摘要： 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 nonconvex 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 nonconvex because they 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 finite-difference time-domain 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 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.