Super-radiant Smith-Purcell radiation from periodic line charges

Smith-Purcell radiation occurs when an electron passes close to the surface of a metallic grating. The radiation becomes coherent when the length of the electron bunch is smaller than the wavelength of the radiation. A train of periodic bunches can enhance the spectral intensity by changing the angular and spectral distribution of the radiation. This is called super-radiant Smith-Purcell radiation, and has been observed in experiments and particle-in-cell simulations. In this paper, we introduce a new method to study this effect by calculating the reflected waves of an incident evanescent wave from periodic line charges. The reflection coefficients are numerically computed, and the spectral distributions of the super-radiant radiation are demonstrated. These analytical results are in agreement with those obtained through part-in-cell simulations.