Fig. 1. Spatiotemporal dynamics of a thin slab target driven by an intense few-cycle circularly polarized (CP) electric field and with thickness in the sub-λ regime (in this case, a foil of thickness d = 0.75λ). The spatiotemporal evolutions of the laser intensity Ey2+Ez2 (representing the incident, reflected, and transmitted light), electron density n_e, and ion density n_i are shown for two different initial plasma densities n₀ representative of the two distinct scenarios. The upper row [(a) and (b)] for n₀ = n_e = 3n_c corresponds to the transmission regime, and the lower row [(c) and (d)] for n₀ = n_e = 9n_c corresponds to complete reflection of light. The blue solid horizontal lines mark the instant t = 48τ when the peak of the pulse envelope interacts with the target surface. (b) and (d) show magnified views of the dashed boxes in (a) and (c), respectively, capturing the distinct signatures of the ion and electron dynamics in the two different regimes during the relativistic interaction. Both targets are initially overdense and reflect the laser (because n₀ > n_c), which interferes with the incident pulse to form a standing wave pattern in front of the target (see the intensity fringes on the front side of the target). (a) Near the peak of the pulse, the target becomes optically thin (dynamically underdense), pushing out electrons from the surface, initiating complex electron dynamics in the ion background, and expanding the ion density distribution shown in (b). (c) The target remains overdense within the laser pulse duration, leading to the synchronous motion of the electron and ion density peaks shown in (d). For both cases, we used a Gaussian laser pulse with normalized peak laser pulse amplitude a₀ = 20 [as in Eq. (1)].

Fig. 2. Temporal snapshots showing profiles of driver laser field magnitude (dashed blue line representing incident, reflected, and transmitted fields), longitudinal electric field (red solid line representing accelerating field), ion density (green solid line), and electron density (black solid line) for the two scenarios: (a) and (b) are for n_e = 3n_c (relativistic transparency regime), and (c) and (d) are for n_e = 9n_c (overdense regime). For each case, the fields and densities are plotted at two different times: (a) and (c) are for time t = 48τ, when the peak of the laser pulse is interacting with the target, and (b) and (d) are for the later time t = 70τ, long after the driving field has ceased to interact with the target. Near the peak of the interaction, the target with lower initial density has already started to transmit the incident laser pulse, which can be seen in the blue shaded part on the right side of the target in (a) and is clearly captured after the interaction in the laser pulse co-propagating to the right along with the electron bunch as seen in (b). No such transmission can be seen in (c) and (d) [note the significantly small scale used for plotting the magnitude on the right axis for the radial electric field presented in (d)].

Fig. 3. Temporal snapshots of ion phase-space distribution and corresponding energy spectra. (a) and (b) Ion phase-space distribution in blow-out regime (n_e = 3n_c) of interaction at two different time instants. (c) and (d) Ion phase-space distribution in opaque regime (n_e = 9n_c) of interaction at the same time instants. At t = 48τ, the laser pulse peak is interacting with the target, whereas t = 70τ is long after the driving field has ceased to interact with the target. The color bar for (a)–(d) represents the number of macroparticles d²N_i/dβ_idx accelerated in the laser propagation direction with velocity (β_i), normalized by the speed of light in vacuum (c), per unit bin in the phase space. (e) and (f) Ion energy spectra dNi/dE corresponding to the interactions represented in (a)–(d). Both targets behave in a reflective manner until near the peak of the laser pulse envelope reached at t = 48τ. At the peak of the driving field, the foil with n_e = 3n_c becomes transparent, whereas the one with n_e = 9n_c remains reflective during the whole relativistic interaction. The two different behaviors representative of the two different regimes of operation are captured very well in this figure. During the reflective regime, the phase-space velocity distributions in (a), (c), and (d) have similar folded shapes and peaky features that become significantly different in (b) once the interaction enters the transparent regime, and the aforementioned similar behavior is a signature in the ion energy spectra. For the reflective part of the interaction [dark shaded curves in (e) and (d)], the ion energy spectra show distinct quasi-monoenergetic behavior that is retained even after the interaction is over, as seen in the light shaded curve in (f), but that erodes away once the target enters the transparent regime even partially, as seen in the light shaded curve in (e).

