On the Drude‐Born‐Fedorov Model for Chiral Electromagnetic Media

Abstract

Electro-magnetic wave propagation in more complex linear materials such as bi-anisotropic media have come to a considerable attention within the last fifteen to twenty years. The Drude-Born-Fedorov model has been extensively studied mostly in the time-harmonic case as a model for chiral media. In the physically relevant time-dependent case the record is much less convincing. In this paper we focus on this case and analyze the Drude-Born-Fedorov model in the light of recently developed Hilbert space approach to evolutionary problems. The solution theory will be developed in the framework of extrapolation spaces (Sobolev lattices). The Drude-Born-Fedorov model starts from a "material relation" of the formwhere ε, µ ∈ R >0 and η ∈ R is the so-called chirality parameter.As it turns out, a general solution theory can be achieved purely in a Hilbert space setting, thus reducing conveniently the conceptual complexity of our considerations, see as a general reference on functional analytical concepts [2,6].Writing · | · 0 for an L 2 -type inner product regardless of the number of components, i.e. E H. To formulate this model in an L 2 (Ω)-setting properly we first need to introducec url as the closure in L 2 (Ω) of curl restricted to C ∞ (Ω) vector fields with compact support (we do not indicate the number of components). Then curl is defined as the adjoint ofc url:Similarly,div is defined as the closure in L 2 (Ω) of div restricted toC ∞ (Ω) vector fields. Thenfor the conceptual details. Containment of a field E in D c url is the proper weak generalization of the classical boundary condition "n × E = 0 on ∂Ω", whereas E ∈ D d iv generalized the classical boundary condition "n · E = 0 on ∂Ω". A suitable boundary condition for the Drude-Born-Fedorov model can now be stated in terms of the range ofc url, R c url , and of the domain ofdiv, D d iv . We require(or equivalently curl E ∈ D d iv and curl E ⊥ H N , where H N denotes the set of harmonic Neumann fields H N := E ∈ D d iv | div E = 0, curl E = 0 ). Condition (1) generalizes the classical boundary condition "n · curl E = 0 on ∂Ω" for simply connected domains to non-smooth boundaries of arbitrary genus and data. We shall denote the operator curl *