Abstract
The classical motion at a conical surface is bounded at one (narrower) side of the cone and unbounded at the other. However, it is shown here that a dielectric cone with a small half-angle gamma can perform as a high Q-factor optical microresonator which completely confines light. The theory of the discovered localized conical states is in excellent agreement with experimental data. It provides both a unique approach for extremely accurate local characterization of optical fibers (which usually have gamma ~10^-5 or less) and a new paradigm in the field of high Q-factor resonators.
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