Radiation by a modulated sheet electron beam propagating parallel to a reflection grating under a finite magnetic field

The authors investigate two-dimensional radiation from a modulated sheet electron beam propagating parallel to a reflection grating under the influence of an externally applied magnetic field. From the wave-theoretic viewpoint, the spontaneous emission can be interpreted as the radiative leakage of the space-charge waves associated with the electron beam. The analysis is reduced to the determination of the complex eigen wavenumber of the space-charge waves perturbed by the presence of the periodic structure. The set of Maxwell equations and relativistic hydrodynamic equations for cold electrons is solved using a perturbation approach based on multiple-space scales. The transport equations governing the radiative leakage of the space-charge waves are systematically derived. Numerical results on the leakage coefficients and the phase change due to the leakage that depend on the applied magnetic field are presented.<>