1. INTRODUCTION
In recent years, we have seen remarkable progresses in laser intensity
improvement (Mourou et al., 1998; Strickland & Mourou, 1985), and new particle
acceleration schemes have been explored, such as electron acceleration in
a vacuum (Hafizi et al., 1997; Kawata et al., 1991; Kong
et al., 2003; Malka & Miquel,
1997), high-energy ion production (Allen et
al., 2003; Nakamura & Kawata,
2003; Passoni & Lontano, 2004; Pommiers & Lefebvre, 2003; Ramirez et al., 2004; Shorokhov & Pukhov, 2004; Wilks
et al., 2001; Chen & Wilks 2005). There was also a remarkable progress in the
simulation of high intensity laser plasma interactions, which were
discussed recently (Lebo et al., 2004;
Limpouch et al., 2004). When an intense
laser irradiates thin slab plasma, electrons are accelerated by the laser
and oscillate around the plasma (Kawata et
al., 2005). Therefore, the electrons accelerated produce an
electron cloud at the sides of the plasma (see e.g., Nakamura & Kawata, 2003). The electron cloud
extracts and accelerates ions. This acceleration mechanism is well known
in laser-plasma interactions.
In this paper, we study an ion focusing mechanism in an interaction of
intense laser and thin slab plasma by using 2.5-dimensional (2.5D)
particle-in-cell (PIC) simulations. The electron cloud generated on the
opposite side of the irradiation surface of laser is formed locally in the
transverse and longitudinal directions. Therefore, a static electric
potential is formed as well by the electron cloud, and ions are focused
and accelerated by the strong static electric potential. Consequently,
even in the slab plasma target, one can expect the progress of ion
focusing effect. In our ion acceleration and focusing mechanism, a
magnetic field influences the electrons and the static electric potential.
We also investigated an influence of the magnetic field on our
mechanism.
The simulation model and parameters are presented in Section 2. In
Section 3, the simulation results of proton focusing and acceleration
processes are presented. In Section 4, the conclusions are described.
2. SIMULATION MODEL
Figure 1 shows the schematic view of the
calculation model. The slab plasma consists of an Al+11 layer
of 0.5 μm thickness with an additional 1.5 μm linearly-changing
density slope, and an H+ layer of 0.5 μm thickness. The
peak density of Al+11 is a solid density and the H+
plasma is in its solid density. At the initial time, the temperature of
plasma ions and electrons are in the Maxwell distribution with 10 KeV and
100 KeV, respectively. The laser intensity is 2 × 1020
W/cm2, the wave length of laser λ is 1.053 μm, the
laser spot size w0 is 2.5λ, and the duration of
Gaussian laser pulse τ is 10 fs. The laser propagates in the
x direction and is polarized in the y direction. The
center of laser is at y = 6λ. In this paper, the target side
irradiated by the laser is called the laser side and the other side is
called the rear side.
A schematic view of calculation model. The slab plasma consists of an
Al+11 layer of 0.5 μm thickness with an additional 1.5
μm linearly-changing density slope and a H+ layer of 0.5
μm thickness. The laser propagates in the x direction and is
polarized in the y direction. The laser center is at y =
6λ.
3. SIMULATION RESULTS
In this section, the simulation results of proton acceleration and
focusing mechanism are presented. In order to clarify a detail of proton
focusing and acceleration, we show in Figure 2
the time developments of distributions of high-energy (>2.0 MeV)
electrons, static electric potential, and magnetic field. The laser peak
reaches the target at t = 27.6 fs. The electrons are accelerated
strongly by the laser and generate the electron cloud at the rear side. In
this paper, we focus on the rear side for practical reasons. Therefore, a
static electric potential is generated, and protons are accelerated by
this static electric potential longitudinally, and obtain a MeV order of
energy in the parameter range employed in this paper. In our calculations,
the spot size of laser is small compared with the target width. Therefore,
the electrons accelerated are localized in the rear side and produce the
static electric field locally as shown in Figure
2a and Figure 2b. To investigate the
detail of localized static electric potential, a precise distribution of
static electric potential is shown in Figure 3.
The localization of the static electric potential can be seen clearly in
Figure 3. The protons are extracted and
accelerated from the rear side by this localized static electric
potential. Consequently, the proton divergence in the transverse direction
is suppressed and protons are focused on the center. A strong magnetic
field is also produced by the electrons accelerated in the rear side as
shown in Figure 2c. The magnetic field may
influence the electrons motion and consequently may have an influence on
the form of static electric potential. The magnetic effect is important in
our ion focusing and acceleration mechanism. The analyses of the magnetic
field effect are described later on this paper. Figure
4 shows the distributions of the proton bunch focused at t
= 28.0 fs, t = 38.0 fs, and t = 58.0 fs, respectively.
