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 photo nic band-gap structures. We introduce the concept of a complex effective in dex for structures of finite length, derived from a generalized dispersion equation that identically satisfies the Kramers-Kronig relations. We then a ddress the conditions necessary for optimal, phase-matched, resonant second harmonic generation. The combination of enhanced density of modes, field l ocalization, 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 p hase velocities, pointing to the existence of two dispersion relations for structures of finite length. [S1063-651X(99)15010-4].