Microscopic theory of refractive index applied to metamaterials: Effective current response tensor corresponding to standard relation $n^2 = \varepsilon_{\mathrm{eff}} \mu_{\mathrm{eff}}$

摘要

In this article, we first derive the wavevector- and frequency-dependent,microscopic current response tensor which corresponds to the "macroscopic"ansatz $\vec D = \varepsilon_0 \varepsilon_{\mathrm{eff}} \vec E$ and $\vec B =\mu_0 \mu_{\mathrm{eff}} \vec H$ with wavevector- and frequency-independent,"effective" material constants $\varepsilon_{\mathrm{eff}}$ and$\mu_{\mathrm{eff}}$. We then deduce the electromagnetic and optical propertiesof this effective material model by employing exact, microscopic responserelations. In particular, we argue that for recovering the standard relation$n^2 = \varepsilon_{\mathrm{eff}} \mu_{\mathrm{eff}}$ between the refractiveindex and the effective material constants, it is imperative to start from themicroscopic wave equation in terms of the transverse dielectric function,$\varepsilon_{\mathrm{T}}(\vec k, \omega) = 0$. At the same time, this resultrefutes again the relation $n^2 = \varepsilon_{\mathrm{r}} \mu_{\mathrm{r}}$ interms of the microscopic response functions, and thus confirms the recentlydeveloped microscopic theory of the refractive index [Optik 140, 62 (2017)]. Onthe phenomenological side, our result is especially relevant for metamaterialsresearch, which draws directly on the standard relation for the refractiveindex in terms of effective material constants. Since for a wide class ofmaterials, the current response tensor can be calculated from first principlesand compared to the model expression derived here, this work also paves the wayfor a systematic search for new metamaterials.