Muhammad Ali Bake,Aimierding Aimidula

(College of Physics Science and Technology,Xinjiang University,Urumqi Xinjiang 830046,China)

0 Introduction

One of the most interesting applications of the ultraintense laser pulses is the acceleration of charged particles to high energies.Plasma-based charged particle acceleration has been massively investigated recently because of its potential for realizing table-top accelerators[1,2].The laser accelerated energetic ions have applications ranging from cancer therapy[3]to fast ignition[4]in inertial conf i nement fusion,and from laboratory astrophysics[5]to high energy density physics[6].

Many researches have demonstrated that the radiation pressure acceleration(RPA)mechanism[7–17]leads to a possibility of reaching very high energy conversion efficiencies and production of quasimonoenergetic proton beams with very high energies.In particular,RPA with circularly polarized lasers suppresses fast electron generation[9].However,it requires a long acceleration distance to achieve high energy,since the proton energy grows slowly with time[8].Because of instabilities,the acceleration length is limited and the proton energy spectrum is broadened.So,it is difficult to obtain the higher energy monoenergetic protons only by simple RPA.Even in simulation it is difficult to overcome the ten GeV barrier for protons using RPA,since the accelerating gradient drops quickly due to the relativistic Doppler ef f ect[9,10,16].

Recently,Macchiet al.[16]and Robinsonet al.[17]studied the ion acceleration from intense laser interacting with solid targets,and gave the clear physical picture of the hole-boring and light-sail RPA processes.Some other authors proposed schemes of proton(ion)acceleration by combination of laser radiation pressure and backside plasma[18–20],which can accelerate protons to GeV energies.

Motivated by these studies,in this paper,we investigate the acceleration of the background plasma protons by relativistic two-dimensional(2D)particle-in-cell(PIC)simulation.We mainly investigate the ef f ects of the foil thickness and backside plasma density on the proton acceleration and bunch generation efficiency.It is found that there exist the optimal foil thickness and backside plasma density,which play key roles to obtain energetic proton bunch with narrow energy spread.The 2D PIC simulation results show that one can accelerate the background plasma protons to beyond tens of GeV using this method.

1 Model

For the completeness of the paper we present brief l y this acceleration scheme.A solid foil with a long backside underdense plasma layer with proper length and density is used.As known in RPA,the laser-driven double layer is accompanied with an intense charge separation f i eld.It is assumed that,the double layer propagates in backside underdense plasma with a constant velocity and acts like a moving electric f i eld.The physical picture is as follows:at the beginning of the laser foil interaction,the laser ponderomotive force pushes the foil electrons forward and an electron-ion double layer is formed during the hole-boring process.Then the double layer acts as a laser piston that moves through the underdense plasma with a constant velocity and maintains its structure for a long time.The backside plasma protons are ref l ected and accelerated by the double-layer electric f i eld to give rise to a monoenergetic proton beam.As the laser pushes the double layer like light sail,which moves with the constant velocity through the plasma channel,the electric f i eld ref l ects the background protons continuously.Accordingly,the proton acceleration continues as long as the laser intensity is high enough to maintain the double-layer structure.Consequently,the background plasma protons can be trapped by this f i eld under proper conditions and accelerated to high energies.We use the formula for an optimized foil thickness[9,19]

where,n0and λ0are the initial foil density and laser wavelength,ncr=meω0/4πe2is the critical plasma density,ω0,meand−eare the laser frequency,electron mass and charge,respectively,a0=eA/mec2is the normalized peak laser amplitude,cis the speed of light in vacuum,andAis the vector potential.Our numerical optimization will refer to the condition for double layer formation after the hole-boring process.Note that for foils thinner thanl,the compressed electron layer will be permanently separated from the foil without forming double layer,since balance between the laser ponderomotive force and the force arising from the charge separation will be violated.

