Ultra-Intense Laser Pulse Absorption and Fast
Particles Generation at Interaction with
Inhomogeneous Foil Target
A.A. Andreev1, T. Okada2, and S. Toraya2
1
Institute for Laser Physics, St.Petersburg, Russia
Tokyo University of Agriculture and Technology, Tokyo, Japan
2
Abstract. The absorption of a short laser pulse with duration 40 fs and the intensity more than
1018 W/cm2 at the interaction with foil targets is analyzed in the theory and particle-in-cell (PIC)
simulations. Initially, foil target density distribution has a smooth gradient with a variable scalelength of plasma density inhomogenity L. Laser absorbed energy transforms into the energy of fast
electrons, which oscillate in the foil and partly go out from the target transforming their energy into
accelerated ions. The analyses of the different mechanisms of ion acceleration in foil plasma, the
influence of L and other plasma parameters on ion acceleration have been done. The angular
distributions of fast electron and ion beams are calculated. The optimum laser plasma conditions for
the maximum ion acceleration are found.
INTRODUCTION
Fast particles generated in laser plasma interaction can be used in many applications
from technology to medicine and even for the initiation of tabletop nuclear reactions.
Fast ion generation by the interaction of an ultra short high intense laser pulse with
plasma has been demonstrated already in some recent theoretical1"4 and experimental5"9
papers, where the maximum ion energy up to 0.5 GeV is observed10. Different schemes
of fast ion generation have been proposed in gas10 and solid11 targets. At the moment,
the maximum conversion efficiency of laser to ion energy of fast ions has been obtained
for foil targets11. It has been shown that energy of a laser pulse could be converted into
fast ion energy with good efficiency. Some simulations1" have shown that under these
conditions the main mechanisms of ion acceleration are the ambipolar field from fast
electrons in a sharp-gradient plasma and the Coulomb explosion. It has been shown that
the main mechanism of ion acceleration from the foil rear side is the ambipolar
acceleration from fast electrons accelerated by a laser field and leaving a foil to create
an accelerated field. Most experimental high power lasers produce a pre-pulse, and for
this reason the foil target transforms into a plasma layer with a smooth gradient. A
collimated ion beam can be achieved by focusing an intense laser onto the surface of a
solid film. Due to the small diameter of the laser spot and the anisotropy of the hot
electron velocity distribution, the ion emittance can be comparable to or even better
than that of electrostatic accelerators. Fast ions accelerate along a normal to foil surface
because the ambipolar force is doing on ion along this direction. It is clear that these
ions can be focused by a curve foil at some point and it has been shown in a numerical
simulation but without any optimization of this process. In the present paper, we make
CP634, Science of Superstrong Field Interactions, edited by K. Nakajima and M. Deguchi
© 2002 American Institute of Physics 0-7354-0089-X/02/$ 19.00
303
an attempt to develop an analytical model to analyse the mechanisms of ion acceleration
and, based on this model and our PIC simulations, to find the optimum foil target.
In this paper we tried to analyze some positive and negative processes for optimum
impedance formation of the ion jet during its acceleration by a shaped laser pulse
interacted with a curve foil target with a possibility to focus these ions at some distance.
PIC SIMULATONS
We apply a PIC method to simulate the interaction of a plasma layer with an intense
ultra-short laser pulse. The method is based on the electromagnetic PIC and it is
appropriate for the analysis of the dynamics of over-dense plasmas created by an
arbitrary polarized, obliquely incident pulse laser. The 2D (in the rectangular Cartesian
coordinate system) relativistic, electromagnetic code is used to calculate the interaction
of an intense laser pulse with an over-dense plasma. The calculation with movable ions
is carried out for a plasma with a variable initial density profile. Simulations are
18
2
performed at wavelengths of 1 fim for a laser intensity / > 10 W/cm . The laser pulse
has a Gauss shape with a duration > 20 fs. The incident angle of the pulse is 0°. The
time step is set to 0.1/#>p, where cop is the initial plasma frequency and the spatial step
0.2c/a) . The number of electrons is 107 and the same for the ions. Initial electron
temperature is 1 keV and ion temperature is 800 eV. The thickness of the layer varies
from 1 jiim to several jum.
Figure 1 shows the schematic view of the simulation model. For simplicity, the
uniformity in the z-direction is assumed. The relativistic equation of motion and the
Maxwell equation are solved for the components x, y, px, py, pz and Ex, Ey and Bz
= qj(E + v XB),
= pJ9
dEldt = -jj + c2rotB,
= - rotE.
