Solitons in waveguide arrays with competing quadratic nonlinearities

Quadratic nonlinearities in waveguide arrays enable ultra-fast all-optical shaping, switching, and routing of optical pulses, taking advantage of photonic band engineering through periodicity. In particular, the quadratic nonlinearity supports parametric interactions involving fundamental wave (FW) and second-harmonic (SH) modes, which can lead to suppression of spatial signal broadening due to diffraction and formation of self-trapped spatially localized states known as discrete quadratic solitons [1,2]. Recently, efficient parametric nonlinear interactions involving one FW and two different SH modes were demonstrated experimentally [3], and it was found that the beam self-focusing can be suppressed due to a competition between parametric interactions. In this work, we provide a theoretical explanation of this phenomenon through the study of the corresponding soliton solutions. Our analysis identifies the appearance of a threshold for nonlinear self-focusing, which can be selected by varying the wavenumber mismatches, and it also predicts an effective nonlinearity saturation effect.