Generating sub-TeV quasi-monoenergetic proton beam by an ultra-relativistically intense laser in the snowplow regime

1 MB·English

by  F. L. Zheng

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, China
2Graduate School, China Academy of Engineering Physics,
P.O. Box 8009, Beijing 100088, Peoples Republic of China
3Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing 100088, China
1 4Fakult¨at f. Physik, LMU Mu¨nchen, D-85748 Garching, Germany
1 (Dated: January 18, 2011)
0
2 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 pressure
n
accelerationcanbetrappedandacceleratedbythelaserplasmawakefield. Thetheorysuggeststhat
a
sub-TeVquasi-monoenergeticprotonbunchescanbegeneratedbyacentimeter-scalelaserwakefield
J
accelerator, drivenbyacircularly polarized (CP)laser pulsewith thepeakintensityof1023W/cm2
7
and duration of 116fs.
1
PACSnumbers: 52.38.Kd,41.75.Jv,52.35.Mw,52.59.-f
]
h
p
Laser-driven ion acceleration has drawn attention for quasi-monoenergeticprotonbunchescanbegeneratedby
-
m manyapplications,e.g.,protoncancertherapy[1],fastig- a circularlypolarized (CP) laser pulse with the intensity
s nition of thermonuclear fusion by protons [2],conversion of 1023 W/cm2 and duration of 116 fs. A combined tar-
a of radioactive waste [3], high energy physics accelerator get in Fig.1 (a) shows a planar hydrogen target that is
l
p [4]and astrophysics [5]. All these applications may be followedbyunderdenseheavy-ionbackgroundgaswitha
. enabledby the introductionofnear-futurehigh-intensity charge-to-massratioof1/3. WhenthefoilthicknessD is
s
c lasers (i.e. ELI), which will be capable of producing equalto 1 nca λ ,amono-energeticprotonbeamisgen-
si pulses with intensity 1022−1024 W/cm2 [6]. eratedfro2mπnae m0ovlingdoublelayer[14],Fig.2(a)suggests
y that protons and electron layer are very close. The rel-
Radiation Pressure Acceleration (RPA)[7] accelerates
h
ativistic protons can quickly overrunthe laser front that
p ions efficiently and theoretical studies show that GeV
results in shorter dephasing length and lower proton en-
[ proton beam may be generated by an ultra-intense laser
with intensities above 1022W/cm2 [8, 10, 11]. Although ergy. If a thinner foil satisfies
2
v the acceleration field is over tens TeV/m, the accel- 1 nc
l <D < a λ , (1)
0 eration length is normally shorter than hundreds mi- 0 2πn 0 l
e
5 crons. Because of the energy scaling, it is difficult to
the ponderomotive force quickly pushes electrons to the
3
further increasethe ionto veryhighenergies(i.e. TeV).
2 rearofthefoil(seeFig.2(b)), theelectronsthatarepiled
Quasi-monoenergetic GeV electron beam can be gener-
. up there reaches density far greater than γn . Mean-
1 atedbyacentimeter-scalelaserplasmawakefieldacceler- c
whileprotonsinthefoilareacceleratedby the separated
0
ator(LPWA) [12]; however, it is not easy to capture and
1 charge field. Here l0,ne,nc, and λl are the plasma skin
continuously accelerate the nonrelativistic ions. Shen et
1 depth, plasma density,critical plasma density, and laser
v: al.[13]reportsalaserpulsewithanultrarelativisticinten- wavelength, respectively. a0 = eA/meωlc is the normal-
sity can excite an electrostatic field, which can capture
i ized laser amplitude and A is the laser vector potential.
X protons from underdense plasma and accelerate them to γ =(1+Iλ2/1.37×1018)1/2,I islaserintensity,λ inunits
r tens GeV. Recently Yu et al. [14] found that the under- l l
ofmicron. The electron-layerinrearofthefoilispushed
a
dense gas behind a thin foil can accelerate the proton
outofthe foilby thelaserpulsesoonbeforedouble-layer
beam to 60GeV by driving high-amplitude electrostatic
(consists of electrons and protons) is formed. The layer
fields moving at a high speed. There is an important
runsinthelowerdensitygasandisfurtherpushedbythe
questionwhetheritispossibletogenerateTeV-levelpro-
laserpulse like light-sailto generate ”Snow-plow”effects
ton beams by the laser plasma wakefield accelerator.
