$k$-dependent optics of nanostructures: Spatial dispersion of metallic nanorings and split-ring resonators

The response of matter on an electromagnetic wave at a certain point and time depends on the field strength prior to this time and at places close to this point. Hence the material parameters are functions of the frequency $\omega$ and the wave vector $\vec{k}$, in general. While the temporal dispersion is common knowledge, spatial dispersion in usually disregarded. However it becomes crucial in the optical response of nanostructures. Here we map the complete $\vec{k}$-dependent optical response of a split-ring-resonator array over a broad frequency range via Mueller-matrix spectroscopic ellipsometry at different angles of incidence and all azimuthal orientations. The comparison with a closed-ring structure elucidates the rule of spatial dispersion in metal-dielectric nanostructures.

Figure 1

Scanning electron micrograph of the investigated closed-ring structures (left) and the split-ring-resonator array (right).

Figure 2

Transmission spectra at normal incidence of the split-ring-resonator (SRR) and closed-ring (CR) arrays. Whereas the CR appears nearly isotropic with a slight shift in the resonances between perpendicular directions, the SRR exhibits a strong dichroism.

Figure 3

(a) MM contour plot for the closed-ring array measured at 0.8 eV in transmission. In spite the isotropic lattice, the off-diagonal elements $m_{13}=m_{31}\ne 0$ clearly indicate a mixing of polarization states at oblique incidence. Note that some matrix elements are enhanced by a factor as indicated; $m_{13}$ and $m_{31}$ are multiplied by a factor of 40, for instance. (b) Simulated MM contour plot in transmission for a 20 nm closed Au film on glass. The upper right and the lower left ($2\times 2$) submatrices are zero; isotropic samples normally do not mix the polarization states at all.

Figure 4

(a) Measured and (b) simulated MM contour plots for the split-ring-resonator array at a given photon energy of 0.6 eV. For the simulations a simple in-plane anisotropy was used with two and three Lorentz oscillators in the two perpendicular directions, respectively. This model describes the overall optical response of the SRR array very well. It is not necessary to assume any permeability $\mu \ne 1$.