Transverse coupled-bunch instabilities in damping rings of high-energy linear colliders.

Abstract

In this paper we present methods for studying and controlling transverse coupled-bunch instabilities. Our primary motivation is the study of damping rings for next-generation linear colliders in which many bunches are damped in the same ring simultaneously; however, the methods presented are also applicable to other situations. The theory developed here treats the motion of the bunch centroids, since the coherent dipole modes of coupled-bunch oscillation are expected to be strongly dominant. A formalism to obtain the $n$ normal modes of oscillation of $n$ bunches is developed. The imaginary part of the frequency of each normal mode determines its stability. However, not only the long-term stability of each oscillation mode, but also the transient behavior of the bunches just after injection, must be considered in damping rings. Two methods of studying the transient behavior are presented: (1) A Laplace-transform method, using the eigenmodes and corresponding eigen-frequencies found by the normal-modes formalism, and (2) computer tracking, using a localized-kick approximation. Examples are given for damping-ring designs appropriate to a linear collider of about 0.5-1.0 TeV center-of-mass energy.