Diffractive dipolar coupling in non-Bravais plasmonic lattices

Honeycomb plasmonic lattices are paradigmatic examples of non-Bravais lattices. We experimentally measure surface lattice resonances in effectively free-standing honeycomb lattices composed of silver nanospheres. By combining numerical simulations with analytical methods, we analyze the dispersion relation and the near-field properties of these modes along high symmetry trajectories. We find that our results can be interpreted in terms of dipole-only interactions between the two non-equivalent triangular sublattices, which naturally lead to an asymmetric near-field distribution around the nanospheres. We generalize the interaction between the two sublattices to the case of variable adjacent interparticle distance within the unit cell, highlighting symmetry changes and diffraction degeneracy lifting associated to the transition between Bravais and non-Bravais lattices.


Analytical Model
Consider a 2-d array of identical, non-magnetic metallic particles, which can be obtained through the translation of a unit cell with N particles by lattice vectors t 1 and t 2 . The position of a particle in the lattice can be determined by r jmn = r j00 + m t 1 + n t 2 , (1.0.1) where j = 1, ...N and r j00 is the position of the j-th particle relative to the cell, and (m, n) = 0, ±1, ±2 . . . . In this work we use a the spectral representation method 1 at the dipole level, and due to the large interparticle separation distances, modify the interaction terms to 1 Electronic Supplementary Material (ESI) for Nanoscale Advances. This journal is © The Royal Society of Chemistry 2020 include long-range radiative terms. The method can be seen as analogous to the coupled dipole approximation. 2 The response of a particle to the local electric field at position r jmn is given via its complex dipole polarizability through equation The electric dipole polarizability of a spherical particle α(ω), can be obtained from Mie theory 3 and is taken as α(ω) = (3ik/2)a 1 (ω), where a 1 (ω) is the first order Mie scattering coefficient. 4 The local electric field E( r jmn ) is the sum of the external field and the field due to the dipoles induced on all other particles in the lattice. The latter can be written using the dipole-dipole interaction matrix, 5 which describes the field generated by a dipole at position r and felt by a particle at r , given by where the external field is taken to be a monochromatic plane wave E ext = E 0 exp(i k· r−iωt).
Notice that Eq. (1.0.6) can be obtained by generalizing the spectral representation 1 in the dipole approximation to take into account the radiative terms in the dipole-dipole interaction.
Due to the periodicity of the system, Bloch's theorem guarantees that solutions to  It is useful to consider the specific case of a non-bravais lattice with honeycomb symmetry, which can be seen as a superposition of two triangular lattices. Notice that matrixG will be of the formG The termsG(A, B) can be written in the form: 7 where α, β = x, y, z, r AB is the vector that joins the particles of the different sublattices, g mn = m g 1 + n g 2 is a vector of the reciprocal lattice and g 1 , g 2 are primitive vectors of the reciprocal space, k is the projection of the incident wavevector onto the array, k g z = to that obtained using a spectral representation in the dipole approximation, 1 with analogous resonant conditions.

Quadrupole Contribution
To further demonstrate the dipolar nature of our SLR and to evaluate the weight of the quadrupole mode of the individual particle, we calculated the extinction spectrum including both dipolar and quadrupolar interaction. Under this approximation, the dipole and quadrupole moments induced on each particle are calculated taking into account dipoledipole, dipole-quadrupole and quadrupole-quadrupole interaction. Eq. (1.0.7) is modified in order to take into account these new interactions, and is of the form:  where p j and ← → Q j are the induced dipole and quadrupole moments on particle j, and I is a 3 × 3 identity matrix. 8 Eq. (2.0.2) is then used to calculate the extinction, making it is possible to separate the dipole and quadrupole contributions. These are shown in Fig. S1.
The extinction maps calculated including quadrupole interaction and s-polarazation are also shown. Figure S1: Extinction spectrum at normal incidence including dipole-dipole, dipolequadrupole and quadrupole-quadrupole interaction. However, the RAs and the SLR studied in this work are very detuned from that resonance. Figure S3: Electric near-field for normal incidence and s-polarization including dipole-dipole, dipole-quadrupole and quadrupole-quadrupole interaction for λ = 477 nm (left) and λ = 660 nm (right). Matched layers (PMLs) behind them. A user-defined plane wave illumination is performed by means of the port located on the air side of the structure. Zeroth-order transmission is evaluated with an opposite port located on the substrate side through the built-in S12 matrix element. The distance between the lattice and the ports must been large enough to avoid any evanescent field within the modeling domain. In our case the height of the substrate and air domains was of 850 nm. We checked that all our far field and near field results are independent on the type of cell used. To do this we compared with a rectangular and a hexagonal cell also using periodic boundary conditions. The former one contains 4 particles, while the latter contains 6 times 1/3 of a particle. We crosschecked that our calculation correctly treats the diffraction orders of the structure by verifying the consistency of the far and near field results when Periodic Ports and built-in wave excitation are employed.  Figure S5: Cross section efficiency calculated by Mie theory (blue curve) and spectral representation method (red curve). The main peak corresponds to the dipolar localized plasmon mode of the nanosphere and it matches well with the one in Fig. 1(b). The only difference between the two calculations relates to the small peak around λ = 400 nm, which is associated to an out of plane quadrupole localized plasmon mode. 9 Figure S6: Simulated s-polarized extinction map, along the Γ − M trajectory, for the honeycomb lattice homogeneously surrounded, i.e, without air-silica interface.   Figure S10: (a) s-polarized extinction spectra calculated at normal incidence for different ∆x. Figure S11: Cut of Fig. 2 (c) and (d) along the axis perpendicular to the the incident electric field. The regions between ±100 nm and ±200 nm corresponds to the position of the nanospheres. The middle panel is obtained after inclusion of the quadrupole interactions, which improves the agreement between the calculation and the simulation. As expected, the only difference between the results obtained under dipole and quadrupole approximation is the intensity in the close proximity of each sphere.