The problem of accurate calculation of eigenfrequencies in resonators of complex geometry is not only fundamental but also has many practical applications. In particular, a possibility for calculating the eigenfrequencies and geometry dependent dispersion of whispering gallery modes is important for optimization of dielectric microresonator-based Kerr frequency combs. In this case, the required anomalous second-order dispersion may be controlled by means of small shape variations of the resonator. Unfortunately, all uniform approximations for the eigenfrequencies do not reach the required precision for this purpose. We propose new approximations for spheroids, quartics, and toroids with better precision, which also allow for the estimation of the second-order dispersion. We also obtain analytical expressions for field distribution in microresonators and investigate the possibility of achieving better approximations by combining analytical and numerical methods.
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