Exciting Pseudospin Dependent Edge States in Plasmonic Metasurfaces
Matthew Proctor, Richard V. Craster, Stefan A. Maier, Vincenzo Giannini, Paloma A. Huidobro
aa r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Exciting Pseudospin Dependent Edge Statesin Plasmonic Metasurfaces
Matthew Proctor, ∗ , † , ‡ Richard V. Craster, † Stefan A. Maier, ¶ , ‡ Vincenzo Giannini, § and Paloma A. Huidobro k , ‡ † Department of Mathematics, Imperial College London, London, SW7 2AZ, UK ‡ Department of Physics, Imperial College London, London, SW7 2AZ, UK ¶ Chair in Hybrid Nanosystems, Nanoinstitut München, Faculty of Physics,Ludwig-Maximilians-Universität München, 80539, München, Germany § Instituto de Estructura de la Materia (IEM), Consejo Superior de InvestigacionesCientíficas (CSIC), Serrano 121, 28006, Madrid, Spain k Instituto de Telecomunicações, Insituto Superior Tecnico-University of Lisbon, AvenidaRovisco Pais 1,1049-001 Lisboa, Portugal
E-mail: [email protected]
Abstract
We study a plasmonic metasurface that supports pseudospin dependent edge statesconfined at a subwavelength scale, considering full electrodynamic interactions includ-ing retardation and radiative effects. The spatial symmetry of the lattice of plasmonicnanoparticles gives rise to edge states with properties reminiscent of the quantum spinHall effect in topological insulators. However, unlike the spin-momentum locking char-acteristic of topological insulators, these modes are not purely unidirectional and theirpropagation properties can be understood by analysing the spin angular momentum ofthe electromagnetic field, which is inhomogenous in the plane of the lattice. The local ign of the spin angular momentum determines the propagation direction of the modeunder a near-field excitation source. We also study the optical response under far-fieldexcitation and discuss in detail the effects of radiation and retardation. Keywords plasmonics, metasurface, nanoparticle array, topological photonics, nanophotonics, pseu-dospinTopological insulators are materials which are insulating in the bulk but which haveconduction surface states protected against disorder (1). The remarkable properties of thesestates in electronic systems has inspired the search for photonic topological insulators (PTIs),which aim to guide and manipulate photons with the same level of control and efficiency(2–6). Systems which possess these effects whilst preserving time reversal symmetry areappealing as they do not require complicated experimental setups such as strong magneticfields or bianisotropic coupling. Motivated by this, a proposal to emulate the quantum spinHall (QSH) effect in photonic crystals was presented by Wu and Hu in Ref. 7. Effectsreminiscent of the QSH phase such as a band inversion between dipolar and quadrupolarmodes, and pseudospin dependent edge states are realised but, rather than relying on thetime reversal symmetric pairs characteristic of electronic systems, they instead rely on thespatial symmetry of the lattice structure. As a result, the edge states have a reduction inbackscattering over trivial ones (8, 9). The method has since been applied to a variety ofbosonic systems (10–14), and has recently experimentally been demonstrated in the visibleregime (15).The combination of topological effects with plasmonics offers the possibility of preciselycontrolling light on the the nanoscale. The strong enhancement and localisation of electricfields due to localised surface plasmon (LSP) resonances (16) is a widely employed platformfor light confinement on the nanoscale (17, 18). Plasmonic metasurfaces can be formed byarranging plasmonic nanoparticles in two-dimensional (2D) lattices, where the LSPs become2elocalised across the whole metasurface as collective resonances. The optical propertiesof metasurfaces are then determined by the individual nanoparticle elements as well as thegeometry of the lattice (19, 20). The tunable optical properties of metasurfaces makesthem versatile tools for the manipulation of light on the nanoscale (21, 22). For instance,appropriately designed plasmonic metasurfaces can host spin dependent directional stateswhich can couple to valley excitons when interfaced to 2D materials (23).One-dimensional (1D) chains of dielectric and metallic nanoparticles were some of thefirst systems used for the investigation of topological phases in nanophotonics (24–30), inparticular systems analogous to the Su-Schrieffer-Heeger (SSH) model, which hosts topologi-cally protected edge states in 1D. In the plasmonic chain, initial studies into these topologicalstates were limited to the quasistatic approximation (QSA) (31) despite the well known ra-diative effects which are of great importance for large enough nanoparticles and retardationwhich is important at large lattice periods (32, 33) or at very small periods, where higherorder multipolar effects occur (34). Indeed, beyond the quasistatic limit, the plasmonic SSHchain becomes non-Hermitian and band structures are distorted compared to the quasistaticmodel, with effects such as polariton splitting at the light line (35). In addition, the ubiq-uitous bulk-edge correspondence of topological insulators has been shown to break downdue to retardation (36). 