Generalized Bloch boundary conditions based on a symmorphic space group and the finite-element implementation in photonic crystal

We studied generalized Bloch boundary conditions and their finite element implementation within the theoretical framework of a symmorphic space group. By combining translation symmetry operations with mirror and rotational symmetry operations, we developed a procedure for implementing generalized Bloch boundary conditions in the finite element method (FEM) for periodic photonic structures. First, we lay out the theoretical foundation and numerical implementation of generalized Bloch boundary conditions in FEM. We illustrate the proposed method via 2D/3D periodic photonic structures. Without a loss of generality, we calculate the band structures of 2D/3D photonic crystals using our proposed generalized Bloch boundary conditions and benchmark the results against the conventional Bloch boundary conditions. The comparisons show that band structure and eigenmode yield excellent agreement with the results obtained from conventional Bloch boundary conditions. However, our method has improved the computational efficiency by at least twofold. We further elaborate the comparisons with computation errors, memory efficiency, and computation times, all of which show that our proposed method outperforms the conventional one due to careful consideration of the mirror and rotational symmetry operation, apart from the translation symmetry. In addition, our method can easily be extended to other methods such as FDTD and transfer matrix.