Whispering gallery modes in optical resonators have received a lot of attention as a mechanism for constructing small, directional lasers. They are also potentially important as passive optical components in schemes for coupling and filtering signals in optical fibres, in sensing devices and in other applications. In this talk it is argued that the evanescent field outside resonators that are very slightly deformed from circular or spherical is surprising in a couple of respects. First, even very small deformations seem to be capable of leading to highly directional emission patterns. Second, even though the undelying ray families are very regular and hardly differ from the integrable circular or spherical limit inside the resonator, a calculation of the evanescent field outside it is not straightforward.
This is because even very slight nonintegrability has a profound effect on the complexified ray families which guide the external wave to asymptopia. An approach to describing the emitted wave is described which is based on canonical perturbation theory applied to the ray families and extended to comeplx phase space.