Effective phase matching conditions for parametric interactions in one-dimensional photonic band gap structures

Summary form only given. In a recent publication, the theoretical enhancement of second harmonic generation at the band edge of a mixed quarter-half wave PBG structure was discussed. The structure was chosen so that both the pump and second harmonic fields were tuned near the first and second order band edges respectively, such that the pump was tuned at the first transmission resonance of the first order gap, and the second harmonic at the second transmission resonance of the second order gap. Second harmonic enhancement, which was predicted to be several orders of magnitude larger compared to an exactly phase matched bulk structure of the same length, was attributed to several factors: an increase in the mode density, and a corresponding decrease in group velocity; an increased interaction time; strong mode overlap. The novelty of the approach outlined above rests on the fact that the active material is not poled in the usual way in order to produce a phase-matched device. Instead, the process relies on the linear, geometrical properties of the structure to significantly enhance second harmonic gain. We examine how phase matching conditions are established in a periodic structure, and how this knowledge can be used for design purposes. We first use the matrix transfer method, combined with previous analytical results, in order to derive a complex effective index of refraction for a generic PBG structure.