Kinetic analysis of the sideband instability in a helical wiggler free-electron laser for electrons trapped near the bottom of the ponderomotive potential

A kinetic formalism based on the Vlasov-Maxwell equations is used to investigate properties of the sideband instability for a tenuous, relativistic electron beam propagating through a constant-amplitude helical wiggler magnetic field (wavelength ${\lambda }_0$=2π/$k_0$ and normalized amplitude $a_w$=eB${^}_w$/${\mathrm{mc}}^2$ $k_0$). The analysis is carried out for perturbations about an equilibrium Bernstein-Greene-Kruskal state in which the distribution of beam electrons $G_s$(γ’) and the wiggler magnetic field coexist in quasisteady equilibrium with a finite-amplitude, circularly polarized, primary electromagnetic wave (${\omega }_s$,$k_s$) with normalized amplitude $a_s$=eB${^}_s$/${\mathrm{mc}}^2$ $k_s$ and constant equilibrium wave phase.