Transfer ABCD Matrix for Time-Varying Media and Time Crystals

This paper introduces a formal description of the transfer ABCD parameters in time-varying electromagnetic systems. The formal description comes after the rearrangement of the electric displacement field <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> and magnetic flux density field <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</i> 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 (ω vs <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i>) of time crystals.