Topological parametric gain and two-color edge states in equidistant lithium niobate waveguide arrays

We analyze parametric χ₂ processes in equidistant finite-size arrays of thin-film lithium niobate waveguides, where the fundamental harmonic (FH) field supports topological edge states due to the specific interplay between inter- and intra-modal couplings of two families of guided modes, while the second-harmonic (SH) field only supports bulk modes. Regimes of topological parametric gain are identified, where the gain only occurs in the edge states of the FH field, regardless of the spatial distribution of the pump SH field. The topological gain of the FH component generally triggers localization of the SH field near an edge of the array in the optical parametric oscillation dynamics. In small-size arrays, parametric gain at both edges can be observed even when pumped at one side. This process can lead to an anomalous "tunneling" of the SH field to the opposite edge. We also analyze the existence and stability of two-color nonlinear edge states (solitons), in which both FH and SH fields are localized at an edge of the array. Depending on the phase-matching condition, such solitons either emerge from the linear FH edge state without a power threshold or exist above a certain power threshold dictated by the coupling strength in the SH field.