Enhancement of collisionless shock ion acceleration by electrostatic ion two-stream instability in the upstream plasma

Ion acceleration in electrostatic collisionless shocks is driven by the interaction of the high-power laser with specially tailored near-relativistic critical density plasma. 2D EPOCH particle-in-cell simulations show that the ion acceleration is dependent on the target material used. In materials with low charge-to-mass ratio h Z=A i , proton beams with high flux and low energy spread are generated. In multi-ion plasmas the ions with different h Z=A i acquire different velocities under a non-oscillating component of electrostatic field in the upstream region. This relative drift between the protons ( h Z=A i ¼ 1 ) and the lower h Z=A i ions leads to the excitation of electrostatic ion two-stream instability. This in turn generates a low-velocity component in the upstream expanding protons. The velocity distribution of the upstream expanding protons is further broadened toward the higher velocity by the electrostatic ion two-stream instability between reflected protons, which results in large number of protons being accelerated by the shock.


I. INTRODUCTION
The development of high-intensity laser systems has opened a new era for laser-driven ions acceleration and there are several promising mechanisms for laser-driven ion acceleration. The energetic ion beams driven by laserplasma interaction have many potential applications such as accelerator physics, cancer therapy, proton radiography, and inertial confinement fusion [1][2][3]. However, producing a proton beam with high energy, high flux, and low energyspread proved to be the major challenge for practical applications [4,5]. Several mechanisms have been studied for laser-driven acceleration. At present, the most widely understood mechanism is Target Normal Sheath Acceleration (TNSA). This mechanism [6] and a related radiation-pressure hybrid-scheme [7] can drive protons to energies approaching 100 MeV. Alternative acceleration schemes, such as radiation-pressure acceleration that can achieve monoenergetic and higher energy suitable for applications impose strict and challenging conditions in experiments, such as the target thickness and laser contrast [8][9][10]. We describe a method to circumvent such constraints whilst providing methods to achieve higher ion energies.
Collisionless shock acceleration (CSA) has been proposed separately by Denavit [11] and Silva [12], with a detailed theoretical investigation complemented by Fiuza [13]. These studies suggest that a special near-critical density profile, N cr , is important in order to control the sheath electric field, E TNSA , at the plasma-vacuum interface. This in turn affects the ion spectrum in CSA. The E TNSA amplitude can be reduced by using an exponentially decreasing density profile on the rear-side of the target [14], finally resulting in a quasimonoenergetic distribution for the CSA ions. CSA experiments using a linearly polarized CO 2 laser with near-N cr gas-jet targets produced 20 MeV proton beam [15]. A number of experiments have been carried out over the last few years to understand and characterize ion acceleration via CSA [16][17][18][19][20][21]. One aspect of CSA that is currently not well understood for accelerating mono-energetic ions to high energies is the effect of the target material used.
In this paper we used the EPOCH particle-in-cell (PIC) open source code [22] to study the generation of electrostatic (ES) collisionless shocks and proton acceleration from shocks formed in plasmas of different material composition. Our results show that a low average charge-to-mass ratio (hZ=Ai) plasma produces a higher proton beam flux with a higher laser-to-proton energy conversion efficiency, and these differences become smaller at higher laser intensities. In the plasma with multi-ion species the ions with lower hZ=Ai compared to the protons (with hZ=Ai¼1) gain different velocities. This results in difference between relative drift velocities of the protons and lower hZ=Ai ions in the upstream region of the shock. This leads to the excitation of an electrostatic ion two-stream instability (EITI) [23], which in turn enhances the number of the shockaccelerated protons [24]. To our best knowledge, most of the previous work has focused on shock formation [25][26][27] and ion heating [28][29][30]. This paper is the first investigation on the material (or hZ=Ai) dependence of EITI on CSA.

II. SIMULATIONS
We used 2D EPOCH simulations to investigate proton acceleration via CSA mechanism by modeling plasmas with different multi-ion compositions used commonly in experiments as target materials. The simulation box is 300 × 6 μm in size and composed of 9000 × 180 cells along the x-and y-axis respectively. Each cell contains 24-30 particles depending upon the material being studied. Open and periodic boundary conditions are used along the x-and y-axis respectively for both fields and particles.
