  Efficient micro-resonators simultaneously require a large quality factor $Q$
and a small volume $V$. However, the former is ultimately limited by bending
losses, the unavoidable radiation of energy of a wave upon changing direction
of propagation. Such bending losses increase exponentially as $V$ decreases and
eventually result in a drop of $Q$. Therefore, circular cavities are generally
designed with radii that are much larger than the optical wavelength. The same
leakage of energy by radiation limits the sharpness of bends in photonic
integrated circuits. In this article, we present a way to reduce bending losses
in circular micro-resonators. The proposed scheme consists of one or more
external dielectric rings that are concentric with the cavity. These rings
alter the field outside the cavity where radial oscillations set in, and thus
control the far field radiation. As a result, the $Q$ factor can be increased
by several orders of magnitude while keeping a small cavity volume.
