Accurate iterative analytic solution of the Kapchinskij-Vladimirskij equations for the case of a matched beam

The well-known Kapchinskij-Vladimirskij (KV) equations are difficult to solve in general, but the problem is simplified for the matched-beam case with sufficient symmetry. We show that the interdependence of the two KV equations is eliminated, so that only one needs to be solved—a great simplification. We present an iterative method of solution which can potentially yield any desired level of accuracy. The lowest level, the well-known smooth approximation, yields simple, explicit results with good accuracy for weak or moderate focusing fields. The next level improves the accuracy for high fields; we previously showed [Part. Accel. 52, 133 (1996)] how to maintain a simple explicit format for the results. That paper used expansion in a small parameter to obtain results of second-level accuracy. The present paper, using straightforward iteration, obtains equations of first, second, and third levels of accuracy. For a periodic lattice with beam matched to lattice, we use the lattice and beam parameters as input and solve for phase advances and envelope functions. We find excellent agreement with numerical solutions over a wide range of beam emittances and intensities.