Topological photonics aims to utilize topological photonic bands and corresponding edge modes to implement robust light manipulation, which can be readily achieved in the linear regime of light-matter interaction. In this talk, I will review some recent results regarding nonlinear interactions of one-way edge-modes in frequency mixing processes in topological photonic crystals (PhCs) and graphene metasurfaces. In particular, I will demonstrate SHG via nonlinear interaction of double topological valley-Hall kink modes in PhCs with hexagonal lattice. I will first show that two topological frequency band gaps can be created around a pair of frequencies, ω0 and 2ω0, by gapping out the corresponding Dirac points. Importantly, I demonstrate that tunable, bidirectional phase-matched SHG via nonlinear interaction of the valley-Hall kink modes inside the two band gaps can be achieved. More specifically, by using Stokes parameters associated with the magnetic part of the valley-Hall kink modes, we introduce the concept of SHG directional dichroism, which is employed to characterize optical probes for sensing chiral molecules. I also show that these ideas can be extended to graphene metasurfaces, where the Kerr-type nonlinearity of graphene can be used to control the light transmission in topological domain-interface waveguides. In the second part of my talk I will illustrate how bound states in the continuum (BICs) of certain silicon nitride PhC slabs, engineered to possess sharp resonances with high quality factors at both the fundamental-frequency and second-harmonic, can be used to achieve an orders-of-magnitude enhancement of the SHG. Certain connections between the topological charge of the BICs and the properties of the nonlinear optical interaction (SHG) are also revealed.
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