and refractive index $n=2.4$ shown in Fig. \[fig:Wavenumber-sweeps\]. The wave number corresponds to the top of the first Q-factor peak, $k\approx13.6$ ($Q=2612$). Waveguide and resonator structure are underlaid as green shading, and the false-color scale represents the electric field. (b) Same deformation and refractive index, but approximately ten times shorter wavelength, $k\approx116.0$ and $Q=9898$. (b) is discussed in section \[sec:Discussion\].\[fig:MEEPmode1\]](Figure9){width="0.8\columnwidth"}

Having convinced ourselves that the same results can be obtained with both numerical approaches (boundary-integral and finite-difference time domain), we proceed with MEEP to look for folded chaotic WGMs in resonators with thinner waveguides, cf. Fig. \[fig:Two-different-modesThinWires\]. The size of the opening in Fig. \[fig:Two-different-modesThinWires\] (a) is comparable to the wavelength, whereas it is approximately twice the wavelength in (b). The whispering-gallery circulation along the boundary, characteristic of the earlier results in Fig. \[fig:Q-factor-versusEps\], shows that the phenomenon is robust not only to variations in the refractive index, but also to changes in waveguide width.

![Two different modes with thinner waveguides, obtained by introducing a vertical offset between the two lobes of Eq. (\[eq:ShapeCurve\]) and adjusting$\epsilon$ to match them smoothly to the horizontal waveguides. The refractive index is $n=2.4$. In (a) $k\approx13.3$ and $Q\approx1580$, in (b) $k\approx22.4$ and $Q\approx2550$. \[fig:Two-different-modesThinWires\]](Figure10){width="0.8\columnwidth"}

