Laser acceleration of ion bunches at the front surface of overdense plasmas

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by  Andrea Macchi

Laser acceleration of ion bunches at the front surface of overdense plasmas
Andrea Macchi,1,∗ Federica Cattani,2 Tatiana V. Liseykina,2,† and Fulvio Cornolti2
1Istituto Nazionale per la Fisica della Materia (INFM) and POLYLAB,
Dipartimento di Fisica “E. Fermi”, Universit`a di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy
2Dipartimento di Fisica “E. Fermi” and INFM, Universit`a di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy
(Dated: February 2, 2008)
5
The acceleration of ions in the interaction of high intensity laser pulses with overdense plasmas
0
0 is investigated with particle-in-cell simulations. For circular polarization of the laser pulses, high-
2 density ion bunches moving into the plasma are generated at the laser-plasma interaction surface.
Asimpleanalyticalmodelaccountsforthenumericalobservationsandprovidesscalinglawsforthe
n
ion bunchenergy and generation time as a function of pulse intensity and plasma density.
a
J
PACSnumbers: 52.38.-r,52.38.Kd,52.50.Jm,52.65.Rr

1
3
The study of the interactions between ultra-intense
]
h laserpulsesandplasmashasprovedtobeaveryrichsoil
p where technological progress and fundamental physics
- meeteachother. Particularlyintriguingistheconceptof
m
laser-plasma based ion acceleration. From astrophysics
s
[1], to medical hadrontherapy [2], from proton radiog-
a
l raphy [3], to nuclear physics [4], from proton imaging
p
techniques [5], to nuclear fusion [6], the problem of ac-
.
s celeratingandmanipulatingchargedparticleswithlaser-
c
i plasma interactions offers a series of challenges ranging
s
fromfundamentalto appliedphysics,thus aclearunder-
y
h standing of the basic mechanisms is mandatory. Several FIG. 1: Ion density ni (top, solid line) electrostatic field Ex
p recent experiments have reported the emission of ener- (top, dotted), and (x,px) phase space projection (bottom)
[ getic ions from solid targets [7]. It is still a matter of at different times (labels) from 1D PIC simulations. Run

2 debate whether the ions are mainly accelerated at the parameters are aL=2, ne/nc =5, ℓ=2λL. E0=meωc/e.
v rear surface of the target (by the field generated by fast
3 electrons escaping in vacuum [8]) or at the front surface
102 ilnesvsoelvleinctgropshteantiocmsehnoacksu[9c,h1a0s,a1c1c],elbeyraatisoonlitbayryawcoalvliesi[o1n2-] bwohuenredanr0y,isotvheerdiennitsiealpelalescmtraonwditehnsnit0y/,nncc==ωmp2e/ωωL2L2/4>πe12,
or by ion trapping in a propagating double layer [13]. is the critical density for a laser with carrier frequency
1
4 Inthisworkweelucidateanevenmorebasicprocessof ωL, ωp is the plasma frequency and me, e are the elec-
0 ionaccelerationincoldplasmas,purelyrelatedtothefor- tron mass and charge. The laser field amplitude will
s/ mation of an electrostatic field due to the action of the be given in units of the dimensionless parameter aL =
c laser ponderomotive force on the electrons and, conse- (eEL/meωLc). In the PIC simulations, the resolution is
i
s quently, onthe ions via space chargedisplacement. This high enough to resolve low density regions, sharp gra-
y investigation shows both the necessity of a kinetic de- dients and the dynamics of both electrons and ions by
h
p scriptionofthisprocessandthe fundamentalroleplayed takingatleast20particlespercellatne =nc andaspa-
: by the laser light polarizationby showing the differences tial resolution better than 0.1dp where dp = c/ωp. For
v between circular and linear one. It will be shown by reference, in all simulations shown the target boundary
i
X particle-in-cell (PIC) simulations that circularly polar- is located at x=0 and the laser impinges on it at t=0.
