On the Nature of Numerical Plane Waves in FDTD

The nature of uniform plane waves in the standard finite-difference time domain (FDTD) grid is investigated. In particular, it is shown that for the waves to be truly planar in the grid, they can only propagate at a countably infinite set of rational angles defined by integer ratios of grid-cells. The plane wave can then be expressed on a one-dimensional (1D) grid with uniform spacing, where each 1D point is directly associated with field locations in the main three-dimensional (3D) grid. Furthermore, it is shown that angles used in the projection of field components that account for the field nonorthogonality are inherently contained in the dispersion equation, and hence, a consistent set of projection operators is defined from it.