Dmitry Khlopin

Frédéric Laux

William P Wardley

Jérôme Martin

Gregory A Wurtz

Jérôme Plain

Nicolas Bonod

Anatoly V Zayats

Wayne Dickson

Davy Gérard
email: davy.gerard@utt.fr

Lattice modes and plasmonic linewidth engineering in gold and aluminum nanoparticle arrays

Lattice modes have been proposed as a means to engineer and control the linewidth and spectral position of optical resonances in arrays of metallic nanoparticles sustaining localized surface plasmon (LSP) resonances. Lattice modes are produced by the interference of LSP-enhanced in-plane scattered light, leading to a Fano-like lineshape with reduced linewidth. In this paper, we study the lattice modes supported by gold and aluminium nanoparticle arrays in the visible and UV, both experimentally and theoretically. The measured and simulated dispersion curves allow us to comprehensively analyze the details of the LSP coupling in the array. We show that when the spectral position of the Rayleigh anomaly, which depends on the period of the array, is slightly blue-shifted with respect to the LSP resonance, the quality factor in the nanoparticle array is significantly increased. We also provide evidence that the formation for the lattice modes, i.e. the coupling between LSPs and the in-plane scattered light, critically depends on the incident light polarization, the coupling efficiency being maximum when the polarization direction is perpendicular to the propagation direction of the grazing wave. The results obtained provide design rules allowing high quality factor resonances throughout visible and ultraviolet spectral ranges, needed for sensing and active nanophotonic applications.

INTRODUCTION

The field of plasmonics has been attracting a lot of attention because it allows light control at the nanoscale [START_REF] Zayats | Nano-optics of surface plasmon polaritons[END_REF]. In many cases, this relies on localized surface plasmon (LSP) resonances sustained by metal nanoparticles. When illuminated at its resonant frequency, the metal's free electron gas strongly interacts with light yielding a significative enhancement of the electromagnetic field in the vicinity of the nanoparticle. This resonance frequency, ω, critically depends on a number of parameters including the nature of the metal, the size and shape of the nanoparticle, and the refractive index of the surrounding medium. As is the case with any resonant phenomenon, the width of the resonance peak is directly related to the associated losses, including scattering and absorption losses. The resonance width ∆ω is often described using the quality factor, Q, as a figure of merit, defined as Q = ω/∆ω. Due to intrinsic losses in metals, the quality factor of LSP resonances is usually modest (in the 1-10 range) and mainly depends on the nature of the metal -noble metals such as silver and gold exhibiting better quality factors than the so-called "poor" metals in the visible and near-infrared spectrum. However, poor metals can be valuable plasmonic metals, especially outside the visible spectrum [START_REF] West | Searching for better plasmonic materials[END_REF]. For instance, aluminum is currently being rediscovered as a relevant material for plasmonics, especially in the ultraviolet (UV) spectrum (see [START_REF] Gérard | Aluminium plasmonics[END_REF] for a review). Actually, the main interest in aluminum as a plasmonic material lies in its broadband nature: apart from a relatively small region around λ = 800 nm (corresponding to the interband transition), aluminum can sustain LSPRs from the deep-UV to the infrared [START_REF] Bisio | Pushing the high-energy limit of plasmonics[END_REF][START_REF] Martin | High-resolution imaging and spectroscopy of multipolar plasmonic resonances in aluminum nanoantennas[END_REF]. Aluminum nanostructures have been used so far both in the UV, for instance to realize an UV-emitting plasmonic laser [START_REF] Zhang | A room temperature low-threshold ultraviolet plasmonic nanolaser[END_REF], and in the visible range -e.g. as a platform for color rendering [START_REF] Tan | Plasmonic color palettes for photorealistic printing with aluminum nanostructures[END_REF][START_REF] Olson | Vivid, fullcolor aluminum plasmonic pixels[END_REF], or for photodetection [START_REF] Zheng | Color-selective and CMOS-compatible photodetection based on aluminum plasmonics[END_REF].

However, the amount of losses in the visible and IR regions is significantly higher than for noble metals, leading to relatively poor quality resonances.

The quality factor of LSP resonances can be improved in geometries where plasmonic particles are arranged in a periodic array, so that electromagnetic coupling between scatterers taking place due to diffractive effects leads to the emergence of so-called lattice modes (see Ref. [START_REF] Ross | Optical properties of one-, two-, and three-dimensional arrays of plasmonic nanostructures[END_REF] for a review). These diffracting lattice modes, sometimes called surface lattice resonances, are generally observed near a Rayleigh anomaly of the array. Rayleigh anomalies correspond to the appearance (or disappearance) of diffracted orders, and are associated with a grazing wave propagating in the plane of the array [START_REF] Bonod | Diffraction gratings: from principles to applications in high-intensity lasers[END_REF]. Lattice modes can be described via a Fano-type process involving a broad resonance (the LSP resonance) and a discrete state associated to the light scattered in the plane of the array at the Rayleigh anomaly position [START_REF] Giannini | Fano resonances in nanoscale plasmonic systems: a parameter-free modeling approach[END_REF]. This coupling yields two hybrid states sharing the properties of its contributing components. The lattice mode is one of these hybrid states, located close to the Rayleigh anomaly position, that offers an enhanced scattering cross-section, based on the LSP contribution, while also demonstrating a high quality factor [START_REF] Giannini | Fano resonances in nanoscale plasmonic systems: a parameter-free modeling approach[END_REF][START_REF] Gallinet | Relation between near-field and far-field properties of plasmonic Fano resonances[END_REF][START_REF] Väkeväinen | Plasmonic surface lattice resonances at the strong coupling regime[END_REF]. This effect was predicted as early as 1985 in the context of rough surfaces for surface-enhanced Raman scattering [START_REF] Meier | Enhanced fields on rough surfaces: dipolar interactions among particles of sizes exceeding the Rayleigh limit[END_REF][START_REF] Carron | Resonances of two-dimensional particle gratings in surface-enhanced raman scattering[END_REF], before being rediscovered almost 20 years later for one-dimensional arrays of silver nanoparticles [START_REF] Zou | Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes[END_REF]. Later, lattice modes in two-dimensional gold nanoparticles arrays were shown to significantly improve the finesse of the resonance [START_REF] Väkeväinen | Plasmonic surface lattice resonances at the strong coupling regime[END_REF][START_REF] Félidj | Grating-induced plasmon mode in gold nanoparticle arrays[END_REF][START_REF] Auguie | Collective resonances in gold nanoparticle arrays[END_REF][START_REF] Giannini | Lighting up multipolar surface plasmon polaritons by collective resonances in arrays of nanoantennas[END_REF][START_REF] Kuznetsov | Laser fabrication of large-scale nanoparticle arrays for sensing applications[END_REF][START_REF] Zhou | Lasing action in strongly coupled plasmonic nanocavity arrays[END_REF][START_REF] Nikitin | Narrow plasmon resonances in diffractive arrays of gold nanoparticles in asymmetric environment: Experimental studies[END_REF][START_REF] Marae-Djouda | Angular plasmon response of gold nanoparticles arrays: approaching the rayleigh limit[END_REF], finding applications in metal-enhanced fluorescence [START_REF] Vecchi | Shaping the fluorescent emission by lattice resonances in plasmonic crystals of nanoantennas[END_REF], non-linear optics [START_REF] Czaplicki | Surface lattice resonances in second-harmonic generation from metasurfaces[END_REF] and the realization of plasmon-based lasers [START_REF] Yang | Real-time tunable lasing from plasmonic nanocavity arrays[END_REF]. Aluminum nanoparticle arrays have been shown to enhance and redirect the emission of nearby dye molecules, with potential applications including solid-state lighting [START_REF] Lozano | Plasmonics for solidstate lighting: enhanced excitation and directional emission of highly efficient light sources[END_REF][START_REF] Lozano | Tailor-made directional emission in nanoimprinted plasmonic-based light-emitting devices[END_REF]. Recently, a plasmonic sensor based on the lattice modes sustained by a one-dimensional aluminum grating has been demonstrated [START_REF] Liu | Fano coupling between rayleigh anomaly and localized surface plasmon resonance for sensor applications[END_REF]. It should also be emphasized that lattice modes can be observed in different periodic systems, such as hole arrays [START_REF] García De Abajo | Site and lattice resonances in metallic hole arrays[END_REF][START_REF] García De Abajo | Colloquium: Light scattering by particle and hole arrays[END_REF], arrays of magnetic nanoparticles [START_REF] Kataja | Surface lattice resonances and magnetooptical response in magnetic nanoparticle arrays[END_REF] and SiC nanocylinders arrays at midinfrared wavelengths [START_REF] Laroche | Tuning the optical response of nanocylinder arrays: An analytical study[END_REF].

