Analysis of laser-proton acceleration experiments for development of empirical scaling laws

Numerous experiments on laser-driven proton acceleration in the MeV range have been performed with a large variety of laser parameters since its discovery around the year 2000. Both experiments and simulations have revealed that protons are accelerated up to a maximum cut-off energy during this process. Several attempts have been made to find a universal model for laser proton acceleration in the target normal sheath acceleration regime. While these models can qualitatively explain most experimental findings, they can hardly be used as predictive models, for example, for the energy cut-off of accelerated protons, as many of the underlying parameters are often unknown. Here we analyze experiments on laser proton acceleration in which scans of laser and target parameters were performed. We derive empirical scaling laws from these parameter scans and combine them in a scaling law for the proton energy cut-off that incorporates the laser pulse energy, the laser pulse duration, the focal spot radius, and the target thickness. Using these scaling laws, we give examples for predicting the proton energy cut-off and conversion efficiency for state-of-the-art laser systems.

Figure 1

Proton cut-off energies as a function of peak laser intensity from several experimental data sets [6, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42]. The linear fit to the experimental data (green line) reveals an average increase in ${\mathrm{E}}_c$ $\propto I_0^{0.47}$. However, deviations from this fit are more than $\pm 400$% suggesting that the peak intensity cannot be used as the only parameter for laser proton acceleration.

Figure 2

Proton cut-off energy from 13 experiments [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], in which an energy scan has been performed. The highest PCEs were measured using laser systems with high pulse energies $>100\mathrm{J}$. Laser systems with lower pulse energies and shorter pulse duration can reach similar intensities, but the observed PCEs were significantly lower.

Figure 3

Proton cut-off energies as a function of laser energy. The data points show a significant decrease in their spread in comparison to Fig. 2. The gray area shows the standard deviation from the fit function $E_c=5.3E_L^{0.89}$ below 2 J and $E_c=7.3E_L^{0.4}$ above. The corresponding publications from which the data were extracted can be found in Fig. 7 in the Supplemental Material [54].

Figure 4

Proton cut-off energies as a function of laser energy using the same data as Fig. 3. Only the shots with the highest PCE for each scan are highlighted representing the maximum performance of the laser system in terms of laser ion acceleration. These data are fitted to find a function to predict the maximum ion energy of a laser system according to their energy. Below 2 J of laser energy, the cut-off energy scales with $E_c=7.5I^{0.75}$ and for higher energies it follows $E_c=12I^{0.3}$. The data points were taken from [6, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42].

Figure 5

Comparison of pulse length scans from six experiments [26, 29, 30, 34, 35, 36]. The shape of the marker groups the data sets. The color represents the laser energy whereby blue corresponds to the highest and red to the lowest pulse energy. The fits characterize the dependency of the PCEs on the intensity caused by changes in the pulse length using $E_c\left(I\right)=A{I}^b$.

Figure 6

Proton cut-off energies (PCEs) as a function of the intensity that has been varied experimentally by a focal spot size scan. Four experiments are included in the analysis [26, 27, 30, 34]. Fits of the intensity scaling were performed, leading to an average scaling of ${\mathrm{E}}_c\propto I^{0.29\left(7\right)}$.

Figure 7

Proton cut-off energies as a function of the target thickness from various experiments [6, 22, 26, 33, 35, 37, 38, 39, 40, 41, 42]. Only experimental data are included that show an increase of the PCE with decreasing target thickness. Red data points have laser energies below 2 J, blue markers are between 2 and 3 J and purple markers are above 5 J. It can be seen that similar laser energies lead to similar scaling behavior.