Effect of the rising edge of ultrashort laser pulse on the target normal sheath acceleration of ions

Laser-driven ion acceleration is theoretically/numerically mostly studied with the assumption of an idealised main ultrashort pulse of the Gaussian temporal shape, where nanosecond/multi-picosecond pedestals and short prepulses preceding the main pulse can be incorporated in the form of modifications in the initial density profile of irradiated ionised targets. This paper shows that the relatively slowly rising edge (also called picosecond ramp) of the main ultrashort pulse, usually neglected in previous studies, can substantially change the efficiency of the target normal sheath acceleration of ions depending on the laser intensity. The rising edge can enhance ion acceleration at mildly relativistic laser intensities, but increases the divergence and reduces the cutoff energy of accelerated ions at highly relativistic intensities relevant to petawatt lasers.


Introduction
The most intense part of the ultrashort (femtosecond) laser pulse is accompanied by a nanosecond-amplified spontaneous emission pedestal, low-energy ultrashort prepulses, and (multi-)picosecond slope of the pulse rising slowly compared to an ideal Gaussian pulse (identified as a coherent pedestal) [1].Although the laser pulse contrast can be substantially enhanced (thus, the intensity of pedestal and prepulses can be substantially reduced below the ionisation threshold) [2][3][4][5][6], the relatively slowly rising edge of the laser pulse 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.
(also called picosecond ramp) is still observed [7][8][9] and can change the interactions between the initially solid target and the main part of the laser pulse (of Gaussian temporal shape).The effect of the picosecond ramp should be more essential with increasing pulse intensities (due to the increasing laser pulse power) when the rising edge of the pulse changes the density profile of the target (creates short scale-length preplasma or causes the pre-expansion of ultrathin foils or nanostructures).
The density profile of ionised targets strongly affects phenomena such as hot electron generation in targets irradiated by the laser pulse [10][11][12] or laser-driven ion acceleration.In the case of ion acceleration, which is the subject of this study, the effects of pulse pedestals or prepulses were broadly investigated in the past [13][14][15][16][17][18][19].Ion acceleration can be enhanced when the optimal scale length of the preplasma is formed on the laser-irradiated side of the target [15,16,19,20].In contrast, the acceleration can be substantially reduced or even destroyed as a shock wave launched by the rapid surface preheating and/or radiative heating from x-rays generated in the focus of a prepulse incident on the target front side induces a rear surface expansion of thin targets [13][14][15].
In recent years, with the increasing power of lasers and improved pulse contrast attenuating the effect of pedestals and prepulses on ion acceleration, several studies on the impact of the rising (leading) edge of ultrashort laser pulses have been published.These studies can be divided into several groups.One of the papers investigated the effects of a more realistic temporal profile based on an experiment with a thin target with nanoholes [7], the target proposed earlier in the simulations with an ideal Gaussian temporal shape of the laser pulse for enhanced ion acceleration [21,22].However, more realistic simulations in [7] have shown that the picosecond ramp of the laser pulse creates plasma that fills nanoholes before the arrival of the main ultrashort pulse.Therefore, no increase in ion energies for foils including nanoholes compared with regular foils was observed in the experiment when the foils were irradiated by a real ultrashort laser pulse.Other papers investigate ion acceleration in the relativistic induced transparency regime, which is reached earlier with the slowly rising edge of the laser pulse due to the foil pre-expansion compared with an ideal Gaussian pulse [23,24].The gradient of intensity increase in time can be even more reduced; thus, asymmetric laser pulse with a slow rise and a sharp fall in time is produced, which can lead to an enhanced ion acceleration as demonstrated in several studies [25][26][27].Picosecond ramps are also found to be responsible for the preferential acceleration of heavier carbon ions by radiation pressure acceleration compared to protons for optimum foil thickness [28] and advantageous for heavy ion acceleration in the target normal sheath acceleration (TNSA) regime [29] in the case of ultrashort (30-40 fs) pulses.
Motivated by the studies mentioned above, the effect of the rising edge of the ultrashort pulse on the widely used TNSA mechanism [30,31] is investigated here, depending on the laser pulse intensity.It is shown that the picosecond ramp is favourable for ion acceleration for relatively lower pulse intensities.However, the ramp reduces the efficiency of acceleration at relatively higher intensities (relevant to petawatt (PW) laser pulses).The rising edge also changes the angular distribution of accelerated ions.The divergence of ions is larger when the rising edge is introduced in the simulations and increases with laser intensity.

