In Depth Investigation of Lattice Plasmon Modes in Substrate

Article pubs.acs.org/JPCC

In Depth Investigation of Lattice Plasmon Modes in SubstrateSupported Gratings of Metal Monomers and Dimers Nabil Mahi,†,‡ Gaeẗ an Lévêque,*,† Ophélie Saison,† Joseph Marae-Djouda,¶,⊥ Roberto Caputo,¶ Arthur Gontier,¶ Thomas Maurer,¶ Pierre-Michel Adam,¶ Benemar Bouhafs,‡ and Abdellatif Akjouj† †

Institut d’Electronique, de Micro-électronique et de Nanotechnologie (IEMN, CNRS-8520), Cité Scientifique, Avenue Poincaré, 59652 Villeneuve d’Ascq, France ‡ Laboratoire de Physique Théorique, Université Abou Bekr BelkaidTlemcen, B.P. 230 Tlemcen, Algeria ¶ Laboratory of Nanotechnology and Optical Instrumentation, UMR 6281 STMR, Technological University of Troyes, 12 Rue Marie Curie, CS 42060, 10004 Troyes Cedex, France ⊥ Ermess, EPF-Ecole d’ingénieurs, 3 Bis Rue Lakanal, 92330 Sceaux, France S Supporting Information *

ABSTRACT: We study theoretically and numerically bidimensional square gratings of monomers and dimers of gold nanocylinders supported on a dielectric substrate, under plane wave illumination as a function of the angle of incidence and of the polarization. The number of parameters investigated makes that system a rich platform for the investigation of how grating coupling, and in particular edge diffraction which corresponds to the grazing propagation of a particular diffracted order, influence the surface plasmons response of nanoparticles. In particular, the considered periods are comparable to the range of incident wavelength, which makes the interpretation of the observed phenomena complex due to the large number of diffraction orders coming into play. In order to analyze those systems, we perform exact numerical simulations using Green’s tensor method, and compare them to a simplified approach based on the coupled-dipole approximation. The systematic identification of the grazing diffracted orders, combined with the computation of the S-matrix components, leads to better understanding of the different types of profiles (sharp maxima or angular minima) observed in the extinction spectra around the Rayleigh wavelengths associated with grazing diffraction in air or glass. The analysis is supported by computation of several electric field distributions computed for selected parameters.

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INTRODUCTION Gratings of metal nanoparticles have been investigated for their optical properties since the beginning of the 1990s.1−4 Compared to their isolated counterparts, which support localized plasmon modes, those gratings present “lattice plasmon modes” which are dispersive and moreover exhibit in their optical spectra very narrow features which can take the shape of sharp maxima or minima. In regular arrays, these features arise when an “edgediffraction” occurs for a wavelength close to localized surface plasmon resonances. Edge diffraction happens at so-called Rayleigh wavelengths, when one of the diffracted order changes from radiative (shorter wavelength) to evanescent (longer wavelength). The transition corresponds to grazing propagation of the diffracted order along the grating, and the perpendicular component of the wavevector cancels. While the occurrence of those features in metal particles gratings has theoretically been known since the beginning of 2000, it has been challenging to observe it as their observation required very regular arrays of a large number of particles and weakly diverging illumination.5−7 In most of the theoretical and experimental configurations, such systems have been investigated at normal incidence,8−13 where the Rayleigh wavelengths solely depend on the period of the grating. However, as the edge diffraction depends as well on the © 2016 American Chemical Society

incidence angle of the illumination, playing with the direction of the incoming plane wave gives the advantage to tune the position of the associated features.14 In a recently published article,15 we have experimentally and numerically studied such a square grating of substrate-supported gold nanocylinders under an oblique illumination. The fact that the lineshapes of spectra around Rayleigh wavelengths can be very narrow makes those kind of systems very suitable for designing plasmon nanolasers,16 or for biosensing applications as the associated figures of merit are expected to be large.10,17−19 For the same reason, they could be used for monitoring of mechanical stress and deformations.20,21 One of the simplest model to describe the optical behavior of uni- or bidimensional periodic systems of metal nanoparticles is based on the “coupled-dipole approximation”,1,22 where the metal nanoparticle is assimilated to a spheroid described by its polarizability tensor. In that model, particles are self-consistently coupled through the electric field that they scatter directly or after reflection onto the substrate, if there is one. In several Received: November 10, 2016 Revised: December 16, 2016 Published: December 30, 2016 2388

DOI: 10.1021/acs.jpcc.6b11321 J. Phys. Chem. C 2017, 121, 2388−2401

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The Journal of Physical Chemistry C

and the (Oz) axis) and the polar angle φ (between the incidence plane and the Oxz plane). The dielectric constant for gold, ϵ(λ), is obtained as a function of the wavelength λ from Johnson and Christy,30 while the data for ITO are derived from König et al.31

studies, the couple dipole approximation has been applied to the conduction of light by linear and curved chains of particles supported on a surface.23−25 For infinite arrays analysis, the strength of that model is to allow separating the response ot the metal particle (its polarizability), and the response of the grating. That latter one is completely included in a matrix “S” that solely depends on the environment, the periods and the illumination, but not on the particle’s parameters. Recently, that approach has been used to compute the dispersion relation of linear chains on a substrate,26 following preceding works about linear chain embedded inside a homogeneous medium.27 Humphrey and Barnes have applied that model to a bidimensional finite grating of metal dimers in a homogeneous medium, where the structure is excited at normal incidence.28,29 However, still missing is a complete angle- and polarization-dependent investigation of the optical response of bidimensional grating of metal nanoparticles supported on a substrate. In this paper, we present a theoretical and numerical investigation of bidimensional square gratings of monomers and dimers of gold nanocylinders supported on a dielectric substrate. Simulations based on Green’s tensor method and the coupled dipole approximation have been compared by taking into account the effect of the polarization and of the incidence angle of the incoming plane wave. Beside the numerous degrees of freedom of those systems, their richness is partly due to the fact that considered periods are comparable to the range of wavelengths investigated, with the consequence that several types of edge diffraction can occur close to localized plasmon wavelengths. Moreover, specific effects are due to the presence of the substrate interface under the particles, compared to gratings in an homogeneous environment. In the first part, we present the plasmonic properties of the isolated systems through the computation of extinction spectra and surface charges distributions of the different localized plasmon modes, and compare the results obtained from Green’s tensor method with calculations based on the coupled-dipole approximation. Then, we investigate in the second part the periodic systems, by essentially computing the extinction spectra with both methods, systematically identifying the different Rayleigh wavelengths as a function of the illumination parameters, and presenting distributions of the electric field in the structures for selected parameters of interest. For the grating of monomers, we additionally investigate the behavior of the S matrix around Rayleigh wavelengths corresponding to different types of edge diffraction. Those behaviors have noticeable consequences on the extinction spectra, depending on the polarization and on whether the diffraction is grazing in air or glass. The dispersion curves of the different lattice plasmon modes both for monomers and dimers gratings are as well computed using an eigenmode decomposition based on the coupled dipole approximation and compared to the exact simulation. In the whole article, the gold nanocylinders (AuNCs) are placed in air (refractive index n1 = 1.0, dielectric constant ϵ1 = n12 = 1.0), have a radius R = 100 nm and a height h = 50 nm, with revolution axis parallel to the (Oz) axis of an Oxyz reference frame. For Green’s tensor simulations, the AuNCs are deposited onto the topmost surface of a substrate consisting in a semi-infinite glass space of refractive index n2 = 1.5 (dielectric constant ϵ2 = n22 = 2.25) coated with a 30 nm-thick ITO film, in order to take the experimental configuration of our previous paper as a reference.15 The structure is illuminated from the glass substrate by a linearly polarized plane wave of incident wavevector kinc, whose direction depends on the azimuthal angle θ (between kinc