2D plasmonic systems, including graphene and arrays of plasmonicnanoparticles have also been considered for hosting topological states. Time reversal brokeneffects reliant on magnetic fields have been proposed for graphene plasmons (37, 38). Hon-eycomb lattices of plasmonic nanoparticles have been investigated in the quasistatic limit,where a direct analogy between the tight binding model in graphene and nearest neighbourapproximation in plasmonics can be made (39–41). More recently, theoretical and experimen-tal investigations have shown how the long range, retarded interactions affect the physicalbehaviour of the system (42, 43). Here, we consider the lattice geometry proposed in Ref.7, shown in Figure 1, to study pseudospin dependent edge states in plasmonic metasurfaces.This scheme has been considered in 2D arrays of plasmonic nanoparticles in the QSA (44).3n this work, we show the importance of going beyond this approximation and study therealisation of electromagnetic modes resembling the QSH effect on a plasmonic metasurface,including full electrodynamic interactions in our description of the system.We use semi-analytical techniques to investigate plasmonic metasurfaces. These consistof a triangular lattice with unit cells containing six nanoparticles arranged in a hexagon,Figure 1a. We begin by outlining the coupled dipole method used to model these arraysand then we investigate the behaviour of the modes supported by the metasurface. Bycalculating the eigenmodes of the infinite lattice, we characterise the band inversion processwhich occurs between a shrunken phase (where nanoparticles in the unit cell are displacedtowards the centre) and an expanded phase (where they are displaced outwards from thecentre). The response of the infinite system is also studied under far-field excitation.We then consider the interface between two phases in a semi-infinite ribbon layout toelucidate the nature of the edge states that emerge due to the band inversion between the twophases. We characterise their pseudospin dependence, showing that these modes are moreappropriately characterized by means of the spin angular momentum of the electromagneticfields, and probe these states in the near-field with a magnetic dipole source. By investigatingthe effect of the source position, we unambiguously show that, unlike the unidirectional edgestates characteristic of topological insulators, the directionality of these edge states dependson the source position. Finally, we highlight the radiative and retardation effects on the edgestates.
Design and set up of the plasmonic metasurface
The metasurface we consider here consists of a 2D array of metal nanorods (modelled asspheroidal nanoparticles) arranged in the lattice shown in Figure 1a. The unit cell containssix nanoparticles of radius r and height h arranged in a hexagon separated by nearest neigh-bour spacing R with lattice vectors a and a , as shown in the figure. In the regime R > r ,4he nanoparticles can be considered as point dipoles and higher order resonances can beneglected (32, 45). A nanoparticle at position d i with polarisability α ( ω ) supports a dipolemoment p i when excited by an external field E i . For a lattice of nanoparticles, a coupleddipole equation can be written which describes the dipole moment of a nanoparticle inducedby an external field plus the sum of all neighbouring dipole moments, α ( ω ) p i = E i + X i = j ˆ G ( d i − d j , ω ) p j , (1)where the dyadic Green’s function ˆ G , describes the interactions between point dipoles, andis given by, ˆ G ( d i − d j , ω ) = k e ikd d (cid:20)(cid:18) ikd − k d (cid:19) ˆ I − (cid:18) ikd − k d (cid:19) n ⊗ n (cid:21) . (2)Here, k = √ ǫ B ω/c is the wavenumber of the surrounding medium, with ǫ B the permittivityof the surrounding environment (which is assumed to be the vacuum throughout this work, ǫ B = 1 ), and d = | d i − d j | is the distance between nanoparticles, with n = ( d i − d j ) / | d i − d j | the unit vector in the direction along the line that joins two nanoparticles.In equation 1, the polarisability, α ( ω ) , describes the optical response of an individualnanoparticle. The static polarisability for a spheroidal nanoparticle is written, α s ( ω ) = V π ǫ ( ω ) −
11 + L ( ǫ ( ω ) − , (3)where ǫ ( ω ) is the dielectric function of the metal, V is the spheroid volume, and L is thestatic geometrical factor which is dependent on the radius and height of the nanoparticle;for a sphere L = (46). The dielectric function of the nanoparticles is given by the Drudemodel, ǫ ( ω ) = ǫ ∞ − ω p ω + iωγ . (4)5 )c) Figure 1: Metal rods are arranged on a plane to form a plasmonic metasurface, and theiroptical response is modelled including radiative and retarded effects. a) Layout of the lat-tice of plasmonic nanoparticles, including a close up of the arrangement of particles in theunit cell, b) The perturbation of the unit cell into shrunken and expanded phases with scal-ing parameter s . s = 1 corresponds to the unperturbed honeycomb lattice. c) Extinctioncross sections σ ext , normalized to geometrical cross section, of silver spheroidal nanoparti-cles showing the splitting of in-plane (blue) and out-of-plane (pink) resonances, d) σ ext ofnanoparticles with increasing radius, showing the radiative broadening and redshifting of theplasmon resonance. 6hroughout the work we consider silver nanoparticles, with ǫ ∞ = 5 , ω p = 8 . eV and γ = 1 / fs ≈ . eV (47). The static polarisability neglects radiative effects which areessential for describing larger nanoparticles. We take this into account by means of themodified long wavelength approximation (MLWA), α MLWA ( ω ) = α s ( ω )1 − k l E Dα s ( ω ) − i k α s ( ω ) , (5)where l E is the spheroid major axis half length and D is a dynamic geometrical factor; D = 1 for a sphere (46). The importance of the radiative correction for spheroidal silvernanoparticles is exemplified in Figure 1c. The extinction cross section σ ext for a spheroidalnanoparticle with radius r = 5 nm and height h = 20 nm, as well as a spherical nanoparticlewith radius r = 5 nm is shown. We normalise to the cross sectional area A perpendicular tothe dipole moment. The nanoparticle supports two resonance modes: one where the dipoleis aligned with the minor axis (in-plane) and one with the major axis (out-of-plane). Theout-of-plane resonance becomes redshifted and well separated in frequency from the in-planeresonance, which allows us to investigate the out-of-plane and in-plane modes separately.The increased radiative effect on a single nanoparticle is demonstrated by the extinctioncross section for larger radii, up to r = 20 nm, where the resonance becomes broader andcontinues to be redshifted, Figure 1d. Whilst the static polarisabilty adequately describes thebehaviour of the smallest nanoparticle sizes, the MLWA incorporates the effects of dynamicdepolarisation and the radiative correction, and is necessary to correctly model particles ofradius above ∼ nm. Spectral response of the metasurface
