  Dielectric optical micro-resonators and micro-lasers represent a realization
of a wave-chaotic system, where the lack of symmetry in the resonator shape
leads to non-integrable ray dynamics. Modes of such resonators display a rich
spatial structure, and cannot be classified through mode indices which would
require additional constants of motion in the ray dynamics. Understanding and
controlling the emission properties of such resonators requires the
investigation of the correspondence between classical phase space structures of
the ray motion inside the resonator and resonant solutions of the wave
equations. We first discuss the breakdown of the conventional eikonal
approximation in the short wavelength limit, and motivate the use of
phase-space ray tracing and phase space distributions. Next, we introduce an
efficient numerical method to calculate the quasi-bound modes of dielectric
resonators, which requires only two diagonalizations per N states, where N is
approximately equal to the number of half-wavelengths along the perimeter. The
relationship between classical phase space structures and modes is displayed
via the Husimi projection technique. Observables related to the emission
pattern of the resonator are calculated with high efficiency.
