Numerical simulations of side-electron emission

Before showing experimental results, we present numerical simulations to illustrate the dynamics of the side-electron emission. We start with a general overview of the mechanism leading to the ejection of electrons at wide angles. We then provide examples of angular distributions and energy spectra of oblique electron beams. We finally discuss the properties of oblique beams for a wide range of laser and plasma parameters.

When a short intense laser pulse travels in underdense plasma, its ponderomotive force pushes electrons away from high intensity regions, leaving approximately spherical ion cavities (“bubbles”) trailing behind the pulse4,5. Snapshots of the first two bubbles are shown in Fig. 1, obtained from two-dimensional particle-in-cell (2D PIC) simulations for a laser that is linearly polarised in the x 3 direction and travelling along the x 1 axis; details of the set-up are given in the Methods. The plots include typical trajectories of electrons ejected at wide angles, with their positions at different times, marked by symbols. Background electrons undergo longitudinal and transverse acceleration as they interact with the bubble fields, but most of them gain little energy and stream backwards relative to the bubble motion, mainly feeding the formation of further cavities13,36,37,38. On the other hand, electrons located off-axis at a distance close to the bubble radius form a dense sheath and experience strong accelerating fields (Fig. 1a). When reaching the bubble base (Fig. 1b), some of these can be trapped inside the bubble and accelerate to energies that can reach GeV levels. Most others, having insufficient longitudinal momenta, are not injected into the bubble, but eject from the plasma at an angle of 30°–60°, depending on the specific energy gain (Fig. 1c,d). For example, electrons (white squares in Fig. 1) streaming around the sheath of the first bubble gain more energy and are ejected closer to the laser propagation axis than those further away (green circles), which can also lose energy due to the decelerating field at the front of the second bubble. Electrons can also be injected into the second bubble, forming a hollow beam that is accelerated to nearly 300 MeV energies with 10 pC charge39. Here, however, we investigate the properties of electrons that are not trapped in the bubble and report an efficient acceleration mechanism for nC-level beams with 1–10 MeV energy.

Figure 1: Snapshots of the electron density distribution produced behind an intense laser pulse propagating along the x 1 direction in a plasma. Curves correspond to sample trajectories of electrons ejected at wide angles, with the line colour representing the electron energy. The symbols show the position at the given time of electrons streaming close (white squares) and further away (green circles) from the accelerating structure (bubble) interior. Full size image

In order to characterise the oblique beam quantitatively, we now consider 3D PIC simulations over a wide range of parameters for a laser linearly polarised in the x 2 (horizontal) direction. The laser intensity I 0 is expressed in terms of the normalised vector potential a 0 = 8.5 × 10−10 λ L [μm](I 0 [W/cm2])1/2, with λ L the laser wavelength. Here, λ L = 0.8 μm. Results show that electrons are ejected over a wide cone with angles as large as 60° from the laser axis, and are often clustered in beamlets with properties that strongly depend on specific laser and plasma parameters. Representative examples are presented in Fig. 2 for electrons produced in a pre-ionised plasma with a density of 2 × 1019 cm−3, comparable with the parameter of experiments described below. Plots in the left column are obtained for a 0 = 2, spot size w 0 = 5 μm and propagation length 0.7 mm. In this case, injection into the bubble does not occur, and therefore no high-energy forward beams are formed. The bubble evolves little during propagation and the oblique electrons are ejected into a hollow cone with 36° mean angle, 7° rms divergence and 3.3 nC total charge. The mean energy is 1.7 MeV with 50% rms energy spread. The oblique beam carries a total energy of 2.8 mJ, corresponding to approximately 4% of the initial laser energy (68 mJ). If the spot size is increased to w 0 = 7 μm (plots in the middle column), injection occurs after 0.2 mm propagation and ejected electrons fill in the cone, since they can gain a larger longitudinal momentum by entering the bubble. The mean angle is 33°, with 12° divergence, 7.4 nC total charge, 2.4 MeV mean energy and 135% energy spread. The total energy is 18 mJ, corresponding to ~13% of the initial laser energy (133 mJ). Also prominent in the plots are two narrow beamlets aligned along the vertical axis and having a mean angle of 26°, with 4.5 MeV mean energy, 90% energy spread and ~200 pC charge. This feature depends strongly on the thermal momentum distribution of the electrons before the arrival of the laser pulse, in particular along x 3 (see Fig. 2c and S.1 and S.2 in the Supplementary Section). Electrons initially at rest cluster along the x 2 axis, which is parallel to the laser polarisation direction, whereas an initial momentum spread causes the formation of patterns at different angles depending on the laser parameters. Particle tracking indicates that these beamlets correspond to short intense bursts around the time of injection and contain only a small fraction of the total charge in the cone, despite the high visibility in the plot. They are attributed to the increased contribution to the wake potential of the longitudinal variation of the electron density during laser evolution (self-focusing and compression) and charge build-up at the bubble base. The same conditions also favour the onset of injection13,36,37,38, explaining the link between ejection angle and injection dynamics, which leads to the cone filling in. Plots on the right column are obtained for w 0 = 7 μm, a 0 = 3 and propagation length 0.5 mm. Again electrons fill the entire cone, but cluster preferentially along x 2 . The mean angle is 38° with 13° divergence, 10 nC total charge, 1.8 MeV mean energy and 116% energy spread. The total energy is 19 mJ, corresponding to 6% of the initial laser energy (300 mJ). It should be noted that the fraction of laser energy driving the plasma wake depends on laser and plasma parameters. For example, about 30% of the initial laser energy is transferred to the plasma after 1 mm propagation for laser a 0 = 3, spot size between 7 and 10 μm and plasma density of 5 × 1018 cm−3. On the other hand, for a plasma density of 1 × 1019 cm−3 the energy transfer is about 80% for the same laser conditions, and at higher densities the loss can be as high as 90% after 0.5 mm propagation.