In this paper, we report on a comprehensive and comparative study of the lattice modes supported by gold and aluminum gratings in the visible and UV regions. Dispersion plots are experimentally retrieved, allowing us to analyze the coupling between LSPs in the array. In the recent review, [START_REF] Ross | Optical properties of one-, two-, and three-dimensional arrays of plasmonic nanostructures[END_REF] the critical role played by the periodicity has been underlined in order to correctly position the Rayleigh anomalies in the spectrum with respect to the LSP and hence observe lattice modes. In the present work, we evidence the role of (i) the dielectric environment of the array and (ii) the polarization of the incident light, showing that the scattering pattern of the individual nanoparticles is the key parameter that controls the formation of the lattices modes. The paper is organised as follows. Section 2 is devoted to the design of our structures using numerical calculations. The role of the dielectric environment is unveiled by plotting the dispersion curves of the arrays. Then in Section 3 we present the experimental methods we used to fabricate and characterize the nanoparticles arrays. Results obtained on gold and aluminum nanoparticles arrays are presented in Sections 4 and 5. Finally, in Section 6 we summarize our results and conclude.

LINEWIDTH ENGINEERING WITH LATTICE MODES

A. Design and modelling

The structure under study is shown in Fig. 1a. It consists of a square array of nanoparticles (nanodisks) deposited onto a dielectric substrate with refractive index n 1 . The period of the array is denoted D, the diameter of the nanodisks is d and their thickness is h. Two cases were considered, depending on the refractive index n 2 of the superstrate (upper medium). In the asymmetric configuration, the upper medium is simply air (n 2 = 1). In the symmetric case, a superstrate whose refractive index is close to that of the substrate (e.g. a polymer layer) is added, so that the nanoparticles are effectively located inside a homogeneous environment.

To design the structures, we performed numerical modelling based on the finite difference time domain (FDTD) method [START_REF] Solutions | FDTD Solutions[END_REF]. 3D modelling was used, with PML as boundary conditions on the top and bottom sides of the simulation box, while periodic boundary conditions are applied on the remaining four sides. An adaptive mesh was used to precisely model the shape of the nanoparticle. Refractive indices were taken from Ref. [START_REF] Palik | Handbook of Optical Constants of Solids[END_REF] and [START_REF] Johnson | Optical constants of the noble metals[END_REF]. The incident light is a plane wave, impinging from the substrate side. Oblique incidence on the periodic structure was described using the approach presented in Ref. [START_REF] Aminian | Spectral FDTD: A novel technique for the analysis of oblique incident plane wave on periodic structures[END_REF]. It is worth noting that due to the peculiar nature of the investigated lattice resonances, a special attention must be paid to the PML delimiting the simulation box. Lattice modes are high Q-factor (long-lived) resonances with a significant spatial extension and propagates in a direction that is parallel to the PML layers. This may yield undesired reflections on the PMLs leading to a wrong estimation of the resonance width. Here, we used the stretched coordinates PMLs as implemented in [START_REF] Solutions | FDTD Solutions[END_REF] and performed exten-(±1,0) (0,±1) (±1,±1) Fig. 2. Normal incidence extinction cross-section spectrum calculated with FDTD for a square array of gold nanodisks with diameter d = 80 nm and thickness h = 50 nm, for different lattice periods and in a homogeneous dielectric environment (n 1 = n 2 = 1.5). Solid lines show the theoretical position of Rayleigh anomalies, and the dotted horizontal line is the position of the dipolar resonance from the isolated gold nanodisk. sive convergence tests.

B. Effect of the dielectric environment

We focus our attention on gold nanoparticles arrays first. Fig. 2 shows a map of the normal incidence extinction cross-section σ ext , as calculated using FDTD, for a square array of gold nanodisks with diameter D = 80 nm and thickness h = 50 nm plotted for different lattice periods and wavelengths. Here we consider a symmetric dielectric environnement, as the refractive indices of the substrate and superstrate are set to n 1 = n 2 = 1.5. Experimentally, this corresponds to a situation where the nanoparticles are deposited onto a substrate and subsequently covered with an index-matching superstrate. For shorter periods, the resonance position of the array, observed as a maximum in the extinction, is very close to the LSP resonance of an isolated particle (dotted horizontal line) but as the period increases, the resonance position red-shifts and simultaneously sharpens significantly. Moreover, the lineshape is also modified, from a relatively symmetric shape to a Fano-like lineshape (see the resonance near λ = 750 nm in Fig. 3, insert). To understand this behavior, we superimposed on the dispersion diagram the theoretical position of the Rayleigh anomalies. According to grating theory, in the most general case of a square grating with period D sitting on the interface between two media with refractive indices n 1 and n 2 , the position of the Rayleigh anomalies is given by:

λ m,p = D n 2 2 (m 2 + p 2 ) -n 2 1 p 2 sin 2 θ ± mn 1 sin θ m 2 + p 2 , ( 1
)
where θ is the angle of incidence, and m and p are integers corresponding to the diffracted order of the anomaly. Here the grating axes are aligned with the Cartesian axes (Ox) and (Oy), and the plane of incidence is the (zOx) plane. Note that in Eq. 1, m and p do not play symmetrical roles, as the symmetry of the grating is broken by the incident light. In the range of wavelengths and periods covered in Fig. 2, the (±1, 0), (0, ±1) and (1, ±1) Rayleigh anomalies can be excited and their position, according to Eq. 1, is shown in Fig. 2 by solid white lines. Note that at normal incidence on a square grating, the (±1, 0) and (0, ±1) diffraction anomalies are degenerated and fall onto the same line. It clearly appears that the wavelength position of the resonance closely follows the position of the (±1, 0) anomaly.

In fact, careful analysis of Fig. 2 shows an anti-crossing between the (1, 0) Rayleigh mode and the LSPR for a period of about 430 nm. A similar effect, although less prominent, occurs between the (1, 1) mode and the LSRP for a lattice period of about 600 nm. This behavior corresponds to strong coupling between the LSP resonance sustained by the nanoparticles in the array and the Rayleigh anomalies, determined by their periodic arrangement. The hybridization of the LSP with the light scattered in the plane of the nanoparticles at the Rayleigh anomaly provides an opportunity to combine the high extinction cross-section of plasmonic modes with the narrow bandwidth of the Rayleigh anomalies to design spectral lineshapes. The sharp and Fanolike lineshape of the lattice mode is more clearly seen on the line cuts shown as inserts in Fig. 3 for a period of 500 nm. The broader resonance around λ = 630 nm corresponds to the other hybrid mode.

More insight on the nature of both hybrid modes can be gained by plotting the normalized electric field at the wavelength corresponding to each of the modes, for different periods. Let us start with a period D = 300 nm, corresponding to the very left part of Fig. 2. The extinction spectrum shows only one relatively broad resonance around λ = 600 nm. The field map associated with that resonance is shown in Fig. 3a and evidences that the electric field is mainly localized around the gold nanodisks, as expected for isolated particles. If the period is increased to D = 500 nm, the extinction spectrum shows two resonances. The field map associated with the lower wavelength, broader resonance (Fig. 3b) evidences field maxima that are localized in the vicinity of the gold nanodisks, similarly to Fig. 3a and linked with the dipolar plasmonic resonance of the isolated nanodisk. On the other hand, the field map corresponding to the sharp resonance (Fig. 3c) evidences a simultaneous excitation of the metal nanoparticles along the polarization direction and a standing grazing wave (Rayleigh anomaly) in the perpendicular (Oy) direction. In other words, the coupling gives birth to two hybrid modes, one with a more "photon-like", delocalized nature, and the other one exhibiting a "plasmon-like", localized nature. As the plasmon-like mode is more localized than the photon-like mode, it experiences more Joule losses resulting in a much broader peak with a low quality factor. This is further confirmed by the field maps computed for an array exhibiting a period of 650 nm (Fig. 3d-e). Fig. 3d corresponds to the plasmonic mode, and Fig. 3e to the photonic mode, evidencing coupling with the (1, 1) Rayleigh anomaly at ±45 • for the axes of the array.

In order to underline the critical role played by the grating period, we computed the quality factor as a function of period for four different nanodisk diameters (Fig. 4a, note the log scale on the vertical axis). Note that in Fig. 4, the range of studied diameters and periods has been chosen to yield lattice modes in the visible and near-infrared spectrum. Extremely high values of Q can be obtained (Q > 1000) for high values of the lattice period, while an isolated gold nanodisk would exhibit Q ∼ 5 -10. Interestingly, the value of Q seems to continuously increase with the period. Also, higher values of Q are observed for smaller nanodisk diameters. The former effect can be understood as follows. When D increases, the lattice mode is getting closer to the position of Rayleigh anomaly (see Fig. 2). As a consequence, the hybrid mode becomes more and more "photonlike" in nature, and exhibits reduced losses. What will ultimately limit the value of the Q-factor is the size of the particles.

As the period increases the spectral position of the lattice mode red-shifts, hence the nanoparticles are becoming smaller with respect to the resonance wavelength. This implies that their scattering efficiency, and thus the diffractive coupling strength between the LSP resonances near the Rayleigh anomaly, should decrease for higher values of the lattice period.

