Betatron oscillations, undulator acceptance and beam conditioning (β∥(r) = const.) taking space charge effects into account

The use of high-current beams of low and medium energy electrons in infrared and optical FELs makes it necessary to take into account minute effects to balance transverse velocities due to undulator field transverse inhomogeneities and betatron oscillations, and/or longitudinal velocity spread due to a potential drop inside the electron beam. One must calculate the reduction of betatron focusing in the undulator as a function of beam current (i.e. lengthening of the period of betatron oscillations). This effect leads also to diminishing geometric or radiation acceptance of the undulator. The simple analytical treatment given in the paper allows one to calculate electron current limits of beam transport in undulator fields for some beam geometries (ribbon-like beams in a plane undulator and round cylindrical beams in a helical one) and the dependence of the betatron oscillation period in an undulator on beam current. Beam conditioning to obtain β∥ = const., taking into account modified betatron oscillations and the potential drop in the beam due to space charge, is discussed.