Theory of topological corner state laser in Kagome waveguide arrays

In comparison with conventional lasers, topological lasers are more robust and can be immune to disorder or defects if lasing occurs in topologically protected states. Previously reported topological lasers were almost exclusively based on the first-order photonic topological insulators. Here, we show that lasing can be achieved in the zero-dimensional corner state in a second-order photonic topological insulator, which is based on the Kagome waveguide array with a rhombic configuration. If gain is present in the corner of the structure, where the topological corner state resides, stable lasing in this state is achieved, with the lowest possible threshold, in the presence of uniform losses and two-photon absorption. When gain acts in other corners of the structure, lasing may occur in edge or bulk states, but it requires substantially larger thresholds, and transition to stable lasing occurs over much larger propagation distances, sometimes due to instabilities, which are absent for lasing in corner states. We find that increasing two-photon absorption generally plays strong stabilizing action for nonlinear lasing states. The transition to stable lasing stimulated by noisy inputs is illustrated. Our work demonstrates the realistic setting for corner state lasers based on higher-order topological insulators realized with waveguide arrays.