We start by considering the optical response of the plasmonic metasurface in the expandedand shrunken phases, Figure 1b. We do so by setting up an infinite lattice of nanoparticles7nd applying periodic boundary conditions to Equation 1 by writing the external electric field E , and dipole moments p , as periodic Bloch functions. The following system of equationscan then be written, (cid:18) α ( ω ) − ˆ H ( k , ω ) (cid:19) · p = E , (6)where the interaction matrix ˆ H ( k , ω ) has elements, H ij = P R ˆ G ( d i − d j + R , ω ) e i k · R i = j P | R |6 =0 ˆ G ( R , ω ) e i k · R i = j , (7)with q , the Bloch wavevector and R , the lattice site positions. We note that the interactionmatrix is a × matrix since we restrict our study to the out-of-plane modes, such thatthere is a single degree of freedom for each particle in the unit cell. The in-plane modes of ahoneycomb lattice of plasmonic nanoparticles have been studied elsewhere (41) and being wellshifted in frequencies, they are completely decoupled from the out-of-plane modes. Finally,we note that sums in the interaction matrix are conditionally convergent due to the slowlydecaying /r term and so additional manipulation is required to converge these expressions(see Methods).Figure 2 shows the spectral response of the plasmonic metasurface under study. Weconsider nanoparticles with radius r = 5 nm and height h = 20 nm, nearest neighbourspacing R = 20 nm and lattice constant a = 60 nm. To calculate the eigenvalues andeigenvectors of the periodic lattice we solve Equation 6 without an incident field. Initially,in order to keep the eigenvalue problem linear we take the QSA and only consider the quicklydecaying /d term in the Green’s function. However, we choose to go beyond the nearestneighbour approximation and include interactions between all particles in the lattice (48).The band structure of a metasurface with the nanoparticles arranged in a honeycomb lattice8 oneycomb, s = 1Shrunken, s = 0.9 Expanded, s = 1.1ShrunkenExpanded p x p y d x − y d xy pdfs spdf a) b)c) d)e)f) fs Figure 2: Exciting plasmonic metasurfaces from the far-field. Nanoparticles have radius r = 5 nm and height h = 20 nm, and the lattice constant a = 60 nm. a) Bulk dispersionrelations of the out-of-plane modes of the honeycomb plasmonic lattice, including all neigh-bours in the quasistatic approximation (QSA), b) Normalised, out-of-plane electric fields ofthe quadrupolar and dipolar modes of the metasurface (at lower and higher energies, respec-tively), including full GF interactions. c) Dispersion relations for the shrunken ( s = 0 . ) andd) expanded ( s = 1 . ) systems. The bands are coloured according to their dipolar ( p ) orquadrupolar ( d ) nature, showing the band inversion at Γ between shrunken and expandedphases. The monopolar ( s ) and hexapolar ( f ) bands are not involved in the inversion. e),f) Extinction cross sections, normalised to the maximum, under excitation with an externalfield including full retarded Green’s function (GF) interactions and radiative polarisabilitiesfor the shrunken and expanded phases. The dots highlight all the modes in the system,some of which are dark in an external field. σ ext at a fixed wavevector (purple dotted line)is shown in the right hand plots. 9s shown in Figure 2a. Instead of using the conventional rhombic unit cell, we take the largerhexagonal cell such that the Brillouin zone (BZ) of the honeycomb lattice becomes foldedand the Dirac points at K and K ′ are mapped onto each other to create a doubly degeneratepoint at Γ (7), as shown in Figure 2a. Whereas the original honeycomb lattice is formedof two triangular sublattices, the system is now formed of six sublattices correspondingto the six nanoparticles in the unit cell; meaning there are six bands present in the bandstructure. In the QSA, assuming nearest neighbour interactions, the Dirac points of thehoneycomb plasmonic lattice occur at the surface plasmon resonance frequency w sp (41) andthe band structure is symmetrical about this frequency (44). This is no longer the casedue to the sublattice symmetry breaking interaction term between particles of the samesublattice in neighbouring unit cells. Figure 2c shows the band structures for metasurfaceswith shrunken (left) and expanded (right) unit cells. The bands are labelled as s, p, d, f ,indicating monopolar, dipolar, quadrupolar and hexapolar characters. This was determinedby calculating the overlap with the eigenstates of an isolated hexagon of nanoparticles. Thedipole and quadrupole modes of the lattice are shown in Figure 2b. In both the shrunken andexpanded phases, we see how the the ordering of the modes is opposite to that obtained inphotonic crystals and other bosonic analogues (7, 13, 14, 49), with the monopolar mode beingthe highest in energy and the hexapolar being the lowest in energy. This is due to the differentelectromagnetic properties of the two systems: while in the photonic crystal the permittivityis positive and constant, in the plasmonic system the permittivity is negative and dispersive.For the metasurface considered here, the bonding or antibonding character of the couplingbetween the plasmonic nanoparticles determines the mode ordering, as we explain in moredetail later. Next, we show that near the centre of the BZ, there is a mode inversion betweenthe shrunken and expanded phases. This is evident from the colour scale, which encodes thedipolar/quadrupolar character of the modes. In the shrunken phase ( s < ), the band abovethe band gap is dipolar and