Calculating Transmission Spectra and Field Distributions of 2D Photonic Crystals Using FDTD Method

The Finite-Difference Time-Domain (FDTD) method is a powerful numerical technique for simulating electromagnetic wave propagation in two or three dimensions. This method is particularly effective for analyzing photonic crystals - periodic structures that manipulate light propagation. The core implementation involves discretizing Maxwell's equations in both spatial and temporal domains using central difference approximations, then solving them iteratively through time-stepping algorithms. In practical code implementation, the FDTD method typically requires defining Yee cell grids where electric and magnetic field components are staggered in space. Key computational steps include: initializing field arrays with proper boundary conditions (such as Perfectly Matched Layer - PML for absorption), implementing update equations for E and H fields using finite differences, and applying source excitation (like Gaussian pulse or continuous wave). The field evolution is tracked through multiple time steps until convergence. For calculating transmission properties of 2D photonic crystals, the implementation involves monitoring the field components at specific detection planes. The transmission coefficient is computed by comparing the Fourier-transformed fields at input and output ports, while field distributions are obtained by storing spatial field data at selected time steps. Post-processing includes performing discrete Fourier transforms to obtain frequency-domain information and visualizing field patterns using contour plots or vector field displays. By analyzing the resulting transmission spectra and field distributions, researchers can understand light behavior in photonic crystals and optimize structural parameters for desired optical properties. The method allows for designing novel photonic devices with tailored bandgap characteristics and light guidance capabilities.