Single-shot emittance measurement

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.

106 6. Measurements of the electron beam emittance more than 4 MeV. This energy range is large enough to t an emittance with good accuracy.

Figures 6.7(a) and 6.7(b) show the tted curve to the raw data for two shots at dierent energies. Due to the larger dispersion of the dipole magnet spectrometer at 245 MeV, the energy range on the YAG:Ce crystal is only 2.5 MeV at this energy. For the 300 MeV shot the tted source size and source divergence are 0.628±0.005µmand 0.383±0.013 mrad, resulting in a normalised emittance of 0.143±0.004π·mm·mrad.

The error bars for all numbers in this paragraph represent a 95% condence interval based on a bootstrap analysis (for a description and comparison to other error bars see below). This shot has a particularly low emittance which can be seen in the context of some shots around it. For the 44 shots taken with similar conditions the average observed emittance value is 0.22±0.02 π·mm·mrad. The 245 MeV shot is also a low-emittance shot with a derived value of 0.12±0.01 π·mm·mrad, the 48 shots taken around it have an average of 0.21±0.03π·mm·mrad. The average values at both energies agree well with the values obtained from the multi-shot methods described earlier in this section.

As the analysis for the energy-scan method relies on tting to electron beam sizes of dierent energy, it is more robust in the 300 MeV than for the 245 MeV case.

This becomes clear when looking at the hypothetical t curves for increased electron beam source parameters. The source size and source divergence which best t to the data are increased by 20% and the expected beam size at the YAG:Ce crystal is calculated. From these curves it becomes clear that the source size mainly inuences the height of the t curve^iv, and the divergence the steepness of the curve ank (see gure 3.13). In gure 6.7(b) it is clear that the source size is well distinguishable from the 20% larger source case but the same can not be said for the divergence.

A bootstrap analysis ^v of the 245 MeV shot shows the 95% condence interval for the divergence is 0.387±0.035mrad, an error of approximately 9% which leads to a similar accuracy for the emittance. The condence interval for the divergence of the 300 MeV shot is0.383±0.011mrad, an error below 3%. For this experiment the

>4 MeV energy window at 300 MeV is therefore a sensible minimum energy range to obtain an accurate t for the divergence and therefore the emittance.

Aside from a bootstrap analysis, the error or accuracy of the t to individual YAG:Ce crystal shots can be estimated by varying relevant experimental parameters (within a reasonable error range) and checking if a t for the data can still be found.

This method is dicult to quantify as it requires subjective choices for what are reasonable errors for the experimental parameters. However, it is a useful analysis to get a feeling for which parameters are particularly important to keep small for

ivAs the system is approximately imaging the electron beam source, a 20% increase in source size results in a∼20% increase in measured beam size at the YAG:Ce crystal.

v2000 resampled residuals, see appendix A.

6.2 Electron beam emittance 107

300 301 302 303 304

Electron energy (MeV) 16

18 20 22 24

xbeamsize(µm)

best fit source +20%

divergence +20%

(a)

244 245 246

Electron energy (MeV) 18

20 22 24 26

xbeamsize(µm)

best fit source +20%

divergence +20%

(b)

Figure 6.7. Electron beam size of two shots at the YAG:Ce crystal. The smaller spectrometer dispersion at higher energies means a larger energy range can be seen on the YAG:Ce crystal at 300 MeV (a) than around 245 MeV (b).

an accurate measurement. The experimental errors for the derived emittance of the 300 MeV shot shown in gure 6.7(a), lead to bounds of±0.03π·mm·mrad, an order of magnitude larger than the bootstrap bounds obtained above. The experimental parameters that were varied to nd these bounds where the drifts between the gas cell and lens 1 (d1), the drift between the lenses (d2), and the drift from lens 2 to the YAG:Ce crystal (d3). The t parameters were the source size and divergence, and the electron beam energy. The drift d2 (∼10 cm) is best determined as is it was measured with calipers. Even with a generous measurement error for d2 of±3 mm, the tted emittance varies by only 1%. For d1 the situation is more complex: on top of the physical measurement error as for d2, there is the additional uncertainty of where exactly the source of the electron beam is. As discussed in section 2.6, the electron beam source may not be at the physical exit of the gas cell due to the gas density downramp. Such a source position shift has not been measured explicitly

108 6. Measurements of the electron beam emittance

2 4 6 8 10 12 14

Accelerator Length (mm) 0.5

0.6 0.7 0.8 0.9 1.0

So ur ce si ze ( µ m )

20 40 60 80 100

Beam charge (arb. units)

Figure 6.8. Accelerator length scan. The plasma density was constant at 5.5×10¹⁸cm⁻³. Each data point is the mean value of at least 36 shots (except at 5 mm, here the mean of 8 shots). The error bars show the 95% condence interval for the mean. The charge data points are slightly oset in the horizontal axis to be distinguishable from the derived source size data. The increase of source size with acceleration length can be explained by heating of the electron beam due to its interaction with the tail of the laser during acceleration. The decrease in charge with acceleration length could be due to the higher charge observed near the high-energy cuto (see the main text); for shorter accelerator lengths, the observed electrons 232 MeV are closer to the cut-o energy.

but simulations indicate shifts of several mm are possible. The upper and lower bounds of±0.03π·mm·mrad for the derived emittance given above are a worst-case consideration and arise from a combined error for d1 and d3. A shift of the electron beam source position of ∆d1 = ±10 mm and a measurement error for the drift to the YAG:Ce crystal ∆d3 =±50 mm, results in this largest possible error.

The t parameter values of this section are summarised in table 6.1 on page 102.