Yee lattice
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:<math>(i+0.5,j+0.5,k+0.5)-\hat\mathbf{e}_\ell / 2</math>. | :<math>(i+0.5,j+0.5,k+0.5)-\hat\mathbf{e}_\ell / 2</math>. | ||
- | In two dimensions, we set <math>\hat\mathbf{e}_3=0</math>; the 2d Yee lattices for the TE and TM polarizations ('''E''' in the ''xy'' plane or in the ''z'' direction, respectively) are shown in the figure at right. | + | In two dimensions, we set <math>\hat\mathbf{e}_3=0</math>; the 2d Yee lattices for the TE and TM polarizations ('''E''' in the ''xy'' plane or in the ''z'' direction, respectively) are shown in the figure below. |
+ | |||
+ | :(TO DO: Figure) | ||
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 <math>(\mathbf{E}^* \cdot \mathbf{D} + |\mathbf{H}|^2)/2</math> or the flux <math>\textrm{Re}\, \mathbf{E}^* \times \mathbf{H}</math>, 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 <math>(i+0.5,j+0.5,k+0.5)</math>: the centers of each grid voxel. | 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 <math>(\mathbf{E}^* \cdot \mathbf{D} + |\mathbf{H}|^2)/2</math> or the flux <math>\textrm{Re}\, \mathbf{E}^* \times \mathbf{H}</math>, 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 <math>(i+0.5,j+0.5,k+0.5)</math>: the centers of each grid voxel. | ||
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In order to discretize the equations with second-order accuracy (in homogeneous regions), FDTD methods store different field components at different grid locations. This discretization is known as a Yee lattice
In general, let a coordinate (i,j,k) in the grid correspond to:
- .
Then, the th component of or (or ) is stored at the locations
- .
The th component of , on the other hand, is stored at the locations
- .
In two dimensions, we set ; the 2d Yee lattices for the TE and TM polarizations (E in the xy plane or in the z direction, respectively) are shown in the figure below.
- (TO DO: Figure)
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 or the flux , 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.