一种处理色散介质问题的通用时域有限差分方法

A general method for finite difference time domain modeling of wave propagation in frequency-dispersive media

色散介质的介电系数是频率的函数,使本构关系在时域成为卷积关系.这就给用时域有限差分方法计算色散介质中波的散射和传播带来了困难.现有算法往往要针对不同色散介质模型推导相应的递推公式,算法的通用性较差.本文完善和发展了移位算子-时域有限差分方法,使之成为一种处理色散介质电磁问题的通用方法.首先,证明了常见的三种色散介质模型(德拜模型、洛伦兹模型和德鲁模型)的介电系数均可以写成适于移位算子法计算的有理分式函数形式.然后,用/t代替jω,过渡到时域,再引入时域移位算子zt代替时间微分算子来处理有理分式函数形式的介电系数,给出离散时域本构关系的表示式,进而导出时域有限差分方法当中电位移矢量和电场强度之间的关系.最后,计算了几种色散介质的电磁散射,数值结果表明了本文方法和程序的通用性和正确有效性.

The analysis of electromagnetic scattering and propagation in dispersive media is complicated in time domain, because its dielectric property is frequency-dependent. A disadvantage of the prevailing algorithms is the need to deduce different formulations for each dispersion model. In this paper, the shift operator finite difference time domain （SO-FDTD） method is developed. First, we prove that the complex permittivity of three kinds of general dispersive media models,i, e. Debye model, the Lorentz model and the Drude model, may be described by rational polynomial functions in jω. By introducing a shift operator zl, the constitutive relation between D and E is derived in discretised time domain. The shift operator method is then applied to the general dispersive medium case. The recursive formulation for D and E available for FDTD computation is obtained. Finally, the scatterings by a dispersive sphere and a PEC object covered with dispersive media are computed. The computed results are in good agreement with the literature and the one obtained by Mie＇s series solution. This illustrates the generality and the feasibility of the presented scheme.