Temporal multi-stage energy pumping

As an important theoretical concept, temporal boundaries provide researchers with new insights for tailoring electromagnetic waves in the time domain. Because a temporal boundary breaks the time translation symmetry, a source is necessary to satisfy energy conservation. In this Letter, we quantify the relationship between refractive index contrast and the required energy exchange. More specifically, to realize a temporal boundary with a large refractive index contrast, a correspondingly large and abrupt energy exchange is required. Considering this practical difficulty, we propose to mimic a large-contrast temporal boundary by staggering a series of small-contrast temporal boundaries separated by carefully designed durations. In this way, the process of energy input/output is distributed over an elongated duration, but their effect can still be cumulative. This process is analogous to a multi-resonant system with a periodic energy input. Based on this design principle, we discuss several scenarios for different temporal profiles of refractive index and their corresponding energy requirements.