Surface solitons in two-dimensional quadratic photonic lattices

Two-dimensional surface solitons have been recently predicted to exist as novel types of discrete solitons localized in the corners or at the edges of two-dimensional photonic lattices. These theoretical predictions were followed by the experimental observation of twodimensional surface solitons in optically-induced photonic lattices and waveguide arrays laser-written in fused silica. Importantly, these two-dimensional surface solitons demonstrate novel features in comparison with their counterparts in truncated one-dimensional waveguide arrays. In particular, in a sharp contrast to onedimensional surface solitons, the threshold power of twodimensional surface solitons is lower at the surface than in a bulk making the mode excitation easier. Surface solitons are usually considered for cubic or saturable nonlinear media. However, multicolour discrete solitons in quadratically nonlinear lattices have been studied theoretically in both oneand two-dimensional lattices irrespective to the surface localization effects. Only Siviloglou et al. studied discrete quadratic surface solitons experimentally in periodically poled lithium niobate waveguide arrays, and they employed a discrete model with decoupled waveguides at the second harmonics to model some of the effects observed experimentally. More elaborated theory of one-dimensional surface solitons in truncated quadratically nonlinear photonic lattices, the so-called two-color surface lattice solitons, has been developed recently by Xu and Kivshar who analyzed the impact of the phase mismatch on the existence and stability of nonlinear parametrically coupled surface modes, and also found novel classes of onedimensional two-color twisted surface solitons which are stable in a large domain of their existence. In this Letter, we extend the analysis of two-color surface solitons to the case of two-dimensional photonic lattices. We study, for the first time to our knowledge, twocolor surface solitons in two-dimensional square photonic lattices with quadratic nonlinear response. We analyze the effect of mismatch on the existence, stability, and generation of surface solitons located in the corners or at the edges of the nonlinear lattice. We consider the propagation of light in a twodimensional photonic lattice of a finite extent imprinted in a quadratic nonlinear medium, which involves the interaction between the fundamental frequency (FF) and second-harmonic (SH) waves. Light propagation is described by the following coupled nonlinear discrete equations