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In the free-electron laser the interaction between the radiation and electrons can result in radiation focusing. The radiation tends to follow the electron beam when the centroid of the electron beam is laterally displaced. Spatial modulation of the electron-beam envelope induces a similar modulation in the radiation-beam envelope. These and other phenomena are studied by use of a novel source-dependent modal representation of the fully three-dimensional radiation field. Among the merits of this approach is that few modes are needed to describe the radiation accurately.
Sprangle et al. (1987) studied this question.