  Chaotic whispering-gallery modes have significance both for optical
applications and for our understanding of the interplay between wave phenomena
and the classical ray limit in the presence of chaotic dynamics and openness.
In strongly non-convex geometries, a theorem by Mather rules out the existence
of invariant curves in phase space corresponding to rays circulating in
whispering-gallery patterns, so that no corresponding modes of this type are
expected. Here we discuss numerical computations of the electromagnetic fields
in planar dielectric cavities that are strongly non-convex becausethey are
coupled to waveguides. We find a family of special states which retains many
features of the chaotic whispering-gallery modes known from convex shapes: an
intensity pattern corresponding to near-grazing incidence along extended parts
of the boundary, and comparatively high cavity Q factors. The modes are folded
into a figure-eight pattern, so overlap with the boundary is reduced in the
region of self-intersection. The modes combine the phenomenology of chaotic
whispering-gallery modes with an important technological advantage: the ability
to directly attach waveguides without spoiling the Q factor of the folded mode.
Using both a boundary-integral method and the FDTD technique, we explore the
dependence of the phenomenon on wavelength in relation to cavity size,
refractive-index contrast to the surrounding medium, and the degree of shape
deformation. A novel feature that distinguishes folded from regular whispering
gallery modes is that a given shape will support high-Q folded chaotic
whispering gallery modes only in certain wavelength windows.
