Novel approach to TNSA enhancement using multi-layered targets—a numerical study

In the context of ion acceleration driven by ultra-high contrast lasers using thin foils, there is a clear trend towards increasing ion energy when the target thickness is reduced. However when the target is too thin and the prepulse strength is not negligible, this trend is reversed due to degradation of the target mainly caused by prepulse-induced shocks, among other effects (thermal plasma expansion, early onset of transparency, etc). In this paper, we propose and motivate the use of multi-layered targets for the purpose of enhancing the target normal sheath acceleration mechanism by means of attenuating the shock waves inside the target. It is demonstrated through hydrodynamic simulations that multi-layered targets, composed of alternating layers of plastic and gold, can significantly delay the time of shock wave breakout, reducing the shock energy that breaks out of the target and shortening the plasma scale-length. This approach paves the way for enhanced laser-driven ion acceleration using thinner targets even for relatively low contrast lasers.


Introduction
The target normal sheath acceleration (TNSA) mechanism has been the workhorse of laser-driven ion acceleration experiments [1][2][3][4][5] over the past decades. The experimental prevalence of this mechanism lies in the fact that it exists over a wide range of target-laser parameters and does not impose strict requirements on target surface density, laser intensity or contrast. For example, TNSA can be observed * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. even when shooting relatively thick solid targets (ranging from hundreds of nanometers up to tens of micrometers) with sub-relativistic intensities (I ∼ 10 16 W cm −2 ) and low contrast. Recent advances in the enhanced TNSA regime worldwide in facilities such as Draco-PW at HZDR or J-KAREN-P at Kansai [6] have shown that there is still significant potential for further optimization of the TNSA mechanism.
In contrast to TNSA, radiation pressure acceleration (RPA) [7][8][9][10][11][12] is an innovative mechanism requiring more intense lasers with better contrast. More specifically, light sail RPA (LS-RPA) can be driven by circularly-polarized pulses incident on ultra-thin (tens of nanometers) targets [13]. This is a promising mechanism because it is able to produce high-energy quasi mono-energetic ion beams, while recent advances in laser technology have allowed its experimental study. Nevertheless, the TNSA mechanism still remains relevant and interesting today because of its robustness, ubiquity and the ability to exist even in hybrid form [14,15].
In the quest for enhancing TNSA ion energies, researchers have tried using lasers with ever increasing peak intensities (currently exceeding I = 10 21 W cm −2 ) and reducing target thickness [16]. It has been documented that ion cutoff energy is a function of many different parameters and is not well described by the intensity alone [17][18][19]. Scaling laws have been discovered for intensity, target thickness, focal spot size, pulse duration and pulse energy [18,20]. It has been found that, in general, ion energy grows with increasing pulse energy and with decreasing pulse duration, focal spot size and target thickness. However, there is a limitation on how thin a target can be before it is destroyed prematurely by prepulses or the main laser pedestal [21,22]. Currently, there is ongoing effort in different directions (such as plasma mirrors [23][24][25] and plasma shutters [26][27][28]) aiming to improve laser contrast. It is worth noting, however, that improving contrast usually comes at the cost of reduced pulse energy. In the case of double plasma mirrors, routinely used in experiments on ion acceleration from ultra-thin targets, transmission of 50% was measured in optimum configuration [29]. On the other hand, the influence of prepulses can also be beneficial for TNSA through the formation of preplasma [17]. The interaction of the main laser pulse with preplasma has been found to enhance laser energy absorption and improve fast electron heating on the front side [30,31], which results in stronger accelerating fields on the rear side. Therefore, it makes sense to design relatively thin TNSA targets that can withstand some level of prepulses while maintaining their structural integrity for the main pulse.
