# Maximal Spontaneous Photon Emission and Energy Loss from Free Electrons

## TL;DR

This paper establishes a fundamental limit on free electron spontaneous photon emission, supported by experiments, predicting new radiation regimes and potential for efficient, compact radiation sources across a broad spectrum.

## Contribution

It derives a universal upper bound on free electron radiation and energy loss, revealing new regimes and the role of bound states in enhancing emission.

## Key findings

- Experimental validation with Smith--Purcell radiation measurements.
- Prediction of stronger radiation from slower electrons at subwavelength distances.
- Identification of divergence in emission probability for lossless materials.

## Abstract

Free electron radiation such as Cerenkov, Smith--Purcell, and transition radiation can be greatly affected by structured optical environments, as has been demonstrated in a variety of polaritonic, photonic-crystal, and metamaterial systems. However, the amount of radiation that can ultimately be extracted from free electrons near an arbitrary material structure has remained elusive. Here we derive a fundamental upper limit to the spontaneous photon emission and energy loss of free electrons, regardless of geometry, which illuminates the effects of material properties and electron velocities. We obtain experimental evidence for our theory with quantitative measurements of Smith--Purcell radiation. Our framework allows us to make two predictions. One is a new regime of radiation operation---at subwavelength separations, slower (nonrelativistic) electrons can achieve stronger radiation than fast (relativistic) electrons. The second is a divergence of the emission probability in the limit of lossless materials. We further reveal that such divergences can be approached by coupling free electrons to photonic bound states in the continuum (BICs). Our findings suggest that compact and efficient free-electron radiation sources from microwaves to the soft X-ray regime may be achievable without requiring ultrahigh accelerating voltages.