A finite element method for the band structure computation of photonic crystals with complex scatterer geometry

Abstract

In this paper, a Petrov–Galerkin finite element interface method (PGFEIM) is proposed to compute the band structures of 2D photonic crystals (PtCs) with complex scatterer geometry, which is formulated as a generalized eigenvalue problem (GEP) for given wave vectors. The key idea of this method is to choose a piecewise linear function satisfying the jump conditions across the interface to be the solution basis, and choose a special finite element basis independent of the interface to be the test function basis to remove the boundary term upon the assertion of Bloch boundary conditions. Non-body-fitting projected grid is employed to implement this approach. Both isotropic and anisotropic materials are considered and discussed for two- and three-component PtCs with square or triangular lattice. Taking advantage of the PGFEIM in dealing with sharp-edged interfaces, PtCs with various peculiar scatterer geometries are studied. Particularly, some distinctive three-component structures with triple-junction points between different materials are fabricated, and the middle frequencies of its absolute photonic band gaps are higher than conventional three-component structures. Numerical examples demonstrate that the bands move to higher or lower frequency regions, which is determined by the component materials, with the increase of number of sharp corners on the surface of entire scatterer.