22.4 Intro E&M Waves via Finite Difference Time Domain

The script introduces the topic of electromagnetic waves and the Finite-Difference Time-Domain (FDTD) algorithm. It emphasizes the importance of understanding the wave equation and the practical applications of electromagnetic problems. The lecturer sets up a problem involving the determination of electric and magnetic fields within a finite spatial region, with given initial conditions for these fields. The script also mentions the significance of boundary conditions and the simplicity of the problem due to the absence of charges or currents, focusing on solving Maxwell's equations for this scenario.

This paragraph delves into the theoretical foundation of electromagnetic waves, which are described by Maxwell's equations. The script simplifies the problem by noting the absence of charges and currents, thus reducing the complexity of the equations. It highlights the initial conditions for the electric and magnetic fields and discusses the transverse nature of these waves. The script also explains how the electric and magnetic fields are related through time and space derivatives, drawing parallels to the advection equation and setting the stage for the FDTD method.

The script outlines the process of implementing the FDTD algorithm to solve for electromagnetic waves. It describes the use of finite differences to approximate derivatives in both space and time, leading to a set of equations that can be solved iteratively. The paragraph discusses the challenges of dealing with coupled fields and the vector nature of the problem, introducing the concept of a staggered grid for electric and magnetic fields to maintain stability and accuracy in the algorithm.

This section provides a detailed look at the FDTD algorithm, explaining the notation and the use of superscripts and subscripts to denote space and time variables. It introduces a normalized electric field to balance the algorithm and simplify stability analysis. The script also explains the parameter beta, which is crucial for determining the stability of the algorithm, and how it relates to the speed of light and the grid size.

The script discusses the importance of stability analysis in the FDTD algorithm, emphasizing the conditions under which the algorithm remains stable. It provides guidelines for choosing the time step and space step to maintain stability, highlighting the inverse relationship between the time step and the stability parameter beta. The paragraph concludes with a simple representation of the algorithm as two coupled first-order difference equations, which form the basis for the numerical solution.

This paragraph sets up the specific problem of solving electromagnetic waves in a finite region of space with given initial conditions. It describes the code implementation of the FDTD algorithm, including the handling of boundary conditions and the visualization of the results. The script provides an overview of the code structure and its components, such as the algorithm itself and the visualization routines, and encourages the user to experiment with the code to understand its behavior.

The script encourages further experimentation with the FDTD algorithm by modifying boundary conditions, testing the stability of the algorithm, and exploring different initial conditions. It suggests adding a dielectric slab to the simulation to observe reflections and transmissions, as well as trying other geometries and initial conditions to see how they affect the wave propagation. The paragraph also proposes investigating resonator modes and circularly polarized waves as extensions of the basic simulation.

The final paragraph introduces the concept of circularly polarized waves and demonstrates how they can be simulated using the FDTD algorithm with specific initial conditions. The script shows an example of a rotating circularly polarized wave and encourages the user to explore other field configurations. It concludes by highlighting the relevance of the FDTD method in modern microelectronics and the potential for the user to contribute to state-of-the-art research in the field.