Yee lattice

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Illustration of Yee lattice in 3d for a single grid voxel.
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Illustration of Yee lattice in 3d for a single grid voxel.

In order to discretize Maxwell's equations with second-order accuracy (for homogeneous regions), FDTD methods store different field components at different grid locations. This discretization is known as a Yee lattice.

The form of the Yee lattice in 3d is shown in the illustration here 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 at the edges of the cube in the corresponding directions, while the components of H are stored at the faces of the cube.

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

\mathbf{x} = (i \hat\mathbf{e}_1 + j \hat\mathbf{e}_2 + k \hat\mathbf{e}_3) \Delta x.

Then, the \ellth component of \mathbf{E} or \mathbf{D} (or \mathbf{P}) is stored at the locations

(i,j,k)+ \frac{1}{2} \hat\mathbf{e}_\ell  \Delta x.

The \ellth component of \mathbf{H}, on the other hand, is stored at the locations

(i+\frac{1}{2},j+\frac{1}{2},k+\frac{1}{2})-\frac{1}{2} \hat\mathbf{e}_\ell  \Delta x.

In two dimensions, the idea is similar except that we set \hat\mathbf{e}_3=0. The 2d Yee lattices for the TE polarizations (E in the xy plane and H in the z direction) is shown in the figure at right.

Yee lattice in 2d for the TE polarization.
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Yee lattice in 2d for the TE polarization.

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. Meep does this interpolation for you—in particular, whenever you compute energy density or flux, or whenever you output a field to a file, it is stored at the locations (i + 0.5,j + 0.5,k + 0.5): the centers of each grid voxel.

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