The divergence of the protons is suppressed by the static electric
potential, and the protons are accelerated and slightly focused on the
center. In this specific case, the maximum proton energy is 8.61 MeV, and
the average proton energy is 0.672 MeV. Figure
5a and Figure 5b shows the time
developments of the intensity distributions of the protons bunch
accelerated in the transverse and longitudinal directions at t =
26.0 fs, t = 36.0 fs, and t = 58.0 fs. The protons are
extracted and accelerated by the localized static electric potential.
Consequently, the intensity of proton beam increases near the center. The
maximum intensity of proton bunch is 1.89 × 1017
W/cm2 at t = 58.0 fs. A diameter of the proton
bunch accelerated at t = 58.0 fs is decreased to 71.2% compared
with t = 22.0 fs. Here the diameter of the proton bunch
accelerated is defined by the intensity FWHM of the proton bunch. Figure 2 to Figure 5 show
clearly that the protons are focused and accelerated by the localized
electron cloud and the generated electric potential. As shown in Figure 2c, one can imagine that the magnetic field can
have an influence on the proton focusing. Figure
6 shows results without the influence of the magnetic field: in
this case, the magnetic field is set to zero in the equation of motion in
order to check the magnetic effect on the proton focusing, though the
Maxwell equation is fully solved. Figure 6a,
Figure 6b, and Figure
6c show the distributions of high-energy electron (>2.0 MeV),
the static electric potential, and the magnetic field at t = 42.0
fs in the case without the magnetic field effect. The detail of the static
electric potential is presented in Figure 7.
Figure 6a show that the electrons are slightly
diffused in transverse, and the magnetic field in the rear side is weak as
shown in Figure 6c. Consequently, the static
electric potential in the rear side is weak and it is also relatively flat
as shown in Figure 6b and Figure 7.
The time development of distribution of (a) the high-energy
(>2.0 MeV) electrons, (b) the static electric potential, and
(c) the magnetic field. Each distributions are corresponding to
the time of t = 30.0 fs, t = 38.0 fs, and t =
42.0 fs, respectively.
Distribution of the static electric potential (a) in the
x-y plane, (b) versus y at x
= 7.85λ, and (c) versus x at y = 6.0λ
at t = 42.0 fs.
The maps of proton bunch focused at t = 28.0 fs, t =
38.0 fs, and t = 58.0 fs.
The intensity distribution of protons bunch accelerated (a)
versus x and (b) versus y at t = 26.0
fs, t = 36.0 fs, and t = 42.0 fs.
The distributions of (a) the high-energy electrons,
(b) the static electric potential, and (c) the magnetic
field at t = 42.0 fs. In this calculation, the magnetic field is
set to zero in order to investigate the effect of magnetic field on the
proton focusing, though the Maxwell equation is fully
solved.
The details of the static electric potential (a) in the
x-y plane, (b) versus y at x
= 7.85λ, and (c) versus x at y = 6.0λ
at t = 42.0 fs.
Figure 8 shows the intensity distribution of
proton bunch accelerated in the y direction. In this specific
case, the protons are not well focused by the static electric potential.
Therefore, the intensity is nearly flat and the peak of intensity is
decreased compared with that in Figure 5. The
peak intensity of proton bunch accelerated is 1.24 × 1017
W/cm2, the maximum proton energy is 8.59 MeV, and the
averaged energy is 0.353 MeV. Figure 6, Figure 7, and Figure 8
demonstrate that the magnetic field influences the distribution of the
electron cloud and the static potential, and consequently the protons are
focused much.
The intensity distribution of proton bunch accelerated versus
y direction at t = 26.0 fs and t = 58.0
fs.
4. CONCLUSIONS
In this paper, the proton bunch focusing and acceleration effects are
clarified and demonstrated by the particle simulations. The proton
focusing effect comes from the localization of the electron cloud and the
static electric potential. In our study, the effect of magnetic field is
also investigated. The strong magnetic field has a great influence on the
proton focusing effect.
ACKNOWLEDGMENTS
The authors would like to extend their thanks to Prof. K. Mima, Prof.
K. Tachibana, Prof. K. Nakajima, Prof. S. Kurokawa, and Prof. N. Yugami
for their valuable discussions and suggestions on this subject. This work
was performed partly under the collaboration program of the Institute of
Laser Engineering, Osaka University and supported in part by the grant in
aid for science research under JSPS (Japan Society for the Promotion of
Science).
