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Optical properties of Fano-resonant metallic metasurfaces on a substrate S. Hossein Mousavi, Alexander B. Khanikaev, and Gennady Shvets∗ Department of Physics and Institute for Fusion Studies, University of Texas at Austin, One University Station C1500, Austin, Texas 78712, USA Three different periodic optical metasurfaces exhibiting Fano resonances are studied in mid-IR frequency range in the presence of a substrate. We develop a rigorous semi-analytical technique and calculate how the presence of a substrate affects optical properties of these structures. An analytical minimal model based on the truncated exact technique is introduced and is shown to provide a simple description of the observed behavior. We demonstrate that the presence of a substratesubstantiallyaltersthecollectiveresponseofthestructuressuppressingWood’sanomalies 2 andspatialdispersionoftheresonances. DifferenttypesofFanoresonancesarefoundtobeaffected 1 differently by the optical contrast between the substrate and the superstrate. The dependence of 0 the spectral position of the resonances on the substrate/superstrate permittivities is studied and 2 thevalidity of thewidely used effective medium approaches is re-examined. n a PACSnumbers: 41.20.Jb,42.25.Fx,78.67.Pt,73.20.Mf J 6 1 I. INTRODUCTION ThesimplestapproachtodescribetheFanoresonances isamechanicaltoymodelofcoupledharmonicoscillators s] Optical nanostructures exhibiting Fano resonances 3,6,7,28. In many cases,such model providessufficient in- c sight into the physics of the system; it allows one to fit have attracted a significant attention of the research i t community in recent years1–12. As classical analogues opticalspectraandassociatespectralfeaturesofthesys- p o of quantum Fano system13, these systems represent el- tem with the known properties of the mechanical Fano model. Nevertheless, such simplified representation of a . egant tabletop tools for testing fundamental principles cs of physics14. From the applications’ point of view, the complex physicalsystemneglects variouseffects relevant for real structures. Among these are substrate effects, i possibility to trap, enhance, and manipulate light in op- s effects ofperiodicity resulting in a collective characterof y ticalFano nanostructuresandmetamaterialsis also very the electromagnetic response of meta-materials, geome- h promisingandhasalreadybeendemonstratedtobeben- p eficial in sensing and bio-sensing15–18, photovoltaics and try/proximityeffects modifyingthe interactionsbetween [ thermo-photovoltaics19–21, and slow-light generation3. meta-molecules. 1 Fano resonances in optical metamaterials originate Anotheranalyticaldescriptionapplicabletosomerela- v from electromagnetic interactions between their con- tivelysimplesystems,suchasarraysofpoint-likedipoles, 6 stituents. According to the type of interaction, Fano is offered by a dipole model5,29–32. Indeed, a few of the 4 resonances can be classified into two groups: (a) coher- above-mentionedphenomena, including collective effects 1 ent Fano resonanceswhich are the resultof interferences andFanoresonances,canbeadequatelydescribedbythis 3 between all of the meta-atoms forming a periodic ar- model33. However, there are situations when the dipole . 1 ray, and (b) local Fano resonances which originate from model is inadequate. For instance, as soon as the scat- 0 the complex local structure of individual meta-atoms. terers or the metamolecules forming the array become 2 While for the first class of these resonances, periodic ar- comparableinsizetothe wavelengthoflight,the dipole- 1 rangement of a large number of meta-molecules is cru- dipole interaction mechanism breaks down because in- : v cial, for the second class, even a single meta-molecule teractions through higher multipoles come into play. In- Xi may exhibit a Fano resonance2. However, in both cases corporating substrate effects also represents a challenge the response of the meta-surface will depend on the di- for the dipole model. To resolve these problems, the r a electric environment. For coherent metamaterials, it is dipole model has been extended by including effects of well-knownthat the presence of a substrate modifies the higher multipoles34, found with the use of numerically far-field interactions22,23, which substantially alters the calculated polarizabilities of metamolecules and by ap- collective response. However, even in the case of lo- plyingthescattering-matrixtechnique35–37. However,af- cal Fano resonances, the presence of a substrate can ter such generalizations, the dipole model can no longer significantly modify the interaction between the meta- be considered analytical. To fully account for all the molecule’s constituents24. Thus, understanding of sub- above-mentioned effects, the approach which is being strate effects in both cases is critical for basic under- widely used is to apply a powerful yet time consum- standing of Fano resonances in experimentally relevant ing ab-initio numericalsolversof the Maxwellequations, systems. orvariousgeneralizationsofhybridnumerical/analytical ThegeometryofthesystemswithFanoresonancescan approaches38. vary from simple designs such as metal nanoparticles2 However, there is another very promising semi- and perforated