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**Example text**

1 Inverse problems: retrieval and constitutive parameters Standard media The previous section was concerned with the determination of the electromagnetic ﬁelds scattered by a slab of metamaterial structure in order to obtain the reﬂection and transmission coeﬃcients. This section is devoted to the other aspect of the problem: how to determine the constitutive parameters of an eﬀective medium once the reﬂection and transmission coeﬃcients are known. In a numerical or experimental setup, the conﬁguration often looks like the one depicted in Fig.

14b) For numerical purposes, the sum in Eq. 14a) is truncated at an upper limit Ns which needs to be determined from convergence considerations. These considerations reveal that Eq. 14a) is slowly convergent, making it somewhat ineﬃcient to calculate the Green’s function (Tsang et al. 2000b). An alternative approach is to convert Eq. 14a) into the spectral domain. 15b) with Im{kz l1 l2 } > 0 and Ω = ax ay is the surface of a periodic unit. The numerical evaluation of this expression shows already a much faster convergence rate than Eq.

5) −∞ where φ(r; t, t ) is called the local response function. We assume here that the polarization that sets in a medium depends on the local ﬁelds – an assumption that can be violated at small lengthscales due to correlations in the polarization over a given volume of the material. , all physical quantities depend only on the elapsed time intervals and the above integral becomes a convolution. 6a) dωD(r, ω) e−iωt . 6b) −∞ Introducing these deﬁnitions into Eq. 7) where ε(r, ω) is the frequency-dependent dielectric function given by ε(r, ω) = 1 ε0 ∞ dτ φ(r; τ )eiωτ .