Probing dynamics of time-varying media: Beyond abrupt temporal interfaces

This work investigates the effects of time-varying media, where optical properties change over time, on electromagnetic wave propagation, focusing on plane waves and free-electron evanescent waves. We introduce a switching parameter, $\tau$, to model ultrafast switching in the femtosecond to nanosecond range. For plane-wave incidence at angular frequency $\omega_0$, we derive a generalized expression for the backward-to-forward flux ratio as a function of $\omega_0$ and $\tau$, aligning with recent experimental data and providing a unified interpretation framework. For free-electron incidence, we observe intensity saturation in temporal transition radiation at $I_{\textrm{max}}$ for $\tau \leq \tau_{\textrm{0}}$, with both $I_{\textrm{max}}$ and $\tau_{\textrm{0}}$ depending on electron speed. These results highlight the importance of precise $\tau$ control in experiments to probe time-varying media effectively.