Inhomogeneous waves and faster-than-light propagation in the Yee FDTD grid

The numerical dispersion relation governing the propagation of homogeneous plane waves in the Yee finite-difference time-domain (FDTD) grid is well. known. We present the dispersion relation governing inhomogeneous plane waves and show that the homogeneous dispersion relation is a special case of this more general relation. It is found that, unlike in the physical world, constant amplitude planes are not necessarily orthogonal to constant phase planes for inhomogeneous plane waves in lossless materials. However, the inhomogeneous dispersion relation does converge to the exact one in the limit as the discretization goes to zero. Additionally, we show that, for very coarsely resolved fields, homogeneous waves will experience exponential decay as they propagate and they may propagate faster than the speed of light. Bounds are established for the speed of propagation within the grid, as well as the highest frequency and the shortest wavelength that can be coupled into the grid.