Refraction laws in temporal media

Abstract: We consider the time dependent Maxwell system in the sense of distributions in the context of temporal interfaces. Just as with spatial interfaces, electromagnetic waves at temporal interfaces scatter and create a transmitted and reflected wave. We provide a rigorous derivation of boundary conditions for the electric and magnetic fields at temporal interfaces with precise assumptions on the material parameters. In turn, we use this to obtain a general Snell's Law at such interfaces. From this, we obtain explicit formulas for the reflection and transmission coefficients. Unlike previous works, we do not make any simplifying ansatz on the solution to the Maxwell system, nor do we assume that the fields are smooth. We also consider material parameters which are not necessarily constant on either side of the temporal interface.