Second-Harmonic Generation at a Fourth-Order Exceptional Point Degeneracy

An anomalous flat-band dispersion provided by a degenerate band edge (DBE) of longitudinal optical modes in a double-grating waveguide is used to enhance second-harmonic generation (SHG). The DBE is a fourth-order exceptional point degeneracy (EPD) in a lossless and gainless waveguide, characterized by the coalescence of four eigenmodes that establish a frozen mode in a cavity. At a DBE resonance, the cavity quality factor scales $Q\propto N^5$, where $N$ is the number of unit cells of the grating waveguide. In our numerical experiments, we observe the peak intensity of the fundamental field in the edge-excited cavity scaling as $I_1\propto N^{3.6}$. This leads to a highly efficient SHG process that is radiated vertically from the cavity (i.e., normal to the grating) without requiring collinear phase matching, with a conversion efficiency scaling as $\eta\propto N^{8.27}$. These results establish DBE-based waveguides as promising platforms for miniaturized efficient nonlinear photonic devices.