Reply to “Comment on ‘Nonlinear quantum effects in electromagnetic radiation of a vortex electron' ”

Abstract

We argue that although the experiment of Remez et al. [Phys. Rev. Lett. 123, 060401 (2019)] is interesting, and its conclusions may well be correct, the observed lack of dependence of the measured angular distributions on the electron's transverse coherence length could have been expected for the parameters chosen. This is because for Smith-Purcell radiation it is the coherence length of a virtual photon ${\ensuremath{\sigma}}_{\ensuremath{\perp}}^{(\ensuremath{\gamma})}\ensuremath{\approx}\ensuremath{\beta}\ensuremath{\gamma}\ensuremath{\lambda}\ensuremath{\lesssim}\ensuremath{\lambda}$ that plays a role of the radiation formation width and not the entire electron's coherence length that can well be orders of magnitude larger than the former. This is a common feature for all the radiation processes in which a photon is emitted not directly by the electron packet, which can be delocalized in space, but rather by a much better localized atom or a conduction electron on a surface. Therefore, in our opinion the results of Remez et al. cannot rule out the alternative hypothesis of the delocalized charge. The question, mainly addressed in the Comment by Karnieli et al., of whether the measurements were performed in the wave zone or not is interesting but somewhat secondary. We emphasize that the measured azimuthal distributions are unusually wide and there exists a family of classical effects that could also have resulted in the measured distributions. Such alternative classical hypotheses include: (i) effects of the beam sizes, of its angular divergence, of the temporal coherence of the radiation process, which is also related to how the wave zone is defined, and (ii) influence of the grating shape and of its material---the effects that are known to be of crucial importance for Smith-Purcell radiation from nonrelativistic electrons. Finally, we propose to repeat the experiment and to measure diffraction radiation from a thin metallic semiplane (or a strip) in which case the aforementioned classical effects play a much smaller role.