## 1 Introduction
In this work, we study the renormalization of a nanoparticle polarizability located near the interface between two dielectrics interspaced with a doped graphene sheet, or with an array of graphene ribbons (see Figure 1). One of the dielectrics is the vacuum and the other acts as substrate for the support of the graphene sheet. In order to keep the analysis simple we shall restrict ourselves to the case of a non-dispersive and non-dissipative substrate, characterized by a frequency-independent and real dielectric constant. We explore the imaginary part of the polarizability in the THz range of the electromagnetic spectrum, a spectral region where graphene supports surface plasmon polaritons [16; 17; 18]. As we will see, the excitation of these polaritons leads to a significant change of the polarizability of both metallic and semiconductor nanoparticles. Indeed, the bare polarizability of a metallic nanoparticle in vacuum is essentially constant in the THz, with a very small imaginary part. However, when located near a graphene sheet, the polarizability undergoes strong renormalization, especially with respect to its imaginary part. Although the problem of modeling the polarizability of a nanoparticle close to a graphene sheet has been considered before by some of the authors of the present paper [19], that work relied on a electrostatic approximation. The present work goes beyond that, taking into account retardation effects, allowing us to correctly describe the imaginary part of the polarizability. It should be noted that the problem of determining the nanoparticle's polarizability in the presence of a homogeneous flat dielectric substrate has also been considered previously both in the electrostatic approximation [20] and in the full electrodynamic approach [6; 21; 22]. The goals of this work are as follows: (1) to extend the study of [19] including retardation effects, thus using a more general formalism; (2) to bring together in a single paper a formalism that is scattered in the literature using many different notations; (3) to introduce a rigorous formulation of the dyadic Green's function formalism that is absent in many papers; and (4) to extend this approach to the case where a nanoparticle has both dipolar electric and dipolar magnetic moments. This paper is organized as follows: in Section 2 we introduce the concept of dyadic Green's function for the electric field as a tool to obtain the electric field in the presence of source currents. In Section 2.1 we study in detail the electric field dyadic Green's function in free-space (or in a homogeneous medium). The Weyl's, or angular spectrum, representation of the dyadic Green's function is introduced in Section 2.2. This representation is well-suited to deal with the problem of radiation scattering at planar interfaces. It is also shown that the dyadic Green's functions can be expressed in terms of the tensor product of the electric field \(s\)- and \(p\)-polarization vectors. In Section 2.3, we focus on the problem of scattering at a planar interface and define the reflected and transmitted Green's functions. In Section 3 we deduce the polarizability of a nanoparticle close to an interface covered by graphene. We start defining and studying the polarizability of a nanoparticle embedded in a vacuum, in Section 3.1. The approach is generalized in Section 3.2 to the case of a nanoparticle close to a planar interface. This general description is then used to describe the renormalization of a nanoparticle's polarizability close to a continuous graphene sheet and to a graphene grating in Sections 3.3 and 3.4. In Section 4 we present a generalization of the formalism to the case where the nanoparticle has both electric and magnetic dipole moments. Such a magnetic moment can be generated, even for nanoparticles formed by a non-magnetic material, due to induced currents inside the nanoparticle [23], and can actually be the main contribution for the polarizability in the case of dielectric NPs [4, 5, 6]. Finally, a set of appendices provides some auxiliary results.