Dispersion formula of a free electron laser with two relativistic electron beams and water-bag distribution function

Theoretical work on a free electron laser (FEL) with two relativistic electron beams is presented. Using the kinetic approach, the dispersion formula (DF) of a helical magnetic wiggler pumped FEL is derived. Stability properties are studied by linearizing the governing equations and DF is obtained and solved numerically. For generality, the momentum spread of electron beams with a water-bag distribution function is considered. Bandwidths of two-beam FEL resonances have been shown and effects of momentum spread have been noticed. It is found that momentum spread causes the pick growth rates to decrease.