films25–27 to structures with complex ge- analytical yet rigorous approach based on the modal- ometries such as oligomers and dolmens9–11,14. matchingtechnique(MMT).MMThasbeenproventobe 2 (a) modeloftheMMTbytruncatingtheelectric-currentba- sisofthefullmodelanddemonstratethateventheresult- inganalyticalmodeliscapabletoquantitativelydescribe ax Py the optical properties of the structures. The main em- ay phasis of the paper is on the effect of the substrate. It is demonstrated that with the model in hand the MMT can predict the spectral positions of the resonances and describe the interference among the different scattering pathways provided by the different metamolecule’s resonances. The rest of the paper is organized as follows. In Sec- (b) tionIIageneralMMTbasedontheRayleighandcurrent expansions is derived. In Section III a minimal model is developedandappliedtostudyaperiodicsingle-antenna Dx metasurface (SAM), shown in Fig. 1a. In Section IV we analyze periodic double-antenna meta-surfaces (DAMs) [Fig. 1b], which are known to exhibit a Fano resonance Px due to the interference of the modes corresponding to symmetricandanti-symmetricchargedistributionsinthe antenna pairs. Finally, in Section V a dolmen structure formed by three antennas (Fig. 1c) and known to exhibit a plasmonic analogue of electromagnetically induced transparency (EIT)6,7 is studied. II. MODAL MATCHING TECHNIQUE: THEORETICAL FORMALISM FIG.1. (Coloronline.) Schematicunitcellofthethreestruc- Inthissectiontheelectriccurrentexpansiontechnique turesconsideredinthispaper: (a)single-antenna,(b)double- is generalized to the case of infinitesimally thin metallic antenna, and (c) dolmen metasurfaces. In (c), two unit cells antennas of finite surface conductivity. Following Ref.39, (separated by a dashed line) are shown. (d) shows the side werelyonthecurrentexpansion,butrequirethecurrent view of the metasurfaces cladded by the superstrate and the in the plasmonic antennas to be defined by the high- substrate. frequencyconductivity ofthe metalandthe electric field right on the antennas’ surface. To make the expressions more compact and keep the formulation more general, very fruitful for describing various electromagnetic sys- the Dirac notation is used. Using the periodicity of the tems ranging from frequency-selective surfaces and an- structuresthetangentialcomponentsofthefieldsareex- tenna arrays in radio frequency domain39 to perforated panded in the superstrate (I) and the substrate (II) in metallic structures40–43. In both cases, the approach is the plane-wave basis, relyingontheexpansionofasystem-specificpolarization (teh.eg.h,othlees)toitnaltchuerrfeonrmtinotfhaeasunpteenrpnaossiotirotnheoffiethldeseinigseidne- EIk = (in,τeikzn,Iz +rn,τe−ikzn,Iz)|n,τi, modes satisfying appropriate boundary conditions. It is Xn,τ important that, because the basis used for the expan- EIkI = tn,τeikzn,IIz|n,τi, sion takes into account meta-molecules’ geometry, this Xn,τ ainpcplurodainchg schaanpeacrceosuonntanfocerstahnedgperoomxeimtriyt-yspeeffceicfitcs4e4ff,4e5c.ts, −zˆ×HIk = YnI,τ(in,τeikzn,Iz −rn,τe−ikzn,Iz)|n,τi, Xn,τ In this paper, the semi-analytical MMT based on the (1) current expansion39 is applied to study Fano resonances in two-dimensional periodic arrays of metallic antennas −zˆ×HIkI = YnI,Iτtn,τeikzn,IIz|n,τi, on a substrate (Fig. 1). The emphasis is on arrays of Xn,τ complex antennas resonant in the mid-IR part of opti- calspectrum,whichisimportantforbio-sensing16,18 and where n,τ = nx,ny,τ represents the in-plane (tan- | i | i thermo-photovoltaic applications19–21. MMT is shown gential) electric field of the n-th diffracted plane wave to capture the meta-molecule geometry as well as the with an in-plane wave-number kn = (kn,kn), a polar- k x y substrate effects and is easily expandable to the cases of ization state τ (s or p polarization) and a wave admit- more complex geometries composed of several spatially tance in the superstrate (substrate) YnI,τ (YnI,Iτ). The extended scatterers per unit cell. We develop a minimal wave admittance of an s-polarized plane wave is Yn,s = 3 n kz/(Z0k0)andthatofap-polarizedplanewaveisYn,p = Eqs. (7a-7b) are obtained by multiplying Eqs. (6a-c) ǫ k /(Z kn). In the above expressions, kn = 2πn /P , (from left) by n,τ and Eq. (7c) is obtained by multi- kryn 0=2π0nyz/Py,andkzn =qǫrk02−(kkn)2 axrerespecxtivelxy pnly,iτngjEiqs.d(e5fi)nh(efdroam|s left) by hjmα |. The scalar product thex,y,andz componentsofthewavevectork. k isthe h | i 0 wavenumberofthewaveinvacuumandǫr isthe relative ax/2 ay/2 exp( iknx ikny) permittivity of the medium in which the wave is propa- n,s j = − x − y (8) h | i Z Z kn P P × gating. Z0 376.73Ω is the impedance of vacuum. The −ax/2 −ay/2 k x y spatial repr≈esentation of n,τ is given by: knj (x,y) knj (x,yp) dxdy, | i { y x − x y } exp(iknx+ikny) ax/2 ay/2 exp( iknx ikny) r n,s = x y knxˆ knyˆ , (2) n,p j = − x − y (9) k| kn P P { y − x } h | i Z Z kn P P × (cid:10) (cid:11) k x y −ax/2 −ay/2 k x y pn n n n p exp(ik x+ik y) k j (x,y)+k j (x,y) dxdy. r n,p = x y knxˆ+knyˆ . (3) { x x y y } k| kn P P { x y } (cid:10) (cid:11) k x y where j (x,y) and j (x,y) are the x and y components x y p of the current profile j . Eqs. (7a-7c) can be solved to The surface current in the plasmonic antennas is ex- | i