To evidence the influence of the dielectric environment, we also computed the quality factor as a function of the period in the case of an asymmetric environment (glass substrate and air superstrate). Results are presented in Fig. 4b for different nanodisks diameters. Inside that range of periods, the observed behavior offers a striking difference with the symmetric case. Optimum values of the lattice period clearly appear that for each diameter, and this optimum value increases with the diameter of the nanodisks. Moreover, the obtained values of Q are more than one order of magnitude lower than in the symmetric geometry. Value in the range Q ∼ 10 -13 are obtained, which is roughly three times the value for an isolated nanodisk. We can also note that in order to obtain an efficient linewidth reduction, larger particles than in the symmetric case have to be considered. In other words, if we set the grating period, the linewidth reduction will be efficient for different diameters in different dielectric environments. Altogether, from a "linewidth engineering" perspective, working in a homogeneous environment is clearly beneficial. However, a significant linewidth reduction can still be observed in an asymmetric environment, provided the grating is properly designed.

The role played by the dielectric environment can be qualitatively understood as follows. In the asymmetric geometry, the radiation pattern tends to be highly asymmetric with most of the light scattered inside the substrate [START_REF] Lieb | Single-molecule orientations determined by direct emission pattern imaging[END_REF]. This leads to a weak angular overlapping with grazing diffracted orders. When the dielectric environment is symmetric, so is the radiation pattern -allowing for a much higher coupling into in-plane scattered light. This is why setting an adapted period for the array is then a necessary, but not sufficient, condition to observe high-Q lattice modes. Further evidence of the central role played by the radiative pattern of the nanoparticles will be given in Section 4.

It must be underlined that while the symmetric configuration yields by far the sharpest resonances, such a configuration is not without drawbacks from an experimental perspective [START_REF] Nikitin | Narrow plasmon resonances in diffractive arrays of gold nanoparticles in asymmetric environment: Experimental studies[END_REF]. First, the nanoparticles are isolated from their environment, limiting potential interactions with emitters or analytes. But most importantly, as the effective refractive index around the nanoparticles is higher in this geometry, the lattice modes are red-shifted with respect to the asymmetric geometry.

Obtaining high energy resonances with the symmetric geometry hence requires significantly smaller nanoparticles whose reliable fabrication is more demanding.

In the following we turn our attention on the experimental verification of our findings on gold and aluminum nanoparticles arrays.

EXPERIMENTAL METHODS

A. Sample fabrication

Arrays of gold and aluminum nanoparticles were made using electron beam lithography, which allows the realization of nanostructures with well-controlled size, shape and period [START_REF] Lérondel | Nanofabrication for plasmonics[END_REF]. The gold nanoparticles are 50 nm thick nanodisks and were deposited onto a glass substrate with a 3 nm thick TiO 2 layer to ensure adhesion of gold onto glass. Two different aluminum nanoparticle samples were made: d = 120 nm nanodisks exhibiting a LSP resonance in the visible and d = 60 nm nanodisks with a LSP in the UV spectrum. The aluminum thickness was set to 50 nm in both cases. Both structures were deposited onto fused silica to ensure transparency in the UV. Note that for aluminum no adhesion layer is required as this metal firmly adheres onto glass [START_REF] Martin | Localized surface plasmon resonances in the ultraviolet from large scale nanostructured aluminum films[END_REF]. The total size of the arrays is 50µm × 50µm.

The quality of the fabricated structures was characterized with both atomic force and electron microscopy. Typical images are shown in Fig. 1b for gold nanodisks and Fig. 1c for d = 120 nm aluminum nanodisks. Regular arrays are observed in both cases, but clearly the gold nanoparticles exhibit a more defined shape than the aluminum ones. This is a well-known difference in the thermal evaporation of both metals, as it is much more difficult to obtain smooth surfaces of aluminum because it tends to nucleate in the early stages of deposition [START_REF] Martin | Fabrication of aluminium nanostructures for plasmonics[END_REF]. After measurement of the arrays' dispersion curves (see below), the nanostructures were spin-coated with PMMA to obtain the symmetric geometry. This ensures that the comparison between the asymmetric and symmetric geometries is made on the same arrays. The thickness of the deposited PMMA layer is ∼ 180 nm. Let us underline that the refractive index of PMMA does not exactly match that of the glass substrate. However, as it is slightly lower, this ensures that no guided modes can propagate inside the PMMA layer and interact with the LSPR.

B. Optical characterization

The optical properties of all samples were measured using an angle-resolved extinction spectroscopy set-up. This setup allows the dispersion curves of the arrays to be retrieved, that is, the extinction plotted as a function of the angle of incidence and the wavelength (or, equivalently, the wavevector and the energy). The set-up works over a broad spectrum, from the UV to the near-infrared. Collimated light from a UV/visible lamp (Avalight-DHS, Avantes) is slightly focused onto the sample, which can be rotated by a controllable angle θ with regards to the incident beam. The light transmitted by the sample is collected using a microscope objective lens (MPlan UV 50x, Mitutoyo) confocally coupled to an optical fiber (solarisation resistant fused silica fiber, core diameter 100µm) to define a collection area corresponding to the sample size. Finally, the output of the fiber is sent towards a spectrometer (IsoPlane SCT320, Princeton Instruments) equipped with a cooled camera (ProEM 1600 2 , Princeton Instruments). The polarization of the incident light is controlled by a broadband polarizer placed in the illumination path. To obtain the dispersion curves shown below, we measured an extinction spectrum for every 4 • , from θ = 0 • up to θ = 48 • . Note that in the following, the arrays are illuminated from the glass substrate and the angle of incidence used as the horizontal axis is measured in air (i.e. before refraction inside the substrate).

GOLD NANOPARTICLE ARRAYS

Fig. 5 presents the dispersion curves of an array of gold nanoparticles for s-polarization (electric field parallel to the yaxis) and p-polarization (electric field in the xz-plane) in the asymmetric geometry, that is, for gold nanoparticles in air on a glass substrate. In the left panels are plotted the experimental results, while the right panels show the corresponding FDTD simulations. On Fig. 5 are also plotted the position of the Rayleigh anomalies, according to Eq. 1. The two polarizations exhibit slightly different behaviors. The p-polarization shows a broad and relatively "flat" dispersion branch corresponding to the plasmon resonance of this branch, while in s-polarization an interaction with the (1, 0) glass/air Rayleigh anomaly appears. The agreement with the numerical simulations is very good.

The most important difference between p and s-polarizations concerns the (1, 0) glass/air anomaly (dotted white line in Fig. 5). In s-polarization, the anomaly has a detrimental effect on the plasmon branch, weakening its intensity for high angles (see Fig. 5d) while in p-polarization the anomaly seems to increase the absorbance. However, globally the effect of the diffraction anomalies is weak, confirming what we saw in Section 2: to obtain significant hybridization, a symmetric dielectric environment is required. Next, we consider the case of the "symmetric" dielectric environment, i.e. after deposition of a ∼ 180 nm layer of PMMA onto the previously measured array. While such an environment cannot be considered as rigorously homogeneous due to the slight refractive index difference between glass (n = 1.52) and PMMA (n = 1.49), this configuration is close enough to the one discussed in Section 2. Experimental results are reported in Fig. 6a for p-polarisation and Fig. 6b for s-polarization. Comparison with Fig. 5 shows a clear effect of the change in the dielectric environment. Notably, for s-polarization a clear bending of the LSP branch is observed as it approaches the (1, 0) symmetric Rayleigh anomaly. The resulting lattice mode presents a sharper profile: the quality factor of the lattice mode is found to be Q = 15 at normal incidence, increasing to Q = 28 at θ = 12 • . This is a clear demonstration of the beneficial influence of generating lattice modes inside a symmetric dielectric environment.

More insight can be gained by comparing the two polarizations. Considering Fig. 6a, it is remarkable that the (+1, 0) anomaly (corresponding to a grazing wave propagating along the x direction) crosses the main dispersion branch with limited perturbation. In contrast, in the other polarization a strong interaction between the same anomaly and the LSPR is evidenced (Fig. 5b), with a bending of the band and the appearance of a Fano lineshape similar to those of Fig. 3. In other words, to be significantly active and hence allow the creation of lattice modes with sensible coupling, an anomaly must be coupled in a direction perpendicular to the incident polarization. This effect can be explained qualitatively by considering how a single nanoparticle scatters light. If we assume the nanoparticle to behave as a dipole, then its far-field radiation will be perpendicular to the induced polarization, which is along the polarization of the incident electric field. In other words, as each nanoparticle scatters light in a direction perpendicular to the incident polarization, it will preferentially couple with grazing diffractive orders along that direction. This can be ascertained by looking again at Fig. 3c, showing the electric field map at the wavelength corresponding to the excitation of a lattice mode. While the polarization is horizontal in the figure, it excites a grazing wave in the vertical direction. It is interesting to note that for non-square gratings, it implies that the position of the Rayleigh anomaly is governed by the period along the direction perpendicular to the incident polarization.

