing su ace ield was p edominan ly o ien ed owa ds he illumina ion side o he nanocylinde s. The e was almos no ield isible inside he dielec ic co e o he nanocylinde , whe eas a s ong ield enhancemen along he su ace be ween he me al and he inne dielec ic co e egion was isible in he egion o he second abso p ion band (only o a coa ed nano od). Figu e 6c depic s he ansmission spec a in he wa eleng h ange, 550 nm <λ<650 nm (second abso p ion band), o ou di e en alues o he e ac i e index o he backg ound medium. I can be obse ed ha he esonance wa eleng hs sensi i ely depended on he e ac i e index and we e hus shi ed o longe wa eleng hs wi h an inc easing index o e ac ion n, which could be exploi ed in he con ex o a e ac i e index senso . Senso s 2019, 19, 3923 7 o 10 second abso p ion band, 550 nm < λ < 650 nm, was o med because o he exci a ion o he su ace plasmons along he bounda y laye be ween he me al and he inne dielec ic co e. Fo be e unde s anding o he p oblem, we also nume ically s udied he nea ield dis ibu ions o he magne ic ield z H wi hin hese abso p ion bands, and he enla ged igu es o he supe cell a e shown as inse in Figu e 6b. F om hese igu es, i can be no ed ha in he i s abso p ion band egion he esul ing su ace ield was p edominan ly o ien ed owa ds he illumina ion side o he nanocylinde s. The e was almos no ield isible inside he dielec ic co e o he nanocylinde , whe eas a s ong ield enhancemen along he su ace be ween he me al and he inne dielec ic co e egion was isible in he egion o he second abso p ion band (only o a coa ed nano od). Figu e 6c depic s he ansmission spec a in he wa eleng h ange, 550 nm < λ < 650 nm (second abso p ion band), o ou di e en alues o he e ac i e index o he backg ound medium. I can be obse ed ha he esonance wa eleng hs sensi i ely depended on he e ac i e index and we e hus shi ed o longe wa eleng hs wi h an inc easing index o e ac ion n, which could be exploi ed in he con ex o a e ac i e index senso . Figu e 5. C oss-sec ional iew o he plasmonic g a ing composed o Ag-coa ed dielec ic nanocylinde s pe iodically a anged on a dielec ic slab wi h hickness d. Ma e ial pa ame e s o he backg ound medium, he coa ing me al, he dielec ic co e, and dielec ic subs a e we e 0 (, ) B ε μ , 01 (, ,) M ε μ , 02 ,(, ) ε μ , and 0 (, ) s ε μ , espec i ely. The exci a ion was p o ided by an impinging H- pola ized plane wa e wi h an inciden angle i ϕ . Ano he impo an issue in he applica ion o e ac i e index sensing conce ned he coupling mechanism be ween he plasmonic g a ing and he unde lying dielec ic slab. When he phase ma ching condi ion [17,18], 2 1 2 2 2 1 an 1 2 2 2 ε λ βλ π επ εβλ βλ ε πε π π − − − −          +=               B B s s s m d (7) 2 ( 1, 2, 3, )==±±±pp h π β , (8) be ween he guided slab mode m and one o he p- h space-ha monics was ul illed a some speci ic wa eleng h, hese wo wa e componen s e ec i ely in e ac ed and a ac ion o he powe o he inciden wa e was esonan ly ans e ed o he guided wa e along he slab wa eguide. I led o a subs an ial dec ease o he ansmi ed powe a hese pa icula wa eleng hs (400 nm < λ < 470 nm), as depic ed in Figu e 6a. In his wa eleng h ange and unde ou p esc ibed geome ical pa ame e s, a phase ma ching be ween he undamen al H-pola ized slab mode and he sca e ed wa e o he i s di ac ion o de s was a ained. We es ima ed he sensi i i y S by