Simulation method and parameters
Since the length of a relatively highly intense picosecond ramp is limited to a few picoseconds, especially after using plasma mirrors (based on data from experiments [7,8]), it enables us to employ particle-in-cell (PIC) simulations for calculating its interactions with the target used for ion acceleration.However, such simulations are quite challenging due to the relatively large duration of the interactions between realistic laser pulses and ionised targets as well as the necessity of large spatial resolution due to the relatively low intensity at the beginning of the interactions of the rising pulse edge with the target.Moreover, collisions between plasma particles should be included in the simulations to assess their role, as they can be important for laser energy absorption at lower laser intensities [32], which further increases computational demands.Therefore, only 2D PIC simulations using the PIC code SMILEI [33] can be performed without 3D extension.Although the maximum energies of accelerated ions are usually overestimated in 2D PIC simulations compared with the 3D simulation (and experiment) [34][35][36][37], the qualitative comparison of two or more studied cases should lead to the same conclusions by using 2D PIC simulations and 3D simulations.Input files for all 2D simulations used for the production of data discussed in the paper are available at Zenodo repository [38].
In our numerical simulations, we assumed 1 µm thick plasma layer with an electron density of 200 n ec , where n ec is the plasma critical density for a laser wavelength equal to 1 µm.The plasma layer comprises electrons and light ions with mass m LI = 200 m e (where m e is the electron rest mass) and charge Z = 1.The main laser pulse length was set to 35 fs at full width at half maximum (FWHM) in the intensity profile.In this study of the influence of the picosecond ramp on ion acceleration, two cases of temporal profiles of laser pulses were assumed.The first case represents an ultrashort laser pulse with its picosecond ramp and it is also called the realistic pulse case in the following text.The time characteristics of the pulse are the same as in [7]: the whole pulse envelope is described by the sum of three Gaussian functions A i * exp (−t 2 /(2σ 2 i )) in the intensity of laser radiation, where A 1 = 0.5, σ 1 = 15 fs, A 2 = 0.5, σ 2 = 90 fs, A 3 = 0.0001, σ 3 = 350 fs.The second case represents a clean ultrashort pulse of the same energy as the whole pulse in the first case.It means that the whole pulse intensity envelope is described by A * exp (−t 2 /(2σ 2 )), where A = 1 and σ = 15 fs.This is also called the idealised pulse case in the following text.
Peak intensities of idealised laser pulses are in the range from 10 19 W cm −2 up to 5 × 10 21 W cm −2 .Note that the peak intensities of realistic pulses are reduced by a factor of 3.5 to keep the same total energy of the pulse in 2D simulations.This also means a reduction of the dimensionless amplitude a 0 by a factor of √ 3.5 in the realistic pulse case.Therefore, we also use the term maximum intensity in the following text and figure captions for simplicity, which is equal to the peak intensity of an idealised laser pulse.In all cases, the laser beam width was set to 3 µm at FWHM in the focus.The beam is incident on the target at an angle of 45 • or normal incidence.All the simulations model plasma irradiated by a laser pulse during 1500 fs, the peak intensity of the pulse enters into the simulation box at time 1000 fs and reach the target at 1120 fs.
The cell size was set to 10 × 10 nm 2 , and the number of cells in the simulation box to 8960 × 7680.In total, 400 electrons and 400 light ions per cell were initialised in the simulation.The total number of cells in the simulation reaches almost 7 × 10 7 and the total number of macroparticles exceeds 3 × 10 8 for each species.Collisions between electrons and electron-ion collisions were included in the simulations with a realistic pulse.In this case, collisions can be important during the interactions of the picosecond ramp with the target as the laser intensity is relatively low for a longer time.However, the simulations demonstrated that the effect of collisions is rather negligible in the later stages of interaction (when the peak intensity of the pulse interacts with the plasma); for example, the maximum difference in the cutoff energies of light ions is only a few per cent in simulations with switched-on/switchedoff collisions.Moreover, the simulations require approximately two times larger computational time with the collisions included.As a result, collisions were switched off in the simulations with idealised pulses, where their negligible effect was expected already from the beginning of the interaction.

