FDTD Algorithm for Calculating 3D Photonic Crystal Bandgaps

This document explores the implementation of the Finite-Difference Time-Domain (FDTD) algorithm using MATLAB for calculating bandgap properties in three-dimensional photonic crystals. The algorithm enables comprehensive analysis of electromagnetic wave behavior within crystal structures and demonstrates how structural modifications can regulate electromagnetic wavelengths. Specifically, we detail the FDTD methodology for simulating wave transmission and reflection phenomena in photonic crystals, including the implementation of Yee's grid scheme for spatial discretization and perfectly matched layer (PML) boundary conditions. The code implementation covers key aspects such as material parameter assignment using dielectric constant matrices, time-stepping procedures with Courant stability conditions, and Fourier transform techniques for frequency domain analysis. We further discuss how geometric parameter variations - including lattice constants, filling ratios, and scatterer configurations - can be programmed to systematically tune photonic bandgap characteristics. The MATLAB implementation incorporates visualization functions for displaying field distributions and band structure diagrams, providing intuitive insights into wave propagation dynamics. Mastering this algorithmic approach facilitates deeper understanding of photonic crystal applications in optoelectronics and photonic devices, enabling researchers to design customized structures for specific wavelength control applications. The code structure emphasizes modular design principles, allowing separate optimization of simulation parameters, material properties, and post-processing routines for efficient bandgap analysis.