Enhanced laser-driven hadron sources with nanostructured double-layer targets

Laser-driven ion sources are approaching the requirements for several applications in materials and nuclear science. Relying on compact, table-top, femtosecond laser systems is pivotal to enable most of these applications. However, the moderate intensity of these systems (I ≲ 1019 W cm−2) could lead to insufficient energy and total charge of the accelerated ions. The use of solid foils coated with a nanostructured near-critical layer is emerging as a promising targeted solution to enhance the energy and the total charge of the accelerated ions. For an appropriate theoretical understanding of this acceleration scheme, a realistic description of the nanostructure is essential, also to precisely assess its role in the physical processes at play. Here, by means of 3D particle-in-cell simulations, we investigate ion acceleration in this scenario, assessing the role of different realistic nanostructure morphologies, such as fractal-like foams and nanowire forests. With respect to a simple flat foil, the presence of a nanostructure allows for up to a × 3 increase of the maximum ion energy and for a significant increase of the conversion efficiency of laser energy into ion kinetic energy. Simulations show also that the details of the nanostructure morphology affect both the maximum energy of the ions and their angular distribution. Furthermore, combined 3D particle-in-cell and Monte Carlo simulations show that if accelerated ions are used for neutron generation with a beryllium converter, double-layer nanostructured targets allow to greatly enhance the neutron yield. These results suggest that nanostructured double-layer targets could be an essential component to enable applications of hadron sources driven by compact, table-top lasers.


Introduction
Ion acceleration driven by ultra-intense laser pulses is a well established reasearch topic [1][2][3]. In state-of-the-art Petawatt-class laser facilities, few nanocoulomb, relatively collimated, broad spectrum ion bunches can be accelerated up to a ∼100 MeV [4][5][6] cut-off energy. Target normal sheath acceleration (TNSA) [7] is arguably the most robust and widely studied ion acceleration scheme. In TNSA an ultra-intense laser pulse interacts with a thin solid foil, heating the electrons of the target up to several MeV energies. The expanding electron cloud generates an intense electric field at the back side of the target. This electric field is then responsible for the ion acceleration process. Historically, the foreseen societal application in radiotherapy [8,9] has been a preeminent driving force behind research efforts on laser-driven ion acceleration. However, existing laser-driven ion sources still fall short of the stringent requirements in terms of energy, monochromaticity and stability [10]. Indeed, up to now, laser-driven ion sources have been routinely used only as a diagnostic tool for transient electromagnetic fields in laser-plasma interaction experiments [11][12][13].
Especially in recent years, materials and nuclear science have emerged as a promising domain of application of laser-driven ion sources. Proof-of-principle experiments [14] and numerical works [15][16][17] support the

Methods
Our work is based on two numerical tools: particle-in-cell and MC simulations. This section describes in detail the methods and codes used in this work and the parameters of the simulations. In the following we will consider a laser wavelength λ=800 nm, as for a Ti:sapphire laser system.