pandedwiththeuseofthebasisfunctions jm satisfying obtainthe amplitude ofthe scatteredfields tn,τ and rn,τ | αi in terms of the antenna current modes cm the boundary condition of vanishing current on the an- α tenna edges. The electric current in the m-th antenna 2YI n,τ jm cm within the unit cellJtamke=s thecfmormjm (4) tn,τ = YnI,τ +n,YτnI,Iτin,τ − Pα,YmnIh,τ +Y|nIα,Iτi α , (10) wherecm istheamplitudeXoαfthαe|α-αthi currentmode. The rn,τ = YYnnII,,ττ +−YYnnII,,IIττin,τ − Pα,YmnIh,τn+,τY|jnImα,Iτicmα , (11) α preciseformofthefunctions jm dependsontheantenna | αi where the amplitude of the α-th current mode in the m- geometry,whichwillbetakenrectangularthroughoutthe th antenna cm is paper. Forthis case,the explicitexpressionsfor jm are α given elsewhere.39 | αi σ−1δ δ +Smm′ cm′ =χm (12) Thefiniteconductivityoftheantennasisincorporated { α,α′ m,m′ αα′} α′ α αX′,m′ into the model using the relation In Eq. (12), Smm′ is the Green’s function that describes Jm =σEm (5) αα′ k the cross-talk between the α-th current mode of the m- where σ =ihǫ k is the effective surface conductivity of th antenna with the α-th currentmode of the m-th an- m 0 ′ ′ the antennas of thickness h and permittivity ǫ 46. The tenna in the unit cell. This coupling is mediated by the m fields in the substrate and the superstrate should satisfy plane waves of all (propagating and evanescent) diffrac- the standard continuity boundary condition over the in- tion orders n as given by Eq. (13a). χmα is the direct terface between them coupling strength of the α-th current mode of the m-th antenna to the external field. EI =EII, (6a) k k jm n,τ n,τ jm′ zˆ HI = zˆ HII. (6b) Smm′ = h α | i | α′ (13a) − × k − × k αα′ Xn,τ YnI,τ +(cid:10)YnI,Iτ (cid:11) Over the antennas, the electric field is continuous and 2YI jm n,τ Eneqt.ic(6fiae)ldstailplpheoalrdss,duheowtoevtehreafidniistceocnutrinreunittyininththeeanmtaeng-- χmα =Xn,τ Ynn,Iτ,τh+αY|nI,Iτ iin,τ. (13b) nas In addition to the scattering characteristics, the eigen- zˆ HI HII =J. (6c) modesofthestructurecanalsobedeterminedbysolving − ×(cid:16) k− k(cid:17) the secular equation Combiningtheseconstraintstogetherwiththemodalex- pansions and using the orthonormalityof the basis func- det σ−1δ δ +Smm′ =0. (14) { α,α′ m,m′ αα′} tions, we obtain a system of three linear equations: In general, Eq. (14) can be satisfied only at complex in,τ +rn,τ =tn,τ, (7a) frequencies ω = ωr +iωi, showing that the eigenmodes YnI,τ(in,τ −rn,τ)−YnI,Iτtn,τ = hn,τ|jmα icmα, (7b) haeveσa =fini0t)e olirfertaimdieatdivuee dtoeceaiyt4h2e,4r3O. hAmircealolsesiegsen(wvahleune αX,m ℜ { } 6 ω may be ideally achieved only in the limit of lossless hjmα |n,τi(in,τ +rn,τ)=σ−1cmα. (7c) meta-surfaces and for kk > k, which ensures absence of Xn,τ absorptionandradiation. Whileinthispaperweassume 4 atwo-dimensionalperiodicarrayoftheantennas,butthe (a) 1 25 ε=12, ε =12 resultscanbegeneralizedtothecaseofanindividualan- I II ε=1, ε =12 tenna. Inthatcase,thesumoverthediscretesetofplane 0.8 I II 20u.) waves in the above expressions, should be replaced a. by an iPntnegral over the continuum R R dkxdky.47 sion0.6 εεII==112, ,ε IεII=I=1122 15de|c|( III. COLLECTIVE RESPONSE OF ansmis0.4 10mplitu SINGLE-ANTENNA METASURFACES r a T nt e r While Eqs.(10-12) fully characterizethe opticalprop- 0.2 5 ur C erties of the system, they have to be solved numerically. However, in the case when the wavelength of the inci- 0 0 6 8 10 12 dent light matches the resonant wavelength of a partic- λ (µm) ular current eigenmode jm , truncation of the current modes’ basis can provide| sαimiple and instructive expres- (b) WA0 1 sions48. Indeed, it has been shown for the case of per- at 0.9 forated metal films that near the cut-off frequency of a n o particular waveguide mode such a “minimal” model can si0.8 s provideaverygoodapproximation41,49. Aswillbeshown mi0.7 s later this situation holds for the antenna geometries in n a r the frequency range studied here. It is worthwhile men- T −10 −5 0 5 10 ε−ε tioning here, however, that as dimensions of antennas I II decrease and they become increasingly subwavelength, the convergence of the MMT deteriorates and more and FIG. 2. (Color online.) (a) Zeroth-order normal-incidence more current modes should be considered. transmission (solid lines) and amplitudes of the fundamental current mode (dashed lines) for the single-antenna metasur- Bylimiting the currentbasisto onemode perantenna face for thesymmetric (ǫI =ǫII, black lines) and asymmetric jm = 2 cos(πy)yˆ we assume a dominant role for | 1 i qay ay (ǫI 6=ǫII, red lines) claddings. (b) Zeroth-order transmission the fundamental (dipolar) antenna-mode. Such a trun- of the single-antenna metasurface at the Wood’s anomaly as cation allows us to get simple analytical expressions for a function of the dielectric contrast ǫI −ǫII(solid line) and the bare (without antennas) interface transmission (dashed the transmission and reflection amplitudes which quali- tatively explain the system’s response yet quantitatively line). The structureparameters are as follows: ax =0.3 µm, match the numerical results. ay =1.5 µm, Px =1.8 µm, Py =1.8 µm, and ǫII=12. Firstweconsiderastructurewithasingleantennaper unit cell with large length-to-width aspect ratio so that where the first term in the square bracket corresponds