Fokker-Planck simulations of bunched beams: high-Q rf modes and response functions

A computer code was developed to extend the method of Warnock and Ellison (1999) for integrating the nonlinear Vlasov-Fokker-Planck equation for bunched beams (The 2nd ICFA Advanced Accelerator Workshop on the Physics of High Brightness Beams, 1999). The code handles arbitrary radio-frequency potentials and high-Q impedances and is used to study instabilities in stretched bunches. This paper describes how high-Q resonant rf modes are incorporated into the code. A method by which this time-domain code is used to calculate beam response functions, which are response functions that include the beam acting back on itself through the ring impedance, is also described and applied to stretched bunches