gives rise to modes that explore the boundary only at isolated points corresponding to the discrete vertices where ray reflections occur. In Ref.  [@Gmachl05061998], $Q>1000$ was obtained at relatively high refractive index contrast of $n\gtrapprox3$, corresponding to a critical angle for total internal reflection of $\chi_{\mathrm{c}}\approx\arcsin\nicefrac{1}{n}\approx0.34$ (where $\chi$ denotes the angle of incidence with respect to the surface normal). For such high index contrast, even Fabry-Perot cavities with near-normal angle of incidence provide similar Q-factors without the need for additional mirrors.

By contrast, all the high-Q modes in our design are localized predominantly in the whispering-gallery region of phase space, much further from the critical angle, even when the index contrast is below $n=2$. The light is then well-confined by total internal reflection. Reminiscent of a figure-eight shape, folded chaotic WGMs have a waist that allows them to avoid two isolated sections of the boundary; but unlike stable periodic orbits such as the bowtie, folded chaotic WGMs explore the remainder of the perimeter in the same way as would be expected for conventional WGMs. In this sense, these types of modes come as close as possible to the circulating ray patterns that are strictly ruled out by Mather’s theorem.

The 