Calculating Defect Modes in Two-Dimensional Photonic Crystals Using Plane Wave Expansion Method

We can employ the plane wave expansion method to calculate defect modes in two-dimensional photonic crystals. This computational approach enables deeper understanding of defect mode formation mechanisms and their physical properties. In the implementation process, we can incorporate various parameters and boundary conditions, such as lattice constants, defect positions, and geometric configurations, to thoroughly investigate the behavior and characteristics of defect modes within photonic crystal structures. The algorithm typically involves Fourier transforming the dielectric function and solving Maxwell's equations in frequency domain using eigenvalue solvers. Key computational steps include setting up the supercell structure with defects, defining the basis set of plane waves, and diagonalizing the resulting Hamiltonian matrix to obtain eigenfrequencies and field distributions. This methodology can be extended to other research domains, including magnetic materials and semiconductor devices, providing valuable insights for performance optimization and material property analysis. The code implementation often utilizes numerical libraries for matrix operations and may include parameter scanning functionality for systematic studies of defect mode dependencies on structural variations.