Fig. 1. Solution of the equations of the relativistic electron spring model for a Gaussian pulse of duration 3 fs and amplitude a₀ = 30 and a homogeneous plasma of density n₀ = 10. (a) Trajectory of the electron boundary x_s(t). (b) Trajectory of the electron boundary in phase space (x, v_x = dx/dt). (c) Temporal profiles of the incident (blue) and reflected (orange) pulses.

Fig. 2. Result of a 1D PIC simulation for a Gaussian laser pulse of duration 20 fs and amplitude a₀ = 27.85 and a homogeneous plasma of density n₀ = 30. (a) Spatiotemporal dynamics of the electron concentration (orange) and reflected radiation (violet). The concentration is normalized to the critical concentration for the carrier frequency of the incident laser pulse N_cr = mω²/4πe². (b) Time profile of the reflected pulse without filtering (red), after filtering in the <0.1μm wavelength range (blue), and after filtering in the <3μm wavelength range (orange). The field is normalized to m_ecω/e.

Fig. 3. Comparison of Fourier spectra for the incident pulse (blue dotted line), the reflected pulse obtained from the relativistic electron spring model (orange dashed line), and the reflected pulse obtained from the 1D PIC simulation (green full line). The inset shows the time profile of the pulse obtained after filtering the reflected signal in the 1D PIC simulation in the range 3–20 µm. The simulation parameters are the same as in Fig. 2.

Fig. 4. Dependence of the Fourier spectrum of the reflected signal obtained in 1D PIC simulation of the plasma density. The data shown are for a laser pulse amplitude a₀ = 27.85 and a duration of 20 fs. The red line corresponds to a Doppler shift induced by a mirror moving at the velocity of the laser front penetrating into the plasma as estimated from the simulations.

Fig. 5. Dependence of the energy efficiency of the conversion of the incident laser radiation into the reflected signal in different ranges for different amplitudes of the pulse. The results are shown for 1D PIC simulations with immobile ions. For comparison, the result for the case of mobile ions is also presented. The laser pulse duration is 20 fs, and its amplitude is given in the legend.

Fig. 6. Result of a 2D PIC simulation of the interaction of a Gaussian pulse of duration 20 fs, width 5 µm, and amplitude a₀ = 27.85 with a plasma of density n₀ = 30, which has a preplasma layer of thickness 2 µm with a linear density gradient. In the center is shown the laser pulse at the beginning of the calculation. On the right are shown the transverse fields and the electron density at time t = 200 fs after the start of the calculation. On the left are shown the low-frequency (in the wavelength range >3μm, red) and high-frequency (in the wavelength range <300 nm, blue) parts of the reflected pulse at time t = 550 fs after the start of the calculation.