The idea presented in this paper involves the use of multilayered targets to attenuate shock waves inside the target. It is worth noting that some work on the topic of multi-layered targets irradiated by a laser pulse has been published previously [32,33]. However, these publications explore different aspects and benefits of multi-layered targets, such as ion heating dynamics and enhanced laser energy absorption on the front side. The new target scheme proposed here could mitigate the prepulse issues associated with the use of low contrast lasers, allowing them to compete very cost-effectively with high contrast lasers. Moreover, this approach can circumvent the need to use double plasma mirrors, thus avoiding substantial loss of laser pulse energy [29].

Motivation
Laser prepulses with intensity above the laser-induced breakdown threshold (varies depending on material and duration of pulse; see e.g. [35,36]) have two main effects on solid targets. Firstly, they can drive the transition from an initial solid state to a plasma state and the subsequent target preexpansion. Secondly, such prepulses can also launch shock waves into the target interior. These high pressure, high density fronts travel with speed greater than the speed of sound in the material. If the shock is strong enough when it eventually reaches the target rear, it can break out into vacuum causing a rear side plasma. This process is known as shock-wave breakout (SWB) [37] and it has been shown to negatively affect ion acceleration from the rear side [38]. In fact, numerous experimental studies have examined TNSA in the presence of rear-side plasma by irradiating the back of the target using a lower intensity pulse [39][40][41][42]. The results of these studies show that a longer plasma scale length on the rear side results in lower ion cutoff energies. To demonstrate the detrimental effect of SWB and the subsequent formation of a plasma gradient, a series of 2D particle-in-cell (PIC) simulations were ran using the PIC code Smilei [34,43]. The outputs of these simulations are shown in figure 1.
A main laser pulse with λ = 800 nm and peak intensity of I = 10 21 W cm −2 was incident normally on a target with preplasma in front of a dense thin foil. For simplicity, a plasma composed of electrons and protons was used. The scale-length of the preplasma density (30 n cr × exp[x/L] for −40 µm < x ⩽ 0 µm) was L = 7 µm, until the overdense part of the target with constant density (plateau) located at 0 µm < x < s was reached, where s is the plateau thickness. A preplasma of the order of tens of micrometers is typical for nanosecond amplified spontaneous emission pedestal [44]. In order to investigate the effect of rear-side plasma (n = 30 n cr × exp[−(x − s)/l]), three cases were studied: firstly the case of a step-like profile (l = 0 µm) and then two short scale-lengths (l = 1 µm and l = 2 µm), where l is the rear-side plasma scale-length [34,45,46].
It is apparent that even a short plasma on the rear-side has a detrimental effect on acceleration by the TNSA mechanism, with the proton cutoff energy decreasing for longer rear scale-lengths. This difference in energy can be attributed to the fact that the plasma on the rear hinders the accelerating field and decreases its maximum value. On the other hand, a thinner target appears to significantly enhance the proton energy cutoff. The simplest explanation of this trend correlates the higher proton cutoff energies obtained from thin targets to the higher hot electron density created, due to the reduced dispersion of hot electrons drifting through the bulk [47]. The effect of hot electron re-circulation should not be so important in this case due to the presence of the front-side preplasma instead of vacuum [48].
In light of what has been discussed above, it is reasonable to attempt to construct a target which is as thin as possible while shielding the rear-side from the shock imparted by the laser on the front side. As a starting point of this work, this paper presents a set of hydrodynamic simulations performed to demonstrate the role of SWB in the final ion acceleration by TNSA. These hydrodynamic simulations are not intended to correspond precisely to any real experimental results, but are rather used as a means to investigate the potential of multilayered targets to enhance TNSA by suppressing the SWB.

