served as an initial condition for ray simulations of the escape directionality, caused by the mixed phase space in the vicinity of the escape window.

It has long been understood that no true WGMs can be sustained in circular domains if *waveguides* are attached, because this creates openings in the boundary that interrupt the whispering-gallery circulation. In phase space, the escape window then depends not only on tangential momentum but on position. For a detailed study of this breakdown in the context of electronic microstructures with otherwise impenetrable walls, see Ref.  [@PhysRevB.54.10652]. This can be viewed as a special case of a theorem by Mather [@MatherInvariant], who proved the non-existence of whispering-gallery invariant curves in planar billiards when the boundary is not everywhere convex. An example for Mather’s theorem is the Bunimovich stadium (two semicircles joined by straight, parallel sides) [@BunimovichStadium], whose ray dynamics does not permit whispering-gallery orbits even if $\ell$ is arbitrarily short, even though the weaker condition of non-concavity still holds. Strict convexity is therefore required in order to sustain WG circulation, but any waveguide openings will necessarily introduce corners where this condition breaks down.

By exploiting the preferential emission directions of chaotic WGMs in the near 