Yee lattice
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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 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 <math>(i,j,k)</math> in the grid correspond to: | In general, let a coordinate <math>(i,j,k)</math> in the grid correspond to: |
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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.