Fig. 1 SEM image of the Au nanoparticles on SiO₂ with the scale bar indicating 500 nm. The image was recorded at 15 kV with a magnification of 50 k.

Fig. 2 Schematic diagram of the measurement geometry and the polarization base. The measurements are performed in specular reflection so that ϕ ₀ = ϕ_s and θ ₀ = θ_s . It is assumed that the coordinate system is oriented so that its x- and y-axis are parallel to the two primitive translation vectors of the square lattice that the particles in Fig. 1 define.

Fig. 3 Contour plots of the elements of the normalized Mueller matrix m for the sample shown in Fig. 1, measured for θ ₀ = θ_s = 55° (where θ_s denotes the polar scattering angle). The photon energy and the azimuthal rotation angle (ϕ ₀) of the incident light represent the radius and the angle in these polar plots, respectively. The inner circle of the plots corresponds to the photon energy 0.73 eV, while the outer corresponds to 5.90 eV. The Rayleigh-lines for the first BZ (upright semi-square), the 2nd BZ (tilted semi-square) in air (white lines), and in the glass substrate (black lines) have been superimposed the m ₂₁ element. In the m ₁₃ and m ₁₄ elements, the extended Rayleigh-lines for air (white lines) and glass (black lines) that were calculated for a 90° symmetry are additionally superimposed, in addition to the LSPR resonance at 2.1 eV (white circles) as estimated from the quasi-static approximation. For reasons of improved visibility, the amplitudes of the elements m ₂₁ and m ₂₂ were reduced for E <2.5 eV; for this energy range, sgn(m _{2 j} )|m _{2 j} |^{1 / 4} was presented for j = 1,2 where sgn(·) denotes the signum (or sign) function. The circles in the schematic inset, replacing the m ₁₁ element (that is trivially one), correspond to photon energy of the incident light from 1 eV (thick inner line) to 5 eV (thick outer line) in steps of 1 eV, while 0, 45, 90, etc. denote the azimuthal rotation angle (ϕ ₀) in degrees.

Fig. 4 Schematic diagram of the reciprocal square lattice, where the experimental azimuthal angle is defined by the angle ϕ ₀ between the vectors k _∥ and ${\mathbf{G}}_{\parallel }^{(10)}$ ; ${\varphi }_0=\angle ({\mathbf{k}}_{\parallel },{\mathbf{G}}_{\parallel }^{(10)})$ . The boundaries of the 1st, 2nd, and 3rd BZs are indicated by dotted lines, and the high symmetry points are shown. The Bragg conditions for the first and the third BZs are indicated for ϕ ₀ ≤ 45°, with Θ _m = ϕ _m − ϕ ₀.

Fig. 5 The energy dependence of ψ_pp in the range 1.5–5.9 eV (left); and Im 〈ε〉 _pp in the range 2.5–5.9 eV (right), for azimuthal angles of incidence from ϕ ₀ = 0° (incidence along Γ-X) to ϕ ₀ = 45° (incidence along Γ-M) in steps of Δϕ ₀ = 5°. The dispersion of the LSPR can be seen in ψ_pp around 2.1 eV. Three approximate Rayleigh-lines are drawn by hand in both sub-figures, while the location of the BZ-2 in air is more ambiguous. For clarity, the ψ_pp -curves were offset by 1.4°κ and the Im 〈ε〉 _pp -curves by 0.2κ where κ = ϕ ₀/Δϕ ₀.

Fig. 6 Contour plots of the variations of Im 〈ε〉 _pp (top) and ψ_pp (bottom) with azimuthal angle ϕ ₀ and photon energy E. In both sub-figures, the Rayleigh-lines corresponding to the 1st and the 2nd BZ in air (white lines), and in the substrate (black lines), have been indicated. The horizontal black dotted lines seen abound 2.1 eV correspond to the position of the LSPR obtained by GranFilm [21]. For better visibility, a scaling has been applied for photon energies below 2.5 eV so that the plotted quantities are Im 〈ε〉 _pp /5.6 (top) and ψ_pp /2 (bottom).

Fig. 7 Contour plots of the ellipsometric angles ψ_ps (top) and ψ_sp (bottom) with azimuthal angle ϕ ₀ and photon energy E. In both sub-figures, the Rayleigh-lines corresponding to the 1st and the 2nd BZ in air (white lines), and in the substrate (black lines), have been indicated. The horizontal black dotted lines seen around 2.1 eV correspond to the energy position of the LSPR obtained by GranFilm [21]. For reasons of improved visibility, ψ_ps and ψ_sp are scaled by 2 for E <2.5 eV.