Fig. 1. Left: fixed-time snapshot of 2D PIC experiment before the pulse hits the target. Right: spatio-temporal electron density distribution zoomed on the plasma surface. The white line is the analytical solution and the red line the numerical solution (1D PICWIG) for a single particle. The parameters are n = 80n_c, a₀ = 5, and a sin²-enveloped laser pulse with a duration of four optical cycles and linear polarization. The inset on the right shows the evolution of the target density in the relativistic case (n = 80n_c, a₀ = 20).

Fig. 2. (a) Field strength and (b) intensity of incident, reflected, and combined fields in the nonrelativistic case for two different densities (10n_c and 40n_c), with a₀ = 0.9. (c) Results of PIC simulations with a laser intensity a₀ = 4 and a sharp step in density of 40n_c. (d) Ultrarelativistic case with a₀ = 20 and 80n_c. The incident and reflected pulse intensities measured at the critical density surface are plotted as solid and dashed green lines, respectively. The combined laser pulse intensity is shown as a solid black line.

Fig. 3. Dependence of the phase shift of the reflected radiation with respect to the incoming radiation (Δφ = φ_r − φ₀) on the plasma density for different values of a₀. Curves are given by Eq. (5). For a ≥ 1, the relativistically corrected plasma density from Eq. (7) is utilized. Smaller and more lightly shaded markers correspond to the cases where the relativistic pulse breaks the too-thin plasma mirror.

Fig. 4. Dependence of the phase shift of the reflected radiation with respect to the incoming radiation (Δφ = φ_r − φ₀) on the value of n₀/L_λ for different values of a₀. Results for a₀ = 0.1 are shown in green and those for a₀ = 1 in black. Curves are given by Eq. (6). For a₀ = 1, the relativistically corrected plasma density from Eq. (7) was used. Circles and crosses are the results of 1D PIC simulations.

Fig. 5. Dependence of the phase shift of the reflected radiation with respect to the incoming radiation (Δφ = φ_r − φ₀) on the value of n₀/L_λ for different values of a₀. Results for a₀ = 10 are shown by the red curve and triangles, those for a₀ = 20 by the brown curve and squares, and those for a₀ = 40 by the blue curve and diamonds. The curves are given by Eq. (6), using the relativistically corrected plasma density from Eq. (7), and the markers with the corresponding colors are the results of 1D PIC simulations.

Fig. 6. (a) Attosecond XUV pulses generated by a three-cycle incident pulse with a₀ = 20 normally incident on a step-like target with n_e = 80n_c. For the ROM, the reflected XUV pulses have been filtered from the 10th to the 30th harmonic. A single XUV pulse is generated only in the case of the sine incident pulse. (b) Generation of attosecond pulse in transmission. Similar to the case of reflection, a sine-shaped incident pulse generates a single XUV pulse in transmission. A cosine-shaped incident pulse generates two distinct transmitted pulses. No filter has been applied to the transmitted pulses.

Fig. 7. (a) Dependence of the intensity of the reflected ROM attosecond pulses on time measured in cycles and CEP phase φ₀ of the incoming laser pulse (color-coded image). Colored curves outline the intensity of the attosecond pulses (harmonics with numbers >10) for four different values of φ₀. The laser pulse has a₀ = 20, a duration of three cycles, and n_e = 400n_c. (b) The same, but in the case when a linear preplasma with length λ_L/10 is added to the target.

Fig. 8. Intensity of reflected pulses and pulses in transmission in 2D, showing that the result is independent of the simulation code and the dimensionality. (a) corresponds to the reflected pulse and CSE pulse of the incident “sine” pulse, and (b) corresponds to the reflected pulse and CSE pulse of the incident “cosine” pulse. Incident pulses had an amplitude of a₀ = 20, and the target density in all cases was n_e = 80n_c.