the one below is quadrupolar (Figure 2c). On the other hand,for the expanded phase ( s > ) they become inverted at Γ (Figure 2d). The degeneracy of10he bands above and below the gap at Γ for the shrunken and expanded lattices suggestslinear combinations of these modes can be taken, p ± = ( p x ± ip y ) and d ± = ( d x − y ± id xy ) ,which correspond to pseudospins; taking positive or negative combinations gives clockwiseor anticlockwise rotations (7).After characterising the eigenstates in the QSA, we now consider the full electrodynamicinteraction between the nanoparticles by including all terms in the Green’s function, andexplore retardation and radiative effects by calculating the extinction cross section of thesystem when excited by an external field. We show the response of the system in theshrunken and expanded phase over the BZ in Figure 2e-f (contour plots). The incidentplane wave is defined as E = ( E x , E y , E z ) , where the field components satisfy Maxwell’sequations and k = k k + k z . Above the light line, the wave is propagating in the z direction, k z = q k − k k and below the light line it is evanescent, k z = i q k k − k , which allows us toprobe the eigenstates in this region of the spectrum. Starting at Γ , the incident field has no z -component meaning out-of-plane dipoles cannot be excited and the extinction cross sectionis zero. As soon as we move away from Γ , we begin to excite out-of-plane polarised modes.The highest energy mode corresponds to a monopolar mode, with all dipole moments in theunit cell pointing in the same direction. The energy ordering can be understood from theordering of bonding and anti-bonding modes in a plasmonic dimer. The bonding mode liesat a higher energy compared to the anti-bonding mode, as has been shown theoretically (50)and experimentally (51). This is a consequence of the relative orientation of the dipoles:in the antibonding mode the dipoles are antiparallel which minimizes the radiated electricfield and hence the energy, whereas in the bonding mode the dipoles are parallel and hencesustain a larger electric field and appear at higher energies. When plasmonic nanoparticlesare arranged in a lattice with more than one particle per unit cell, this ordering is mantained,as has also been shown elsewhere (42, 52). Notably, this is not an effect of retardation andwill arise in any near-field coupled ensemble of plasmonic nanoparticles, as shown by ourquasistatic results (Figure 2c,d) and in Refs. (42, 50, 52).11n the infinite lattice, the bonding nature of the monopolar mode across the whole systemcauses it to be highly radiant and so dominates the response of the lattice for propagatingwaves. Nevertheless, we are still able to examine the peaks in extinction cross section at lowerfrequencies. By symmetry arguments, only the dipolar mode is excitable by a plane wave.The extinction cross section of the system agrees with the characterisation of the modes inthe QSA. In the shrunken phase, the higher energy dipolar mode is visible in the extinctioncross section whereas in the expanded phase a band inversion occurs. This inversion betweendipolar and quadrupolar modes close to the BZ centre constitutes a signature to distinguishboth metasurface phases, since it can be detected experimentally by far-field measurements(49). In the right panels of Figure 2e-f we show the extinction spectrum at a fixed incidentmomentum, k : the resonance peak corresponding to the dipolar mode is visible for theshrunken structure at higher energies than for the expanded structure. In Figure 2e-f, we plotthe loci of peaks in the spectral function based on the effective polarisability (see Methods),to make all of the modes visible, even those which are dark in an external field. These peaksqualitatively agree with the QSA dispersion relation with only a slight redshift due to theradiative correction given that all the length scales in the system are very subwavelength.Retardation effects are most apparent in the highest energy band. The coupling of theplasmonic mode with free photons causes the strong polariton-like splitting at the light line(33, 48). Since the monopolar (bonding) mode is strongly radiant, it shows the largestinteraction with the light line. We also observe how the radiative broadening of the modegrows larger going from Γ towards the light line; this has been observed in 1D plasmonicchains and the effect is much greater in this 2D lattice due to constructive interferencebetween dipole moments in the monopolar mode (42). The polariton splitting and radiativebroadening importantly do not affect the band inversion at Γ . Finally, the lowest energyband is a hexapolar mode where all dipole moments are in anti-phase meaning such a modecannot be excited by a plane wave, and it is not visible in the extinction cross section formost of the BZ. 12 seudospin edge states in finite systems After characterising the modes in the metasurface, we now look at the edge states betweenregions of different phases. As discussed earlier, the band inversion between different phasesis reminiscent of the QSH effect and thus pseudospin dependent, directional edge statesare anticipated. In order to study the edge states, we consider an interface between thetwo different phases of the metasurface. In the calculations, a finite ribbon of a region inthe expanded phase is cladded with regions in the shrunken phase so that we have a finitestructure along the direction a , and we apply Bloch periodicity along a . Figure 3a showsthe band structure of the ribbon, calculated with retardation (black lines), and also in theQSA (grey lines). In the case of full electrodynamics interactions, the Green’s function waslinearised by letting ω = ω sp and the static polarisability was assumed (see Methods). Twoedge states can be seen clearly within the band gap (we colour the retarded solution, as wewill explain below). Importantly, the edge states