explain the directional emission from free-standing asymmetric cavities [@NoeckelChapter2002] by predicting at what positions the condition for total internal reflection is first violated for a chaotically diffusing trajectory.

![(a) Stable and unstable manifolds around an unstable periodic orbit. The zoomed-in region shown in (a) corresponds to the top left quadrant of the Husimi projection in (b). The real-space wave intensity of the mode in (b) is shown in the inset to (a). It is the same mode as shown in Fig. \[fig:Field-intensities-forPeaks\] (b). Shaded regions in (b) mark the waveguides, which were treated as escape windows in (a). The thin horizontal lines mark $\left|\sin\chi_{c}\right|=\nicefrac{1}{2.4}$. Refractive escape is not considered in the ray simulation for (a), to get a more complete picture of the manifolds. Note the correspondence in shape between the phase space structure and the areas of large Husimi weight.\[fig:HusimiPhaseSpace\]](Figure11){width="\columnwidth"}

To expose the relationship between the observed modes and the underlying ray phase space structure exemplified in Fig. \[fig:Poincare-surface-of-section\], it is useful to project the numerical wave results onto the phase space by means of the Husimi function. Choosing $\phi$ and $\sin\chi$ as the coordinates for this projection, only the boundary fields are needed, and this is just what 