Yee Lattice#

In order to discretize Maxwell's equations with second-order accuracy for homogeneous regions where there are no discontinuous material boundaries, FDTD methods store different field components for different grid locations. This discretization is known as a Yee lattice.

The form of the Yee lattice in 3d is shown in the schematic above for a single cubic grid voxel with dimensions \(\Delta x \times \Delta x \times \Delta x\). The three components of \(\mathbf{E}\) are stored on the edges of the cube in the corresponding directions, while the components of \(\mathbf{H}\) are stored on the cube faces.

More precisely, let a coordinate \((i,j,k)\) in the grid correspond to:

where \(\hat{\mathbf{e}}_k\) denotes the unit vector in the k-th coordinate direction. Then, the \(\ell\)^th component of \(\mathbf{E}\) or \(\mathbf{D}\) (or \(\mathbf{P}\)) is stored for the locations:

The \(\ell\)^th component of \(\mathbf{H}\), on the other hand, is stored for the locations:

In two dimensions, the arrangement is similar except that we set \(\hat{\mathbf{e}}_3=0\). The 2d Yee lattice for the \(H_z\)-polarization (\(\mathbf{E}\) in the \(xy\) plane and \(\mathbf{H}\) in the \(z\) direction) is shown in the figure below.

The consequence of the Yee lattice is that, whenever you need to access field components, e.g. to find the energy density \((\mathbf{E}^* \cdot \mathbf{D} + |\mathbf{H}|^2)/2\) or the flux \(\textrm{Re}\, \mathbf{E}^* \times \mathbf{H}\), then the components need to be interpolated to some common point in order to remain second-order accurate. Meep automatically does this interpolation for you wherever necessary — in particular, whenever you compute energy density or Poynting flux, or whenever you output a field to a file, it is stored at the centers of each grid voxel: \((i+0.5,j+0.5,k+0.5)\).

In a Meep simulation, the coordinates of the Yee lattice can be obtained using a field function.