Calibration method for complex permittivity measurements using s-SNOM combining multiple probe tapping harmonics

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 complexity of 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 analyzers, allow the extraction of the permittivity without detailed electromagnetic modeling 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 periodically modulated to differentiate the near-field interaction from the far-field background via lock-in detection of the harmonics of the periodic motion. This paper proposes an improved black-box calibration method that takes account of the effects of the probe tapping, including its multiple harmonics, and far-field background. The method is validated for an s-SNOM operating in the mid-infrared spectral range by applying it to spectroscopic measurements of silicon microstructures of different but well characterized doping.