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RESULTS AND DISCUSSION Isolated Systems. In this first section, we use Green’s tensor method (GTM) to compute the extinction spectra of isolated monomers and dimers of gold nanocylinders (AuNCs), and implement the coupled-dipole approximation (CDA) using an equivalent system consisting in oblate particles floating above a simple air-glass interface. In order to completely elucidate the plasmonic modes of the single monomer and dimer of AuNCs, spectra have been computed for a large set of incident planewaves. For the monomer, the rotation invariance around the vertical axis (going through the center of the particle) allows to restrict the study to the effect of θ and of the polarization. However, in the dimer we must separate the longitudinal illumination, where the incidence plane contains the dimer’s axis (φ = 0°), and the transverse illumination where the incidence plane is perpendicular to the dimer’s axis (φ = 90°). Isolated Monomer. Let us consider first the single AuNC pictured in Figure 1a. The evolution of the associated GTM

Figure 1. (a) Cross-section of the isolated AuNC configuration. (b) Extinction spectra of the isolated AuNC, as a function of the angle of incidence θ, for the two polarizations p and s. The colored dashed lines indicate the wavelengths of the four localized modes seen in the spectra, with corresponding distributions of the surface charges at their maximal amplitude.

extinction spectra is shown on Figure 1b as a function of the incidence angle θ of the plane wave, for the two polarizations p (incident magnetic field parallel to the substrate interface) and s (incident electric field parallel to the substrate interface). The inplane quadrupolar mode Q and dipolar mode P are clearly identified at respectively λ = 630 nm and λ = 805 nm, recognizable by their surface charges distributions which experience changes of sign parallel to the substrate, and are essentially localized at the bottom of the nanoparticle. The extinction cross-section of the dipolar mode P decreases with θ for p polarization, due to the fact that the incident field, transmitted through the glass-air interface, becomes perpendicular to the substrate at the critical incidence angle θl = asin(n2 sin(θ)/n1) and then cannot couple with the horizontal mode. However, the incident field is always parallel to the interface in s polarization, and is proportional to the transmission coefficient which increases close to θl: for that reason, the extinction of the 2389


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The Journal of Physical Chemistry C in-plane dipolar mode increases in s polarization with θ. The quadrupolar mode Q is however not excited at normal incidence due to obvious symmetry reason, and, as for the P mode, disappears for large values of θ in p polarization. An additional maximum is observable in p polarization around λ = 525 nm for large incidence angles. Its line-shape is not exactly symmetric, and hide a weaker mode in its red wing. As shown in the Supporting Information, the main resonance at 525 nm is due to a D mode, whose position is almost insensitive to the substrate refractive index. It is characterized by a field distribution mainly localized at the “top-surface” of the AuNC, and results from the high-energy combination of the in-plane dipolar mode supported by the AuNC in air with its out of plane (meaning the surface charges distribution experiences changes of sign perpendicular to the substrate) quadrupolar mode, the coupling being induced by the reflection of the electric field on the substrate.32−34 In the red shoulder of the D mode, a V mode appears for a wavelength slightly larger than 525 nm. It is red-shifted when the refractive index of the substrate increases, can only be excited with a large vertical component of the electric field, and shows a scattering cross-section dominating the absorption cross-section (see Supporting Information): for those reasons, that mode is probably a vertical dipolar mode, although its surface charges distribution is very deformed. In the CDA, the nanocylinder is replaced by an oblate nanoparticle whose center is located in air at a distance z1 from a glass-air interface, see Figure 2a. Its degenerated semiaxes a are

Factors Lj are the well-know depolarization factors35 which describe the shape of the nanoparticle, whose expressions can be found in the Supporting Information. Size effects are taken into account in order to extend the model beyond the electrostatic approximation, through the addition of two corrections in k2 and k3, the latter describing the increased losses due to radiation. The corresponding coefficients Dj and lj are axis-dependent and their expression have been taken from reference36 (see Supporting Information). The dipole p created inside the particle due to the incident field is proportional to the local field Eloc at the particle location through p = α̅ Eloc. The local field is here simply equal to the sum of the incident field at the location of the spheroid, Einc, and of the field scattered by the particle itself after reflection onto the substrate interface: p = α [Einc + Gs(z1)p] ⇒ p = [α −1 − Gs(z1)]−1 Einc (3)

where Gs is the surface dyadic Green’s function,37,38 which generally depends on the source location r′ and on the destination location r. In the case where the source and the destination share the same position, Gs only depends on z1. Often, as the distance to the interface is small compared to the wavelength, the electrostatic approximation can be applied, and Gs takes a simple analytical expression.40 However, that approximation cannot be applied here as the particles in the gratings are separated by arbitrarily large distances. For that reason the exact value of Gs must be evaluated numerically by integration in the Fourier space.38 Once the dipole p is known, the extinction is obtained with Cext(λ) = k 0 0[p. E*inc], where 0 designates the imaginary part and ∗ the complex conjugation. It will be convenient to introduce at this point the polarizability − s modified by the substrate, defined by α̅ −1 surf = α̅ 1−G (z1), which leads to p = α̅ surf Einc. In that simple configuration, α̅ surf is diagonal. We obtain a good agreement between GTM and CDA spectra using parameters a = 100 nm, b = 28 nm, and z1 = 42 nm, as shown on Figure 2b.39 The CDA spectra only present dipolar V and P modes at respectively short and long wavelengths, as the CDA does not take into account the quadrupolar response of the spheroids (see refs 41 and 42 for multipolar response). The fact that the oblate particles must be located floating in the air at a certain distance from the interface is explained by the limitation of the dipolar model in the presence of a dielectric interface, which leads to much faster red-shifts when going to contact than in real systems: close to the substrate, the quadrupolar response of the spheroid cannot be neglected anymore.43 Isolated Dimer. We consider now a dimer composed of two of the previous AuNCs with center-to-center distance Δ = 240 nm, which corresponds to an interparticle gap of 40 nm. The centers of the bottom faces of two AuNCs are located at positions l1 = −(Δ/2) ex (for particle 1) and l2 = (Δ/2) ex (for particle 2). The direction defined by the dimer axis breaks the rotation invariance around the (Oz) axis, and we need to distinguish transverse and longitudinal modes which are respectively antisymmetric and symmetric compared to the plane (Oxz). The symmetry property here is the one associated with the electric field or the surface charges distribution. Notice that the longitudinal/transverse definition for plasmon modes is different (and independent) from the longitudinal/transverse definition for the illumination direction. Hence, as a basic set of modes of each AuNC, we need to consider the four possible combinations transverse/

Figure 2. (a) Cross-section of the geometry used in the dipolar approximation. (b) Comparison of the extinction spectra computed with the GTM (solid line) and the CDA (dashed line) for θ = 0° (black), θ = 20° (red), and θ = 35° (green), with parameters z1 = 42 nm, a = 100 nm, and b = 28 nm and for p and s polarizations.

parallel to the (Ox) and (Oy) directions, and the semiaxis b < a is parallel to the (Oz) direction. The response of the nanoparticle is given by its tensorial polarizability α̅, which is diagonal in the chosen frame (Oxyz), with diagonal elements αj=x,y,z following ϵ ⎛ Dj 1 1 2 ⎞ = − 1 ⎜⎜k 2 + i k3⎟⎟ , k = n1k 0 4π ⎝ l j 3 ⎠ αj(λ) α0, j(λ)

(1)

where k0 = 2π/λ and α0, j is the polarizability in the electrostatic limit α0, j(λ) =

ϵ(λ) − ϵ1 4 3 πr 3 ϵ1 + Lj[ϵ(λ) − ϵ1]

(2) 2390


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with global vectors for dipole 7 = [p1p2], incident field , inc = [E inc,1E inc,2], and with global polarizability ( and coupling . matrices:

longitudinal dipolar/quadrupolar modes, for which we adopt notations of Figure 3.