Fig. 4. Correlation between interaction regime and nature of resulting ion energy spectra: step-density targets. (a) Fraction of laser energy transmitted through target with varying thickness (d/λ) and peak electron density (n_e/n_c) at simulation time 70τ. The 2% (black solid line) and 40% (black dashed line) transmitted energy fractions are shown, with the 40% black dashed line indicating the threshold target density (n_th) for relativistically induced transparency (RIT) for varying target thickness. The analytically predicted threshold target density [Eq. (2)] is shown with a white dashed line and is consistent with the 40% iso-line obtained from particle-in-cell (PIC) simulations. The color bar shows the laser pulse energy transmission coefficient (Et) as in Eq. (3). The target density n_e = 5n_c and thickness d = 1.15λ are marked with horizontal and vertical gray lines in (a). In (b), the ion energy spectral map with varying target thickness is presented for the threshold density of n_e = 5n_c [along the horizontal iso-density gray dashed line in (a)]. In (c), the ion energy spectral map for the target thickness of d = 1.15λ is presented with varying target density [along the vertical iso-thickness gray dashed line in (a)]. Note that the color map for ion energy is plotted in logarithmic scale. The spectral map in (b) and (c) shows unequivocally that as the interaction enters from transparency into the opaque regime [across the 2% iso-transparency black solid curve in (a)], the accelerated ion energy spectra goes from exponential to quasi-monoenergetic peak structures irrespective of whether it is along the iso-density line or along the iso-thickness line. This establishes a consistent correlation between the regime of interaction and the nature of ion energy spectra over a wide range of parameter space. To probe this point further, we plot the ion energy spectral map along different iso-thickness lines with varying peak target density. (d) Shows the ion energy spectral map iso-thickness lines through the red semitransparent circles in the black curve separating the transparent and opaque regimes in (a). The black dotted lines in (b)–(d) represent the cutoff ion energy in both regions, whereas the orange dashed contour lines in (d) mark the peak ion energy in the relativistic transparency region. The black arrows in the ion energy spectral maps in (b)–(d) mark the interaction conditions corresponding to the (n_e/n_c, d/λ) pairs on the black solid curve in (a) [identifying with the target conditions indicated by each circle shown in (a)] emphasizing the transition phase between the regimes, i.e., from transparent to opaque conditions through the 2% iso-transmission curve. A clear correlation can be seen between Et and d. The color bar above (b) shows the number of ions accelerated and represents (b)–(d). All the spectra are for time 70τ.

Fig. 5. Correlation between interaction regime and nature of resulting ion energy spectra: foils with varying plasma gradient scale lengths. (a) Fraction of transmitted laser energy for varying plasma scale length (L/λ) and density (n_e/n_c). The color bar shows the energy transmission coefficient (Et) as defined in Eq. (3). The black solid line corresponds to 2% transmitted energy fraction, and the black dashed line corresponds to 40% transmitted energy fraction. (b) and (c) show the ion energy spectral maps with varying scale lengths for n_e = 3n_c [dashed white line (a)] and 9n_c [dashed green line (a)], respectively. The black dashed contour lines in (b) and (c) mark the maximum or cutoff ion energy in each case. The color bar on top of (b) represents the number of particles accelerated per energy bin at the corresponding energy.

