Transfer ABCD Matrix for Time-Varying Media and Time Crystals

Abstract: This paper introduces a formal definition of the transfer ABCD parameters in time-varying electromagnetic systems. The formal definition comes after the rearrangement of the fields $D$ and $B$ at the inputs and outputs of the temporal system based on the time-varying boundary conditions. Then, we derive the ABCD parameters of a temporal transmission line, i.e., a temporal slab, and compute the associated scattering parameters (reflection and transmission coefficients). The results presented here open up an alternative way, based on network theory, to analyze multilayer temporal configurations. Moreover, we show that the ABCD parameters can be used to compute the dispersion diagram ($\omega$ vs $k$) of time crystals.