Plasmonic lattice Kerker effect in UV-Vis spectral range
V.S. Gerasimov, A.E. Ershov, R.G. Bikbaev, I.L. Rasskazov, I.L. Isaev, P.N. Semina, A.S. Kostyukov, V.I. Zakomirnyi, S.P. Polyutov, S.V. Karpov
LLattice Kerker Effect Goes Plasmonic
V.S. Gerasimov,
1, 2, ∗ A.E. Ershov,
1, 2
R.G. Bikbaev,
1, 3
I.L. Rasskazov, I.L. Isaev, P.N. Semina, A.S. Kostyukov, V.I. Zakomirnyi,
1, 2
S.P. Polyutov, † and S.V. Karpov
1, 3 Siberian Federal University, Krasnoyarsk, 660041, Russia Institute of Computational Modeling SB RAS, Krasnoyarsk, 660036, Russia L. V. Kirensky Institute of Physics, Federal Research Center KSC SB RAS, Krasnoyarsk, 660036, Russia The Institute of Optics, University of Rochester, Rochester, New York 14627, United States (Dated: August 28, 2020)The Kerker effect, generally requiring co-existence of electric and magnetic responses, is difficultto observe in conventional non-magnetic plasmonic nanostructures. We found that delicate interplaybetween localized surface plasmon resonances and collective lattice oscillations in arrays of plasmonicAl nanostructures leads to the emergence of lattice Kerker effect. Variation of Al arrays geometryallows to tailor suppressed reflection throughout UV and visible wavelength ranges, which is hardto achieve on using other plasmonics or all-dielectric materials.
The concept of backscattering suppression of light bya single spherical particle has been proposed over threedecades ago by Kerker et al. [1] The essential requirementfor this effect is the equivalence of permittivity and per-meability of a sphere, ε = µ , which implies a co-existenceof pronounced electric and magnetic responses at thesame frequency. This exciting idea did not receive a lot ofattention in the past because such materials are close toimpossible to find in nature. However, the situation haschanged drastically with the emergence of all-dielectricnanophotonics [2]. Optically induced magnetic momentsmediate so-called “artificial magnetism” [3], which sur-passes the requirement for the material to exhibit con-ventional magnetic properties. Experimental evidenceof the Kerker effect has been provided for single high-index Si [4] or GaAs [5] nanoparticles (NPs). After thesepioneering experimental observations, the Kerker effecthas been demonstrated in numerous setups [6–16] with apromising applications in a variety of endeavors such assensing [17], imaging [18] and others [19]. These advancesof all-dielectric nanophotonics left plasmonics much in ashadow due to inherently weak magnetic (either artificialor natural) response of metal NPs, which together withlosses [20] immediately make the plasmonic NPs unsuit-able for the Kerker effect. Nonetheless, different combi-nations of plasmonic materials with all-dielectric [21, 22]or magnetic [23] structures may satisfy the Kerker con-dition.On a larger scale, i.e., in arrays of NPs, the Kerker ef-fect can be implemented via collective lattice resonances(CLRs) [24, 25]. CLRs are high-quality modes originat-ing from the coupling between Rayleigh anomaly andMie resonance, which emerge in arrays of both plas-monic [26–29] and all-dielectric [30, 31] NPs. By tailoringthe configuration of the array and geometry of the indi-vidual constituents, the lattice Kerker effect can be ob-served [32–35]. Lattice Kerker effect originates from the ∗ [email protected] † [email protected] interaction between lattice modes (i.e., CLRs) and reso-nances in a single NP, while conventional Kerker effect isbased on resonances in a single NP. We emphasize, thatthe lattice Kerker effect has been reported only for all-dielectric nanostructures, with strong magnetic dipole (orquadrupole) resonances. Rapidly developing aluminumplasmonics [36–38] provides a solid ground for CLRs [39–43], but most of