⎛Gs(z1) G21 ⎞ ⎛α 0 ⎞ ⎟ (=⎜ ⎟ , . = ⎜⎜ ⎟ s ⎝0 α ⎠ G G ( ) z ⎝ 12 1 ⎠ Figure 3. Notation for the quadrupolar and dipolar modes of a AuNC, depending on their symmetry compared to the (Oxz) plane. If the AuNC is isolated, P and P′ are degenerated, as are Q and Q′.

We note Gij = G0(ri, rj) + Gs(ri, rj), where G0 is the direct (for homogeneous medium) dyadic Green’s function.44 As for the monomer, we introduce the surface polarizability of the dimer as 1 (−surf = (−1 − . , which gives

7 = (surf ,inc

In the dimer, those eight individual modes (four modes per particles) must combine into eight collective modes, which are symmetric and antisymmetric combinations of either the dipolar or the quadrupolar modes. Some of them are bright (strongly radiative) if they follow the symmetry of a dipole, the other being “dark” (weakly radiative). Figure 4 shows the extinction spectra and the shape of the in-plane modes of the dimer for the different parameters of the illumination, with bright modes indicated with a bold font. It is worth noting that, contrary to the monomer for which quadrupolar modes cannot be excited by a plane wave at normal incidence, the excitation of the bright mode Q′1 − Q′2 is allowed by an incident planewave polarized along the (Oy) axis, while the bright mode Q1− Q2 can be excited by a plane wave at normal incidence, polarized along the (Ox) axis. However, only the last one is clearly observed at θ = 0° on Figure 4, the first one being hidden in the blue-wing of the dipolar mode P1+P2. The extension of the CDA to the spheroids dimer is made by expressing the induced dipoles p1 and p2 in the first and second spheroids, located at respectively r1 = l1 + z1ez and r2 = l2 + z1ez, for which we must now take into account, in the local field, the electric field scattered by the other spheroid, either directly or after reflection onto the interface. This leads to 7 = ([,inc + GP] ⇒ 7 = [(−1 − .]−1 ,inc

(5)

(6)

Again, the extinction spectrum is simply given by Cext(λ) = k 0 0[7·,*inc]. Using the previously determined dimensions for the spheroids, a good agreement between GTM and CDA spectra is obtained taking Δ = 250 nm, close to the expected value of 240 nm; see Figure 5a. Again, only the modes resulting from the combinations of dipoles are observed in the CDA. The six eigenmodes of the dimer of spheroids are obtained by diagonalizing the inverse surface polarizability ( surf −1. The corresponding eigenvectors πμ, as for the GTM results, closely look like the symmetric and antisymmetric combinations of the single particles dipoles along each of the direction x (longitudinal modes), y (transverse modes), and z (vertical modes). We will only focus in the following on the four lowest energy modes corresponding to in-plane modes (parallel to the substrate interface), which are π1 ⇔ P1 + P2, π1′ ⇔ P1−P2, π2 ⇔ P′1 − P′2, and π2′ ⇔ P′1 + P′2. Let us note that π1 and π1′ have a small z component induced by the interface, which will be neglected in the following. The obtained eigenvalues Λ0μ behave as an equivalent of inverse polarizability for each eigenmode (the word polarizability is here ambiguous because some of those modes are dark and weakly radiating). Figure 5b shows the partial spectra 0(1/Λ0μ) associated with each of the four in-plane modes.

(4)

Figure 4. Extinction spectra of the isolated dimer of AuNCs, as a function of the incidence angle θ, for the two polarizations p and s and for longitudinal (φ = 0°) and transverse (φ = 90°) illuminations. On the right side are plotted the distributions of the surface charges at their maximal amplitude for the different extinction maxima obtained in the four illumination configurations beyond 600 nm. The corresponding wavelengths are indicated by a green dashed line, on which the blue solid line emphasizes the values of θ, φ and polarization for which the mode is excited. Bright modes are written with bold characters. 2391


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Figure 6. Geometrical parameters of the grating of monomers (a) and dimers (b).

0 = ϵik 0

2

2 ⎛ ⎞2 ⎛ x π 2π ⎞ 2 y − ⎜kinc + m ⎟ − ⎜⎜kinc + n ⎟⎟ Lx ⎠ Ly ⎠ ⎝ ⎝

x

(7)

y

where kinc = n2k0 sin(θ) cos(φ) and kinc = n2k0 sin(θ) sin(φ). The corresponding Rayleigh wavelengths will be marked with symbols on spectra, which are coded in shape (depending on the integer associated with the diffraction out of the plane of incidence) and in color (depending on the integer associated with the diffraction along the plane of incidence). For longitudinal illumination (φ = 0°, the plan of incidence is (Oxz)), symbols are indicated on Figure 7. For transverse

Figure 5. (a) Comparison between the extinction spectra computed with the GTM (solid line) and the CDA (dashed line) for θ = 0° (black), θ = 20° (red) and θ = 35° (green), for the two polarizations p and s and for longitudinal (φ = 0°) and transverse (φ = 90°) illuminations. The spheroids parameters are the same as for the monomer, and the interparticle distance is Δ = 250 nm. (b) Partial spectra 0(1/Λ0μ), associated with each of the four in-plane eigenvectors πμ of the dimer of spheroids.

