Analysis of Cyclotron-Type Electric Lenses and the Effects of Posts

The focusing properties of cyclotron-type electric lenses with time-varying potentials were investigated by analytical and numerical methods. First-order focusing formulas were derived which include the effects of "liner" and of focusing grids. The main emphasis of the numerical studies was on the aspects of lens effects which had not been previously investigated: (a) The behavior of the lenses at large transit times or rf frequencies for which the approximations of analytical theory are not valid. (b) The effects of posts (or grids) placed at one or both sides of the gap. The lens parameters, i.e. focal lengths and location of principal planes, were computed numerically for several geometries using electric field data obtained by computer with relaxation method. Computations were carried out for several two-dimensional and three-dimensional gap geometries at three different frequencies of the gap voltage as a function of kinetic energy and rf phase of the particles at the center of the gap. A criterion for the validity of first-order theory was established. The application to cyclotrons is discussed briefly.