The bending of the plasmonic branch happens after interaction with the symmetric (1, 0) Rayleigh anomaly. In principle, in a homogeneous dielectric environment the asymmetric glass/air anomalies should not affect the LSP resonance. However an effect can be seen on the experimental measurements for both polarizations (dotted line in 6a-b). To understand this unintuitive behavior, we performed additional numerical simulations. In Fig. 6c and 6d are plotted the simulated dispersion curves for a glass substrate (n 1 = 1.52) covered with a semi-infinite PMMA superstrate (n 2 = 1.49). A number of discrepancies with the measurements are obvious: the position of the LSP branch is red-shifted, and the high-angle behavior is totally different, especially in s-polarization where the secondary branch, experimentally observed, is not predicted by the simulations. In order to better match the experimental geometry, we performed additional simulations by replacing the semi-infinite superstrate by a finite-thickness PMMA layer. Results are presented in Fig. 6e-f. It can be seen that now in s-polarization a branch around the (1, 0) glass/air anomaly is visible. However there is still a significant red-shift in the position of the resonance when compared to measurements. Such a shift could be explained if the nanoparticles are experiencing a lower effective refractive index in the experimental configuration, for instance in the case of the presence of air inclusions in the polymer film.

To check the validity of this hypothesis, we assumed that, when spin-coated the polymer did not perfectly fill the space between nanoparticles, but rather formed a film on top of the nanoparticles, with air between each nanoparticle. Results are shown on Fig. 6g-h, evidencing a good agreement with the experiments for both polarizations: the resonance position is close to the experimental one. Now we can sum up our results concerning the effects of both polarization and dielectric environment into the following rule of thumb: the coupling strength between LSPs near the Rayleigh anomalies is mainly governed by the radiation pattern of the single scatterers, and how it spatially and angularly overlaps with the grazing diffractive orders. More precisely, the Fano splitting (i.e., the coupling strength) is governed by three physical parameters: (i) the spectral overlap between the initial modes, (ii) modal (i.e., spatial) overlap between the initial modes, and (iii) the initial linewidth.

ALUMINUM NANOPARTICLE ARRAYS

In this Section we apply our previous findings to the LSP sustained by aluminum nanoparticle arrays. As pointed out in the introduction, aluminum has a plasmonic response extending over a broad spectral region but the amount of losses, relative to gold, increases significantly for energies lower than the UV. Here, we discuss the possibility of using lattice resonances to reduce the amount of losses in two different spectral ranges: the first sample is designed to work in the visible range, the second sample is designed for the blue-UV range.

A. Lattice modes in the visible

The first sample has been designed to exhibit resonances in the visible spectrum. We fabricated arrays of aluminum nanodisks with a diameter of 120 nm and three different periods: D = 250, 300 and 350 nm. Since a symmetric dielectric environment yields the sharpest resonances, the sample was covered with PMMA. The measured dispersion curves are presented in Fig. 7 for both polarizations. As the diameter of the metallic nanoparticles is kept constant while the period is changing, we are effectively moving the spectral position of the Rayleigh anoma- lies for a fixed LSP resonance. The effect is clearly seen for spolarization comparing Fig. 7a, b, andc. In Fig. 7a the anomaly is located far from the LSP, and a weak interaction is observed with the broad LSP resonance. At normal incidence the observed linewidth is quite similar to the one of an isolated aluminum nanoparticle (Q ≃ 2.5). When the period is increased to D = 300 nm, the position of the (1, 0) Rayleigh anomaly shifts toward higher wavelengths and a stronger coupling of the LSP resonances is observed as a bending of the plasmon branch and linewidth reduction when the angle of incidence is increased. At normal incidence, the linewidth is significantly lower than previously (Q ≃ 5) and the resonance further sharpens when the angle of incidence increases (Q ≃ 9.5 for θ = 32 • ). Finally, for D = 350nm, the anomaly is very close to the LSP resonance, leading to a stronger coupling and the apparition of a sharper lattice mode (Q ≃ 9 at normal incidence). For p-polarization (Fig. 7d-e), we obviously observe the same linewidth at normal incidence due to the symmetry of the nanoparticles, but the angular dependence is different, as no bending and linewidth reduction are observed when θ increases. This is due to the weak coupling efficiency between the LSPs excited in p-polarization near the (1, 0) Rayleigh anomaly. We also note that similarly to what happened with gold nanoparticle arrays, the glass/air Rayleigh anomaly influences the dispersion curve (see Fig. 7e), while it should be absent in a homogeneous environment. As previously, this is likely due to the finite thickness of the PPMA layer and the presence of air inclusions.

The obtained quality factors can be compared to the quality factor for isolated Al nanoparticles, which is generally Q ≈ 3 -4 in the visible range [START_REF] Martin | High-resolution imaging and spectroscopy of multipolar plasmonic resonances in aluminum nanoantennas[END_REF]. Hence, the linewidth engi- neering enabled by lattice modes allows Q-factors to be reached which are comparable to isolated gold nanoparticles in the visible. It is also interesting to compare Fig. 7b and7c. In the latter, the anomaly is very close to the LSP spectral position, leading to a sharp resonance at normal incidence whose intensity dramatically decreases when the angle of incidence increases. In contrast, in Fig. 7b, the normal incidence linewidth is higher (due to the higher frequency mismatch between the LSP and the anomaly) but as the angle is increased the resonance sharp-ens without being too detrimental to the lattice mode intensity. Consequently, if sharp lattice modes at normal incidence are sought, then the Rayleigh anomaly should be designed to appear as close as possible from the LSP resonance. But if angular tolerance is required, then a minimal spectral mismatch should be introduced between the anomaly and the LSP.

B. Lattice modes in the blue-UV

Finally, we present results obtained on aluminum nanodisk arrays that have been designed to exhibit a LSP resonance in the blue and near-UV region. As explained in the introduction, aluminum can sustain LSP resonances down to deep-UV wavelengths, however the particle diameter is a strong practical limitation. We made d = 60 nm diameter nanoparticles as a trade-off between exhibiting UV resonances and fabrication constraints. The period of the array is D = 260 nm. We chose the superstrate to be air to avoid any red-shift of the plasmon resonance with the dielectric environment.

The experimental dispersion curve for s-polarization is presented in Fig. 8. A clear interaction between the LSP resonance around λ = 420 nm and scattered light near the (1, 0) Rayleigh anomaly is observed, leading to hybridization and the appearance of a sharper resonance. Under normal incidence, we obtain Q ≃ 8, a value that should be compared to the Q ≈ 4 -5 obtained for isolated aluminum particles [START_REF] Martin | High-resolution imaging and spectroscopy of multipolar plasmonic resonances in aluminum nanoantennas[END_REF].

CONCLUSION

In summary, we have reported a study of the lattice modes, arising from diffractive coupling between localized plasmon modes sustained by individual nanoparticles near Rayleigh anomalies of the periodic array. Our results evidenced the crucial role played by the dielectric environment and the incident polarization on the coupling strength and subsequent formation of lattice modes. It was shown that the hybridization between the localized plasmon resonances and the formation of the lattice modes is mainly controlled by the radiation pattern of the individual metal scatterers forming the arrays. If carefully designed, nanoparticles arrays can exhibit lattice modes with remarkably high quality factors. For arrays of gold nanoparticles, it is possible to reach Q-factors of several hundreds. In the case of aluminum nanoparticles used in the visible spectrum, Q-factors of the same magnitude as those obtained for isolated gold nanoparticles were obtained, in spite of the relatively higher intrinsic losses of Al compared to gold. Moreover, it should be noted that the quality of the lithographed aluminum nanostructures was lower than the gold ones, which might have played a detrimental role on the Q-factor. As mentioned earlier, this is mainly due to the surface roughness of the evaporated Al film, which is difficult to master [START_REF] Martin | Fabrication of aluminium nanostructures for plasmonics[END_REF]. However this limit can be lifted by using, for instance, the newly developed epitaxial aluminum films [START_REF] Cheng | Epitaxial growth of atomically smooth aluminum on silicon and its intrinsic optical properties[END_REF] to obtain high-quality Al nanostructures exhibiting low scattering and absorption losses. Altogether, we believe our results are a further demonstration of the potential of aluminum as a broadband plasmonic material.

FUNDING INFORMATION

CNRS and The Royal Society joint grant "Alu3D". DK is supported by Région Champagne-Ardenne. Financial support of Nano'Mat (www.nanomat.eu) by the "Ministère de l'enseignement supérieur et de la recherche", the "Conseil régional Champagne-Ardenne", the "FEDER fund" and the "Conseil général de l'Aube" is acknowledged. This work was also partially supported by the HPC Center of Champagne-Ardenne ROMEO. W.P.W., W.D. and A.V.Z. work is supported by EPSRC (UK). A.V.Z. acknowledges support from the Royal Society and the Wolfson Foundation.

Fig. 1 .

 1 Fig. 1. (a) Schematic of the considered geometry: an array of metal nanoparticles deposited onto a dielectric substrate of refractive index n 1 . The superstrate can be air or a dielectric with refractive index n 2 . (b) AFM image of a square array of gold nanodisks with D = 480 nm, d = 160 nm and h = 50 nm. (c) SEM image of an array of aluminum nanodisks with D = 300 nm, d = 120 nm and h = 50 nm.

