  Electromagnetic interaction between a sub-wavelength particle (the `probe')
and a material surface (the `sample') is studied theoretically. The interaction
is shown to be governed by a series of resonances corresponding to surface
polariton modes localized near the probe. The resonance parameters depend on
the dielectric function and geometry of the probe, as well as the surface
reflectivity of the material. Calculation of such resonances is carried out for
several types of axisymmetric probes: spherical, spheroidal, and pear-shaped.
For spheroids an efficient numerical method is developed, capable of handling
cases of large or strongly momentum-dependent surface reflectivity. Application
of the method to highly resonant materials such as aluminum oxide (by itself or
covered with graphene) reveals a rich structure of multi-peak spectra and
nonmonotonic approach curves, i.e., the probe-sample distance dependence. These
features also strongly depend on the probe shape and optical constants of the
model. For less resonant materials such as silicon oxide, the dependence is
weak, so that the spheroidal model is reliable. The calculations are done
within the quasistatic approximation with the radiative damping included
perturbatively.
