Degenerate eigenfrequency splitting by gyrotropic sample in spherical microwave resonator

When a spherical microwave resonator contains a spherical isotropic semiconductor sample, precisely at the centre of the cavity, the lowest TM modes are triply degenerate in frequency. External DC magnetic field breaks the complete rotational symmetry, by inducing anisotropy in the sample. This results in the frequency splitting of degenerate eigenmodes. We calculate the frequency splitting for the lowest TM modes in a K-band resonator. It is shown that the splitting is a measurable effect, which gives information about the material parameters of the sample. Furthermore, in special cases it is possible to determine both the permittivity and permeability tensors, by mere resonant frequency measurements in spherical microwave cavities.