Particle-in-cell simulations
Particle-in-cell simulations are a standard numerical technique to investigate laser-plasma interaction [71,72]. For our 3D simulations we relied on the open source, massively parallel piccante code [73]. The numerical box was 100λ×60λ×60λ, with periodic boundary conditions. The spatial resolution was 40 points per λ (ppλ) in each direction, while the temporal resolution was 98% of the Courant condition. The total duration was 80 λ/c. The laser beam had a Gaussian spatial profile with a waist equal to 5 λ and a cos 2 temporal profile with a FWHM for the fields equal to 15 λ. These parameters model an idealized, high contrast, Ti:sapphire laser system with tens of TW power. The beam was linearly polarized and the angle of incidence was 0°. The normalized intensity was a 0 =4 (a 0 =eA/m e c 2 , where e is the elementary charge, A is the vector potential, m e is the electron mass and c is the speed of light).
Concerning the target, ions were initialized perfectly cold, while electrons were initialized with a very small temperature (few eV). The substrate-i.e. the flat solid foil or bare target-was 0.5 λ thick, with an electron density of 40 n c . We decided to use an electron density of only 40n c for the substrate because 3D simulations with a more realistic density for a highly ionized target (∼100 n c ) would have required a huge computational effort. However, we performed convergence tests with 2D simulations to assess the appropriateness of our choice for both the solid foil density and the resolution. We let the density of the solid foil vary among 20 n c , 40 n c and 80 n c , while the resolution varies between 40 ppλ and 60 ppλ. The main results of these 2D tests are shown in figure 2, where the protons spectra at time 80λ/c are shown for (a) a simple 0.5 λ-thick solid foil and (b) a DLT modeled with a uniform near-critical layer with 2.14 n c density and 4 λ thickness. Increasing the electron density of the substrate from 20 n c to 80 n c did not result in any appreciable change of the ion energy spectrum for the DLT. Instead, a strong reduction of the maximum ion energy was observed for the simple flat target under the same density variation. Nonetheless, when doubling the density from 40 n c to 80 n c only a slight reduction is obtained. These results confirm that using a substrate density of 40 n c in our simulations should not significantly affect the results presented in this manuscript. Overall, a solid density of 40 n c is found to be a good compromise: largeenough to be overdense, but low-enough to let us resolve the skin depth of the laser with a resolution of 40 ppλ. In any case, to ascertain the goodness of such a resolution value, we also tested a higher resolution of 60 points per λ in 2D geometry and we observed negligible differences in the proton spectra (see figure 2). Scanning electron microscope picture of a nanostructured low-density foam similar to those used in [48,49]. (d) 3D particle-in-cell simulations are used to model laser-driven ion acceleration (reproduced with permission from [70]). (e) Laser-accelerated ions are fed into the Geant4 Monte Carlo code to simulate neutron generation in a beryllium converter. 40 macro-electrons per cell and 2 macro-ions with Z/A=0.5 per cell were used. The hydrocarbon contaminant layer was 0.05 λ thick with an electron density of 7 n c , a proton density of 1 n c and a fully ionized carbon density of 6 n c (64 macro-particles per cell per species were used). The low-density layer was chosen to be 4 λ thick. Three different morphologies (an idealized uniform plasma, a fractal-like nanostructured foam and a random nanowires forest) and two different average densities (1.5 n c and 2.29 n c ) were considered as well as the case of a simple flat target for comparison, for a total of seven simulations. Both the realistic morphologies considered here are representative of low-density nanostructured materials used in laser-plasma interaction experiments [48,49,74,75].
The foam layer is generated through an extension of the diffusion-limited cluster-cluster aggregation model (DLCCA), designed to simulate the structure of fractal aggregates, such as colloids and soot [76] (a detailed description of the method is provided in [66]). The building blocks of the fractal aggregates are nanospheres having a local electron density of ∼40 n c , arranged in such a way that the filling factor is 5.7% and 3.8% for the two density values. The radii of the nanospheres are 0.05λ for the higher density foam and 0.047 λ for the lower density foam. Nanowires forests are modeled as a collection of cylinders, with a radius of 0.12 λ and an inclination angle extracted randomly from a uniform distribution between −30°and +30°with respect to laser axis. The nanowires have a local electron density of ∼40 n c and their number is chosen so that the filling factor is either 5.7% or 3.8%. For the nanostructured layers 40 macro-electrons per cell and 2 macro-ions with Z/ A=0.5 per cell were used, while for the uniform layers 4 macro-electrons per cell and 1 macro-ion with Z/ A=0.5 per cell were used. The main properties of the near-critical layers are summarized in table 1. All the considered near-critical layers are coupled to a substrate with 0.5 λ thickness and 40 n c electron density.
We point out that the nanostructures show large local fluctuations of their average density on scales comparable with the laser waist (5λ).

