Nonlinear δfδf particle simulations of collective effects in high-intensity bunched beams

Collective effects with strong coupling between the longitudinal and transverse dynamics are of fundamental importance for applications of high-intensity bunched beams. The self-consistent Vlasov–Maxwell equations are applied to high-intensity bunched beams, and a generalized δfδf particle simulation algorithm is developed for bunched beams with or without energy anisotropy. Numerically, the distribution function is spit into a reference distribution and a perturbed part. The perturbed distribution function is represented as a weighted summation over discrete particles, where the particle orbits are advanced by the equations of motion in the focusing field and self-generated fields, and the particle weights are advanced by an equation equivalent to the nonlinear Vlasov equation. The nonlinear δfδf method exhibits minimal noise and accuracy problems in comparison with standard particle-in-cell simulations. Systematic studies are carried out for the particle dynamics under conditions corresponding to strong 3D nonlinear space-charge force. The simulations showed that finite bunch-length effects on the collective excitations become insignificant when the aspect ratio (zb/rb)(zb/rb) is larger than 10 for a moderately intense beam with normalized intensity sb=ωpb2/2ωβ2=0.27. For bunched beams with energy anisotropy (T∥/T⊥<1)(T∥/T⊥<1), a reference state has been constructed and a dynamic equilibrium is established in the simulations. Collective excitations relative to the dynamic equilibrium have also been successfully simulated by the generalized δfδf algorithm.