Evolution of an Overlapped Bandgap and Topological Flat Bands in a Higher-Order Valley Photonic Insulator Based on Dendritic Structure

Robust edge states and corner states in photonic topological insulators provide effective ways to manipulate the propagation of electromagnetic waves. Bandgaps of the previously reported photonic topological insulators are independent of each other. In this paper, a higher-order valley photonic insulator composed of arrays of dendritic structure is designed. The band structure shows an overlapped bandgap is observed, and the overlapped bandgap divides the band structure into three bandgaps. The evolution of the overlapped bandgap is investigated by changing the geometric parameters and increasing the fractal to break the C_3v symmetry drastically. Besides, it is notable that the band structure of the proposed valley photonic insulator is flat bands. It is demonstrated that undistorted transmission can be observed as a plane electromagnetic wave transmits through the proposed valley photonic insulator. The interface consisting of two valley photonic structures with distinct topological nontrivial phases is constructed. Edge states and corner states with strong energy localization are obtained in multi-band frequencies. Remarkably, two triangular structures with different Wannier center configurations both have corner states in the same bandgap, which does not obey the valley selectivity. The phenomenon is caused by the weak valley locking property due to the overlapped bandgap. The proposed valley photonic insulators are expected to benefit applications in optical devices such as topological lasers.