Figure 7. Symbols used to mark the positions of the different Rayleigh wavelengths (case φ = 0°) on the next extinction spectra.

illumination (φ = 90°, the plan of incidence is (Oyz)), the code is the same but m is coded in shape and n is coded in color. Larger symbols indicate grazing diffraction in silica, while smaller symbols are for grazing diffraction in air. Grating of Monomers. In the CDA, the spheroids are located at positions rmn = lmn + z1ez. Under plane wave illumination, the dipoles pmn obey the Bloch’s theorem pmn = p00 exp(ik∥inclmn), with k∥inc = kincxex + kincyey, and the extinction only depends on the dipole p00 of the particle at the center of the lattice: Cext(λ) = k 0 0[p00 ·E*inc], where Einc is the incident field seen by the central particle. Compared to the case of the isolated monomer, the local field at r00 must include the electric field scattered from the particles of all the other unit cells, directly or after reflection onto the interface:

Clearly, the bright modes π1 and π2′ are wider than the dark ones, π1′ and π2, and the splitting is slightly larger for transverse modes than for longitudinal modes, which is due to the fact that the transverse coupling is dominated by slowly decaying components (in 1/r) while the longitudinal coupling results from faster decaying near-field components (in 1/r2 and 1/r3). Periodic Systems. As a difference with the previous isolated structures, which support nondispersive localized surface plasmon modes, the periodic systems sustain dispersive modes called lattice plasmon modes (LPM), which depend on the periods and the direction of the illumination. The periods of the grating are Lx and Ly in the x and y directions, see Figure 6. The center of each unit cell is lmn = mLxex + nLyey, where m and n are integers. The bottom faces of monomers, Figure 6a, are located at the center of each unit cells, while for dimers, Figure 6(b), the bottom face of each of the two particles in each unit cell are located at positions lmn − (Δ/2)ex (particle 1) and lmn + (Δ/2)ex (particle 2). The full numerical simulations are performed with a periodic version of the GTM.45 For a complete analysis, we need to identify in the extinction spectra the Rayleigh wavelengths for the different diffraction orders. Keeping in mind that each diffraction order can be grazing either in air (ϵ1) or in glass (ϵ2), two families of Rayleigh z wavelengths exist, defined by ki,mn = 0 with i = 1, 2, which gives

αsurf −1p00 = Einc + [

∑

ei k incl mnGmn]p00

mn ≠ 00 0

(8)

s

where Gmn = G (r00, rmn) + G (r00, rmn). If the environment is homogeneous (no substrate), only the direct contribution G0 exists, thus the summation on the right side of the equation can be in principle evaluated by adding the matrices G0(r00, rmn), expressed in the direct space, until limit is reached to a chosen precision. However, the convergence is very slow when the particles are at the same distance from the (Oxy) plane, and it is 2392


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Figure 8. Extinction spectra for a planewave at normal incidence on a monomers grating with Ly = 400 nm and Lx varying from 250 to 1000 nm by steps of 50 nm, for p polarization (a) and s polarization (b). The left (respectively, central) column shows the extinction spectra computed with the GTM (respectively CDA), while the right column shows the evolution of 9(Sμμ) corresponding to each direction μ = x, y, z. The green dashes show 9(1/αsurf, μμ) where it intersects the S matrix component. Field maps on the left show the distribution of the electric field amplitude above the substrate, for wavelengths indicated by the colored stars (the color bar applies to all maps).

required to work simultaneously with the real space and the Fourier-space expansion in (kx, ky) of G0 in order to speed up the process.45 Besides, in the presence of a substrate, only the Fourier expansion of the surface contribution Gs is known analytically, and for that reason it is more convenient to transpose the sum into the reciprocal space, making use of the expansion of Green’s tensors in (kx, ky), see Supporting Information: ̃ − Gs(z1)]p αsurf −1p00 = Einc + [∑ Gmn 00 mn

k mn

⎛ kinc x + m2π /Lx ⎞ ⎟ ⎜ =⎜ y ⎟ ⎝ kinc + n2π /Ly ⎠

̃0

where G̃ mn = G (kmn , x

kmny)

+

p00 = [αsurf −1 − S]−1 Einc

(10)

̃s

G (kmnx,

kmny).

We finally have (11)

with S = ∑mn G̃ mn − G (z1). Normal Incidence. We consider here the case where θ = 0°, taking (Oxz) as the plane of incidence: in s (respectively p) polarization, the incident electric (respectively magnetic) field is s

(9)

with: 2393


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This is due to the constructive and nonconverging interference of fields radiated by in-phase dipoles polarized along y and located along the (Ox) axis, as the radiated field decays with the inverse of the distance (see Figure 9a). The same happens for 9[Szz]. On

parallel to the (Oy) axis. In this particular situation, the matrix S is diagonal and the positions of the lattice plasmon modes (LPMs) are given by, provided that the imaginary part varies slowly around the resonance ⎛ 1 ⎞ − Sii⎟⎟ = 0, i = x , y , z 9⎜⎜ ⎝ αsurf , ii ⎠

(12)

where 9 designates the real part. Figure 8 corresponds to a grating of monomers excited in normal incidence from glass, with period Ly = 400 nm, and Lx varying from 250 to 1000 nm by steps of 50 nm, from bottom to top. Figure 8a corresponds to p polarization, while Figure 8b is related to s polarization. For both polarizations, the left panel shows the extinction spectra computed with the GTM, and the central panel the spectra given by the CDA. The right panel shows the evolution of 9[Sxx] (a), (respectively, 9[Syy] (b)), together with the trace of 9[1/αsurf, xx] (a) (respectively, 9[1/αsurf, yy] (b)) around the intersection between the two curves in order to indicate the position of the LPM (see eq 12). Finally, electric field distributions (amplitude) computed above the substrate’s interface are shown for selected wavelengths indicated by the colored stars. Because of the fact that the periods are on the order of magnitude or larger than the wavelength of the incident light, a lot of edge diffraction occurs at Rayleigh wavelengths indicated by symbols. It appears clearly that the CDA, despite the simplicity of the description, gives very similar spectra to the ones obtained with the GTM, in term of profile and position of resonances. We briefly describe their main characteristics in the following. For periods smaller than 400 nm, which corresponds to the situation where Rayleigh wavelengths do not yet overlap with the dipolar mode, we essentially observe a blue shift of the LPM for p polarization and a red-shift for s polarization when Lx increases. This fact is completely coherent with the evolution of the wavelength of the modes supported by a chain of nanoparticles parallel to the (Ox) axis. For that system, it has been shown that the transverse mode experiences a red-shift when the interparticle distance increases while the longitudinal mode experiences a blue-shift, provided that the interparticle distance is smaller than λLSP/n2, where λLSP indicates the resonance wavelength of the isolated particle in air.46 This is similar to what happens to the bright modes of a dimer for an interparticle distance smaller than the resonance wavelength, where a red-shift with decreasing distance is observed for the low-energy combination of the longitudinal dipolar modes. On the contrary, the transverse symmetric dipolar mode corresponds to the high energy combination and then experiences a blue shift when the distance decreases. For larger periods, the results are complicated by the fact that the line-shapes of the plasmon modes are strongly modified by the grazing diffracted waves excited close to the Rayleigh wavelengths. However, useful insight is brought by the plot of 9[Sii] and 9[1/αsurf, ii] as a function of the wavelength. We need to mention first that the substrate interface greatly modifies the shape of the S matrix components around the Rayleigh wavelength, compared to a grating embedded in an homogeneous environment. In particular, those components never diverge for the system investigated in this paper, which is not the case without interface. Let us consider for instance a grazing diffracted order propagating along the (Ox) axis. Without interface, 9[Syy] exhibits a singularity at the Rayleigh wavelength.

