Nonlinear Photonic Crystals

Nonlinear frequency conversion in 2D ${\chi }^{\left(2\right)}$ photonic crystals is theoretically studied. Such a crystal has a 2D periodic nonlinear susceptibility, and a linear susceptibility which is a function of the frequency, but constant in space. It is an in-plane generalization of 1D quasi-phase-matching structures and can be realized in periodic poled lithium niobate or in GaAs. An interesting property of these structures is that new phase-matching processes appear in the 2D plane as compared to the 1D case. It is shown that these in-plane phase-matching resonances are given by a nonlinear Bragg law, and a related nonlinear Ewald construction. Applications as multiple-beam second-harmonic generation (SHG), ring cavity SHG, or multiple wavelength frequency conversion are envisaged.