Modal decompositions of the local electromagnetic density of states and spatially resolved electron energy loss probability in terms of geometric modes

We present universal modal decompositions of the quasistatic electromagnetic local density of states (EMLDOS) of nanoparticles in the presence of dissipation and for arbitrary materials. This relies on a generic and universal description of the optical eigenmodes in arbitrary structures. In this description, already developed in various former theories, the eigenmodes are independent of the energy, scale invariant, and depend only on the structures shapes. For these reasons, we call the modes geometric eigenmodes. A direct analogy with the well-known modal decomposition of the EMLDOS in the case of nondissipative photonic modes is drawn in the special case of a material described by a Drude's model. Moreover, we show that this formalism is suitable to describe the electron energy loss spectroscopy and some scanning near optical field microscopy experiments. The link between such experiments and the mapping of the geometric mode is analyzed. In particular, this allows us to show that the delocalization of the inelastic signal can be interpreted as a convolution of the surface eigencharges at the boundary of the particle with the Coulombian interaction that arises in both experimental set up. A local density of states for the geometric eigenmodes depending only on the geometry of the particle is introduced, by analogy with the well-known EMLDOS for the photonic eigenmodes. This density of states and the related Green's functions, which have a very simple and concise form, are shown to be capable of generating all relevant quantities in the quasistatic approximation. Finally, we discuss the impact of the energy dispersion of the dielectric functions on the loss of spatial coherence of the geometric eigenmodes.