onenhancementofelectronnumber. Theelectronsingas
Thisletterreportsanultra-relativisticallyintenselaser plasma drew away from the foil generate a large electric
can excite the plasma wakefield that traps the relativis- field,thentheprotonsthathavebeenacceleratedtoGeV
tic proton beam and accelerates it over a few centime- level start long march in gas by the wakefield accelera-
ters. An analytic model is derived for proton acceler- tion. This acceleration scheme is named the snowplow
ation that extends the applicability of Esarey’s classic regime. We alsonotice aconnectionofthe aboveplasma
plasma wakefield theory [15] and has been confirmed by acceleration with ”snowplow” acceleration in Ref. [16].
ourPICsimulations. Ourscalinglawshowsthatsub-TeV In order to grasp the main physics and to construct a
2
N = n /n = 20. Behind the hydrogen foil is the tenu-
0 c
ousheavy-ion-gaswithacharge-to-massratioof1/3and
a plasma density ofn =0.01n .Here the simulationbox
e c
has a size of 12000 λ in x plane and a space resolution
l
of 20 cells/λ. Each cell is filled with 10 macroparticles
l
for gas plasma and 20000macroparticles 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 through
the foil and piles electrons in the underdense gas plasma
FIG.1: (coloronline)(a)Acceleration scheme,theinitialden- asFig.1(b)shows,whichisoneofmostimportantandre-
sityofthehydrogenfoiln0/nc =20,thicknessD=0.5λl;the markablefeaturesinthesnowplowregime. Heretheelec-
plasmadensityofthetenuousgasne/nc =0.01,thelengthof tron layer and the positive electrostatic field are called
the gas plasma is 12000λl; (b)Sketch map for the snowplow
thesnowplowlayerandsnowplowfield,respectively. The
process,thedynamicaldensityofionsisni andelectronden-
sity in the snowplow layer is ncom. The position B indicates snowplow layer can continuously trap electrons in the
thelaserpulsefrontandAisanarbitrarypointforcalculating snowplow process, so the acceleration field exists stably
thelocal electrostatic field in thesnowplow region. for a long distance. In the snowplow region, it is found
that the length of the snowplow region is equal to l =
s
(γn /n )1/2λ and the snowplow velocity approximately
c e l
equals to the group velocity of laser pulse in the under-
physical model , in this letter one-dimensional(1D) the- dense plasma. First we estimate the electrostatic field
ory is derived and testified by 1D and 2D PIC simula- E atanarbitrarypointx inthesnowplowregion. The
A A
tions. componentfromionsisE′ =4πen x −4πen (l −x ).
A i A i s A
Considering the classic 1D plasma wakefielddriven by Thecontributionfromthesnowplowlayeratpositionx
A
a laser pulse [15], two typical features should be no- is E′′ = 4πenels. The longitudinal electrostatic field at
A 2
ticed in the ultra-relativistic intensity : (i)Lengthening x is E = E′ +E′′ = 4πen (2x −l /2). Therefore,
of the plasma wavelength ls = (γnc/ne)1/2λl; thAe maxAimumAsnowpAlow field aet theAlasesr front xB is
(ii)Enhancement of the nonlinear wavebreaking field
E = 3π(γn /n )1/2E , where the critical plasma den- E =3π(γn /n )1/2E . (2)
B e c 0 B e c 0
sity n = πm c2/e2λ2. The highly nonlinear wake-
c e l
field wavelength l and wavebreaking field E are in- SincetheinjectedGeVprotonsarestillslowerthanthe
s B
creased by a factor of γ1/2 over the non-relativistic ver- laser pulse at the beginning, the ion beam should be in-
sion, where the electric field strength is normalized by jectedintothefrontsideofsnowplowregion. Afterwards
E = m ω c/e. In this regime we adopt a CP laser protons are gradually accelerated and overrun the laser
0 e l
pulse with a wavelength λ = 1µm and a peak intensity pulse ( see Fig.4(a)). The optimized distance between
l
I =1.7125×1023W/cm2,correspondingtopeakdimen- injected proton beam and laser pulse front is defined by
0
sionless laser amplitude a0 = 250. When the foil thick- linj. It is approximately 14ls in our simulation. Once
ness D satisfies Eq.(1), the relativistic protons can be the proton beam is injected into the wakefield, its maxi-
captured by the excited wakefield in the underdense gas mumenergydepends onthelaserpumpdepletionlength
and be stably acceleratedin the snowplow regime over a and proton dephasing length. The laser pump depletion
longdistance. Ananalyticmodelisdevelopedtoestimate length Lpd is estimated [15]: Ex2 ·Lpd ≃ El2 ·Ll, where
the acceleration field, dephasing length, pump depletion Ex is the longitudinal electrostatic field driven by laser
length, and maximum proton energy. The acceleration pulse in the snowplow wakefield and Ll is the length of
length in snowplow regime is proportional to a3/2 and laser pulse. Assuming the group velocity of laser pulse
maximumprotonenergyscaleswitha2. Itshowst0hatsub is extremely close to light velocity in free space, pump
0 depletion length can be defined as
TeVmonoenergeticprotonbeamisgeneratedinthelaser
intensityof1023 W/cm2. Toourbestknowledge,itisthe L
best efficient proton acceleration mechanism in terms of Lpd ≃ l ·c, (3)
v
etch
acceleration length and maximum energy.
