| The use of nonlinear focusing in particle accelerators has been proposed in a variety of applications. This work proposes and studies yet another application and analyzes the dynamics associated with nonlinear focusing. To begin with, it is proposed that beam halos can be controlled by combining nonlinear focusing and collimation, which is verified by numerical simulations. The study relies on a one dimensional, continuous focusing Particle-in-Cell (PIC) model and a Particle-Core model. Results from the PIC simulations establish the importance of reducing the mismatch of the beam in order to reduce halo formation. It is then shown that nonlinear focusing leads to damping of the beam oscillations thereby reducing the mismatch. This damping is accompanied by emittance growth causing the beam to spread in phase space. To compensate for this, the beam is collimated, and further evolution of the beam shows that the halo is not generated. The use of the idealized, one-dimensional, continuous focusing model is justified by analyzing nonlinear alternate gradient focusing systems. The Lie Transform perturbation theory is used to derive an equivalent continuous focusing system for the alternate gradient focusing channel by canonically averaging over the lattice or fast oscillating time scale. The analysis shows the existence of a condition in which the system is azimuthally symmetric in the canonically transformed, slowly oscillating frame. Numerical results show that this condition leads to reduced chaos and improved confinement in the charged particle motion. The Lie Transform analysis is then extended to include space charge effects which enables one to calculate a near equilibrium distribution function which is azimuthally symmetric in the nonlinear lattice. |
