We present a formal theory of single quantum-dot coupling to a planar
photonic crystal that supports quasi-degenerate cavity modes, and use this
theory to describe, and optimize, entangled-photon-pair generation via the
biexciton-exciton cascade. In the generated photon pairs, either both photons
are spontaneously emitted from the dot, or one photon is emitted from the
biexciton spontaneously and the other is emitted via the leaky-cavity mode. In
the strong-coupling regime, the generated photon pairs can be maximally
entangled, in qualitative agreement with the simple dressed-state predictions
of Johne {\em et al.} [Phys. Rev. Lett. vol. 100, 240404 (2008)]. We derive
useful and physically-intuitive analytical formulas for the spectrum of the
emitted photon pairs in the presence of exciton and biexciton broadening, which
is necessary to connect to experiments, and demonstrate the clear failure of
using a dressed-state approach. We also present a method for calculating and
optimizing the entanglement between the emitted photons, which can account for
post-sample spectral filtering. Pronounced entanglement values of greater than
80% are demonstrated using experimentally achievable parameters, even without
spectral filtering.Comment: 5pages 4 fi