Figure 1

(Color online) (a) Schematic representation of the system under study. An infinite periodic two-dimensional square array of gold nanoparticles is situated in a semi-infinite, homogeneous medium of refractive index $n_1$ at a distance $d$ above a substrate of refractive index $n_2$. The center-to-center separation between particles is $a$. Light is incident from the superstrate medium with a direction normal to the plane of the array and polarization along a lattice vector $\left(\widehat{x}\right)$. (b) Elastic scattering cross section of an individual gold sphere immersed in water $\left(n=1.33\right)$ and in glass $\left(n=1.46\right)$, as obtained from Mie theory. (c) Same as (b) for a particle array of period $a=500\text{nm}$ rather than an individual particle, also including the asymmetric configuration in which the spheres are in water with their surfaces located at a distance of 1 nm from a glass substrate. Vertical dashed lines indicate the position of the $⟨0,1⟩$ and $⟨1,1⟩$ diffraction conditions in the different media. All gold nanospheres are taken to have a radius of 35 nm.Reuse & Permissions

Figure 2

(Color online) (a) Dispersion diagrams showing the reflectance of square arrays of 35-nm-radius gold particles immersed in water (left), in glass (center), and in water at a surface-to-surface distance of 1 nm from a water-glass interface [right, see Fig. 1a]. Here, $k_1$ is the wave vector in the host medium, $k_{\parallel }$ is the parallel wave vector along a principal lattice direction, and $a=500\text{nm}$ is the lattice period. The incident polarization is TE. (b) Scheme showing the first Brillouin zone (shaded area) and its vicinity. The dashed lines (light line and diffracted light line) correspond to the condition of grazing low-order diffracted beams. The circles signal the three beams contributing to the $⟨0,1⟩$ diffraction spectral anomaly under normal incidence via $\left(0,\pm 2\pi /a\right)$ wave-vector transfers from the lattice (double-arrowed horizontal lines). (c) Calculated normal-incidence specular reflectance of the square array in the asymmetric environment considered in (a). Rigorous numerical solutions of Maxwell’s equations including multipoles with orbital angular momentum up to $l=4$ are compared with a simplified solution using only dipoles $\left(l=1\right)$. These results, which require a large number of diffracted beams $\sim 20$ to get convergence, are compared with our analytical model for dipoles [Eq. (9)] using only one beam [zero order, central circle in (b)] or three beams [all circles in (b)].Reuse & Permissions

Figure 3

(Color online) Calculated normal-incidence specular reflectance of a square array of 60-nm-radius gold spheres immersed in water and placed above a glass substrate [see Fig. 1a]. The lattice period is 500 nm. The distance from the bottom of the spheres to the substrate is varied from 1 nm to $2\mu \text{m}$ (see labels). Dashed curves: analytical model of Eq. (9). Solid curves: fully converged multiple-scattering numerical solution.Reuse & Permissions

Figure 4

(Color online) (a) Calculated normal-incidence specular reflectance of a square array of 60-nm-radius gold spheres and period equal to 500 nm immersed in a fluid and placed at a surface-to-surface distance of 1 nm from a glass substrate [see Fig. 1a]. The refractive index of the fluid is varied in the range $n_1=1.2–1.5$ around the glass index $n_2=1.46$. [(b) and (c)] Maximum of the reflectance spectra as a function of $n_1$ for two values of the array period and three different sphere radii (see text insets). All spectra are calculated with the analytical model of Eq. (9).Reuse & Permissions

Figure 5

(Color online) (a) Scattering cross section of gold prolate ellipsoids with increasing long-axis radius $R_2$ (see labels) and short axis $R_1=50\text{nm}$. The polarizability is derived from Ref. 33. (b) Reflectance of ellipsoid arrays calculated from the analytical model of Eq. (9), using the polarizabilities of (a). The ellipsoids are immersed in water at a surface-to-surface distance of 1 nm from a glass substrate [see Fig. 1a]. The incident electric field is along the long axes of the particles $\left(R_2\right)$, which are in turn oriented along a principal lattice direction. The diffraction edges associated with the water and glass media are indicated as vertical dashed lines.Reuse & Permissions

Figure 6

(Color online) Calculated zero-order transmittance, reflectance, and absorbance spectra for a square array of gold spheres with increasing radius as indicated in the legend [dielectric function from Johnson and Christy (Ref. 24)]. The array has pitch 550 nm and is placed inside water (refractive index 1.33). The sphere surfaces are separated 1 nm from a glass substrate (refractive index 1.46).Reuse & Permissions

Figure 7

Real and imaginary parts of the lattice sum $G$ as a function of wavelength $\lambda$ normalized to the period $a$ for normal incidence on a square array. The wavelength is given relative to the medium in which the lattice is immersed.Reuse & Permissions