Figure 9. Superposition of the field scattered by transverse (a) and longitudinal (b) dipoles in a chain.

the contrary, as a dipole does not radiate along its own axis, the sum of the electric fields emitted by the same dipoles polarized along x converges because the field amplitude decays at least with the inverse of the square of the distance (see Figure 9b). Now, when placing a dipole above and close to a substrate with a larger index of refraction, the electric field radiated transversely at long distance along the interface interferes destructively with the same field reflected onto the interface, due to the π phase difference experienced by the electric field at grazing reflection: when placed above a substrate with larger (than air) refractive index, the dipole, whatever its orientation, does not transversely radiate in the direction parallel to the interface. This implies that, whatever the component of Sii considered, the summation on all the dipoles always converges at every Rayleigh wavelengths. However, their profile around the Rayleigh wavelength depends on the polarization and whether the diffraction is grazing in air or in glass. For example, considering on Figure 8 the degenerated orders (±1, 0) grazing in glass (large red circle), 9[Sii] has a “tipshape” for both s and p polarizations, while for the same orders grazing in air (small red circle), 9[Sii] has a tip-shape for p polarization but a smoother “S-shape” for s polarization. Those differences have noticeable consequences on the extinction spectra. In particular, striking features on the extinction spectra of parts a and b of Figure 8 are the very narrow peaks occurring in the extinction spectra for specific values of Lx. Those peaks appear for both s and p polarizations when Lx ≈ 550 nm, at the Rayleigh wavelength associated with (±1, 0) diffraction grazing in glass (large red circle), λRA = n1Lx = 825 nm. However, when the same diffraction order is grazing in air (small red circle), the narrow maximum is only obtained in p polarization (Lx = λRA ≈750 nm), as for s polarization we observe a small angular minimum in between two poorly defined maxima, coherent with the low field enhancement on the field map at that wavelength. The reason for those differences clearly originate in the profiles of 9(Sxx) and 9(Syy) around the corresponding Rayleigh wavelengths. Indeed, the narrow maxima appear when 9(Sii) has a tip-shape and 9(1/αii) intersects exactly 9(Sii) at the Rayleigh wavelength. However, for s polarization and (±1, 0) diffraction grazing in air (small red circle), 9(Syy) has a S-shape around the associated 2394


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Figure 10. Extinction spectra for a planewave in oblique incidence on a monomers grating with Lx = Ly = 300 nm, θ = 0°,..,40°, for p polarization (a) and s polarization (b). The first (respectively second) panel on the left shows the extinction spectra computed with the GTM (respectively CDA). For p polarization the evolution of 9(Sxx) and 9(Szz) are shown, while for s polarization only 9(Syy) is plotted. Again, the green dashes are the trace of 9(1/αsurf, μμ) where it intersects 9(Sμμ). Field maps on the left show the distribution of the electric field amplitude above the substrate, for wavelengths indicated by arrows which colors match the color of the map’s frame (colorbar applies to all maps). (c) Dispersion curves of the two in-plane dipolar modes for p polarization in red and s polarization in black, computed with 9(Λμ) = 0. The pale pink dots are for p polarization, computed with 9(1/αsurf, xx − Sxx) = 0. The trace of the grazing diffracted order (−1, 0) in glass is plotted in blue, together with the light line in glass, and the lines kx = n1k0 sin θ are plotted in black for θ = 20° and 35°.

Rayleigh wavelength and a minimum is obtained in extinction. As shown in the Supporting Information, the difference of profiles in the 9(Sii)’s can be retrieved from an expansion of Green’s tensor components around the Rayleigh wavelength: transverse (compared to the direction of propagation of the grazing diffracted order) components are always characterized by a Sprofile (for instance 9(Syy) for (±1, 0) grazing diffraction in air), longitudinal component by a tip-shape profile (9(Sxx) for (±1, 0) grazing diffraction in glass). However, grazing diffraction in glass always leads to a tip-shape profile in every component of the S matrix. Oblique Incidence. We investigate now the effect of the angle of incidence, supposing the direction of the illumination is parallel to the (Oxz) plane, see Figure 6a. The periods of the grating are fixed to Lx = Ly = 300 nm. Rigorously speaking, due to the nonzero incidence angle, the matrix S is no longer diagonal, as a vertical dipole is able to create, by reflection onto the interface, a parallel electric field component in the neighboring particles. Because of the phase differences between the unit cells imposed by the incident wave, this field does not destructively

interfere. Then, the position of the LPM must be found by diagonalizing the matrix α̅surf−1 − S. For the grating of monomers, this procedure gives three distinct eigenvalues Λμ, μ = 1, 2, 3. However, despite the symmetry break due to the oblique incidence, the corresponding eigenvectors closely resemble those of the single monomer (that is ex, ey, and ez). Similarly, the resonance wavelengths of the LPM, which are rigorously given by 9(Λμ) = 0, μ = 1, 2, 3, closely match the positions given by 9(1/αsurf, μμ − Sμμ) = 0, μ = x, y, z. Figure 10 shows the evolution of the extinction spectra with the incidence angle θ (from 0° to 40° by steps of 5°). Part a (respectively, part b) is calculated for p (respectively, s) polarization. For both polarizations, the two panels on the left compare the spectra computed with the GTM and with the CDA. For p polarization, 9(Sxx) and 9(Szz) are plotted with their respective intersections with 9(1/αsurf, xx) and 9(1/αsurf, zz), shown by a green dash to mark the position of the LPMs. For s polarization, 9(Syy) and 9(1/αsurf, yy) are displayed. Again, selected electric field distributions computed above the substrate 2395


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Figure 11. Dispersion curves of the four in-plane dipolar modes for longitudinal illumination (φ = 0°) (a) and transverse illumination (φ = 0°) (b). The gray/black lines shows kx/y = n1k0 sin θ for θ = 0°, ..., 90°, every 5°. The trace of the grazing diffracted orders are indicated in red for glass and in blue for air.

Figure 10a, magenta arrow). This asymmetry results from the interaction of the quadrupolar mode with the grating, which appears quite clearly on the field distributions for p polarization, Figure 10a, as for θ = 25°, red arrow, and θ = 30°, blue arrow, the electric field is localized essentially in between two particles adjacent along the x direction, at the Rayleigh wavelengths. In contrast, the intensity of light between particles is lower at the extinction maxima indicated by the magenta and cyan arrows. Similarly, the field distributions for the s polarization, on Figure 10b, show an enhanced field in between the nanocylinders at the Rayleigh wavelengths corresponding to the same order (−1, 0) (large magenta circle) close to the quadrupolar modes position (see the distribution of electric field for the red and blue arrows) compared to the field at the extinction maxima (see the distribution of light for the magenta and cyan arrows). Similar behavior is obtained for θ = 40° with both polarizations. The second effect put into evidence by the CDA model is the origin of the small maximum in p polarization for θ = 25−40° around 550−600 nm, marked by a brown arrow on Figure 10a, whose position is close to the resonance wavelength of the vertical plasmon mode of the single particle. Indeed, if the origin of the maximum between 700 and 800 nm is clearly due to the dipolar LPM, as evidenced by the crossing between 9(Sxx) and 9(1/αsurf, xx), the CDA calculations show however that no crossing occurs between 9(Szz) and 9(1/αsurf, zz) in that range of incidence angles. As a consequence, the maxima which appear around 550−600 nm are probably not linked to the excitation of a vertical dipolar mode, but more likely to the proximity of the (−1, 0) diffracted order grazing in air (small magenta circle), which gives strong S-shape profile in 9(Szz) and tip-shape profile in 9(Sxx ). The corresponding field distributions on Figure 10a,

