Numerical Dispersion Relation for the 2-D LOD-FDTD Method in Lossy Media

A closed-form expression is derived for the numerical dispersion relation of the 2-D locally one-dimensional finite-difference time-domain (LOD-FDTD) method in lossy media. In contrast to the lossless formulation, we found that transverse-electric (TEz) and transverse-magnetic (TM
_z
) waves in lossy media exhibit different numerical dispersion relations. Moreover, when the material relaxation-time constant is not well resolved by the integration time-step, the TM
_z
case shows much worse accuracy than the TE
_z
case. To remove this limitation, a split-field LOD-FDTD formulation for TM
_z
waves is then considered, which exhibits the same dispersion relation as the LOD-FDTD method for TE
_z
waves. The validity of the theoretical results is illustrated through numerical simulations.