100 MeV protons from nanostructured hemispherical target using PIC simulations

The improvement of laser-driven proton energy with the use of nano-structured hemispherical targets of 100 nm thickness over conventional flat foil has been reported in this work. The curvature of the target is found to result in focussed particle density at the center of the hemispherical target followed by emergence of energetic ions due to combined action of sheath electric field and ambipolar expansion. The presence of nano-rods on the curved hemispherical target further increases the laser energy absorption by the electrons, thus resulting in increase in the maximum proton energy. Use of hemispherical target embedded with nanorods is possibly reported here for the first time that may generate protons with energy 92 MeV by using linearly polarised laser of intensity 1021 W cm−2 and pulse duration of 30 fs. At this laser intensity, the energy gain by the protons is much higher compared to the conventional flat foil targets. The maximum proton energy can be increased further to 103 MeV by using truncated hemispherical target of similar parameter.


Introduction
The ongoing development of high-power, ultra-short laser facilities has paved the way for numerous applications over the past few decades [1][2][3][4][5].Laser driven acceleration of electrons and ions is one of such areas, which has received a lot of attention because of its cost effective and compact nature in comparison to the conventional accelerators.The localised nature of laser energy makes it possible for the electrons to get accelerated to MeVs of energy within micro-meter long distance owing to its strong accelerating field, to attain which, the conventional accelerators require km long beamlines.
Applications in medical fields, nuclear physics research, material science studies, etc require the accelerated charged particles to have certain characteristics like high energy, low divergence, mono-energetic nature, stability, repeatability, etc.To attain these, researchers work with various types of targets and irradiate them with different kinds of laser pulses.Based on the target and laser parameters, different electron and ion acceleration regimes come into play.Strategically capitalising the different ion acceleration mechanisms makes it feasible to optimise and tailor the desired characteristics of the accelerated ions, making it applicable for practical purposes.
The interaction of intense laser pulse with micron-thick foil targets has been explored experimentally and protons with energy of multi MeVs were first reported in the year 2000 [6][7][8].Since then, with the improvement in laser technology and the increase in energy of the laser pulse, the energy of accelerated protons has increased manifold experimentally.The understanding of laser energy absorption and acceleration of the electrons and ions has developed with the development of basic mechanisms of electrons and ion acceleration and is continuing [9][10][11].
Over time, several different target geometries have been explored, the most popular of which is the flat foil target.Flat foils of thickness ranging between a few tens of nanometers to a few micrometers are irradiated with laser pulses of duration between a few femtoseconds (fs) to picoseconds (ps) [12,13].Spherical targets of the dimension of the order of the laser spot size have been found to help in improving the laser energy absorption by the electrons and ions [14].Simulation of spherical targets of tailored gaussian density gradient has shown the ability to produce monoenergetic bunches of proton that have resulted due to shock acceleration near the center of the spherical target [15].Experimentally, isolated micron-sized spherical targets have been found to generate quasi-monoenergetic proton and heavier ion energy spectra [16].There has been a growing interest in hollow spherical targets, which are found to facilitate effective trapping of laser energy [17][18][19].This capability has been harnessed not only for the study of ion acceleration but also for the exploration of neutron generation.However, laser-driven ion acceleration is yet to be understood when a curved surface is exposed to the intense laser.Several new features may emerge that may be useful in designing experiments that can produce quality ion beams with high energy using a laser of relatively lower intensity.
Various advanced targets like structured flat foils with nanorods/nanowires/micropillars/nanospheres/ nanometer-sized foam structures etc are used by many researchers with the aim of improving laser energy absorption by the electrons and ions [20][21][22][23][24][25][26].Nanostructured targets help in achieving the near-critical density regime for the laser used, which is known to have the highest laser energy absorption.
We have used a thin nanostructured hemispherical shell target, which has not been explored earlier to the best of the authors' knowledge.In this paper, we report the improvement in the energy of accelerated protons on the use of thin hemispherical shell target.Kaymak et al reported protons accelerated with energy ∼8.5 MeV on use of laser of intensity 1 × 10 20 W cm −2 from thin hemispherical targets [27].Another study of thin hemispherical target by Qiao et al found meagre improvement in proton energy in comparison to conventional flat foil on use of picosecond long laser pulse [28].Here, we examine the use of nano-structures embedded on thin hemispheres with an aim to uncover new knowledge from the previously unexplored target regime.By using 2D particle-in-cell (PIC) simulation, we investigate the mechanism of ion acceleration involved in accelerating the protons to a higher energy in comparison to flat foil targets.In the part following the introduction, the simulation scheme of the work is reported.The results and discussion of the findings about the processes involved in acceleration of ions from the plain and nanostructured hemispherical target are reported in section 3. The results are summarised in the conclusion at the end.