are plotted for wavelengths indicated by the colored arrows. Figure 10c shows the dispersion curves of the two horizontal dipolar LPMs as obtained from 9[Λμ(k 0 , kx)] = 0 (black: s polarization, red: p polarization), and computed with 9(1/αsurf, xx − Sxx) = 0 (pink dots). The red and pink dispersion curves, associated with p polarization, stay close together as long as they are inside the light cone of air. We have estimated how close the eigenvectors of the grating are to the eigenvectors of the isolated monomer by computing the largest weight of each grating eigenvector on the isolated monomer ones: on the ensemble of points displayed on Figure 10(c), 97% have a weight larger than 0.9, with an average 0.985. First, despite obvious differences due to the quadrupolar mode only appearing in the GTM spectra, the agreement between the GTM and CDA spectra is very good. Moreover, the comparison between both methods provides a useful insight in the origin of the narrow dip which occurs at the Rayleigh wavelength for the (−1, 0) diffraction order grazing in glass (large magenta circle), in p and s polarizations, at λ = 675 nm for θ = 30° (indicated by the blue arrow). In the GTM spectra, the (−1, 0) diffraction order seems to create nearly a zero in the line-shape of the dipolar mode of the isolated AuNC, similarly to what would be observed with the same grating embedded in a homogeneous medium.13 However, this cannot be the case as such an effect is linked to divergences in the S matrix components, which does not occur for a grating of particles placed above a substrate with larger refractive index. Hence, the extinction dip is more likely due to the simultaneous lowering (without cancellation) of the bluepart of the dipolar line-shape (visible on the CDA spectra) and a steepening of the red-wing of the line-shape of the nearby quadrupolar mode, which is clearly asymmetric (see for instance 2396


Figure 12. Extinction spectra for a planewave in oblique incidence on a dimers grating with Lx = Ly = 600 nm, θ = 0°, ..., 40°. The column (a, c) is for p polarization, while the column (b, d) is for s polarization. The line (a, b) is for longitudinal illumination (φ = 0°) and the line (c, d) is for transverse illumination (φ = 90°). In each subplot, the left (respectively right) spectra are computed with the GTM (respectively CDA). Field maps on the sides show the distribution of the electric field amplitude above the substrate, for wavelengths indicated by the colored stars (color bar applies to all maps).

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modes πμ of the isolated dimer the corresponding LPM is the closest. The distance between Πμ and the corresponding πμ is the largest very close to the intersection of the associated dispersion curve with the lines figuring the Rayleigh wavelengths. In Figure 12 are shown the extinction spectra and selected electric field distributions of the dimers grating for varying incidence angle θ, under different illumination configurations. Parts a and c (respectively, parts b and d) are for p (respectively s) polarization. Parts a and b (respectively, parts c and d) are for longitudinal (respectively, transverse) illumination. The Rayleigh wavelengths for grazing diffraction in air and glass are indicated, following the labeling defined in Figure 7. In each subfigure, spectra on the left have been computed with the GTM while spectra on the right result from the CDA. Again, the agreement between GTM and CDA calculations is quite good, as long as quadrupolar modes in the GTM spectra are left aside. However, the spectra are more difficult to interpret than for the grating of monomers, for the reason that the incident field vector , inc is generally not proportional to any of the eigenvectors Πμ, except for the transverse illumination for which , inc(φ = 90°, p) is only coupled to Π2′ and , inc(φ = 90°, s) is only coupled to Π1. For the longitudinal illumination, , inc(φ = 0°, p) is decomposed onto Π1 and Π1′, while , inc(φ = 0°, s) is decomposed onto Π2 and Π2′. Several remarkable features can be noticed. Let us start with the longitudinal illumination (φ = 0°), p polarization (Figure 12a). We mostly observe, exactly as for the isolated dimer (Figure 5), a clear switch between the low-energy Π1 mode, which is excited predominantly for θ < 25°, and the high energy Π1′ mode, whose positions are coherent with a jump from the blue curve to the black curve around θ = 25° in Figure 11a. This behavior is confirmed by the field maps, where the mode excited at low angle appears as “bounding” (Π1), as the field is large in the gap of the dimers, while the mode at large angle is “antibounding” due to the repulsing configuration of the surface charges in the gap of the dimer for the Π1′ mode which tends to expel the field from that region. However, the mode Π1, despite being radiative and very wide in the isolated dimer configuration (see π1 mode in Figure 5), is much narrower within the grating, actually of width comparable with the one of the dark mode Π1′. This effect is due to the fact that part of the radiative losses, which occur here mostly transversally to the dimer (along the (Oy) direction), are recoupled into particles of the grating when a grazing diffraction occurs close to the mode resonance wavelength in the transverse direction (here essentially the order (0, 1), grazing in glass (black diamond)). Through that mechanism, the losses are lowered down to the metal inner absorption. Second, similarly to the monomer case, a grating resonance due to an S-shape Rayleigh profile occurs (more visible on the CDA spectra) for large incidence angles in the p polarization close to the Rayleigh wavelength associated with the order (−2, 0), grazing in air (small cyan circle), as well observable for the transverse illumination on Figure 12c. For the s polarization (Figure 12b), a second switch is expected to happen between the two lowenergy Π2 and high-energy Π2′ transverse modes with the incidence angle. The effect is clear enough on the CDA calculation, where at normal incidence the Π2′ mode is excited at about λ = 720 nm, while for θ = 40° the maximum at 810 nm corresponds to the excitation of the Π2 mode, the transition occurring again at θ ≈ 20°/25°. However, the comparison between the CDA and the GTM shows some discrepancies, especially at normal incidence. This is to be partly expected as the

indicated by black and brown arrows, show that the electric field have a similar intensity at the Rayleigh wavelength, black arrow, and at the wavelength of maximum extinction, light brown arrow, the only difference being the position of the maxima of the stationary wave due to the superposition of the incident wave with the (−1, 0) diffraction order. For the s-polarization, a similar effect due to the S-shape profile of 9(Syy) happens for a wavelength slightly lower than the Rayleigh wavelength, (see CDA spectrum for θ = 40°) but it is completely hidden in the GTM by the excitation of the quadrupolar mode. Extension to a Grating of Dimers. In this final part, we investigate the evolution of the optical properties of a grating of dimers in the view of what we have previously discussed. Only the effect of the incidence angle is studied here, the periods being fixed at Lx = Ly = 600 nm. We must emphasize the fact that the period is twice the one of the grating of monomers, and a lot of diffraction orders come into play, which are all very dependent on the incidence angle. As for the isolated dimer, we investigate two orientations of the plane of incidence, see Figure 6b: the longitudinal illumination corresponds to φ = 0° (the incidence plane contains the dimer axis) while the transverse illumination corresponds to φ = 90° (the incidence plane is perpendicular to the dimer axis). As for the monomer, the CDA of the isolated dimer presented in the first section can be adapted to the grating of dimers by adding to the local field of each of the two spheroids composing the central dimer the contribution from the other unit cells: (surf −1700 = ,inc + SP00

(13)

where ⎛ S11 S21 ⎞ ⎟ , Sij = :=⎜ ⎝ S12 S22 ⎠

∑ eik mn

mn(l i − l j)