thefundamentalcurrentmode j j1 withthecurrent | i≡ 1 to the s-polarized diffraction orders and the second along the long antenna dimension(cid:12)(cid:12)do(cid:11)minates. In this term corresponds to the p-polarized diffraction orders. case Eq. (12) can be analytically solved for the current sinc(t) sin(t)/t can be approximated by unity for thin amplitude ≡ antennas (t=k a /2 0). We emphasize that this sum x x ≈ χ (k ,ω)E converges much faster compared to that in the dipole c(kk,ω)≡c11(kk,ω)= σ−1(1ω)+k S (kext,ω), (15) model. Intheframeworkoftheminimalmodelthetrans- 11 k mission and reflection coefficients of the single-antenna whereχ (k ,ω)=2YI /(YI +YII ) j 0,τ isthecou- meta-surface assume a simple form: 1 k 0,τ 0,τ 0,τ h | i pling efficiency of the antenna to the external field, the Esuemxt =S11i0(k,τk,aωnd) t≡heSr11e11s(tkokf,ωth)eaisn,τd’esfianreedaisnsuEmqe.d(t1o3ab)e, tn,τ = YnI,2τY+nI,YτnI,Iτin,τ − YnhI,nτ,+τ|YjniI,Iτc, (17) zpIneorltoah.reizInmatisinoeincmtiiasolnamlIoIonIdgaetlnh,deSIlVonigsweedxiampslseiucnimstiloyentghoivafettnhtehbeya:nintecnidneanst. rn,τ = YYnnII,,ττ −+YYnnII,,IIττin,τ − YnhI,nτ,+τ|YjniI,Iτc. (18) 11 S11 = 4πZ20Pa2xPay sinc2(a(x1kxn/(2a)(k1n+/πc)o2s)(2aykyn))× Ethqes.d(ir1e7c,t1t8r)anexsmpliiscsitiolyns(hroeflwetcwtioons)catthtreoruingghptahtehdwiaelyesc:tr(iic) x y Xn − y y interfacebetweenthesubstrateandthesuperstratewith- n n 2 n n 2 out interacting with the antennas and (ii) the field radi- k k /k k /k  0(cid:16) x k(cid:17) + (cid:16) y k(cid:17) , atedbytheantennasduetothe excitationofthe current kn,I+kn,II ǫ k0/kn,I+ǫ k0/kn,II mode. These two scattering pathways and the presence z z I z II z   of the Wood’s anomaly50,51 lead to a Fano interference (16) resulting in an asymmetric lineshape typically observed 5 in antenna arrays (Fig. 2a) and systems with extraordi- (a) naryopticaltransmission. Theconditionofthevanishing denominator in the Eq. (15), σ−1(ω)+S (k ,ω) = 0, 11 k approximatelydescribestheeigenmodesdispersionofthe antenna array (cf. Eq. (14) for the exact expression). 0.7 COMSOL 0.6 Minimal model Full model 0.5 n o i s0.4 s i m s Dipole n0.3 a r T (b) 0.2 0.1 0 5 6 7 8 9 10 11 12 13 14 λ (µm) FIG. 3. (Color online.) Direct transmission at normal incidence calculated using three methods: (i) first-principles COMSOL simulations (circles), (ii) full MMT model (solid line),and(iii)theminimalMMTmodel(dashedline). Struc- tureparameters: same as in Fig. 2. Note that Eq. (15) strongly resembles that of the effective polarizability of an array of dipoles in the dipole model32. In our case, m 1/σ(ω) plays the role of FIG. 4. (Color online.) Angular-resolved zeroth-order (di- ℑ { } “plasmonicpolarizability”of the antennas while the role rect) transmission through thesingle-antenna metasurface in of S11(kk,ω) is similar to that of the dipole lattice sum. (a)symmetric(ǫI=ǫII)and(b)asymmetric(ǫI =1,ǫII=12) Notethatbecausethemodal-matchingtechniqueisbased cladding. Theparameters ofthestructurearethesameasin on current expansion, the polarizability and the lattice Figs. 2 and 3. sum have π/2 phase shift as compared to the dipole model. It can be shown that when the dipole sum of the dipole model is written in the reciprocal space, the merical error due to spurious singularities at sharp cor- sums of both models have similar terms diverging at the ners. Wood’s anomalies. However, in contrast to the dipole model, the antenna model fully considers the substrate, antennashape andfinite conductivity. Herewe limitour A. Substrate effects on Wood’s anomalies consideration to the mid-IR domain where antennas can be considered as perfectly conducting (1/σ=0). Effect Thesubstrateeffectappearsduetothefinitedielectric of the finite conductivity of the antennas on Fano reso- contrast ∆ǫ = ǫ ǫ between the superstrate and sub- I II − nances will be considered elsewhere. strate claddings, and in the MMT it is described by the Figure 3 compares the results obtained with the use wave admittances (Yn,τ) entering the expression for the of the minimal model, the full MMT with a sufficient sum S [Eq. (13a)]. Predictions of the minimal model 11 number of current modes (such that the convergence is andtheexactcalculationsclearlyshowthatthepresence reached), and the full-wave COMSOL Multiphysics sim- of a substrate strongly affects the scattering character- ulations. The minimal model shows very good agree- istics of the antenna array (Fig. 2). This effect can be ment with both of the exact techniques. The inclusion fully understood from the analytical expressions of the of the higher-order modes only insignificantly changes minimal model. Let’s first consider the case of a Wood’s the spectra. Some discrepancy between the full MMT anomaly that indicates the onset of a diffraction order andCOMSOLresults is observedinthe longwavelength and appears in the transmission spectrum as a maxi- rangewhichis attributedto the rounding ofthe antenna mum (for symmetric cladding, ǫ = ǫ ) or as a kink (in I II corners that was done in COMSOL to minimize the nu- the general case) and represents the sharpest feature of 6 (a) 1 the spectrum. Finite optical contrast between the substrate and the superstrate makes the sum S converge 11 at the onset of the s-polarized diffraction orders which 0.8 manifests as the suppression of the