Simulation scheme
The simulations in this work has been performed using the 2D-3V simulation code EPOCH [29].We have used a laser pulse of wavelength 1 µm having gaussian profile in both longitudinal (x) and transverse (y) directions.In the transverse direction, the full-width at half maxima (FWHM) is 4.24 µm and the temporal FWHM, i.e. pulse duration taken is 30 fs.The intensity of the peak of the pulse (I 0 ) taken is 6 × 10 20 W cm −2 , unless mentioned otherwise.
The simulation box has a resolution of 8.5 nm in the x-direction and 13.33 nm in the y direction.In each cell, the no. of macro-particles is chosen to be 150 for each species.
In this study, the target taken is (a) a hemispherical shell and (b) a hemispherical shell with nanorods attached on the convex laser irradiated surface.The target is made up of plastic (CH) consisting of carbon (C 6+ )ions, electrons (e − ) and protons (H + ).The density of electrons in the target is 270n c , where n c = 1.12 × 10 21 cm −3 , making the target an overdense one for the laser used here.
We have used a linearly polarised laser of I 0 = 6 × 10 20 W cm −2 , having normalised vector potential a 0 = 20.8, and irradiated it on a hemispherical shell of thickness = 200 nm and outer diameter = 8 µm, which is greater than the laser spot size of 4.24 µm.

