Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions

We discuss the linear dispersive properties of finite one-dimensional photonic band-gap structures. We introduce the concept of a complex effective index for structures of finite length, derived from a generalized dispersion equation that identically satisfies the Kramers-Kronig relations. We then address the conditions necessary for optimal, phase-matched, resonant second harmonic generation. The combination of enhanced density of modes, field localization, and exact phase matching near the band edge conspire to yield conversion efficiencies orders of magnitude higher than quasi-phase-matched structures of similar lengths. We also discuss an unusual and interesting effect: counterpropagating waves can simultaneously travel with different phase velocities, pointing to the existence of two dispersion relations for structures of finite length.