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Generating sub-TeV quasi-monoenergetic proton beam by an ultra-relativistically

intense laser in the snowplow regime

F.L.Zheng,1, 2 H.Y.Wang,1 X.Q.Yan,1, 3, ∗ J.E.Chen,1 Y.R.Lu,1 Z.Y.Guo,1 T.Tajima,4 and X.T.He1, †

1Center for Applied Physics and Technology, State Key Laboratory of

Nuclear Physics and Technology,Peking University, Beijing 100871, China2Graduate School, China Academy of Engineering Physics,

P.O. Box 8009, Beijing 100088, Peoples Republic of China3Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing 100088, China

4Fakultat f. Physik, LMU Munchen, D-85748 Garching, Germany

(Dated: March 2, 2022)

Snowplow ion acceleration is presented, using an ultra-relativistically intense laser pulse irradi-ating on a combination target, where the relativistic proton beam generated by radiation pressureacceleration can be trapped and accelerated by the laser plasma wakefield. The theory suggests thatsub-TeV quasi-monoenergetic proton bunches can be generated by a centimeter-scale laser wakefieldaccelerator, driven by a circularly polarized (CP) laser pulse with the peak intensity of 1023W/cm2

and duration of 116fs.

PACS numbers: 52.38.Kd, 41.75.Jv, 52.35.Mw, 52.59.-f

Laser-driven ion acceleration has drawn attention formany applications, e.g., proton cancer therapy[1], fast ig-nition of thermonuclear fusion by protons [2],conversionof radioactive waste [3], high energy physics accelerator[4]and astrophysics [5]. All these applications may beenabled by the introduction of near-future high-intensitylasers (i.e. ELI), which will be capable of producingpulses with intensity 1022 − 1024 W/cm2 [6].

Radiation Pressure Acceleration (RPA)[7] acceleratesions efficiently and theoretical studies show that GeVproton beam may be generated by an ultra-intense laserwith intensities above 1022W/cm2 [8, 10, 11]. Althoughthe acceleration field is over tens TeV/m, the accel-eration length is normally shorter than hundreds mi-crons. Because of the energy scaling, it is difficult tofurther increase the ion to very high energies (i.e. TeV ).Quasi-monoenergetic GeV electron beam can be gener-ated by a centimeter-scale laser plasma wakefield acceler-ator(LPWA) [12]; however, it is not easy to capture andcontinuously accelerate the nonrelativistic ions. Shen etal.[13] reports a laser pulse with an ultrarelativistic inten-sity can excite an electrostatic field, which can captureprotons from underdense plasma and accelerate them totens GeV. Recently Yu et al. [14] found that the under-dense gas behind a thin foil can accelerate the protonbeam to 60GeV by driving high-amplitude electrostaticfields moving at a high speed. There is an importantquestion whether it is possible to generate TeV-level pro-ton beams by the laser plasma wakefield accelerator.

This letter reports an ultra-relativistically intense lasercan excite the plasma wakefield that traps the relativis-tic proton beam and accelerates it over a few centime-ters. An analytic model is derived for proton acceler-ation that extends the applicability of Esarey’s classicplasma wakefield theory [15] and has been confirmed byour PIC simulations. Our scaling law shows that sub-TeV

quasi-monoenergetic proton bunches can be generated bya circularly polarized (CP) laser pulse with the intensityof 1023 W/cm2 and duration of 116 fs. A combined tar-get in Fig.1 (a) shows a planar hydrogen target that isfollowed by underdense heavy-ion background gas with acharge-to-mass ratio of 1/3. When the foil thickness D isequal to 1

2πnc

nea0λl, a mono-energetic proton beam is gen-

erated from a moving double layer [14], Fig.2(a) suggeststhat protons and electron layer are very close. The rel-ativistic protons can quickly overrun the laser front thatresults in shorter dephasing length and lower proton en-ergy. If a thinner foil satisfies

l0 < D <1

2π

nc

nea0λl, (1)

the ponderomotive force quickly pushes electrons to therear of the foil (see Fig.2(b)), the electrons that are piledup there reaches density far greater than γnc. Mean-while protons in the foil are accelerated by the separatedcharge field. Here l0,ne,nc, and λl are the plasma skindepth, plasma density,critical plasma density, and laserwavelength, respectively. a0 = eA/meωlc is the normal-ized laser amplitude and A is the laser vector potential.γ = (1+Iλ2

l /1.37×1018)1/2, I is laser intensity, λl in unitsof micron. The electron-layer in rear of the foil is pushedout of the foil by the laser pulse soon before double-layer(consists of electrons and protons) is formed. The layerruns in the lower density gas and is further pushed by thelaser pulse like light-sail to generate ”Snow-plow” effectson enhancement of electron number. The electrons in gasplasma drew away from the foil generate a large electricfield, then the protons that have been accelerated to GeVlevel start long march in gas by the wakefield accelera-tion. This acceleration scheme is named the snowplowregime. We also notice a connection of the above plasmaacceleration with ”snowplow” acceleration in Ref. [16].In order to grasp the main physics and to construct a

