Maser gain determined by the spatial profile of work done by an average electron and the wavelength of density fluctuation driven by the maser field in multi-photon free-electron two-quantum Stark radiation

Abstract

We find that the electron in an electron–cyclotron maser (ECM) of Nc = n1/3λ ≫ 1, where n and λ are the electron density and the maser wavelength, respectively, can only lower its energy through masing transition. From this fact and the application of Heisenberg's uncertainty principle on photon emission, we infer that until the electron energy becomes lower to pass through the width of uncertainty in the electron energy, the interval time Tint between two successive radiative transitions is zero. Hence, we find that if the number Nt of radiative transitions during the laser period T under the assumption of Tint = 0 is far larger than the number Nu of radiative transitions required to pass through the half-width ΔE of uncertainty in the electron energy, the radiation power from an electron is equal to ΔE/T. We deduce that the shift in the energy level of an average electron is predominantly produced by the density-deviation mode driven by the laser field so as to be spatially sinusoidal with period λw and amplitude $\mathcal W$. We recognize that the uncertainty in the z position of an electron emitting a laser photon through free-electron two-quantum Stark (FETQS) radiation is the wavelength λe of the electric wiggler. Thus, if λw ≪ λe, then ΔE is equal to $\mathcal W$. Based on the above findings, we identify electron–cyclotron masing in a high-density ECM as a gyration-driven FETQS radiation whose power is given by P = ΔE/T, where ΔE is not caused by gyration but rotation around the waveguide axis. The gain calculated based on this identification agrees with the measured one.