A method that calculates the frequency of the lowest order mode in a cylindrical isotropic dielectric is extended to higher order modes in a anisotropic crystal. Four different axial match equations are derived depending on whether they are quasi TE or quasi TM, and have an odd or even axial mode number. A general radial match equation is also derived. Combining it with the relevant axial equation forms a set of two coupled transcendental equations that can be solved numerically. The theory is confirmed by room temperature measurements in two sapphire crystals of different aspect ratios, and in cryogenic sapphire resonators used in high stability fixed and tunable oscillators. The sensitivity of mode frequency to dimensional and permittivity perturbations is analyzed.
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