2

FIG. 1: (color online)(a)Acceleration scheme, the initial den-sity of the hydrogen foil n0/nc = 20, thickness D = 0.5λl; theplasma density of the tenuous gas ne/nc = 0.01, the length ofthe gas plasma is 12000λl; (b)Sketch map for the snowplowprocess, the dynamical density of ions is ni and electron den-sity in the snowplow layer is ncom. The position B indicatesthe laser pulse front and A is an arbitrary point for calculatingthe local electrostatic field in the snowplow region.

physical model , in this letter one-dimensional(1D) the-ory is derived and testified by 1D and 2D PIC simula-tions.

Considering the classic 1D plasma wakefield driven bya laser pulse [15], two typical features should be no-ticed in the ultra-relativistic intensity : (i)Lengtheningof the plasma wavelength ls = (γnc/ne)

1/2λl;(ii)Enhancement of the nonlinear wavebreaking fieldEB = 3π(γne/nc)

1/2E0, where the critical plasma den-sity nc = πmec

2/e2λ2l . The highly nonlinear wake-

field wavelength ls and wavebreaking field EB are in-creased by a factor of γ1/2 over the non-relativistic ver-sion, where the electric field strength is normalized byE0 = meωlc/e. In this regime we adopt a CP laserpulse with a wavelength λl = 1µm and a peak intensityI0 = 1.7125×1023W/cm2, corresponding to peak dimen-sionless laser amplitude a0 = 250. When the foil thick-ness D satisfies Eq.(1), the relativistic protons can becaptured by the excited wakefield in the underdense gasand be stably accelerated in the snowplow regime over along distance. An analytic model is developed to estimatethe acceleration field, dephasing length, pump depletionlength, and maximum proton energy. The acceleration

length in snowplow regime is proportional to a3/20 and

maximum proton energy scales with a20. It shows that subTeV monoenergetic proton beam is generated in the laserintensity of 1023 W/cm2. To our best knowledge,it is thebest efficient proton acceleration mechanism in terms ofacceleration length and maximum energy.

First we carried out simulation using a fully relativis-tic particle-in-cell code (KLAP) [8, 9]. The laser pulsehas a trapezoidal shape longitudinally with 15 λl flattop and 20λl ramp on the rise side. The ramp profileis a = a0sin

2(πt/40). At time t=0 it is normally inci-dent from the left on a uniform, fully ionized hydrogenfoil of the thickness D = 0.5µm and normalized density

N = n0/nc = 20. Behind the hydrogen foil is the tenu-ous heavy-ion-gas with a charge-to-mass ratio of 1/3 anda plasma density of ne = 0.01nc.Here the simulation boxhas a size of 12000 λl in x plane and a space resolutionof 20 cells/λl. Each cell is filled with 10 macroparticlesfor gas plasma and 20000 macroparticles for foil plasma.The initial electron and ion temperature are 0.1eV.After the proton beam is accelerated to GeV level in

the laser foil interactions, the laser pulse passes throughthe foil and piles electrons in the underdense gas plasmaas Fig.1 (b) shows, which is one of most important and re-markable features in the snowplow regime. Here the elec-tron layer and the positive electrostatic field are calledthe snowplow layer and snowplow field, respectively. Thesnowplow layer can continuously trap electrons in thesnowplow process, so the acceleration field exists stablyfor a long distance. In the snowplow region, it is foundthat the length of the snowplow region is equal to ls =(γnc/ne)

1/2λl and the snowplow velocity approximatelyequals to the group velocity of laser pulse in the under-dense plasma. First we estimate the electrostatic fieldEA at an arbitrary point xA in the snowplow region. Thecomponent from ions is E

′

A = 4πenixA−4πeni(ls − xA).The contribution from the snowplow layer at position xA

is E′′

A = 4πenels2 . The longitudinal electrostatic field at

xA is EA = E′

A + E′′

A = 4πene(2xA − ls/2). Therefore,the maximum snowplow field at the laser front xB is

EB = 3π(γne/nc)1/2E0. (2)

