Numerical modeling of a proof-of-principle experiment on optical stochastic cooling at an electron storage ring

Cooling of beams circulating in storage rings is critical for many applications including particle colliders and synchrotron light sources. A method enabling unprecedented beam-cooling rates, optical stochastic cooling (OSC), was recently demonstrated in the Integrable Optics Test Accelerator (IOTA) electron storage ring at Fermilab [J. Jarvis et al., Nature (London) 608, 287 (2022)]. This paper describes the numerical implementation of the OSC process in the particle-tracking program elegant and discusses the validation of the developed model with available experimental data. The model is also employed to highlight some features associated with different modes of operation of OSC. The developed simulation tool should be valuable in guiding future configurations of optical stochastic cooling and, more broadly, modeling self-field-based beam manipulations.

Figure 1

(a) A TTOSC system with pickup and kicker undulators, a bypass chicane, an optical line, and (b) the position of a particle and its own undulator radiation at the exit of the pickup undulator, and at the entrance and exit of the kicker undulator. The particle slips behind the radiation one wavelength each undulator period and the bypass introduces a delay so that the particle arrives at the kicker at the front of the radiation wavepacket. The delay can be tuned such that the subsequent energy exchange in the kicker produces corrective kicks that cool the beam.

Figure 2

Diagram of the IOTA storage ring at Fermilab (a) with a enlarged view of the OSC line (b) and horizontal and vertical betatron functions around the ring (c). In (a), (b) blue, red, and green shapes correspond, respectively, to dipole, quadrupole, and sextupole magnets. In (c), the shaded area corresponds to the location of the OSC section and the labels $\mathrm{M}i\mathrm{R}$ ($i=1$, 2, 3) indicate the location of the first three out of eight bending magnets [see (a)] where beam synchrotron radiation monitors are available. The beam circulates in the clockwise direction in the ring (i.e., $\mathrm{SQ}1\to \mathrm{M}1\mathrm{R}\to \mathrm{M}2\mathrm{R}\to \mathrm{M}3\mathrm{R}\to \mathrm{OSC}\to \mathrm{SQ}1$).

Figure 3

Total energy transferred to the particle in the KU depending on the path length difference. The shaded region represents the envelope which dictates the maximum energy that can be exchanged due to the shift in arrival time.

Figure 4

Waterfall plot of the temporal evolution of longitudinal (a)–(c), horizontal (d)–(f), and vertical (g)–(i) beam distribution. OSC is turned on instantaneously at $t=0\mathrm{ms}$ (indicated with the vertical dash lines). The “1D,” “2D,” and “3D” columns respectively correspond to lattices configured for longitudinal-only, longitudinal-horizontal, and three-dimensional cooling.

Figure 5

Waterfall plot showing the evolution of longitudinal projections measured (a) and simulated with elegant (b) as the optical delay varies within the range $\delta \ell \in [0,1.25{\lambda }_r]$.

Figure 6

Comparison of the experimental (blue) and simulated (green) equilibrium distributions in longitudinal (a), (b), horizontal (c), (d), and vertical (e), (f) directions for the cooling (${\psi }_0=0$, left column) and heating (${\psi }_0=\pi$, right column) modes. The shaded area in plots (c), (d) represents the regions where the measurements are accurate (see text for details). The development of a tail toward negative values of $x$ is a measurement artifact; see text for details.

Figure 7

Evolution of macroparticles from their initial positions (black dots) in ($a_p\text{−}{a}_x$) amplitude space due to OSC heating in the 2D-cooling configuration (a). The blue dots labeled “1” represent macroparticles associated with the high longitudinal-amplitude attractor, while the green dots labeled “2” correspond to macroparticles in the high horizontal-amplitude attractor. The equilibrium distribution is shown in longitudinal (b) and horizontal phase space (c). The spatiotemporal distribution $(s,x)$ of particles is shown in (d), where the axes correspond to the spatial components of (b) and (c).

Figure 8

Streak camera measurement of the spatiotemporal distribution $(s,x)$ at M3R for the 2D-cooling configuration ($s-x$ coupling) operated in the heating mode (a) and corresponding distribution simulated with elegant (b) along with the simulated measurement obtained by smearing the simulated distribution with a Gaussian point-spread function along the $x$ axis (c).

Figure 9

The total energy kick $\mathrm{\Delta }E\equiv \mathrm{\Delta }\mathcal{E}+\mathrm{\Delta }\tilde{\mathcal{E}}$, including the coherent and incoherent contributions, each particle receives relative to the maximum kick $\mathcal{K}$ (a) and the standard deviation of the incoherent contribution for a single turn normalized to the maximum kick strength (b).

Figure 10

Effect of amplification and incoherent OSC on OSC damping rates for a beam of $N={10}^5$ particles. (a) The longitudinal emittance evolution for optical gains $G\in [20,70]\mathrm{dB}$ and (b) cooling rate of amplified OSC versus the optical gain $\sqrt{G}$.

Figure 11

Effects of angular misalignment on damping (a) and equilibrium longitudinal distribution (b).