Fig. 3 .

 3 Fig. 3. Normalized electric field modulus maps for different lattice periods. The insets on the top of each of the columns are line cuts of Fig. 2, showing the extinction cross-section σ ext vs. wavelength for three different periods: D = 300 (left column), D = 500 nm (center column) and D = 650 nm (right column). The arrows indicate the wavelengths where the field maps have been calculated. (a) FDTD maps of the normalized electric field modulus computed in a plane located at mid-height of the metal nanoparticles, for an array with D = 300 nm, at λ = 630 nm. The black arrow denotes the incident polarization. The same field maps have been computed for: (b) D = 500 nm and λ = 630 nm; (c) D = 500 nm and λ = 750 nm; (d) D = 650 nm and λ = 630 nm; (e) D = 650 nm and λ = 700 nm.

Fig. 4 .

 4 Fig. 4. Linewidth engineering using lattice modes. (a) Calculated quality factor of the lattice mode of arrays of gold nanodisks as a function of the lattice period, for different disk diameters, at normal incidence. The dielectric environment is symmetric (n 1 = n 2 = 1.5). (b) Same, with air as the superstrate (n 2 = 1).

Fig. 5 .

 5 Fig. 5. Experimental (a, c) and simulated (b, d) dispersion curves of the gold nanodisks arrays for s (a, b) and p (c, d) polarizations. The period of the array is D = 480 nm, the diameter of the nanodisks is d = 180 nm. White lines represent the position of the Rayleigh anomalies computed using Eq. 1 with n 1 = 1.52 and n 2 = 1.52 (solid lines) or n 2 = 1 (dotted lines).

Fig. 6 .

 6 Fig. 6. Experimental (a-b) and simulated (c-h) dispersion curves of the PMMA-covered gold nanodisks arrays for s and p polarizations. The period of the array is D = 480 nm, and the diameter of the nanodisks is d = 180 nm. White lines represent the position of the Rayleigh anomalies computed using Eq. 1 with n 1 = 1.52 and n 2 = 1.52 (solid lines) or n 2 = 1 (dotted lines). The simulations were performed using three different geometries as pictured in the insets: (c-d) a semi-infinite PMMA superstrate; (e-f) a 180-nmthick PMMA layer onto the top of the substrate, (g-h) a 180-nm-thick PMMA standing onto the top of the nanoparticles.

Fig. 7 .

 7 Fig. 7. Experimental dispersion curves of PMMA-covered aluminum nanodisk arrays for: (a-c) s-polarization, and (d-f) ppolarization. The diameter of the nanoparticles is d = 120 nm. The period of the array is: (a,d) D = 250 nm; (b,e) D = 300 nm; and (c,f) D = 350 nm. White lines represent the position of the glass/glass (solid lines) or glass/air (dotted lines) Rayleigh anomalies computed using Eq. 1.

Fig. 8 .

 8 Fig. 8. Experimental dispersion curve of the d = 60 nm aluminum nanodisk arrays in the ultraviolet for s-polarization. The period of the array is D = 260 nm. White lines represent the position of the glass/glass (solid lines) or glass/air (dotted lines) Rayleigh anomalies computed using Eq. 1.

ACKNOWLEDGMENTS

This work was performed during D.G.'s stay at King's College London as a Visiting Senior Research Fellow, thanks to support from the CNRS and the Région Champagne Ardenne. We are most grateful to Loïc Le Cunff for his help with FDTD calculations. The data access statement: all the data supporting this research are presented in full in the results section and supplementary materials.


P Vallée

T Robert

Influence of the strategies and biomechanical parameters on the capacity to handle balance perturbation: a numerical assessment

Keywords: Balance, fall, strategies, ageing

Introduction

Fall is a major issue especially in the communitydwelling elderly population. Consequences can be extremely severe (fractures or even death) and lead to a loss of autonomy (World Health Organization, 2008). Identifying balance recovery (BR) parameters which strongly influence the success of BR process would be a very interesting track in order to train elderly on their identified physical weaknesses and consequently improve their ability to restore their balance. BR process has been primarily investigated using an experimental approach. It allowed, for example, to define the BR characteristics such as the step length, ageing effect, timing, etc., for different population and experimental conditions [START_REF] Hsiao-Wecksler | The effect of step length on young and elderly women's ability to recover balance[END_REF][START_REF] Mcilroy | Age-related changes in compensatory stepping in response to unpredictable perturbations[END_REF][START_REF] Wojcik | Age and Gender differences in single step recovery from a forward fall[END_REF]. However, it is extremely complex to experimentally manipulate the different neurological or biomechanical parameters influencing the BR. As such, there still is a lack of knowledge about their role and influence on the BR process and on the outcome of a given perturbation. To fill this gap, there is a need of a BR model that: 1/ includes several BR strategies, in particular the change of support type of strategy (e.g. protective steps); 2/ describes the use of these strategies by parameters that are linked to neurological and biomechanical factors; 3/ is able to predict the outcome of a perturbation;4/ is validated against human experimental data.

We recently proposed such a model and two sets of parameters representing young and elderly population group. This model matched experimental observations about the maximal release angle that can be handled using the fixed support strategies and a protective step [START_REF] Robert | Assessment of a numerical model of balance recovery by stepping[END_REF][START_REF] Vallée | Possible recovery or unavoidable fall?A model to predict the one step balance recovery threshold and its stepping characteristics[END_REF]. Through this paper we propose to use this model to investigate the influence of the parameters describing the recovery actions on the maximal initial release angle that can be handled. The analysis will be performed at the level of the BR strategies (ankle, hip and protective stepping) and, through a sensitivity analysis, at the level of the neurological (timings) and biomechanical parameters (strength, ability to step, BoS limits) describing the use of these strategies. .For concision, these analyses will only be presented for the set of parameters representing the elderly population.

Methods

Model, recovery reactions and BR condition

The human body is represented as an Inverted Pendulum (IP) mounted on a massless foot-like Base of Support (BoS), with a concentered mass at the Center of Mass (CoM) and with a flywheel (FW) centered at the CoM.The three main BR strategies are thus represented and supposed to be used at their maximum: 1/ After a reaction time (RT) the Center of Pressure (CoP) is instantaneously shifted toward the edge of the BoS (ankle strategy); 2/ Also after the RT the FW is launched following a bang-bang acceleration profile constrained by its maximal torque (T max ) and its maximal angle of rotation (α max ). This profile represent the segments' angular momentum effects (hip strategy); 3/ After an additional delay from the RT called Step Preparation Time (SPT), a protective step can be triggered. It represents a BoS extension which is bounded by the swing foot maximal acceleration (A max ) and, if reached, by its maximal final position (L step ). The parameters describing the use of these strategies for the elderly population group are given in Table 1.

In this study we foccus on tether-release perturbations: the model is released from an initial forward leaning posture. The BR threshold is the maximal release angle from which the model could restore its balance.

Parameters assessment

Firstly we determine the BR threshold for different combinations of the three recovery strategies (ankle, hip and protective stepping strategy can be cancelled by setting the Functional BoS, T max or L step to zero, respectively).

*Corresponding author. Email: thomas.robert@ifsttar.fr

Then we performed a sensitivity analysis on the BR parameters. Each parameter is varied from 50% to 150% of its initial value and the obtained thresholds (θ) are compared to the reference threshold θ ref . A linear regression is then fitted between the percentages of variation of the studied parameter and the induced percentages of variation of θ. The slopes (K) represent the influence of each parameter on the BR threshold. The stepping strategy is from far the most influent strategy (see Figure 1). The influence of the ankle and the hip strategy appears to be slightly equivalent. The sensitivity analysis confirmed this first approach. Parameter variations are linearly linked to the induced variations of the BR threshold (see R² in Table 1). The maximal step length (L step )and the SPT are the most influent parameters with a K coefficient of 0.8 and -0.5, respectively. An increase of L step by 10cm or a decrease of SPT by 10ms induces an increase of the BR threshold of 2° and 0.7° respectively. Other parameters as the RT, α max , A max and functional BoSsize have a smaller influence on the threshold evolution with a K value between -0.26 and 0.3. This model showed that the most important strategy to be able to recover a large TR perturbation is the ability to perform a recovery step. This step has to be the furthest possible but also has to be initiated as soon as possible with a reduction of the SPT corresponding to anticipatory postural adjustments (APA) needed to trigger the step. The influence of the RT is less important because its duration is already short compared to SPT (see Table 1).

Results and discussion

Conclusions

To sum up, we propose an investigation of biomechanical parameters of a numerical model previously validated with experimental data from the literature. First results confirm that triggering a step is a very effective way to handle large perturbations (more than using fixed-support strategies). But this model also provided a quantitative evaluation of other BR parameters which cannot be investigated experimentally. Further developments have to be done in order to integrate more complex and realistic perturbations in various directions but also other BR reactions as for example grasping. Nonetheless these results highlight the importance of the protective stepping on the ability to avoid a fall and provide interesting insight into the detection and care of the person at higher risk of fall.