Fig. 6. A simplified picture illustrating the influence of laser chirp. Upper panels: temporal profiles of a circularly polarized (a) negatively chirped (ζ = −5), (b) unchirped (ζ = 0), and (c) positively chirped (ζ = +5) laser pulse. The projected transverse field components a_y(t) and a_z(t) are presented as gray solid lines plotted on the respective transverse planes. The variation in time-dependent frequency of the laser pulse as shown in the color bar [Ω(t) as defined in Eq. (5)] is encoded on the color variation across the circularly polarized laser field. The pulse peak here is located at t = 0. Ω(t) = 1 represents the carrier frequency, and negative t indicates early in the interaction. Lower panels: color maps of ratio of time-dependent laser pulse amplitude a(t) and parameter ξ(t) [defined in Eq. (6)] during first half of interaction (the time axis covers the range from the start of the laser pulse at −8τ to the peak of the pulse intensity at t = 0) for the step target case over a range of target densities for (d) negatively chirped, (e) unchirped, and (f) positively chirped pulse scenarios. The black contour lines are plotted at a(t)/ξ(t) = 0.7 (in opaque region), 1.0 (in threshold region), and 1.5 (in transparent region) for the fixed density case of n_e = 6n_c (black horizontal solid line). As is evident, if the target remains unchanged during the interaction, then in (d) for the negatively chirped case, before the peak of the laser pulse intensity, the interaction at an initial target density of n_e = 6n_c enters the transparent regime of operation. (e) The transparent regime is approached near the peak of the pulse for ζ = 0, and (f) the same target meets the transparency condition after the peak of the laser pulse for ζ = +5.

Fig. 7. Upper row: laser pulse transmitted energy fraction [E(t)] for (a) negatively and (b) positively chirped pulse with varying target density (n_e/n_c) and scale length (L/λ). Similar to the above figures, the iso-lines for transmitted laser energy fraction are marked at 2% (black solid) and 40% (black dashed). The shaded squares (black square with gray fill or the reverse) on the colormap in (a) and (b) indicate the specific transparency conditions corresponding to the primary target electron density. The point marked with the black circle with gray fill in (a) and (b) identifies the target conditions at which the transparency difference due to positive and negative chirp is maximum. Lower row: ion energy spectrum from (c) primary layer (PL) with peak density of 5.6n_c and scale length of 0.7λ and (d) secondary layer (SL) with thickness of 0.2λ and density of 0.1n_c for unchirped (maroon solid), positively chirped (yellow solid), and negatively chirped (black solid) pulses. The spectra were obtained at a simulation time of 70τ.

Fig. 8. Ion energy spectra from double-layer target composed of deuterium (PL) and thin (0.2λ) low-density (0.1n_c) hydrogen (SL). (a)–(d) show the ion energy spectra from both layers for a PL with a peak electron density of 3n_c and 9n_c, respectively, and with a scale length of 0.4λ in all cases. The colors of the traces indicate the laser chirp conditions: positively chirped (yellow solid), unchirped (maroon solid), and negatively chirped (black solid). The shaded squares (black square with gray fill or the reverse) on the top left corner of each sub-figure correlates with those marked in Figs. 7(a) and 7(b) identifying the transparency conditions corresponding to the primary target electron density.

Fig. 9. Target ion density maps for PL (deuterium) and SL (hydrogen) are shown in (a) for n_e = 3n_c (transparency regime) and (d) n_e = 9n_c (reflection regime). The color bars represent the ion densities on a logarithmic scale for both species, in yellow-green for D⁺ and yellow-red for H⁺. (b) and (e) Ion energy spectrum corresponding to central line outs for n_e = 3n_c and 9n_c, respectively (PL: deuterium layer in blue and SL: hydrogen layer in orange). The angular energy distribution of the predominant species is shown with the polar plots for both cases in (c) H⁺, 3n_c and (f) D⁺, 9n_c. The color bar represents the number of ions accelerated at a certain angle with a specific energy in megaelectronvolts. The ion energy 2D spatial distributions and the spectra were obtained at t = 48.8τ, i.e., 24τ after the peak interaction of the laser pulse with the target, similar to the condition in the 1D simulations.

Table 1. Peak and cutoff ion energies obtained in relativistic transparent region (n_e = 3n_c) and overdense region (n_e = 9n_c) from double-layer target configuration (PL comprising D⁺ and SL comprising H⁺) with three different driving focal-spot radii (1/e² of the radius of the intensity spatial envelope at focus), i.e., 4, 8, and 12 μm. Figure 9 summarizes the results for the intermediate focal-spot radius of 8 μm.