Enhancement of laser-driven ion acceleration in non-periodic nanostructured targets

Using particle-in-cell simulations, we demonstrate an improvement of the target normal sheath acceleration (TNSA) of protons in non-periodically nanostructured targets with a thickness of a few microns. Compared to standard flat foils, an increase in the proton cutoff energy by up to a factor of two is observed in foils coated with nanocones or perforated with nanoholes. The latter nano-perforated foils yield the highest enhancement, which we show to be robust over a broad range of foil thicknesses and hole diameters. The improvement of TNSA performance results from more efficient hot-electron generation, caused by a more complex laser-electron interaction geometry and increased effective interaction area and duration. We show that TNSA is optimized for a nanohole distribution of relatively low areal density and that is not required to be periodic, thus relaxing the manufacturing constraints.


I. INTRODUCTION
Laser-driven ion acceleration has become a well established technique to produce compact, high-energy ion beams, owing to the ultra-strong accelerating fields that can be achieved at the surfaces of solid targets [1,2]. Such ion sources show great potential for a number of applications ranging from radiography [3] to nuclear photonics [4] and proton therapy [5]. However, even though proton energies close to 100 MeV have been demonstrated in recent experiments using petawatt-class laser facilities [6,7], the few tens of MeV energies that are routinely attained using multi-terawatt-class laser systems are insufficient for many of the foreseen applications, which therefore limit the applicability of laser-driven ion sources. This spurs the development of novel schemes yielding significantly increased proton energies.
The most robust, and extensively investigated, acceleration scheme is the so-called target-normal-sheath acceleration (TNSA) [8,9], whereby surface ions are driven outwards by the charge separation field set up by the laser-accelerated relativistic electrons escaping into vacuum. Because of their large charge-to-mass ratio, the protons that are naturally present due to hydrogen containing contaminants at the target surfaces respond the fastest to the electric sheath field, and reach the highest velocities. Their final energy spectrum has typically the form of a decreasing exponential with a sharp high-energy cutoff.
Different strategies have been explored in recent years to increase the proton cutoff energies resulting from TNSA. With micrometric foil targets, this requires enhancing the fast electron generation at the target front * illia-thiele@web.de † julien.ferri@chalmers.se side. One option is to manipulate the laser temporal profile so as to create a preplasma with an optimal scale length [10,11], or to induce an optimal electromagnetic interference pattern [12]. Another option is to modify the target properties. Reduction of the target thickness [13,14] or transverse size [15] results in higher proton energies and numbers. An alternative, which is addressed in the present paper, is to employ nano-(or micro-)structured targets. While twofold increase in proton energy has been reported from foils with periodic surface structures [16,17], little attention has been paid so far to the potential of non-periodic structures [18]. However, relaxing the constraint on the periodicity would enable simpler and more robust target fabrication methods [19,20] as is required to bring laser-driven proton sources closer to applications. In this paper, we investigate by means of particle-incell (PIC) simulations, the potential of non-periodically structured targets to enhance TNSA. Two target types are considered, consisting of a flat foil either coated on the front surface with randomly positioned nanocones ("nanocone targets") or perforated by nanoholes ("nanohole targets"). In Sec. II, we describe the physical and numerical setup and in Sec. III we compare the results obtained with the structured targets to the flat-foil case and investigate the origin of proton energy enhancement. In Sec. IV, a parametric study of the nanohole targets is presented. Finally, in Sec. V we summarize our results.

II. PHYSICAL AND NUMERICAL SETUPS
We will investigate how the aforementioned two types of structured foils behave with respect to proton acceleration by means of 2D PIC simulations with the code SMILEI [21]. The considered structures are visualized in ness d = 100 nm. It is composed of gold atoms, assumed to be 11-times ionized, with an ion number density of 5.85 × 10 22 cm −3 . Simulations with different ionization states of gold atoms (up to 29-times ionized) were also performed but did not result in significant differences in proton spectra. A 10 nm thin proton-electron-plasma layer with a number density of 1.74 × 10 23 cm −3 is added on the foil surfaces to model the hydrogen contaminants.
The nanocone target, sketched in Fig. 1(b), is composed of the above-mentioned flat foil coated with a distribution of cones, each having an opening angle of 44 • and a variable base size of b (specific values will be set below). The nanohole target, displayed in Fig. 1(c), consists of the reference foil pierced by holes of the same width and location as the above nanocones. The surfaces of both types of structured targets are coated with a proton layer.
For the initialization of the nanocones' positions, we choose the following model: Given the position y 1 of the first nanocone, the average distance D = 2.24b and the distance spread parameter s = D/2, the position of the ith nanocone is set iteratively to y i = y i−1 +D+(u−1/2)s as long as y i < y max , where y max is the maximal position and u is a uniformly distributed random number taking values from 0 to 1. This leads to an average cone coverage density of ρ = b/D = 44%. The targets are located at x = 20 µm.
The p-polarized laser pulse has a wavelength of λ = 0.8 µm and a maximum intensity of 5 × 10 19 W/cm 2 , corresponding to normalized vector potential a 0 = 4.8 and a peak electric field of 194 GV/cm. It has Gaussian space and time profiles with a focal spot of 5 µm FWHM and a duration of 38 fs FWHM. It is incident on the target from the lower left-hand side, at 45 • from the surface normal. The peak intensity of the laser pulse reaches the foil after 175 fs in the simulation. In the following, we will set this as the time reference t = 0.
The numerical discretization of the simulations is δx = δy = 5 nm and δt = 11.7 as. We use 100 macro-particles per cell and per species in the bulk plasma, while the surface proton-electron layers are represented by 1000 macro-particles per cell and per species.