Hemispherical shell target
The laser pulse is allowed to hit the target normally at the center of the convex front surface of the target, as shown in the schematic figure 1. Figure 2(a) demonstrates the electron density plot after the laser hits the target, at around 80 fs simulation time.It is observed from electron phase plot of figure 2(b) that bunches of electrons are emitted from the rear side of the target.From the electron phase plots (figure 2(b)), it is seen that the bunches are emitted with spatial periodicity of λ/2, which is a characteristic of J × B heating mechanism.
The comparison of the electron energy spectra of the flat and hemispherical target shows a higher electron cutoff energy along with a higher 'hot electron temperature' in the hemispherical target case  (figure 2(c)).The hemispherical target has a larger surface area of interaction with the incident laser in comparison to the flat foil.The interplay of J × B heating and vacuum heating may be another reason for increased laser energy absorption by the electrons.The laser is incident obliquely on the edges of the target that may allow vacuum heating of the electrons.The oblique laser incidence at the surface near the edge of the target may allow vacuum heating to occur.The characteristic bunches with spatial periodicity of λ may be superimposed with the bunches emitted due to the J × B heating.As a result, the absorption of laser energy by the electrons is found to be higher in the hemispherical target case.The laser energy absorbed by electrons in the hemispherical target improves to 4.8% from 3.04% in the flat foil.
The energy absorbed by the electrons is then transferred to the ions via various mechanisms.The proton density plot (figure 3(a)) shows expansion of protons from the hemispherical target.When the laser is incident on an overdense target, the electrons are pushed inwards due to the radiation pressure of the incident laser, creating a dense layer of electrons.This electron layer pulls the ions which are left behind and a hole-like deformation is created in the target.A strong longitudinal electric field (E x ) is generated at the front surface of the target due to the charge separation region created by the two layers of electrons and protons and the ions are then accelerated along the direction of laser propagation via the radiation pressure  acceleration (RPA) mechanism.Another strong longitudinal electric field is generated at the rear surface of the target which is governed by the TNSA mechanism.This field is formed because of the formation of a sheath of electrons at the target rear, after the hot electrons generated at the front surface of the target travel to the target rear.Figure 3(b) shows these two fields at the front and rear surface of the target at a time when the peak of the laser pulse has already hit the target, i.e. at around 80 fs simulation time.The RPA field at the front surface accelerates the ions up to the pulse duration.The TNSA field ionises the atoms at the rear surface and accelerates the ions at a direction normal to the rear surface of the target.
The proton dynamics is further studied and it is observed that the protons experience a momentum rise at simulation time of around 110 fs.The rise in momentum can be clearly seen from the proton phase plot (figure 4(a)).This momentum rise of protons is not observed when we performed the simulation on a target without carbon ions, i.e. a pure hydrogen plasma of same electron density (figure 4(b)).This establishes the role of slow moving heavier ion (in this case C 6+ ) in creating a longitudinal electric field, which exerts a push on the fast moving protons.However, this momentum rise is feeble in case of flat foil (figure 4(c)).The density of the front layer of carbon ions accelerated from the flat foil is observed to be less than the density of carbon ions accelerated from the hemispherical target.This is because of the curvature in case of the hemispherical target, creating a density increase in the accelerated carbon ions as they move at a direction normal to the rear side of the target, i.e. towards the center of the hemisphere.Figure 4(d) shows a longitudinal electric field created by the presence of carbon ions, which is stronger in the hemispherical target case.This field exerts a force on the protons and the momentum rise is observed, which is more distinct in hemispherical target in comparison to the flat foil target, owing to the stronger field in hemispherical case.A branch of accelerated protons is also seen in the proton phase plots, which arises from the edges of the hemisphere.These protons are accelerated at a direction normal to the flat surface at its two ends, indicating the TNSA mechanism of acceleration.
A second rise in proton momentum is observed when they converge towards the center of the hemisphere as shown in figure 6(a).Due to the curved inner surface of the hemisphere, the accelerated protons are focussed near the center of the hemisphere (figure 5).These protons are driven by the TNSA field formed directed normal to the target rear surface, i.e. towards the center of the hemisphere.At around 150 fs of simulation time, the protons are found to gain momentum after they converge near the centre of the hemisphere as shown in figure 6(b).Flat foils do not allow the protons to converge and hence this momentum rise is not observed in flat targets.At this central region, the formation of ambipolar field is observed as seen in figure 6(c).
Figure 7(a) illustrates the variation of proton cutoff energy with time for both hemispherical and flat foil.A rise in energy of protons is observed in the hemispherical target case in comparison to the flat foil at around 150 fs, which is the time when the ions converge at the center of the hemisphere.This brings an improvement in the proton energy in the hemisphere compared to flat foil target.The laser energy absorbed by protons in the hemispherical target improves to 2.6% from 2.49% in the flat foil.Figure 6(d) shows the density difference between the negative and positive charges, indicating the formation of a ion rich region at the center of the hemisphere.Although this region is ion rich, some electrons are still present in the vicinity of the central region.This indicates the possibility of ambipolar expansion at the center of the hemisphere.

The energy due to ambipolar expansion is given by
), where T h is the hot electron temperature and Λ = R t λ Dh .Here R t and λ Dh are the size of the electron cloud and Debye length of the plasma at the center of the hemisphere after the electrons and ions converge [30].The function W(x) ≈ x for x ≪ 1 and W(x) ≈ ln(x/ln(x)) for x ≫ 1.From the simulation, we obtained the hot electron temperature T h ∼ 1.88 MeV.R t and λ Dh are found to be ~700 nm and ~152 nm respectively.Here R t is taken to be equal to half of the size of the electron cloud where the density falls to 1/e times the peak density at a time just after they converge to the centre.So, the expected energy provided by the ambipolar expansion is ~5.64 MeV, which matches with the observed energy difference of ~5 MeV as seen in figure 7. The observed improvement of the proton energy from hemispherical target in comparison the flat foil may thus be accounted for the ambipolar expansion, which is facilitated by the geometry of the target.The electrons after getting converged to the central region of the target evolves into a non-spherical shape due to elongation along the laser propagation direction.The electrons emitted from the middle part of the target where the peak of the laser hits, travel faster than the electrons emitted away from this region.This results in slight  deviation from the exact spherical shape of the plasma cloud.However, this does not affect the estimation of energy of the protons.According to Murakami's model, the proton energy remains consistent across all geometries including linear, cylindrical, spherical shapes [30].
With an objective of obtaining the optimum target parameters like diameter of the hemisphere and its thickness, the diameter of the hemisphere is varied, keeping the thickness constant at 200 nm. Figure 8(a) shows the dependence of the maximum energy of the accelerated protons on the diameter of the hemisphere.It is observed that the proton cutoff energy first increases and then decreases with increase in the diameter with the maximum value of the proton energy at 8 µm.The thickness of the target is varied in the range 100-500 nm keeping the diameter fixed at 8 µm.The energy is found to be maximum at 100 nm thickness.The thickness however is not further decreased considering the fact that further decrease in the thickness might bring difficulty in manufacturing for practical uses.

