or far field, light can be coupled into and out of the resonator without contacting (and thereby perturbing) the boundary geometry directly. However, monolithically attached waveguides offer some distinct advantages from an engineering point of view. In laser applications, they allow both efficient optical pumping and more complete collection of the emission. Although directional emission patterns from planar cavities can be made highly directional *in the plane* [@Nockel:94], losses by out-of-plane diffraction can be significant [@backes:VerticalAndNotches; @VerticalLossWithWG; @VerticalSpreadingDisk; @LimaconDirectionality]. With attached waveguides, such losses are minimized. Moreover, waveguides afford precise control over the number of input/output channels.

This has motivated several recent proposals to integrate chaotic cavities with waveguides [@ChaoticChannelingPRL; @EndFireInjection; @MicrospiralWG; @SquareMicrolasers; @TriangleAndSquareWG; @octagonWithLead]. Because of Mather’s theorem, however, almost all such proposals involve modes that are *not* of whispering-gallery type, achieving good confinement instead with states localized on periodic orbits: Stable and unstable periodic ray orbits whose reflections occur far enough away from any waveguide openings will be insensitive to the boundary shape at the waveguide apertures. These periodic orbits exist with or without the presence of attached waveguides, and they make lasing possible even in the free-standing Bunimovich stadium [@Fukushima:07].

The appearance of wavefunction scarring due to 