Physical background
The idea of constructing a target made up of multiple layers is based on the knowledge that the shock wave energy is partially reflected at an interface between a low impedance and a high Figure 1. Proton spectra from 2D particle-in-cell simulations of hydrogen plasma, shown at 500 fs after the start of the simulation. The initial plasma profile is irradiated with a 30 fs laser pulse of peak intensity 10 21 W cm −2 . Inset plot (top-right): cross-section of initial electron density profile. The preplasma scale-length is the same for all cases (7 µm), while the target thickness (s) and the rear breakout plasma scale-length (l) are changed [34].
impedance material. The formula for the shock impedance is given by the expression Z s = ρ 0 u s = ρ 0 (c 0 + αu p ), where u s is the shock speed, ρ 0 is the initial density, c 0 is the speed of sound, u p is the particle speed and α is a material constant [49]. In the weak shock limit, the impedance can be approximated by the expression Z s ≃ ρ 0 c 0 . In general, an interface between a low density, low bulk modulus material and a high density, high bulk modulus material is required in order to maximize reflection efficiency. Gold and tungsten are both good candidates for high density materials with experimentallydetermined impedances (in the weak shock limit) of 58.0 and 77.5 GPa (km s −1 ) −1 , respectively. In contrast, an acrylic material with lower density, like Plexiglas, has impedance value 3.1 GPa (km s −1 ) −1 [50].
The initial shock speed depends on many factors, including material properties, laser parameters and absorption efficiency and is, in general, dependent on the strength of the shock imparted on the material. The shock speed is related to shock pressure by the relation [38,51] where P s is the shock pressure and ρ 0 is the initial density of the material. In principle this relation is strictly valid for an ideal gas. Nevertheless, in the high pressure range (Megabar), all materials approach the perfect gas state. Finally, the shock pressure generated by prepulse ablation scales as [52] P s (M bar) ∝ ( I 10 14 where the prepulse intensity I is measured in W cm −2 , the laser wavelength λ in µm, and A and Z are the atomic weight and the atomic number of the material. This equation assumes adiabatic target compression. By introducing shock reflections inside the target, it is theoretically possible to reduce the shock energy and pressure (and thus the shock speed) in subsequent layers. This potentially offers a two-fold benefit: firstly the SWB can be delayed and secondly the shock reaching the target rear surface will be weaker and slower. Ideally the shock will not be able to break out at all. Therefore, a target with enough layers should be able to attenuate the shock better, delay the SWB time and reduce the breakout strength and energy (i.e. either no breakout or shorter plasma scale).

Hydrodynamics and methodology
In order to investigate the potential benefits of multi-layered targets in the context of TNSA, a number of 2D hydrodynamic simulations were designed and performed using the FLASH code [53][54][55]. The FLASH code [55] is a high performance computing, multi-physics application code. Among many other modules, it includes physics solvers for hydrodynamics and laser energy deposition. FLASH is an Eulerian code with adaptive mesh refinement (AMR) grid implementation. The laser energy deposition is based on geometric optics with inverse Bremsstrahlung (IB). Moreover, FLASH supports any multi-material equation-of-state (EOS) provided in tabulated form as an additional file. Therefore, this is a capable code that is well-suited for studying the effect of laser energy deposition on solid targets in the picosecond and nanosecond timescales.
For this investigation, the Frankfurt equation-of-state (FEOS) computer code package is used for EOS table generation [56,57]. The FEOS package, provides all the The laser pulse chosen for this study has the following parameters: 50 µJ, 1.5 ps full width at half maximum (FWHM) Gaussian pulse focused down to 3 µm FWHM on target at normal incidence, with intensity I ≈ 10 14 W cm −2 . These parameters do not correspond to any specific laser but rather try to simulate a generic picosecond prepulse of a low-contrast laser. The laser energy is partially absorbed at the critical density surface using a constant absorption coefficient of 20%, while the rest is reflected back. This is necessary since the IB model implemented in FLASH is not suitable for short picosecond pulses and steep density gradients [59]. For heat conductivity, the Lee-More model is used [60].
The computational domain was a square with side-length 10 µm (the computational domain is larger than the target on the front side to avoid any boundary effects; the y-boundary of the simulation was placed at −1 µm). The maximum level of block refinement for AMR was set to 4, with a grid of 32×32 computational cells per sub-block (regardless of subblock size). The maximum spatial resolution of the domain was 8 nm in the y-direction (laser propagation direction) and 40 nm in the x-direction. The premature hydrodynamic expansion of solid material is prevented by artificial inner boundaries, which are successively released when material temperature is increased due to laser absorption or heat conduction.
