  The quasi-bound modes localized on stable periodic ray orbits of dielectric
micro-cavities are constructed in the short-wavelength limit using the
parabolic equation method. These modes are shown to coexist with irregularly
spaced "chaotic" modes for the generic case. The wavevector quantization rule
for the quasi-bound modes is derived and given a simple physical interpretation
in terms of Fresnel reflection; quasi-bound modes are explictly constructed and
compared to numerical results. The effect of discrete symmetries of the
resonator is analyzed and shown to give rise to quasi-degenerate multiplets;
the average splitting of these multiplets is calculated by methods from quantum
chaos theory.
