Yee Lattice

In order to discretize Maxwell's equations with second-order accuracy for homogeneous regions where there 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 illustration above for a single cubic grid voxel ($\Delta x \times \Delta x \times \Delta x$). The basic idea is that the three components of E are stored for the edges of the cube in the corresponding directions, while the components of H are stored for the faces of the cube.

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

The consequence of the Yee lattice is that, whenever you need to compare or combine different 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 flux, or whenever you output a field to a file, it is stored for the locations $(i+0.5,j+0.5,k+0.5)$: the centers of each grid voxel.