Energy threshold in Smith–Purcell radiation

Smith–Purcell radiation has emerged as a crucial platform for investigating light-matter interactions and developing compact, tunable light sources that span from microwaves to X-rays. In classical theory, it is believed that Cherenkov radiation exhibits an energy threshold for electrons, while Smith–Purcell radiation is considered free of such a threshold. Although quantum theory suggests there is an emission cutoff in Smith–Purcell radiation, the behavior of this radiation near the threshold remains understudied. In this article, we address this gap by examining the behavior of Smith–Purcell radiation near the threshold from quantum perspectives. Specifically, we derive a quantum energy threshold based on energy-momentum conservation, providing a rigorous limit for the onset of Smith–Purcell radiation. Furthermore, we find that around the threshold the incident electron emits a photon and subsequently reverses its direction of motion. Additionally, we establish a classical energy threshold—below which the classical theory breaks down—by applying the Duane-Hunt limit to Smith–Purcell radiation. Accordingly, quantum theory is required when the electron energy falls between the classical and quantum thresholds. Our findings enrich the understanding of Smith–Purcell radiation and provide valuable insights for developing low-energy-driven and heralded quantum light sources.