Localization of ionization-induced trapping in a laser wakefield accelerator using a density down-ramp

The inner length of the cell, i.e. the distance between the inner surfaces of the front and back windows, is set to 2 mm and a sequence of shots is recorded while the backing pressure supplied to the gas cell is varied using either pure hydrogen or a gas mixture containing 1% of nitrogen.

As shown in figure 2(a), the electron number density threshold for trapping of electrons is significantly lower for the gas mixture than for pure hydrogen. This is expected, since the threshold for ionization-induced trapping is lower than for self-trapping [19]. Furthermore, it is observed that the threshold for trapping is much sharper in the case of self-trapping in pure hydrogen than for ionization-induced trapping in the gas mixture. Using the gas mixture at low electron number density, $4\times {{10}^{18}}~\text{c}{{\text{m}}^{-3}}$ to $7\times {{10}^{18}}~\text{c}{{\text{m}}^{-3}}$, the amount of charge in the beams of accelerated electrons scales almost linearly with the electron number density in the plasma, indicated by the red dashed line in figure 2(a). The shot-to-shot fluctuations in charge, as deviations from a fitted linear dependence on the density, is only $1.2~\text{pC}$ here. At higher electron number densities the charge increases more rapidly with density but the relative shot-to-shot fluctuations in charge remains low, and is on average 14.5% at each selected value of electron number density.

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A typical image of a dispersed electron beam accelerated when the cell is filled with the gas mixture to provide an electron number density of $9\times {{10}^{18}}~\text{c}{{\text{m}}^{-3}}$ is shown in figure 2(b). This image shows the reproducible features of the beams of accelerated electrons trapped by ionization, with a large energy spread up to a cut-off energy of approximately $150~\text{MeV}$ and a rather large divergence that is increasing with electron energy (≈12 mrad at $100~\text{MeV}$). The beam charge, of electrons with energy above $50~\text{MeV}$, is in this case $38~\text{pC}$.

Beams with similar amount of charge are produced using pure hydrogen at a higher electron number density, $13\times {{10}^{18}}~\text{c}{{\text{m}}^{-3}}$. A typical image of a dispersed electron beam accelerated under these conditions is shown in figure 2(b). Due to oscillations of the peak normalized vector potential, a₀, these beams typically contain multiple, peaked, components as self trapping occurs at multiple locations along the plasma and the energy and number of peaks are fluctuating from shot to shot. Each of these components has an energy spread of about 10% and a divergence smaller than $10~\text{mrad}$.

The length of the cell is varied in a sequence of shots with the cell filled with the gas mixture of hydrogen and $1\%$ of nitrogen and the typical spectral shapes are studied. The electron number density is set to $12\times {{10}^{18}}~\text{c}{{\text{m}}^{-3}}$ to operate the accelerator well above the threshold for ionization-induced trapping [20]. At this density, the power of the laser pulse exceeds the critical power for self-focusing, $P/{{P}_{\text{c}}}\approx 4$. The peak normalized vector potential is therefore expected to increase rapidly as the laser pulse enters the gas cell and reaches approximately the value for a matched pulse, for which the transverse size of the laser pulse is approximately equal to the transverse size of the electron void. For a laser pulse containing $585~\text{mJ}$ of energy this corresponds to a₀  =  3.2, according to the theory by Lu et al [21]. Under these conditions, the dephasing length is ${{L}_{\text{d}}}\approx 0.6~\text{mm}$. Thus, for regular ionization-induced trapping, broad electron energy spectra, without peaked features are expected.

In figure 3(a), spectra are shown for five shots at different inner cell lengths. For inner cell lengths of above $1.8~\text{mm}$, the electron energy spectra are wide and monotonically decreasing with energy. For cell lengths of $1.8~\text{mm}$ and lower, the charge distribution is modified and a peak builds up at energies around 100 MeV. The visibility of the peak is largest for a cell length of $1.3~\text{mm}$ as made clear in figure 3(b), in which the spectra are normalized to the charge at $50~\text{MeV}$.

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For cell length below 1.8 mm, it is observed that the shape of the distribution is not sensitive to the cell length and with a peak in the range 100–110 MeV, which suggests that the distance from the point of trapping to the end of acceleration is unaffected by changes in the cell length. This is consistent with the assumption that these electrons are trapped in the density down-ramp at the back of the gas cell.

For a cell length of 0.7 mm, after re-optimizing both longitudinal laser focus position, using the deformable mirror, and pulse compression by adjusting the grating separation, and after decreasing the electron number density to $9.6\times {{10}^{18}}~\text{c}{{\text{m}}^{-3}}$, the peaked features in the spectrum are even more clear (see figure 4(a)). In addition, another peaked feature at higher energy is observed. The peak at lower energy is reproducible and appeared in 73 of 100 shots in a sequence. For these shots the energy of the peak is $90~\text{MeV}\pm 10\%$ and the energy spread is $20~\text{MeV}\pm 48\%$. For the remaining 27 shots, a component at similar energy and divergence is still observed, but is not contributing to a peak in the spectrum but rather a wide distribution of energies up to $\approx 120~\text{MeV}$. In both cases the divergence was only approximately $3~\text{mrad}$.

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The peak at higher energy is observed on all shots in this sequence at an energy of $150~\text{MeV}\pm 7\%$, with an energy spread of $50~\text{MeV}\pm 18\%$. This second peaked feature has a larger divergence (≈10 mrad) than the component at lower energy. The total amount of detected charge is on average $9~\text{pC}\pm 28\%$ in this series of shots.

Under the same conditions, but switching to pure hydrogen, only the peak at $150~\text{MeV}$ remain, as shown in figure 4(b). In this series of shots, the amount of charge is on average $1.0~\text{pC}\pm 29\%$, but increases to $2.0~\text{pC}$ when moving the cell only $0.25~\text{mm}$ along the optical axis.