The hydrodynamic simulations were designed in order to compare the effect of laser-induced shocks in different targets. Firstly, the propagation of shock waves was studied in various single material targets (plastic) and then this was expanded to multi-layered targets. The characteristics of the selected targets are listed in table 1. Plastic was chosen because of its low density. The plastic used had equal parts carbon and hydrogen. Aluminum and gold were also tested, with gold being used in the multi-layered versions of the targets due to its high density.
The methodology of comparing different targets for their resistance to shock wave propagation and SWB is not a trivial one. The process of breakout is a complicated physical process to analyze in detail because there are many factors involved. Assessing whether a breakout should happen or not is difficult and, thus, it is challenging to quantify a definitive threshold for breakout. However, a more heuristic approach can be constructed in order to assess the strength of breakout on the rear target side, based on the concept of SWB energy. This is a derived quantity calculated by summing the total energy in all cells with target material behind the initial position of the target rear surface. This quantity has value zero at the start of each simulation before SWB. Eventually, the rear surface is disrupted by the shock wave, target material expands behind the original target rear and the SWB energy increases. Consequently, SWB energy is a quantity which should (to some extent) reflect the strength of the shock reaching the rear side. This is used extensively in the following section. Figures 2 and 3 illustrate the simulation setup as well as the density evolution of both a single-material target and a multilayered target. In all cases, the target front side (laser side) is located at y = 0 µm and the rear side at y = 2 µm. The laser maximum intensity impacts the target 2.5 ps after the start of the simulation. Figure 2 shows the shock propagation inside a plastic target and the subsequent breakout. Note how the shock wave also spreads in the x-direction inside the target. Figure 3 shows the shock propagation inside a multi-layered target composed of four alternating layers of plastic and gold. The top panel shows the shock wave reflection in plastic and   plasma. The free-surface velocity, u fs , can be approximated using the particle flow velocity behind the incident shock as u fs = 2u p [49,61]. Therefore it is advantageous to have a high atomic mass material as the final layer on the rear surface because the particle velocity will naturally be lower.

Results and discussion
The first set of simulations was carried out using plastic targets of various thicknesses, with the objective to understand the shock wave propagation and SWB in a pure material. Figure 4 summarises the outcome of these simulations. From the inset plot, it is clear that the shock speed decreases with propagation deeper into the target bulk. This can be explained by the fact that, as the shock wave propagates and expands in radius, the energy spreads inside the target in all dimensions. Moreover, heat is conducted away in the target interior. Therefore the shock pressure drops whichaccording to equation (1)-leads to decrease in the shock speed. The main SWB energy plot in figure 4 also illustrates how the shock spreads in the transverse direction. SWB energy is, naturally, a measure of the shock energy moving in the longitudinal (target normal) direction. Therefore, the observation that this quantity is significantly reduced for thicker targets suggests that the shock front spreads more and a larger portion of the shock energy remains inside the target, moving in a direction along the target surface rather than perpendicular to it. Figure 5 displays the quantity of SWB energy as a function of time for various pure and multi-layered targets of thickness 2 µm. It is estimated from the simulations that only roughly 10% of the total energy is internal energy. In fact, most of the energy is kinetic, with internal energy being approximately constant in time. The intersection of the curves with the time axis marks the SWB time for each target type. Moreover, the initial slope of these curves is related to the velocity and energy of the breakout. As demonstrated by the breakout times for plastic, aluminum and gold targets of thickness 2 µm (table 1 and figure 5), shock waves propagate slower in materials with high density and high atomic mass. This is consistent with the discussion on shock impedance (section 3). In fact, the shock velocities in plastic, aluminum and gold scale with ρ 0 as expected from equation (1) (P s is much less dependent on materialequation (2)). Moreover, heavier materials attenuate more of the shock energy during propagation since SWB energy is lowest for gold.