Figure 1 :

 1 Figure 1: One step BR threshold angle according to strategies enabling (A: Ankle; H: Hip; ø: No strategy)

Table 1 :

 1 Elderly parameter set from Vallée et al. 2015. Coefficient of determination is displayed as R² value and the influence on the BR threshold is represented as the K value.

		Reference	R²	K
		value
	RT (ms)	80	0.99 -0.26
	SPT (ms)	180	0.98 -0.52
	A max -²)	145	0.98 0.28
	L step (% body height)	55	0.99 0.74
	Functional BoS (% BoS) 42	0.99 0.15
	T max (N.m)	150	0.99 0.15
	α max (rad)	π/4	0.86 0


Frédéric Laux

Nicolas Bonod

Davy Gérard
email: davy.gerard@utt.fr

Single emitter fluorescence enhancement with surface lattice resonances

Surface lattice resonances (SLRs) are high quality factor resonances arising in periodic arrays of nanoantennas that offer unique opportunities to enhance light-matter interactions. SLRs have been successfully used to enhance non-linear signals, magnetooptical effects and also to enhance the fluorescence signal from quantum emitters. However, the high potential of SLRs to tailor light emission of single photon sources is still unexplored. In this study, we report numerical investigations of light emission from a single emitter coupled to SLRs. We demonstrate that the excitation of SLRs with an impinging plane wave allows for the enhancement of the local field intensity by several orders of magnitude, much higher than what can be achieved with localized resonances.

We also study the alteration of the decay rate of the emitter (Purcell effect) inside the array, evidencing an emission enhancement similar to the one of the localized resonance of a single antenna. Finally, we plot the far-field radiation patterns of the arrays and show that they are controlled by the period of the arrays and the emission wavelength, enabling photon sorting with the emission angle. Altogether, our study demonstrates a new route to design high-brightness sources based on single emitters.

Introduction

Optical antennas are nano-sized devices transducing optical energy from the near-field (localized energy) to the far-field (propagating waves), and vice-versa. 1 This unique ability to control light at the nanoscale has been massively used to engineer the radiative properties of emitters. [2][3][4] Nanoantennas are often based on metallic nanostructures sustaining localized surface plasmon resonances (LSPRs), optical modes that are confined within a few nanometers of the metal surface. 5 Light emitters, such as quantum dots or fluorescent molecules, coupled to nanoantennas have been reported to exhibit directional emission, 6 high brightness, 7,8 ultrafast emission 9 and high quantum yield. 10 While most of the reported studies deal with individual nanoantennas, arrays of nanoantennas are of special interest. If the nanoantennas are arranged into a periodic array whose period is commensurate with the LSPR's wavelength, diffractive coupling between the antennas can be observed. 11 In this case, the existence of the so-called Rayleigh anomalies, corresponding to diffraction of light in the plane of the array, 12 allows for long distance communication between the nanoantennas. As a consequence, the localized and spectrally broad plasmon resonance of each nanoantenna can interfere with in-plane scattered light in a narrow spectral range near the position of the Rayleigh anomaly, leading to a Fano lineshape. [13][14][15][16][17][18][19][20] This resulting mode is known as a surface lattice resonance (SLR), or lattice plasmon mode. SLRs are spectrally narrower than LSPRs, which means that they exhibit lower inherent losses.

To understand the interest of SLRs for fluorescence enhancement, let us consider the canonical model of a quantum emitter inside a cavity. In this case, the Purcell factor, the physical quantity driving the acceleration of the total decay rate of the emitter, writes: 3,21

F P = 3 4π 2 Q V λ 2n 3 , (1)
where V is the effective volume of the cavity mode, 22,23 and Q the quality factor of the cavity resonance, defined as Q = λ/∆λ. Eq. 1 indicates that high Q-factor plasmonic reso-nances should lead to high Purcell factors and are thus extremely appealing for fluorescence enhancement. The coupling of fluorescent emitters to arrays of nanostructures has been widely studied, [24][25][26][27] including arrays sustaining SLRs. [28][29][30][31][32][33] However, those works focused on the case of ensembles of emitters. Studying the emission from single emitters is of critical importance to design efficient single-photon sources based on SLRs. 4 Herein, we study how a periodic array of plasmonic resonators sustaining high quality factor SLRs affects the fluorescence enhancement of a single emitter. Two geometries of metallic nanoantennas are considered: (i) the simplest case of single nanocylinders, and

(ii) dimers of nanoparticles, as this system has been reported to yield extremely large fluorescence enhancement factors. 7 Using numerical calculations based on the finite-difference time-domain (FDTD) method, we show that SLRs allow to reach significantly higher values for the excitation enhancement (due to higher local field intensity). At the same time, we find that the emitter's emission rate (as modeled by an electric dipole) reaches a value comparable to what is obtained with an isolated nanoantenna. Altogether, we demonstrate that SLRs brought a significant advantage to increase the fluorescence brightness of single emitters in comparison to isolated nanoantennas.

Results

We consider a square array of gold nanoantennas. The period of the array is P , the height of the nanoantennas is h. We consider two different nanoantenna geometries. The first geometry, depicted in Figure 1a, is the nanodisk antenna, corresponding to the canonical case of an emitter coupled to a single gold nanodisk, whose diameter is denoted d. The emitter is located on the side of the nanocylinder, 10 nm from the metal surface, at mid-height, and is perpendicular to the metal surface. The emitter-surface distance and the emitter's orientation have been chosen because they correspond to a very favorable case for emission enhancement. 34,35 The second geometry (Figure 2a) consists of a dimer of nanodisks, also known as a gap antenna: two gold nanodisks (with the same diameter d) are separated by a 20-nm-gap. This geometry is known to yield an extremely large field enhancement inside the gap between the two nanodisks. In this configuration the emitter is located in the center of the gap, at mid-height, parallel to the dimer axis. For the sake of simplicity, for both cases the array is supposed to be in a homogeneous dielectric environment with a refractive index n = 1.5. It has also been demonstrated that a homogeneous environment is the most favorable geometry to obtain sharp lattice modes. 20 All computations were performed using a commercial finite difference time-domain (FDTD) method (see Methods for details).

To evidence the presence of surface lattice resonances, we start by computing the extinction spectrum of both nanoantenna arrays. Normal incidence extinction spectra have been computed for an infinite array of gold nanodisks with diameter d = 100nm and height h = 50nm, for periods ranging from 300 to 600 nm. The resulting dispersion curve is shown in Figure 1b. On Figure 1b are also plotted as solid white lines the positions of the Rayleigh anomalies, obtained by the following expression:

λ m,p = nP m 2 + p 2 (1 -sin 2 θ) ± m sin θ m 2 + p 2 , ( 2
)
where θ is the angle of incidence, n is the refractive index of the embedding medium, and m and p are integers corresponding to the diffracted order of the anomaly. 20 For shorter periods, the dispersion curve evidences a broad resonance around λ = 700 nm corresponding to the (dipolar) LSPR of the nanodisks. When the period increases, the resonance shifts toward the position of the Rayleigh anomaly and significantly sharpens. This effect results from the coupling between the LSPR and the in-plane scattered light 20 and is best seen in the line cuts presented in Fig. 1c. On the same figure is also plotted the extinction spectrum for an isolated nanodisk, evidencing the remarkable sharpening of the resonance peak obtained with lattice modes. The quality factor of the resonance, as well as its spectral position, can be tuned by changing the period. Fig. 1d shows the value of the quality factor Q as a function of the period (blue dots). It appears that the quality factor obtained with a SLR can be two orders of magnitude higher than the one of an isolated nanodisk. Similar effects are observed for an array of gap antennas (see Figure 2b-d). However, in comparison with nanodisks, gap antennas yield lower Q-factors as near-field coupling between the particles forming the dimer broadens the resonance.

