  Scattering-type scanning near-field optical microscopy (s-SNOM) enables
sub-diffraction spectroscopy, featuring high sensitivity to small spatial
permittivity variations of the sample surface. However, due to the near-field
probe-sample interaction, the quantitative extraction of the complex
permittivity leads to a computationally demanding inverse problem, requiring
further approximation of the system to an invertible model. Black-box
calibration methods, similar to those applied to microwave vector network
analysers, allow the extraction of the permittivity without detailed
electromagnetic modelling of the probe-sample interaction. These methods,
however, are typically designed for stationary setups. In contrast, the
distance between the sample and the probe tip of the s-SNOM is slowly
modulated, which is required for the lock-in detection used to extract the
near-field interaction buried in the far-field background. Here we propose an
improved calibration method that explicitly takes probe tapping into account.
We validate our method for an s-SNOM operating in a mid-infrared spectral range
by applying it to measurements of silicon microstructures of different but well
characterised doping.
