This study delves into the exploration of wave propagation in spatially homogeneous systems governed by a Klein-Gordon–type equation with a periodically time-varying cutoff frequency. Through a combination of analytical calculations and numerical simulations, intriguing and distinctive features in the dispersion diagram of these systems are uncovered. Notably, the examined configurations demonstrate some remarkable transitions as the modulation frequency increases. These transitions encompass a transformation from a frequency gap to a wave-number (q) gap around q=0, with the transition point corresponding to a gapless Dirac dispersion with an exceptional point of degeneracy. Subsequently, the q gap undergoes a bifurcation into two symmetric gaps at positive and negative wave numbers. At this second transition point, the dispersion diagram takes the form of an imaginary Dirac dispersion relation and exhibits an isolated exceptional point at the center of the q=0 gap. These findings contribute to a deeper understanding of wave dynamics in periodically modulated media, uncovering tunable phenomena.

Metamaterials are a fascinating class of photonic materials since they allow us to control optical responses (largely) at will. Besides being an intellectual challenge, adding time variations into spatial metamaterials increases the degrees of freedom to tune their effective response, which motivates their exploration. However, to exploit such materials in the future design of functional devices, we may wish to treat them at the effective level to avoid considering all the mesoscopic details. To permit such effective treatment, we describe here an eigenmode-based approach to homogenize spatiotemporal metamaterials composed of a periodic arrangement of scatterers made from a time-varying material. Practically, we consider the periodic arrangement of spheres within one layer. In our two-step homogenization scheme, we first temporally homogenize that metasurface using the eigenmodes of the bulk time-varying material. Following this, we perform spatial homogenization by inverting the Fresnel coefficients of a slab made from a stationary material. These steps effectively describe the optical response of the spatiotemporal metasurface as a homogeneous slab. We validate our results by comparing the optical observables, i.e., reflectivity and transmissivity, of the metasurface with those of the homogenized slab, and we assess the limitations of the homogenization.