Synchronization of Synchrotron Radiation Bursts during a spatio-temporal Instability in accelerator-Based source

TL;DR

This work investigates synchronization of bursts from a spatio-temporal microbunching instability in an accelerator-based light source. By applying a sinusoidal modulation of the RF cavity amplitude, the authors demonstrate phase locking of terahertz CSR bursts to the external drive, revealing Arnold tongues at the fundamental, harmonic, and subharmonic of the natural bursting frequency $f_b$ and observing phase-slip near the synchronization threshold. The approach is supported by Vlasov–Fokker–Planck simulations and a proof-of-principle SOLEIL experiment, showing that a small modulation $A_{RF}$ can substantially influence bursting dynamics while leaving the micro-structure evolution largely intact. The results suggest a path toward controlled CSR emission and potential synchronization of bursts across multiple bunches, with implications for tailoring THz radiation in storage-ring facilities.

Abstract

Synchronization is a fundamental phenomenon in dynamical systems, occurring in a wide range of contexts such as mechanical, chemical, biological, and social systems. In this work, we explore a novel manifestation of synchronization in accelerator-based light sources, specifically in storage rings where relativistic electron bunches circulate and emit synchrotron radiation, used for user experiments. In such systems, a systematic spatio-temporal instability arises when the bunch contains a large number of electrons. This instability is characterized by the spontaneous formation of microstructures within the bunch, which appear with a bursting behavior. We demonstrate that these bursting events can be synchronized with an external sinusoidal signal by modulating the electric field in a radiofrequency (RF) cavity. This external modulation induces typical synchronization features such as Arnold tongues at fundamental, harmonic, and subharmonic frequencies of the natural bursting rate, as well as phase-slip phenomena near the synchronization threshold. The synchronization mechanism is analyzed using numerical simulations based on the Vlasov-Fokker-Planck equation, and a proof-of-principle experiment is conducted at the SOLEIL synchrotron facility.

Figures (4)

• Figure 1: Schematic representation of the system and of the synchronization phenomena (based on numerical simulations of the Vlasov-Fokker-Planck equation, see Appendix for details). (a) An electron bunch circulates in a storage ring at relativistic velocity and emits synchrotron radiation. (b) A portion of this radiation interacts with other electrons in the bunch (in particular in bending magnets), leading to the microbunching instability when the number of electrons exceeds a threshold. This spatio-temporal instability is characterized by the spontaneous appearance of structures in the longitudinal distribution of the bunch [see electron bunch phase-space in (g) and (j)]. (f) These structures appear by bursts, resulting in intense bursts of coherent synchrotron radiation (CSR). Note that the burst period $T_b$ is much larger that the revolution period of the bunch in the ring $T_r$ (for example at SOLEIL, $T_b\simeq~2$ ms and $T_s\simeq~1~\mu$s). d) To synchronize theses bursts, an external modulation signal $\Delta V_{RF}$ is applied to a RF cavity, to modulate the amplitude of the RF signal. It permits to modify the longitudinal dimensions of the bunch (which are directly related to the instability gain). i) example of the emitted CSR when the modulation shown in h) is applied. j) associated distribution of the electron bunch in the longitudinal phase-space.
• Figure 2: Illustration of the synchronization phenomena from numerical simulations (from the integration of the Vlasov-Fokker-Planck equation). a-c) temporal evolution of the modulation of the RF signal amplitude (a), the emitted THz power (b) and the bunch length (c), when the RF modulation is turn ON at $t=0$, with $T_{RF}\simeq 0.65~$ms and $A_{RF}=0.05$ (the spontaneous burst period is about 0.73 ms). d) THz power (same data than in b) in a map representation, where each horizontal line corresponds to a period of the RF signal modulation (the white dot indicates the time at which the modulation is applied). e-f) Details of two bursts, where a modulation can be observe, which is due to the evolution of the micro-structures in the phase-space Roussel2014.
• Figure 3: Synchronization phenomena from experimental signals. a-d) Time evolution of the THz power (recorded with a bolometer), when (a) there is no modulation ($A_{RF}=0$) and the burst period $T_b$ is about of $2.19$ ms (the period used for constructing the map is the same as in the next column, 2.25 ms) ; (b) the RF modulation period $T_{RF}$ is near the spontaneous burst period $T_b$ ($T_{RF}=2.25~$ms, $A_{RF}=0.03$); (c) $T_{RF}$ is near the half of $T_b$ ($T_{RF}=4.29~$ms, $A_{RF}=0.04$); and (d) $T_{RF}$ is near twice $T_b$ ($T_{RF}=1.07~$ms, $A_{RF}=0.02$). (e-h) associated RF modulation signal $\Delta V_{RF}$. In these figures, vertical grey lines indicates the RF signal period $T_{RF}$. i-l) THz power in the map representation [as in the FIG. 2(d)]. m-p) average value of the THz power (in the map representation) over the number of periods (in the last case, the average is taken every two periods of the RF signal, when there is a burst). Grey lines are the superposition of all the bursts. In all theses plots, a transient of 20 ms (after the application of the RF modulation) is not shown.
• Figure 4: a) Fluctuations of the THz power $\Delta P$ in function of the frequency modulation $f_{RF}$ (in logscale), and in function of the modulation amplitude $\Delta V_{RF}$, in the map reprensentation (with a transient of xx ms removed). With this representation, three Arnold tongues appears (for the sub-harmonic case centered around $f_b/2$, for the fondamental case centered around $f_b$ and for the harmonic case, centered around $2 f_b$, with $f_b$ the spontaneous burst frequency (i.e. without modulation). ). Positions of the four cases of the FIG. 3 are indicated by the white circle. b-e) THz power in the map representation, for four different frequency modulations ($f_{RF}=432.9$ Hz,$444~$Hz, $455~$Hz, $466$ Hz) crossing the border of the fundamental arnold tongue and for a fixed amplitude modulation $\Delta V_{RF}=0.025$. Positions of these four cases are indicated in green circles in the figure a). In these figure, the transient is displayed (the modulation is activated at $t=0$ of the first period). todo: check burst frequency (fb) for deltaV=0.