do not join the bulk modes, rather theyreconnect with each other at the edges of the BZ. This is a fundamental distinction withtopological edge states since it means it is possible, by a continuous change of parameters,to remove these states from the band gap; which indicates that these edge states are notresistant to backscattering (53). In Figure 3b, we also plot the spectral function of thesystem with radiative corrections in the polarisability, which accounts for the frequencyshift (in this plot we let the Drude losses γ = 0 . eV to improve the visibility of the edgestates). Similarly to the bulk dipolar and quadrupolar bands, we see the edge states donot interact strongly with the light line although the radiative broadening and redshift isapparent. Since the pseudospin effect is reliant on the C v symmetry of the bulk systemwhich is necessarily broken at the interface between the shrunken and expanded region therewill always be a ‘minigap’ between the edge states at Γ which will never fully close. Theamount of perturbation between shrunken and expanded phase determines the size of thebulk band gap and as well as the size of the minigap. To minimise the gap, the perturbationalong the edge can be graded; the edge states are then excitable across the whole band gap1311, 44).To investigate the nature of the edge modesand to elucidate their excitation under pointsources we now look at the two eigenstates of the expanded/shrunken interface with oppositegroup velocity, and at a frequency ω in the upper band. These are shown as points s and s inFigure 3a. We first plot the time averaged Poynting vector S = Re ( E × H ∗ ) along with thenormalised out-of-plane electric field E z , in the plane of the metasurface z = 0 , in Figure 3c.This demonstrates that the edge modes are confined to the interface between the two regions.The Poynting vector (arrows) characterises the flow of electromagnetic energy, which is inopposite directions for wavevectors with opposite sign. In inhomogeneous, dispersive mediait can be argued that the Poynting vector does not necessarily accurately characterise a spinor pseudospin (54). More appropriately, we also consider the spin angular momentum inthe plane of the lattice, T = Im ( E ∗ × E + H ∗ × H ) (55). Since the dipole moments arealigned out of the plane, at z = 0 the field components are E z , H x , H y and the spin angularmomentum is then T z = Im ( H ∗ × H ) . This spin angular momentum quantifies the degreeof elliptical polarization of the magnetic field and its handedness, similar to the studies forphotonic crystals in Refs. (56, 57). We plot the normalised spin over the same region asthe Poynting vector in Figure 3c, where its inhomogenous character across the lattice isevident, with varying magnitude and sign across the interface. To determine the pseudospindependence of the edge states, we integrate the spin angular momentum T over the regionshown in red in Figure 3d, for wavevectors k k across the whole BZ, and we plot the edgestates with this colour code in the dispersion relation in Figure 3a. At the edges of the BZthere is maximum mixing between pseudospins and the integral of T = 0 , but as we movetowards the centre we see either edge state acquires an opposite pseudospin. This pattern ofopposite pseudospins travelling in opposite directions is akin to the QSH effect. Again, wenote that since in this system the two edge modes are linked through a pseudo-time reversaloperator (7), which is only rigourously defined at Γ , there is not complete protection againstbackscattering. 14 ) k x k x b)c) d) − − E z T z Figure 3: Pseudospin modes along the interface between metasurfaces in expanded andshrunken phases. a) Dispersion relations of a semi-infinite ribbon with a 28 unit cell regionin the expanded phase ( s = 1 . ) cladded with two 6 unit cell regions in the shrunken phase( s = 0 . ). Results obtained in the QSA including all neighbours are shown in grey andresults including full GF interactions are shown in black, with both in the non-radiativeregime. We highlight the edge modes in this case with colours according to their pseudospin,characterised by spin angular momentum T . b) Spectral function of the retarded, radiativesystem with Drude losses γ = 0 . eV. The frequency shift in the edge states compared tothe QSA in a) is due to the radiative effects, c) Normalised out-of-plane electric field E z for s and s , in the metasurface plane. The arrows show the time averaged Poynting vector S ,demonstrating the directionality of the edge states, d) Normalised spin angular momentum T for modes s and s , marked in a). T is integrated in the red region colour the edge statesin a) 15e now consider the excitation of edge modes with a localised source, and its relation tothe spin angular momentum. The sign of the spin angular momentum T is positive in theregion within the hexagon of nanoparticles immediately above the edge for the mode s , asshown by the purple area in Figure 3d, left panel. However, at a point directly on the edge,shown by the black line in Figure 3d, the sign of T switches. Importantly, this means that thelocal handedness of the elliptical polarization of the magnetic field for each of the edge modesis inhomogeneous in the plane of the array, as expected for a confined mode in a complexenvironment. This position dependence of the spin angular momentum has implicationson the excitation of unidirectional modes by point sources, which we show by modelling afinite array of nanoparticles with an interface between an expanded and shrunken region.To fully understand how the propagation along the edge is dependent on the source andits position, we consider an interface with zero material losses; the edge state is preventedfrom reflecting off the hard boundary with the vacuum by slowly increasing material lossesat these boundaries. We place a right circularly polarised magnetic dipole source withmagnetic