When an optical cavity is deformed even very slightly from perfect circular or spherical symmetry, tunnelling emission cannot be determined directly from (complexified) ray dynamics due to the appearance of natural boundaries. It has been established however, that for sufficiently small deformations, perturbative approaches to the complexified ray families can give a good quantitative description of the exterior evanescent wave, even when the exact ray families are ruined by the intervention of natural boundaries. These perturbative solutions are especially interesting in the neighbourhood of resonances of the internal ray dynamics. Here the emitted wave of a cavity provides a laboratory for the exploration of solutions of the Schroedinger equation in complex space, including for example, the ability to observe Stokes transitions in physically measurable quantities.