Wood’s anomalies. Indeed, as can be seen from the expression for n S (k ,ω), in the case of a symmetric cladding (ǫ =ǫ ) o 11 k I II i0.6 s there is a divergent term corresponding to the onset of s i m a s-polarized diffraction order n,s ; when the diffraction order experiences a transit|ion bietween the evanes- ans0.4 n r claetntticaensdumproSpag(kat,inωg)rdeigviemrgeessYans,s1∼/kknz. Tvahnisisghievsesanridsethtoe T ε=1,ε =12 11 k z I II theWood’sanomalyandvanishingofthecurrentmodein 0.2 ε=ε =ε =6.5 I II eff theantennasc(k ,ω) 0,asillustratedbyblackdashed k → n=n =n =2.32 curveinFig.2a. Thus,attheWood’sanomaly,antennas I II eff are inactive and invisible (t0,τ = 1 and r0,τ = 0) as can 0 4 6 8 10 12 be seen from Eqs. (17-18). λ(μm) In the case of an asymmetric cladding, ǫ = ǫ , the (b) I II 6 same term of the lattice sum that was divergent for the 0 symmetric case, assumes the form 1/(kn,I +kn,II) and, z z since kn,I = kn,II, the lattice sum S (k ,ω) never di- −0.02 z 6 z 11 k verges. AsshowninFig.2abyreddashedcurve,evenat 12 −0.04 theWood’sanomaly,thecurrentinthewiresc(kk,ωWA), m) datoieosn.not vanish and the antenna arrayscatters the radi- /λ−0.06 (µλr 10 λ The fact that antennas remain polarized and con- ∆−0.08 8 tributetoscatteringattheWood’sanomalyforanyfinite −10 −5 0 5 10 −0.1 value of the optical contrast∆ǫ is illustrated by Fig. 2b, ε−ε I II where the transmission at the wavelengthcorresponding −0.12 n ε toWood’sanomalyoftheantennaarrayonthesubstrate eff eff (solidline) isplottedalongwiththatofthe bare(i.e., no −0.14 antennas) interface (dashed line). The two curves in- −10 −5 0 5 10 tersect only for the case of a symmetric cladding and ε −ε I II the discrepancy can be significant for a large contrast. Thus, the peak transmissionofthe antenna arrayon the FIG. 5. (Color online.) (a) Comparison between the exact substrate never reaches that of the bare interface. It is treatmentofthesubstrateandtheresultsofthehomogenized also remarkable that the transmissionof the antenna ar- antenna’senvironmentaccordingtotheeffectivemediumthe- ray appears very asymmetric with respect to the sign of ories: (i) ǫeff = (ǫI + ǫII)/2 (dashed-dotted line) and (ii) the contrastandthedifferencecanreachtensofpercent. neff = (nI+nII)/2 (dashed line). (b) Deviation of the res- This asymmetryiscausedbythe presenceofthe Wood’s onance spectral position (∆λ/λr) predicted by the effective anomaly of the second medium whose diffraction order medium approaches from the exact result λr shown in the dominates in the regime of ǫ >ǫ . inset. I II B. Substrate’s influence on collective antenna tranearthe resonanceinFig. 3indicates thatourmodel resonances fully accounts for the modified polarizability of the antennas caused by the presence of the substrate. The spectral position of the collective dipolar antenna The presence of the substrate and suppression of the resonance corresponds to the minimum of the transmis- Wood’sanomaliesdramaticallyaffecttheangulardisper- sion or the maximum of the reflection spectra. It can sion of the resonance and change its nature. Figure 4(a) also be found from the zeros of the imaginary part of shows that for the case of a symmetric cladding we ob- the denominator in Eq. 15. In this paper, the length of servetheeffectof“dragging”ofthe modebytheWood’s the antennas are such that the observed collective mode anomaly32,33, which results in its strong spatial disper- stems from the λ/2 resonance of an individual antenna sion. This strongly dispersive character of the mode is (as can be seen from the charge distribution plotted in a manifestation of its collective origin. Indeed, the long- the inset to Fig. 3). As can be seen from dashed curves rangeinteractionamongtheindividualmetamoleculesre- in Fig. 2a, it also corresponds to the maximum of the sultsintheappearanceofthe collectivemodeswhichare currentamplitude cexcitedintheantennas. Notethata spectrally displaced and sharper as compared to those good agreement between the MMT and COMSOL spec- observed in the individual metamolecules32,33. However, 7 for the case of an asymmetric cladding, shown in the COMSOL Fig. 4b, the dragging effect is diminished and the reso- Minimal model nancecrossestheWood’sanomalies,everytimereducing 0.5 Full model its quality factor due to the opening of additional ra- Dipole diative channels. The resulting non-dispersive behavior n0.4 andlowerqualityfactorofthemodeimplythatthereso- o i nancehaslostits collectivecharacterandrepresentlocal ss i excitationmodifiedbythecomplexelectromagneticenvi- m0.3 s ronment. Notethatfortheconfinedmodeoftheperiodic n a Quadrupole metasurfaces, i.e. non-leaky spoof plasmons, presence of r T0.2 thesubstrateandassociatedsuppressionofthecollective behavior may result in the disappearance of the mode36. 