field H = H x + iH y at the various positions shown in Figure 4a. We choose theexcitation frequency from Figure 3b as ω = 2 . eV. The emission of the magnetic dipolesource placed in the plasmonic lattice will be modified due to the surrounding environment.This is characterised by the Purcell factor P F , the ratio of emitted power of a magneticdipole to the emitted power in free space P M (58), P M = µ π ω | m | c , (8)where µ is the vacuum permeability and m is the magnetic dipole moment. The Purcellfactor as a function of source position is shown in Figure 4b (dotted orange line). Theenhancement is greatest when the source is close to the metallic nanoparticles in the lattice,but we note it is generally modest due to the distance between the source and nanoparticles.To characterise the directionality of the energy flow along the edge, we integrate the16 xpandedShrunken Figure 4: Exciting pseudospin edge states with near-field probes. a) A right circularlypolarised magnetic dipole is placed at positions along the path shown to excite the edgestate between expanded and shrunken regions in the metasurface, b) The fraction of powertravelling left and right, P L/R , as well as the Purcell factor P F are dependent on sourceposition across the edge. Although the source is right circularly polarised, it will excite anedge state in the opposite direction for some positions along the edge. (Regions where thecoupled dipole approximation does not hold are shaded), c) Pseudospin edge state excitedby a source at the optimal position for a right travelling mode. The middle panel showsthe dipole moments in the plane of the metasurface ( xy plane). The left and right panelsshow the normalised electric field intensity | E | perpendicular to the metasurface ( yz plane),showing the mode is strongly confined in the out-of-plane direction z . d) A source with thesame polarisation as in c) is placed at the least optimal position showing excitation in theopposite direction. 17oynting vector through a plane perpendicular to the interface and metasurface, and we plotthis as solid lines in Figure 4b; with purple and green corresponding to the fraction of flow tothe right ( P R ) and left ( P L ), respectively. Regions in which the coupled dipole approximationdoes not hold are shaded. Starting at r , and looking at the power flow for the right circularpolarisation, the flow of energy is predominantly to the right ( P R , purple line) as expectedfrom the polarisation of the source, which couples to the right-propagating pseudospin mode.Importantly there is still a fraction of energy travelling to the left ( P L , green line) whichdemonstrates the existence of pseudospin mixing. As the source is moved towards r theflow is completely to the right before quickly flipping to the opposite direction. From r to r the energy flow again changes sign as the source moves from an area with negativespin angular momentum to an area with positive spin angular momentum. The propagationdirection of the excited edge state can then be predicted from the interplay between thesource polarisation and the local handedness of the polarisation of the mode, given by thespin angular momentum in real space, rather than by the pseudospin. More details on thedirectionality of the modes excited by dipole sources are given in the Supporting Information,Figure S1). We emphasise that we have used a right hand polarised source throughout, whichsuggests one should always expect energy flow in the right direction.In Figure 4c-d we show the stark difference in directionality along the edge depending onthe position of the source (magenta star). We consider the same right-circularly polarizedsource placed at two different positions. In Figure 4c we choose the optimal placement forexcitation in the expected direction, and in Figure 4d the least optimal placement. Allbosonic systems with these lattice symmetries, such as the photonic crystal, will also possessthese position dependent directional modes when excited with a circularly polarised source(56). In the left and right panels, we show the electric field intensity, | E | , at plane cutperpendicular to the metasurface in the left and right directions. We note that the edge stateis not only confined to the edge in the plane of the metasurface but also out of the plane atsubwavelength scales. Finally, we stress that in this discussion we have considered sources18laced in the plane of the metasurface. For sources above the nanoparticles, the excitationof directional edge modes will be determined by, first, the value of the Purcell factor, and,second, the interplay between the source polarization and the distribution of spin angularmomentum. (Plots of the spin angular momentum in planes above the metasurface are givenin the Supporting Information, Figure S2.) Retardation and Radiative Effects
We finally discuss in detail the effect of retardation and radiation in the pseudospin edgestates. While so far we have considered a very subwavelength period and nanoparticle size,we have already seen the effect of retarded interactions which cause the band structure tobe altered with respect to the QSA, in particular close to the light line, and the radiativebroadening and redshifting of resonances. Radiative effects become more apparent for largernanoparticles, with very large broadenings and shifts as in the single particle extinction crosssection shown in Figure 1d; effects which are not captured in the QSA. On the other hand, itis important to note that retardation can have striking consequences not only on bulk bandstructures but also on the properties of edge states. It has been shown for example howin the 1D plasmonic SSH model, retardation can result in the breakdown of bulk boundarycorrespondence and the disappearance of edge states (36). To investigate the effects ofretardation in the 2D system studied here, we excite an interface between expanded andshrunken