moni o ing he wa eleng h shi o he ansmission peak om ai λ ( esonance wa eleng h a he coupling be ween he plasmonic g a ing and he slab in he ai ) o B λ ( esonance wa eleng h a coupling be ween he plasmonic g a ing and Figu e 5. C oss-sec ional iew o he plasmonic g a ing composed o Ag-coa ed dielec ic nanocylinde s pe iodically a anged on a dielec ic slab wi h hickness d. Ma e ial pa ame e s o he backg ound medium, he coa ing me al, he dielec ic co e, and dielec ic subs a e we e (εB , µ0) , (εM , µ0 , 1) , (ε , µ0 , 2) , and (εs , µ0) , espec i ely. The exci a ion was p o ided by an impinging H-pola ized plane wa e wi h an inciden angle ϕi. Senso s 2019,19, 3923 8 o 11 Senso s 2019, 19, 3923 8 o 10 he slab in he backg ound medium) as /=Δ Δ =Sn λ ()/(1)−− ai Bn λλ . A 1< n < 1.4, he sensi i i y was equal o 72, and a n > 1.4, he sensi i i y was 84. When n2 was close o pe mi i i y s ε o he dielec ic slab, he ield con inemen inside he slab became weake and he dielec ic slab lacked in suppo ing any guided modes. Hence, no guided mode esonance was obse ed in he ansmission spec a a n > 1.58, as depic ed in Figu e 6a. In o de o p o ide a deepe insigh in he o ma ion o he esonance due o he coupling be ween he g a ing and he slab, we in es iga ed he dependence o he ansmission spec a e sus he hickness o he slab d when he backg ound was a ee space. Two egions o he esonance peaks showed an almos linea beha io , ma ked as Equa ions (7) and (8), and a e displayed in Figu e 7. The i s one ep esen ed he esonance peak a ibu able o he coupling o he i s e anescen wa e ( he i s space-ha monic) wi h he undamen al m = 0 guided mode, whe eas he second one demons a ed he esonances a ibu able o a coupling o he i s e anescen wa e wi h he m = 1 guided mode inside he slab. The simula ed nea ield dis ibu ions a λ = 399 nm (associa ed wi h he coupling o he i s e anescen wa e o he g a ing wi h he m = 1 guided mode) and λ = 451 nm (associa ed wi h he coupling o he i s e anescen wa e o he g a ing wi h he m = 0 guided mode) o a ixed slab hickness d = 390 nm a e shown as inse s in Figu e 7. Ou analysis showed ha hese sha p esonances we e sensi i e o he backg ound e ac i e index and can be conside ed as one o he componen s o a design o e ac i e index senso s. Finally, we no ed ha he senso cha ac e is ics shown in he manusc ip needed some imp o emen s. We hope ha he as and igo ous nume ical me hod we de eloped will help enginee s and esea che s o e ec i ely analyze a wide class o plasmonic g a ings in o de o design e y highly sensi i e plasmonic senso s. Figu e 6. Plasmonic g a ing as e ac i e index senso : Con ou plo s o ansmission spec a (a) and abso p ion spec a (b) e sus e ac i e index 0 / εε =B no he backg ound medium (Figu e 4) a 1 = 40 nm, 2 = 20 nm, h = 300 nm, 0 /6.5= εε , d = 200 nm, 0 /2.5= s εε and 0 90= i ϕ . The egions o he i s and second abso p ion bands a e ma ked by he whi e lines. Powe ansmission spec a in he second abso p ion band o ou di e en alues o he e ac i e index o he backg ound medium (c). Figu e 6. Plasmonic g a ing as e ac i e index senso : Con ou plo s o ansmission spec a ( a ) and abso p ion spec a ( b ) e sus e ac i e index n=pεB/ε0 o he backg ound medium (Figu e 4) a 1=40 nm , 2 =20 nm, h=300 nm, ε/ε0= 6.5, d=200 nm, εs/ε0= 2.5 and ϕi= 90 ◦ . The egions o he i s and second abso p ion bands a e ma ked