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Reply to Comment on "Nonlinear quantum effects in electromagnetic radiation of a vortex electron" by A. Karnieli, R. Remez, I. Kaminer, et al

We argue that while the experiment of Remez et al. is interesting and its conclusions may well be correct, the observed lack of dependence of the measured distributions on the electron's transverse coherence length should have been expected for the parameters chosen. This is because for Smith-Purcell radiation it is the coherence length of a virtual photon 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. This is a common feature for all the radiation processes in which a photon is emitted not directly by the electron, 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 secondary. We emphasize that the measured distributions are unusually wide and neither the original paper nor the recent comment fully discusses and rules out all alternative hypotheses that could have led to the same distributions. On the contrary, there exists a family of classical effects that could also have resulted in the measured distributions and that were neither discussed nor even mentioned by the authors. Such alternative hypotheses include (i) effects of the beam sizes, of its angular divergence, of the temporal coherence of the process, and (ii) influence of the grating shape and of its material. Finally, we propose to repeat the experiment and to measure diffraction radiation from a thin metallic semi-plane or Compton emission in a laser pulse. In these cases, the classical effects play a much smaller role and the results of such measurements would have higher credibility.