Theoretical models for longitudinal coupled-bunch instabilities driven by harmonic cavities in electron storage rings

We present a theoretical framework for analyzing longitudinal coupled-bunch instabilities in double-rf systems with even filling patterns, accounting for potential-well distortion and multiple azimuthal modes. The linearized Vlasov equation is solved in the frequency domain for an arbitrary rf potential to derive the Lebedev equation. We unified different formulations, obtaining results from recent publications as particular cases. Applications to Robinson dipole-quadrupole mode coupling and the periodic transient beam loading (PTBL)/mode-1 instability are presented. Notably, for the first time, theoretical predictions of the mode-1 thresholds show excellent agreement with experimental data. The analysis reveals that the PTBL instability is triggered when coherent focusing is lost for the dipole motion of the coupled-bunch mode-1. We also confirm that this instability is dependent on azimuthal mode interactions and resistant to Landau damping, providing new insights into its mechanism. The methods are implemented in the open-source package , offering a useful semianalytical tool for studying instabilities in electron storage rings with harmonic cavities.