La noción de justicia en John Rawls

the dependece of the resulting electron current density J on electron energy E is characterized by a power-law

J(E) =−J0(1−E/E0)α. (6.10)

This scenario exhibits similarities with the situation of laser-driven ion acceleration from ultrathin foils, where the laser heats and pushes a cloud of electrons out of the target. The authors fitted above function to the electron current density obtained from 1D PIC simulations around the optimum foil thickness and determined the in- dexα, which is referred to as coherence parameter, toα ≈3. Tracking the evolution of electrons, ions and electrostatic field self-consistently in time, an expression for the maximum energy of accelerated ions of chargeqi =Zeis obtained in its simplest form for short pulses and an ab initio transparent target

Emax = (2α+ 1)ZE0. (6.11)

In the case of long pulses where thicker targets are optimal that are not transparent from the start, the authors follow the arguments of relativistic transparency and resulting strong ion acceleration between timest1 andt2 as given here. Treating the

process again in self-similar fashion yields

Emax = (2α+ 1)ZE0

h

(1 +ωL(t2−t1))1/2α+1−1

i

, (6.12)

where the characteristic electron energyE0is evaluated over the time interval (t1, t2)

and a laser pulse duration of 2τ > t1, t2 is assumed. By use of this semi-analytical

approach, the authors achieve good agreement with the experimental results pre- sented here.

6.5

Increasing the Intensity

As a part of my PhD work, a follow-up series of measurements was carried out at the upgraded TRIDENT facility delivering increased peak intensities. The above described lead-off campaign relied on a double plasma mirror setup to suppress ASE and prepulses and thus ensure interaction with an ultrahigh contrast laser pulse, however giving rise to energy losses of around ∼ 45%. For the subsequent experiment, the TRIDENT frontend was redesigned to incorporate idler generation in a short-pulse pumped OPA stage [146]. An in-depth description of the upgraded

100 6. Enhanced Ion Acceleration in the Transparency Regime

frontend type 1 is presented in section3.2.3. Thus, the use of plasma mirrors became obsolete and the mentioned energy losses were avoided. Consequently, energies of

∼ 90 J at high contrast on target were achieved. In addition to making plasma mirrors dispensable, in the upgrade the pulse duration was successfully narrowed down further, yielding ∼ 550 fs as compared to the previous value of ∼ 700 fs. By virtue of those measures, the resulting peak intensity on target could be increased to ∼2×1020_W/cm2_.

While the DLC foils of the first TRIDENT campaign were fabricated by V. Liecht- enstein at Kurchatow Institute in Moscow using a modified glow discharge technique, for the follow-up measurements DLC targets could be produced for the first time at the newly established target fabrication laboratory at LMU employing cathodic arc deposition. For details on the respective target fabrication method refer to sec- tion 4.1.2. With cathodic arc deposition, the DLC foils exhibit a higher fraction of sp3 bonds amounting to ∼75%, and consequently also higher density of 2.7 g/cm3. Self-suporting targets of thicknesses spanning from 2.5 nm up to 58 nm were suc- cessfully manufactured. Thus, a broader range of target parameters was available for investigation in the second campaign. Foils thicker than ∼60 nm turned out to be mechanically unstable, owing to internal stress in the material arising from the deposition process.

The damage threshold of DLC was tested at TRIDENT upon controlled irradi- ation with an attenuated 1.2 ns pulse simulating the extended ASE pedestal and a 500 fs pulse modeling short prepulses. Target degradation was observed at respective intensities ratios of∼2×10−12and ∼5×10−10 in relation to the peak intensity on a full system shot, giving an estimate of the contrast on target employing the new OPA frontend. On actual target shots, a full aperture backscatter (FABS) diagnos- tic was used to monitor light backscattered from the foil surface and thus ensure target integrity on interaction with the main pulse [205].

Two Thomson parabola spectrometers were employed, one orientated along tar- get normal, the second located at an angle of 8.5◦orthogonal to the laser polarization direction. For characterization of the proton beam profile detector stacks composed of RCF and CR39 were used. In addition, a magnetic electron spectrometer was placed at an angle of 6.5◦ orthogonal to the laser polarization direction. All diag- nostics were operated in parallel in single-shot mode.

6.5 Increasing the Intensity 101 0 100 200 300 400 500 600 0 10 20 30 40 50 60 0 100 200 300 400 500 600 protons carbon C6+ 105 106 107 108 109

target thickness (nm) energy (MeV)

particles (MeV

-1 msr -1)

max. energy (MeV)

b) a)

C6+ _(58nm)

Fig. 6.7. (a) Maximum proton (blue squares) and carbon C6+ _{(red squares) ion energies over}

target thickness observed at increased intensities of2×1020_W/cm2_{. The data shown was obtained}

from a Thomson parabola spectrometer placed at an angle of8.5◦ with respect to target normal, orthogonal to the laser polarization. As expected by the developed analytical model, the optimum foil thickness shifts to larger values. Using a 58 nm thick target, the measured carbon C6+ ion spectrum extended beyond 0.5 GeV (b), representing the current world record for a laser-driven accelerator.

6.5.1

Measured Ion Beams

From the analytical model specified above, the optimum foil thickness for maxi- mizing the energy of accelerated ions is expected to be shifted to larger values of

∼80 nm due to the increase in intensity [202]. Indeed, at the upgraded laser param- eters of ∼ 90 J pulse energy, ∼ 550 fs pulse duration and ∼ 2×1020W/cm2 peak intensity the largest ion energies were experimentally observed at the maximum available target thickness of 58 nm (see figure6.7a). Here, the spectrum of fully ion- ized carbon reached out all the way up to 525 MeV, while the proton cut-off energy amounted to 43 MeV. The corresponding C6+ _{ion spectrum is illustrated in figure}

6.7b. This result represents the current world record in carbon ion energies gen- erated via laser-driven acceleration, superseding the all-time high of 185 MeV C6+

presented beforehand and by far outreaching the previous top mark of ∼ 40 MeV for carbon C5+ _{seen at TRIDENT [5].}

For targets as thin as 5 nm, the measured carbon C6+ _{ion energies are down by}

around one order of magnitude with respect to the optimum. However, for such thin foils a distinct quasi-monoenergetic electron spectrum peaked at an energy of 30 MeV was observed [97]. This might be a first experimental indication of the generation of highly dense electron bunches as predicted in theory when the laser ponderomotive force exceeds the restoring force of the induced charge separation field [95, 96].

102 6. Enhanced Ion Acceleration in the Transparency Regime