Equilibrium of an Arbitrary Bunch Train in the Presence of Multiple Resonator Wake Fields

TL;DR

This paper extends the analysis of bunch train equilibrium in synchrotron light sources by including multiple resonator wake fields, demonstrating their impact on bunch profiles and lifetimes with a new iterative compensation scheme.

Contribution

It introduces a generalized framework for coupled Ha"issinski equations to incorporate multiple resonator wake fields, enhancing the modeling of beam dynamics in accelerators.

Findings

Multiple resonator wake fields significantly affect bunch profiles.

Compensation of main cavity beam loading influences Touschek lifetime.

Short range wakes modeled by low-Q resonators impact bunch shape.

Abstract

A higher harmonic cavity (HHC), used to cause bunch lengthening for an increase in the Touschek lifetime, is a feature of several fourth generation synchrotron light sources. The desired bunch lengthening is complicated by the presence of required gaps in the bunch train. In a recent paper the author and Venturini studied the effect of various fill patterns by calculating the charge densities in the equilibrium state, through coupled Ha\"issinski equations. We assumed that the only collective force was from the beam loading (wake field) of the harmonic cavity in its lowest mode. The present paper improves the notation and organization of the equations so as to allow an easy inclusion of multiple resonator wake fields. This allows one to study the effects of beam loading of the main accelerating cavity, higher order modes of the cavities, and short range geometric wakes represented by low-$Q$ resonators. As an example these effects are explored for ALS-U. The compensation of the induced voltage in the main cavity, achieved in practice by a feedback system, is modeled by adjustment of the generator voltage through a new iterative scheme. Except in the case of a complete fill, the compensated main cavity beam loading has a substantial effect on the bunch profiles and the Touschek lifetimes. A $Q=6$ resonator, approximating the effect of a realistic short range wake, is also consequential for the bunch forms.