First we carried out simulation using a fully relativis- where vetch is the depletion velocity of laser pulse. The
tic particle-in-cell code (KLAP) [8, 9]. The laser pulse etching velocity of laser pulse reads vetch ≃ 9aπ02nncec. It
has a trapezoidal shape longitudinally with 15 λl flat shows the etching velocity of laser pulse scales as ne/a0.
top and 20λ ramp on the rise side. The ramp profile Thedephasinglengthofprotonisgivenapproximately
l
is a = a0sin2(πt/40). At time t=0 it is normally inci- by Ldp ≃ vp−(vlgin−jvetch) · vp, where linj is the injected
dent from the left on a uniform, fully ionized hydrogen length between proton beam and laser pulse front, v
g
foil of the thickness D = 0.5µm and normalized density is the laser group velocity, v is the proton velocity. It
p
3
(a) (b) (a) (b)
8 600 120
ωeE/mc,n/nxelec11055000 nnep ωeE/mc,n/nxelec 0246 nEnepx p(mc)xi234500000000 Arb.Unit104680000
100 20
−2
0
0 10 20x/λl30 40 50 20 25 30 x3/5λl 40 45 50 10.19 1.1925 1x.1/λ9l5 1.1975 x 1014.2 00 100 200 energ30y0 (GeV4)00 500 600
(c) (d)
20 15
n×2.5 n×2.5 FIG. 3: (Color Online) Longitudinal phase space for protons
c,n/nec 11055 Enepx×40 c,n/nec10 Enepx×40 psixm/umlaicti(oan)saantdt=en1e2r0g0y0Tsple.ctrumoftrappedprotons(b)from
ωl ωl 5
me 0 me
eE/x −5 eE/x 0 and 437GeV, respectively. There are 1.72% of the to-
−10 tal protons located inside this energy window. The pro-
−15 −5 ton beam is compressed in the phase space and the final
4900 4925 4950 4975 5000 1.19 1.192 1.194 1.196 1.198 1.2
x/λl x/λl x 104 quasi-monoenergetic beam is shorter than 100 µm.
As Fig.4(a) shows,protonbeams movebackwardwith
FIG. 2: (Color online) Electron density ne/nc(blue solid respecttothesnowploughreferencesincethelasergroup
line),protondensitynp/nc(theredsolidline)andlongitudinal velocity is greater than the proton beam. As a result,
electrostaticfieldeEx/meωlc(theblackdashedline).(a)When
D = 21πnncea0λl double-layer is formed at t=50Tl. When tchreeasdeissttaonrceeacbhettwheeemnapxriomtuomn bleaminathnedinlaitsiearlfinrojencttiionn-
D=0.5µm snapshots are taken at time (b) 50Tl, (c) 5000Tl, inj
(d)12000Tl. stage. Later it begins to decrease due to the erosion of
the laser pulse front.The slopes of the solid lines repre-
sent the etching velocities of the laser pulse for different
can be further simplified when both vg and vp are very gas densities. The maximum longitudinal electrostatic
close to lightspeed, which means vetch ≫(vp−vg). The field, dephasinglength,andmaximumprotonenergyare
proton dephasing length is: plotted versus the density in Fig.4(b,c,d). It shows the
maximum energy of the proton beam is increasing with
1
Ldp ≃ 36π2(a0nc/ne)3/2λl. (4) plasmadensitydecreasing. Thedottedlinesindicatethat
the pulse depletion length is shorter than the dephasing
In case of Ll > 41(a0nc/ne)1/2λl the laser pump deple- length if the plasma density is smaller than 0.01nc. In
tion length is greater than the dephasing length , the orderto increase the protonenergy,we may increase the
maximum energy proton beam can be given as intensity of lase pulse and reduce the gas density, which
canincreaseboth the dephasing lengthandpump deple-
1
W ≃ (a2n /n )m c2. (5) tion length.
