Plane waves and localized modes in quadratic waveguide arrays

In this paper, we examine theoretically the system of periodically poled LiNbO3 waveguide arrays that feature a quadratic nonlinearity and that have recently become accessible to experimental studies. Motivated by these earlier works, we provide a detailed analytical study of the existence, stability, and dynamics of plane wave delocalized solutions, as well as that of strongly localized modes consisting of a few sites. The linear stability of both classes of solutions is quantified as a function of the system parameters, such as the wave vector mismatch parameter or the interchannel coupling strength, using experimentally accessible ranges. Our findings are, in all the cases, corroborated by numerical bifurcation analysis results; furthermore, when the solutions are found to be unstable, typical examples of the instability evolution are shown.