Analysis of colinear quasi-phase-matching in nonlinear photonic crystals

Standard quasi-phase-matching (QPM) schemes assume exact momentum relations between the interacting beams, thus necessitating the direct use of a reciprocal lattice vector to satisfy momentum balance. Usage of finite width beams for colinear interactions permits to use a projection of the reciprocal lattice vector along the propagation direction of the beams. We analytically derive this result and exam the new options given by this projection-based QPM for the analysis and design of nonlinear optical devices.