The laser pulse and target in the PIC simulations use the parameters achievable at the Institute of Laser Engineering, Osaka University. An experiment would use the high intensity LFEX for the main interaction and the Gekko XII laser to preionize the target rear surface and create the necessary density profiles for CSA mechanism. The high intensity beam irradiated on the target front surface is linearly p-polarized with a temporal Gaussian profile of 1.5 ps full-width at half-maximum and peak intensity of 1.4 × 10 19 W=cm 2 (a 0 ¼ 3.35). The target consists of a tailored density profile increasing exponentially for 30 μm with 5 μm density scale length at N cr on laser side, a 5 μm region of constant density at the relativistic critical density a 0 N cr , and an exponentially decreasing density with 30 μm scale-length on the rear extending for 125 μm. The initial density profile is shown in Fig. 1(a), t ¼ 0. The corresponding ion densities are configured to ensure quasineutrality. The density profile and laser parameters are the same for all materials. The different material compositions and associated hZ=Ai used in this study are summarized in Table I. The simulations are timed to reach peak irradiance at 1.5 ps and have an initial electron and ion temperature of T e ¼ T i ¼ 500 eV. Figure 1(a) shows the temporal evolution of N H =N cr in C 2 H 3 Cl for a number of times after the peak of the laser pulse, which enter the particle grid at 1.5 ps. During the time from t ¼ 0 to t ¼ 1.75 ps the laser interaction with plasma results in uniform electron heating via J × B [31]. This compresses the plasma in front of the exponentially decreasing density on the rear target side, as shown by the peak in the density profile in Fig. 1(a). As time evolves, the density on the target rear side is expanding, resulting in the formation of the ES collisionless shock. The shock propagates at velocity V sh in the forward direction. The exponentially decreasing density profile in the upstream plasma, the region ahead of the shock results in a uniform ES sheath field ahead of the shock, E TNSA ¼ T e =eL g , where T e is the upstream electron temperature, e is the electric charge, and L g is the exponentially decreasing rearside scale-length [14]. This E TNSA field gives a uniform velocity V 0 to the upstream expanding protons.

A. Collisionless shock formation
For an ES collisionless shock to accelerate protons by ion reflection the ES potential ϕ at the shock must satisfy the relation Zeϕ ≥ 1 2 Am p ðV sh − V 0 Þ 2 , where Z ¼ A ¼ 1 for protons, and m p is the proton mass [32]. Shock reflected protons have a velocity V r ¼ 2V sh − V 0 . ϕ can be estimated by subtracting the non-oscillating component of E TNSA from the overall ES field (E x ) and integrating along the , and the velocity of the shock reflected protons V r at t ¼ 4.0 ps. shock reflection occurs. By t ¼ 4.0 ps, ϕ is gradually dissipating and reflection of protons becomes negligible. Figure 1(b) compares the hZ=Ai dependence of a number of velocities relative to the speed of light, c. Here V ϕ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2eϕ=m p p and V L ¼ V sh − V ϕ is the lower threshold for the proton reflection. The velocities V r , V sh , V L , V ϕ and the ion acoustic speed C S show a general increase with hZ=Ai. The increasing velocities result from a ffiffiffiffiffiffiffiffi ffi Z=A p dependence in the hole-boring velocity q with the nonrelativistic limit [33]. This is a good approximation for the low a 0 used. Here m e is the electron mass, a 0 and N e are the same for all the materials. The shock front moves forward at velocity V sh , which is powered by the laserdriven hole boring process at the target front. As there is a rear surface density drop, V sh increases in time. Protons with velocities V 0 that lie between V L and V sh are accelerated via shock reflection. A 2D-relativistic Maxwellian fðEÞ ∝ E expð−E=T e Þ is used for the electron energy spectrum in the upstream region. For all materials, T e ∼ 2.0 MeV. This implies that the ion acoustic speed is given by C s ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ZT e =Am p p is only a function of hZ=Ai as shown in Fig. 1 show the proton phase-space at 4.0 ps for each of the materials shown in Table I. The red solid line overlaid on each of the plots corresponds to the proton velocity distribution in the upstream region ahead of the shock front. This is integrated from the velocities within the vertical box in the phase-space plots, which spans across a range of Δx ¼ 3 μm. The horizontal lines are the material dependent velocities V r , V sh , and V L . Protons that satisfy the inequality V L ≤ V 0 ≤ V sh are shock reflected. For target materials with lower hZ=Ai, such as C 2 H 3 Cl and CH, a significantly larger number of protons satisfy this inequality when compared to He 3 H and H. As a result, a larger number of protons are shock accelerated. This is deduced by comparing Figs. 2(a) and 2(b) with Figs. 2(c) and 2(d). The proton velocity distribution is broader at lower hZ=Ai due to the presence of the lower hZ=Ai ions.
