Geometric scaling of laser-driven proton focusing from hemispherical foils

TL;DR

The paper investigates how proton focusing from laser-irradiated hemispherical targets scales with target diameter, using mesh radiography to extract both virtual and physical focal properties across a range of dimensionless geometries $Ψ = D_{hemi}/D_{Laser}$. By collecting over 70 high-repetition-rate shots, the study reveals that small hemispheres ($Ψ \,=\,6.1$) focus near the geometric center, while larger hemispheres ($Ψ \,=\,14.6$) exhibit degraded focusing with the focal location shifting deeper inside the hemi; the inferred virtual focal spot size is consistently around $9 \,±\,3\,\\mu$m. A hyperbolic-trajectory model links virtual observations to a physical focus, finding $\\Delta z_{phys}/R_{hemi}$ ranges from ~0.92 down to ~0.32 as $D_{hemi}$ increases, indicating a systematic shift of focus with geometry. The work highlights the trade-off between focusing quality and pointing stability, provides a robust dataset for scaling toward ignition-relevant conditions, and offers benchmarks for validating PIC simulations and guiding fast-ignition target designs. These insights are crucial for advancing proton fast ignition and warm dense matter experiments at higher driver energies.

Abstract

We systematically characterize the focusing behavior of laser-driven proton beams from hemispherical targets of various diameters using mesh radiography. The proton focal location is inferred to be near the geometrical center for the smallest tested hemisphere ($Ψ=D_{hemi}/D_{Laser}=6.1$). However, larger hemispheres ($Ψ=14.6$) degrade the focusing behavior and behave more like flat foils with focal location significantly inside the hemisphere. We also infer a tight virtual focus of $9\pm3~μ$m through a mesh transition analysis.

Figures (9)

• Figure 1: Experimental parameter space of dimensionless focusing geometry $\Psi=D_{hemi}/D_L$ and laser intensity from Refs. patel_isochoric_2003patel_integrated_2005snavely_laser_2007offermann_characterization_2011kar_ballistic_2011bartal_focusing_2012chen_focusing_2012 and this work. Refs. kar_ballistic_2011chen_focusing_2012 used semi-cylinders. The markers illustrate the different experimental characterization techniques of heating a secondary sample, mesh radiography, and side-on proton radiography.
• Figure 2: (a) Mesh radiography setup used to characterize the proton focusing. $p_0$ is the mesh pitch and $p_d$ is the magnified pitch on the detector. (b) RCF images from a single shot showing different layers and energy resolution.
• Figure 3: Virtual focal location $\Delta z$ as the mesh distance $L_m$ is varied. For reference, $\Delta z=0$ corresponds to the front surface of the target where the laser hits (hemi apex or flat foil) as shown in Fig. \ref{['fig:setup_data']}(a). The flat target is shown in black triangles, while the hemis of diameter 525, 325, and 220 $\mu$m are shown in blue diamond, red square, and yellow circle, respectively. RCF layers 2, 3, and 4 are shown. The dashed lines correspond to the geometric center of each hemi target. The stars on the right side are the inferred physical focus from fitting hyperbolic trajectories to the data with errorbars indicating the fit uncertainty. A small amount of horizontal spreading is applied to the datapoints for visibility.
• Figure 4: Sample proton trajectories for each target type. The blue dots show the mesh pitch normalized to the measured detector pitch $(p_0/p_d)$ at different mesh positions $L_m$. The data are plotted this way to compare self-similar proton trajectories. A hyperbolic fit is given for each target type (solid line). The red star is the physical focal position based on the minimum of the hyperbolic trajectory. A small amount of horizontal spreading is applied to the data for visibility. Note that the hemis are depicted with a circular aspect ratio but should actually be stretched in the radial direction from the normalization.
• Figure 5: Normalized focal location $\Delta z/R_{hemi}$ for different hemi diameters. Virtual focal locations are shown by shaded ellipses and errorbars containing one standard deviation. The virtual focii have slight horizontal offsets corresponding to different mesh distances $L_m$ and culminating in a star referring to the inferred physical focal, or equivalently the virtual focus as $L_m \to \infty$. The $\Psi=D_{hemi}/D_{Laser}$ value for each target is given in the secondary x-axis.