Time domain study of the Drude–Born–Fedorov model for a class of heterogeneous chiral materials

Abstract

Abstract. We deal with the well-posedness of the transient Maxwell equations in a particular class of heterogeneous chiral material modeled by the Drude-Born-Fedorov constitutive relations. A new formulation of the underlying evolution problem allows us to correct a previous result establishing the existence and uniqueness of the electromagnetic fields in a homogeneous medium.1. Introduction. Chiral materials are examples of media which respond with both electric and magnetic polarization to either electric or magnetic excitations. Thanks to this peculiar behavior, they have been studied extensively by the electromagnetics community, for a wide range of applications, including prospective ones.In this work, we consider the time-dependent Maxwell equations in a heterogeneous, isotropic chiral medium filling a bounded domain surrounded by a perfect conductor. Although their use is mainly restricted to time-harmonic applications, the Drude-Born-Fedorov constitutive relations (see for instance [10]) are used to model the behavior of the chiral material, within the scope of what is referred to in the literature as the optical response approximation [9]. Recent mathematical investigations dealing with electromagnetic waves in chiral media and involving the Drude-BornFedorov relations are the main topics of a number of articles, e.g. [1,3] and references therein for the time-harmonic case, [8,11] and references therein for the time domain case. Here, the well-posedness result erroneously stated in [8] is reexamined (and corrected) through a different mathematical interpretation of the evolution system in the case where the electric permittivity and magnetic permeability of the medium are both possibly non-constant and proportional to the chiral admittance. Our main result asserts the existence of a unique solution to the problem under a spectral condition involving the admittance. Its proof is based on the invertibility property of the so-called Drude-Born-Fedorov operator, some characterizations of the orthogonal of its range and the introduction of an appropriate invariant subspace of its inverse.More precisely, we consider the following problem. Let Ω be a bounded subset of R 3 with a Lipschitz boundary Γ, no assumption being made on the simple connectedness of Ω, nor on the connectedness of Γ. We consider the time-dependent Maxwell equations, supplemented with boundary and initial conditions to close the system (the reader is referred to [8] for a justification of this model),

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.…”

A non‐autonomous model for the evolution of electromagnetic fields in complex media

“…Maxwell's equations supplemented with the previous initial‐boundary conditions, under the non‐autonomous constitutive relations , lead to the following initial‐boundary value problem for E and H : As we can see from the set of equations and the assumptions on the source term, a suitable functional space in which a corresponding Cauchy problem that deals with both E and H can be posed, is the closed space H = W × W , where W = H (div0; Ω) ∩ H 0 (div; Ω) . In fact H , as a closed subspace of L 2 (Ω) 3 × L 2 (Ω) 3 , is also a Hilbert space.…”

“…As we can see from the set of equations (3) and the assumptions on the source term, a suitable functional space in which a corresponding Cauchy problem that deals with both E and H can be posed, is the closed space H D W W, where W D H.div0; / \ H 0 .div; / [9,11,12]. In fact H, as a closed subspace of L 2 .…”

“…This approximation has been extensively used in the modeling of chiral media, especially in the time harmonic case. However, as we can see in [1,2,5,[7][8][9][10][11][12][13], DBF-like approximations have been used for the study of electromagnetic fields in chiral media in the time domain also. DBF-like relations, in the time domain, can be formally justified when the˛, e , m are localized functions of their arguments and attention is confined in the optical response region, assuming instantaneous responses in the material (see [1,7,8] and for a more detailed analysis [5]).…”

Direct and inverse electromagnetic scattering problems for bi-anisotropic media

“…The topic of electromagnetic propagation and scattering in complex media has recently received an increasing amount of interest from mathematicians. Theoretical analysis for the direct problem in both the time domain and frequency domain can be found in [1,3,4,10,11,31,32,36,37]. We particularly refer to the book [37] for an account of the most recent mathematical results on complex media electromagnetics.…”