by he whi e lines. Powe ansmission spec a in he second abso p ion band o ou di e en alues o he e ac i e index o he backg ound medium ( c ). Ano he impo an issue in he applica ion o e ac i e index sensing conce ned he coupling mechanism be ween he plasmonic g a ing and he unde lying dielec ic slab. When he phase ma ching condi ion [17,18], λ d 1 π εs−βλ 2π2 an−1 εs εB u u u u u βλ 2π2−εB εs−βλ 2π2 +mπ 2 =1 (7) β=2π hp(p=±1, ±2, ±3, · · · ), (8) be ween he guided slab mode mand one o he p- h space-ha monics was ul illed a some speci ic wa eleng h, hese wo wa e componen s e ec i ely in e ac ed and a ac ion o he powe o he inciden wa e was esonan ly ans e ed o he guided wa e along he slab wa eguide. I led o a subs an ial dec ease o he ansmi ed powe a hese pa icula wa eleng hs (400 nm < λ <470 nm), as depic ed in Figu e 6a. In his wa eleng h ange and unde ou p esc ibed geome ical pa ame e s, a phase ma ching be ween he undamen al H-pola ized slab mode and he sca e ed wa e o he i s di ac ion o de s was a ained. We es ima ed he sensi i i y Sby moni o ing he wa eleng h shi o he ansmission peak om λai ( esonance wa eleng h a he coupling be ween he plasmonic g a ing Senso s 2019,19, 3923 9 o 11 and he slab in he ai ) o λB ( esonance wa eleng h a coupling be ween he plasmonic g a ing and he slab in he backg ound medium) as S=∆λ/∆n= (λB−λai )/(n− 1 ) . A 1 <n<1.4, he sensi i i y was equal o 72, and a n>1.4, he sensi i i y was 84. When n 2 was close o pe mi i i y εs o he dielec ic slab, he ield con inemen inside he slab became weake and he dielec ic slab lacked in suppo ing any guided modes. Hence, no guided mode esonance was obse ed in he ansmission spec a a n>1.58, as depic ed in Figu e 6a. In o de o p o ide a deepe insigh in he o ma ion o he esonance due o he coupling be ween he g a ing and he slab, we in es iga ed he dependence o he ansmission spec a e sus he hickness o he slab dwhen he backg ound was a ee space. Two egions o he esonance peaks showed an almos linea beha io , ma ked as Equa ions (7) and (8) , and a e displayed in Figu e 7. The i s one ep esen ed he esonance peak a ibu able o he coupling o he i s e anescen wa e ( he i s space-ha monic) wi h he undamen al m=0 guided mode, whe eas he second one demons a ed he esonances a ibu able o a coupling o he i s e anescen wa e wi h he m=1 guided mode inside he slab. The simula ed nea ield dis ibu ions a λ=399 nm (associa ed wi h he coupling o he i s e anescen wa e o he g a ing wi h he m=1 guided mode) and λ =451 nm (associa ed wi h he coupling o he i s e anescen wa e o he g a ing wi h he m=0 guided mode) o a ixed slab hickness d=390 nm a e shown as inse s in Figu e 7. Ou analysis showed ha hese sha p esonances we e sensi i e o he backg ound e ac i e index and can be conside ed as one o he componen s o a design o e ac i e index senso s. Finally, we no ed ha he senso cha ac e is ics shown in he manusc ip needed some imp o emen s. We hope ha he as and igo ous nume ical me hod we de eloped will help enginee s and esea che s o e ec i ely analyze a wide class o plasmonic g a ings in o de o design e y highly sensi i e plasmonic senso s. Senso s 2019, 19, 3923 9 o 10 Figu e 7. Plasmonic g a ing as e ac i e index senso : Con ou plo s o ansmission spec a e sus hickness o he slab d a n = 1. Simula ed nea ield dis ibu ions o z H a λ = 451 nm and λ = 399 nm o a ixed