Phase-Space Shaping in Wakefield Accelerators due to Betatron Cooling

Abstract

Plasma-based accelerators are beginning to employ relativistic beams with unprecedented charge and ultrashort durations. These dense driver beams can drive wakes even in high-density plasmas ($\gtrsim10^{19}$ cm$^{-3}$), where betatron radiation becomes increasingly important and begins to affect the dynamics of the accelerated beam. In this Letter, we show that betatron cooling leads to a strong, structuring of the phase space of the beam. This gives rise to bunched, ring-like structures with positive radial position and momentum gradients, \emph{i.e.}, population inversion of the amplitude of oscillation. We derive the characteristic timescales for this process analytically and confirm our predictions with multi-dimensional Particle-in-Cell simulations. The radiation-dominated regime of beam dynamics fundamentally alters the acceleration process and produces self-structured beams capable of triggering coherent betatron emission in ion channels.

Figures (7)

• Figure 1: Phase-space evolution governed by Eq. (\ref{['eq:general_sol']}), shows how initially smooth Gaussian distributions evolve into ring structures at $t = 4t_r$ (here $\gamma=25$ and $\omega_\beta=10^5\,t_r^{-1}$). Column (a) corresponds to an emittance-matched beam with $\sigma_r = 20\, c/\omega_{p}$ and $\sigma_p = 20 \sqrt{\gamma/2}\, m_e c$; column (b) shows a mismatched beam with $\sigma_p = 10 \sqrt{\gamma/2}\, m_e c$. The top row (1) shows the transverse phase-space distribution $f(x, p_x)$, with white lines indicating the projected radial and momentum profiles; dashed lines mark the initial distributions. The second row (2) shows the distribution of oscillation amplitudes $f(A)$, highlighting bunching into a narrow peak (solid line). All initial profiles are shown with dashed lines in each panel.
• Figure 2: FACET-II–like electron beams develop a transversely bunched profile when propagating through high-density plasma ($n_0 = 1.6e19cm^{-3}$): a) under the action of radiation reaction, and b) while no bunching appears without radiation reaction. A dense driver beam ($1nC$, $\sigma_x = 5µm$, $\sigma_z=2µm$) creates a blowout cavity, followed by a trailing beam ($0.1nC$, $\sigma_z=1µm$, $\sigma_x=5µm$) that undergoes wakefield acceleration. Both beams have a normalized emittance $\varepsilon_n = 300mmmrad$ and an initial energy of $10GeV$. The plasma density has a $2mm$ upramp (1), $13mm$ flat-top (2), and $2mm$ downramp (3).
• Figure 3: Synthetic diagnostics of the luminescent screen (100 cm downstream from the plasma) and spectrometer analysis demonstrate that phase-space bunching features are observable after background subtraction (The background is defined as the contribution from driver-only shots, which is subtracted from the combined driver–witness shots to isolate the witness-induced signal.). (a.1) Image with radiation reaction enabled shows visible ring structures. (a.2) Image without radiation reaction shows no ring formation. (b.1) Divergence versus energy with radiation reaction reveals clear correlation with ring features. (b.2) Divergence versus energy without radiation reaction shows no such structure.
• Figure 4: Parameter scan showing the formation and evolution of ring structures in PIC simulations confirm the scalings for $t_r$. Rows correspond to scans over a single parameter: initial beam energy (top), background plasma density (middle), and initial transverse width (bottom). The left column shows the temporal evolution of the ring radius for each case. The right column displays the ring formation time as a function of the scanned parameter. The observed trends confirm the model predictions: $t_r \propto n^{-2}$ (top), $t_r \propto \sigma_\perp^{-2}$ (middle), and $t_r \propto E^{-1}$ (bottom), as indicated by the overlaid reference slopes.
• Figure 5: PIC simulations showing transversethe formation of ring-shaped, phase-bunched profiles in a single FACET-II–like electron beam propagating through high-density plasma. The front of the beam drives the blowout cavity, while the rear undergoes betatron cooling. Row (a) includes radiation reaction; row (b) does not. Columns correspond to different propagation stages: upramp (1), uniform plasma (2), and downramp (3). Ring formation occurs only when radiation reaction is included.