Two Dimensional Aspects of Regenerative BBU in Two-Pass Recirculating Accelerators

In this paper, I present the formula, describing a threshold of the regenerative multi-pass Beam Breakup (BBU) for a single dipole higher order mode with arbitrary polarization in a two-pass accelerator with a general-form, 4×4 recirculation matrix. Then, I present a mathematical relation between transfer matrices between cavities of the accelerating structure and recirculation matrices for each cavity, which must be satisfied in order to successfully suppress the BBU by reflection or rotation in several cavities.