Transient analysis of free-electron lasers with discrete radiators

In the linear regime before saturation, we solve the one-dimensional, classical free-electron laser equations, maintaining the longitudinal discreteness of the electrons throughout the analysis. We then take the limit in which the beam of discrete electrons is approximated by a continuum fluid. In the continuum limit, we recover the Green function used by Wang and Yu in their treatment of self-amplified spontaneous emission (SASE). For a bunched electron beam, we discuss both incoherent and coherent SASE. We also discuss the field radiated from a bunch whose length is short compared to the radiation wavelength.