Electrodynamics of metamaterials and the Landau–Lifshitz approach to the magnetic permeability

We discuss a relationship between the traditional framework of the frequency-dependent dielectric permittivity $\varepsilon \left(\omega \right)$ and magnetic permeability $\mu \left(\omega \right)$ in the electrodynamics of continuous media and the spatial dispersion framework utilizing the dielectric tensor ${\varepsilon }_{ij}\left(\omega \text{,}\text{k}\right)$ depending both on the frequency $\omega$ and wavevector $\text{k}$. For electromagnetic waves, the latter approach includes the former as a specific limiting case for small k within the $k^2$ accuracy. While the dispersion of the transverse electromagnetic waves in this approximation is captured by the $\varepsilon \left(\omega \right)–\mu \left(\omega \right)$ phenomenology, the dispersion of the longitudinal electric waves would be missed. The general ${\varepsilon }_{ij}\left(\omega \text{,}\text{k}\right)$ framework also accommodates more complex situations such as excitonic resonances and additional electromagnetic waves. We also review the well-known Landau–Lifshitz arguments on the physical meaning of $\mu \left(\omega \right)$ at sufficiently high frequencies. In that context, the need is discussed for the effective medium response to include contributions from the spatial dispersion of the electric-dipole polarization and from the electric-quadrupole polarization on an equal footing with contributions from the magnetic-dipole resonances.