Herpin equivalence in temporal metamaterials

In analogy with spatial multilayers, we put forward the idea of Herpin equivalence in temporal metamaterials characterized by step-like time variations of the constitutive parameters. We show that, at a given frequency, an arbitrary temporal multistep exhibiting mirror symmetry can be replaced by an equivalent temporal slab with suitable refractive index and travel-time. This enables the synthesis of arbitrary values of the refractive index, in a way that differs fundamentally from the effective-medium approach, and adds new useful analytical machinery to the available toolbox for the study and design of temporal metamaterials, with potentially intriguing applications to anti-reflection coatings and filters.