Three electron beams from a laser-plasma wakefield accelerator and the energy apportioning question

Laser-wakefield accelerators are compact devices capable of delivering ultra-short electron bunches with pC-level charge and MeV-GeV energy by exploiting the ultra-high electric fields arising from the interaction of intense laser pulses with plasma. We show experimentally and through numerical simulations that a high-energy electron beam is produced simultaneously with two stable lower-energy beams that are ejected in oblique and counter-propagating directions, typically carrying off 5–10% of the initial laser energy. A MeV, 10s nC oblique beam is ejected in a 30°–60° hollow cone, which is filled with more energetic electrons determined by the injection dynamics. A nC-level, 100s keV backward-directed beam is mainly produced at the leading edge of the plasma column. We discuss the apportioning of absorbed laser energy amongst the three beams. Knowledge of the distribution of laser energy and electron beam charge, which determine the overall efficiency, is important for various applications of laser-wakefield accelerators, including the development of staged high-energy accelerators.


Numerical simulations
The interaction of a high-power laser with a helium gas jet is simulated both in 3D and 2D geometry with the particle-in-cell (PIC) code OSIRIS 1 . The laser propagates along the x 1 direction, with a longitudinal profile taken as a sin 2 function with full width at half maximum in the range between 20 and 30 fs. The transverse profile is Gaussian and the beam radius w 0 is varied between 5 and 10 µm. The focal plane is located at the entrance of a pre-ionized plasma with constant density profile and linear up-ramp of length 40 µm in 3D and 60 µm in 2D. The laser is linearly polarised in the horizontal (x 2 ) direction for 3D simulations and in the x 3 direction for 2D simulations. The intensity I 0 is expressed in terms of the normalised vector potential a 0 = 8.5 × 10 −10 λ L [µm](I 0 [W/cm 2 ]) 1/2 , with λ L the laser wavelength. Here λ L = 0.8 µm. Simulations are performed in a box moving at the speed of light. In 3D the grid size is 50 µm × 40 µm × 40 µm with 1560 × 160 × 160 cells, corresponding to a resolution of λ L /25 in the longitudinal (x 1 ) direction and λ L /3.2 in the transverse direction. In 2D several configurations are employed to study the effect of grid size and resolution on the properties of side electrons. Energy and angle are found to change by only a few percent when going from a resolution of (λ L /25, λ L /3.2) to (λ L /72, λ L /24). Unless otherwise specified, results presented here are obtained for a 60 µm × 80 µm moving window with 3450 × 2400 cells, corresponding to a resolution of λ L /46 in the longitudinal (x 1 ) direction and λ L /24 in the transverse direction. When the properties of wide-angle electrons are calculated, high-energy forward electrons and low-energy plasma electrons are filtered out from all simulations by selecting only electrons with longitudinal momentum 1.4 < p 1 /mc < 50 and transverse position r > 5 µm. Comparison with 2D simulations performed in a fixed 1030 µm × 300 µm box shows that this choice of filtering and moving window offers a good compromise between accuracy and computational time.
Although the non-linear evolution of laser and plasma cannot be fully reproduced in 2D geometry 2 , 2D simulations allow quick scans of a large number of parameters and offer some guidance on the qualitative behaviour of the system. This is because the power contained in a pulse slice with transverse size w is proportional to a 2 0 w 2 in 3D, and a 2 0 w in 2D 3 . The reduced power in 2D geometry results in underestimating the effect of laser self-focusing, which is a key process governing system evolution 4 . To compensate for this power discrepancy, 2D simulations are performed with a scaled a 0 . It will be shown that with this correction 2D and 3D simulations predict similar energies and ejection angles for oblique electrons, in addition to similar maximum energies for the forward accelerated electron beams. Fundamental differences, however, still remain, especially for high plasma densities and laser intensities, when bubble size and shape changes significantly during propagation. For example, 2D simulations often predict a large growth in bubble size, whereas in 3D the accelerating structure becomes asymmetric and quickly collapses.
The properties of oblique electrons obtained from 3D and 2D simulations after 0.5 mm interaction are presented in Figure S.1 for a pre-ionised plasma with density 2 × 10 19 cm −3 (left and middle column) and for He gas with neutral density 1 × 10 19 cm −3 (right column). Plots in the left column have been obtained for laser a 0 = 2 and spot size w 0 = 10 µm and show a typical distribution with electrons clustering along the laser polarisation plane. A similar pattern is observed in the middle column, which is obtained for laser a 0 = 3 and spot size w 0 = 7 µm, corresponding to Figure 2d in the main paper. The clustering along the polarisation plane is stronger in neutral gas, as shown by the plots in the right column, obtained for the same laser parameters a 0 = 3 and w 0 = 7 µm. In Osiris, electrons released through ionisation have zero temperature, although the ADK model predicts a momentum distribution with Gaussian shape and widths of a few eVs transversely to the laser polarisation direction 5,6 . Experimental and theoretical studies suggest that electrons are born with a non-zero momentum spread also in the direction parallel to the laser electric field 6,7 . Further heating in the laser field can increase the temperature to tens of eV 8 . Similar temperatures are also found in pre-formed plasmas. For example, hydrogen-filled capillary waveguides have plasma temperature around 5 eV 9 . Figure  The emission rate is presented in Figure S.3, which shows that high charge is achievable over short propagation distances, especially for large laser a 0 and high plasma densities.

