Gridless Quasistatic Model for Efficient Simulation of Plasma-based Accelerators

Abstract

The accurate modeling of plasma-based accelerators relies on costly numerical simulations due to the complexity of laser-plasma and beam-plasma interactions. Several strategies can highly reduce the computational cost compared to 3D first-principles particle-in-cell simulations, such as exploiting the near axial symmetry and quasistatic nature of plasma wakefields in many practical cases. Here, we propose a quasistatic algorithm that enables the modeling of axially symmetric plasma wakes without the need of a numerical grid. The gridless approach allows extremely fine features to be resolved without a dramatic increase in computational cost. This is critical, e.g., for the design of future plasma-based colliders with nanometer emittance beams. The proposed model has been implemented in the Wake-T code, where it is coupled to a laser envelope solver and a particle beam pusher to enable the efficient simulation of laser- and beam-driven plasma accelerators.

Figures (5)

• Figure 1: Plasma response to (a) an electron and (b) laser beam as calculated with the gridless quasistatic model. The trajectory of each plasma electron macroparticle is shown in blue. These trajectories can be used to calculate the resulting electromagnetic fields at any location using Eqs. (\ref{['eq:b_theta_plasma']}--\ref{['eq:dpsi_dzeta']}). In this case, the $E_x$ and $E_z$ fields are shown for both the beam- and laser-driven cases in panels (c)-(f).
• Figure 2: Simplified view of the main Wake-T particle tracking loop (left) and the implementation of the gridless model (right). The main loop evolves each of the elements in the simulation from their initial state up to a maximum final time $t_\mathrm{max}$. Since each element can have a different time step, every iteration begins by determining the next element to evolve by comparing the current simulation time $t_\mathrm{sim}$ with the current time $t_e$ and time step $\Delta t_e$ of each element. When updating the plasma fields with the gridless model, the loop shown on the right is executed. The plasma macroparticles are initialized at the right boundary $\zeta=0$ and are iteratively evolved until the left boundary $\zeta_\mathrm{min}$ with a step $\Delta \zeta$. Every iteration over longitudinal slices involves sorting the macroparticles radially, computing the required quantities at the location of the macroparticles, and evolving their radial motion. A grid is used to communicate between the plasma and the laser and macroparticle beams, as explained in the main text. The steps that involve communication with a grid are highlighted with a blue gridded background.
• Figure 3: Use of adaptive grids (AGs) for gathering the plasma fields into the beam macroparticles. The grids dynamically adapt to the size of the macroparticle beams at (a) $z=0$ and (b) $z=1.5cm$. For improved visibility, the grids in (a) and (b) show only a fraction of the grid resolution used in the simulation. The AGs allow for high-resolution field sampling only where needed (i.e., in the region the macroparticle beams), as seen in (c) and (d). The trajectories of the plasma electrons and ions are shown in blue and red, respectively. There, plasma is initialized with more macroparticles close to the axis to capture more accurately the ion motion due to the witness beam.
• Figure 4: Simulation results of the laser-driven study. (a) Side-by-side comparison of the highest-resolution results obtained from FBPIC (top) and Wake-T (bottom) at the same propagation distance in the simulation. The colored lines at the back to the wake indicate the shape of the blowout sheath for different radial resolutions. The inset shows the final longitudinal phase space of the witness as obtained from both simulation codes. The bottom axes show the convergence of the final beam energy spread, measured as the median absolute deviation (MAD), in terms of (b) radial and (c) longitudinal resolution.
• Figure 5: (a) Simulation of a beam-driven plasma stage with collider-relevant parameters performed with Wake-T. The use of adaptive grids (AGs) is advantageous here due to the small transverse size of the witness beam. The insets in (a) show a detailed view of the witness beam and its AG. The fields in the AG feature a much higher resolution than in the rest of the domain, and are able to resolve the ion motion cone. A transverse lineout of $W_x = E_x - B_y$ in the base grid and the AG is shown in the top right inset. (b) Final emittance growth for different effective radial resolution as obtained from Wake-T (with and without AGs) and HiPACE++ (with and without MR), including a case where ion motion is disabled. (c) Comparison of the total runtime of the Wake-T simulations with and without AGs.