We show that manipulation by a spatial profile of the refractive index of a circularly symmetric dielectric cavity results in a novel way of fine tuning frequency separations as well as spatial localizations of high-Q whispering-gallery modes excited in the cavity. The method permits dispersion compensation in the modes (spectrum equalization), diminishes the quality-factor limitation by surface roughness and contamination, and allows critical coupling to ultra-high-Q modes without maintaining an air gap with evanescent couplers.