Scattering of electromagnetic waves from a cone with conformal mapping: Application to scanning near-field optical microscope

We study the response of a conical metallic surface to an external electromagnetic (EM) field by representing the fields in basis functions containing integrable singularities at the tip of the cone. A fast analytical solution is obtained by the conformal mapping between the cone and a round disk. We apply our calculation to the scattering- based scanning near-field optical microscope (s-SNOM) and successfully quantify the elastic light scattering from a vibrating metallic tip over a uniform sample. We find that the field-induced charge distribution consists of localized terms at the tip and the base and an extended bulk term along the body of the cone far away from the tip. In recent s-SNOM experiments at the visible-IR range (600nm - 1$\mu m$) the fundamental is found to be much larger than the higher harmonics whereas at THz range ($100 \mu m-3mm$) the fundamental becomes comparable to the higher harmonics. We find that the localized tip charge dominates the contribution to the higher harmonics and becomes bigger for the THz experiments, thus providing an intuitive understanding of the origin of the near-field signals. We demonstrate the application of our method by extracting a two-dimensional effective dielectric constant map from the s-SNOM image of a finite metallic disk, where the variation comes from the charge density induced by the EM field.