Fractional Derivative Based FDTD Modeling of Transient Wave Propagation in Havriliak-Negami Media

In this paper, an accurate finite-difference time-domain (FDTD) scheme for modeling time-domain wave propagation in arbitrary dispersive biological media is proposed. The main drawback occurring in the conventional FDTD implementation for such materials is the approximation of the fractional derivatives appearing in the relevent time-domain permittivity model. To overcome this problem, we propose a novel FDTD scheme based on the direct solution of the time-domain Maxwell equations by using the Riemann-Liouville operator for fractional differentiation. The feasibility of the proposed method is demonstrated by simulating the transient wave propagation in general bulk and slab dispersive materials with dielectric spectrum described by Cole-Cole, Cole-Davidson, and Havriliak-Negami formulas. In particular, the comparison between the numerical results and those evaluated by using an analytical method based on the Fourier transformation and the matrix formulation for lossy layered media demonstrates the accuracy of the proposed FDTD scheme in a broadband frequency range.