Fractional Derivatives Based Scheme for FDTD Modeling of $n$th-Order Cole–Cole Dispersive Media

Abstract

A finite-difference time-domain (FDTD) modeling of the wave propagation in a Cole-Cole dispersive media is presented. Since the empirical Debye and Lorentz models are not accurate for the representation of the frequency dependence of some dispersive media terms, the Cole-Cole dispersion relation was used to model the electromagnetic properties of biological tissues. The main problem in time-domain modeling of the Cole-Cole model is the approximation of the fractional derivatives that appear in the model equation. Researchers face this problem by approximating the Cole-Cole terms (poles) by a sum of Debye terms or by a sum of decaying exponentials or by polynomials. The accuracy of these models depends on the number of terms needed to model each Cole-Cole term, which may consume large amounts of time and memory. In this letter, all the FDTD fields are approximated by a linear function of time that has a closed form for its fractional derivative. The proposed scheme is considered the more general scheme that has the capability to model nth-order Debye and Cole-Cole models. The scheme is a straightforward extension that can deal with other models such as Lorenz, Drude, and the chiral media. Promising results are observed when calculating the reflection coefficient at an air/muscle material interface. The SAR distribution within a Cole-Cole equivalent brain spherical material excited by an infinitesimal dipole is calculated and compared to the normal FDTD at 900 MHz.