Beam breakup with transverse focussing

The large transient beam displacement which occurs in a high current linear accelerator consisting of identical sections can be reduced by including transverse focusing. In the work conducted at SLAC, the focusing was considered weak compared to the beam breakup ''defocusing''. More recently, Yokoya has shown that there is some amplification of the transverse oscillations due to beam breakup even if the external focusing is strong. The authors started with the difference equations for cumulative beam breakup and have obtained expressions for the transient behavior of the transverse displacement in both limits. Simulations are then used to connect these limits through the region of intermediate transverse focusing strength