  We present a general framework for cavity quantum electrodynamics with
strongly frequency-dependent mirrors. The method is applicable to a variety of
reflectors exhibiting sharp internal resonances as can be realized, for
example, with photonic-crystal mirrors or with two-dimensional atomic arrays
around subradiant points. Our approach is based on a modification of the
standard input--output formalism to explicitly include the dynamics of the
mirror's internal resonance. We show how to directly extract the interaction
tuning parameters from the comparison with classical transfer matrix theory and
how to treat the non-Markovian dynamics of the cavity field mode introduced by
the mirror's internal resonance. As an application within optomechanics, we
illustrate how a non-Markovian Fano cavity possessing a flexible photonic
crystal mirror can provide both sideband resolution as well as strong heating
suppression in optomechanical cooling. This approach, amenable to a wide range
of systems, opens up possibilities for using hybrid frequency-dependent
reflectors in cavity quantum electrodynamics for engineering novel forms of
light-matter interactions.
