A quasilinear theory of the self‐consistent evolution of the electron‐beam distribution and radiation field within a free‐electron laser is presented. The relativistic electron beam propagates through an ideal helical wigglerB0=−Bw(cos kwzeˆx+sin dwzeˆy), for which the transverse gradients ∂/∂x=0=∂/∂yhave been neglected. The transverse variation of both the transverse‐polarized radiation field and the electron‐beam distribution have also been neglected. Additionally, the electron beam is assumed to be sufficiently tenuous that the longitudinal space‐charge potential &dgr;&fgr; is negligible and the Compton approximations are applicable. The resulting asymptotic expression for the spectral energy density is used to evaluate the efficiency and saturated radiation power in a variety of parameter regimes relevant to experiments.