r ized light gives rise to a “pulsed”acceleration and pro- The essentialfeatures ofthe generationofionbunches
a
duces ion bunches directed into the target. A simple canbeseenfromone–dimensional(1D)simulations. The
analytical model is used to explain the acceleration dy- laser pulse is circularly polarized, incident from the left
namics and for the deduction of scaling laws that relate sideandpropagatingalongthexaxis;itsamplituderises
the interactionparameterstothe energyacquiredbythe for6T (whereT =2π/ω isthelaserperiod)uptothe
L L L
ions. With respect to other concepts for laser ion accel- value a = 2 and then remains constant. The plasma
L
eration, the present mechanism with circularly polarized has a slab profile with n0 = 5nc and thickness ℓ = 2λL.
lightleadstoveryhighdensitiesinthebunches,asmight Electrons and ions are assumed to be cold, T =T = 0,
e i
be of interest for problems of compression and accelera- and ion charge and mass number are Z =1, A=1.
tion of high–density matter. Fig. 1 shows the ion density profiles and (x,p ) phase
x
We consider a laser pulse impinging on a cold, step– space projections at different times. At t =8T a sharp
L
2
and E en l /2≃2I /c (due to the balance between the
0 0 s L
total radiation and electrostatic pressures).
The electrostatic field on the ions starting at initial
positionsx <disaconstantoverthe trajectoriesofthe
0
ions, E = E (x ), and increases with x . Thus, these
x x 0 0
ions will never reach those with x > d, the ion density
0
will decrease and a ion “shelf” will be formed, the field
onthe leftmostionlayerbeing zero. Asforthe ionswith
initialpositioninthecompressionregiond<x <d+l ,
0 s
assuming that the electrostatic field is a function of x
0
for these ions too is consistent with the assumption that
the electrons follow a quasi–equilibrium dynamics and
the ponderomotive force equals the electrostatic field at
FIG. 2: (Color online) a): Schematic of the model profiles any time, the total pressure being always 2I /c. Being
L
of the ion density (green, thick) the electron density (blue, the electrostatic field a linear function of x , all ions will
0
thin) and the electric field (red, dashed) at the initial stage
reach the point B at the same time and the ion density
when the electrons are in equilibrium and the ions have not
willassumeaninfinitevaluethere,i.e. thehydrodynamic
moved yet. The label A indicates the electron front where
description breaks down. The ion bunch is thus due to
the laser evanescence starts, B indicates the point where the
ponderomotive force vanishes so that AB =ls. b): Compar- accelerated ions that cross the point B and are injected
ison between the ion bunch velocity observed in simulations intotheunperturbedplasmaregion,wheretheycanmove
(data points) and the model prediction (lines), as a function ballistically provided that the charge unbalance is neu-
of the laser pulse amplitude aL and the normalized density tralizedbyelectronsaccompanyingthe bunch. Thetotal
N = ni0/nc. The cases studied are: N = 100, Z/A = 1/2 number of accelerated ions per unit surface is n l .
(red, stars); N = 20, Z/A = 1 (blue, crosses); N = 5, p0 s
Note that the model assumes the point B, where the
Z/A=1/2 (green, diamonds).
field vanishes, not to move during compression, and pre-
dicts ions to have a a flat–top velocity distribution ex-
tending between 0 and v and thus the density to
max
peak of the ion density reaching up to seven times the be uniform in the region behind the ion front. Ac-
initial value is shown which moves inwards the plasma tually, since for the evanescence length l we expect
s
with an average velocity of about 0.013c leaving behind l ≃d ∼n−1/2, during the compression of the ion fluid
s p e
a low–density shelf with velocity linearly decreasing. At
the field will tend to penetrate deeper into the plasma
t = 12T , after the density at the peak has reached the
L keepingthefieldatthesurfaceandthetotalelectrostatic
maximumvalueofabout7n andthevelocityhasreached
0 pressureconstant. Thus,ionsbeyondpointAwillbe ac-
the value of 0.02c, a second density peak has appeared
celeratedby afielddecreasingintime andwillgetto the
on the right of the first peak and moves into the plasma
breaking point later. This effect causes the ion bunch to
at the speed of about 0.04c with the (x,p ) distribution
x be more localized both in coordinate and velocity space.
now clearly a two–valued function of x.
