Computation of 3D Photonic Crystal Band Structure Using Plane Wave Expansion Method

To compute the band structure of 3D photonic crystals, we employ the plane wave expansion method which utilizes a complete set of plane waves as basis functions for expanding the electromagnetic field components. The implementation involves discretizing Maxwell's equations in Fourier space, where the periodic dielectric function is expanded in Fourier series. The key computational steps include constructing the Helmholtz eigenvalue problem matrix by calculating Fourier coefficients of the inverse dielectric function through fast Fourier transforms (FFT). The algorithm requires careful selection of reciprocal lattice vectors and energy cutoff parameters to ensure convergence. Typical implementations involve solving a large-scale eigenvalue problem H(k)ψ = ω²ψ where H represents the discretized wave operator in momentum space. The code structure generally includes: 1) Brillouin zone discretization using k-point sampling, 2) Matrix diagonalization using numerical libraries like LAPACK, and 3) Band structure visualization through high-symmetry points in the irreducible Brillouin zone. The plane wave expansion method provides an efficient approach for predicting photonic band gaps and dispersion relations. Computational accuracy depends critically on the number of plane waves used in the expansion, with typical calculations requiring 100-1000 plane waves per wavevector. This method enables detailed analysis of photonic crystal properties including band gap widths, defect mode frequencies, and transmission characteristics through post-processing of eigenfrequency results.