Two-Dimensional Finite-Difference Time-Domain (FDTD) Algorithm

The two-dimensional Finite-Difference Time-Domain (FDTD) algorithm is a fundamental numerical method in computational electromagnetics that discretizes Maxwell's equations in both space and time domains to simulate electromagnetic wave propagation through media. The core principle involves staggered arrangement of electric and magnetic field components on grid points, with alternating updates at each time step. In code implementation, this typically requires defining separate arrays for E-field (Ez) and H-field (Hx, Hy) components, with update equations derived from the discretized curl equations using central differencing schemes.

Perfectly Matched Layer (PML) serves as a widely-used absorbing boundary condition that theoretically absorbs all incident waves without reflection through special artificial media layers at computational domain boundaries. For 2D FDTD implementation, PML requires field splitting where each component (e.g., Hx split into Hxy and Hxz) is assigned distinct attenuation coefficients to handle waves incident from different directions. This is typically coded using additional auxiliary arrays and conductivity profiles that increase gradually toward the boundary.

For source excitation, Gaussian pulses are frequently employed due to their well-defined spectral characteristics and easy controllability. By adjusting the temporal width parameter (sigma) in the Gaussian function exp(-(t-t0)^2/sigma^2), one can control the excited frequency bandwidth. Plane wave incidence, commonly used for far-field radiation simulation, is typically implemented through the Total-Field/Scattered-Field (TF/SF) technique. This method partitions the computational domain into total-field and scattered-field regions, with incident fields introduced as equivalent sources at the interface using additive source terms in the update equations.

Algorithm implementation must adhere to the Courant stability condition, requiring the time step (dt) to be smaller than the time for electromagnetic waves to cross the smallest grid cell (dt < dx/(c√2) in 2D). The numerical dispersion effect becomes pronounced with larger grid sizes, necessitating appropriate spatial discretization (typically λ/10 to λ/20) based on simulation accuracy requirements. Code implementation should include validation checks for these parameters during initialization.