By introducing shock reflections from multiple layers, the effectiveness of targets at weakening the SWB is improved. This enhancement is clearly visible in figure 5, where multilayered targets significantly outperform single material targets both in terms of SWB energy reduction and SWB delay. The only exception to this is target CHAUx1 = , which has an earlier SWB time than AU2. This is partially expected because the shock wave travels fast through the first half of the target. However, the shock reflection on the interface still provides a significant reduction in SWB energy compared to the AU2 target. Thus, by cleverly designing multi-layered targets (both in terms of number of layers/interfaces and proportion of low-Z s to high-Z s material) it is possible to achieve both: (1) SWB delay and (2) weaker SWB. These effects are unmistakably demonstrated by the rest of the multi-layered targets in figure 5.
Finally, another important parameter for assessing the impact of the shock wave on the target is the extent of the breakout plasma on the rear. According to the PIC simulations in figure 1, targets with shorter rear plasma scale-length are expected to perform significantly better in accelerating ions by the TNSA mechanism. Figure 6 shows the 1D SWB plasma density cross-section on the rear side 100 ps after SWB for the same targets considered in figure 5. Based on these profiles, it is evident that multi-layered targets have definite potential to enhance TNSA acceleration in the case of lower-contrast lasers.
The question of finding the optimal target and layer thickness, the ideal low-Z s to high-Z s ratio in the target and the best layer distribution to delay and weaken SWB is more complex. The optimal design may vary depending on the specific laser temporal characteristics but the key ideas remain the same: thin µm-level targets, low ratio low-Z s :high-Z s material, high impedance mismatch with as many layers as possible. However, these positive effects seem to be diminishing as the number of layers is increased and the thickness of layers is decreased. The hypothesis is that going down to very low thickness layers can degrade the ability of the structure to shield the rear surface, which leads to some kind of performance ceiling. Therefore, there is likely some minimum layer thickness (minimum low-Z s :high-Z s ratio, maximum number of layers) that still retains all the benefits without compromising the target performance.

Conclusions and perspectives
In the context of laser-driven ion acceleration, multi-layered targets have been used for a long time in order to increase the energy absorption and the conversion efficiency of the laser energy into plasma and fast particle kinetic energy [62,63]. The new idea presented in this paper is to combine the properties of multiple layers in order to minimize the impact of the laser-induced shock on the rear side and, thus, enhance the efficiency of the TNSA mechanism from the rear surface. This numerical hydrodynamic study demonstrates that it is possible to both delay SWB and reduce the scale-length of breakout plasma by alternating layers of low and high impedance in order to reflect part of the shock wave and slow it down.
The great advantage of such an approach is that the targets can be easily manufactured using a range of materials and densities. Physical vapor deposition (PVD) describes a variety of vacuum deposition methods which can be used to produce thin films and coatings on substrates [64,65]. Some examples of PVD methods include thermal evaporation [66], pulsed-laser evaporation [67] and magnetron sputtering [68]. A well-designed experiment should be able to confidently prove the effectiveness of multi-layered targets for TNSA. Such an experiment would require a high-power laser system with well-characterized temporal profile before the main pulse. Then a thickness scan of various materials can be performed to determine the minimum thickness before breakout occurs. Proton spectra from TNSA can be compared for thin plastic targets and also multi-layered targets of the same thickness (with the first layer being the same type of plastic). If the experiment is designed optimally (sufficiently thin targets to allow SWB given the laser parameters), it should be possible to observe the enhancement of proton energies in the multilayered case.
In summary, this theoretical work paves the way for optimizing the TNSA mechanism in foils with thickness ≳1µm, while reducing the need for contrast cleaning techniques. The authors strongly believe that the use of multi-layered targets resistant to SWB is an idea worth further investigation and could significantly improve the ion acceleration performance of relatively low-contrast lasers, allowing them to compete cost-effectively with the highest contrast lasers.

Data availability statement
The data cannot be made publicly available upon publication because they are not available in a format that is sufficiently accessible or reusable by other researchers. The data that support the findings of this study are available upon reasonable request from the authors.