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

Abstract

In this paper we present methods for studying and controlling transverse cou--pled 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 Laplace transform method, using the eigenmodes and corresponding eigenfrequencies found by the normal modes formalism, and (2) computer tracking, using a localizedkick approximation. Examples are given for damping ring designs appropriate to a linear collider of about 0.5 to 1.0 TeV center of mass energy.

Time evolution of coupled-bunch modes from beta function variation in storage rings

“…Even when it is known to vary strongly, it has been hoped that the rapid oscillations of the beta function can be averaged out when assessing their effect on the beam [4]. In Ref.…”

“…Reference [4] shows that, for small wake forces, it is sufficient to compute an instantaneous change in momentum, or ''kick,'' from the wake forces once every turn around the damping ring. The kick is computed by taking the product of the wake force at a fixed point on the ring and the time of revolution, T 0 .…”

“…For example, there exist analytic formulas for the growth rates that describe the increase in the amplitudes of the normal modes resulting from wakefield coupling [5]. Another example is analytic formulas to describe the time evolution of each bunch [4]. These results are important for the design of feedback systems needed to suppress any instabilities.…”

“…Feedback systems are commonly installed in modern, high intensity storage rings to maintain bunches on the design trajectory by providing precisely timed kicks [7,8]. In the case of the damping rings of a linear collider, strong wake forces can occur during the injection process because of the large offsets of these bunches from the design trajectory [4,6], and there are concerns that these forces could induce jitter in the damped bunches ready for extraction.…”

Accelerator physics challenges in future linear colliders

“…Because beams are being injected and extracted at the collider repetition rate, one must be sure that all injection transients damp to levels small compared to the extracted beams by the time of extraction. This is complicated because most rings damp multiple bunch trains at once and the long-range wakefields can couple an oscillation from the most recently injected bunch train to drive a damped train to large amplitude even if the ring is 'stable' [23].…”