Nonlinear topological photonics, which explores topics common to the fields of topological phases and nonlinear optics, is expected to open up a new paradigm in topological photonics. Here, we demonstrate second-harmonic generation (SHG) via nonlinear interaction of double topological valley-Hall kink modes in all-dielectric photonic crystals (PhCs). We first show that two topological frequency bandgaps can be created around a pair of frequencies, $omega_0$ and $2omega_0$, by gapping out the corresponding Dirac points in two-dimensional honeycomb PhCs. Valley-Hall kink modes along a kink-type domain wall interface between two PhCs placed together in a mirror-symmetric manner are generated within the two frequency bandgaps. Importantly, through full-wave simulations and mode dispersion analysis, we demonstrate that tunable, bi-directional phase-matched SHG via nonlinear interaction of the valley-Hall kink modes inside the two bandgaps can be achieved. In particular, by using Stokes parameters associated to the magnetic part of the valley-Hall kink modes, we introduce a new concept, SHG directional dichroism, which is employed to characterize optical probes for sensing chiral molecules. Our work opens up new avenues towards topologically protected nonlinear frequency mixing and active photonic devices implemented in all-dielectric material platforms.
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