Mathematical models for dispersive electromagnetic waves: an overview <s> On the equivalence between the different notions of passivity <s> A passive system is one that cannot produce energy, a property that naturally poses constraints on the system. A system in convolution form is fully described by its transfer function, and the class of Herglotz functions, holomorphic functions mapping the open upper half-plane to the closed upper half-plane, is closely related to the transfer functions of passive systems. Following a well-known representation theorem, Herglotz functions can be represented by means of positive measures on the real line. This fact is exploited in this paper in order to rigorously prove a set of integral identities for Herglotz functions that relate weighted integrals of the function to its asymptotic expansions at the origin and infinity. The integral identities are the core of a general approach introduced here to derive sum rules and physical limitations on various passive physical systems. Although similar approaches have previously been applied to a wide range of specific applications, this paper is the first to deliver a general procedure together with the necessary proofs. This procedure is described thoroughly and exemplified with examples from electromagnetic theory. <s> BIB001 </s> Mathematical models for dispersive electromagnetic waves: an overview <s> On the equivalence between the different notions of passivity <s> We prove that well-known speed of light restrictions on electromagnetic energy velocity can be extended to a new level of generality, encompassing even nonlocal chiral media in periodic geometries, while at the same time weakening the underlying assumptions to only passivity and linearity of the medium (either with a transparency window or with dissipation). As was also shown by other authors under more limiting assumptions, passivity alone is sufficient to guarantee causality and positivity of the energy density (with no thermodynamic assumptions). Our proof is general enough to include a very broad range of material properties, including anisotropy, bianisotropy (chirality), nonlocality, dispersion, periodicity, and even delta functions or similar generalized functions. We also show that the "dynamical energy density" used by some previous authors in dissipative media reduces to the standard Brillouin formula for dispersive energy density in a transparency window. The results in this paper are proved by exploiting deep results from linear-response theory, harmonic analysis, and functional analysis that had previously not been brought together in the context of electrodynamics. <s> BIB002
It is natural to wonder whether the reciprocal of theorem 2.7, namely "any physically passive material is passive in the sense of definition (2.4)", is true. Such a property seems to be commonly or implicitly admitted in the literature. However, it is far from obvious, as this is mentioned in  for instance. Note that, as a consequence of (24) , the definition of physical passivity is equivalent to assuming that for vector fields E, H, P and M related by (22) and also by Maxwell's equations (20) . Let us introduce a third notion of passivity, clearly stronger than physical passivity:  The strong physical passivity property is the one that is most often used in the literature (see  , BIB002 ). The proof of theorem 2.7 shows in fact that passivity implies strong physical passivity. The converse is also true under additional assumptions. To demonstrate this, we will need a density lemma. Theorem 2.10. Assume that for each x ∈ R 3 , Im ω ε(x, ω) and Im ω µ(x, ω) are bounded functions of ω. Let χ e (x, ω) and χ m (x, ω) be Fourier-Laplace functions of L 1 causal functions t → χ e (x, t) and t → χ m (x, t). Assume furthermore that Then, the strong passivity assumption (31) implies passivity. Proof. Notice that in (31) , (E, P) and (H, M) are not connected by Maxwell's equations. Hence, it suffices to show that (31) for (E, P) implies Im ω ε(x, ω) ≥ 0 in C + (the proof is identical for µ instead of ε). We start from identity (27) . Since t → χ e (x, t) belongs to L 1 (R + ), the function ω → χ e (x, ω) extends continuously to the real axis. Thus, this also holds for the function ω → ε(x, ω), which is thus continuous and bounded (thanks to (32) ) along the real axis. Thanks to these properties, using Lebesgue's dominated convergence theorem, we can pass to the limit in (27) when η tends to 0 to obtain This being true for any T and any E ∈ L 2 (R), using the density lemma 2.9, we get from which we immediately infer that To extend this positivity result to the half-space C + , let us set, for any R > 0, Ω R = {ω ∈ C + / |ω| < R}, that we identify to an open set of R 2 . Let By analyticity of ε(x, ω) in C + , u x is harmonic in R 2 + so that, in Ω R , the minimum of u(x, y) is attained on iθ , θ ∈ (0, π) . Since u x is non-negative on the real axis, we get On the other hand, due to (32) , u x L ∞ (Γ R ) → 0 when R → +∞. Thus, for any δ > 0 (arbitrarily small), there exists R δ > 0, with R δ → +∞ when δ → 0 such that sup (x,y) ∈ Γ R δ |u x (x, y)| < δ, thus min (x,y) ∈ Ω R δ u x (x, y) ≥ − δ, and one easily concludes by making δ tend to 0 that u x (x, y) ≥ 0 Remark 2.11. The result of theorem 2.10 is likely to be valid under much weaker assumptions (removing in particular the L 1 assumption for χ e or χ m ), as stated in the book  and used e.g. in BIB001 , BIB002 . 3 Local dispersive materials