Semi-analytical model
The electromagnetic fields produced by a ionic sphere moving at constant velocity v b = v bx in a plasma with electron density n e can be derived analytically in the quasi-static approximation to obtain 10 with ω p = n e e 2 /(m e 0 ) the plasma frequency, e the magnitude of the electron charge, m e the electron mass, 0 the vacuum permittivity and c the speed of light in vacuum. Terms of order 1/γ 2 b , with γ b = 1/ 1 − v 2 b /c 2 the relativistic factor, are neglected. It is assumed that the cavity is surrounded by a thin electron sheath of width d, which is modelled by multiplying the electromagnetic fields by the shape function 10

Energy and angle
The dynamics of electrons initially at rest at x = 15 µm, z = 0 and interacting with a single non-evolving bubble is investigated for different bubble parameters, which are independently varied, although, as it will be shown below, this is not always possible in practice. The dependence of the electron beam mean energy and angle on the bubble radius is presented in Figure S.5a and S.5b for different bubble speeds and a fixed accelerating gradient, which corresponds to a plasma density n e = 2 × 10 19 cm −3 . Higher energies are achievable for larger bubble sizes and higher bubble speeds, when the acceleration lengths are longer, but the difference is small, due to the onset of injection, which removes high energy particles. The accelerating gradient is proportional to the density (equation 1), therefore the beam energy increases with density for fixed bubble radius and speed, but the growth is capped by injection, as shown in Figure  closer to a linear growth (Figure S.6c). The speed of the back of the bubble is shown in Figure S.6e and S.6f, with the curves representing quadratic fits. In first approximation, the bubble speed is given by the laser group velocity in plasma v g = c 1 − ω 2 p /ω 2 , but for high intensities the non-linear evolution of the laser beam causes it to slow down 11 . Figure S.5a indicates that side-electron mean energies beyond 5 MeV should be achievable for high bubble speeds. However, Figure S.6f shows that these conditions are not realisable in practice, since high speeds require low density plasmas, where the low accelerating gradient limits the energy to the 1-2 MeV range ( Figure S.5e). A small energy boost can be obtained by increasing the laser spot size, and therefore the bubble radius, but stable propagation would require increasing a 0 to satisfy the condition r b ≈ w 0 ≈ 2 √ a 0 c/ω p , which minimises bubble size oscillations 11 .
As discussed above, increasing a 0 reduces the bubble speed and therefore the beam energy. The narrow range of achievable energies makes side-electron beams relatively insensitive to laser and plasma parameters, as observed in the PIC simulations presented here and in Figure 2 and 3 of the paper.

Charge
Here we describe a rudimentary model to estimate the side-electron beam charge using the results of the previous section. As shown in Figure S.4, obliquely ejected electrons originate from a cylindrically symmetric region located approximately a bubble radius off-axis. The beam charge is estimated to be Q = q e n e V , with q e the electron charge, n e the plasma density and V the volume of the hollow cylinder, which has a thickness determined by the laser and plasma parameters. However, it should be noted that this method has many limitations, such as an incomplete modelling of the field and bubble evolution, which also leads to inaccurate predictions of the injection threshold. Furthermore, the laser field is not included, even though it can push electrons off-axis to positions where the energy gain is larger. This model also does not account for beam-loading by accelerated beams in the bubble, which will also affect the flow of current around the bubble. Nevertheless, the dependence of the side-electron beam charge on laser a 0 and plasma density obtained from 3D PIC simulations (Figure 3e and 3f of the main paper) can be reproduced.  The scaling of the beam charge with a 0 is estimated by assuming that the cylinder outer radius r o is given by the transverse bubble radius plotted in Figure S.6c, which is proportional to a 0 for the parameter range considered here. The cylinder inner radius r i is also assumed to grow linearly with a 0 , although at a slower rate, to match the increase in source size observed in Figure S.7b. With these assumptions, the base of the hollow cylinder has area π(r 2 o − r 2 i ) and grows quadratically with a 0 . For a constant height and density, the cylinder volume, and therefore the beam charge, also grows quadratically with a 0 , recovering the results of Figure 3e in the main paper, which is reproduced in Figure S.8a together with a curve obtained for a cylinder with thickness growing from 0.2 µm for a 0 = 3 to 0.9 µm for a 0 = 8 and with 0.4 mm height, matching the emission length in Figure S.3 over the first 0.5 mm propagation distance.
The scaling of the beam charge with the plasma density n e is estimated by assuming that the cylinder outer radius r o is given by the transverse bubble radius r b = r 0 1 + n e,0 /n e plotted in Figure S.6d. According to Figure S.7b, the source width should decrease with increase in density because of the smaller bubble radius. However, Figure S.7c and S.7d indicates that this effect is balanced by the larger accelerating gradient, which causes the width to remain approximately constant over a large range, with a drop only at low densities. Therefore, the volume of ejected electrons changes little and the beam charge is proportional to the density, recovering the results of Figure 3f in the main paper, which are also reproduced in Figure S.8b together with a curve obtained for a cylinder with thickness increasing from 0.1 µm at 5 × 10 18 cm −3 to 0.25 µm at 4 × 10 19 cm −3 and with 0.4 mm height.