Typical spectra from the two cases, normalized at the peak at high energy, are shown in figure 4(c). Here it can be observed that the energy spread of the component at $150~\text{MeV}$ is approximately $50~\text{MeV}$ (FWHM) in both cases. The component at low energy, for this shot at $90~\text{MeV}$, has a significantly smaller energy spread of only $15~\text{MeV}$.

Simulations are performed using the fully relativistic three-dimensional particle-in-cell code CALDER-CIRC [22], which is exploiting the fact that only a low number of azimuthal Fourier components are required to model laser wakefield acceleration. In the simulations the time step is set to 52 attoseconds and a window following the laser pulse, with a size of $50~\mu \text{m}$ longitudinally and $75~\mu \text{m}$ radially, is used. The spatial grid size is set to $16~\text{nm}$ in the longitudinal direction and $190~\text{nm}$ in the radial direction, and three azimuthal Fourier modes are used.

The laser pulse is defined in the simulations to have a pulse duration of $34~\text{fs}$ and a transversal size of $17~\mu \text{m}$ (FWHM) when focused in vacuum, with a peak normalized vector potential of 1.25, approximately corresponding to laser parameters in the experiment. The waist of the laser beam, when not affected by the propagation in the plasma, is chosen in the simulations to be located at the beginning of the density plateau inside the cell. Furthermore, the shape of gradients of the density distribution in the entrance and exit of the cell are fitted as piece-wise linear functions, consisting of three parts, from the results from the characterization of the gradients using computational fluid dynamics. The length of the gradient is chosen to be $500~\mu \text{m}$, in the fall from $90\%$ to $10\%$ of the plateau density, which corresponds to the shortest possible gradient determined using computational fluid dynamics.

First, simulations are performed at different electron number densities to determine the threshold of self trapping in the density plateau. A simulation is then performed for a cell with an inner length of $0.8~\text{mm}$ at an electron number density of $9.1\times {{10}^{18}}~\text{c}{{\text{m}}^{-3}}$ to avoid self-trapping in the density plateau. The results from this simulation are summarized in figure 5(a). Here, the evolution of the electron energy spectra are shown for the electrons released by ionization from $\text{H}$ and $\text{N}-{{\text{N}}^{4+}}$ in black (upper) and the electrons released by ionization from ${{\text{N}}^{5+}}$ and ${{\text{N}}^{6+}}$ green (lower). Also, the evolution of the peak normalized vector potential inside the plasma is shown in red and the electron number density is shown in blue. Each of these curves are normalized to the corresponding maximum value of the simulation. Furthermore, in figures 5(c) and (d), the electron distribution is shown, both in real space and longitudinal phase-space, close to the laser pulse before (b), in the middle (c) and in the end (d) of the density down-ramp together with the corresponding electron energy spectrum at each position.

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Under these conditions, ionization-induced trapping occurs continuously in the plateau between the input and exit of the cell and provide a wide, but very weak, background in the electron energy spectrum.

An increased rate of trapping of electrons is observed in the density down-ramp, both for electrons from the background plasma and for electrons ionized from ${{\text{N}}^{5+}}$ and ${{\text{N}}^{6+}}$. The electron energy spectrum at the end of the density down-ramp contains two slightly separated peaks, consistent with the experimental results. The peak at higher energy is identified to contain electrons from ionization of hydrogen and from the first ionization states of nitrogen ($\text{N}-{{\text{N}}^{4+}}$). These electrons are released into the plasma, by ionization from their corresponding atoms or ions, far from the peak of the laser pulse, and the simulation shows that these electrons are injected and trapped in the density down-ramp at the exit of the gas cell by density down-ramp injection. The spectral peak at lower energy is due entirely to electrons from ionization of ${{\text{N}}^{5+}}$ and ${{\text{N}}^{6+}}$, which occur much closer to the peak of the laser pulse. The simulation shows that these electrons are also trapped in the density down-ramp at the exit from the gas cell.

The simulation further reveals a longitudinal separation of the electrons that build up the two components after they have been trapped in the plasma wave. During the subsequent acceleration in the density down-ramp, the electrons from ionization of ${{\text{N}}^{5+}}$ and ${{\text{N}}^{6+}}$ are located closer to the center of the electron void, and thus experience a lower accelerating field than the trapped electrons ionized from $\text{H}$ and $\text{N}-{{\text{N}}^{4+}}$, which are located further to the back of the electron void.

The reason for this separation is a fundamental difference in the injection and trapping mechanism. The electrons, already released from the atoms far from the peak of the laser pulse, are injected at the back of the electron void following the laser pulse, as the size of the bubble is growing in the density down-ramp. In contrast, ionization of ${{\text{N}}^{5+}}$ and ${{\text{N}}^{6+}}$ releases electrons into the plasma closer to the peak of the laser pulse and they are able to pass through the rest of the laser field into the electron void. The released electrons are then accelerated by the longitudinal electric field and catch up with and exceeds the speed of the laser pulse (corresponding to a relativistic factor ${{\gamma}_{l}}\approx 14$).

Overall, the final electron energy spectrum of the simulation agrees qualitatively well with the experimental findings under similar conditions. However, the energy of the two spectrally peaked features are lower than observed in the experiments. We attribute those differences to discrepancies in the parameters of the laser pulse and/or density distribution between the experiment and simulation. In particular, the effect of a longer density ramp is two-fold: (I) the effective length of acceleration becomes longer. (II) The plasma wavelength is increasing more slowly, which means that the dephasing of the electrons due to the expansion of the plasma wave is slower.