  We present a quantum model of two-level atoms localized in a 3D lattice,
based on the Hopfield theory of exciton polaritons. In addition to a
polaritonic gap at the exciton energy, a photonic bandgap opens up at the
Brillouin zone boundary. Upon tuning the lattice period or angle of incidence
to match the photonic gap with the exciton energy, one obtains a combined
polaritonic and photonic gap as a generalization of Rabi splitting. For typical
experimental parameters, the size of the combined gap is on the order of 25
cm^{-1}, up to 10^5 times the detuned gap size. The dispersion curve contains a
branch supporting slow-light modes with vanishing exciton probability density.