Quantum yield enhancement

Now we turn our attention to the fluorescence emission enhancement. We model the emission by considering a single emitter (electric dipole) located inside a finite array of nanoantennas. The use of a finite array is due to numerical reasons: infinite arrays being modeled with periodic boundary conditions on the sides of the simulation box, this will lead to the computation of the emission from an infinite array of emitters oscillating in phase. Hence, we compute the emission from a single dipole, coupled to an antenna located at the center of a 25 × 25 antenna array. Then, we compute the radiative enhancement factor f r , the non-radiative factor f nr and the quantum yield enhancement factor f η = η/η (0) (see Methods). The emitter is assumed to exhibit an initial quantum yield η (0) = 1% (corresponding to the quantum yield inside the homogeneous medium without any antenna). Results are summarized in Fig. 3 for a finite array of gold nanodisk antennas, and in the S.I. (Figure Figure 3a shows the non-radiative factor f nr as a function of the wavelength and the period of the array. It evidences clearly that, whatever the period, the coupling with the array generates non-radiative losses in the spectral range around λ = 520 nm. This wavelength corresponds to the onset of interband transitions in gold, showing that non-radiative processes are mostly governed by the nature of the metal and are not affected by the geometry of the array. A closer inspection of Figure 3a evidences a slight increase of the non-radiative factor in the range 550-650nm, weakly depending on the period, which corresponds to the excitation of higher-order LSPRs (quadrupolar resonance). The radiative rate (Figure 3b) is enhanced in a relatively broad spectral range corresponding to the position of the dipolar LSP resonance. Looking closely at Figure 3b, it appears that the spectrum is slightly modified when the period is changed. However, the amplitude of the variation is limited: for instance the value of f r oscillates between 25 and 20 at λ = 700 nm. The quantum yield enhancement (Figure 3c) exhibits a very similar behavior: changing the period of the array does not imply dramatic effects on the decay rate of a single emitter. The amplitude of variation is further illustrated in the spectra from Figure 3d, showing as a black line the quantum yield enhancement factor for a single emitter coupled to a single nanoantenna, and in red the superimposition of the spectra for arrays with periods from 300 to 600 nm. Figure 3d demonstrates that the array does not bring any real beneficial effect with respect to a single nanoantenna -the maximum relative gain in quantum yield is less than 1%. Very similar results are observed for the gap antennas (see S.I.). Figure S1d shows that arrays of gap-antennas do not bring any gain in quantum yield when compared to a single gap antenna.

(±1,0) (0,±1) (±1,±1)
After what was observed in the previous Section for the excitation enhancement, this result might appear as a surprise. It was shown previously that the field monitored at a given wavelength λ in the vicinity of a nanoantenna could be significantly enhanced providing those antennas were arranged into an array. Conversely, it seems that an emitter located at the same position, and emitting at the same wavelength λ inside the same array, emits light at roughly the same rate it would do near a single antenna. In other words, while SLRs can significantly enhance the excitation rate of a single emitter, they appear to be useless for the enhancement of its emission. However, the array significantly impacts the decay rates in the case of an array of emitters coupled to nanoantennas, all emitting in phase. 29,30 Such a situation is considered in Figure 4, where we compare the radiative factor of a single emitter (blue line) and of an array of coherent emitters (red line), both coupled to the same array of gold nanodisks. A striking difference is observed between the curves. For the array of emitters, one retrieves the Fano shape, which is characteristic of the extinction spectrum of SLRs (see Figure 1c), associated with high values of the radiative factor (f r ≃ 140). In contrast, the single emitter exhibits a broad spectrum similar to the extinction spectrum of a single gold nanodisk, with limited values of the radiative factor (f r ≃ 20).

Generally speaking, the quantum yield enhancement factor critically depends on the value of the intrinsic quantum yield of the emitter. 36 However, we underline that our results do not depend on our arbitrary choice of η (0) = 1%. Computations have been performed for different values of η (0) and show no clear influence of the array, even for extremely high or low values of the intrinsic quantum yield (see S.I., Figure S2).

Beaming and radiation pattern

The modification of the quantum yield is not the only effect that can affect fluorescence emission. Interaction with a nanoantenna can also modify the emitter's radiation pattern, leading to light beaming or redirection. 6 In the case of arrays of nanoantennas, it has been shown recently that the beaming effect can be reinforced 37 or give rise to structured emission. 17,38,39 Structured emission is well-known in antenna phased arrays and corresponds to higher emission in specific directions where the scattered light from each antenna interferes constructively, due to the phase relation between each scatterer. [START_REF] Balanis | Antenna theory[END_REF] Therefore, even if the emission rate is not impacted by the lattice mode, the collection efficiency of the emitted light will be improved if beaming effects exist.

To ascertain the existence of beaming, the far-field radiation patterns were calculated for different sizes of the array. We started from the emitting dipole alone, then the emitter coupled to a single antenna, and then coupled to arrays with increasing sizes (3 × 3, 5 × 5, 9 × 9 and 15 × 15 nanoantennas). We chose to plot the antenna directivity D(θ, ϕ), which measures the ability of an antenna to direct and concentrate light in a given direction (θ, ϕ). [START_REF] Novotny | Principles of Nano-Optics[END_REF] Results are presented in Fig. 5 for arrays of nanodisk antennas with a period P = 500 nm, and an emission wavelength λ = 750 nm. It appears that the formation of the array strongly affects the emission pattern, the fluorescence light being gradually beamed into two emission lobes as the size of the array is increased. emitter is located in the center of the gap, are perfectly symmetric (see Figure S3). To understand the formation of the far-field radiation pattern, it is interesting to plot the position of the diffraction orders in the Fourier space, or wavevector diagram. This approach has been previously successfully applied by Rigneault and coworkers to model dipole emission inside a grating for light extraction applications. 24 The position of the first orders of diffraction (for a plane wave under grazing incidence) has been plotted as white lines in Fig. 6a-e. In the approximation where the periodic structure is only weakly perturbing the dipole radiation (which is obviously not the case here), the far-field radiation would be along the directions defined by those arcs. 24 In our case, the emission lobes are located near the intersection of several diffraction orders, and the polarization of the emitting dipole has to be taken into account.

The far-field radiation patterns evidence that the angle of emission is controlled by the dispersion diagram. Furthermore, for a given period of the array, the angle of emission depends on the emitting wavelength. This is evidenced in Fig. 7, showing the far-field emission for different wavelengths and a constant period P = 500 nm. For the sake of simplicity, we plot in Fig. 7 the results for a gap antennas, as they yield symmetrical patterns. Fig. 7a shows polar plots of the far-field emission (directivity) for different wavelengths and a period P = 500 nm. All patterns have been normalized to unity, and the far-field emission of a dipole inside a single antenna is also shown for reference. In the relatively narrow range of wavelengths explored in Fig. 7a, very different emission angles are observed, from 0 to 30 • . Moreover, for each wavelength the emission cone is narrow, meaning that color-dependent photon sorting is possible. [START_REF] Aouani | [END_REF] Fig. 7b is a color plot of the emission angle as a function of the wavelength, and shows that tuning of the emission angle is possible by changing the wavelength. A transition from a two-lobe emission pattern to a pattern exhibiting one main lobe is observed around 780 nm.

Discussion

In this Section we assess the interest of using SLRs in the context of single emitter fluorescence enhancement, by comparing our results with what was previously reported in the literature -keeping in mind that the enhancement factor is emitter-dependent. 36 Assuming a two-level quantum emitter excited well below its saturation intensity, the total fluorescence (i.e., emitter brightness) enhancement factor f low can be written as:

f low = f exc • f η • f col (3)
where

f exc = |E/E 0 | 2 is the enhancement of the excitation, f η = η/η (0)
is the enhancement of the quantum yield, and f col is the enhancement of the collection efficiency. 43 In order to clearly evidence the advantage given by surface lattice resonances, we summarized in low Stokes shift emitter. The period of the array (P = 440nm) has been chosen to maximize the directivity and hence f col . We also assume that the fluorescence is collected using a low numerical aperture (NA = 0.3) microscope objective.

Table 1 unambiguously evidences the advantage brought by the SLR, both for a bare nanodisk antenna or for gap antennas. We emphasize that the studied geometries are easy to make with a standard electron-beam lithography process. The enhancement factors can be compared to the performance metrics given in the recent Perspective article by Koenderink. 4 Obviously, the huge enhancement factor obtained for an array of gap antennas (f low = 2.4 × 10 5 ) will be mitigated in a practical experiment, by fabrication imperfections, by the emitter orientation, position and quantum yield, and for the excitation enhancement by the numerical aperture of any excitation lens. It is interesting to compare our results on the single gap antenna with the experiment reported by Kinkhabwala and coworkers, 7 who measured a total enhancement factor of 1340 using bow-tie gap antennas and low quantum yield (2.5%) emitters. Kinkhabwala et al. relied on numerical simulations to infer the excitation and emission enhancements, and obtained f exc = 180 and f η = 9.3, with no collection enhancement. Their lower value for the emission enhancement is likely due to the orientation of the emitter and to the higher value of the initial quantum yield.

Using an emitter with η (0) = 1% increases the value of f η to 21.7, reasonably close to our value. This comparison shows that the theoretical performances reported in Table 1 are experimentally achievable, even if the numbers should be understood as upper limits.

Moreover, this comparison underlines that there is a lot of room for further improvement of the fluorescence brightness, keeping the same bow-tie antenna geometry arranged into a periodic array. Interestingly, as SLRs also exist in arrays of dielectric optical antennas, 44,45 arrays of dielectric antennas also offer avenues for the enhancement of fluorescence of single emitters with low-loss materials. [46][47][48][49]

Conclusion

To sum up, we theoretically demonstrated that arrays of plasmonic nanoantennas sustaining surface lattice resonances are highly beneficial for the enhancement of the excitation rate of single emitters. On the one hand, the array does not bring specific advantage on the spontaneous emission rate compared to a single nanoantenna, whatever the geometry of the nanoantenna or the intrinsic quantum yield of the considered emitter. On the other hand, the array both harvests the incident light into nanometric volumes and generates structured emission in the far-field that can increase the collection efficiency of fluorescence.

