# 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 . The three components of **E** are stored on the *edges* of the cube in the corresponding directions, while the components of **H** are stored on the cube *faces*.

More precisely, let a coordinate in the grid correspond to:

where denotes the unit vector in the *k*-th coordinate direction. Then, the ^{th} component of or (or ) is stored for the locations:

The ^{th} component of , on the other hand, is stored for the locations:

In two dimensions, the arrangement is similar except that we set . 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 access field components, e.g. to find the energy density or the flux , 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: .