regions in a ribbon with a plane wave at a finite wavevector above the light line, k x = 0 . π/a , and calculate the extinction cross section for increasing lattice constants a .In Figure 5a, we show the extinction cross section for a metasurface with an interfacebetween the two phases for silver nanoparticles with radius nm and height nm. We let theDrude losses γ = 0 . eV, as in Figure 3b, and highlight the edge states with white dots forvisibility. As in the infinite lattice, in the quasistatic and nearest neighbour approximations,the edge states are expected to be symmetrical about the plasma frequency ω sp (dotted purple19ine) (44). In contrast, when radiative effects are taken into account, there are substantialshifts in the edge state frequencies. Initially, for a = 60 nm the edge states are well separatedfrom the bulk but as the lattice constant increases up to nm the bulk modes close up andthe edge states are lost. We calculate the full width at half maximum (FWHM) of the highestenergy edge state at a = 60 nm where FWHM = 0 . eV. As an aside, this edge statehas a larger cross section as along the interface the nanoparticles form a bonding-like state(Figure 5c, top) whereas the lower energy edge state has an anti-bonding distribution whichleads to lower cross section (Figure 5c, bottom). In Figure 5b we show the cross section fornanoparticles with radius nm and height nm, with lattice constant varying from to nm. Here, the edge states become significantly redshifted far below ω sp . Again wemeasure the FWHM, at a = 120 nm, FWHM = 0 . eV demonstrating the broadeningof the mode for larger nanoparticles. We note that these widths are smaller than the Drudelosses since the lattice structure modifies the optical response by increasing the quality factor(17, 18). Conclusion
In this work we have presented a study of spin dependent edge states in a plasmonic meta-surface. These states rely on lattice symmetries and are similar to the quantum spin Halleffect in topological insulators. By going beyond the quasistatic approximation and includ-ing retardation and radiative effects, we model the plasmonic system appropriately and showhow the long range interactions result in a more complex band structure than the QSA. Thebands involved in the band inversion are not greatly affected by retardation, and they providea signature for far-field measurements. We note that the ordering in energy of the modesof the plasmonic metasurface are opposite to what is seen in photonic crystals (7), withthe dipolar modes lying at higher energies than the quadrupolar modes for the non-invertedbands. 20 = 5nm r = 10nm a)b) c) Figure 5: Far-field excitation of pseudospin edge states showing radiative and retardationeffects. Extinction cross section for ribbon system under plane wave excitation at k x =0 . π/a for increasing lattice constants a with Drude losses γ = 0 . eV. The edge statesare highlighted (white dotted lines), a) For particle radius r = 5 nm and height h = 20 nm.The surface plasmon frequency ω sp is shown as the purple dotted line, b) For particle radius r = 10 nm and height h = 40 nm. The bands are shifted to lower frequencies, far below ω sp ,c) Normalised out-of-plane electric field, in the metasurface plane, for the upper and lowerenergy edge states for the parameters in a). The upper mode is bright as it has a bondingsymmetry along the edge, allowing it to be excited by a plane wave, while the lower modeis dark. 21mportantly, we determine the spin angular momentum of the edge modes, and showthat this is the quantity that characterises the directionality of the modes. Remarkably, thisleads to a more complex behaviour which can be opposite to what would be expected froma direct analogy with the QSH effect. We emphasise that these conclusions are valid notonly for the plasmonic metasurface but for any bosonic system with this lattice (56). Weprobe the spin angular momentum by looking at the excitation of the edge states in the near-field by a magnetic dipole source. By varying the position of the source, we highlight howthe location is essential in exciting a purely unidirectional state. In agreement with the spinangular momentum characterisation of the edge in the ribbon we show that in some positionsa source will excite a mode in completely the opposite way to the expected direction.Finally, we have also considered the optical response of the plasmonic metasurface and theedge modes under far-field excitation, showing the effect of radiative and retardation effectsfor the larger radius nanoparticles and larger lattice constants. Although the edge states per-sist for larger nanoparticles, we observe a radiative broadening and shift in frequency whichis naturally not captured in a quasistatic approximation. For increasing lattice constants,the bulk modes close up and eventually the edge states are indistinguishable, which providesa parameter regime for which these edge states could be experimentally observed. Methods
Lattice sums
The summations in the interaction matrix in Equation 7 are conditionally convergent. Firstly,we will consider sums including the origin term and split these into long range and short/mediumrange terms, S incl = X R e i k · R ˆ G ( r , ω ) = S L + S SM , (9)22here S L = k P R e i k · R e ikd d and S SM = k P R e i k · R e ikd (cid:0) ikd − d (cid:1) . The slowly converging S L term is handled by using Ewald’s method (59). This splits the real space sum into twoand then takes the Fourier transform of one part using Poisson’s summation, resulting ina sum over the reciprocal lattice. The sum is optimised with a Ewald parameter to ensurethe real space and reciprocal space sum converge within approximately the same numberof lattice constants. The S SM term converges rapidly above the light line. Outside of thisregion the first few terms within a radius R min are added and then the rest of the sum iscalculated numerically by approximating the summation as an integral (48), S SM ≈ k R min X R =0 e i k · R e ikd (cid:18) ikd − d (cid:19) + k Z ∞ R min e i k · R e ikd (cid:18) ikd − d (cid:19) . (10)For sums excluding the origin, S excl = X R =0 e i k · R ˆ G ( R , ω ) , (11)the integral method from Ref. 48 is used. For ribbons which are infinite in only one direction,the summations converge easier and so no techniques are used to speed up convergence (41).In the quasistatic approximation, we only consider the short range /d term whichconverges quickly when including all neighbours, in both the infinite lattice and semi-infiniteribbon. We note that this method of including all neighbours results in a kink at Γ in oneof the modes of the infinite lattice which is due to the group velocity necessarily being zeroat the centre of the BZ (48). Linearised Green’s Function
