2D FDTD and Plane Wave Expansion Methods for Optical and Microwave Computations

In computational electromagnetics, 2D Finite-Difference Time-Domain (FDTD) and Plane Wave Expansion methods are two widely used numerical techniques for optical and microwave simulations. These approaches offer distinct advantages for different scenarios, enabling efficient analysis of electromagnetic wave propagation and interactions in two-dimensional space through specialized algorithmic implementations.

The FDTD method is a time-domain technique that discretizes Maxwell's equations in both space and time, solving electromagnetic field evolution through iterative time-stepping. In code implementation, 2D FDTD typically utilizes Yee's grid arrangement where electric and magnetic field components are staggered in space and calculated alternately using leapfrog time integration. This method requires relatively smaller computational resources and is suitable for simulating transient electromagnetic phenomena in complex dielectric structures such as waveguides, photonic crystals, and metamaterial optical responses. Since it operates directly in the time domain, FDTD can intuitively capture pulse propagation, reflection, and scattering processes, and easily accommodates nonlinear effects through appropriate material model integration in the update equations.

The Plane Wave Expansion method is a frequency-domain approach primarily used for calculating band structure characteristics of periodic structures like photonic crystals. In 2D simulations, this method expands electromagnetic fields in reciprocal space as superpositions of plane waves, analyzing Bloch modes and bandgap structures by solving eigenvalue problems. The numerical implementation involves constructing a Hamiltonian matrix from Fourier components of the dielectric function and diagonalizing it to obtain eigenfrequencies. This method proves particularly efficient for computing photonic bandgaps and mode distributions, though it shows weaker adaptability for non-periodic structures and broadband response analysis due to its frequency-domain nature.

These two methods complement each other in practical research. FDTD excels in time-domain dynamic simulations with implementations that handle complex boundaries and material variations through careful discretization of curl equations, while Plane Wave Expansion specializes in frequency-domain characteristic analysis with efficient matrix diagonalization techniques. By combining both approaches through appropriate data exchange interfaces, researchers can more comprehensively investigate the working mechanisms of optical and microwave devices, such as designing novel waveguides, optimizing photonic crystal structures, or analyzing antenna radiation characteristics using both temporal and spectral perspectives.