0.1 By followingthe resonanceonecanalsotrackits spectral position as a function of the dielectric contrast ∆ǫ. Inset to Fig. 5b shows the corresponding spectral shift 0 5 6 7 8 9 10 11 12 13 14 of the resonance. The frequency of the resonance shows λ (µm) nearly linear dependence on the contrast. We emphasize that the effect of the substrate on the antenna polariz- FIG. 6. (Color online.) Comparison of the zeroth-order ability cannot be rigorously described by a basic dipole transmission spectra of a double-antennametasurface on the model, but can be phenomenologicallyincorporatedinto ◦ substrate for 50 angle of s-polarized incidence calculated it. One of the most common approaches to include the withtheuseofCOMSOLMultiphysics,andtheminimaland opticalcontrastistouseaneffectivemediumapproach52 full MMT models. The structure parameters are as follows: and approximate the permittivity or the refractive in- ax =0.3µm,ay =1.5µm,Px =Py =1.8µm,Dx =0.6µm, dex of the background medium by ǫeff =(ǫI+ǫII)/2 or ǫI=1, and ǫII=12. n = (n +n )/2. The validity of such approxima- eff I II tionscanbetestedbycomparingthe spectralpositionof the resonancecalculatedforthe antennasembeddedinto Using the unitary transformation the effective medium with the exact MMT result. Fig- ure5ashowsthedeviationsoftheresonancefrequencyas c =c exp(ik D /2) c exp( ik D /2), sub 1 x x 2 x x − − predicted by the effective medium theories from the ex- c =c exp(ik D /2)+c exp( ik D /2), (20) sup 1 x x 2 x x act result. One can see that the agreement between the − effective medium approaches and the exact calculations the basisofthe twofundamentalelectriccurrentscanbe is rather good. The homogenization of the permittivity changed to the more instructive basis of the sub-radiant ǫ = (ǫ +ǫ )/2 is closer to the exact result and pro- eff I II and super-radiant current modes. In the new basis, the videsagoodestimateevenforthecaseofalargecontrast. system is described by the matrix equation However, it can be seen from Fig. 5a that the effective mediumapproachfailsinpredictingthetransmissionand S +∆ iκ c 2 reflection, especially close to the Wood’s anomaly. (cid:20) 11iκ S11 ∆(cid:21)(cid:20)cssuupb(cid:21)=χEext(cid:20)0(cid:21), (21) − − where ∆= 1/2[S exp(ik D )+S exp( ik D )] and 12 x x 21 x x − − κ=i/2[S exp(ik D ) S exp( ik D )]. Thesemodes IV. FANO RESONANCE IN 12 x x − 21 − x x are the result of the radiative coupling of the dipolar DOUBLE-ANTENNA META-SURFACES resonances of the antenna pairs3,7,33. The super-radiant andsub-radiantmodeshavedistinctsymmetricandanti- Next, the MMT for the double-antenna meta-surface symmetricchargedistributions,while thecurrentscorre- (DAM) shown in Fig. 1b is considered. In this case, the spondingtothemarecollinearandanti-collinear,respec- truncationofthebasistothefundamentalmodesinboth tively, as illustrated by the inset to Fig. 6. The modes antennas, results in a minimal model with a 2x2 matrix interaction makes them blue-shifted (super-radiant) and equation for the amplitudes c1 and c2: red-shifted (sub-radiant) with respect to the electric- dipolar resonance of SAM. Another consequence of such S11 S12 c1 =χE e−ikxDx/2 , (19) hybridization and different symmetry of the modes is (cid:20)S21 S11(cid:21)(cid:20)c2(cid:21) ext(cid:20)e+ikxDx/2(cid:21) their different radiative coupling efficiency. The sub- radiantmodeisnotdirectlycoupledtotheincidentlight. where S Smm′ represents the effective Green’s The latter fact is reflected in the mode’s name and di- functionmgmiv′en≡by E11q. (13a), and χ=2Y0I,τ/(Y0I,τ +Y0I,Iτ) rectly follows from the zero radiative coupling strength on the RHS of Eq. (21). exp(ik D /2) j(1) 0,τ is the coupling strength of the x x WiththeuseofEqs.(10,11)weobtainasetofminimal- D | E incident light( 0,τ ) to the fundamental currentmodes. modelexpressionsfor the transmissionandreflectionco- | i 8 (a) 1 0.05 sym asym 0.8 0 n 14 o i0.6 −0.05 iss /λ m) 12 m λ µ ns ∆ λ(r10 a0.4 −0.1 Tr ε , q mode 8 eff n , q mode −10 0 10 0.2 −0.15 eff ε−ε ε , d mode I II eff n , d mode eff 0 −0.2 7 8 9 10 11 12 13 14 −10 −5 0 5 10 λ (µm) ε −ε I II (b) 30 |c | FIG. 8. (Color online.) Deviation of the spectral position sub,sym 25 |csup,sym| (∆λ/λr) of the dipolar (red lines with circle markers) and |c | quadrupolar(bluelineswithtrianglemarkers)resonancespre- sub,asym |c | dictedbytheeffectivemediumapproachesfrom theexactre- sup,asym 20 sults λr shown in the inset. ) u. a. 15 ( transformation:33 c| | ∆+X iκ 10 d= c + c , sup sub 2X 2X iκ ∆+X 5 q = c + c , (24) sup sub 2X 2X 0 where X = √∆2+κ2. The modal coefficients d and q 7 8 9 10 11 12 13 14 are related to the incident field through λ (µm) 1+∆/X FIG.7. (Coloronline.) (a)Zeroth-orders-polarizedtransmis- d= χEext, Sd =S11+X, S sion spectra of thedouble-antennameta-surface for thesym- d iκ/X metric (ǫI=ǫII=12, blue line) and asymmetric (ǫI=1, ǫII=12, q = χE , S =S X, (25) redline)claddings. (b)Amplitudeofthesub-radiant(dashed S ext q 11− q lines) and super-radiant (solid lines) current modes. The pa- rametersofthestructureandtheincidenceanglearethesame In the particular case of normal incidence, the as in Fig. 6. quadrupolar mode q exactly coincides with the sub- radiantmode c since its coupling to the super-radiant sub mode c vanishes (κ 0 as k 0). However, sup k → → at finite angles of incidence, this mode acquires a fi- efficients: nite electric-dipolar moment and its radiative coupling and bandwidth gradually increase. The dipolar mode, tn,τ = YnI,2τY+nI,YτnI,Iτin,τ − Dn,τ|j(1Y)nEI,τex+pY(−nI,IiτkxDx/2)csup, iins scpoencttrraasltly, ibsroaalwdaaytsasntyroinngcliydernacdeiaatnivgelley. cEoqusp.l(e2d2-a2n3d) describe a Fano resonanceof a single ”continuum” inter- (22) acting with two ”discrete” dipolar (d) and quadrupolar rn,τ = YYnnII,,ττ +−YYnnII,,IIττin,τ − Dn,τ|j(1Y)nEI,τex+pY(−nI,IiτkxDx/2)csup. (isq)vTerhreeysodtnriaffannecrseemns.itssHdiuoonewtesovpeterhc,ettrihruemdibffaoenfrdeDnwtAidrMtahdifoaoftritvhaeencolaousbptlliitqnwugoe. (23) incidence of 50◦ calculated using the minimal model, full MMT and COMSOL solver are plotted alongside in Note that because of their coupling, c and c Fig.6. Asbefore,thedifferenttechniquesshowverygood sup sub are not the true eigenmodes of the system. The agreement,which implies the applicability of the analyt- true eigenmodes which will be referred to as dipo- icalminimalmodel evenfor the descriptionoflightscat- lar d and quadrupolar q can be obtained using the teringbymorecomplexmetasurfacessuchasDAM.Two 9 resonances,abroadoneduetotheexcitationofthedipo- shownthat the transmissionpeak corresponds to the di- lar mode and a narrow one due to the excitation of the vergence of c and zero of c (Fig. 7b). Because only sub sup quadrupolarmode,areclearlyseeninthespectra. Asex- the super-radiant mode (c ) is coupled to the radia- sup pected, the narrow resonance has a strongly asymmetric tion, the antennas do not radiate at this frequency and shape typical for Fano resonances. the transmission acquires a universal value of the transmission of the bare interface between the substrate and the superstrateasseenfromEqs.(22,23). Note thatthis A. Dipolar mode behavior is different from that found for another type of Fanoresonances,Wood’sanomaliesinSAM,whenforan Because the dipolar and quadrupolar resonances are asymmetriccladding configuration,the radiativecurrent spectrally separated and the quadrupolar resonance is mode always had some finite amplitude. spectrally narrow,first we can consider the dipolar reso- In the previous section, it was demonstrated that the nance individually. It has been recently shown that the effectivemediumapproachaccuratelypredictedthespec- collectivedipolarresonanceinDAMappearstohavespa- tral position of the dipolar resonance for SAM. Here tial dispersion [ω(k )] very different from that found in we test this approach for both cases of the dipolar and k SAM 33. Using the basic dipole model, this difference quadrupolar resonances in DAM. At first, the inset to Fig. 8 shows the spectral position of the resonances cal- was shown to be due to the suppression of the long- culated by the full MMT approach as a function of range interactions in DAMs. MMT provides analogous the dielectric contrast, and Fig. 8 shows the deviation results. In the present case, however, the role of the from the exact result. The effective theories match well substrate and the antenna geometry are fully taken into with the exact results for both quadrupolar and dipo- account. In DAM, the Wood’s anomalies would have lar resonances, while the permittivity homogenization correspondedto the divergences(or local maxima for an ǫ = (ǫ +ǫ )/2 is more accurate than the index ho- asymmetric cladding) of the effective Green’s function eff I II mogenization n = (n + n )/2. However, just as in S = S + X playing the role of the lattice sum in eff I II d 11 thecaseofSAM,noneofthehomogenizationapproaches the dipole model or the Green’s function S of SAM. 11 succeedinpredictingthetransmission/reflectionspectra. However, it can be shown that exactly at the frequency where the Wood’s anomaly is expected, the diverging Finally, we observe that in contrast to the dipolar terms in S and X exactly cancel out eliminating the mode,thequadrupolarmodeinDAMisstronglyaffected 11 divergence, thereby suppressing the spectral features as- by the Wood’s anomaly. Fromthe expressionfor its am- sociatedwith the Wood’s anomaly33. Thus, inthe DAM plitude Eq. (25) one can see that for the quadrupolar case,Wood’s anomaly appears to be suppressedevenfor modethedivergenceintheGreen’sfunctionS11 X does − a symmetric cladding (∆ǫ=0). Therefore for the asym- take place in the case of a symmetric cladding ǫI = ǫII. metric cladding, the Wood’s anomaly is suppressed by The effect of the Wood’s anomaly on the quadrupolar two mechanisms: one due to substrate/superstrate con- resonanceoftheDAMisexpectedtobeespeciallysignif- trast just as in the case of SAM, and another one due icant for large angles of incidence, when they approach to disconnected topology of DAM. Note, however, that each other. This results in a strong spatial dispersion despite such suppression, one might still observe some of the quadrupolar mode observed earlier both in mid- variations of reflectivity in the vicinity of the expected and near-IR plasmonic DAMs 33. For the case of an Wood’sanomalies,whichisaresultoftheopeningofthe asymmetric cladding the effect of the Wood’s anomaly diffraction channel and change of the mode’s radiative onthe quadrupolarresonanceis againsuppresseddue to lifetime.33. the mismatch of the wave admittances in the substrate andthesuperstrateandalltheargumentsaboutthesup- pression of the Wood’s anomalies used in the previous B. Quadrupolar mode section for SAM are applicable here. Nowwefocusonthe quadrupolarresonance. The narrow spectral region where the transmission undergoes a V. ELECTROMAGNETICALLY INDUCED TRANSPARENCY rapid variation is especially interesting because of the strong field enhancement and slow light regime reported earlier3,4. The asymmetric shape of the spectrum is the In this section a periodic dolmen structure