A broad velocity distribution results in a variation of the Mach number (M). To account for this we define M ¼ðV sh − V mean Þ=C s , where V mean is a mean value of V 0 in the distribution. In our calculations, we obtain M ¼ 1.6-1.7 for all materials. This is in agreement with the critical Mach number (M cr ¼ 1.6), which has been previously validated in PIC simulations for linearly polarized laser pulse [11,13]. Therefore, the particle acceleration, and thus shock dissipation, is expected when the ions are being reflected. The maximum velocity of the reflected protons from the shock front is expected to reach 2V sh − V L . Simulations show that the velocity distribution just ahead of the shock peaks around this maximum velocity. In the upstream plasma, protons with higher hZ=Ai are preferentially accelerated in the x-direction by the ES field E TNSA , which in turn drives the EITI. Figure 3(a) shows the energy distribution of accelerated protons in the whole upstream region. We find that the energy spread increases with increasing hZ=Ai. Simulations show that this occurs at early times when ϕ is relatively weak. This is due to hole-boring on the front side of the target. These protons have a different velocity distribution and lower energies compared to the shock reflected protons, which results in the larger energy spread. The differences are apparent in Figs. 2(c) and 2(d). The energy distribution of the shock reflected protons multiplied by the peak energy in the distribution, E 0 ,i s shown as a function of hZ=Ai in Fig. 3(b). The laser-toproton energy conversion efficiency (CE) is compared to the distribution in the same figure. Material with the lowest hZ=Ai (C 2 H 3 Cl) show a CE almost an order of magnitude higher than a pure proton target with the highest hZ=Ai. In comparison, the peak energy of the distributions is increasing by a comparatively small factor of ∼1.4 from C 2 H 3 Cl to H as shown in Fig. 3(a).

Figures 4(a) and 4(b)
show the x-y spatial profile of the ES field, E x , for the C 2 H 3 Cl upstream plasma. The simulation box size is y ¼ 6 and 20 μm, respectively. Both cases show a wavelike structure propagating in the x-direction, with increasing wavelength for larger x. The E x profile along the x-axis is averaged along the y-axis and overlaid on the phase-space of the corresponding proton velocity distribution in the upstream shown in Fig 4(c). The modulation in E x shows comparative correlation with the proton phase-space distributions. These modulations are caused by the EITI in the upstream plasma. This is excited by the relative drift velocity (v d ) between the two ion populations in plasmas with multi-ion species [23].O u r simulations show that this relative drift is driven by the E TNSA in the upstream plasma which results in slower moving heavier ions with low hZ=Ai and faster moving protons with high hZ=Ai. Simulations done by Zhang [26] have shown that in the transverse size-reduced simulation EITI propagates in the x-direction, whereas in the large transverse size simulation it propagates obliquely. In our case, since the propagation directions of EITI are in the x-direction both for the small and the large transversesize simulations [Figs. 4(a) and 4(b)], the instability is not oblique but longitudinal.
