Parametric localized modes in quadratic nonlinear photonic structures

We analyze two-color spatially localized nonlinear modes formed by parametrically coupled fundamental and second-harmonic fields excited at quadratic (or ${\chi }^{\mathrm{}\left(2\right)\mathrm{}})$ nonlinear interfaces embedded in a linear layered structure—a quadratic nonlinear photonic crystal. For a periodic lattice of nonlinear interfaces, we derive an effective discrete model for the amplitudes of the fundamental and second-harmonic waves at the interfaces (the so-called discrete ${\chi }^{\mathrm{}\left(2\right)\mathrm{}}$ equations) and find, numerically and analytically, the spatially localized solutions—discrete gap solitons. For a single nonlinear interface in a linear superlattice, we study the properties of two-color localized modes, and describe both similarities to and differences from quadratic solitons in homogeneous media.