̃ Gmn

(14)

with l1 = −(Δ/2) ex and l2 = (Δ/2) ex. Again, neither ( surf nor : matrices are diagonal, because of the nonzero incident angle, the interaction between particles (1) and (2) within the central cell unit and from the other unit cells. Then, in order to find the wavelengths and eigenvectors corresponding to the LPMs, the matrix ( surf −1 − : must be diagonalized. Once the eigenvalues Λμ are calculated, the position of the LPMs for a given value of k∥inc can be found again by searching the zeros of 9(Λμ). Figure 11 shows the obtained dispersions curves for the four in-plane LPM modes Πμ. For an easier comparison with the following extinction spectra, the lines kx/y = n1k0 sin θ have been added in gray and black to the dispersion diagrams, for values of θ every 5° between 0° and 40°. Those modes are again strongly dispersive; however, the principal directions of the grating being along the symmetry axis of the dimer, it seems reasonable to think that the eigenvectors Πμ of ( surf −1 − : will resemble those of the isolated dimer, πμ: for a proper labelization of the lattice eigenvectors Πμ, we expect that |Πμ·πν| ≪ |Πμ·πμ|for μ ≠ ν, and |Πμ·πμ|≲ 1. This is an intuitive guess which turns to be wrong for lower symmetry systems like triangular trimers of nanoparticles arranged in a square grating (not shown). We have verified, similarly to the grating of monomers, that 79% (respectively 99.8%) of the eigenmodes corresponding to longitudinal (respectively transverse) illumination have a maximal weight larger than 0.9 on the eigenvectors of the isolated dimer. Hence, the color of each point on Figure 11 has been attributed according to which one of the four in-plane 2398


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The Journal of Physical Chemistry C mode Q′1 − Q′2, which is transverse and radiative, can be excited by the s polarized plane wave at normal incidence, as for the Π2′ mode (even if only that last mode is seen in the case of the isolated dimer, Figure 4). The quadrupolar mode is clearly visible on Figure 12b for θ = 0° as a maximum close to the Rayleigh wavelength of the (1, 1) order, grazing in glass (red diamond), which is about the wavelength of 620 nm obtained with the isolated dimer. However, the second maximum is found close to 900 nm, very much red-shifted if compared to the wavelength of 730 nm predicted by the CDA model. That discrepancy could be attributed to an unexpected grating-induced coupling between the very wide P1′ + P2′ and the bright Q1′ − Q2′ modes supported by the isolated dimer. Again, field maps bring useful informations: the shape of the mode at 620 nm clearly differs from the one of Q1′ − Q2′ (see Figure 4), as the light is essentially expelled from the gap region in the grating configuration. The interaction between the two modes between different dimers of the grating is completely plausible because both are dipoles oriented in the same direction and the width of the P1′ + P2′ overlaps the Q1′ − Q2′ mode. In order to precise this effect, we have simulated the transition from finite to infinite grating of dimers. Figure 13 shows the transition from finite (from one

11b) at its intersection with the red line representing the grazing propagation in air for order (0, −1). This is coherent with the conclusion of the previous discussion as grazing diffraction in air gives a tip-shape profile of the S-matrix component for longitudinal polarization. The corresponding distribution of electric field (indicated by a red hollow star) shows both an enhanced intensity close to the particles and clear secondary maxima of light in between the lines of dimers along the x direction. For larger angles, the extinction spectra are very similar to what has been obtained with the monomer grating, the lineshape of the dipolar P1′ + P2′ and quadrupolar Q1 + Q2 modes being similarly modified by the (0, −2) diffraction order, grazing in glass (large cyan circle). Let us notice that, contrary to the dipolar LPM which still is Π2′ whatever θ, the mode excited around 620 nm switches from Q1′ − Q2′ at normal incidence to Q1 + Q2 for large angle of incidence, as confirmed by comparing the field maps marked with a full green star and a blue hollow star. For s polarization (Figure 12d), the position of the extinction maxima from CDA again correctly follows the associated blue curve on Figure 11b. When compared to p polarization (still for φ = 90°), the particularly striking difference is that the narrow maximum in p polarization for θ = 10° turns into an angular minimum for s polarization at the Rayleigh wavelength of the (0, −1) diffraction order, grazing in air (small magenta circle)), resulting in two weak maxima at about 750 and 900 nm for θ = 15°−20°. Again, this phenomenon must be attributed to a Sshape of the transverse component of the S-matrix. Concerning the quadrupolar modes, the small shoulder around 600 nm observable on the GTM spectra at normal incidence corresponds to the bright quadrupolar and longitudinal Q1 − Q2 mode, but the clearer maximum for larger incidence angle is more probably due the transverse Q1′ + Q2′ dark mode: this shift is again clearly verified on the corresponding field maps, indicated by respectively a hollow green star and a hollow magenta star.

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CONCLUSIONS In this work, we have presented an in-depth analysis of bidimensional square gratings of monomers and dimers of gold nanocylinders supported by a transparent substrate, in a configuration never completely investigated up to now. We have taken into account effects of the substrate, direction and polarization of the plane wave illumination. The investigated range of periods is comparable or larger than the incident wavelength, with the consequence that several diffraction orders strongly interact with the localized plasmon modes supported by the nanostructures when they are propagating closely parallel to the interface, either within the air or the glass half-space. The comparison between exact Green’s tensor simulations and simplified calculations based on the coupled dipole approximation, together with the systematic identification of grazing diffracted orders, allowed us to put into evidence subtle effects that cannot be explained uniquely using the GTM. First, two types of profiles (narrow maxima or angular minima) are obtained in the extinction spectra close to Rayleigh wavelengths, which are relied to the behavior of the S matrix. In most of the situations, narrow maxima arise from a tip-shape profile of the real part of a particular S matrix component, when the lattice plasmon mode is exactly excited at the Rayleigh wavelength; however, smoother S-shape profile is the origin of angular minima, when the diffraction is grazing in air and the polarization of the diffracted wave is transverse and parallel to the interface. Second, despite what seems to be seen on several spectra obtained from Green’s tensor method, no cancellation of the

Figure 13. Evolution of the extinction spectra of a finite size grating of AuNCs dimers, for N × N dimers, and compared with the infinite grating. Extinction is normalized to the number of particles.

dimer to an 8 × 8 grating) to infinite grating of dimers, in the case of normal incidence with polarization along the (Oy) axis. The left graph (a) shows the exact numerical simulation with GTM, where we observe the clear shift of the dipolar mode P′1 + P′2 to the red, while the mode Q′1 − Q′2 appears more clearly when the grating becomes larger. However, the CDA simulation (Figure 13b), without quadrupolar modes, show only a very small shift of the P′1 + P′2 mode to the blue. We interpret those contradictory results between the two simulations as the result from an interaction between the quadrupolar Q1′ − Q2′ and dipolar P1′ + P2′ modes within the grating, not described in the CDA model. Finally, for transverse illumination (φ = 90°), the coupling of the p and s polarized incident plane waves are essentially realized respectively with the modes Π1 and Π2′, so the phenomena are expected to be simpler than for the longitudinal polarization. For p polarization, (Figure 12c), the maxima of extinction computed with the CDA simulation correctly follows as expected the green dispersion curve of Figure 11b. The most noticeable feature on the extinction spectra is the narrow resonance which occurs for θ = 10° at the Rayleigh wavelength of the (0, −1) diffraction order, grazing in air (small magenta circle) (λRA ≈ 756 nm), which corresponds exactly to the conditions under which the line ky = n1 k0 sin(θ) crosses the dispersion curve of mode Π2′ (see Figure 2399


