Calculating Band Structure of 2D Photonic Crystals Using Plane Wave Expansion Method

The Plane Wave Expansion (PWE) method enables computational analysis of band structures in two-dimensional photonic crystals. This computational approach, rooted in Maxwell's equations, solves electromagnetic wave propagation problems to derive photonic band structures and optical characteristics. The PWE algorithm's key advantage lies in its inherent handling of periodic lattice structures, making it applicable to various photonic crystal geometries including square, triangular, and hexagonal arrangements. Implementation typically involves Fourier transforming dielectric constants and solving eigenvalue problems through matrix diagonalization. The method's computational framework can incorporate material dispersion relations and loss factors using complex permittivity tensors, which is crucial for practical photonic crystal device simulations. Code implementation generally requires reciprocal space discretization, where convergence depends on the number of plane waves used in the expansion. Advanced implementations may include optimization techniques for handling anisotropic materials and boundary conditions through specialized eigenvalue solvers.