  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.
