FDTD-Based Electromagnetic Modeling of Dielectric Materials with Fractional Dispersive Response

Abstract

The use of fractional derivatives and integrals has been steadily increasing thanks to their ability to capture effects and describe several natural phenomena in a better and systematic manner. Considering that the study of fractional calculus theory opens the mind to new branches of thought, in this paper, we illustrate that such concepts can be successfully implemented in electromagnetic theory, leading to the generalizations of the Maxwell’s equations. We give a brief review of the fractional vector calculus including the generalization of fractional gradient, divergence, curl, and Laplacian operators, as well as the Green, Stokes, Gauss, and Helmholtz theorems. Then, we review the physical and mathematical aspects of dielectric relaxation processes exhibiting non-exponential decay in time, focusing the attention on the time-harmonic relative permittivity function based on a general fractional polynomial series approximation. The different topics pertaining to the incorporation of the power-law dielectric response in the FDTD algorithm are explained, too. In particular, we discuss in detail a home-made fractional calculus-based FDTD scheme, also considering key issues concerning the bounding of the computational domain and the numerical stability. Finally, some examples involving different dispersive dielectrics are presented with the aim to demonstrate the usefulness and reliability of the developed FDTD scheme.

On the Application of Fractional Derivative Operator Theory to the Electromagnetic Modeling of Frequency Dispersive Media

“…Maxwell's equations are a set of coupled partial differential equations (PDEs), which, along with the Lorenz force, governs classical electrodynamics, although these equations only employ integer calculus. On the other hand, heuristic measurement-based models describing the relaxation of electromagnetic waves in frequency-dispersive materials fit very well with fractional calculus [7].…”

“…Most methods try to recursively update the convolution integral by approximating the time series by a truncated sum of decaying exponentials [67], with the coefficients found by some means of optimisation or fitting, like enhanced weighted quantum particle swarm optimisation [22] or the damped least-squares method [68], for example. But, other methods also exist, thereby approximating the complex permittivity function using a general series expansion [69], a binomial series [70], or by using a fractional polynomial series [7].…”

Approximation of Electric Field in Biological Tissue

“…Study [10] refers to an overview of Fractional Calculus (FC) methods in electromagnetic theory, as well as their application to dielectric relaxation modeling. Various numerical techniques related to the dispersive dielectric media are discussed through simulations based on FDTD methods.…”

“…Homogenization consists of replacing the entire complex structure with a simplified model that has effective physical properties. In some studies [10,12,13], different approaches to the homogenization procedure of the multilayer structure are proposed.…”

A Nonstandard Path Integral Model for Curved Surface Analysis

“…To evade these difficulties, an assortment of proficient techniques has been proposed (and summarized in [8]), ranging from curvilinear or nonorthogonal FDTD variations to modified Cartesian and conformal algorithms. Nonetheless, the research on this significant topic is constantly escalating, thus leading to various schemes which offer treatment to many contemporary applications from the microwave and optical regime [19][20][21][22][23][24][25][26][27][28][29][30][31] and paving the way to the trustworthy analysis of many future state-of-the-art scientific fields [32][33][34][35][36][37][38][39][40][41][42][43]. Therefore, the manipulation of arbitrarily-curved surfaces or material interfaces via a staircase model deteriorates the reliability of any FDTD-based approach and, particularly, of the NS-FDTD scheme, whose stencils must be very carefully selected.…”