Electrodynamics in 2D

Setup

The simulation solves Maxwell's equations in two dimensions (2D) in a transverse electric (TE) configuration where the electric field is in-plane \(\mathbf{E} = E_x\mathbf{\hat{x}}+E_y\mathbf{\hat{y}}\) and the magnetic field is perpendicular to it \(\mathbf{H} = H_z\mathbf{\hat{z}}\). Maxwell's equations in this configuration read: $$ \begin{align*} \frac{\partial E_x}{\partial t} &= \frac{1}{\epsilon}\left(\frac{\partial H_z}{\partial y} - J_x - \sigma E_x\right) \\ \frac{\partial E_y}{\partial t} &= -\frac{1}{\epsilon}\left(\frac{\partial H_z}{\partial x} + J_y + \sigma E_y\right) \\ \frac{\partial H_z}{\partial t} &= \frac{1}{\mu}\left(\frac{\partial E_x}{\partial y} - \frac{\partial E_y}{\partial x}\right) \end{align*} $$ where \(\epsilon = n\) is the permittivity, \(\mu = n/c^2\) is the permeability, \(\sigma\) is the (Ohmic) conductivity, and \(\mathbf{J} = J_x\mathbf{\hat{x}} + J_y\mathbf{\hat{y}}\) is an external current density source and \(c=1/2\) is the speed of light. The refractive index \(n\) and conductivity \(\sigma\) can vary spatially to represent different materials. A region at the edge of the simulation is absorbing to prevent reflections (5% of the region).

Method

We use a finite-difference time-domain (FDTD) method with a staggered Yee grid. The time step is set so the Courant-Friedrichs-Lewy number is \(c \Delta t/{\rm min}(\Delta x,\Delta y) = 0.1\). The fields are updated in time using a leapfrog scheme and the spatial derivatives are approximated using central differences. The index of refraction is stored at cell centers and interpolated to obtain the permittivity at the cell edges. The conductivity is included using a semi-implicit approach to ensure stability. Line current sources oscillate harmonically following \(\mathbf{J}(t) = A \sin(2\pi f t + \phi) \mathbf{\hat{n}}\).