In Figure 3, we linearise the Green’s function to investigate the spin angular momentum semianalytically. For the retarded Green’s function and radiative polarisabilities the eigenvalue23roblem is non-linear and non-Hermitian, (cid:18) ˆ H ( k , ω ) − α ( ω )ˆ I (cid:19) · p = 0 . (12)To avoid the computational complexity of searching for complex ω solutions we linearisethe Green’s function by making the approximation ω = ω sp , the surface plasmon frequency.In spherical nanoparticles, ω sp = ω/ √ ǫ ∞ + 2 and for spheroids, ω sp = ω/ p ǫ ∞ − /L .For out of plane modes in the system we consider, ω sp = ω/ . . This is valid for the sizeof nanoparticles considered here since ω varies faster in the polarisabilty term than in theGreen’s function. As particle size increases the approximation becomes less valid close tothe light line. We can then rewrite Equation 13 as, (cid:18) ˆ H ( k , ω sp ) − α ( ω )ˆ I (cid:19) · p = 0 , (13)for which eigenvalues λ and eigenvectors p are found at each point in the BZ, k , and bandstructures are calculated by rearranging λ = 1 /α ( ω ) to find ω , for the static polarisability. Extinction Cross Section
To calculate the extinction cross section, σ ext , we use the following system of equations fora non-zero external field, (cid:18) ˆ H ( k , ω ) − α ( ω )ˆ I (cid:19) · p = E inc . (14)From Maxwell’s equations, we have k k E k + k z E z = 0 and we assume E k = 1 . Rearranging,we can then write, E z = − k k k z = − k k k − k k . (15)24he total incident field on each particle includes an additional phase due to the posi-tion within the unit cell d , E inc = E exp ( i k · d ) . After calculating dipole moments usingEquation 14, the extinction cross section is given by the optical theorem, σ ext = 4 πk P i Im ( p i · E ∗ inc ) | E | . (16) Spectral Function
The spectral function method relies on an effective polarisability formulation of the system(41, 48). We rewrite the system of equations for a non-zero external field as p = α eff E . Theeffective polarisability α eff = 1 /λ for eigenvalues of M , λ . The spectral function is analogousto the extinction cross section but rather than describing the system when excited by a welldefined external field instead characterises all modes in the system in the retarded, radiativeregime, regardless of whether they are bright or dark modes. This corresponds to the forcedoscillation of each mode of the lattice at some driving frequency ω and Bloch wavevector k .The spectral function is defined, σ spectral = 4 πk X i Im ( α ( i ) eff ) , (17)where the sum is over the number of elements in the unit cell in the infinite lattice or thesuper cell in the semi-infinite ribbon. Peaks in the spectral function will correspond to thereal part of the band structures from the linearised Green’s function. Acknowledgement
We acknowledge fruitful discussions with A. García-Etxarri and Mehul P. Makwana. M.P.,R.V.C. and P.A.H. acknowledge funding from the Leverhulme Trust. P.A.H. also acknowl-edges funding from Fundação para a Ciência e a Tecnologia and Instituto de Telecomuni-25ações under project CEECIND/03866/2017. S.A.M. and R.V.C. acknowledge funding fromEPSRC Programme Grant “Mathematical Fundamentals of Metamaterials” (EP/L024926/1).S.A.M. additionally acknowledges the Lee-Lucas Chair in Physics. We thank Jude Marcellafor the TOC figure.
Supporting Information Available
The following files are available free of charge. Position dependence on directionality of theedge states, with left and right polarised magnetic dipole sources. Spin angular momentumcalculated at planes above the metasurface. This material is available free of charge on theACS Publications website.
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Exciting Pseudospin Dependent Edge States in Plasmonic Metasurfaces
Matthew Proctor, Richard V. Craster, Stefan A. Maier,Vincenzo Giannini, Paloma A. Huidobro
Comprised of:2 pages,2 figures,0 tables osition dependence of magnetic dipole sources on directionality
RH source, centre of unit ce lRH source, edge of un t cellLH source, edge of unit cellLH source, centre of unit ce la)b)c)d)
Figure S1: The position dependent directionality of edge states. We excite a mode with a magnetic dipole sourcewith the same properties as in the main text at the centre of the unit cell next to the edge and on the edge itself.a) When the source is at the centre of the cell, directionality is as we expect from a right handed source, b) Fora source at the edge directionality is opposite, c) Expected directionality from a left handed source placed at thecentre of the cell, d) Directionality is opposite for a left handed source placed at the edge.
Spin angular momentum above the plane of the metasurface S1 = 5 nm z = 10 nm z = 20 nm z = 40 nm -0.016 0.016 -1.9E-3 1.9E-3 Figure S2: Spin angular momentum T z calculated at planes above the metasurface, for mode s in the main text.For z = 5 nm and 10 nm we normalise to the maximum value at z = 0. The inhomogeneity pattern of T z is stillpresent for z = 5 nm, but soon becomes less visible for z = 10 nm. For larger z , T zz