Parametric resonances in intense one-dimensional beams propagating through a periodic focusing channel

We study parametric resonances in arbitrary periodic lattices, incorporating the effect of space-charge self-fields. The linearized Vlasov approach is employed to provide an analytic description of resonant beam instability. Neglecting the momentum dispersion and image charges, we obtain an eigenvalue equation that determines the frequencies of collective modes in a one-dimensional waterbag beam. Approximate formulae for the resonance stopbands, growth rates, etc. are given. The theoretical predictions are compared with multi-particle simulation results. The effect of error fields is also analytically explored in the present paper.