Macroscopic Maxwell's equations and negative index materials
We study the linear phenomenological Maxwell's equations in the presence of a polarizable and magnetizable medium (magnetodielectric). For a dispersive, nonabsorptive, medium with equal electric and magnetic permeabilities ε(ω) and μ(ω), the latter can assume the value of −1 (+1 is their vacuum value) for a discrete set of frequencies ±mathn, i.e., for these frequencies the medium behaves as a negative index material (NIM). We show that such systems have a well-defined time evolution. In particular, the fields remain square integrable (and the electromagnetic energy finite) if this is the case at some initial time. Next we turn to Green’s function G(x,y,z) (a tensor), associated with the electric Helmholtz operator for a set of parallel layers filled with a material. We express it in terms of the well-known scalar s and p ones. For a half space filled with the material and with a single dispersive Lorentz form for ε(ω) = μ(ω), we obtain an explicit form for G. We find the usual behavior for NIMs for ω = ±math, there is no refection outside the evanescent regime and the transmission (refraction) shows the usual NIM behavior. We find that G has poles in ±math, which lead to a modulation of the radiative decay probability of an excited atom. The formalism is free from ambiguities in the sign of the refractive index.