Energy spectra of accelerated ions
The energy spectra of accelerated light ions are compared for realistic (case I) and idealised (case II) ultrashort pulses of the same energy in the intensity range from 10 19 W cm −2 to 5 × 10 21 W cm −2 for oblique (45 • ) and normal (0 • ) incidence of laser beam on target (figure 1).At the lower maximum intensities, simulations show higher cutoff energies of accelerated light ions in case I (up to intensity 10 20 W cm −2 ) and higher laser-to-ion energy conversion efficiency, that is a higher number of accelerated ions (up to intensity 10 21 W cm −2 ).In contrast, the cutoff energies of accelerated ions (and conversion efficiencies) are significantly reduced at maximum intensities exceeding 10 21 W cm −2 , relevant for PW lasers, in realistic case (I) compared with idealised case (II).The reduced acceleration efficiency can be explained by the target pre-expansion before the main pulse arrival, leading to reduced accelerating fields after the interaction with peak pulse intensity (see details below).In contrast, this target pre-expansion is favourable for the higher absorption of laser pulse energy on the front side of the target (in fact, it is enhanced absorption of laser pulse energy in short scale length preplasma [39]), thus, to better laser-to-ion conversion efficiency in the case with picosecond ramp.
At 10 19 W cm −2 and oblique incidence angles, the difference in the proton energy spectra is most pronounced.In contrast, the spectra are similar for realistic and idealised pulses at the normal incidence for such maximum intensity.It corresponds to a very low absorption of laser pulse energy in the target for normal incidence at non-relativistic intensities as there is no perpendicular component of the laser wave field to vacuum-plasma interface [11].Thus, the target pre-expansion does not occur in this case.With increasing intensities leading to the relativistic motion of electrons in the field of laser wave, the importance of ⃗ v × ⃗ B term in the Lorentz force leads to the absorption of laser pulse energy through ⃗ j × ⃗ B heating of electrons [40].At the maximum intensity of 5 × 10 21 W cm −2 , the observed cutoff energies for realistic pulses are closer to experimental values.However, 2D simulations usually overestimate cutoff energies [34][35][36][37], which can explain still higher values of cutoff energies compared with experiments using laser pulses of comparable parameters in the TNSA regime ( [41]).
One can also estimate the scaling of cutoff energies on maximum intensity based on the presented data.In case I (realistic pulse), the scaling well corresponds to the averaged scaling of the experimental data [42], that is the cutoff energy is proportional to ∼ I 0.5 .In case II (idealised pulse), the scaling is more favourable with cutoff energy proportional to ∼ I 0.65-0.7 .This case is closer to the linear scaling suggested for ultrashort pulse durations, short preplasma, and pulse intensity below a certain threshold at the same time [43][44][45].

Angular distribution of accelerated ions
Target pre-expansion due to picosecond ramps strongly affects the angular distribution of accelerated ions at high laser intensities (figure 2).In case I (realistic pulse), a broader angular distribution can be observed.In case II, ions with higher energies are more focused and deflected from the target normal direction when the laser beam is incident obliquely on the target (figure 2(a)).Deflection of the most energetic ions from the target normal direction can be seen for both cases (figures 2(c) and (d)) and corresponds to the experimental results reported in [45].This deflection does not occur at normal incidence.However, a broader angular distribution persists for realistic pulses, with a double-peak structure.This 2D structure can also be seen in some experimental results as a ring structure in 3D (e.g., figure 3 in [46]).It can be explained by the shape of the ion front on the rear side, which is curved by the target preexpansion due to the rising edge of the ultrashort pulse.The shape of the ion front at the moment when the peak intensity of the laser pulse interacts with targets in both cases is shown in figure 3. The flat surface is kept in the case of the idealised pulse, whereas a bell-shaped surface can be observed for the realistic pulse case.Note that the double-peak structure of the angular distribution discussed above can be seen from higher pulse intensities from 10 21 W cm −2 .Since the curved shape of the ion front is more pronounced with increasing pulse intensity, it also leads to an increasing divergence of accelerated ions.
One can also see irregular structure in the angular distribution with multiple minor peaks.This structure can be attributed to instabilities during hot electron propagation through the target leading to electron filamentation (e.g., the Weibel instability) [47][48][49] and separate bunches of hot electrons that propagate through the target [11].These electron filaments and separate bunches result in local structures in the accelerating fields for ions on the rear side of the target, as can be observed in figures 5(a) and (b).

