Rigorous analysis of the influence of the aspect ratio of Yee´s unit cell on the numerical dispersion property of the 2-D and 3-D FDTD methods

In this paper, the influence of the aspect ratio of Yee´s unit cell on the numerical dispersion errors [in terms of the physical phase-velocity error (PVE) and the velocity-anisotropy error (VAE)] of two-dimensional (2-D) finite-difference time-domain (FDTD) and three-dimensional (3-D) FDTD methods is comprehensively investigated. Numerical results reveal that, for a fixed mesh resolution, the physical PVE and the VAE of both the 2-D and 3-D FDTD methods converge to certain limits for higher aspect ratio. Most importantly, it is found for the first time that for the 2-D and 3-D cases the converged dispersion errors (i.e., the limits) are, respectively, about 2.0 and 1.5 times of the corresponding square and cubic unit cells; and the validity of the above theoretical prediction is verified through numerical tests. The investigation carried out in this paper certainly confirms, from the numerical dispersion point of view, that very accurate numerical results can still be obtained even when the aspect ratio of the cells is higher. Consequently, it gives design engineers more freedom and confidence to use the FDTD methods, especially when the aspect ratio of the cells has to be greatly adjusted due to the special requirement of structures under study.