Longitudinal dynamics of an electron pulse in SASE FELs

The motion of a one-dimensional electron pulse in a helical undulator has been investigated. In the analysis, electrons are considered monoenergetic, moving along a semi-infinite undulator under the influence of Lienard-Wiechert fields. The equations of motion of such electrons are derived and solved numerically for the special case of a square pulse. The spatial evolution of electrons energy and density along the pulse are calculated. The initial distribution of electrons positions within the pulse is considered to follow Poisson statistics. The energy, the density, and the density correlation functions along the pulse are identified as functions of pulse position in the undulator. It is shown that these functions are modulated by the fundamental undulator radiation wavelength. The obtained results are compared to previous works on longitudinal electron correlation function in SASE FELs. The growth rate of shot noise microbunching reveals a positive correlation between electrons.