curve, and we generally found good convergence up to wavenumbers of $k\approx60$ by choosing $580$ points. For additional analysis of the exponential convergence of the method with discretization density, see Ref.  [@HeiderBIM].

![Small-cavity limit of the folded chaotic WGM, showing only low Q factors as the transverse width of the mode is comparable to the cavity size. Below $k\approx7$, the wavelength is too long to observe the WGM like concentration of intensity near the top and bottom of the cavity. \[fig:Small-cavity-limit\] The horizontal waveguide stubs are attached in a way that smoothly matches the shape given by Eq. (\[eq:ShapeCurve\]) for the vertical lobes. The refractive index is $n=2.4$ inside and $n=1$ outside, and the vertical lobes are described by Eq. (\[eq:ShapeCurve\]) with $\epsilon=0.444$. In this and the following plots, one can discern even and odd parity with respect to reflections at the horizontal axis. Although we intentionally break reflection symmetry across the vertical axis by making the horizontal waveguide lengths unequal, the modes still show approximate antinodes (left) or nodal lines (right) along the vertical axis. This is because the cavity supporting most of the intensity is still left-right symmetric.](Figure4){width="0.6\columnwidth"}

![Wavenumber sweeps of the Q factor for different refractive indices 