Temporal evolution of the upstream ion velocity distribution dN=dV x in the C 2 H 3 Cl plasma is compared in Figs. 5(a) and 5(b). These plots were captured just ahead of the shock at t ¼ 2.0 ps and 4.0 ps respectively. Initially, v d is small and the instability has not established. As time evolves, v d increases, which leads to a growth in the instability. This results as shown in Fig. 5(b), in a highvelocity tail in the C-ion and a low-velocity component to the proton distributions. We note that the chlorine distribution does not broaden in time. This observation suggests that the broadening of proton distributions discussed in relation to Fig. 2 results from the EITI. Temporal evolution of the velocity at the peak of the dN=dV x distribution for Cl-ions, V Cl (from the C 2 H 3 Cl target) is plotted in Fig. 6. V Cl shows a logarithmic dependence with time indicating that V Cl is determined by the upstream E TNSA . By taking a derivative of a fit to the V Cl distribution with hZ=Ai¼0.429, we infer that the time dependent E TNSA evolves with a 1=t dependence as shown in Fig. 6. The temporal variations for V Cl and E TNSA relationship described by Mora [34]. The time dependence of the C-ion velocity (V C ) at the peak of dN=dV x agrees well with calculations of V C based on the inferred E TNSA and hZ=Ai¼0.5 (V C−TNSA ). At t ¼ 2.5 ps, a high-velocity tail (V H C ) appears in V C . This results from acceleration by an electric field associated with the EITI. The calculated velocity for H-ions from E TNSA (V H−TNSA )withhZ=Ai¼1, which gives the maximum possible velocity for H-ions caused by E TNSA , is included for the comparison in Fig. 6. We approximate the simulated velocity distributions for protons as a sum of the three 1D-shifted Maxwellian's. These velocities, the lower (V L H ), mean (V M H ), and higher (V H H ) velocity Maxwellian distributions are shown in Fig. 6. We note that V H C and V L H represent the EITI accelerated and decelerated C-ion and proton populations, respectively, and these populations have similar velocities. This occurs as an additional EITI develops due to the relative drift between the shock reflected proton population moving at approximately 2V sh − V 0 , through the upstream expanding proton population moving at V 0 . This further increases the number of protons satisfying the condition for CSA.

B. Electrostatic ion two-stream instability
In the cold plasma approximation, the dispersion relation for EITI is given by [30] 1 þ 1 where k x is the wave number in the x-direction, λ De is the Debye length, ω p1 and ω p2 are the proton and heavy ion plasma frequencies, and v d is the relative drift between the two ion populations. In multi-ion plasmas, ω p1 is the proton plasma frequency ω pp , and ω p2 is the heavy-ion plasma frequency, as in, ω p2 is the carbon plasma frequency ω pc for C 2 H 3 Cl and CH. For C 2 H 3 Cl at t ¼ 4.0 ps, the maximum amplitude of the modulation in E x in the region of x ¼ 135-145 μm occurs at k x ¼ 4.7 × 10 6 m −1 [dashed line in Fig. 4(d))], and λ De ¼  which agrees relatively well with the value of k x ¼ 4.7 × 10 6 m −1 extracted from simulation. We refer to this as heavy-ion EITI.
At t>2.5 ps, the number of protons reflected by the shock front increases, which contributes to the excitation of the EITI between the expanding and the reflected protons. This is referred to as reflected-proton EITI. This leads to the generation of a high-velocity tail in V H H in the upstream plasma. In the case of reflected-proton EITI, ω p1 ¼ ω pp and ω p2 is the reflected proton plasma frequency ω pref . As shown in Fig. 2(a), the number of reflected protons is smaller than that of the expanding protons even at t ¼ 4.0 ps. Hence, ω pref < ω pp .T h e maximum growth rate of reflected-proton EITI [30] is The relative drift between the reflected and expanding protons is v r d ¼ 0.064ca tx ¼ 140 μm and t ¼ 4.0 ps. Therefore, k x ∼ 6.8 × 10 5 m −1 , which is comparable to the k x value for the maximum growth rate of heavy-ion EITI at x ¼ 140 μm by a factor of 0.31. As a result, this suggests both that heavy-ion and reflected-proton EITIs occur with similar growth rates and wave numbers.
