Exact Solution of the Envelope Equations for a Matched Quadrupole-Focused Beam in the Zero Space-Charge Limit

摘要

The Kapchinskij-Vladimirskij (KV) equations describe the evolution of the beam envelopes in a periodic system of quadrupole focusing cells and are widely used to help predict the performance of such systems. When the focusing fields are moderately strong, numerical solutions are easily obtained. Problems can arise with very strong fields, especially when the phase advances are larger than 180. Analytic solutions are often useful in the latter case and can provide helpful insight at any focus strength. Thus, there have been numerous analytic papers describing approximate solutions with varying degrees of accuracy. In this paper, we present an exact solution for a matched beam (periodic envelope with the same period as the focusing lattice) for arbitrary phase advance. We treat a problem examined by Courant and Snyder in their classic paper, except that we assume a linear rather than circular machine. They discussed the case in which the focus and defocus sections each had uniform focusing strength with no intervening gaps. They called this the CLS configuration.