Quantitative Evaluation of Trapping and Overall Efficiency for Simple Models in One-Degree of Freedom

A key ingredient for the Multi-Turn Extraction (MTE) at the CERN Proton Synchrotron is the beam trapping in stable islands of transverse phase space. The control of the trapping process is essential for the quality of the final beam in terms of intensity sharing and emittance. In this paper, a method allowing an analytical estimation of the fraction of beam trapped into resonance islands as a function of the Hamiltonian parameters is presented for a very simple model of the dynamics (pendulum) and is extended to the case of the interpolating Hamiltonian of the H´enon model, the latter being a good 2D model of theMTE dynamics. The analytical results are compared with numerical simulations. Additional numerical simulations are presented for the minimum trapping amplitude and a fitted model is proposed. Results are discussed in detail.