The electromagnetic fields and eigen-frequencies of triaxially ellipsoidal resonators have been calculated. The method used is modal superposition, based on the decomposition of the fields inside the resonator into a series expansion, with unknown coefficients, of the eigen-states of a spherical resonator. The coefficients and eigen-frequencies have then been calculated by minimizing the rms electric field at the resonator surface using software designed within MatLab~(~R). Accuracies of parts in 10~(10) have been obtained for the eigen-frequencies, and the dependence of the calculations on the ellipsoidal eccentricities and the resonator modes has been analysed. The results are compared with those of Mehl (2009 Metrologia 46 554-9), who used a second-order approximation. The method of modal superposition has the potential for solving many similar problems in resonators of arbitrary shapes.
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