Wilson loop and topological properties in a three-dimensional woodpile photonic crystal

We numerically study the first- and the second-order topological states of electromagnetic waves in the three-dimensional (3D) woodpile photonic crystal (PhC). Recent studies on 3D PhCs have mainly focused on the observation of the topological states. Here we not only focus on finding the topological states but also propose a numerical calculation method for topological invariants, which is based on the Wilson loop. For the 3D woodpile PhC, the topological states emerge due to the finite difference in the winding number or partial Chern number. The selection rule for the emergence of topological hinge states is also pointed out based on the topological invariants. Our numerical calculation results are essential and take a step toward the experimental realization of topological waveguide in 3D PhCs.