2 Simulation parameters

Accordingly,we carry out 2D PIC simulation by using the fully relativistic electromagnetic code to the show dynamical acceleration process.In the simulation,the circularly polarized laser pulse is transversely fourth-order super-Gaussian,i.e.,withr0=20λ0,t0=12.5Tand λ0=0.8µm,whereTis the laser period.The normalized laser amplitude isa0=223,which corresponds to the laser intensityI=2.14×1023W/cm2.The simulation box is 64µm(x)×144µm(y),which corresponds to a moving window with 2000×900 cells and there are 40 super particles per cell for the solid foil and 4 super particles per cell for the backside plasma.The solid foil of density 80ncris located in 15µm≤x≤15.4µm and −72µm≤y≤72µm,andncr=1.72×1021/cm3for λ0=0.8µm.The massive and high-Zions are used for the foil in order to obtain a stronger charge separation f i eld.The backside plasma of density 0.1ncris initially located in 15.4µm ≤x≤ 800µm and −72µm ≤y≤ 72µm.The left vacuum is 15µm.The laser pulse enters the simulation box from the left boundary att=0,and it is normally incident on the solid foil target.The initial electron and proton(and ion)temperatures are assumed to be so small that their ef f ects can be ignored.The transverse and longitudinal boundary conditions are periodic and absorbing,respectively.For convenience the electric fi eld and momentum are normalized bymeω0c/eandmpc,receptively,wherempis the proton mass.

3 Results and discussion

Fig 1 shows the longitudinal electric f i eld and the phase space of the backside plasma protons att=80fs and 860fs.From Fig 1(a)it can be seen that a very strong shock-like structure,with normalized electric f i eld amplitude above 100,is formed,which acts like a relativistic piston to background protons.Behind the shock-like piston a strong charge separation f i eld is induced by the double layer.Therefore,the protons in backside plasma are f i rst ref l ected by the piston and then trapped by the strong electric f i eld,as shown in Fig 1(a).

Fig 1 The 2D PIC simulation results of the longitudinal electric f i eld and protons phase space portrait(x,px)at(a)t=80fs and(b)t=860fs,respectively.The simulation parameters are shown in text

Fig 2 (a)Energy spectra of the protons at t=500fs,700fs and 860fs,respectively.(b)Time scaling of protons energy from 2D PIC simulations and theoretical analysis.Other parameters are the same as in Fig 1

Since the double layer moves through the backside underdense plasma channel with a constant piston velocity,the electric f i eld moves with the same velocity and maintains its prof i le a long time.As can be seen in Fig 1(b),the amplitude of the moving electric f i eld att=860fs is still above 25.So once the protons are ref l ected from the shock-like piston and trapped by the moving electric f i eld,it can be stably accelerated for a long time to high energies.

For the purpose of some special applications,the importance are not only the maximum energy but also the energy spread of the accelerated protons.The energy spectra of the accelerated protons att=500fs,700fs and 860fs(fora0=223)are shown in Fig 2(a).It can be seen from this f i gure that,att=500fs,the proton energy spectrum has a low peak energy with broad energy spread.However,the peak energy increases with time and the energy spectra show lower energy spread,but at the expense of smaller proton number.We can also f i nd from Fig 2(a)that the maximum energy of the accelerated protons att=860fs is above 20GeV with∼12.5GeV peek energy and∼7.5%energy spread.The peak energy is much higher than that from simple RPA.The time evolution of the energy until 1000fs is presented in Fig 2(b).The proton energy f i rst varies almost linearly in time,and then more rapidly.Aftert=860fs it begins to saturate.This f i gure also shows that in this scheme the acceleration time is longer and the maximum energy is higher than that in simple RPA.It is worth to note that we have also done theoretical analysis which is similar to that in Refs.[10]and[17].The theoretical result is also shown in Fig 2(b)by the black solid curve.It is shown that the 2D PIC simulation results agree very well with the simple theoretical analysis.