Nanostructured hemispherical shell target
One important approach that may result in improved ion energy spectra is target nanostructuring.Rods of the same material as the hemispherical shell are attached at the front (convex) surface of the target.Figure 9(a) shows the electron density plot showing generation of hot electrons at the front surface and emission of bunches of electrons from the rear side of the target.Figure 9(b) shows the protons converging at the center of the nanostructured hemisphere.From figures 8(c) and (d), it is found that nanorods with diameter and gap-500 nm, length-300 nm give best results in terms of proton energy with the present configuration.
A comparison of proton energy spectra is shown in figure 10(a), which demonstrates an enhancement in the proton energy after attaching nanostructures on the laser irradiated side of the target.The proton energy increases from ~51 MeV in the plain hemisphere case to ~72 MeV in the nanostructured hemisphere target using the best nanostructure parameters as obtained from figures 8(c) and (d).
In order to understand the physics behind the improvement in proton energy, the role of nanostructures on the different mechanisms of laser energy absorption by electrons and ions in the target is examined.Figures 10(c) and (d) show the proton phase plots in case of hemispherical target with and without nanostructures.It can be seen that the protons in the nanostructured target have gained a higher momentum in comparison to the plane hemisphere target at a time when the peak of the laser hits the target.This early rise in proton energy in nanostructured hemisphere can be explained by the improved laser energy absorption by electrons and higher hot electron temperature in the nanostructured target case as shown in figure 10(b).The laser energy absorbed by electrons in the nanostructured hemisphere is 17.7% and the laser energy absorbed by the protons is 4.83%.(c) and (d) shows the proton phase plots for the nanostructured and plain hemipshere targets respectively around the time when the peak of the pulse has hit the target.
For further understanding of the role of nanostructures in the proton acceleration process, a plot showing the longitudinal electric field at around 80 fs simulation time in both the nanostructured and plain hemisphere target is obtained.Figure 11(a) shows the formation of a stronger electric field in the rear surface (at the region behind the gaps between the nanostructures) of the nanostructured target in comparison to the field in the rear surface of the plain hemisphere target.This electric field drives the ions from the rear side of the target at a direction normal to the target rear surface via the TNSA mechanism of ion acceleration.These accelerated electrons and ions converge near the center of the hemisphere (figure 10(b)) and form a strong ambipolar-like field.This field at the centre of the nanostructured hemisphere target is compared with the plain hemisphere target and it is seen that the strength of the field is slightly stronger in the nanostructured case as shown in figure 11(b).In order to investigate the ambipolar expansion process at the center of the nanostructured hemisphere, the energy attained by the protons is calculated.The hot electron temperature is found to increase from 1.88 MeV in the plain hemisphere case to 2.63 MeV in the nanostructured hemisphere case.However the density of electrons is found to decrease to 3n c at the center of the nanostructured hemisphere, which was 4n c in the plain hemisphere case.The hot energetic electrons move very fast giving less scope of convergence, resulting in the observed decrease in electron density at the center in the nanostructured case.This provides us with an expected energy of ~6.2 MeV from the ambipolar expansion, which was 5.6 MeV in the plain hemisphere case.It implies that the contribution of nanostructures in the ambipolar expansion at the center of the target is meagre.However, the increase of proton energy in the nanostructured hemisphere in comparison to the plain hemisphere is ∼41%.Figure 11(c) shows the proton energy variation with time in the three targets.In both the nanostructured targets, i.e. the flat and hemispherical targets, the proton energy increase in comparison to the flat foil starts at an early time, implying the high laser energy absorption by the electrons and ions in the nanostructure case.However the increase in the proton energy in the nanostructured hemisphere in comparison to the nanostructured flat foil begins from around 150 fs, i.e. the time of convergence of the electrons and ions at the center of the hemisphere, which was also seen the targets without nanostructure.
