Multi-Bunch Dynamics in RF Photoinjectors Through an Envelope Equation   Approach
  Free Electron Lasers driven by Super-Conducting Linacs require the generation, acceleration and transport of trains of bunches with high charge, high repetition rate and low emittance. A numerical model already developed for the modeling of beam dynamics in SC Linacs has been now extended to treat RF guns with proper description of the photocathode bunch generation. The model algorithm is based on a coupled integration of Newton and Maxwell equations under a slowly varying envelope approximation for the time evolution of mode amplitudes in the accelerating structure and an envelope equation description for a set of bunch slices. In this way beam loading effects and higher order modes excitation can be studied. The application to the TTF-FEL injector as a particular design is presented and the optimization according to the invariant envelope concept is discussed, in the frame of single bunch calculations compared to the results of other multi-particle codes. 
M. Ferrario, L. Serafini