Linear theory of superradiance in a free-electron laser

This study is motivated by the analytical solutions of superradiance in a high-gain free-electron laser obtained by Bonifacio, Maroli, and Piovella, using the technique of Laplace transforms [Opt. Commun. 68, 369 (1988)]. An error in these analytical solutions is remedied by a correct treatment of the boundary conditions on the electron beam, and the earlier theory is extended to allow for both electron shot noise and an optical pulse in the initial state. It is shown that, when the optical pulse is shorter than the electron pulse, superradiant behavior can also occur at the leading edge of the optical pulse.