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Here we will derive the general theory of the beam-breakup instability in recirculating linear accelerators, in which the bunches do not have to be at the same rf phase during each recirculation turn. This is important for the description of energy recovery linacs where bunches are recirculated at a decelerating phase of the rf wave and for other recirculator arrangements where different rf phases are of an advantage. Furthermore it can be used for the analysis of phase errors of recirculated bunches. It is shown how the threshold current for a given linac can be computed and a remarkable agreement with tracking data is demonstrated. The general formulas are then analyzed for several analytically solvable cases, which show (a) why different higher order modes (HOMs) in one cavity do not couple so that the most dangerous modes can be considered individually; (b) how different HOM frequencies have to be in order to consider them separately; (c) that no optics can cause the HOMs of two cavities to cancel; (d) how an optics can avoid the addition of the instabilities of two cavities; and (e) how a HOM in a multiple-turn recirculator interferes with itself. Furthermore, a simple method to compute the orbit deviations produced by cavity misalignments has also been introduced. It is shown that the beam-breakup instability always occurs before the orbit excursion becomes very large

Beam-breakup instability theory for energy recovery linacs