Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics

Abstract

Electromagnetic complex media are artificial materials that affect the propagation of electromagnetic waves in surprising ways not usually seen in nature. Because of their wide range of important applications, these materials have been intensely studied over the past twenty-five years, mainly from the perspectives of physics and engineering. But a body of rigorous mathematical theory has also gradually developed, and this is the first book to present that theory. Designed for researchers and advanced graduate students in applied mathematics, electrical engineering, and physics, this book introduces the electromagnetics of complex media through a systematic, state-of-the-art account of their mathematical theory. The book combines the study of well posedness, homogenisation, and controllability of Maxwell equations complemented with constitutive relations describing complex media. The book treats deterministic and stochastic problems both in the frequency and time domains. It also covers computational aspects and scattering problems, among other important topics. Detailed appendices make the book self-contained in terms of mathematical prerequisites, and accessible to engineers and physicists as well as mathematicians.

The factorization method for the Drude-Born-Fedorov model for periodic chiral structures

“…In the applied math community, the forward problem for chiral media has excited interest during the past years, see for example [1,2,5,12,13,26,[34][35][36]. In particular, we refer to [37] for recent mathematical developments on chiral media. However, to the knowledge of the author, the number of works related to the inverse problem for chiral media is relatively small.…”

Linear stochastic degenerate Sobolev equations and applications†

“…The propagation of electromagnetic waves in complex media is the subject of many studies, and numerous references are available in the literature (see [30] and references therein). The mathematical description of these media is done through the consideration of the constitutive relations for the Maxwell equations in a region Ω ⊂ R 3 , t > 0:…”

The definition and measurement of electromagnetic chirality

“…Such laws are given by the Drude‐Born‐Fedorov model. () This model has also been used in previous studies() when studying the formal mathematical solution of a scattering problem for a homogeneous chiral medium by boundary integral equations and in Heumann in an analysis of the application of the factorization method to a chiral scattering problem. The material inside a bounded open domain is characterized by an electric permittivity ε D >0, a magnetic permeability μ D >0, and a chirality , where we have set ε r = ε D / ε 0 and μ r = μ D / μ 0 .…”