Figure 2 3D PIC simulations of spatial (a–c), angular (d–f) and spectral (g–i) distribution of oblique electrons on a screen 10 mm downstream from the plasma. The laser intensity and spot size are a 0 = 2, w 0 = 5 μm (left column), a 0 = 2, w 0 = 7 μm (middle column) and a 0 = 3, w 0 = 7 μm (right column). Dashed curves in Figures (d–f) show the energy dependence on angle. Further examples are given in the Supplementary Section. Full size image

As shown by the dashed curves in Fig. 2d–f, the electron energy is angle dependent, increasing from a mean value of about 1 MeV at large angles to almost 10 MeV on-axis. The associated energy spectra are presented in the bottom row of Fig. 2, showing typical energies of 1–2 MeV. Since electrons are continuously ejected during laser propagation, the bunch duration is typically picoseconds and depends on the interaction length. Simulations of neutral helium gas, where the ionisation process is included using the ADK model40, present similar beam properties (see Supplementary Section). In addition, the ejected electrons are also shown to produce strong magnetic fields as they cross the transverse plasma boundaries, in analogy to the flow of charged particles from an ionised to a non-ionised interstellar medium41,42.

The properties of oblique beams for a wide range of laser and plasma parameters are presented in Fig. 3. The dependence of mean angle and energy on laser strength a 0 are shown in Fig. 3a and c for plasma density 2 × 1019 cm−3, laser spot size w 0 = 7 μm and 0.5 mm propagation length. At almost all the intensities, the ejected electrons fill the full 60° cone. There are no large changes in the energy and angle, since the non-linear laser evolution eventually causes the bubble to slow down, capping the energy gain (Figures S.5 and S.6 in the Supplementary Section). For the given range, the charge grows quadratically with a 0 (Fig. 3e), which is expected if the laser expels a hollow column of plasma electrons with thickness determined by the bubble transverse size, which grows linearly with a 0 (see Figures S.6 and S.7 in the Supplementary Section and the associated discussion). The dependence on plasma density is presented in Fig. 3b and d for a 0 = 3, w 0 = 7 μm and 0.5 mm propagation length. Again, beam properties change little, since the higher accelerating gradients achievable at high densities are countered by the shorter acceleration length and lower bubble speed (Figures S.5 and S.6 in the Supplementary Section). Higher energies are observed close to 1 × 1019 cm−3 due to short bursts of high energy electrons emitted during injection. At 5 × 1018 cm−3 there is no injection within the given propagation length, but injection occurs for laser spot size w 0 = 10 μm, when a mean energy of 4.9 MeV is observed. At 1 × 1018 cm−3 there is no injection also for w 0 = 10 μm and the mean energy is 1.8 MeV with 1 nC charge after 1 propagation. The charge increases linearly with density (Fig. 3f), as the volume of the hollow column of ejected electrons remains approximately constant (see Figure S.7 in the Supplementary Section and the associated discussion). Nevertheless, for low densities the accelerating structure evolves more slowly and electrons can be produced over longer distances. For example, the charge at 1 × 1019 cm−3 is 3.9 nC after 0.5 mm and 10.3 nC after 1 mm (Figure S.3 in the Supplementary Section).