Here qj and m/ indicate the charge and mass of a particle, respectively and jr} is the
current density.
Target
r
Vacuum
Vacuum
(a)Target
(b)Simulation Box
FIGURE 1. The simulation geometry.
304
FAST ELECTRON GENERATION
Critical for ion acceleration is the efficiency of the laser-energy conversion into a
high-energy electron component, since the latter can produce the requisite strong
electrostatic fields. Numerical simulations show that for / > 1019W/cm2 intensity range,
the absorption coefficient becomes independent of the angle of incidence, and the
absorption is about 10 % without pre-pulse. The laser pre-pulse increases the scale of
plasma inhomogenity L and the absorption coefficient 77. It was shown that in the range
1018-1020 W/cm2 there is 77 °c L. Follow11 we will use the scaling of rf(LJ) as:
,0.7
rj(LJ)=(0.
(1)
where / - radiation intensity in units 1018 W/cm2, L = LcoL/c. L « cs tpi . Here cs - ion
sound velocity and tpi- pre-pulse duration. In Figure 2 we see the simulation result,
which confirms our model approximation.
wLt=133
5.5
5
4.5
4
3.5
3
2.5
0.5
1
1.5
2
2.5
gradient length x=LoJL/c
FIGURE 2. Electron energy dependence on plasma density gradient L.
We make an estimate of the number of fast electrons accelerated by laser pulse as nef
= Ke(I)£L/£e, where £L is the laser pulse energy, Ke(I) is the transformation coefficient of
laser energy into fast electron energy, and ee is the energy of an electron. It was shown
that in the range 1018-1020 W/cm2 , Ke(I) has a linear rise from 0.03 up to the 77 for the
subpicosecond laser pulses. At a laser intensity of more than 1020 W/cm2 , Ke(I) ~ 77.
This means that all the absorbed energy being assumed to be transferred to the motion
of high-speed electrons. The average energy of an individual fast electron is obviously
specified by the wave field strength inside the skin-layer
(2)
The maximum energy of a fast electron accelerated by a laser field in a plasma corona
can be calculated by this formula5:
mec2 (po/mc)(vE/cf,
305
(3)
here VE = eElmo), initial electron impulse - po ~ WIVE , it determined by acceleration in
laser field on L In Figure 3 we see the simulation results of the energy of fast electrons
(solid line) and ions (dotted line), which looks like the dependence on (2). As we
consider / >1019 W/cm2 for fast electron energy we take next expression: Eeh = Neh£eh ~
T]L£L where £eh ~ ilJlcneh , and for fast electron number: Neh=TjL£i/ seh. In Figure 4, we
see on the front side of the foil plasma target the deepness of the electron density from
the laser ponderomotive pressure and accelerated electrons from the surface are
cumulated on the laser axis. On the foil rear we see the "fountain" of electrons, which
go out and come back to the foil surface from the electrostatic field influence. This
structure looks like a ring of electron density on the rear foil surface, which has been
observed in the experiment. To understand this effect in more detail, we consider the
analytical model:
On the initial stage, when electrons are separated from ions (Ko^/^c/v/), a laser beam
produces on a foil the electrostatic field of the spot with the electric charge Q:
E = 2no = 2Q/R2
here R is laser focal spot radius.
The time of movement in such a field of fast electron of a velocity ve is determined by
this next formula:
t = 2vemyleE = vemjR2/eQ
here y is electron Lorentc factor.
The "fountain" effect produces a ring of non-compensated electric charge on the foil
surface with internal radius /?, external radius /?+/ and height h, which are determined
by the next formulas:
h = ymve2/2eE = seR2!2eQ
I = ve (0-AO)t = 0(l-pe2)(2eeR2/eQ)= 0(2eeR2/eQ^).
A magnetic field from electron current produces a deviation angle:
AO- eveH9tlcmve ~ 0 v2lc2
here 9< 1 - angle of electron scattering in a foil target.
The initial can be calculated from the scattering model as follows:
<tf> = (21MeV/se)2 (4/137)Z2r02niLfoilln(183Z ~113).
(4)
In the late stage of the interaction we see from Figure 5 an electron density perturbation
because instability develops and its increment is big enough increment
2>
STC m
2-rii
306
that at this time there is such a modulation of density. As a plasma surface at this
moment has not clear curvature the electrons do not acumulate on the laser axis as
before and focused jet disappear. These electrons propagate through foil with a
scattering on some angle <0>, but the main part reaches the rear foil side and goes out.
