Focusing properties of linear undulators

Abstract

This paper investigates the focusing properties of linear magnetic undulators, i.e., devices characterized by weak defocusing properties in the horizontal (wiggling) plane and strongly focusing in the vertical plane. The problem of identifying the conditions that ensure the existence of the electron beam eigenstates in the undulator lattice for a given working point of electron beam energy E_{b} and resonant wavelength λ_{r} is studied. For any given undulator lattice, a bandlike structure is identified defining regions in the (E_{b},λ_{r}) plane where no periodic matching condition can be found, i.e., it is not possible to transport the electron beam so that optical functions are periodic at lattice boundaries. Some specific cases are discussed for the SPARC FEL undulator.