Spatial dispersion and negative refraction of light

Negative refraction occurs at interfaces as a natural consequence of the negative group velocity of waves in one of the interfacing media. The historical origin of this understanding of the phenomenon is briefly discussed. We consider several physical systems that may exhibit normal electromagnetic waves (polaritons) with negative group velocity at optical frequencies. These systems are analyzed in a unified way provided by the spatial dispersion framework. The framework utilizes the notion of the generalized dielectric tensor $\varepsilon_{ij}(\omega,\mathbf k)$ representing the electromagnetic response of the medium to perturbations of frequency $\omega$ and wave vector $\mathbf k$. Polaritons with negative group velocity can exist in media (whether in natural or in artificial meta-materials) with a sufficiently strong spatial dispersion. Our examples include both gyrotropic and nongyrotropic systems, and bulk and surface polariton waves. We also discuss the relation between the spatial dispersion approach and the more familiar, but more restricted, description involving the dielectric permittivity $\varepsilon(\omega)$ and the magnetic permeability $\mu(\omega)$.