III. ENHANCEMENT OF ELECTRON HEATING AND ION ACCELERATION
Periodic cone structures have been shown to enhance the proton acceleration due to a modification of the electron trajectories maximizing the laser absorption. The enhancement of proton acceleration has already been discussed for periodic nanohole targets [22], periodic nanobrush targets [23] and grating surfaces [24,25]. Thus, it is interesting to investigate whether relaxing the restriction of periodicity would impact the acceleration process. Here, we consider a non-periodic arrangement of the structures. The cone and hole base size in this section is fixed to be b = 300 nm.
The strength of the rear-side sheath field that underpins TNSA, and which therefore determines the efficiency of the latter, is controlled by the energy density of the laser-generated hot electrons [1,2]. Figure 2(a) plots the energy spectrum of the electrons located in the vacuum region behind the rear side of the target, recorded at t = 175 fs. Those electrons mainly account for the generation of the sheath electric field in the early stages of TNSA, when the approximation of the 1D plasma expansion holds. Compared with the flat foil, both structured targets lead to a significantly increased amount of relativistic electrons above 7 MeV, with the nanohole target yielding the largest enhancement -by about an order of magnitude.
A similar behavior is found for the rear-side proton energy spectra, as plotted in Fig. 2(b) at t = 455 fs. Both nanocone and nanohole targets give rise to much enhanced TNSA: the best performance is observed using nanoholes, with an almost doubled proton cutoff energy compared to that from the flat foil (∼ 30 MeV protons vs ∼ 17 MeV). Figure 3 demonstrates that the electron and proton spectra are almost the same for periodic and non-periodic targets. It is thus to be expected that the individual structuring units, rather than their periodic arrangement (possibly leading to the excitation of surface plasma waves), are responsible for the enhancement. This is a favorable result from an experimental perspective since it reduces the target manufacturing constraints. Moreover, Figs. 3(a,b) present the electron and rearproton energy spectra obtained for a flat foil (blue solid line) of reduced thickness (d = 55 nm) such that it contains the same total amount of matter as the d = 100 nm nanohole targets, whether periodic or not. This thinner flat foil produce particle energy spectra very similar to the d = 100 nm foil; hence, the enhanced performance of the nanohole targets cannot be ascribed to the direct effect of their reduced volume (or area in 2D) on the electron kinetic energy density (which would naturally increase assuming the same amount of laser energy is converted into hot electrons), but rather results from a strongly modified hot-electron dynamics.
In order to gain insight on the electron energization process in the nanostructured target, we record the maximum Lorentz factor γ reached by each electron during the simulation, and plot its locally averaged value as a function of the initial electron position [ Fig. 4(a)-(c)]. In the case of the flat foil [ Fig. 4(a)], the resulting map shows that, as expected, most of the accelerated electrons originate from a ∼ 25 nm-thick surface layer at the directly irradiated front-side of the target (with mean energies ∼ 6 MeV being reached), and that rearside electrons undergo negligible acceleration. In the case of the nanocone [ Fig. 4 Fig. 4(c)] targets, part of the highest energy electrons is stemming from additional regions, namely the nanohole walls and the nanocone sides. The nanostructuring of the target surface therefore increases the effective interaction area, leading to a larger number of hot electrons. The mean energy (∼ 10 MeV) reached by these electrons is also significantly larger than in the flat foil. These effects are supported by the local enhancement of the electrostatic field which appears when using nanostructured targets. The electrostatic field is indeed strongly enhanced at the corners of the nanoholes and at the tips of the nanocones, which correspond to the surfaces from which the most energetic electrons arise. Note that the value of the elec-trostatic field in these regions becomes of the same order as the laser field [ Fig. 4(d)-(f)], which is also favorable for electron acceleration [26]. Interestingly, when the target is nanostructured, the change of the laser field pattern on the front side of the target indicates a decrease in the laser reflection compared with the flat foil. In the case of the nanohole target, laser fields are propagating in the nanoholes, leading to partial transmission.

