2.2 Ion Acceleration
2.2.2 PIC Modeling
The ion acceleration in the laser focus discussed above hence can not be treated adequately in a simple analytical model, so numerical methods have to be em- ployed. The main goal was to model the 3-D ion momentum space after the laser pulse and compare it to the experimental data. By Alexander Pukhov’s courtesy I was able to use his 3D-particle-in-cell (PIC) code VLPL-3D [30] to model the interaction of a relativistically intense laser pulse impinging on a steep (few µm) plasma density gradient in front of a slab of solid material. The code runs on the CRAY T3E-816 supercomputer at the Rechenzentrum Garching and uses 256 processors for the simulations performed for this work. A typical simulation run takes 6 hrs and produces a data output of around 10 GB. The code solves the rela- tivistic equations of motion and Maxwell’s equations simultaneously on the nodes of a three-dimensional grid, which samples the space volume (”simulation box”)
under observation. The principle of PIC codes in general and especially VLPL are described in detail elsewhere [30] and will not be repeated here.
The simulation box in most of the runs performed for this work was x= 19.2× y= 16×z = 16µm3 in size and consisted of 640×160×160 cells, leading to a cell spacing of 30 nm in x-direction and 100 nm in y- and z-direction. This ensures that both the skin depth
λs = (c √¯γ)/ωp ≈130nm, (2.25)
which defines the penetration depth of the light field at the critical surface, and the Debye length
λD = s
kT 0
ne2 = 130 nm (for T = 1 MeV), (2.26)
which determines the length over which charge fluctuations in a plasma are shielded, are resolved at the relativistic critical surface. The laser either propagated into positive x-direction or under 45◦ to it.
We can now look a bit more in detail into the ion acceleration at the critical surface. In the following simulation a laser pulse with a duration 80 fs was hitting a preformed plasma with a exponential gradient of 3µm scalelength in front of a target with a maximum density of 16 ncr. The focal spot was Gaussian-shaped in space and time and had a diameter of 4µm. and the intensity 3×1019W/cm2, corresponding to the parameters of the Jena 10 TW laser. It is not easily possible to extract the acceleration fields directly from the simulation output, so in the following pictures (see Fig. 2.2) the difference of the electron and ion densities (ne−ni) was plotted in a plane perpendicular to the polarization direction of the laser for times of 10, 20, 30, ..., 70 laser cycles (cycle duration 2.7 fs for 800 nm laser wavelength). At t=0, the laser pulse maximum is at -30 µm left of the simulation box, and is moving to the right. Electron excess is coded blue, while ion excess is marked red. Strong fields exist in places where the gradient (blue-red) is steep.
At 10 laser cycles, the laser starts to push electrons into forward direction, and a bow-shaped cusp of high electron density is formed around the head of the laser pulse. This leads to strong acceleration fields, but the ions are still stationary, and the structure is only caused by modifications of the electron density. The onset of modulation in the electron density by the oscillating laser field is already visible behind the head of the pulse. As the pulse propagates further (20 cycles), the rapid electron density modulation is fully developed. Still the ions are virtually immobile. As they start to move and the laser stops at the
Figure 2.2: 3-D PIC results: Difference between elec- tron and ion density in the x-z-plane (⊥to laser polar- ization) at times of 10, 20, ...,70 laser cycles (1 cycle = 2.7 fs). Regions with an excess of electrons or ions are plotted blue and red, respectively. The formation of a double layer is evident between the maxima of the Poynting vector in the early stages and around the head of the pulse at later times.
relativistic critical surface (at 30 cycles), the rapid modulation breaks up and a more bowl-shaped electron density enhancement is developed. Note that now at the boundary of disturbed and undisturbed plasma, a very narrow double layer of electrons preceding the ions is formed (40 and 50 cycles), which leads to even stronger fields and efficient ion acceleration. At 60 and 70 cycles, the laser intensity drops so far that now the electrons oscillate back through the ion sheet, and the polarity of the boundary layer is reversed. Since the fastest ions going towards higher plasma density have already outrun the double layer, this reversed polarity causes a backwards acceleration as well. At 70 cycles, the driving force of the laser has vanished and the double layer neutralizes very fast.
In this case, the time between the formation of the double layer and the decrease in laser intensity is very short, so efficient ion acceleration is prevented, and the
ion energies stay far below the value inferred from equation 2.24. Also the pulse bores deeply into the plasma, leading to less forward directed push as in a 1-D case, distributing the available energy among more ions around the circumference of the focal region.