Fig. 1. Plasma density profile used in the study, where the laser pulse propagates along the positive z direction.

Fig. 2. Left: beam current distribution at the plasma exit for n_high = 5 × 10¹⁸ cm⁻³, n_low = 2.5 × 10¹⁸ cm⁻³ and downramp length L = 40 µm. The plasma density and the evolution of the laser vector potential a₀ are shown in the inset. Right: plasma charge density of the wakefield structures at positions where the rear part of the bubble corresponds to the middle of the density downramp.

Fig. 3. (a)–(c) Particle distributions of injected particles in the z_i–r_i plane at their initial positions before being disturbed by the laser for each simulation. (d)–(f) Particle distributions in the space constructed from their initial longitudinal position z_i and the phase position in the wakefield after the injection ξ.

Fig. 4. Beam sliced currents for fixed n_high = 5 × 10¹⁸ cm⁻³ and different downramp lengths L and downramp positions Pos_d.

Fig. 5. Beam charge Q for fixed n_high = 5 × 10¹⁸ cm⁻³ with different downramp lengths L and downramp positions Pos_d.

Fig. 6. (a) Evolution of beam energy with propagation distance for a simulation with n_high = 5 × 10¹⁸ cm⁻³, Pos_d = 300 µm, and L = 40 µm. (b)–(d) Beam profiles in longitudinal phase space for different propagation distances, where the red lines show the on-axis acceleration forces. (e) and (f) Wakefields for two different propagation distances.

Fig. 7. (a) and (b) Evolution of δE and E, respectively, for fixed Pos_d = 300 µm and n_high = 5 × 10¹⁸ cm⁻³ and different values of L. The black dots outline the optimized zones. (c) Plasma density and evolution of the laser a₀. (d) Beam currents for three of the values of L. (e) Relationships between minimum energy spread, energy, and injected beam charge for different values of L (increasing in the direction of the arrow), with the charge values encircled for each point.

Fig. 8. (a) Energy spread as a function of the propagation distance for six values of the downramp gradient. (b) Corresponding beam currents. (c) Relative energy spread and energy as functions of the propagation distance for a set of four values of the downramp gradient.

Fig. 9. Energy spread and energy evolution for Pos_d = 400 µm and four values of the downramp density gradient.

Fig. 10. (a) Energy spectrum. (b) and (c) Longitudinal phase space at two different propagation distances. The initial plasma parameters are n_high = 5 × 10¹⁸ cm⁻³, L = 25 µm, and Pos_d = 500 µm. Significant particle loss is observed after 3 mm of propagation.

Fig. 11. Energy spectrum for the simulation with initial plasma parameters n_high = 1 × 10¹⁹ cm⁻³, L = 25 µm, and Pos_d = 225 µm. The inset shows the beam current for this simulation.

Table 1. Plasma parameters and final beam parameters for the optimum cases that give the smallest δE/E for each group of simulations.