Since the injected GeV protons are still slower than thelaser pulse at the beginning, the ion beam should be in-jected into the front side of snowplow region. Afterwardsprotons are gradually accelerated and overrun the laserpulse ( see Fig.4(a)). The optimized distance betweeninjected proton beam and laser pulse front is defined bylinj . It is approximately 1

4 ls in our simulation. Oncethe proton beam is injected into the wakefield, its maxi-mum energy depends on the laser pump depletion lengthand proton dephasing length. The laser pump depletionlength Lpd is estimated [15]: E2

x · Lpd ≃ E2l · Ll, where

Ex is the longitudinal electrostatic field driven by laserpulse in the snowplow wakefield and Ll is the length oflaser pulse. Assuming the group velocity of laser pulseis extremely close to light velocity in free space, pumpdepletion length can be defined as

Lpd ≃Ll

vetch· c, (3)

where vetch is the depletion velocity of laser pulse. The

etching velocity of laser pulse reads vetch ≃9π2ne

a0ncc. It

shows the etching velocity of laser pulse scales as ne/a0.The dephasing length of proton is given approximately

by Ldp ≃linj

vp−(vg−vetch)· vp, where linj is the injected

length between proton beam and laser pulse front, vgis the laser group velocity, vp is the proton velocity. It

3

0 10 20 30 40 500

50

100

150

x/λl

eE

x/m

eω

lc,n

e/n

c

(a)

ne

np

20 25 30 35 40 45 50−2

0

2

4

6

8

x/λl

eE

x/m

eω

lc,n

e/n

c

(b)

ne

Ex

np

4900 4925 4950 4975 5000−15

−10

−5

0

5

10

15

20

x/λl

eE

x/m

eω

lc,n

e/n

c

(c)

ne×2.5

Ex

np×40

1.19 1.192 1.194 1.196 1.198 1.2

x 104

−5

0

5

10

15

x/λl

eE

x/m

eω

lc,n

e/n

c

(d)

ne×2.5

Ex

np×40

FIG. 2: (Color online) Electron density ne/nc(blue solidline),proton density np/nc(the red solid line) and longitudinalelectrostatic field eEx/meωlc (the black dashed line).(a)WhenD = 1

2π

nc

nea0λl double-layer is formed at t=50Tl. When

D=0.5µm snapshots are taken at time (b) 50Tl, (c) 5000Tl,(d)12000Tl.

can be further simplified when both vg and vp are veryclose to light speed, which means vetch ≫ (vp − vg). Theproton dephasing length is:

Ldp ≃1

36π2(a0nc/ne)

3/2λl. (4)

In case of Ll >14 (a0nc/ne)

1/2λl the laser pump deple-tion length is greater than the dephasing length , themaximum energy proton beam can be given as

Wmax ≃1

6(a20nc/ne)mec

2. (5)

It implies that the maximum proton energy in the snow-plow regime scales with a20 and n−1

e . This is quite anefficient acceleration scaling law.The snapshots of electron density, proton density, and

electrostatic field are plotted in Fig.2. The proton beamhas been continuously accelerated until laser pulse iscompletely depleted at t=12000Tl. When the protonbeam approaches the laser front, the beam loading causesa substantial reduction of the electrostatic field as shownin Fig.2(c) and 2(d). The theoretical values of the lon-gitudinal electrostatic field Ex = 14.9 in the unit of E0,the dephasing length lpd = 11125µm,the maximum en-ergy of proton beam Wmax = 532 GeV and the etchingvelocity vetch = 0.0036c,respectively. They are consis-tent with the simulation results. Fig.3 shows the energyspread (FWHM ) of the beam is less than 20 %. Themaximum proton energy and averaged one are 540GeV

1.19 1.1925 1.195 1.1975 1.2

x 104

0

100

200

300

400

500

600

x/λl

px(m

ic)

(a)

0 100 200 300 400 500 6000

20

40

60

80

100

120

energy (GeV)

Arb

.Un

it

(b)

FIG. 3: (Color Online) Longitudinal phase space for protonspx/mic (a) and energy spectrum of trapped protons (b) fromsimulations at t=12000Tl.

and 437GeV, respectively. There are 1.72% of the to-tal protons located inside this energy window. The pro-ton beam is compressed in the phase space and the finalquasi-monoenergetic beam is shorter than 100 µm.