Our model thus gives a scenario in qualitative agree-
A simple model for this ion bunch generation can be ment with the numerical observations and also provides
inferred. When the laser pulse impinges on the plasma quantitative estimates and scaling laws. Denoting as
surface, electrons are quickly pushed inwardby the pon- x0 = ζ0 +d the initial position of an ion with mass mi
deromotiveforce, i.e. the steady part ofthe v×B force. and charge qi in the compression region, (0 < ζ0 < ls),
(Noticethattheoscillatingpartofthev×Bforceiszero the force acting on this ion is given by Fi = qiE0(1−
forcircularpolarization;this isthereasonwhy,asshown ζ0/ls). Thus, The velocity of an initially immobile ion
below, the interaction regime is completely different for is vi = (qiE0/mi)(1−ζ0/ls)t and the position is xi =
linearpolarization). Theelectronspileupleavingbehind (qiE0/2mi)(1− ζ0/ls)t2 +x0. The “breaking” time at
a charge depletion layer and giving rise to an electro- which all ions reach point B is τi = 2lsmi/qiE0. The
static field Ex back–holdingthem. We assume that they maximum velocity is thus vmax = p2lsqiE0/mi = 2va,
quickly reachan equilibrium position whereEx balances beingva =ls/τi theaverageionvelopcity,i.e. thevelocity
the ponderomotive force exactly. Let us take for sim- of the laser front. By relating the model parameters to
plicity a linear profile of Ex both in the depletion layer the laser intensity and plasma density we obtain
(E = E x/d for 0 < x < d) and in the compression
x 0
region(E =E [1−(x−d)/l ]ford<x<d+l ),which v Z m n 1 Am
x 0 s s a e c p
= a and τ ≃ (1)
implies a uniform electron density np0 in this region, see c sAmpne L i ωLaLrZ me
Fig. 2. The parameters E , n , d and l are related by
0 p0 s
the equations E = 4πen d (due to Poisson equation), where l ≃d has been assumed. The predicted velocity
0 0 s p
n (d+l ) = n l (due to global charge conservation), scaling has been tested via numerical simulations, yield-
0 s p0 s
3
FIG. 3: a) Ion bunch acceleration from a thin (ℓ = 0.1λL),
solid–density(n0/nc =100)targetirradiatedbyashortlaser
pulse with amplitude a = 5 and duration of 6TL (FWHM).
The ion density profile (thick line) and the (x,px) phase
space distribution (dotted) are shown at t = 13TL. b) Ion
density (top) and phase space (x,px) projection (bottom) at
t = 86TL, for the same simulation parameters of Fig. 1, but FIG.4: (Coloronline)Contoursofiondensityni/nc from2D
simulationsatthreedifferenttimes. Thelaserpulseimpinges
linear polarization of thelaser pulse.
fromtheleftandhasatransverseFWHMof4λL. Noticethe
different scales on thex and y axis.
ing a rather good agreement with the model, see Fig.
2. For instance, for the parameters of Fig. 1 the model
foam targets, collisional ion stopping is even less impor-
predicts v /c = 0.021 and τ ≃ 3.4T ; in the simulation
a i L
tant. Collectiveionstoppingintheplasmaisincludedin
the breaking of the ion front occurs at t ≈ 10T which,
L
the PIC simulation, and is found not to affect ion trans-
taking into account the pulse rise time of 6T , is in fair
L
port in the regimes addressed by our simulations.
agreement with the model prediction.
