unstable periodic orbits in [@Fukushima:07] illustrates that wave solutions of the Helmholtz equation can defy ray-optics predictions. This is also true for the predictions of Mather’s theorem, and again the stadium billiard is a case in point: Numerical solutions for closed (hard-wall) cavities [@Prange2001] show that a form of chaotic WGM exists if the straight sides are sufficiently short.

![\[fig:Q-factor-versusEps\]Spatial structure of a folded chaotic whispering-gallery mode (insets), and Q factor of the mode versus deformation parameter $\epsilon$ at refractive index $n=2.4$. The resonator deformation is defined in Eq. (\[eq:ShapeCurve\]); attached waveguides of finite lengths are shown horizontal; their lengths are slightly unequal to remove reflection symmetry. False color represents field intensity. The real part of $k$ (not shown) decreases approximately linearly from $14$ to $12.5$ as $\epsilon$ increases.](Figure1){width="0.9\columnwidth"}

We will describe whispering-gallery type behavior with a different topology, which can be described as *folded* chaotic whispering-gallery modes, cf. Fig. \[fig:Q-factor-versusEps\]. This refers to the fact that the direction of whispering-gallery ray circulation underlying the modes reverses due to a self-intersection near the center of the cavity. The bowtie orbit of Ref.  [@Gmachl05061998] also exhibits a self-intersection, but it – like all other such orbits we are aware of – 