Signal Propagation in Electromagnetic Media Modelled by the Two-Sided Fractional Derivative

Abstract

In this paper, wave propagation is considered in a medium described by a fractional-order model, which is formulated with the use of the two-sided fractional derivative of Ortigueira and Machado. Although the relation of the derivative to causality is clearly specified in its definition, there is no obvious relation between causality of the derivative and causality of the transfer function induced by this derivative. Hence, causality of the system is investigated; its output is an electromagnetic signal propagating in media described by the time-domain two-sided fractional derivative. It is demonstrated that, for the derivative order in the range [1,+∞), the transfer function describing attenuated signal propagation is not causal for any value of the asymmetry parameter of the derivative. On the other hand, it is shown that, for derivative orders in the range (0,1), the transfer function is causal if and only if the asymmetry parameter is equal to certain specific values corresponding to the left-sided Grünwald–Letnikov derivative. The results are illustrated by numerical simulations and analyses. Some comments on the Kramers–Krönig relations for logarithm of the transfer function are presented as well.

Analytical Methods for Causality Evaluation of Photonic Materials

“…As it is shown in Lemma 2 in [26], for this transfer function, as well as for ν ∈ (0, 1) and 1 2 (Θ − 1)ν ∈ Z, the following two equalities hold true:…”

“…This attitude is used in the following example: Example 22. In [26], the transfer function induced by the two-sided fractional derivative introduced by Ortigueira and Machado (see [72,73]) is considered. It results in the following transfer function: .…”

“…Their validity and satisfaction of additional assumptions imply causality of the considered function. Then, the formulated theory is applied to the analysis of electromagnetic media characterized by power-law frequency dispersion [25] and fractional-order (FO) models [26]. However, in our opinion, it might still be unclear for community members how to apply the developed mathematical tools to causality evaluation.…”

“…Therefore, we have decided to set in order the theory of causality and prepare the review of various analytical techniques allowing for evaluating causality of photonic materials. It includes the classical methods and models known from electrodynamics handbooks [7], as well as recently published results on causality evaluation [25,26]. Furthermore, we also consider various causality tests, which allow for establishing minimal limits for losses, for metamaterials [27,28].…”

Editorial for Special Issue “Fractional Behavior in Nature 2019”

“…Jacek Gulgowski et al [7] use the two-sided fractional derivative to model an electromagnetic wave propagation in fractional media. This involves causality problems that are investigated and numerically illustrated.…”