Figure 3: Dependence on laser a 0 and plasma density of mean angle, energy and charge of oblique electrons from 3D (squares) and 2D (circles) simulations for spot size w 0 = 7 μm and propagation length 0.5 mm. The error bars represent rms divergence and energy spread. (a,c,e) have been obtained for n e = 2 ×1019 cm−3 and Figures (b,d,f) for a 0 = 3. Curves in (a–d) show the mean energy and angle predicted by Kostyukov’s model43 for one (dotted line K 1 ) and two (dashed line K 2 ) bubbles. Curves in Figures (e,f) are quadratic and linear fits, respectively. Full size image

The weak dependence of energy and angle on laser and plasma parameters is a consequence of the small range of momenta acquired by electrons not injected into the bubble. This is confirmed by comparing the above PIC simulation results with the predictions from Kostyukov’s semi-analytical reduced model43, where the bubble is modelled as a non-evolving ionic sphere moving in a plasma at a constant velocity and surrounded by a thin electron sheath. The trajectories of displaced background electrons have been calculated both for a single bubble and for two coalesced bubbles. Here, 50,000 sample electrons initially at rest and uniformly distributed are considered and the bubble radius and velocity are estimated from 3D PIC results (Figure S.6 in the Supplementary Section). The final beam properties are plotted in Fig. 3 and Figures S.4–S.5 in the Supplementary Section. Electrons initially close to the laser axis or located at a distance larger than the bubble radius gain little energy and are scattered at large angles or left unperturbed. Only electrons displaced by about the bubble radius are accelerated to MeV energies, capped by the onset of injection. Higher energies are possible for larger bubble sizes, and therefore longer acceleration lengths, but typically have exponential spectra, since the mean energy is still dominated by the bulk of less energetic electrons. Furthermore, the curves obtained for two attached bubbles indicate that the decelerating field at the front of the second bubble reduces the longitudinal energy gain and leads to larger ejection angle. A complete description of the process should include dynamic bubble evolutions (changes in size and velocity, density variations, asymmetries in shape and differences between successive buckets), which is only possible with 3D PIC simulations.

Numerical simulations of backward electron emission

Backward emission of low-energy electrons, opposite to the laser propagation direction, has also been observed, mostly during the initial stages of the laser-plasma interaction. Figure 4 shows sample trajectories obtained from 2D PIC simulations in the laboratory-frame for electrons located in the proximity of the vacuum boundary. The plasma starts at x 1 = 0 with a 60 μm long linear up-ramp to a density of 2 × 1019 cm−3. Electrons close to the boundary (white squares) are ejected immediately on laser arrival, whereas electrons deeper inside the plasma (green circles) are accelerated and then ejected backwards by the plasma fields, with the formation of the bubble structure. Some electrons also stream back from further inside the plasma. The energy distribution of these electrons is exponential, with mean energy of 0.3–0.4 MeV, depending on laser intensity. Here only electrons with energies above 0.1 MeV have been considered. The ejection angle can reach 90°, but typically electrons form a 40°–50° cone around the laser axis. An accurate calculation of the charge requires 3D simulations in the laboratory frame over the full interaction distance, which are very computationally intensive. However, 3D simulations using a moving window indicate that charges of the order of 0.5 nC can be produced over the first 60 μm of interaction for a 0 = 3, n e = 2 × 1019 cm−3 and w 0 = 7 μm, giving a total energy of the backward electrons of about 0.13 mJ. Simulations for different plasma densities show no significant change in mean energy and ejection angle, whereas the charge varies from about 0.15 nC at 1 × 1019 cm−3 to 0.8 nC at 4 × 1019 cm−3. The charge also depends on the laser a 0 , obtaining 1.8 nC with a mean energy of 0.4 MeV for a 0 = 5, n e = 2 × 1019 cm−3 and w 0 = 7 μm. In contrast, for a 0 = 2 the charge is 0.1 nC with a mean energy of 0.3 MeV.