  We present a method for extracting high-spatial resolution dielectric
constant data at microwave frequencies. A scanning near field microwave
microscope probes a sample and acquires data in the form of the frequency and
quality factor shifts of a resonant cavity coupled to the sample. The approach
reported here is to calculate the electromagnetic fields by the finite element
method in both static and time-dependent modes. Cavity perturbation theory
connects the measured frequency shifts to changes in the computed energy stored
in the electromagnetic field. In this way, the complex permittivity of the
sample is found. Of particular interest are thin-film materials, for which a
method is reported here to determine the dielectric constant without the need
to use any fitting parameters.
