Nonlinear Interactions in Periodic and Quasi-Periodic Nonlinear Photonic Crystals

Abstract

Nonlinear photonic crystals (NLPC) are materials in which the second order susceptibility χ (2) is modulated in an ordered fashion, usually in one or two dimensions, while the linear susceptibility remains constant. This modulation can be done, for example, by periodic electric field poling of ferroelectric crystals such as LiNbO3 or KTiOPO4, or by orientation pattering of semiconductors such as GaAs. NLPC are significantly different than the more common photonic crystals, in which the linear susceptibility is modulated, bringing forth optical processes involving several frequencies. For example, one dimensional (1D) modulation of the nonlinear susceptibility is widely used nowadays for quasi-phase-matched nonlinear frequency conversion. The extension to two-dimensional (2D) modulation allows further design flexibility, and is useful for non-collinear second harmonic generation (SHG), for simultaneous wavelength interchange, for third and fourth harmonic generation, and proposed for realization of all optical effects, e.g. all optical deflection and splitting. Note that other types of nonlinear interactions are possible in photonic crystals, including third-order (χ (3)) nonlinearities in “standard” photonic crystals, as well as photonic crystals in which both the linear and the second-order nonlinear susceptibilities are modulated. These interactions will not be considered in this chapter.