Approximation of Grünwald–Letnikov Fractional Derivative for FDTD Modeling of Cole–Cole Media

A finite-difference time-domain (FDTD) method for modeling wave propagation in dispersive Cole-Cole media is proposed. The main difficulty in time-domain modeling of a Cole-Cole medium is that the polarization relation that describes its electromagnetic behavior is a differential equation of fractional order. By definition, the fractional derivative of a function is a nonlocal operator and its computation at a time instant involves all previous function values. Thus, the memory demands of a typical FDTD scheme for Cole-Cole media would be high. However, by an appropriate approximation of the Grünwald-Letnikov definition of the fractional derivative, we can implement an FDTD scheme with reasonable memory demands. This is achieved by means of sums of decaying exponentials used to approximate the coefficients that appear in the Grünwald-Letnikov definition. As a result, the FDTD scheme requires the additional storage of a limited number of auxiliary vectors only. The proposed scheme has been applied successfully to the simulation of the excitation of Cole-Cole media by a wideband Gaussian electromagnetic pulse.