  Confining light to sharp metal tips has become a versatile technique to study
optical and electronic properties far below the diffraction limit. Particularly
near-field microscopy in the mid-infrared spectral range has found a variety of
applications in probing nanostructures and their dynamics. Yet, the ongoing
quest for ultimately high spatial resolution down to the single-nanometer
regime and quantitative three-dimensional nano-tomography depends vitally on a
precise knowledge of the spatial distribution of the near fields emerging from
the probe. Here, we perform finite element simulations of a tip with realistic
geometry oscillating above a dielectric sample. By introducing a novel Fourier
demodulation analysis of the electric field at each point in space, we reliably
quantify the distribution of the near fields above and within the sample.
Besides inferring the lateral field extension, which can be smaller than the
tip radius of curvature, we also quantify the probing volume within the sample.
Finally, we visualize the scattering process into the far field at a given
demodulation order, for the first time, and shed light onto the nanoscale
distribution of the near fields and its evolution as the tip-sample distance is
varied. Our work represents a crucial step in understanding and tailoring the
spatial distribution of evanescent fields in optical nanoscopy.
