Free-electron laser growing modes and their bandwidths

The behavior of free-electron laser amplification in the small signal linear regime can be understood by studying the dispersion relation derived from the electron energy distribution. A thorough understanding of the growth modes in this regime is of great value in understanding numerical results obtained using simulations. In this paper we show that for a typical bell-shape energy distribution in the electron beam there is not more than one growing mode. We also derive an analytical expression, which determines the bandwidth of the free-electron laser. We also discuss the limitation on the number of growing modes for the case of beam energy distributions with multiple peaks.