MC simulations
We simulated the process of proton-to-neutron conversion via MC simulations. To this purpose, we employed the Geant4 (Geometry and Tracking) toolkit [67], which is widely used to simulate particle transport through matter in a variety of scenarios. Here, we are interested in a reliable description of neutron generation from nuclear reactions induced by moderate energy protons (∼few MeVs). The main physical processes, i.e. the hadronic processes, were included via the G4HadronPhysicsQGSP_BIC_AllHP physics list, which was found to be the most reliable in a previous benchmarking work [68]. Due to the hadronic nature of the process, the primary bunch is only made of protons, whereas we disregarded electrons and photons coming from the prior laser-plasma interaction process. The proton momentum distribution is obtained from the 3D PIC simulations results. Indeed, 3D PIC simulations allow one to retrieve the full momentum distribution of any species. Therefore, we sampled the accelerated protons momentum distribution from the PIC output with the Inverse Transform Sampling method. Such sampling procedure is used in the MC code to initialize the energy and propagation direction of the primary proton bunch. The proton source is spatially point-like and is positioned 1.5 cm away from the converter. In each case, 10 10 protons were shot. The converter is made of pure 9 beryllium with thickness equal to the range of the most energetic protons (0.009-0.06 cm). The main nuclear reactions at play are (p, n * ).

Results
In this section we present the results of our numerical investigation, with a particular focus on ion acceleration and neutron generation. A brief discussion of the physical processes at play in ultra-intense laser-plasma interaction is provided as well.

Laser absorption into hot electrons
Laser absorption into electron kinetic energy in a near-critical plasma is a complicated, multifaceted process [52][53][54]77]. In the case of a nanostructued plasma, the absorption process is further complicated by the inhomogeneities of the density profile, which are likely to be maintained during the interaction, especially because the ion dynamics is slower than the tens-of-fs laser temporal duration [65,66]. In our scenario, the foam targets lead to an increased laser absorption into electrons. The flat target allows for ∼10% conversion of laser energy into electron kinetic energy, while the remaining portion of the laser is mostly reflected. On the other hand, the DLTs allow for a maximum electron absorption of ∼40%-60% (except for the denser homogeneous foam, which is too opaque for the considered pulse, for which it is ∼20%). Depending on the morphology of the nanostructure, the absorption process leads to different features of the hot electron population, as shown in figure 4. In particular, the presence of a near-critical nanostructured layer leads to significantly higher electron energies and a broader angular distribution. Concerning the homogeneous foams instead, the relatively dense 2.29 n c uniform layer is barely transparent for a a 0 =4 laser beam. This means that the absorption efficiency is not significantly higher than for the flat target. The uniform foam with a density of 1.5n c and the generation of hot electrons is improved, but still significantly below what is obtained with the nanostructured foams. These differences translate into different strengths and extensions of the quasi-static electrostatic fields at the back side of the target. Since the ion acceleration process is TNSA-like, these features have a profound influence on the properties of the accelerated ions. However, a detailed study of the absorption process requires an extensive, dedicated, investigation, which is not the focus of this manuscript and has been partially addressed elsewhere [55,65,66]. Figure 5 shows the salient features of the accelerated ions obtained with the 2.29 n c foam-coated targets. The results for a simple flat target are reported as well for comparison. With respect to a simple flat target, the presence of the homogeneous foam leads only to a moderate enhancement of the cut-off energy of the accelerated ions (from ∼2 to 3 MeV). On the contrary, the target coated with a fractal-like DLCCA foam attains ∼4.5 MeV and the target coated with a random nanowire forest allows to obtain slightly more than 6 MeV, a 3×increase with respect to the simple flat target. These results are in agreement with what was observed for the electron population, since in a TNSA-like scheme a higher efficiency of laser-to-electrons coupling generally results into higher cut-off energies of the accelerated ions.