In other words, the array acts as a far-to near-field transducer, that is to say, an antenna.

The array of nanoantennas can hence be seen as a "meta-antenna", further increasing the efficiency of its constituting elements. This effect is actually the nano-optical analogue of the enhanced directivity observed in arrays of micro-wave antennas. [START_REF] Balanis | Antenna theory[END_REF] Arrays sustaining SLRs can therefore act as on-chip lenses, allowing efficient concentration and extraction of light without requiring the use of high numerical aperture microscope objectives.

Methods

The whole system is modeled using a commercial FDTD software. [START_REF]Lumerical Solutions[END_REF] The refractive index of gold has been taken from Ref. [START_REF] Johnson | [END_REF] A constant mesh size of 2 nm was set to define precisely each antenna, while a non-uniform mesh was used outside the antennas. To model an isolated particle or a finite array, we used perfectly absorbing layers (PMLs) on the six sides of the computation box. To model an infinite array, we use PMLs on the top and bottom sides, while periodic boundary conditions (PBCs) are applied for the four remaining sides. Let us underline that when dealing with high-Q factor resonances, special attention must be paid to the PML delimiting the simulation box in order to avoid any undesired reflection. Here, we used stretched coordinates PMLs with the 'steep angle' profile as implemented in [START_REF]Lumerical Solutions[END_REF] and performed extensive convergence tests.

Computation of the quantum yield enhancement

We use the methods described in Ref. 7 The emitter's initial quantum yield (taken in the medium with refractive index n) is:

η (0) = γ (0) r γ (0) r + γ (0) nr , (4)
where γ (0) r is the emitter's initial radiative emission rate and γ (0)

nr its non-radiative rate. When the emitter is coupled to the nanoantenna, its apparent quantum yield writes:

η = γ r γ r + γ nr , (5)
In Eq. 5, γ r is the radiative rate of the coupled system and γ nr = γ (0) nr + γ abs its non-radiative rate. For the latter, γ abs accounts for all the new absorption decay channels that are brought by the nanoantenna. It is then possible to compute the radiative rate enhancement factor f r as:

f r = γ r γ (0) r = P r P (0) r , (6)
where P r is the power radiated by the emitter coupled with the nanoantenna and P (0) r is the power radiated by the emitter into vacuum. [START_REF] Novotny | Principles of Nano-Optics[END_REF] The non-radiative rate factor f nr can be calculated as:

f nr = γ abs γ (0) r = P abs P (0) r , (7)
where P abs is the power absorbed by the antenna. By combining Eq. 6 and 7 into Eq. 4 and 5, one obtains the expression for the quantum yield enhancement factor:

f η = η η (0) = f r (f r + f nr -1)η (0) + 1 . ( 8
)

Computation of the antenna radiation pattern

In antenna theory, the ability of an antenna to concentrate light in a given direction (θ, ϕ) is quantified by the directivity D(θ, ϕ), defined as the ratio between the the far-field intensity scattered in the direction (θ, ϕ) and a reference isotropic radiator, that is:

D(θ, ϕ) = I(θ, ϕ) I 0 . (9)
To compute the scattered intensity I(θ, ϕ), we first perform a near-field to far-field projection onto the surface S of a half-sphere exhibiting a 1 meter radius, using a procedure implemented in. [START_REF]Lumerical Solutions[END_REF] This projection yields the electric field in a given direction on the surface of the sphere, denoted E(θ, ϕ). Assuming that in far-field, we are dealing with plane waves, it is possible to write the intensity as:

I(θ, ϕ) = n 2 ε 0 µ 0 |E(θ, ϕ)| 2 .
I 0 being a reference isotropic radiator, it is easily computed as the total scattered power equally distributed over all possible directions, I 0 = P rad 4π .

Hence, the directivity corresponds to the angular power density normalized by a reference isotropic radiator. [START_REF] Novotny | Principles of Nano-Optics[END_REF] Directions where D > 1 corresponds to directions with enhanced scattered intensity.

Figure 1 :

 1 Figure 1: Surface lattice resonances in nanodisks arrays. (a) Schematic of the studied geometry: a square array (with period P ) of gold nanodisks (d = 100 nm, h = 50 nm) inside a homogeneous medium with refractive index n = 1.5. (b) Computed extinction cross-section spectra as a function of the period P of the array, under normal incidence. The white solid lines correspond to the position of the Rayleigh anomalies. (c) Extinction spectra for an array with P = 500 nm (green line), P = 600 nm (red line), and for an individual antenna (blue line). (d) Quality factor (blue dots) and normalized field intensity |E/E 0 | 2 computed at resonance (red dots) of the lattice mode as a function of the period of the array. (e) Normalized electric field modulus |E/E 0 | as a function of the wavelength and the period, under normal incidence. (f) Normalized field modulus |E/E 0 | spectra (calculated at the emitter's position) for an array with P = 500 nm (green line), P = 600 nm (red line) and for an individual antenna (blue line). For (d-f), the electric field is computed 10 nm away from the nanoparticle, at mid-height.

Figure 2 :

 2 Figure 2: Same as Fig. 1, for arrays made of gap antennas.

Figure 3 :

 3 Figure 3: Lattice effect on the emission for gold nanodisks (d = 100 nm, h = 50 nm). (a) Non-radiative factor f nr as a function the period P and the wavelength λ. (b) Same, for the radiative factor f r . (c) Same, for the quantum yield enhancement factor. (d) Quantum yield enhancement factor spectra, for periods ranging from 400 to 600 nm (red shaded area) and for a single antenna (black solid curve).

Figure 4 :

 4 Figure 4: Radiative factor for an array of coherently emitting dipoles coupled to an array of gold nanodisks (P = 500nm, d = 100nm, h = 50nm). The initial quantum yield of each emitter is η (0) = 1%.

Figure 5 :

 5 Figure 5: Far-field radiation pattern of finite arrays of nanodisk antennas, for different sizes of the array. The period of the array is P = 500 nm. The diagrams are plotted for the emission wavelength corresponding to the peak value of the emission rate. (a) Emitter only, λ = 707 nm. (b) Emitter coupled to a single antenna, λ = 707 nm. (c) Emitter in a 3 × 3 array, λ = 723 nm. (d) Emitter in a 5 × 5 array, λ = 725 nm. (e) Emitter in a 9 × 9 array, λ = 717 nm. (f) Emitter in a 15 × 15 array, λ = 715 nm.

Figure 6 :

 6 Figure 6: Far-field radiation pattern of a single dipole inside a finite array of nanodisk antennas, for different positions (P, λ) in the dispersion diagram. All arrays consist of 15×15 gold nanodisks. The solid white lines represent the position of the diffraction orders. (a) P = 400 nm, λ = 700 nm. (b) P = 440 nm, λ = 713 nm. (c) P = 480 nm, λ = 741 nm. (d) P = 500 nm, λ = 717 nm. (e) P = 580 nm, λ = 701 nm. (f) Dispersion diagram of the radiative enhancement factor as a function of P and λ. The positions where the radiation patterns have been calculated are shown as white dots.

Figure 7 :

 7 Figure 7: Far-field radiation pattern of a in a 15 × 15 array of gap antennas, for a constant lattice period P = 500 nm. (a) Polar plot showing the normalized directivity D for three values of the wavelength : λ = 730 nm (blue), λ = 775 nm (green) and λ = 795 nm (red). The emission pattern of the dipole inside a single gap antenna is shown for reference as the light blue shaded area. (b) Directivity as a function of wavelength and emission angle.

Table 1 :

 1 Table various fluorescence enhancement factors for an isolated nanoantenna and foran array of nanoantennas. The excitation (resp. emission) wavelength has been chosen as the wavelength yielding to the highest value of the excitation (resp. emission) enhancement factor. The collection enhancement factor has been computed at the same wavelength than f η . The resulting wavelengths are given in Table1and corresponds to realistic values for a Comparison of the fluorescence enhancements for isolated antennas and arrays of antennas, for two antenna geometries (nanodisks and gap antennas). The period of the array is P = 440nm. The excitation enhancement factor is computed at λ exc , while all the emission-related enhancement factors are computed at λ emi . The initial quantum yield is η (0) = 1%.

		f exc	λ exc	f η	f col	λ emi	f low
	Nanodisks	Single antenna 4.97 709nm 18.2 1.03 709nm 15 × 15 array 230 707nm 18.2 3.3 715nm 13803 93
	Gap antennas	Single antenna 156 734nm 54.7 1.06 744nm 15 × 15 array 1149 734nm 52.8 3.97 748nm 240776 9075

Acknowledgement

This work was partially supported by the HPC Center of Champagne-Ardenne ROMEO.

Supporting Information Available

The following files are available free of charge.

• SuppInfo: Enhanced fluorescence with dimers of gold nanocylinders (Figure S1). Influence of the intrinsic quantum yield of the emitter on the emission enhancement (Figure S2). Radiation patterns for dimers of gold nanocylinders (Figure S3). Equation describing the orders of diffraction (Eq. S1).

This material is available free of charge via the Internet at http://pubs.acs.org/.