While some results are similar in simulation (section VI.2) and experiment (section VI.1), it is obvious that major features of the acceleration dynamics differ strongly.
One aspect that is well modeled by the simulation is the maximum electron energy of 350 MeV (sim) vs. 380 MeV (exp). Furthermore the simulations predict a regime where electrons are continuously trapped. From the high background in the experimental elec- tron spectra it can be concluded that there the injection is continuous as well. The sim- ulation shows that a plasma wave is driven already during self-focusing in the density up-ramp and injection into the wakefield starts before the minimum laser spot size is reached. During further acceleration the spot size oscillates (cf. fig. VI.14). The, espe- cially in the beginning, strongly varying focal size influences the injection dynamics. This effect is enhanced by the changing plasma wavelength in the rising edge of the density profile.
However, not all elements of the simulation can be found in the experiment. Especially, the amount of trapped charge and its effects on the wakefield structure seems to be over- estimated in the simulations. The bunch charge of > 450 pC is more than 10 times the
9The total bunch charge can not be compared since in the simulation the injection center at the rear of the bubble leaves the simulation box before the acceleration ends. This does not influence the characteristics of the spectral peak sitting in the front of the trapped bunch, but the total bunch charge is underestimated. Still, the peak in the simulation contains roughly 10 times the charge of the entire electron bunch in the experiment (above 100 MeV)
experimentally measured charge. In the simulation beamloading heavily distorts the elec- tric field within the blowout region. Even before the head of the electron bunch can reach the zero-crossing of the electric field strength (dephasing) the entire electric field is deteri- orated and decelerates the bunch as a whole. However, in the experiment the beamloading effects seem to be less severe. Electrons are accelerated (and even decelerated again) over a length that corresponds well to the theoretical dephasing length for this pressure. The spectral electron density peaks around the dephasing point, as one would expect if the head of the bunch is decelerated while the tail is still in the accelerating phase. The high- energy cut-offof the electron spectrum changes in good approximation quadratically with the propagation distance suggesting an undisturbed linear acceleration field.
Another difference is seen in the scale of the evolution: in the simulation the high-energy peak forms after only 2.2 mm of propagation, in the experiment it takes roughly twice this distance.
These discrepancies can be partially explained by the slightly differing parameters: The duration of the driver pulse in the simulation is almost 30% shorter than in the ex- periment. According to (III.58) this will also decrease the energy depletion length by 30%. In the simulation the laser therefore is more intense in the beginning, thus driving a wakefield with a higher electric field. More electrons are trappped until the beamloading field cancels the electric field of the bubble (cf. sections III.4 and III.5.3 or e.g. [141]). At the same time it depletes faster than in the experiment, leading to an eventually weaker electric field facing the initial high amount of trapped charge. This can explain both the shorter acceleration length in the simulation and the strongly distorted electric field. In the simulation a Gaussian spatial beam profile is assumed before the focus, while the real beam profile exhibits hot-spots and irregular intensity variations outside the Rayleigh range. The simulations show, that self-focusing already starts in the density up-ramp be- fore the geometrical focus position. As well known from high-power laser systems in that case small-scale self-focusing can occur [142], where the pulse breaks into several fila- ments. It is not obvious whether this really could be completely avoided in the experiment by choosing a long focal length. M.D. Feit [66] claim that, if the power in these filaments is still above Pcr they will eventually coalesce and form a single (cavitated) channel that
contains most of the initial beam power. Even if there is no filamentation, the final beam profile can be asymmetric or otherwise distorted from inhomogeneous self-focusing. A large fraction of the differences between simulation and experiment, however, could result from numerical errors. Cormier-Michel et al. [143] state that ”[n]umerical errors can lead to errors in the macro-particle orbits in both phase and momentum. These errors [...] can be large enough to result in unphysical trapping in the plasma wake. The result- ing numerical heating in intense short-pulse laser-plasma interactions grows much faster and to a higher level than the known numerical grid heating of an initially warm plasma in an undriven system.” This numerical heating is caused by the discretization of the fields and the small number of macro-particles (in our case 1 per cell). To a certain degree it can be reduced by appropriate smoothing and increasing the number of macro-particles, if computationally possible. Additionally, it must considered that the transverse size of
the trapped electron beam is small and only marginally resolved by the grid. According to Cowan et al. [144] this leads to artificial emittance growth. They also mention that a low transverse grid resolution results in wrong numerical dispersion leading to a too low group velocity of the laser pulse and thus too rapid dephasing.
Ideally, a warm plasma (few tens of eV) is initialized with both longitudinal and trans- verse resolutions of the same order and such that high that numerical heating is well below the physical temperature. However, with conventional PIC codes, this is not pos- sible with the currently available computational power. With the availability of Lorentz- boosted-frame codes (e.g. [145]) and adaptive mesh refinement approaches (e.g. [146]) for laser-wakefield acceleration this kind of simulation will be easier in the future.
Wakefields from Tilted Driver Pulses
The experiment described in this chapter is dedicated to the analysis of the impact of a laser pulse front tilt (PFT) or (equivalently) angular chirp (section II.1.4.2) on the accel- erated electron bunch. The necessity to pay more attention to this laser parameter arose when, unexpectedly, in the experiment the electron pointing direction was stable but re- producibly deviated from the laser axis by a few mrad. This was especially problematic as further devices, such as magnetic lenses, an undulator and X-ray diagnostics, were care- fully aligned to the laser axis. As this steering hinted to an asymmetry in the setup, several possible causes were analyzed and a non-vanishing PFT in the laser pulse was detected. After removing the PFT the electron bunch propagated along the laser axis again. These deviations in electron pointing could be observed only due to the excellent stability and reproducibility of the electron bunch parameters in a steady-state flow gas cell [18]. It, consequently, also allowed for a systematic study of laser-wakefield acceleration with a tilted driver pulse with meaningful statistics.
This comprehensive measurement with different intentionally introduced pulse front tilts will be described in this chapter. It is structured as follows: For a better understanding of the experimental problems and measurement results, first the basic characteristics and evolution of an angularly chirped laser pulse are analyzed. In the second section the mea- surement details and results will be shown. The last section supports the interpretation of the experiment by 3D PIC simulations.