  Weyl semimetals possess unique electrodynamic properties due to a combination
of strongly anisotropic and gyrotropic bulk conductivity, surface conductivity,
and surface dipole layer. We explore the potential of popular tip-enhanced
optical spectroscopy techniques for studies of bulk and surface topological
electron states in these materials. Anomalous dispersion, extreme anisotropy,
and the optical Hall effect for surface polaritons launched by a nanotip
provides information about Weyl node position and separation in the Brillouin
zone, the value of the Fermi momentum, and the matrix elements of the optical
transitions involving both bulk and surface electron states.
