orbits persists for a wide range of deformations. (c) is obtained from (b) by adjusting the initial conditions to generate eight instead of three reflections in the top half. For the rectangular unstable periodic orbit in (d), the increasing proximity to the corners of the waveguide aperture is illustrated for increasing $\epsilon$. (e) A ray trajectory launched in the cavity (green dot) and escaping through the waveguide opening (red dot). It retains the folded character of the orbits in (a), (b) for a short time. \[fig:Two-self-intersecting-periodicOrbits\]](Figure3){width="0.85\columnwidth"}

Figure \[fig:Two-self-intersecting-periodicOrbits\] illustrates the non-convex cavity shape in which the long-lived modes are confined. The openings in the boundary are attached to waveguides, but for the purposes of the ray dynamics they constitute escape windows, in addition to the refractive escape mechanism that sets in when the condition for total internal reflection at the dielectric interface is violated. The three self-intersecting periodic orbits shown in Fig. \[fig:Two-self-intersecting-periodicOrbits\] (a - c) have incident angles satisfying $\sin\chi\ge0.6$, which means they are confined by total internal reflection for refractive indices $n>1.7$. Half of the reflections occur with opposite sense of circulation, corresponding to the reversal of angular momentum (with respect to the center) taking place at the 