Charged particle accelerators are an important tool for many studies. Their development is an essential part of science in general. The history of classical radio-frequency systems goes back about a hundred years and has made tremendous progress. However, the acceleration field with this technique is limited by the electrical breakdown, which leads to the necessity of building large facilities to achieve the required parameters of the accelerated beams. Therefore, the development of alternative particle acceleration schemes is a hot topic of research, and one such scheme is plasma wakefield acceleration [1]. In this technique, a charged bunch or laser pulse excites a Langmuir wave in the plasma, where the amplitude of the accelerating fields can be orders of magnitude larger than in classical radio-frequency cavities. Properly phased charged particles can be effectively accelerated to high energies.

The complex phenomena occurring in plasma wakefield accelerators can only be analyses analytically in simplest approximations, which is why numerical simulations are the main method of theoretical research [2]. The problem has very different temporal and spatial scales, ranging from the laser wavelength (about a micron) to the full acceleration length (up to hundreds of meters), which complicates the simulations.  It is therefore important to develop computationally efficient models, and the quasistatic approximation (QSA) is one such model. It can speed up simulations by several orders of magnitude compared to particle-in-cell (PIC) codes based on first-principle equations. The QSA relies on the fact that the properties of investigated objects change much more slowly in the co-moving coordinates than in the laboratory frame. This approach is suitable for many scenarios, but faces the problem of accurately describing cases where waves with non-zero group velocity are present, or plasma particles are accelerated to nearly the speed of light in highly nonlinear waves, or plasma properties change rapidly.

We present a further development of the QSA model for a more accurate study of plasma wakefield acceleration. Advanced quasistatic approximation [3] retains the main advantage over traditional PIC codes - the speed of simulation. Since the information exchange between adjacent plasma layers manifests itself as a small parameter in the equations of plasma motion in quasistatic coordinates, it is omitted in the classical QSA. The new model takes this interaction into account, with the equations for fields and particles changing in a self-consistent way. This allows for a more accurate study of particle acceleration in strongly nonlinear waves and propagation of free radiation along the acceleration path. The advanced QSA can be incorporated into the quasistatic code with the same efficiency as the classical one. When simulating problems inaccessible to the classical QSA, the results of the new model are consistent with analytical predictions and simulation results of a traditional PIC code.

[1] F. Albert, M.E. Couprie, A. Debus, et al., New J. Phys. 23, 031101 (2021).

[2] J.-L. Vay and R. Lehe, Rev. Accel. Sci. Technol. 9, 165 (2016).

[3] P.V. Tuev, R.I. Spitsyn, and K.V. Lotov, Plasma Physics Reports 49, 229 (2023).