Enhanced laser-driven ion acceleration
Concerning the angular distribution of the accelerated ions, the presence of a nanostructured layer seems to lead to a broader angular spread. Again, this is in agreement with what was observed for the electron population. Moreover, the angular distribution is much less homogeneous, with 'holes' and denser stripes appearing in the 'synthetic radiochromic film' diagnostic. These features are essentially irrelevant for the application of laserdriven proton sources considered here (neutron generation). However, they might be a severe hindrance if such a source is used for proton radiography, where having a smooth angular distribution is essential. Figure 6 reports also the proton energy spectra obtained with foams having a lower average density (1.5 n c ). The most significant difference concerns the homogeneous foam: the lower density leads to a much greater laser absorption and therefore a higher maximum energy of the ions. For the structured foams, the lower average density has only a mild effect, slightly beneficial for the DLCCA foam and slightly disadvantageous for the nanowires foam. From the point of view of applications requiring stable ion fluxes, this mild dependence on the density could be a considerable advantage. The average density of nanostructured targets might change from sample to sample, due to unavoidable fluctuations in the manufacturing process. Our results suggest that the properties of an ion source based on these targets would remain relatively stable, even with large shot-to-shot variations of the target properties. This is further supported by results obtained in a previous work [66], where we considered laser interaction with nanostructured foams having quite different morphologies but the same average density, observing only a mild effect on laser absorption. Figure 6 represents the cross sections for the (p, n * ) reaction in 9 Be, together with the energy spectra of the accelerated ions for all the simulated cases. Figure 7 illustrates the properties of the emitted neutrons in all the simulated cases, while figure 8 summarizes the proton-to-neutron conversion rate (top) and the total neutron yield (bottom). A first crucial observation is that accelerating protons at least up to 2 MeV is absolutely necessary to obtain neutron conversion, since this is the threshold for the nuclear reaction to take place. When the simple flat target is used for ion acceleration, the energy cut-off is barely above the 2 MeV threshold for neutron conversion (see figure 6), with extremely few ions having enough energy to have a non-zero chance of generating a neutron. This leads to a very low p-to-n conversion rate (∼10 −8 neutrons/protons), hence low neutron yield (∼10 2 neutrons/shot), as shown in figures 7 and 8. The presence of a near-critical layer leads to more ions having  enough energy for a (p, n * ) reaction (see figure 6). This leads to a very strong enhancement of the p-to-n conversion rate. Assuming that 10 10 protons are accelerated above 2 MeV with the flat foil and taking into account the relative number of protons accelerated to energies above the reaction threshold, we also obtain a very strong enhancement (from 4 to 7 orders of magnitude) of the neutron yield with respect to the simple flat target, as shown in figure 8. Among the DLTs, the denser homogeneous foam is the worst performing case (∼10 6 neutrons/shot), while all the other cases fall within the 10 8 -10 9 neutrons/shot range.

Enhanced neutron generation
It is also worth highlighting that the cross section for (p, n * ) reaction in 9 Be peaks at ∼6 MeV (see figure 6, gray curve). Therefore, the proton energies obtained with the nanostructured targets are already enough to obtain optimal neutron conversion.
The left panel of figure 7 shows the energy spectra of the neutrons emitted in a 15°cone around the target normal in the forward direction. The distributions are quite broad, with a peak at 1-2 MeV and a cut-off extending up to 4 MeV, which means that the neutron source described here generates fast neutrons. A cut-off of ∼4 MeV is coherent with a maximum ion energy of ∼6 MeV and a negative Q-value of the (p, n * ) reaction of ∼−1.85 MeV. Neutrons are emitted in all directions, as shown in the right panel of figure 7. However, the angular distribution is not uniform: about one order of magnitude more neutrons are emitted in forward direction. This could be beneficial for applications, also possibly mitigating radio-protection concerns.

