resilient even to large waveguide openings. For practical applications, much narrower waveguides will typically be desirable because the input and output should be single mode. We show results for large openings because that is the regime in which the coexistence of whispering-gallery modes with an apparent violation of Mather’s theorem is clearest. In particular, the openings are much wider than the wavelength, so they cannot be treated as small perturbations. We also observe the non-monotonic distribution of $Q$-factors shown in Fig. \[fig:Wavenumber-sweeps\] at $\epsilon=0.444$ for other deformations and waveguide widths. Figure \[fig:Q-factor-scan-e52\] shows this for a higher deformation of $\epsilon=0.52$.

![Q-factor scan versus interior wave number at deformation $\epsilon=0.52$ and refractive index $n=2.4$, showing modes with decay rate $\kappa<0.05$. \[fig:Q-factor-scan-e52\]](Figure6){width="0.8\columnwidth"}

Figure \[fig:Field-intensities-forPeaks\] depicts the wave intensities of the highest-Q folded chaotic WGMs in Fig. \[fig:Wavenumber-sweeps\], with wavenumbers ranging from $k\approx13.6\ldots44$ at $n=2.4$. Each image represents a single quasibound state, not a superposition. The nodal structure that decorates the underlying figure-eight patterns becomes more complex with increasing $k$, in much the same way that transverse nodes appear in higher-order Gaussian beams or radial nodes appear in whispering-gallery modes. The difference between those examples and the folded chaotic WGMs is that the 