Simulations are conducted by altering the peak intensity of the laser while keeping the remaining laser parameters constant and the scaling of the maximum proton energy (E) with intensity is obtained.The intensity is varied in the range 6 × 10 19 W cm −2 -10 21 W cm −2 and the maximum proton energy is found to vary as E ∝ I 0.54 0 .Protons with energy ~100 MeV are obtained for laser intensity (I 0 ) ∼ 10 21 W cm −2 .The energy-intensity scaling is also obtained for a flat foil target as seen in figure 11(d).The energy of the protons accelerated in the nanostructured hemisphere target is higher than the flat foil case for all intensities in this study.The difference between the energies in the two target cases is higher for lower intensities and the difference drops for higher intensities.For the intensity of 6 × 10 19 W cm −2 , the energy increases from 8 MeV in the flat foil target to 20.4 MeV in the nanostructured hemisphere target.The proton energy increases by a factor of 2.55 in the nanostructured hemisphere target in comparison to the flat foil target.In the flat foil case, E ∝ I 0.77 0 .The improved scaling in the flat foil target in comparison to nanostructured  hemispherical target may be explained by the mechanism of acceleration of protons in the two cases.The use of nanostructures is known to improve the energy and count of hot electrons generated at the front surface of the target, which helps in improving the energy of the protons accelerated through the TNSA mechanism of ion acceleration.The theoretical scaling constant in the TNSA regime of ion acceleration is 0.5, which closely matches with the observed scaling in the nanostructured target.In case of flat foil target, the RPA mechanism of ion acceleration can play a greater role in comparison to the nanostructured hemisphere target.The short pulse laser encounters a flat surface, which is known to allow greater radiation pressure transfer in comparison to the other target.The scaling constant is 1 in case of RPA mechanism of acceleration.In our flat foil target, the observed scaling constant of 0.77 indicates an interplay of both the TNSA and RPA mechanism of acceleration.Figure 11(d) also indicates that for intensity above 5 × 10 21 W cm −2 , the nanostructured hemisphere does not help in increasing the energy of accelerated protons beyond ~100 MeV.At very high intensities, the pedestrial of the laser destroys the nanostructures before the peak of the pulse reaches the target, which may be the reason for the absence of improvement in the nanostructured hemisphere in comparison to the plain hemisphere for laser intensity beyond 5 × 10 21 W cm −2 .The divergence of protons from the nanostructured hemispherical target has also been studied and compared with the proton divergence from conventional flat foil of same thickness.Figure 13 shows the divergence of the protons from three different targets and it can be seen that the divergence of the protons with normalised momentum (p x /m p c) > 0.25 is lesser in the nanostructured hemispherical target case in comparison to the flat foil.The proton population accelerated with higher momentum in the hemispherical target case is accelerated via the ambipolar expansion mechanism, after converging at the center of the hemisphere.We have also simulated the laser interaction with a spherical shell with an opening of 90 • in the rear side (figure 12(b)).The divergence of the accelerated protons from this target is compared to the hemispherical target, which has an opening of 180 • on the rear side.From figure 13(c), we can see that there is some improvement in the angular divergence, although the momentum of the protons has decreased a little.
A simulation is also performed by irradiating the laser on a section of the spherical shell (figure 12(c)) covering less than 180 • , which is the hemisphere.For the target with angular size of 100 • with the thickness of 100 nm and nanostructures attached, the energy of the accelerated protons is found to increase from 72 MeV in the hemisphere case to 78 MeV in the 100 • angular size case.In comparison to flat foil of same thickness, the proton energy increases by around 66% for this target.The increase in energy may be It is important to note that 2D simulation can reliably help us in predicting and explaining the electron and ion dynamics in laser plasma interaction.However, 3D simulation can give a more comprehensive understanding of the physical process because of inclusion of all spatial dimensions as well as more accurate modelling of 3D geometry of the target.A 3D simulation of the hemispherical target has been performed with an objective to reproduce the proton energy spectra and thus investigate the main quantitative observables.For the 3D simulation, the target electron density taken is 80n c , which is less than the density considered in 2D simulation.Although the density is reduced, the interaction mechanisms remain the same as they are dictated by the same highly overdense nature of the target density.The 42 µm × 40 µm × 40 µm simulation space is divided into 2100 × 1600 × 1600 cells in the x, y, z directions respectively.The number of macroparticles of electrons, protons and carbon ions in each cell is 30, 12, 8 respectively.The thickness of the hemispherical shell is taken to be 100 nm and the diameter taken is 8 µm.Figure 14(a) shows the electron density plot at a time when the protons converge at the center of the hemisphere.The 3D proton density plot at the central planes (x = 0 and z = 0 planes) is shown as the projections along with the protons with y < 0 in the 3D space is shown in figure 14(b).The proton energy spectra is also obtained, which is found to be in close agreement with the spectra obtained from the 2D simulation (figure 14(c)).

