  It is shown that when a circular resonator is deformed in a nonintegrable
way, a symmetry breaking of escaping rays occurs which can dramatically
modulate the outgoing wave even for small perturbations. The underlying
mechanism does not occur in integrable models for which the ray families can be
computed exactly and is described in this Letter on the basis of canonical
perturbation theory. Emission from deformed resonators is currently of immense
practical interest in the context of whispering-gallery optical resonances of
dielectric cavities and the approach outlined here promises simple analytical
characterisations in the important case of small deformations.