the studies are traditionally limitedto purely electric interactions, either dipole or dipole-quadrupole [39]. In our recent work [44], we have shownthat plasmonic arrays of Al NPs, very much similar toall-dielectric nanostructures, support magnetic dipole orquadrupole CLRs. We take the advantage of this prop-erty and for the first time demonstrate the Kerker effectin plasmonic Al metasurface. It is worth to notice that Alis the only plasmonic material which manifests localizedsurface plasmon resonance in important UV wavelengthregion.The lattice Kerker effect in regular arrays of NPs canbe understood as follows. Consider a plane wave withfrequency ω and a wavevector | k | = k = √ ε h ω/c nor-mally incident on a regular infinite array of NPs embed-ded in a homogeneous medium with permittivity ε h (seethe sketch in Figure 1). In this case, each NP becomesindistinguishable from each other in terms of induced (cid:96) -th multipoles. We shall limit the discussion to (cid:96) = 1 and (cid:96) = 2 fundamental modes, i.e., electric dipole (ED), d ,magnetic dipole (MD), m , electric quadrupole (EQ), D ,and magnetic quadrupole (MQ), M . Higher-order (cid:96) ≥ E = − ω c r (cid:18) [[ d × n ] × n ] − [ m × n ] − iω c ([[ D × n ] × n ] − [ M × n ]) (cid:19) e ikr . (1)Here r is a distance to the observation point and n is the unit vector pointing to the observation point.Cartesian components of EQ and MQ are D a = D ab n b , M a = M ab n b (summation over repeating indices is im- a r X i v : . [ phy s i c s . op ti c s ] A ug FIG. 1. (a),(f) Reflectance spectra, and (b)–(e), (g)–(j) multipole (ED, MQ, MD, EQ) decomposition of extinction efficiencyfor arrays with fixed h x = 240 nm and different h y (top), and with fixed h y = 280 nm and different h x (bottom). Al NPs withradius R = 60 nm have been considered in all cases. Notice a suppression of reflection which follows [0;1] RA, λ RA = √ ε h h y (seeFigure 2a for details). The order of the multipoles appearance is chosen for the sake of consistency with eq. (5). CommercialFinite-Difference Time-Domain (FDTD) package [45] is used for calculations. Standard numerical protocols are implementedto mimic infinite 2D periodic structures [40, 44, 46, 47]. Multipole decomposition of the extinction efficiency is calculated fromthe spatial electromagnetic field distribution as described in ref 48. Tabulated values of Al permittivity from ref 49 are used insimulations. plied), where a, b = x, y, z . In a particular case of in-cident electric field polarized along Y axis, d x , m y , D xz and M yz are all equal zero. Thus, after introducing short-hands d = d y , m = m x , D = D yz , and M = M xz , thereflected field is E ref = ω c r (cid:20) d − m + iω c ( − D + M ) (cid:21) e ikr . (2)Dipole and quadrupole moments from the equation abovecan be found as: d = ˜ α d E inc , m = ˜ α m H inc , D =˜ α D ∇ z E inc , M = ˜ α M ∇ z H inc , where ˜ α are the respec-tive effective polarizabilities which depend on the geom-etry of the array. Effective polarizabilities also capturecross-interactions between dipoles and quadrupoles [50].For a planewave considered here, H inc = ε h E inc and ∇ z E inc = ikE inc , thus eq. (2) can be rephrased as E ref = ω c r (cid:20) ˜ α d − ˜ α m + kω c ( − ˜ α D + ˜ α M ) (cid:21) E e ikr . (3)It can be seen from eq. (3) that the reflection is sup-pressed if the expression in square brackets is zero. Weemphasize that each polarizability ˜ α in the equationabove depends on the properties of the individual con-stituent and on the lattice geometry. Thus, by appro-priately tailoring both NPs properties and their arrange-ment, the lattice Kerker effect may emerge.We demonstrate this effect in regular arrays of Al NPsembedded in a homogeneous environment with ε h = 2 .