Conclusion
In this work, we have demonstrated the improvement of energy of accelerated protons by using hemispherical targets in comparison to conventional flat foil targets.The laser energy absorption by the electrons is found to be improved in the hemispherical target.An experimental study by Kaymak et al using a hemispherical target of thickness 110 nm and diameter of 8.8 µm using laser with I 0 = 10 20 W cm −2 reported protons accelerated with energy of 8.5 MeV, which is 3.9 MeV higher than the protons accelerated from a flat foil of same thickness [27].In the hemispherical target considered in our work, the curved geometry of the target rear surface allows the accelerated ions to converge at the center of the target, which creates the condition for ambipolar expansion of the converged ions.The expected energy improvement of the protons after ambipolar expansion is found to match the observed energy difference in comparison to the flat foil target.The central takeaway from the study is that the primary factor driving the observed enhancement in proton energy from the center of the hemispherical target, in comparison to the flat foil is the ambipolar expansion.To the best of our knowledge, the evidence of ambipolar expansion at the center of the hemispherical target has not been reported earlier.
Further improvement in the energy of the protons is observed when nanostructures of suitable dimension are attached to the front of the hemispherical shell target, giving us an improvement of around 40% when compared with the energy of protons accelerated from flat foil of the same thickness on use of a laser of I 0 = 6 × 10 20 W cm −2 .The addition of nanostructures generated a stronger field of from the target rear surface along with enhanced generation of hot electrons from the laser-irradiated side of the target.There has been a very limited study on the acceleration of ions from nanostructured solid spheres or spherical shell targets.A recent simulation work by Mehrangiz [31] has also reported enhanced proton energy by using a gold nanoparticle layer on a hollow spherical target.The increased count of energetic electrons and elevated ion energy are facilitated by the preplasma generated due to the presence of gold nanoparticles preceding the impact of the laser peak on the target sphere.We believe that the improvement in proton energy observed in our work using nano-structured hemispherical shell target with laser in the intensity range 6 × 10 19 W cm −2 -10 21 W cm −2 has been explored for the first time.The energy of the protons is found to further increase when the angular size of the shell is decreased from 180 • (in the case of the hemisphere) to 120 • and the angular divergence of the accelerated protons is found to decrease when the angular size of the target is increased to 270 • .The nanostructured hemispherical target used in our work can improve the energy of accelerated ions for a wide range of the intensity of the laser.The energy of the protons can reach 103 MeVs with the use of a laser of I 0 = 10 21 W cm −2 from the nanostructured hemispherical target.These results may help provide new insight into the physics of ion acceleration from novel targets and pave the way for experiments and practical applications thereafter.

