Probing dynamics of time-varying media beyond abrupt temporal interfaces

Time-varying photonic media enable nonreciprocal wave control, ultrafast switching, and frequency conversion, yet most theoretical studies assume idealized abrupt temporal interfaces. Here, we investigate electromagnetic wave propagation in media undergoing finite-time permittivity switching, addressing both plane-wave excitation and free-electron–induced evanescent waves. A switching parameter, τ, is introduced to describe temporal transitions from femtoseconds to nanoseconds. For free-electron excitation, temporal transition radiation is predominantly emitted in the forward direction, and its intensity saturates at Imax when τ≤τ0, with both Imax and τ0 depending on the electron velocity. For plane-wave excitation at frequency ω0, we derive the backward-to-forward flux ratio as a function of ω0 and τ. We further demonstrate that the temporal shape of the switching critically influences the spectrum; while smooth tanh-like transitions produce limited broadening, piecewise linear ramps with sharp edges generate rich, slowly decaying spectra and greatly enhance temporal reflection, even under relatively slow switching. Together, these results expand the accessible parameter space for probing time-varying electromagnetic media and underscore the importance of engineering not only the switching speed but also the temporal waveform for practical experimental implementations, thereby accelerating experimental progress across different platforms.