25. NPs are arranged in 2D lattice with periods h x and h y .Narrowband suppression of the reflection can be observedin Figure 1a,f. We elaborate on this observation by plot-ting the contributions of each (cid:96) -th mode (electric andmagnetic) to the reflection in Figure 1b–e,g–j. To doso, we consider the extinction efficiency which is propor-tional only to the real part of the expression in squarebrackets in eq. (3). We justify this choice as follows. Onusing well-known expressions for polarizabilities [50]˜ α d = i ε h k ˜ a , ˜ α M = i ε / h k ˜ b , ˜ α m = i ε h k ˜ b , ˜ α D = i ε / h k ˜ a , (4)where ˜ a (cid:96) and ˜ b (cid:96) are expansion coefficients which takeinto account the interaction between NPs (not to be con-fused with the expansion coefficients for a single NP),and recalling that extinction efficiencies for electric andmagnetic (cid:96) -th mode are proportional to real parts ofthe expansion coefficients, Q e ext; (cid:96) ∝ (2 (cid:96) + 1) (cid:60) (˜ a (cid:96) ) and Q m ext; (cid:96) ∝ (2 (cid:96) + 1) (cid:60) (˜ b (cid:96) ), the Kerker condition can be re-formulated as: Q e ext;1 + Q m ext;2 − Q m ext;1 − Q e ext;2 = 0 . (5)Keeping in mind that both real and imaginary parts ofthe expression in square brackets in eq. (3) are antici-pated to be zero for a case of suppressed backscattering,we shall limit the discussion only to a real part capturedby the extinction efficiency, eq. (5).Figures 1b–e,g–j show that spectral properties of 2Darrays are tailored by varying one of the periods whilekeeping the other one constant [31, 51]. In particular,the variation of h x keeping h y = const yields in controlof [1; 0] Rayleigh anomaly (RA) λ RA = √ ε h h x and ad-justs the position of ED and MQ modes. The variationof h y and h x = const adjusts [0; 1] RA: λ RA = √ ε h h y which controls the MD and EQ modes. Thus, Fig-ure 1b–e clearly shows that ED and MQ resonances arecoupled to [1; 0] RA, while MD and EQ are coupled to[0; 1] RA. Noteworthy, cross-interaction between differentmodes [50, 52] results in the emergence of additional min-ima and maxima in Figure 1b–e,g–j. Full suppression ofreflection occurs when the spectral position of resonancescorresponding to different anomalies coincide with eachother and the total contribution of MD and EQ modes isequal to the contribution of ED and MQ modes.It can be clearly seen from Figure 2a that the reflectionis completely suppressed at wavelengths close to [0; 1]RA. Moreover, lattice Kerker effect can be also achievedfor NPs with different R arranged in 2 D lattices withproperly chosen h x and h y . Thus, a complete suppres-sion of backscattering can be tailored across UV andvisible wavelength ranges, as shown in Figure 2a. Toget more insights, we plot in Figure 2b the amplitudeof electric field in ZY plane for the unit cell of the ar-ray with h x = 240 nm, h y = 280 nm, at λ = 420 . λ = 417 . Z axis according to eq. (2), in contrast to the case of astanding wave at λ = 417 . λ = 420 . single lossy NPs [20] it is in principle impossibleto achieve. A complete suppression of the backscatter- ing can be tuned within the UV and visible wavelengthranges by varying geometry of arrays, i.e. radius of NPs FIG. 2. (a) Reflectance for arrays with different geometri-cal parameters (
R, h x , h y ) as marked in the legend. Verticaldashed lines show respective spectral positions of [1; 0] and[0; 1] RAs for each array. (b) Electric field distribution inthe ZY plane for (60 , , λ = 417 . λ = 420 . h y and (b) h x , and(c) for array with h x = 240 nm and h y = 280 nm. and the distance between them. High absorption andstrong electric field localization are observed at the fre-quency which corresponds to the lattice Kerker effect.The latter property is of critical importance for ultra-narrowband absorption [55] and surface-enhanced Ra-man spectroscopy [56, 57]. Funding
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