Three-dimensional theory of the free-electron laser in the collective regime

A fully self-consistent theory of the free-electron laser is derived in the collective regime which includes all transverse variations in the wiggler field as well as the effects of a finite waveguide geometry. A general orbit theory is derived by perturbation about the steady-state trajectories in a configuration which consists of an axial guide field in addition to the helical wiggler field, and used to obtain the source current and charge density for the Maxwell-Poisson equations. By this means, a set of coupled differential equations is found which describes an arbitrary radial beam profile. A dispersion equation is obtained under the assumption of a thin monoenergetic beam, and solved numerically for the growth rates of the T${\mathrm{E}}_{11}$ and T${\mathrm{M}}_{11}$ modes in a cylindrical waveguide. A selection rule is found by which the ${\mathrm{TE}}_{\mathrm{lm}}$ or ${\mathrm{TM}}_{\mathrm{lm}}$ modes are resonant at the $l\mathrm{th}$ free-electron-laser Doppler upshift.