Abstract
•A quantum mechanical treatment is developed to describe the low-gain slow-wave free-electron laser.•The quantum nature of electrons has a significant effect on the gain.•The gain follows two different behaviors basing on the ratio between the spreading width of electron and laser wavelength.•There is an optimum wave packet width at which the maximum gain is reached.•The results of this paper can be used in resolving the particle-wave duality.
A quantum mechanical treatment is presented to describe slow–wave free–electron lasers (FELs) operating in the low–gain regime. Here, we address the quantum effect of electrons on the FEL operation where each electron is represented as a Gaussian wave packet with an arbitrary width. It is shown that this quantum effect should be described in two different regimes. The first regime is realized when the initial width of the electron wave packet σz0 is comparable to (larger or smaller than) the radiation wavelength λL. In this regime, the gain degrades significantly as σz0 increases and approaches λL. The latter regime is realized when σz0 is quite smaller than λL (i.e., approximately when σz0<λL/3) where the behavior of the gain is opposite to that in the first regime since it decreases with decreasing σz0. Therefore, it is predicted that there is an optimum initial width of the electron wave packet at which the gain is maximum.