Atthe“breaking”oftheionprofilewiththeformation Themechanismofionbunchformationoutlinedabove
of the bunch, the equilibrium between the electrostatic works cleanly for circular laser polarization. For lin-
and the ponderomotive forces on electrons is lost and, if ear polarization the oscillating part of the v×B force
thelaserpulseisnotover,electronsrearrangethemselves causes strong electron heating (up to MeV energies)
to provide a new equilibrium. The process of bunch for- which changes the regime of ion acceleration qualita-
mation might then restartalthough with different initial tively. Fig. 3 shows results from a simulation with pa-
conditions,sothatthesimplemodelisnotadequateany- rametersidenticaltoFig.1butthelaserpolarizationthat
moreforquantitativepredictions. InFig.1asecondwide waschangedtolinear. Noclearpropagatingionbunchas
bunchactuallyappearsatt≃15T andthenpropagates that of Fig. 1 is observed. The hot electrons form a De-
L
into the plasma. At later times, bunches with shorter byesheathleadingtoionaccelerationtowardsvacuumat
duration are generated at a higher rate. the backsurfaceassoonastheyreachitwhile afew ions
Forpotentialapplications,itisinterestingtostudythe are also accelerated from the front surface towards vac-
possibilitytoproduceasinglebunchfromathinsolidtar- uuminasimilarway(Fig.3). Nearthelaserpulsefront,
get. Results for the case of an n /n = 100, Z/A = 1/2 the phase space structures look similar to those gener-
0 c
targetwith a thickness ℓ=0.1λ and a short pulse with atedby the shock accelerationinvestigatedinRef.[10]at
L
intensity a = 5 are shown in Fig. 3. The temporal en- higher intensity. For the present parameters, sheath ac-
L
velopeforthe fieldisgivenbysin2(πt/τ) withτ =12. A celeration dominates at the time shown. In contrast, for
singlebunchofionswithvelocity≈0.017cisproducedat circular polarization, it is found that most of the elec-
t≈5.5T and leaves the target from the back side. The trons are heated up to energies of a few keV behind the
L
ion bunch charge is completely neutralized by electrons ion bunch and at the laser–plasma interaction surface.
both inside and outside the target. Notice that in the case of shock acceleration (see e.g.
Collisionalionstopping(notincludedinthePICcode) Ref.[10]) the laserpulse acting asa pistondrivesa shock
should not affect ion bunch generation strongly for thin, waveintotheplasmawhichreflectstheionsthusacceler-
low–Z targets. In fact, taking for the example of Fig.3 ateduptovelocitiestwicetheshockspeed. Inthepresent
λ = 0.5 µm, so that n = 4×1023 cm−3, the stopping case the fastest ions have twice the piston velocity, i.e.
0
power in the cold plasma given by the Bohr formula [14] the velocity of the laser reflection front at the breaking
is≈0.19Z2MeV/µmforionswithv =0.017candcharge time, and they come from behind the front. It is also
Z =A/2, and the stopping length is thus ≈1.4Z−1 µm, worthtonoticethatthepresentmechanismofionbunch
lowerthanthe targetthicknessintheexampleifZ <28. formation is of electrostatic and kinetic nature while a
Forlowerdensities,typicalofpreformedplasmasor,e.g., purely hydrodynamic description is not adequate.
4
tureofthelaserpulse. Theripplingis“imprinted”inthe
iondensityatlatertimes,buthasnonoticeableeffecton
the dynamics of ion acceleration. We conclude that 2D
effects do not qualitatively affect ion bunch formation.
Inconclusion,aregimeoflaserionaccelerationincold
overdenseplasmasbycircularlypolarizedpulseshasbeen
characterizedby means of PIC simulations and a simple
model. The dynamics is qualitatively different from the
caseoflinearpolarizationwhereionaccelerationisdriven
by fast electrons. With the present scheme one may ob-
tain moderate ion energies but very high ion densities
and low beam divergence.
We acknowledge useful discussions with S. Atzeni and
F. Pegoraro. Part of the numerical simulations were
performed on the Linux cluster at the CINECA facility
FIG. 5: (Color online) The (y,px) and (px,py) phase space (Bologna, Italy), with the support of the INFM super-
projections [a)-b)] and the energy spectrum [c)] of ions from computing initiative.
the2D simulation of Fig. 4, at t=47TL.
Weperformed2Dsimulationstoshowthatthemecha-
nismforbunchformationisalsoatplayinmorethanone ∗ Electronic address: [email protected]
dimension, and in particular in the case of a laser pulse † OnleavefromInstituteforComputationalTechnologies,
withafinitetransversewidththatbendstheplasmasur- Novosibirsk, Russia
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(2002).
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