Topological insulators are unique physical structures that are insulators in their bulk, but support currents at their edges which can be unidirectional and topologically protected from scattering on disorder and inhomogeneities. Photonic topological insulators can be crafted in materials that exhibit a strong nonlinear response, thus opening the door to the exploration of the interplay between nonlinearity and topological effects. Among the fascinating new phenomena arising from this interplay is the formation of topological edge solitons — hybrid asymmetric states localized across and along the interface due to different physical mechanisms. Such solitons have so far been studied only in materials with Kerr-type, or cubic, nonlinearity. Here the first example of the topological edge soliton supported by parametric interactions in χ(2) nonlinear media is presented.
Such solitons exist in Floquet topological insulators realized in arrays of helical waveguides made of a phase-matchable χ(2) material. Floquet edge solitons bifurcate from topological edge states in the spectrum of the fundamental frequency wave and remain localized over propagation distances drastically exceeding the helix period, while travelling along the edge of the structure. A theory of such states is developed. It is shown that multicolor solitons in a Floquet system exists in the vicinity of (formally infinite) set of linear resonances determined by the Floquet phase matching conditions. Away from resonance, soliton envelopes can be described by a period-averaged single nonlinear Schr¨odinger equation with an effective cubic nonlinear coefficient whose magnitude and sign depend on the overall phase-mismatch between the fundamental frequency and second harmonic waves. Such total phase-mismatch includes the intrinsic mismatch and the geometrically-induced mismatch introduced by the array, and its value reveals one of the genuine effects exhibited by the Floquet quadratic solitons. Our results open fundamental new prospects for the exploration of a range of parametric frequency-mixing phenomena in photonic Floquet quadratic nonlinear media.