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(3) Zou, S.; Janel, N.; Schatz, G. C. Silver Nanoparticle Array Structures that Produce Remarkably Narrow Plasmon Lineshapes. J. Chem. Phys. 2004, 120, 10871−10875. (4) Zou, S.; Schatz, G. C. Narrow Plasmonic/Photonic Extinction and Scattering Line Shapes for One and Two Dimensional Silver Nanoparticle Arrays. J. Chem. Phys. 2004, 121, 12606−12612. (5) Chu, Y.; Schonbrun, E.; Yang, T.; Crozier, K. B. Experimental Observation of Narrow Surface Plasmon Resonances in Gold Nanoparticle Arrays. Appl. Phys. Lett. 2008, 93, 181108. (6) Auguié, B.; Barnes, W. L. Collective Resonances in Gold Nanoparticle Arrays. Phys. Rev. Lett. 2008, 101, 143902. (7) Kravets, V. G.; Schedin, F.; Grigorenko, A. N. Extremely Narrow Plasmon Resonances Based on Diffraction Coupling of Localized Plasmons in Arrays of Metallic Nanoparticles. Phys. Rev. Lett. 2008, 101, 087403. (8) Lin, L.; Zheng, Y. Engineering of Parallel Plasmonic−Photonic Interactions for On-Chip Refractive Index Sensors. Nanoscale 2015, 7, 12205−12214. (9) Vitrey, A.; Aigouy, L.; Prieto, P.; García-Martín, J. M.; González, M. U. Parallel Collective Resonances in Arrays of Gold Nanorods. Nano Lett. 2014, 14, 2079−2085. (10) Špačková, B.; Homola, J. Sensing Properties of Lattice Resonances of 2D Metal Nanoparticle Arrays: An Analytical Model. Opt. Express 2013, 21, 27490−27502. (11) Nikitin, A. G.; Nguyen, T.; Dallaporta, H. Narrow Plasmon Resonances in Diffractive Arrays of Gold Nanoparticles in Asymmetric Environment: Experimental studies. Appl. Phys. Lett. 2013, 102, 221116. (12) Teperik, T. V.; Degiron, A. Design Strategies to Tailor the Narrow Plasmon-Photonic Resonances in Arrays of Metallic Nanoparticles. Phys. Rev. B: Condens. Matter Mater. Phys. 2012, 86, 245425. (13) Nikitin, A. G.; Kabashin, A. V.; Dallaporta, H. Plasmonic Resonances in Diffractive Arrays of Gold Nanoantennas: Near and Far Field Effects. Opt. Express 2012, 20, 27941−27952. (14) Zhou, W.; Odom, T. W. Tunable Subradiant Lattice Plasmons by Out-of-Plane Dipolar Interactions. Nat. Nanotechnol. 2011, 6, 423−427. (15) Marae-Djouda, J.; Caputo, R.; Mahi, N.; Lévêque, G.; Akjouj, A.; Adam, P.-M.; Maurer, T. Angular Plasmon Response of Gold Nanoparticles Arrays: Approaching the Rayleigh Limit. Nanophotonics 2017, 6, 279−288. (16) Yang, A.; Hoang, T. B.; Dridi, M.; Deeb, C.; Mikkelsen, M. H.; Schatz, G. C.; Odom, T. W. Real-time tunable lasing from plasmonic nanocavity arrays. Nat. Commun. 2015, 6, 6939. (17) Saison-Francioso, O.; Lévêque, G.; Akjouj, A.; Pennec, Y.; DjafariRouhani, B.; Boukherroub, R.; Szunerits, S. Search of Extremely Sensitive Near-Infrared Plasmonic Interfaces: a Theoretical Study. Plasmonics 2013, 8, 1691−1698. (18) Thackray, B. D.; Thomas, P. A.; Auton, G. H.; Rodriguez, F. J.; Marshall, O. P.; Kravets, V. G.; Grigorenko, A. N. Super-Narrow, Extremely High Quality Collective Plasmon Resonances at Telecom Wavelengths and Their Application in a Hybrid Graphene-Plasmonic Modulator. Nano Lett. 2015, 15, 3519−3523. (19) Offermans, P.; Schaafsma, M. C.; Rodriguez, S. R. K.; Zhang, Y.; Crego-Calama, M.; Brongersma, S. H.; Gómez Rivas, J. Universal Scaling of the Figure of Merit of Plasmonic Sensors. ACS Nano 2011, 5, 5151−5157. (20) Maurer, T.; Marae-Djouda, J.; Cataldi, U.; Gontier, A.; Montay, G.; Madi, Y.; Panicaud, B.; Macias, D.; Adam, P.-M.; Lévêque, G.; et al. The Beginnings of Plasmomechanics: Towards Plasmonic Strain Sensors. Frontiers of Materials Science 2015, 9, 170−177. (21) Cataldi, U.; Caputo, R.; Kurylyak, Y.; Klein, G.; Chekini, M.; Umeton, C.; Bürgi, T. Growing gold nanoparticles on a flexible substrate to enable simple mechanical control of their plasmonic coupling. J. Mater. Chem. C 2014, 2, 7927. (22) Markel, V. A. Divergence of Dipole Sums and the Nature of NonLorentzian Exponentially Narrow Resonances in One-Dimensional Periodic Arrays of Nanospheres. J. Phys. B: At., Mol. Opt. Phys. 2005, 38, L115−L121.

extinction happens in the investigated system (the grating is above a substrate with a larger refractive index) because no divergence occurs in the real part of the S-matrix components at Rayleigh wavelengths. However, some narrow and low minima are obtained in particular configurations (for example with the monomer grating in p polarization and oblique incidence) due to the interaction of grazing diffraction with both the dipolar and the quadrupolar modes. Finally, interesting effects have been observed in GTM and CDA calculations for the grating of dimer. The first concerns the width of the resonance of modes identified as bright in the isolated structures: the associated lattice plasmon modes can be much narrower in the grating configuration due to a recoupling of the scattered field with the nanoparticles. The second effect concerns the interaction between bright quadrupolar and dipolar modes when the dimers are arranged in gratings. This effects appears clearly in the GTM simulations, where the transition between finite size and infinite systems shows that the dipolar and quadrupolar modes seem to combine into modes which move aside for larger gratings, while the CDA calculations shows almost no shift of the main dipolar mode. We expect this work to be of interest for the community of scientists concerned with the design of plasmonic periodic systems for a wide range of sensing applications.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.6b11321. Discussion about the D and V modes of the single AuNC, the Fourier components of Green’s tensors, and expansions of those components around Rayleigh wavelengths (PDF)

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AUTHOR INFORMATION

Corresponding Author

*(G.L.) E-mail: [email protected]. ORCID

Gaëtan Lévêque: 0000-0003-1626-8207 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was partially supported by VisionAIRR project “PolarEP” of Région Nord-Pas-de Calais (France). Financial support of NanoMat (www.nanomat.eu) by the “Ministère de l’enseignement supérieur et de la recherche”, the “Conseil régional Champagne-Ardenne”, the “Fonds Européen de Développement Régional (FEDER)”, and the “Conseil général de l’Aube” is acknowledged. T.M. and R.C. thank the Labex ACTION project (contract ANR-11-LABX-01-01) for financial support. The authors would like to acknowledge networking support by the COST Action IC1208 and MP1302.

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REFERENCES

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NOTE ADDED AFTER ASAP PUBLICATION This paper was published on January 13, 2017, before all final corrections were made. The revised version was re-posted on January 19, 2017.

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In Depth Investigation of Lattice Plasmon Modes in Substrate

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