with three result of the Fano interference between the quadrupolar antennas in its unit cell6,53,7,9,3 [shown in Fig. 1c] is resonance and a background provided by two transmis- studied. This design was first introduced to mimic EIT sion channels: (i) the direct transmission without inter- inplasmonicstructuresfornormallyincidentx-polarized actionwith the array,and(ii) the radiationscattered by light6. Two vertical antennas on the dolmen structure the dipolar resonance. This background is nearly con- are responsible for formation of the quadrupolar mode stant and is featureless over the spectral width of the analogoustothatdescribedintheprevioussection. This quadrupolar resonance where the transmission changes modeplaystheroleofthe sub-radiantmode ofEIT.The from its maximal value to zero. It can be analytically horizontal antenna provides a spectrally broad dipolar 10 0.8 response. Its length is chosen in such a way that the D =0.0 P/2 frequency of the dipolar resonance is matched to the y frequency of the sub-radiant (quadrupolar) mode. The 0.7 Dy=0.1 P/2 position of the horizontal antenna within the unit cell D =0.2 P/2 0.6 y defines the degree of symmetry breaking of the meta- D =0.3 P/2 y molecule and the intensity of coupling between the two n0.5 modes. The sub-radiant mode is completely decou- o si pled from x-polarized incident radiation if the antenna mis0.4 is placed equidistantly between the two nearest vertical ns a double antennas, Dy = 0. However, as soon as the hor- Tr0.3 izontal antenna is displaced vertically from this position (Dy = 0), the sub-radiant mode couples to the incident 0.2 6 radiationindirectlythroughthedipolarmodeofthehor- izontal antenna. This coupling to the sub-radiant mode 0.1 results in a transmission/reflection spectrum typical for the EIT: a sharp transmission resonance embedded into 0 14 14.5 15 15.5 16 a region of highly reflecting background provided by the λ(µm) dipolar antenna mode. Note that for the polarization understudy the dipolarmode oftwo-verticalantennasis FIG. 9. (Color online.) Substrate effect on the normal- notexcitedsincethere is noy-componentofthe incident incidence zeroth-order transmission spectra of the dolmen electricfield. Inprinciple,this mode couldbe excitedin- metasurfacefordifferentvaluesofsymmetry-breakingparam- directlyprovidedthatthesymmetryofthemetamolecule eter Dy. Horizontal antenna’s dimensions are ax = 1.79µm, is reduced further by displacing the horizontal antenna ay =0.6µm. Verticalantennasareidenticalwithax =0.6µm in the x-direction3. However, this situation will not be and ay = 1.7µm and separated by Dx = 1.2µm. ǫI = 1, considered here. ǫII=12, and Px=Py=4µm. Within the minimal model, limiting the current basis to only fundamental current mode for each antenna thehorizontalantennaisplacedsymmetrically(inwhich and neglecting finite conductivity of the metallic anten- case κ=0), the modes D and Q are not eigenmodes of nas (1/σ = 0), the EIT structure can be described by a x the EIT structure, since they are coupled to each other 3x3 matrix equation relating the current amplitudes to through the off-diagonal matrix elements in the LHS of the incident electric field: Eq.(27). This coupling gives rise to hybridizationof the S11 S12 S13 c1 0 sub-radiantmodeQandthesuper-radiantmodeDx and S12 S11 S13c2=χEext0, (26) formation of new mixed states – quasi-dipolar D˜x and S S −S c 1 quasi-quadrupolar Q˜. Both of these modes are coupled 13 13 33 3  −     to the incident electromagnetic field, but have very dis- where S Smm′ represents the effective Green’s parate coupling efficiency. The true eigenmodes of the functionmgmiv′e≡n by11Eq. (13a), and χ =< j 0 > structure can be found through an eigen-decomposition 3 2Y0I,τ/(Y0I,τ +Y0I,Iτ) is the coupling strength betwee|n the procedure, D˜x = 1/(1+η2)Dx+η/(1+η2)Q and Q˜ = incidentlightandthehorizontalantenna(labeledas“3”). 1/(1+η2)Q−η/(1+η2)Dx whereη =(SQ−SDx−∆)/2κ TheRHSofEq.(26)reflectsthefactthattheverticalan- [that tends to zero in the case of a vanishing coupling tennas are not coupled to the x-polarized incident light. κ 0] and ∆ = (S S )2+4κ2. Then the equa- → Q− Dx On the other hand, the presence of off-diagonal matrix tions for the eigenpmodes’ amplitudes assume the form elements in the LHS of this equation indicates that they 2/(1+η2) are coupled to the horizontal antenna. It follows from D˜ = χE , (28) x ext the symmetry of the structure that the basis of three SQ+SDx −∆ currents is redundant for the case of x-polarized normal 2η/(1+η2) incidence, under which the dipolar mode of the antenna Q˜ = − χEext, (29) S +S +∆ pair D cannot be excited. By performing the unitary Q Dx y transformation Dy = c1+c2, Q = c1 c2 and Dx = c3, and the reflection/transmission coefficients of the struc- − the non-interacting Dy mode can be eliminated. The ture can be found from the following equations: truncatedgoverningequationforthesuper-radiantmode Dx and the suSb-radiκant mQode Q assume0s the form: rn,τ = YYnnII,,ττ +−YYnnII,,IIττ + YhnIn,τ,+τ|Yj3nIi,Iτ Dx, (30) Q =χE , (27) (cid:20) κ SDx(cid:21)(cid:20)Dx(cid:21) ext(cid:20)1(cid:21) tn,τ = YnI,2τY+nI,YτnI,Iτ + YhnIn,τ,+τ|Yj3nIi,Iτ Dx, (31) whereS =1/2(S S ), S =S ,andthecoupling Q 11− 12 Dx 33 D =D˜ ηQ˜. (32) betweenthemodesisgivenbyκ=S . Note,thatunless x x 13 −