Temporal and spatial evolution of electron bunching in a one-dimensional pulse

A generalized form of bunching parameter for a one-dimensional electron pulse is defined and the formula is simplified into a symmetric form by considering completely elastic one-dimensional collisions. Using a two-parameter Poisson distribution, for electrons arrival times at the undulator beginning and for their initial positions within the pulse, the temporal and spatial evolution of mean value of bunching parameter are evaluated. For the special case, where the distribution parameters, t and x, are set to zero the generalized bunching parameter is reduced to the similar form used in previous free electron laser studies.