Temporal photonic crystals: Causality versus periodicity

Photonic crystals are spatially periodic structures that affect the propagation of electromagnetic waves. Depending on their wavelength, electromagnetic waves may or may not propagate through a photonic crystals. The range of wavelength where propagation is not possible is called a band gap and is a direct result of the periodicity of the crystal. On the other hand, a temporal photonic crystal, which is a structure that alters its properties periodically in time, does not exhibit band gaps in its dispersion relation. This asymmetry in the properties of spatial and temporal photonic crystal is due to causality, which prohibits electromagnetic waves from reflecting back in time, while permitting them to reflect in space. In this work, we explore the effect of causality on the fundamental properties of temporal photonic crystals.