Strong spatial dispersion in time-modulated dielectric media

We present an effective medium description of time-modulated dielectric media. By taking the averaged fields over one modulation period, the relationship between them is derived, therefore defining the different constitutive parameters. In the most general situation, it is found that the effective material is described by means of a spatially and temporally dispersive transverse dielectric function and a constant longitudinal dielectric function. It has been also found that the frequency dependence in the former is weak, in comparison with its wavenumber dependence (spatial dispersion). Different physical consequences of this spatial dispersion are discussed, with special emphasis on the weak dispersion approximation and the limit in which it is found that the effective material behaves as a resonant and isotropic magnetodielectric medium with no additional longitudinal mode, as it is commonly found in spatially dispersive materials. Time-dependent media therefore opens an alternative way of designing dynamically tunable metamaterials.