Now let us turn to see the dependence of the efficiency of proton acceleration on the foil thicknessland backside plasma densitynbackand the optimization for acceleration.In order to determine the appropriate parameters,we carry out more 2D PIC simulations for the dif f erentlandnback.The numerical optimization results is shown in Fig 3.Figure 3(a)displays the accelerated proton energy spectra forl=0.2,0.4 and 0.45µm att=860fs fora0=223.It shows that integrating three features of the maximum proton energy,the peak height and peak width with increase ofl,the energy spectrum forl=0.4µm has a relative good peak structure.We can clearly see from Fig 3(b)that there exists the optimal foil thickness,which results in an enhancement of the peak energy and good energy spread.We f i nd that for a thinner foil target the charge-separated force is smaller than the light pressure exerted on the compressed electron layer so that the balance condition to form a double layer will be failure.Because the electron layer in the rear of the foil is pushed out of the foil by the laser pulse soon before the double layer is formed.This normally results in suppression of the efficiency of acceleration.On the other hand for a thicker foil the most of the laser energy will be depleted through the transfer to foil ions during the hole-boring process,this causes the weaker double layer electric f i eld during the light-sail process,f i nally it results in also the decreasing of the accelerated proton energy and total proton number.

The accelerated proton energy spectra for the samel=0.4µm but dif f erentnback=0.05,0.1 and 0.2ncrare displayed in Fig 4(a)att=860fs fora0=223.Whennback=0.1ncra good peak in energy spectrum is observed compared to that fornback=0.05ncrandnback=0.2ncr,as shown in Fig 4(a).For example,in the case ofnback=0.2ncra peak with larger energy∼17GeV has the broad energy spread compared to the casenback=0.1ncr.So for the good energy spread we have to choose the highest peak as comparison,however,its peak energy is only about a half of that ofnback=0.1ncr.Thus,one can conclude from Fig 4(a)that the generation of a high-quality proton depends also the appropriate choosing of the background plasma density.In our study the optimal value ofnback,given by 2D PIC simulations,is shown in Fig 4(b).

Fig 3 (a)Energy spectra of the protons for different foil thickness at t=860fs.(b)The proton peak energy versus the foil thickness l.Other parameters are the same as in Fig1

Fig 4 (a)Energy spectra of the protons for different backside plasma density nbackat t=860fs.(b)The proton peak energy versus the backside plasma density nback.Other parameters are the same as in Fig 1

We also observe that,when the density of background plasma is too low,because of the higher double layer velocity and few protons in the background plasma,the chance of plasma protons trapping by double layer electric fi eld is decreased.As a result,the accelerated proton energy and total proton number are decreased.For the higher background plasma densities,the velocity of the double layer in the plasma is slowed due to high density and the double layer electric f i eld enhanced by laser snow-plowed plasma electrons,this results in the higher proton energy and total proton number.However,as the further increasing of the plasma density the velocity of double layer decreases more quickly,this also results in suppression of the efficiency of acceleration and broadening of the energy spread,as shown in Fig 4(a).

From discussions mentioned above it seems a easy way to judge the conditions that makes the protons acceleration optimal by observing the features of the highest peak of energy spectra for each set of chosen parameters when other parameters are f i xed.

4 Conclusion

In summary,by means of 2D PIC simulations,we consider a moving double layer in an underdense plasma caused by the laser radiation pressure to trap and accelerate the immobile background plasma protons.Since laserdriven double layer is accompanied with an intense charge separation f i eld and keeps its prof i le in the backside plasma so a long time that it will result in the higher energy protons.2D PIC simulations show that the maximum protons energy can be reached about 20GeV with energy spread 7.5%.Moreover,we also investigate the ef f ects of the foil thickness and backside plasma density on the proton acceleration and bunch generation efficiency.We observe that there exist optimization parameters of foil target thickness and underdense plasma density for the proton acceleration.

It should be noted that,the PIC code used in the simulations dose not contain radiation reaction ef f ects although the laser intensity is very high.However,according to our earlier results[20]and related works[21,22],the radiation reaction ef f ect should not inf l uence the proton acceleration in the studied cases.On the other hand our study indicate that for the higher accelerated protons energy the long-time acceleration time would cause the transverse instability so that the spatial divergence would become problem.Therefore it seems to need 3D PIC simulations for the purpose of realistic application.However,because of limitation of our current computation sources,we have not considered 3D simulation in present paper.

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