  Optical mushroom shaped billiards offer a unique opportunity to isolate and
study non-dispersive, marginally unstable periodic orbits. Here we show that
the openness of the cavity to external fields presents unanticipated
consequences for the far-field distribution, including directional emission.
This is mediated by interactions of marginally unstable periodic orbits with
chaotic modes and is potentially relevant for the design of microcavity lasers.
We also show that such orbits are robust against perturbations to the shape of
the cavity, despite the non-generic character of the classical counterpart and
the lack of structural stability of the corresponding classical orbits.
