self-intersection.

The geometry also permits a rectangular periodic orbit whose angle of incidence has the fixed value $\sin\chi\approx0.707$, but Fig. \[fig:Two-self-intersecting-periodicOrbits\] (d) shows an important distinction to the self-intersecting orbits: as $\epsilon$ increases, the corners of the waveguide aperture encroach on the orbit and eventually touch it. The resulting corner diffraction [@ISI:000323333000016] will degrade the lifetime of any modes based on this orbit. The numerical computations to be described in section \[sec:wavecalc\] have revealed high-Q modes in cavities of the shape (\[eq:ShapeCurve\]), that do show enhanced intensity near the rectangle orbit, but never exclusively on that orbit. Instead, the folded chaotic states consistently show high intensity overlapping with the folded orbits over a wide range of deformations $\epsilon$ and attached waveguide widths.

![\[fig:Poincare-surface-of-section\]Poincar[é]{} surface of section with the curve parameter $\phi$ as position and $\sin\chi$ as momentum variable. Here, $\chi$ is the angle between incident rays and the surface normal, so that $\sin\chi=1$ corresponds to grazing incidence. Ray trajectories show up as point clouds in the chaotic sea, or as one-dimensional lines for regular motion. Shown as insets are a stable bouncing-ball orbit (left) and an unstable periodic orbit (right). Arrows point to the corresponding phase space locations; in particular 