Energy dependence

Derived source size and divergence

Divergence dependence on electron energy

This section shows experimental results of the lens-scan method which was de-scribed in section 3.4.1. The method measures the electron beam size at the YAG:Ce crystal for a range of lens positions. The electron beam parameters can be tted to this data which was done for beams with energies of 245, 270, and 300 MeV at a plasma density of 6 ×10¹⁸cm⁻³. The retrieved source divergence will be com-pared with the measured free-drift value, and also the eect of space-charge on the measurement will be discussed.

Figure 6.5 shows a scan of the z-position of lens two and the resulting measured beam size at the YAG:Ce crystal positioned behind the dipole magnet such that electrons with an energy of 245 MeV are observed. Each data point is the mean rms beam width of 15 or more shots; the width of each shot is evaluated for a small integrated energy bandwidth (±0.05 MeV) around the observed energy. The error bars correspond to a 95% condence interval for the mean. The parameter t gives a source size of 0.93±0.11µmand source divergence of 0.44±0.04mrad, resulting in a normalised emittance of 0.20^+0.01_−0.02 π·mm·mrad. The error limits are 95% condence intervals obtained by bootstrapping procedure using 2000 samples (see Appendix A). The accuracy of the method is also illustrated by the expected dependence for a 20% larger emittance by increasing the inferred source size or divergence.

This measurement was repeated for electron energies of 270 and 300 MeV. For

6.2 Electron beam emittance 103

60 65 70 75 80

z position lens 2 (mm) 25

30 35 40 45 50 55 60

xbeamsize(µm)

best fit source +20%

divg. +20%

simulation (strong space charge)

Figure 6.5. Lens position scan for 245 MeV electrons. The black squares show the mean RMS beam sizes for various z-positions of lens two. The error bars correspond to a 95% condence interval for the mean value. The t curve (blue) neglects space charge and gives a normalised emittance of0.20^+0.01_−0.02π·mm·mrad and inferred RMS source size and divergence of0.93µmand 0.44 mrad, respectively. The inuence of space charge is shown by the yellow line which shows the expected beam sizes for a high space-charge beam. The discrepancy in shape and position along the horizontal scale of the yellow and blue lines indicate that space charge is negligible.

The initial source size and divergence in the space-charge simulation were chosen to be 0.25µm and 0.45 mrad respectively to obtain a similar curve as obtained from the measurements.

270 MeV electrons the t routine gives a source size of 0.92^+0.07₋0.11µmand source diver-gence of 0.34^+0.08_−0.04mrad, resulting in a normalised emittance of0.17^+0.02_−0.01π·mm·mrad.

Similarly for 300 MeV electrons the tted source size is 0.91±0.07µmand the source divergence is 0.35^+0.09₋0.06mrad, resulting in a normalised emittance of0.19^+0.03_−0.01π·mm·mrad.

As above, the error bars represent 95% condence intervals. The normalised emit-tance remains relatively constant which supports the expected linear focusing forces in the wakeeld during acceleration as discussed in section 2.5, and which has been previously observed at lower electron energies of <20 MeV [Sears et al. 2010a]. To make a more conclusive claim about a constant normalised emittance would require a larger range of energies to be measured. One reason is that the margin of error of

∼10%in the calculated emittance is comparable to the energy range that is scanned here.

The derived divergences are compared with the free-drift divergences measured behind the spectrometer with a lanex screen in gure 6.6. Below about 250 MeV the lanex and the lens-scan divergences agree well. For higher energies the larger

diver-104 6. Measurements of the electron beam emittance

150 200 250 300

Electron energy (MeV) 0.3

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Divergence (mrad)

Lanex 245MeV 270MeV 300MeV

Figure 6.6. Comparison of divergence measurements. The plot shows the measured electron beam divergence using the lens scan method (squares). The measured divergences are extrapolated for other energies by assuming the source divergence scales as θ(γ)/γ = θ(γ₀)/γ₀ as would be expected from an electron beam source undergoing adiabatic damping during acceleration (solid blue, green, and red lines). The measured divergence with the lanex screen at S2 are shown in black (crosses: raw data, pluses: raw data deconvoluted with aσ =400µmGaussian to account for the resolution of the lanex measurement).

gences often seen around the spectral peak of individual shots lead to a attening of the average divergence of many shots as observed by the lanex screen (see gure 5.5).

As the spectral peak is at a dierent energy from shot to shot, the lanex divergence attens out for >250 MeV. During the lens-scan measurements the spectral peaks of the electron bunches were well beyond 300 MeV being observed by the YAG:Ce crystal, and hence the larger divergence did not inuence the measured divergence.

There is therefore no discrepancy between the lanex divergence and the lens-scan divergence.

As touched upon in section 3.3.3, the space-charge of the beam aects the beam dynamics during propagation and the position and shape of the lens-scan measure-ments. As a simple test a beam without space-charge was tracked [GPT] for the dierent lens positions as in the experiment. The simulated lens scan without space charge perfectly matches the best t line in gure 6.5 which is a strong indication that space charge is not relevant in the data over many shots as in this scan. To illustrate how a lens scan would look for a high-charge case, a tracking simulation of 3000 macro-particles including point-to-point space charge for an

ini-6.2 Electron beam emittance 105 tially mono-energetic 245 MeV, 50 pC beam with duration 4.5 fs was carried out (see gure 6.5). The gure shows a clear deviation from the measurements in terms of shape and position. To focus a beam including space charge requires a stronger focusing lens system to compensate the repulsion of the electrons. In the present work stronger focusing is achieved by increasing thez-position of lens two, resulting in the shift of the beam size curve to the right as seen in the gure. Such a shift could also be caused by the source position being further away from the rst lens (source position upstream of the physical exit of the gas cell, inside the gas cell).

A source position inside the gas cell seems unlikely as the depletion length of the laser and gas cell under similar conditions was measured to be∼7mm in the exper-iments described in Popp [2011]. Hence the electron beam is still contained within the wakeeld forces for the entire length of the gas cell which was 5 mm for these experiments. Furthermore, as is discussed in section 6.2.7, the plasma-to-vacuum transition rather leads to a shift of the source position closer to the rst lens (source position downstream of the physical exit of the gas cell, outside of the gas cell). De-spite these arguments, as the electron source position is not precisely known, the position of the measured curve of a lens scan is not conclusive about the strength of the space charge aecting the beam. A stronger indicator seems to be the shape of the measured curve. Anderson and Rosenzweig [2002] showed that quadrupole scans of beams with signicant space charge display an asymmetry about their minimum, with the stronger focusing side (larger lens two z-position) showing a steeper ank.

As the measured data does not show a signicant asymmetry (or shift), space charge eects must be much less signicant in the experiment than in the simulation in the gure. As the charge is reduced in the simulation, the yellow curve moves towards the measured data, becomes symmetric, and eventually matches the blue t curve.

The measurement accuracy benets from the spectral splitting of the beam behind the dipole magnet which reduces the charge density and hence the space charge repulsion of the beam. The measured data therefore conrms that space charge is negligible in the present experiment.