Evolution of bunch envelope and emittance in photo-injectors with application to the AWAKE experiment

We investigate on time evolution of envelopes and emittance of bunched beams in photo-injectors. We carry out a detailed beam dynamic study with focus on calculations of space charge forces. We take the average of the forces over the whole ensemble of particles. This results in space charge coefficients that are largely independent of the bunch distribution. By implementing these coefficients in the beam envelope equations we find a set of modified equations as a model for describing the evolution of the bunch envelope. Our equations crucially decrease the consumed time in particle-in-cell codes for designing today’s photo-injectors. The feasibility and accuracy of the model have been benchmarked against the simulation results from existing particle-in-cell codes with very good agreement. We apply our model for the design of the ultra-short bunch photo-injector of the AWAKE experiment. Moreover, by using assumption of linear external fields we develop the governing equations of the bunch emittances in photo-injectors. These equations allow us to partly quantitatively and partly qualitatively study the effects of external and space charge fields on emittance growth phenomena.