These electrons accelerate ions, but some part of it comes back to the foil surface as we
see in Figure 4, if its energy is not enough to overcome the electrostatic barrier:
eeh > ezneh nLcd(l-2Lcz/dz)
(5)
where d is laser spot size and Lc plasma density gradient.
700 i——————————————————————————
FIGURE 3. The electron and ion temperature dependence on laser intensity.
at t=ioo
300
250
g.150
II k.
. v;-;-;:.:|
s»
fe-
•g^jj
100
'••'•.- 'Sj
|;^;. -'-•
so
n
50
100
150 . 200
250
300
350
400
50
100
150
250
300
350
400
450
OJLX/C
0
200
FIGURE 4. Initial stage of laser pulse interaction with foil surface.
307
450
100
120
140
160
180
200
220
200 205 210 215 220 225 230 235 240 245 250
FIGURE 5. Spatial distribution of electron density. Laser beam parameters : diameter 8 /z m,
/=1019 W/cm2, £=40 fs. Maximum electron density ne = 4nc, plasma density gradient L=l n m,
plasma slab length 2 # m.
FAST ION PRODUCTION
It is already well known that ions are accelerated on the front side of the foil target
and its rear. In the case of the front foil side ion generation and ion acceleration volume
is: V = n(d!2)2Lc and fast ions get electrostatic energy: ecp~£eh from charge separation:
2
i2 1/2
£ih ~Z*e(p ~3Z* me ( TfjJis dLc//i ) .
//^\
(o)
The fast proton number Ni we get from the flow of an impulse conservation low:
The average energy of fast proton is calculated from the formula :
This dependence is close to the ion temperature dependence (dotted line) from Figure 3.
The total number of fast protons, we can get from quasi-neutrality: Nt « Neh ~ WI^L
/<£e>. It is two times less than the number of ions than the above formula because in
this case we have ion movement into two sides. Then the total energy of fast ions is: Et
~ (nc/np) 7/L8L, where from simulations nc/np~0.5. In Figure 6, we get the ion space
distribution by the PIC simulation. The rate of transversal velocity to the longitudinal
one (along laser axis) decreases with the increasing of laser intensity in Figure 7. Ion
acceleration on the rear side of a foil target is the most important point for jet
formation. In this process we supposed that fast electrons with a concentration of neh
move from ions in the distance of rd = veh/G)ph- Then these electrons create the electric
charged plane with the electric field:
308
Eam=2mneh veh/a)ph = E
We suppose that foil plasma has a sharp boundary because it has a short laser pulse and
no time for a thermal wave to get to this side. The velocity of an ion accelerated by this
ambipolar force is:v/ = (ZIM)EamtL, and their energy:
:
= Th (nhlnc)(melmi)
(7)
From (7) we see that ion energy is proportional to the electron average energy, which
means that the dependence of this energy from the plasma gradient is approximately the
same as in Figure 2 (see Figure 8). The number of fast electrons we can estimate as
before: n^=r]LeiJ eeh. The number of fast protons HI can be estimated again from the
pulse stream conservation of electrons and ions and total ion energy is: Ej = nte^
0^1=333
FIGURE 6. Spatial distribution of ion density at laser intensity /= 1019 W/cm2, plasma
density gradient L=l # m and plasma slab length 2 JJL m.
,.0
1.2 i,
1.1
1
1 0.9
^ 0.8
>4
0.7
0.6
0.5
''"a"'-..
0.4
I
0.3
18
18.5 19
19.5 2<
log I
FIGURE 7. The fast ion velocity rate dependence at laser intensity.
309
6ULt=133
3.5
3
^ 2.5
I
2
CO
1.5
1
0.5
1
1.5
2
2.5
gradient length x=LcuL/c
FIGURE 8. Fast ion energy dependence on plasma density gradient.
CONCLUSIONS
Focused before the foil target electron bunch is produced by a curve surface of
plasma, a critical density from a ponderomotive laser beam pressure at an early stage of
the process occurs. After instability developed the surface is modulated and the bunch
disappeared. A laser beam produces a beam of fast electrons and fast ions from the rear
surface of the foil target. The average energy of accelerated electrons and ions depends
on the laser intensity as a square root. The angle of velocity of the ion at a maximum
energy decreases on the laser intensity approximately as a square root. The increase of
scale-length of plasma in-homogeneity on some interval increases the laser absorption,
particle energy and decreases the angle distribution of the generated ion bunch from the
rear side of a foil target.
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