Modeling and understanding localized nonlinear waves in photonic bandgap structures

Summary form only given. We review recent progress made to analyze and describe the nonlinear behavior of these structures. Emphasis is placed on the role of localized soliton-like structures. These constitute indeed the key to understand the fascinating behavior of different types of structures sharing the main characteristics of being both periodic and nonlinear. First, we focus on 1D photonic band-gap (PBG) structures with linear periodic properties, e.g. Bragg gratings, which are known to support gap-solitons in Kerr media, first observed in fiber gratings. In this context, a full description of gap-soliton properties is emerging which account for observed experimental features and serve as a guideline for the next generation of experiments, ultimately aimed at tracking the evolution of short pulses along PBG materials. Equally important is the fact that nonlinearities allow for the coupling of different (higher-orders) bandgaps of the same physical structure, a possibility which is prevented at low intensity. Quadratic nonlinearities responsible for second-harmonic generation are best suited for this purpose, and will lead to multicolor gap-solitons with peculiar properties.