Acceleration Limits

In both measurements, at 50 mbar and 130 mbar, the electron energy does not change sig- nificantly after roughly 7 mm of propagation. The gas cell length was extended beyond this length up to 13 mm to verify the stagnation of the cut-off energy. If the energy of the electrons does not change, they are, obviously, not exposed to a longitudinal electric field. In our setting, this happens if the laser stops driving a wakefield. As will be eluci- dated below, in the 130 mbar case this is well explained by energy depletion of the pulse after≈ 7 mm, in the 50 mbar run, on the contrary, self-guiding cannot be sustained long enough, the laser diffracts.

If the laser does not excite a plasma wave anymore, the electron bunch propagates through the plasma and drives its own wakefield. This interaction is responsible for the low en- ergy decrease that is still observed for gas cell lengths of 7 mm to 13 mm. It has to be noted that, in principle, the energy loss of an electron bunch that is driving its own wake- field can also be large. However, the efficiency of the process depends on the current

and thus on the electron density in the bunch. For an electron bunch with a non-constant density-distribution, for example a Gaussian, only the high-current core will lose energy driving the plasma wake, the low-current front/rear parts stay unaffected [133, 134]3. In the discussed plot only the high-energy cut-offof the spectrum is considered. Due to the characteristics of the acceleration process and according to simulations, a laser-wakefield accelerated electron bunch is temporally chirped with the high energies in the leading edge. The fact that the energy of these electrons only slowly decreases within the last millimeters of the gas cell suggests, that this leading high-energy part does not contribute notably to the excitation of a plasma wave.

Self-Guiding

In the low-density run acceleration stops after roughly seven millimeters and the achieved electron energies are comparable to the high-density case (fig. VI.9). The two most evi- dent explanations are that the laser pulse driving the wakefield either depletes or diffracts. The power necessary to sustain self-guiding increases with lower densities, yet energy depletion is less severe for lower densities. The parabolic shape of the energy evolution in the high-density run, with acceleration and subsequent deceleration, indicates that in this case the depletion length is much larger than the dephasing length. For the low-density run the laser should deplete even later. Still, clearly, the dephasing point is not reached, indicating that diffraction limits the acceleration.

Without self-focusing the laser diffracts with a Rayleigh length of lR = 1.37 mm. The

critical power (III.30) needed for self-focusing to occur is 12 TW in the 50 mbar case and 4.6 TW at 130 mbar. Lu et al. [58] also give an estimate of how much power is needed initially in excess of Pc in order to sustain self-guiding over the entire dephasing length

despite of the power loss due to energy depletion during propagation (cf. (III.64); exper- imental verification in [67]). For this 3.6Pc would be needed initially in the high-density

case, 10.2Pc are available. This confirms the assumption that in that run, self-guiding is

not the limiting factor. However, in the low-density case initially 6.3Pc would be neces-

sary and only a power of 3.9Pc is provided. This means that for the 50 mbar run self-

guiding can not be sustained over the entire usable acceleration length. This explains the characteristics of the 50 mbar energy-gain curve. A wakefield is driven and electrons are accelerated over roughly 7 mm. This is less than half the estimated dephasing length (see section VI.1.4) and thus the electrons have not reached the maximum possible energy yet. At longer distances, the laser is diffracted so strongly, due to the lack of self-guiding, that the intensity is not sufficient to drive a plasma wave. From 7 to 14 mm the electron bunch propagates through the plasma and loses some energy due to the interaction with the background electrons and by driving its own plasma wave [135].

3_{Low-current parts of the electron bunch BEHIND the high-current part that is driving a wakefield can also} suffer significant energy losses as they sit in the decelerating phase of the electric field in the blowout region.

run density dephasing length max. el. field max. energy (from fit)

(1018cm−3) (mm) (GV/m) (MeV)

130 mbar 6.42 4.9 162 GV/m 383

50 mbar 2.47 16.0 46 GV/m 841

Table VI.1.:Parameters deduced from the experiment

run density linear deph. length wavebreaking field max. energy

(1018cm−3) (mm) (GV/m) (MeV)

130 mbar 6.42 3.6 244 GV/m 439

50 mbar 2.47 14.9 152 GV/m 1132

Table VI.2.:Parameters predicted by theory

Energy Depletion

As already indicated above the 130 mbar run is limited by energy depletion of the driver pulse rather than deffraction. This can be deduced from the density scaling of the two processes. Self-focusing is stronger or needs less power at higher densities (III.30), (III.31), however, energy depletion is higher for higher densities (III.51), (III.58). Up to approximately 7 mm the electrons are strongly accelerated and decelerated again. The parabolic course of the acceleration and the high acceleration gradient indicate that also the deceleration still takes place within the high electric field of the laser-driven bubble. Only after 7 mm this curve flattens, as it is expected, when the laser depletes and eventu- ally cannot sustain the wakefield anymore. As in the low density case the electrons then only slowly loose energy to the plasma.

As already described in paragraph III.2.4.4, a high-intensity laser pulse with a Gaussian temporal envelope can well contain half of the energy after propagating the dephasing length. Assuming an initiala0 ≈ 4 after self-focussing, the vector potential would then

only be decreased toa0 ≈ 2.8, which is still enough to sustain a wakefield in the blowout

regime. From this consideration and the corresponding measurement that shows that the electron energy indeed is rapidly decreased again beyond the dephasing length it becomes clear that it is important to carefully adjust the acceleration length.

VI.1.5

Comparison to theory

Table VI.2 summarizes different parameters deduced from the measurement and gives the corresponding theoretical values. The linear dephasing length Ldeph,lin as in (III.49)

(a0 < 1) and the cold wavebreaking fieldEmax(III.56) from the 1D non-linear theory are

listed. Assuming a linear electric field, the maximum energy is then simply calculated by:

Wmax =e Ldeph1/2Emax.

The theoretical linear dephasing length is Ldeph,lin = 3.6 mm for 130 mbar and 14.9 mm

for 50 mbar. These values are slightly smaller than the measured values of 4.9 mm and 16.0 mm, respectively. This is expected since fora0 > 1 the plasma wavelength increases,

which consequently leads to a longer dephasing length. Esarey et al. [82] scale the linear dephasing length with a0·

√

2π−1 _{for a}2

0 >> 1 (III.59) and with 2a

−2 0 fora

2 0 << 1.

4 _In

the presented experimenta0is≈2 in the beginning and theoretically close to 4 after self-

focusing until energy depletion becomes significant. Assuming the measured dephasing length and applying this non-linear scaling one can extract an average a0 of 2.9 for the

130 mbar run. The measurement is thus in good agreement with the predictions made by the 1D non-linear theory.

The cold wavebreaking field III.56 from the 1D theory is E0 = 244 GV/m (130 mbar)

and 152 GV/m (50 mbar), the measurement yields 162 GV/m and∼ 46 GV/m, respec- tively. Again the reduced measured field can be explained by beamloading which is not considered in the theory.