Analysis of radiation focusing and steering in the free-electron laser by use of a source-dependent expansion technique

In the free-electron laser the resonant interaction between the radiation field and electron beam can result in radiation focusing (optical guiding). If the centroid of the electron beam is transversely displaced off axis, the radiation field, under certain conditions, will follow and be steered by the electron beam. The effect of a spatial modulation on the electron-beam envelope can also modify the propagation characteristics of the radiation. These and other phenomena are analytically and numerically studied using a novel source-dependent Laguerre-Gaussian modal representation of the fully three-dimensional radiation field. Unlike the vacuum Laguerre-Gaussian modal expansion, the longitudinal spatial dependence of the radiation waist and curvature are determined and characterized by the source term in the wave equation. Among the advantages of this general source-dependent expansion approach is that few modes are needed to accurately describe the radiation. Hence, fast and accurate numerical solutions of the fully three-dimensional free-electron laser problem can be obtained over distances of many Rayleigh lengths. Furthermore, this expansion enables us to drive an envelope equation for the radiation beam as well as an expression for the centroid of the beam.