Development of accelerating fields for energetic ions
Figure 4 shows the temporal development of accelerating (longitudinal) electric fields in the position of accelerated ions.Randomly selected ions reaching the highest final energies and about 70% of the cutoff energy have been tracked in the simulations.The creation of the accelerating fields is related to the    interaction of the laser pulse with the target.In case I (realistic pulse), the field that accelerates ions starts to rise earlier compared with case II (idealised pulse) due to the interaction of the target with the picosecond ramp of the ultrashort pulse that generates hot electrons.For realistic pulses, the accelerating field sharply rises up to a time moment approx.950 fs.Next, the field is almost constant for the ions with the highest final energies or decreases to a local minimum for less energetic ions at approximately 1050 fs.The decrease of the field can be explained by the position of less energetic ions, which are located behind the ion front (figure 5(a)) and the sheath accelerating field is partially shielded from them at this time moment.However, the region of a high accelerating field is broadened with the start of the interaction of the main part of the ultrashort pulse with the target (figure 5(c)).The main part of the ultrashort pulse generates a higher number of hot electrons with higher energies on the front side of the target, which propagates through the target, compared with the picosecond ramp.In contrast, the target is already pre-expanded and the density gradient on its rear side is reduced.Thus, the peak accelerating field reaching about 1170 fs is substantially reduced compared to the idealised case (peak amplitude 0.7 vs. 3 in dimensionless units).
One can observe a higher field strength at the position of the initial rear surface of the target (at x = 1 µm) than at the position of the high-energy ions in figures 5(a) and (c).Thus, ions are already moved away from the target when the longitudinal field at the rear surface peaks.This situation is similar to the analysis of ion acceleration by longer laser pulses (120 fs) in [43].
When the interaction of the maximum pulse intensity with the target is around 1130 fs, the ions with different final energies are clearly separated in figure 5(a) (realistic pulse case), whereas they are at the same distance from the initial position of the rear surface of the target (at x = 1 µm) in figure 5(b) (idealised pulse case).In the case of idealised laser pulses, there is a steep increase in the accelerating field experienced by tracked ions in the time interval between approximately 1060 fs and 1110 fs, followed by a short decrease with a local minimum at 1130 fs (figures 4(b) and (d)).The field then reaches a second local maximum at 1170 fs, which is more pronounced for tracked ions with the highest final energies.This behaviour can be explained by electron dynamics during laser-target interactions.When the interaction starts, the intensity of the laser field is steadily increasing and the first bunches of hot electrons propagate through the target, generating an accelerating field on its rear surface.When the maximum intensity of the laser pulse interacts with the target (around 1120 fs), the field created behind the target has an irregular structure (figure 5(b)) as the bunches of hot electrons propagate far from the target.The interacting laser field rapidly decreases very shortly after this time moment (after 20-30 fs) and a stable sheath field is created which accelerates the ions located there (figure 5(d)).This sheath field is gradually attenuated as the target expands, but it is not further violated by newly incoming bunches of hot electrons.The sheath field is peaked in a much thinner region compared with case I (figures 5(c) and (d)).All the tracked ions with different ion energies are located almost at the same distance from the rear side of the target, but they differ in their angular spread.The most energetic ions are located on the central axis of the sheath (at position |y| = 0), whereas the less energetic ions are also accelerated at larger |y|-axis positions where the sheath field is lower as can be seen in figure 5(d).

Conclusions
The rising edge (picosecond ramp) of an ultrashort laser pulse can substantially change the energy spectra of laseraccelerated ions and their angular distribution.This picosecond ramp can exert a positive effect on laser-driven ion acceleration at mildly relativistic peak laser intensities (up to 10 20 W cm −2 ) as it can enhance the absorption of laser pulse energy in the preplasma generated by the ramp.However, it reduces the acceleration efficiency for highly relativistic laser intensities (from 10 21 W cm −2 ), relevant for PW laser pulses, as the rear side of the target substantially expands during the interaction of the target front side with the rising edge of the pulse.This target pre-expansion before the interaction with the peak laser intensity leads to a broader accelerating sheath with a substantially lower maximum accelerating field and curved surface of the sheath, which results in reduced cutoff energy and broader angular distribution of high-energy ions compared with an idealised clean ultrashort pulse.The acceleration of ions also occurs for a longer time with the rising edge leading to a clear separation of the ions in the space depending on their energy when the peak laser intensity interacts with the target.These results demonstrate that the rising edge of ultrashort laser pulses substantially changes the dynamics of laser-driven ion acceleration and leads to reduced cutoff energies in the TNSA regime for PW class lasers.

Figure 2 .
Figure 2. Angular distribution of ions accelerated in the forward direction with kinetic energy > 5 MeV by (a) realistic/(b) idealised laser pulse with maximum intensity 10 21 W cm −2 incident on target at 45 • and normal incidence.Energy-angular distributions of ions accelerated in the forward direction by the same pulses ((c) realistic, (d) idealised) incident on target at 45 • .

Figure 3 .
Figure 3. Densities of light ions at the time moment when the peak intensity of (a) realistic/(b) idealised laser pulse interacts with the target (maximum intensity: 10 21 W cm −2 , incidence angle: 45 • ).The densities in the colorbar are normalized to the plasma critical density nec.Targets are irradiated from the left (bottom) side, initial position of the front side of the target is at x = 0.

Figure 4 .
Figure 4. Development of longitudinal electric fields seen by selected ions (with final kinetic energies specified in the figure legend) and accelerating them in the forward direction for maximum intensity 10 21 W cm −2 and normal incidence of the laser pulse on the target.(a), (b) for the realistic pulse; (c), (d) for the idealised pulse.

Figure 5 .
Figure 5. Longitudinal electric fields Ex and positions of selected ions accelerated by realistic ((a), (c))/idealised ((b), (d)) laser pulses for maximum intensity 10 21 W cm −2 and normal incidence of laser pulse on the target.The fields in the colorbar are in dimensionless units Exe/(meωc), where e is the elementary charge and ω is the laser angular frequency.The positions of ten tracked ions with the highest final energies are shown in black, and the positions of ten tracked ions with final kinetic energies reaching about 70% of the maximum energy are shown in grey.