Matching bunched beams to alternating gradient focusing systems

A numerical procedure for generating phase-space distributions matched to alternating gradient focusing systems has been tested. For a smooth-focusing system a matched distribution can be calculated. With a particle tracing simulation code such a distribution can be followed while adiabatically deforming the focusing forces until an alternating gradient configuration is reached. The distribution remains matched; that is, the final distribution is periodic with the structure period. This method is useful because it can produce distributions matched to nonlinear forces. This is a feature that elliptical distributions, with ellipse parameters obtained from the Courant-Snyder theory, do not have. External nonlinearities, including nonlinear couplings, were included in our examples but space charge was not. This procedure is expected to work with space charge but will require a three-dimensional space charge calculation in the simulation code