Steady regime of radiation pressure acceleration with foil thickness adjustable within micrometers under a 10–100 PW laser

Quasimonoenergetic GeV-scale protons are predicted to be efficiently generated via radiation pressure acceleration (RPA) when the foil thickness is matched with the laser intensity, e.g., $L_{\mathrm{mat}}$ of several nm to 100 nm for ${10}^{19}-{10}^{22}\mathrm{W}{\mathrm{cm}}^{-2}$ available in laboratory. However, nonmonoenergetic protons with much lower energies than predicted were usually observed in RPA experiments because of too small foil thickness which cannot support insufficient laser contrast and foil surface roughness. Besides the technical problems, we here find that there is an upper-limit thickness $L_{\mathrm{up}}$ derived from the requirement that the laser energy should dominate over the ion source energy in the effective laser-proton interaction zone, and $L_{\mathrm{up}}$ is lower than $L_{\mathrm{mat}}$ with the intensity below ${10}^{22}\mathrm{W}{\mathrm{cm}}^{-2}$, which causes inefficient or unsteady RPA. As the intensity is enhanced to $\ge {10}^{23}\mathrm{W}{\mathrm{cm}}^{-2}$ provided by 10–100 PW laser facilities, $L_{\mathrm{up}}$ can significantly exceed $L_{\mathrm{mat}}$, and therefore RPA becomes efficient. In this regime, $L_{\mathrm{mat}}$ acts as a lower-limit thickness for efficient RPA, so the matching thickness can be extended to a continuous range from $L_{\mathrm{mat}}$ to $L_{\mathrm{up}}$; the range can reach micrometers, within which foil thickness is adjustable. This makes RPA steady and meanwhile the above technical problems can be overcome. Particle-in-cell simulation shows that multi-GeV quasimonoenergetic proton beams can be steadily generated and the fluctuation of the energy peaks and the energy conversation efficiency remains stable although the thickness is taken in a larger range with increasing intensity. This work predicts that near future RPA experiments with 10–100 PW facilities will enter a new regime with a large range of usable foil thicknesses that can be adjusted to the interaction conditions for steady acceleration.

Figure 1

(a) The target thickness $L$ for efficient RPA as a function of the laser amplitude $a_0$, where the pink rectangles and blue rectangles correspond to 2D and 3D PIC results, respectively, the black line is $L_{\mathrm{mat}}$ calculated from Eq. (1), and the red and green dashed lines show $L_{\mathrm{up}}$ calculated from Eq. (2) with $v_i$ estimated as $\frac{2v_p}{1+v_p^2}$ and $v_g$, respectively. The target thickness for efficient RPA is counted when a quasimonoenergetic proton beam is generated in PIC simulation. (b) The corresponding energy peaks ${\varepsilon }_{\mathrm{peak}}$ of the quasimonoenergetic proton beams are displayed by pink rectangles and blue rectangles for 2D and 3D PIC results, respectively. The green line with circles is the fluctuation of the peak energies $\frac{\mathrm{\Delta }\varepsilon }{{\varepsilon }_{\mathrm{peak}}}$ obtained from the 2D-PIC simulations.

Figure 2

Temporal evolution of the laser wavefront velocity $v_f$ (red dotted line) and tracked proton velocity $v_i$ (green line), where we track 100 protons gaining high energies finally and take five typical ones. Evolution of the maximum of the longitudinal electric field $E_{x-\max }$ (corresponding to the right $y\mathrm{axis}$) is also displayed by the blue line. Here the laser amplitude $a_0$ and target thickness $L$ are taken as (100, $0.18\lambda$), (300, $0.56\lambda$), (500, $1.1\lambda$), and (700, $1.7\lambda$) in (a)–(d), respectively.

Figure 3

(a) Evolution of the plasma electron temperature and (b) the energy spectra of protons at $t=70T_0$, where different lines in (a) and (b) represent different ($a_0$, $L$) corresponding to the parameters taken in Figs. 2, respectively. The plasma temperatures are calculated with the electrons in the compressed density layer and normalized by that in the case of $a_0=100$.

Figure 4

(a1)–(d1) Spatial distributions of energy densities of the protons (top half) and the laser (bottom half); the laser energy and the ion energy integrated over the same longitudinal space are also plotted by the red curve and blue curve of the subfigures (a2)–(d2), respectively, where (a) and (b) are the cases with $a_0=100/300$ at $t=30T_0$, and (c) and (d) are the cases with $a_0=500/700$ at $t=50T_0$. The units of the color scales are normalized by the energy density in the case with $a_0=300$. (e), (f) Temporal evolution of the energy ratio between the protons and the laser, where the ratio is calculated by the energy peaks of the protons and the laser.