Mathematical models for dispersive electromagnetic waves: an overview <s> A representation of electric permittivity and magnetic permeability in passive media <s> Abstract It is now well known that the homogenization of a periodic array of parallel dielectric fibers with suitably scaled high permittivity can lead to a possibly negative frequency dependent effective permeability. However this result based on a two-dimensional micro resonator problem on the section of the fibers holds merely in the case of polarized magnetic fields, reducing thus its applications to infinite cylindrical obstacles. In this Note we propose a full 3D extension of previous asymptotic analysis based on a new averaging method for the magnetic field. We evidence a vectorial spectral problem on the periodic cell which accounts for micro-resonance effects and leads to a 3D negative effective permeability tensor. This suggests that periodic bulk dielectric inclusions could be an efficient alternative to the very popular metallic split-ring structure proposed by Pendry. To cite this article: G. Bouchitte et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009). <s> BIB001 </s> Mathematical models for dispersive electromagnetic waves: an overview <s> A representation of electric permittivity and magnetic permeability in passive media <s> We analyze the time harmonic Maxwell equations in a complex three-dimensional geometry. The scatterer $\Omega\subset\mathbb{R}^3$ contains a periodic pattern of small wire structures of high conductivity, and the single element has the shape of a split ring. We rigorously derive effective equations for the scatterer and provide formulas for the effective permittivity and permeability. The latter turns out to be frequency dependent and has a negative real part for appropriate parameter values. This magnetic activity is the key feature of a left-handed metamaterial. <s> BIB002
The representation of lossless passive local media as generalized Lorentz media (cf. theorem 3.21) is, as we shall see, representative of general passive (even lossy) media. This is a consequence of a following well-known representation theorem for Herglotz functions, known as the Nevanlinna's representation theorem. In this section, we assume the familiarity of the reader with basics of measure theory on R  . Lemma 4.1. [ Nevanlinna's theorem] A necessary and sufficient condition for f to be a Herglotz function is given by the following representation: where α ∈ R + , β ∈ R and ν is a positive regular Borel measure for which Moreover, α, β and ν are related to f via the following formulas and the measure ν is given by Remark 4.2. The reader will easily check that the integrand in the right hand side of (70) is in O(ξ −2 ) for large ξ so that (71) ensures the existence of the integral. If, in addition,  Proof. We give the proof for ε. It is obviously the same for µ. By passivity, f (ω) = ω ε(ω)/ε 0 is a Herglotz function. Thus, using lemma 4.1 and the high frequency condition (HF), we can write By the reality principle (RP), f (i) is purely imaginary, and thus β e = ε By the reality principle (RP) again, one has ε(ω) = 1 2 ε(ω) + ε(−ω) . Thus, using (76), we compute: The symmetry of ν e follows from (73) since, by (RP), Imf (−ω) = Imf (ω). One recovers generalized Lorentz materials (52) with finite sums of Dirac measures (in symmetric form) : Other similar passive materials are obtained with non-compactly supported discrete measures such as where ω e, and ω m, are two sequences of positive real numbers satisfying The functions ε(ω) and µ(ω) are not rational, but meromorphic functions with poles ± ω e, and ± ω m, , 1 ≤ ≤ +∞, defined by the following series, whose convergence (outside poles) is ensured by (80) : In the particular case where ω e, = ω m, = 2 − 1 2 π, Ω  Such functions appear naturally in the mathematical theory of metamaterials via high contrast homogenization BIB001 , BIB002 ,  (see remark 4.6 for a concrete and relatively simple example). Measures with an absolutely continuous part (related to losses, see section 4.4) will be considered later. Remark 4.6. Example of high contrast homogenization. Let us consider the case of a 2D transverse magnetic medium (the magnetic field is a 2D scalar function) and heterogeneous non-dispersive Maxwell's equations (see (2)). Let us study a family of problems depending on a small parameter δ > 0, given by The scalar magnetic field H δ is solution of the time harmonic model at a given frequency ω: completed for instance with absorbing boundary conditions (omitted here) on ∂D. The function ε δ (x) : + is δ-periodic and piecewise constant, with high contrast. More precisely, where the reference cell is made of two parts Then it can be shown that , where H hom satisfies the homogenized model can be shown to be of the form (78). 