A discrete Vlasov approach to coherent collective instabilities in FEL-injectors via perturbation theory and semi-Lagrangian simulation

In free-electron laser injectors the so-called microbunching instability can severely degrade the beam quality of electron bunches and therefore negatively impact the performance of the free-electron laser. Precise understanding of the microbunching process is required to develop mitigation strategies that minimize the negative impact of the instability.In this work, two approaches of investigating the microbunching instability are presented - one analytical and one numerical. Analytically, a perturbation theory for the instability is derived which is based on a Fréchet-Taylor expansion of the phase-space density propagation operators - the so-called Perron-Frobenius operators - with respect to their dependence on the initial condition of the phase-space density. With this perturbation theory, important problems regarding the microbunching instability can be treated, such as the impact of two-color density modulations, multi-stage bunch compression, and the generation of higher-harmonics due to non-linear effects. Generalized microbunching gain functions for the contributions of the first and higher-order perturbation terms are derived from this theory, which depend only on normalized beam- and machine parameters. Numerically, the microbunching instability is simulated using the semi-Lagrangian code SelaV${}_{1D}$, which was developed during the course of this work. SelaV${}_{1D}$ employs tree-based domain decomposition to represent phase-space densities numerically on a grid. With this, it is possible to efficiently simulate the exotic phase-space densities prevalent in free-electron laser injectors, which exhibit strong non-linear correlations in the longitudinal phase-space, using the grid-based semi-Lagrangian method. Both approaches are applied to study the microbunching instability for the FLASH2020+ upgrade project, in particular with respect to the microbunching mitigation potential of a laser-heater and different bunch compression schemes.