Beam loading optimized filling patterns for partially filled storage rings

A method of selecting a favorable, with respect of the beam loading effects, multi-bunch (M bunches) filling pattern for a partially filled ring (harmonic number N) is presented. Starting with a standard set of wakefield-coupled equations of motion, the beam loading effects (higher order mode loss) are evaluated analytically (using contour integration technique) for a partially filled ring (M<N). Resulting analytic formula describe the mode loss experienced by a given bunch within a train, as a function of the resonant frequency and the quality factor of the coupling impedance, Q. The analytic formula reveals filling pattern dependent resonant frequency regions, where the beam loading effects are highly suppressed. A possible application of the presented formalism is that for a given configuration of cavity resonances one can design the optimum bunch filling pattern