Intrabunch motion

Impedance-driven (but not only) coherent beam instabilities are usually studied analytically with the linearized Vlasov equation, ending up with an eigenvalue system to solve. The eigenvalues describe the beam oscillation mode-frequency shifts, leading in particular to intensity thresholds defined by the longitudinal mode coupling instability in the longitudinal plane and by the transverse mode coupling instability in the transverse plane in the absence of chromaticity. This can be directly compared to measurements in particular for the lowest modes and in the absence of tune spread. In the presence of nonlinearities or when higher-order modes are involved, this becomes quite difficult, if not impossible, and the coupling between the modes cannot be directly measured (or simulated) anymore. Another important observable is the intrabunch motion, which can be also accessed analytically thanks to the eigenvectors. To the author’s knowledge, until now, the intrabunch signal has only been explained theoretically for independent longitudinal or transverse beam oscillation modes, i.e., when the bunch intensity is sufficiently low compared to the mode coupling threshold. It was never explained theoretically in detail when two (or more) modes are involved. For instance, no answers were already given to these questions: is (are) there some fixed point(s) when the transverse mode coupling instability starts? If yes, where is it (are they)? And what happens in the presence of mode decoupling? Any number of modes can be treated with the general approach discussed in this paper, which is based on the galactic Vlasov solver (which was previously successfully benchmarked against the pyheadtail macroparticle tracking code as concerns the beam oscillation mode-frequency shifts). However, to be able to clearly see what happens when the bunch intensity is increased, the simple case of two modes is discussed in detail. The purpose of this paper is to describe the different regimes, below, at, above the transverse mode coupling instability and also after the mode decoupling (as it happens sometimes), using a simple analytical model (where two modes are considered together), which helps to really understand what happens at each step. Better characterizing an instability is the first step before trying to find appropriate mitigation measures and push the performance of a particle accelerator. The evolution of the intrabunch motion with intensity is a fundamental observable with high-intensity high-brightness beams.