the 6 black dots indicate the location of the 6 bounce points for the periodic orbit. Shaded boxes indicate the intervals of $\phi$ which describe the attached waveguides. Rays entering these regions will escape the resonator. The thin horizontal lines mark the critical angle for total internal reflection, $\left|\sin\chi_{c}\right|=\nicefrac{1}{n}$ ($n=2$ here). Here and in all subsequent results, $n$ can be viewed as the interior refractive index while the exterior refractive index is unity.](Figure2){width="0.8\columnwidth"}

Figure \[fig:Two-self-intersecting-periodicOrbits\] (e) illustrates a ray in the chaotic interior region that escapes into the waveguide opening after several reflections. By folding a WGM over itself and attaching waveguides in the avoided regions of the boundary, two distinct escape windows in phase space are created: one bounded by the critical angle for refractive escape, and the other by the real-space locations of the waveguide openings. This is illustrated in the Poincar[é]{} surface of section of Fig. \[fig:Poincare-surface-of-section\], depicting the phase space of a billiard parametrized in polar coordinates by $$r(\phi)\propto1+\epsilon\,\cos\!\left(4\phi\right)\label{eq:ShapeCurve}$$ The curve parameter $\phi$ is recorded on the horizontal axis, and $\sin\chi$ is a measure of the conjugate momentum variable.

For the numerical wave calculations of Fig. \[fig:Q-factor-versusEps\], waveguide segments were connected to the open sections of 