Q factors of such modes will increase *monotonically* with increasing cavity size, or equivalently with shorter wave length. To address this dependence, we fix the cavity geometry at $\epsilon=0.444$ to identify all the modes with lifetimes above a threshold of $\left|\mathrm{Im}(k_{\text{QB}})\right|=\kappa<0.05$ with wavenumber $k\lesssim60$. The particular choice of deformation corresponds to the maximum in Fig. \[fig:Q-factor-versusEps\] at $n=2.4$, but the same phenomena are observed at other values of $\epsilon$.

Figure \[fig:Wavenumber-sweeps\] shows that the Q factors display a *non-monotonic* peak structure. The data are plotted for four different resonators with refractive indices of $n=1.6$, $1.8$, $2.0$ and $2.4$. The peaks are more pronounced for larger refractive index because the critical angle for total internal reflection decreases with $\sin\chi_{\mathrm{c}}=1/n$. By choosing the horizontal axis to display $nk$ instead of the free-space wavenumber $k$, the peak positions for different $n$ moreover line up to a good approximation. Since $nk$ is the wave number inside the cavity, this indicates that the high-Q peak *locations* are determined by the interior wave patterns, not the monotonically $n$-dependent coupling to the surrounding free space. The absolute length scale of the cavity drops out of $Q$ as per Eq. (\[eq:QfactorDef\]).

The folded chaotic whispering-gallery modes are surprisingly 