hickness o he slab d = 390 nm a e shown as inse s. 4. Concluding Rema ks A igo ous sel -consis en o mula ion was e icien ly applied o a wide class o plasmonic g a ings composed o me al and me al-coa ed nanocylinde s. In pa icula , sca e ing by mul ilaye ed plasmonic c ys als wi h and wi hou de ec s was analyzed. The coupling mechanism o he plasmonic g a ing wi h he dielec ic slab was s udied om he iewpoin o applica ion as a e ac i e index senso . The me hod was compu a ionally e y as and hus ep esen s an e ec i e ool ha is ap o designing and op imizing ailo ed senso s (pa icula ly wi h op imized sensi i i y), such as o ad anced biomedical applica ions. The p oposed o malism can also be applied o analyze he in e ac ion o he plana plasmonic g a ing wi h non-iso opic media, such as, o example, a magne o-op ical [18] o magne ized e i e [19] slab. Gene aliza ion o he desc ibed me hod owa ds he mul ilaye ed sandwiched s uc u es was s aigh o wa d. Such a combina ion enables one o achie e a mul iple inc ease in sensi i i y when compa ed o a senso s uc u e ha ing jus one me al g a ing (o laye ). Ano he poin wo h no icing is ha ou o malism can be easily applied o mul ilaye ed me al- dielec ic plasmonic nanocylinde s wi h eccen ic con igu a ion pe uni cell. Eccen ic con igu a ions o me al-dielec ic plasmonic nanocylinde s a e being in es iga ed in he amewo k o eme ging Fano esonances [20]. Au ho Con ibu ions: De elopmen o heo e ical app oach, K.Y. and V.J.; nume ical analysis, V.J.; w i ing manusc ip , V.J., J.P., D.E. All au ho s analyzed he esul s o nume ical expe imen s, discussed hem and edi ed he manusc ip . Funding: V.J. kindly acknowledges he suppo o he Alexande on Humbold Founda ion. D.E. kindly acknowledges he suppo o he Deu sche Fo schungsgemeinscha in he amewo k o DFG CRC/TRR 196 MARIE (p ojec M03). Con lic s o In e es : The au ho s decla e no con lic o in e es . Re e ences 1. K a e s, V.; Schedin, F.; G igo enko, A. Ex emely na ow plasmon esonances based on di ac ion coupling o localized plasmons in a ays o me allic nanopa icles. Phys. Re . Le . 2008, 101, 087403. 2. Luk’yanchuk, B.; Te no sky, V. Ligh sca e ing by a hin wi e wi h a su ace-plasmon esonance: Bi u ca ions o he Poyn ing ec o ield. Phys. Re . B 2006, 73, 235432. 3. Nguyen-Huu, N.; Cada, M.; Pis o a, J. Impe ec ly geome ic shapes o nanog a ing s uc u es as sola abso be s wi h supe io pe o mance o sola cells. Op . Exp ess 2014, 22, A282–A294. Figu e 7. Plasmonic g a ing as e ac i e index senso : Con ou plo s o ansmission spec a e sus hickness o he slab da n=1. Simula ed nea ield dis ibu ions o |Hz| a λ =451 nm and λ =399 nm o a ixed hickness o he slab d=390 nm a e shown as inse s. 4. Concluding Rema ks A igo ous sel -consis en o mula ion was e icien ly applied o a wide class o plasmonic g a ings composed o me al and me al-coa ed nanocylinde s. In pa icula , sca e ing by mul ilaye ed plasmonic c ys als wi h and wi hou de ec s was analyzed. The coupling mechanism o he plasmonic g a ing wi h he dielec ic slab was s udied om he iewpoin o applica ion as a e ac i e index senso . The me hod was compu a ionally e y as and hus ep esen s an e ec i e ool ha is ap o designing and op imizing ailo ed senso s (pa icula ly wi h op imized sensi i i y), such as o ad anced biomedical applica ions. The p oposed o malism can also be applied o analyze