Refraction Laws in Temporal Media

The time-dependent Maxwell system in the sense of distributions is considered in the context of temporal interfaces. Just as with spatial interfaces, electromagnetic waves at temporal interfaces scatter and create a transmitted and reflected wave. A rigorous derivation of boundary conditions for the electric and magnetic fields at temporal interfaces is provided with precise assumptions of the material parameters. In turn, this is used to obtain a general Snell’s Law at such interfaces. From this, explicit formulas for the reflection and transmission coefficients are obtained. Unlike previous works, there is no simplifying ansatz on the solution to the Maxwell system made, nor is it assumed that the fields are smooth. Material parameters which are not necessarily constant on either side of the temporal interface are also considered.