V H−TNSA or the maximum possible velocity of H-ions accelerated by E TNSA in C 2 H 3 C li ss h o w ni nF i g .6.W e infer that a large number of protons is accelerated by the wave electric field of reflected-proton EITI and obtain the expansion velocity V 0 larger than V H−TNSA .T h i s acceleration generates a significant number of expanding protons with the velocities larger than V L , and results in the enhancement of the shock accelerated protons. Simulation studies done by Grassi et al. [35],haveshown that the development of ion turbulence in H plasma leads to the heating of upstream ions, which allows a fraction of ions to exceed the threshold for ion reflection from the shock front.
To confirm the effect of hZ=Ai on EITI, PIC-simulations with the same laser and density properties used throughout are compared to fully ionized carbon deuterium (CD) plasma. As shown in Fig. 7(a), since both C and D-ions have the same hZ=Ai¼0.5, both ions are accelerated by E TNSA in the upstream plasma, and there is no relative drift between them, as in, v d ¼ 0. Therefore, no instability is excited, and no broadening of the upstream ions occurs. In multi-ion plasmas, a high-velocity tail and a low-velocity component appear in the smaller and larger hZ=Ai ions, respectively. Figure 7(b) represents upstream ion velocity distribution for CCl 2 plasma with fully ionized C and Z Cl ¼ 10. A high-velocity tail in Cl-ions and a low-velocity component of C-ions appear because of heavy-ion EITI.

III. DISCUSSION AND SUMMARY
We further generalize our results by finding the dependence of the peak energy of the reflected protons and proton flux on the laser intensity corresponding to the a 0 ¼ 3.35, 10, and 33. In order to compensate for relativistically induced transparency, the initial electron density is varied  for different laser intensities by a 0 N cr for all plasmas. Figure 8 represents the peak energy (E 0 ), conversion efficiency (CE), and number (dN=dE at E 0 ) of the shock accelerated protons for different laser intensities corresponding to a 0 ¼ 3.35, 10, and 33 in C 2 H 3 Cl and H plasmas. For C 2 H 3 Cl plasma, E 0 , dN=dE, and CE increase with increasing laser intensity. However for H plasma, E 0 and dN=dE increase but CE first increases and then decreases with increasing laser intensity. The E 0 , dN=dE, and CE are always larger for C 2 H 3 Cl plasma compared to H plasma. As shown in Fig. 8(b), the increment in the dN=dE and CE for the higher laser intensity is different from the lower laser intensity, whereas dN=dE and CE of the shock accelerated protons are always higher in lower hZ=Ai (C 2 H 3 Cl) plasma than pure H plasma. At lower laser intensity (a 0 ¼ 3.35), these increments are found to be ∼8.6 and ∼4, while at higher laser intensity (a 0 ¼ 33), these increments are found to be ∼2 and ∼1.3.
We have performed the 2D PIC simulations in a plane wave approximation with an infinite spot-size. In the 3D scenario a smaller amplitude TNSA field will appear on the rear-side of the target according to Xiao et al. [36], which results in a quasimonoenergetic distribution for the CSA ions [14]. Even with this small amplitude TNSA field, the ions with different hZ=Ai will be accelerated to the different velocities. This leads to the development of EITI in multiion species plasma.
In summary, we have presented a PIC simulation study of the ES collisionless shock proton acceleration in different target materials. Protons with high flux and narrower energy spread can be produced using the lower hZ=Ai materials. The excitation of heavy-ion EITI leads to a large velocity spread in the upstream region of protons in multiion materials. The velocity distribution of the upstream expanding protons is further broadened toward the higher velocity by the EITI between reflected protons. As a result, a larger number of protons are driven by collisionless shock acceleration mechanism. This work supports future experimental work and investigations to a complete understanding of EITI and collisionless shocks using accessible experimental parameters.