max 6 0 c e e
The density ranges over more than three orders of
It implies thatthe maximum protonenergyin the snow- magnitude in 1D simulations, it is difficult to do a full
plow regime scales with a2 and n−1. This is quite an scale 2D simulations (e.g. TeV protons). A small sim-
0 e
efficient acceleration scaling law. ulation box 40×1000λ2 is chosen with a spatial resolu-
l
The snapshots of electron density, proton density, and tion of 5 cells/λ in y plane and 20 cells/λ in x plane.
l l
electrostatic field are plotted in Fig.2. The proton beam Each cell contains 4 particles for each species of gas
has been continuously accelerated until laser pulse is plasma, 400 particles for foil plasma. The foil is the
completely depleted at t=12000T. When the proton same as in 1Dsimulations while the gasdensity is 0.2n .
l c
beamapproachesthelaserfront,thebeamloadingcauses The laser pulse is temporally and transversely super-
asubstantialreductionofthe electrostaticfieldasshown Gaussian, I = I exp(−(r/r )4 − [(t − t )/τ]8), where
0 0 0
in Fig.2(c) and 2(d). The theoretical values of the lon- r =10 µm, t =55fs, andτ=55 fs aretaken. Fig.5 shows
0 0
gitudinal electrostatic field E = 14.9 in the unit of E , thesnowplowlayer,electrostaticfield,andphasespaceare
x 0
the dephasing length l = 11125µm,the maximum en- similar with 1D results while the maximum proton en-
pd
ergy of proton beam W = 532 GeV and the etching ergy reaches 50GeV that agrees with 1D simulations.
max
velocity v = 0.0036c,respectively. They are consis- Since protons gain energy mainly from the electrostatic
etch
tent with the simulation results. Fig.3 shows the energy field [14], the return current and magnetic field in 2D
spread (FWHM ) of the beam is less than 20 %. The simulations are not important for proton acceleration.
maximum proton energy and averaged one are 540GeV Furthermore, the lost fraction of electrons in snowplow
4
predict the proton dephasing length, pump depletion
(a) (b)
25 50 length, and maximum proton energy is derived, which
20 40 is verified by 1D and 2D PIC simulations. It shows
c
λl/injl1105 ωE/mxel2300 tbheatgesnuebr-aTteedVbqyuaasi-cmenotniomeenteerrg-sectaicleplraosteornwbauknecfiheelsdcaacn-
e
5 10 celerator, excited by a laser pulse with the intensity of
0 0 1023W/cm2 and duration of 116fs. The final proton
0 5000 10000 15000 0 0.02 0.04 0.06 0.08 0.1
t/Tl ne/nc beam is shorter than 100 µm and may be used to excite
106 (c) 104 (d) the plasma wakefield to accelerate electrons to hundreds
GeV, as Cadwell proposed [17].
V) 103
λL/dpl104 W(Gemax102 ZhWeneg,thHa.nkZhGan.Mg,ouBr.ouL,iuZ,.aMn.dShHen.Cg,.WCu.Tfo.Zrhuosue,fuCl.dYis-.
cussions and help.This work was supported by Na-
102 101 tional Nature Science Foundation of China (Grant Nos.
0 0.05 0.1 0.15 0.2 0.05 0.1 0.15 0.2
ne/nc ne/nc 10935002,10835003,11025523) and National Basic Re-
search Program of China (Grant No. 2011CB808104).
FIG.4: (ColorOnline)(a)Distancebetweenprotonbeamand XQY would like to thank the support from the Alexan-
thelaserpulsefront. Itvarieswithtimefordifferentgasdensi- der von Humboldt Foundation. TT is the holder of the
ties,whereblue,black,redandgreencurvesrepresentthegas Blaise Pascalchair of Ecole Normale Superieure.
densities of 0.01nc, 0.025nc, 0.05nc and 0.1nc, respectively;
(b)Longitudinal electrostatic field eEx/meωlc;(c)Dephasing
length; (d) Maximum proton energy. Here the solid lines
are theoretical curves and simulation results are marked by
different symbols(e.g. star,diamond and squre).
∗ [email protected]
† [email protected]
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