Experimental results
Experiments to investigate the properties and possible applications of wide-angle electrons have been conducted at the Advanced Laser-Plasma High-energy Accelerators towards X-rays (ALPHA-X) beam line 12 . A Chirp Pulse Amplification (CPA), Ti:sapphire laser system delivers 35 fs, 800 nm pulses with on-target energy of 900 mJ. The laser beam is focused on a gas jet by an f /18 spherical mirror to a vacuum spot size of 20 µm (radius) at the 1/e 2 intensity point. A supersonic He gas jet is produced by a 2 mm diameter nozzle with plasma density 1-3 × 10 19 cm −3 . The laser peak intensity is 2 × 10 18 W/cm 2 , corresponding to a normalised vector potential a 0 ∼ 1, which grows to a 0 > 3 due to relativistic self-focusing of the beam causing self-compression to a ∼5 µm radius channel. The laser polarisation direction is horizontal (x). Low charge (∼5 pC), low divergence (∼3 mrad) quasi-monoenergetic electron beams with 100-200 MeV mean energy are produced along the laser propagation direction, simultaneously with the emission of high-charge, low-energy oblique electron beams. As shown in Figure 5 (main paper), a scintillating LANEX screen wrapped in Al foil and placed 60 cm from the gas jet along the laser propagation direction is used to measure the spatial distribution of forward electrons. A second LANEX screen placed at a distance of 7.5 cm from the gas jet at a 55 • angle with respect to the laser propagation axis is used to characterise oblique electrons. These off axis electrons accelerated outside the bubble could easily be mistaken for electrons accelerated inside the bubble.
The angular variation of the oblique electron beam for 200 consecutive shots is presented in Figure S.9, registering a mean ejection angle of (41 ± 1) • , with rms horizontal divergence of (11.0 ± 0.5) • and 20% variation in charge. Sample energy spectra of the side electrons measured with the magnetic spectrometer are presented in Figure S.10. The energy variation for 5 runs containing a total of 984 shots is presented in Figure S.11, registering a mean energy of (1.10 ± 0.25) MeV, with energy spread of (0.52 ± 0.03) MeV and 34% variation in charge.
Measurements made on a LANEX screen behind a multi-layer Al filter ( Figure S.12a) show that the thickest layer (1.25 mm at the top) blocks most of the side-electrons, whereas 90% of the beam is transmitted by the thinnest layer (0.25 mm at the bottom). An exponential fit for 5 layers provides an energy of 1.1 MeV. The mean energy of backward electron is obtained from an exponential fit of 10 shots recorded on a Gafchromic film placed behind 1 to 4 layers of 50 µm Al foils positioned above the beam axis ( Figure S.12b). An exponential fit provides an energy of 200 keV.
To demonstrate a possible application of off-axis beams from LWFAs, we have utilised the wide-angle electrons to image a 6 × 4.5 cm 2 hard drive controller board with 1 mm thick base ( Figure S.13a), placed at a distance of 5 cm from the nozzle and orthogonal to the side electron beam. Figure S.13b is obtained for 20 shots on a Gafchromic film and Figure S.13c is obtained for a single shot on an imaging plate (Fuji BAS-SR), which has higher sensitivity.

Video Legends
Side-electron video Video corresponding to Fig. 1 (main paper), showing results of a 2D OSIRIS simulation of the emission of low-energy electrons obliquely to the laser propagation direction.

Back-electron video
Video corresponding to Fig. 4 (main paper), showing results of a 2D OSIRIS simulation of the emission of low-energy electrons opposite to the laser propagation direction.