Contents - Max-Planck-Institut für Physik komplexer Systeme

Figure 1: (a,b,c) Snapshots of the electron density (gray scale), carbon contour plot (blue) and proton contour plot (red) at: t = 4 τL (a);
t = 14 τL (b) and t = 30 τL (c). In this simulation, wL = 62.8 and nl = 6. (d-g) Temporal evolution of: (a) the accelerating electrostatic field
see by the protons; (b) the proton mean energy; (c) the proton energy dispersion and (d) the proton angular aperture. For nl = 6, wL = 10 λL
(blue, square), nl = 12, wL = 10 λL (green, triangle) and nl = 6, wL = 20 λL (black, circle). Dashed curves account for analytical predictions.
This early stage of proton acceleration ends once the
distance between the electron cloud and the bare heavy
ion layer becomes similar to their transverse size w⊥, at
t ∼ 8 τL for wL ∼ 10λL and t ∼ 13 τL for wL ∼ 20λL.
The electrostatic field seen by the protons then strongly
decreases due to geometrical effects (Fig. 1d) and the
acceleration process slows down (Fig. 1e). Ion acceleration
in this later stage occurs in the heavy ion field in a
way similar to directed Coulomb explosion (DCE) [5,6].
Assuming Zh/mh ≪ Zl/ml, the ion energy gain is simply
the remaining proton potential energy:
ǫDCE ∼ Zl ah w⊥/4 − ǫlinPA/2. (3)
Energy and angular dispersions also slightly increase
(Figs. 1f,g) but do not exceed 10% (for nl = 6) and
3◦ , respectively. Figure 1g also shows that angular dispersion
now depends on both wL and nl thus suggesting
that both the transverse inhomogeneity of the accelerating
field and Coulomb repulsion in the proton
bunch are responsible for it. Energy dispersion on the
contrary remains mainly sensitive to nl: it follows primarily
from Coulomb repulsion between protons. This
effect can be estimated by approximating the second
ion layer by a uniformly charged sphere with radius
R ∼ wl/2 expanding due to its own charge Ql (in units
of enc/k 3 L ). The final energy dispersion can then be derived
as a function of the final ion energy ǫ:
∆ǫDCE ∼ 4 √ ǫǫp , (4)
[1] E. Lefebvre et al, Nucl. Fusion 43 (2003) 629.
[2] M. Grech, S. Skupin et al., New J. Phys. 11 (2009) 093035.
[3] M. Grech, S. Skupin et al., Nucl. Instr. Meth. Phys. Res. A 620 (2010) 63.
[4] A. Macchi et al., Phys. Rev. Lett. 94 (2005) 165003.
[5] T. Zh. Esirkepov et al., Phys. Rev. Lett. 89 (2002) 175003; E. Fourkal et al., Phys. Rev. E 71 (2005) 036412.
[6] H. Schwoerer et al., Nature 439 (2006) 445; B. M. Hegelich et al., Nature 439 (2006) 441.
where ǫp = Zl Ql/(2π wl) is the initial potential energy
of a light ion on the outer shell of the sphere. As a result,
small energy dispersions can be obtained only by
limiting the total proton charge.
Now, comparing Eqs. 1 and 3 allows one to infer which
of the two stages is dominant in the acceleration process.
One obtains that ions with the final energy ǫ
ml/8 gain most of their energy during the later (DCE)
stage. For these ions, the final energy is ǫ ∼ Zl ah w⊥/4
and ∆ǫ/ǫ ∼ 4 ǫp/ǫ. Moreover, since w⊥ ∼ wL, the
condition for electron removal, aL > ah, defines a
threshold for the laser power PL ∼ w2 L a2 √ L and the light
ion energy scales as ǫ ∝ Zl PL/4. On the contrary,
energetic ions (in particular relativistic ions) quickly
separate from the heavy ion layer. They gain most
of their energy during the early (linPA) stage. Writing
tlinPA ∼ w⊥ ∼ wL the duration of this stage and
considering aL > ah, Eq. (1) can be rewritten as a
function of PL. One then obtains a threshold power,
PL > m2 l /Z2 √ l , above which relativistic ions with energy
ǫ ∝ Zl PL are expected.
In summary, this new mechanism of ion acceleration
allows for the generation of high-quality ion beams
which properties can be controlled by a careful target
design. This makes it an interesting candidate
for applications such as hadron-therapy. Scaling laws
presented here suggest that this particular application
might be considered on petawatt laser systems [3].
2.18. High-Quality Ion Beams from Nanometric Double-Layer Targets Using Petawatt Lasers 77