Finite-Difference Time-Domain Simulation of Wave Propagation in Space-Time-Varying Media: Derivation of schemes, extended framework, and illustrative examples

We present a comprehensive tutorial on the finite-difference time-domain (FDTD) modeling of space, time, and space-time-varying media, building upon our previous review by offering a detailed, step-by-step guide for numerical simulations. This manuscript emphasizes practical computational methodologies and examines the dynamic behavior of both TE and TM electromagnetic fields under oblique incidence on space-time-modulated gratings, in which electrical permittivity, magnetic permeability, and conductivity vary in both space and time. The article details the derivation of the FDTD schemes, including the Courant-Friedrichs-Lewy (CFL) stability condition, explicit formulations for TE and TM wave illuminations, and the implementation of appropriate boundary conditions. In addition, we extend the FDTD framework to incorporate nonlinear effects, thereby broadening its applicability to advanced applications in wireless communications, quantum technologies, and radars. Engaging illustrative examples are provided to demonstrate the versatility and practical relevance of the approach.