The time-dependent Maxwell’s equations can be solved numerically by an iterative time-marching algorithm called the Finite Difference Time Domain (FDTD) algorithm. FDTD shows the interaction of light with optical media in real time. Even in case of optics problems where only steady-state behavior of light is studied, FDTD solvers prove to be easier to use over their frequency-domain counterparts.
To derive an FDTD algorithm, we need to model the differential equation by replacing the derivatives with finite difference (FD) expressions. We review the different kinds of FD expressions and the introduce a FD model of the wave equation, and then the FDTD algorithm. We analyze its accuracy and numerical stability, and find that what is called the nonstandard (NS) FD model can greatly reduce the error. We provide training both on in-house FDTD algorithms implemented using Python and commercial software.
Key concepts covered include:
- Finite differences and finite difference models of differential equations
- Finite difference model of the Maxwell’s equation
- The finite difference time domain (FDTD) algorithm
- Algorithm error, stability, grid sampling per wavelength (Nyquist rate)
- Standard vs Non-standard FDTD
- NS-FDTD in dispersive media
- NS-FDTD in nonlinear media
- Coding in Python/Matlab and simulation training with FDTD
- Hands-on training on commercial software