As Fig.4(a) shows, proton beams move backward withrespect to the snowplough reference since the laser groupvelocity is greater than the proton beam. As a result,the distance between proton beam and laser front in-creases to reach the maximum linj in the initial injectionstage. Later it begins to decrease due to the erosion ofthe laser pulse front.The slopes of the solid lines repre-sent the etching velocities of the laser pulse for differentgas densities. The maximum longitudinal electrostaticfield, dephasing length, and maximum proton energy areplotted versus the density in Fig.4(b,c,d). It shows themaximum energy of the proton beam is increasing withplasma density decreasing. The dotted lines indicate thatthe pulse depletion length is shorter than the dephasinglength if the plasma density is smaller than 0.01nc. Inorder to increase the proton energy, we may increase theintensity of lase pulse and reduce the gas density, whichcan increase both the dephasing length and pump deple-tion length.

The density ranges over more than three orders ofmagnitude in 1D simulations, it is difficult to do a fullscale 2D simulations (e.g. TeV protons). A small sim-ulation box 40×1000λ2

l is chosen with a spatial resolu-tion of 5 cells/λl in y plane and 20 cells/λl in x plane.Each cell contains 4 particles for each species of gasplasma, 400 particles for foil plasma. The foil is thesame as in 1D simulations while the gas density is 0.2nc.The laser pulse is temporally and transversely super-Gaussian, I = I0exp(−(r/r0)

4− [(t − t0)/τ ]

8), wherer0=10 µm, t0=55 fs, and τ=55 fs are taken. Fig.5 showsthe snowplow layer,electrostatic field,and phase space aresimilar with 1D results while the maximum proton en-ergy reaches 50GeV that agrees with 1D simulations.Since protons gain energy mainly from the electrostaticfield [14], the return current and magnetic field in 2Dsimulations are not important for proton acceleration.Furthermore, the lost fraction of electrons in snowplow

4

0 5000 10000 150000

5

10

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20

25(a)

t/Tl

l inj/λ

l

0 0.02 0.04 0.06 0.08 0.10

10

20

30

40

50

ne/n

c

eEx/m

eωlc

(b)

0 0.05 0.1 0.15 0.210

2

104

106

ne/n

c

Ldp

/λl

(c)

0.05 0.1 0.15 0.210

1

102

103

104

ne/n

c

Wm

ax(G

eV)

(d)

FIG. 4: (Color Online)(a)Distance between proton beam andthe laser pulse front. It varies with time for different gas densi-ties, where blue, black, red and green curves represent the gasdensities of 0.01nc, 0.025nc, 0.05nc and 0.1nc, respectively;(b)Longitudinal electrostatic field eEx/meωlc;(c)Dephasinglength; (d) Maximum proton energy. Here the solid linesare theoretical curves and simulation results are marked bydifferent symbols(e.g. star,diamond and squre).

x/λl

y/λ l

(a)

0 200 400 600 800 1000

−20

−10

0

10

20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

900 925 950 975 1000−10

−5

0

5

10

15

20

25

x/λl

eEx/m

eωlc

(b)

800 850 900 950 1000−10

0

10

20

30

40

50

60

x/λl

px(m

ic)

(c)

0 10 20 30 40 500

2

4

6

8

x 106

γ

Arb

.Uni

t

(d)

FIG. 5: (Color Online)2D simulation results at t=990Tl.(a)Electron density distribution; (b)Electrostatic field onthe axis ;(c)Proton phase space; (d)Proton spectrum.

layer can be compensated from gas plasma and snowplowlayer is still kept at the end of simulations. The disad-vantageous multi-dimensional effects (e.g. hole boring,Rayleigh-Taylor instability) in laser foil interactions [7]are not observed in our simulations. These effects lead tovery stable proton acceleration in the multi-dimensionalsituation.

In conclusion, snowplow ion acceleration, using anultra-relativistically intense laser pulse irradiating on thecombination target, is presented. A 1D model that can

predict the proton dephasing length, pump depletionlength, and maximum proton energy is derived, whichis verified by 1D and 2D PIC simulations. It showsthat sub-TeV quasi-monoenergetic proton bunches canbe generated by a centimeter-scale laser wakefield ac-celerator, excited by a laser pulse with the intensity of1023W/cm2 and duration of 116fs. The final protonbeam is shorter than 100 µm and may be used to excitethe plasma wakefield to accelerate electrons to hundredsGeV, as Cadwell proposed [17].

We thank G.Mourou, Z.M.Sheng, C.T.Zhou, C. Y.Zheng, H. Zhang, B. Liu, and H.C.Wu for useful dis-cussions and help.This work was supported by Na-tional Nature Science Foundation of China (Grant Nos.10935002,10835003,11025523) and National Basic Re-search Program of China (Grant No. 2011CB808104).XQY would like to thank the support from the Alexan-der von Humboldt Foundation. TT is the holder of theBlaise Pascal chair of Ecole Normale Superieure.

∗ X.Yan@pku.edu.cn† xthe@iapcm.ac.cn

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