(b)] and nanohole [
This general behavior can be complemented by examining individual electron trajectories. We focus on those high-energy electrons breaking through the target rear side and therefore contributing to the accelerating sheath field. For this reason, Figure 5 plots the trajectories of a sample of the most energetic electrons in both the flat and the nanohole targets -with a lower energy cutoff for the selection of 12 MeV in the nanohole target [ Fig. 5(a)] and 5 MeV in the flat target case [ Fig. 5(b)]. Only those electrons that are located at a longitudinal position x ≥ 48 µm at t = 175 fs are selected. The color of each trajectory is indexed on the rate of change of electron energy in the local electromagnetic fields. It can be seen that, on average, higher values are attained in the nanohole target. While the selected electrons mainly exhibit acceleration in the front-and rear-side vacuum regions, sizable energy transfer is also seen within the cavities, which is consistent with the laser partially transmitted through the nanoholes [see Fig. 4(e)].
The sub-micron dense regions that make up the nanohole targets effectively behave as mass-limited targets [15,27], which lead to efficient electrostatic confinement of the hot electrons. This is evidenced by the single particle trajectories displayed in the insets of Fig. 5: the nanohole target causes the electron to recirculate in both longitudinal (across the front and rear sides of the target) and transverse (across the nanohole walls) directions. A favorable consequence of this is a longer effective laserelectron interaction time. Also, the laser-electron interaction occurs under various geometrical conditions, and so with increased degrees of freedom. This results in a more complete exploration of the phase space, which ultimately allows the electrons to be accelerated to higher energies. Finally, being prevented from leaving the laserirradiated region, the hot electrons are able to sustain a strong sheath field over longer times. The confinement of the hot electrons leads to a reduced transverse extent of the sheath field, and therefore of the expanding proton cloud. This can be seen in Figure 6, which shows the boundary of the proton cloud as resulting from the three target types: the nanohole target gives rise to a more narrow proton distribution.

IV. PARAMETRIC SCAN FOR THE NANOHOLE TARGETS
We now perform a parametric scan where we vary the foil thickness d and hole diameter b. Henceforth, we consider only the cases where the hole diameter is at least as large as the foil thickness, i.e., b ≥ d. Figure 7(a) presents the spectra of the rear-side electrons for a foil thickness from 100 nm to 600 nm and various hole diameters. One can see that the electrons from the flat foil (gray line) reach lower energies than electrons from perforated foils, whatever the hole diameter. Especially above 5 MeV, perforated targets produce more high-energy electrons. The enhancement is very similar for hole sizes from 100 nm to 600 nm.
While we are mostly interested in the protons accelerated from the target backside, we also plot for completeness the energy spectra of the protons originating from the front side. Figures 7(b,c) present the corresponding front (b) and rear (c) proton spectra. In both cases, the structuring significantly enhances the proton-cutoff energies. While a 100 nm hole size already enhances the proton-cutoff energy by about 40-50%, the largest enhancements are reached for 300 to 600 nm wide holes.
When increasing the gold foil thickness, the trends remain the same [see ]: The presence of the holes increases the amount of high-energy electrons and boosts the proton energy by about a factor of 2. The thin-ner the gold foil, the higher the proton energies. However, while this nanohole structuring can be used for all target thicknesses, the relative improvement is slightly more pronounced for thicker foils.
The PIC simulation results suggest that as long as the parameters are in the range leading to enhancement as described above, they can be freely chosen as it is suitable from the point of view of other experimental constraints, e.g. ease of handling and fabrication.
Note that our simulations consider sharp-gradient targets, meaning that they do not address the possible influence of finite preplasmas caused by laser prepulses. The generation of a significant preplasma would modify the picture: the holes might be filled with electrons before the arrival of the main pulse, thus suppressing the benefit of the target structuring. Larger hole diameters may then be preferable in actual experimental conditions.
The areal density of the holes can be expected to play a role in enhancing TNSA. Figure 8 displays the energy spectra of the rear-side electrons and protons obtained from 100 nm-thick nanohole targets of nanohole density varying in the range 0 ≤ ρ ≤ 67%. An optimum electron heating and proton acceleration is reached when almost half of the surface area is covered by the holes. However, already for a density of about 20% a notable increase in the cutoff energy is reached.
The optimum is likely formed by two counter-acting effects. On the one hand, when increasing the hole density, the mechanisms described in Sec. III further develop: more electrons can get an energy boost due to the increase in the interaction surface and recirculation. On the other hand, raising the hole density causes a decrease in the effective target volume, eventually resulting in higher laser transmission (not shown here), ultimately reducing the laser-target coupling. To mitigate the latter effect, one could use thicker foils instead of the considered 100 nm thin foils. However, this is limited by the fact that the proton cut-off energy tends to be reduced for the nanohole target -in the same manner as for the standard flat foil target -when the nanohole target thickness is increased.

V. SUMMARY
In summary, we investigate laser-driven proton acceleration using non-periodic structured foil targets. With 2D PIC simulations we demonstrate significant proton cut-  off energy increase for various structured targets, with a focus on nanohole targets. We show that the enhancement originates from a modification of the interaction surface between the laser and the target, which allows a higher number of electrons to be accelerated. Besides, the hot electrons can interact with the laser pulse both longitudinally on the front side of the target and trans-versely in the nanoholes, enabling them to fully explore the phase-space, and be accelerated to higher energies. A large parameter space in terms of hole diameter, foil thickness and hole areal density yielding significant improvement of the accelerated proton spectra is identified. We show that the production of structured targets for improved ion acceleration can be relaxed to non-periodic structures with a relatively low density of structuring units.