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 th-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.Index Terms-Cole-Cole model, dispersive media, finite-difference time domain (FDTD), fractional derivatives.

Fractional‐Calculus‐Based Electromagnetic Tool to Study Pulse Propagation in Arbitrary Dispersive Dielectrics

“…Alternative approaches based on the application of fractional derivatives and recursive evaluated integrals are suggested in ref. . The accuracy of the method illustrated in ref.…”

Fractional Calculus-Based Modeling of Electromagnetic Field Propagation in Arbitrary Biological Tissue

“…Mathematical Problems in Engineering 3 where 0 is the permittivity of free space, ∞ is the relative permittivity at high frequency limit, and is the electric susceptibility. FDTD methods have been developed for incorporating Debye, Cole-Cole, Cole-Davidson, and Havriliak-Negami electric susceptibility models [27][28][29][30]. The common approach is to approximate the relative complex permittivity by means of rational or polynomial functions resulting in auxiliary differential equation FDTD models.…”

Fractional Derivative Modification of Drude Model

“…Usually, a direct numerical implementation of fractional derivatives is problematic. For the so-called Cole–Cole model, which is a popular choice for the description of soft tissue, multiple simplifications have been proposed [ 18 , 19 ]. Another similar relation is the Havriliak–Negami medium, which is also applied to soft tissues as well as to liquid dielectrics and polymers [ 20 ].…”