On the analysis of resonators using finite-difference time-domain techniques

Because a resonator with perfect electrically conducting (PEC) walls has no complications with absorbing boundary conditions and, for canonical geometries, the resonant frequencies are trivial to find, resonators are often used for analyzing the performance of finite-difference time-domain (FDTD) methods. However, when testing the performance of boundary implementations in an FDTD scheme, one should compare to the resonant frequencies of a "perfect" discretized resonator (not to the mode frequencies in the continuous world). On the other hand, when testing the dispersion properties of a method, the resonant frequencies for some structures can be obtained directly from the dispersion relation, thus obviating the need for any simulation. Here, we demonstrate how the dispersion relation can be used to obtain all the resonant frequencies of a rectangular resonator modeled with the Yee algorithm. Furthermore, it is shown that modes that are degenerate in the continuous world can split into distinct modes in FDTD resonators, while modes that are separate in the continuous world can combine in FDTD resonators, thus yielding extra or missing modes. Analytic results are verified using numerical simulations.