Longitudinal dynamics with rf phase modulation in the Brazilian electron storage ring

Abstract

In the Brazilian synchrotron light source (LNLS-Laboratório Nacional de Luz Síncrotron), we observed that modulating the phase of the accelerating fields at approximately twice the synchrotron frequency suppressed remarkably well a longitudinal coupled-bunch mode of the beam driven by a higher order mode in one of the radiofrequency (rf) cavities. In this work, we present the results of a set of systematic measurements, in single and multibunch mode, aimed at characterizing the effects of rf phase modulation on the beam. We compare those experiments with the results of tracking simulations and of a theoretical model in which Landau damping is the stabilizing mechanism that explains the suppression of the longitudinal coupled-bunch instability. We also measure the frequency of the stable islands created in longitudinal phase space by phase modulation and the longitudinal beam transfer function as a function of the modulation frequency and amplitude. The experimental results are in good agreement with theoretical expectations.

Detuning Properties of RF Phase Modulation in the Electron Storage Ring KARA

“…As already discussed in the preceding studies [11][12][13][14][15][16], the phase modulation scheme can improve the beam quality by improving the beam lifetime and stabilise the coupled bunch instabilities. In parallel, however, the energy spread can oscillate under the phase modulation.…”

“…The experimental results and the simulations have shown that the longitudinal distribution of the electrons tends to have the quadrupole oscillation mode accompanied by the bunch lengthening. In the Brazilian synchrotron light source, they experimented with suppressing the longitudinal coupled bunch instability by the rf phase modulation [13]. In this previous study, they have discussed the beam dynamics based on the Hamiltonian formalism and existence of 2 or 3 stable fixed points on the longitudinal phase space from the rf phase modulation.…”

Steady-state solutions of split beams in electron storage rings

“…Neglecting the localized nature of the RF cavity, the Hamiltonian for the longitudinal phase space with a single RF cavity and active RF phase modulation reads 8 – 10 with the normalized momentum coordinate and the phase angle being measured with respect to the synchronous phase angle . Here, h is the harmonic number, is the slippage factor, is the synchrotron tune, and is the energy deviation from the nominal energy .…”

“…In a first step, a transformation into action-angle coordinates is performed via the generating function leading to the substitutions , . Since this still does not eliminate the time dependence of the Hamiltonian, a second transformation into a rotating coordinate system is performed via the generating function 8 , 9 , 11 of second kind to achieve the substitutions and . After time averaging to eliminate time-varying terms and a small-angle approximation with regard to the perturbation term, one reaches the final time-independent Hamiltonian 8 , 11 with the scalar constant Note that the absolute value of c ( n ) drops rapidly with increasing n .…”

“…Figure 2 a,c show examples in which the stable fixed points are marked in green while the unstable fixed points are indicated in red. The fixed points for various resonances are well known 8 – 10 , 12 . For for example, they are located at the actions 8 , 11 These expressions indicate a bifurcation condition for the islands to exist, i.e., .…”