## 5 Conclusions  
In this paper we have studied the influence of two plasmonic structures on the effective polarizability of a nanoparticle made of either a metal (with a nearly dispersionless bare polarizability) or a polar dielectric or semiconductor (with a resonant polarizability due to polar optical phonons). The two studied structures are a continuous graphene sheet and a plasmonic graphene-based grating. In both cases a significant enhancement of the imaginary part of the polarizability has been observed. The two media possess plasmonic resonances which, however, occur at different frequencies. In the particular case of the grating, the resonance is tunable in two different ways: by adjusting the gate voltage and by changing the geometric parameters of the grating. In this case, it is possible to scan the resonance from the THz to the mid-IR range, whereas for the continuous graphene sheet the resonance is always in the THz range for the currently achieved values of electronic doping using a gate. The approach pursued here was to model the nanoparticle by a point-like dipole. The main motivation for this approach lies in its ability to make analytic progress. However, in real systems, one has a finite-size particle which can be modeled as an assembly of many point-like dipoles. These are determined by the coupled dipole equations [49]. In this case, the particle, even a spherical one, has other multipole resonances that can couple to the incoming radiation and contribute to the extinction cross-section. The two lowest multipoles, besides the electric dipole, are the magnetic dipole and the electric quadrupole. It can be shown numerically that for semiconductor nanoparticles such as spheres, cubes, pyramids, disks, and cylinders, the extinction cross-section has a strong magnetic-dipole resonance [4; 5; 6; 23]. We note, however, that for semiconductor nanoparticles, if we consider interband transitions, that is, exciton resonances that are characteristic of semiconductors, the relevance of higher multipole resonances depends much more on the underlying band structure than on the shape. The formalism used in this paper to describe the renormalization of the electric dipole resonances can be extended to include the problem of magnetic dipole resonance [6], as we have seen in the previous section. The contribution to the extinction cross section of the magnetic dipole is given by \(\sigma_{\text{net}}^{\text{m}}=\frac{\omega\mu_{\text{f}}}{2S_{\text{inc}}} \Im([\mathbf{H}_{0}^{*}(\mathbf{r}_{0})\cdot\mathbf{m}(\mathbf{r}_{0})]\), where \(S_{\text{inc}}\) is the power per unit area of the incoming radiation and \(\mathbf{m}(\mathbf{r}_{0})=\frac{\omega\mu_{\text{f}}}{\alpha}_{MM}\mathbf{H}_{ 0}(\mathbf{r}_{0})\), with \(\frac{\omega\mu_{\text{f}}}{\alpha}_{MM}\) being the effective magnetic polarizability of the particle and \(\mathbf{H}_{0}\) the incoming magnetic field. The effective magnetic polarizability can be derived as done before for the electric dipole case. To that end, we will need the dyadic magnetic Green's function which can be obtained by writing the wave equation for the magnetic field using the procedure outlined in Section 4. This study will be pursued in a forthcoming paper.