Threshold for loss of Landau damping in double-harmonic rf systems

TL;DR

This paper investigates the conditions under which Landau damping is lost in double-harmonic RF systems, providing analytical thresholds, experimental validation, and insights into the effects of impedance and phase configurations.

Contribution

It extends Landau damping loss analysis to double-harmonic RF systems, deriving analytical thresholds and validating them with experiments and simulations.

Findings

Loss of Landau damping occurs at a vanishing threshold with inductive impedance.

Analytical equations for LLD threshold incorporate impedance cutoff frequency.

Experimental results agree with theoretical predictions and simulations.

Abstract

Landau damping is a natural stabilization mechanism that mitigates coherent beam instabilities. In the longitudinal plane, loss of Landau damping (LLD) occurs when a coherent mode of oscillation emerges from the incoherent band of the bunch synchrotron frequencies. This work extends the recent LLD studies to the relevant case of double-harmonic rf systems. Specifically, it is shown that in the bunch shortening mode (both rf systems in phase at the bunch position for a non-accelerating bucket), inductive impedance above transition energy results in a vanishing LLD threshold for a binominal particle distribution, similar to the single-harmonic rf case. In this configuration, refined analytical estimates of the synchrotron frequency distribution enabled the derivation of an analytical equation for the LLD threshold by introducing an upper cutoff frequency to the impedance. The LLD threshold is studied through the concept of van Kampen modes and takes into account the effect of the voltage ratio, as well as the relative phase between the two rf systems for an inductive impedance above transition energy (or capacitive below). The validity of the theoretical studies is supported by extensive beam measurements conducted under different bucket-filling conditions in two synchrotrons, the PS and the SPS at CERN. Beyond the analytical estimates, the observations are moreover compared with the semi-analytical code MELODY and macroparticle tracking simulations in BLonD.