Bloch theory of entangled photon generation in non-linear photonic crystals

We present a quantum mechanical description of parametric down-conversion and phase-matching of Bloch-waves in non-linear photonic crystals. We discuss the theory in one-dimensional Bragg structures giving a recipe for calculating the down-converted emission strength and direction. We exemplify the discussion by making explicit analytical predictions for the emission amplitude and direction from a one-dimensional structure that consists of alternating layers of Al0.4Ga0.6As and Air. We show that the emission is suitable for the extraction of polarization-entangled photons.