  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.