Figure 1 .
Figure 1.Schematic diagram of (a) the laser incident on the hemispherical target, (b) 2D nanostructured hemispherical target taken in the simulation.

Figure 2 .
Figure 2. (a) Electron density plot at 80 fs simulation time, showing the bunches of electrons with spatial periodicity of λ/2, emitted from the front part of the target.(b) Electron phase plot showing the periodic bunches emitted at twice the laser frequency.(c) Electron energy spectra of both both the flat foil and hemispherical target showing the hot electron temperatures (T h ) in both the cases.This is obtained at 80 fs simulation time, i.e. at a time when the electron energy reaches the maximum.

Figure 3 .
Figure 3. (a) Proton density plot at 90 fs simulation time.(b) Longitudinal electric field plot of the central axis (y = 0) of the target showing two peaks generated due to RPA and TNSA mechanisms of ion acceleration.

Figure 4 .
Figure 4. Proton phase plot at 110 fs simulation time from the front part of the target in (a) target with carbon ions, (b) target without carbon ions and (c) flat foil target.The momentum rise seen in (a) is absent in target without carbon ion and feeble in flat foil target.(d) Longitudinal electric field plot at the central axis (y = 0) for hemispherical and flat foil target.The carbon density is also plotted for the hemispherical target, showing formation of a peak in Ex corresponding to the position of carbon density peak.The peak in Ex at this position is stronger for hemispherical target.The arrows in figures (a) and (b) shows the protons accelerated from the edges of the hemisphere.

Figure 5 .
Figure 5. Schematic diagram of the hemispherical target showing the electrons and ions from the rear surface of the target converging at the center of the hemisphere.

Figure 6 .
Figure 6.(a) Proton phase plot at 150 fs, corresponding to the time of convergence of protons at the center, showing the momentum increase after reaching the center.(b) Proton density plot of the central axis (y = 0), showing the formation of high proton density region at the center (inset shows the complete proton density plot).(c) Longitudinal electric field (Ex) plot showing the formation of a ambipolar-like field at the center.(d) Charge density plot( ne − n i nc ) showing formation of ion rich region at the center.

Figure 7 .
Figure 7. (a) Proton cutoff energy vs time plot for flat and hemispherical target, showing rise in proton energy in hemispherical target at around 150 fs, corresponding to the time of focussing of electrons and protons at the center.(b) Proton energy spectra at final time of simulation.

Figure 8 .
Figure 8.(a) Maximum proton energy vs hemisphere diameter plot showing highest energy for 8 µm diameter.The thickness of hemisphere is fixed at 200 nm.(b) Proton energy variation with thickness of hemisphere for hemisphere diameter of 8 µm.(c) Maximum proton energy variation with varying nanorod length.The diameter and gap between the nanorods are fixed at ~700 nm.(d) Maximum proton energy variation with varying diameter of rods.Here the length of rods is fixed at 300 nm and the gap between the rods is taken equal to the diameter of the rods.

Figure 9 .
Figure 9. (a) Electron density plot of the nanostructured target at ~80 fs, when the peak of the pulse hits the target.(b) Proton density plot showing the convergence of protons at the center of the hemisphere at ~160 fs simulation time.

Figure 10 .
Figure 10.(a) Proton energy spectra of plain and nanostructured hemisphere targets at the end of simulation, after the energies saturate.(b) Electron energy spectra of Plain and nanostructured hemipshere showing the increased cutoff energy and temperature of the hot electrons in the nanostructured target.Figures(c) and (d)shows the proton phase plots for the nanostructured and plain hemipshere targets respectively around the time when the peak of the pulse has hit the target.

Figure 11 .
Figure 11.(a) Longitudinal electric field plot for the plain and nanostructured hemisphere target at ~80 fs simulation time.(b) Longitudinal electric field plot at ~160 fs showing the formation of the ambipolar-like field at the center of the hemisphere in both the targets.(c) Proton cutoff energy vs time plot for flat foil, nanostructured flat foil and nanostructured hemisphere.(d) Maximum proton energy vs intensity plot, showing the scaling between the two parameters.

Figure 12 .
Figure 12.(a) Plot showing proton density at 90 fs simulation time for (a) hemispherical target, (b) spherical shell with an opening of 90 • at the rear side.(c) Section of hemispherical shell with angular size of 100 • .

Figure 13 .
Figure 13.(a) Plot showing the normalised longitudinal momenta and angular divergence of the protons at the end of simulation for (a) flat foil target, (b) Hemispherical target.(c) Spherical shell with an opening of 90 • at the rear side.

Figure 14 .
Figure 14.(a) Electron density plot at z = 0 plane and (b) 3D proton density plot showing the central planes (x = 0 and z = 0 planes) as the projections and the protons with y < 0 are shown in the 3D space at 150 fs of the hemispherical target from the 3D simulation when they converge at the center of the hemisphere.(c) Proton energy spectra of the 2D and 3D simulation at the end of the simulation.