Coupled electromagnetic modes between a corrugated surface and a thin probe tip

A self‐consistent formalism is presented in order to determine the dispersion equation of the coupled electromagnetic modes between a dielectric probe tip of arbitrary shape and a rough surface. A microscopic picture of the matter is used for describing the dielectric response of the tip, and the coupling with the substrate is introduced from a dynamical matrix expressed in terms of propagators. Retardation effects may be included without formal difficulty through an appropriate response function describing the surface of the solid (local or nonlocal). An advantage of such a calculation lies in the possibility of simulating dielectric tips of arbitrary shape without introducing boundary conditions at the surface of the probe. Connection with atomic force microscopy and near field detection by local probe will be discussed.