Discussion
We presented a scheme of an ion source able to accelerate protons to an energy of few MeVs, driven by a moderately intense laser thanks to the use of nanostructured DLTs. We addressed the issue of setting up a description that takes into account non-trivial features, such as the details of the target nanostructure, the effect of a density fluctuation and the direct coupling with a p-to-n converter. Including these features in the description should be crucial if one expects some kind of quantitative agreement with experiments or aims at designing an actual source.
As discussed in the introduction, the few MeVs energy range is suitable for several applications in materials science. There exists a subset of applications requiring only a modest particle flux, such as the so-called proton induced x-ray emission spectroscopy [35], an ion beam analysis technique. This technique could be an ideal application of laser-driven ion sources [17], since few 10s of laser shots could be enough to complete a measurement. Our work represents a step forward in making the aforementioned moderate-energy applications of laser-driven ions a reality.
Adding a beryllium converter, we estimate that the laser-driven neutron source described in this work could provide ∼5×10 5 neutrons per laser shot if 10 10 protons are accelerated to energies above the reaction threshold. Adopting targetry solutions analogous to those described in [70,78] should allow operation of such a neutron source in a repetitive regime, for at least ∼10 3 laser shots. Since compact 50 TW laser systems able to attain a repetition rate of ∼10 Hz are commercially available [79], we estimate a neutron flux of ∼5×10 6 n s −1 sustained for ∼10 2 s to be within reach of existing technology. This seems doable from the practical point of view. In the most common configuration the target consists of a perforated holder upholding the DLT. In this case, the main limitation comes from the target transverse size and consequently from the number of holes that can be drilled in the holder. No actual limitation comes from having to grow the foam layer on a large substrate (in the worst case different targets may be juxtaposed). In previous experiments holders with ∼200 holes were employed, so that shooting hundreds of times at 1 Hz repetition rate seems doable. After that, a new target needs to be installed.
Our estimation for the neutron flux suggests that it could be comparable with that of other proposed compact laser-driven neutron sources reported in the literature, where for 'compact' we mean relying on tabletop Ti:sapphire lasers not exceeding ∼100 TW. For instance, the interaction of a few mJ laser at kHz frequency with a heavy-water jet [80,81] has been reported to lead to an average neutron yield of 2×10 5 , while 2×10 6 neutrons per shot have been obtained via photoneutron generation with wakefield-accelerated electrons driven by a 0.5 J, 10 TW laser at ∼10 −2 Hz [82]. Concerning schemes similar to that proposed here, where a target for ion acceleration and a separate converter for neutron generation are used, a yield of ∼10 6 neutrons per steradian per shot can be obtained with few J lasers, but relying on deuterated targets (see [29] and references therein). In any case, it is worth remarking that the estimation of the yield of our source depends on our specific choices for the parameters of the contaminant layer, which are unfortunately not well characterized from the experimental point of view.
Conventional, portable, neutron sources based on deuterium-deuterium or deuterium-tritium fusion reaction can attain neutron fluxes two orders of magnitude higher than the source described here [83]. An improvement of the neutron yield for our source could be obtained using deuterated targets for ion acceleration, a strategy explored in several works [25-28, 32, 34]. Indeed, the 9 Be(d, n * ) cross section is significantly higher than 9 Be(p, n * ) cross section and the energy threshold for the projectile is lower [84]. This could allow to partially close the gap. However, the main strength of laser-driven neutron sources is not the average flux, for which competing with conventional sources seems to be a daunting task. Rather, the main advantages of a laser-driven sources lie in their pulsed nature and in their extremely short duration. A pulsed source could be beneficial to reduce signal-to-noise ratio in several applications (e.g. radiography or spectroscopy), since temporally gated detectors could be synchronized with the laser pulse. Indeed, 'flash neutron radiography' has been suggested in previous works as an application especially suited for laser-driven neutron sources [85]. Overall, our modeling approach could be used to guide the design and the optimization of these kinds of applications.

Conclusion
We have performed a numerical simulation campaign of a laser-driven ion acceleration scheme based on an advanced targetry solution, and of a neutron source based on this scheme.
DLTs consisting in a near-critical layer coupled to a solid foil can enhance the efficiency of the ion acceleration process, leading to more ions being accelerated up to higher energies. This results in a strong enhancement of the neutron yield when these ion sources are coupled with a suitable converter. Since one promising way to realize a near-critical layer consists in exploiting low-density nanostructured materials, we assessed the role of different realistic nanostructure morphologies on the ion acceleration process. We found that a realistic modeling of the nanostuctured target is essential to provide reliable indications for the design of experiments and for the development of laser-driven ion sources.
Our results suggest that nanostructured DLTs could be a crucial component to enable applications of hadron sources driven by compact, table-top lasers and our work provides essential tools for the numerical modeling of these schemes. ORCID iDs L Fedeli https:/ /orcid.org/0000-0002-7215-4178 M Passoni https:/ /orcid.org/0000-0002-7844-3691