Quasimonoenergetic electron beams produced by colliding cross-polarized laser pulses in underdense plasmas

The interaction of two laser pulses in an underdense plasma has proven to be able to inject electrons in plasma waves, thus providing a stable and tunable source of electrons. Whereas previous works focused on the"beatwave"injection scheme in which two lasers with the same polarization collide in a plasma, this present letter studies the effect of polarization and more specifically the interaction of two colliding cross-polarized laser pulses. It is shown both theoretically and experimentally that electrons can also be pre-accelerated and injected by the stochastic heating occurring at the collision of two cross-polarized lasers and thus, a new regime of optical injection is demonstrated. It is found that injection with cross-polarized lasers occurs at higher laser intensities.

The interaction of two laser pulses in an underdense plasma has proven to be able to inject electrons in plasma waves, thus providing a stable and tunable source of electrons. Whereas previous works focused on the "beatwave" injection scheme in which two lasers with the same polarization collide in a plasma, this present letter studies the effect of polarization and more specifically the interaction of two colliding cross-polarized laser pulses. It is shown both theoretically and experimentally that electrons can also be pre-accelerated and injected by the stochastic heating occurring at the collision of two cross-polarized lasers and thus, a new regime of optical injection is demonstrated. It is found that injection with cross-polarized lasers occurs at higher laser intensities.
Plasma-based accelerators can sustain high electric fields in excess of 100 GV/m, which is approximately 1000 times beyond what can be achieved in conventional accelerators. In the plasma medium, an intense laser pulse can drive a longitudinal wave, called wakefield, travelling with a phase velocity v p = v g , where v g is the group velocity of the laser, and can be close to the speed of light c. This high phase velocity enables ultra relativistic acceleration [1], but also requires that the initial speed of the particles is high enough to allow trapping in the plasma wave. Whereas in a linear regime, an electron with no initial velocity is not trapped, in a more nonlinear regime, transverse wave breaking effects [2,3] can result in the self-trapping of electrons in the so-called "bubble regime" [4]. In this case, the injection of electrons occurs in a short space-time volume and leads to the production of narrow energy spread electron beams as observed in Ref. [5,6,7]. Nevertheless, in this scheme, self-injection and acceleration depend on the precise evolution of the laser pulse. Therefore, fine control over the output electron beam is hard to achieve. The precise control of electron injection can be achieved by all-optical external injection schemes. These schemes use two pulses with the same polarization that will collide either head on [8,9] or with an angle [10,11]. The collision of the pulses heats the electronic population and some of the electrons can be trapped. Previous publications [12,13] have shown that this external injection scheme with two linearly polarized lasers allows a fine control over the injection and therefore brings tremendous flexibility to laser-plasma accelerators. In particular, it makes it possible to tune the output charge and energy of the electron beam in a stable way.
In this injection scheme, the collision of the two laser fields creates a slow ponderomotive beatwave φ b ∝< a 0 .a 1 > where a 0 and a 1 are respectively the normal-ized vector potentials of the main and injection pulses. For the circular polarizations case the motion of electrons is integrable and if both lasers have the same frequency ω 0 , the maximum energy gain is 2 √ a 0 a 1 . This preacceleration then allows the electrons to be trapped in the main plasma wave and to be further accelerated. The ponderomotive beatwave acceleration can be conversely seen as a heating process. The motion of electrons in two parallel polarized laser fields is not analytically tractable since there is not only the slow ponderomotive beatwave but also a fast varying component at 2ω 0 . For a colliding (head-on) geometry and above a certain threshold, the motion becomes stochastic [14,15]. In this case, the collision of the laser pulses leads to an even more efficient heating of the electrons. In the case of cross-polarized lasers a 0 .a 1 = 0 and there is no ponderomotive beatwave, thus no beatwave injection. However, we will show experimental and numerical evidence that, under some conditions, injection of high quality electron beams is still possible with cross-polarized laser pulses and that their features are comparable with the beatwave injected beams.
The experiment was performed on the "Salle Jaune" laser system at LOA. It delivers 720mJ (pump pulse) and 250mJ (injection pulse) at λ 0 = 0.8 µm on target with Full Width Half Maximum (FWHM) duration 30fs. The pump pulse is focused by a 1 m focal length spherical mirror to an intensity of I 0 = 3.4 × 10 18 W.cm −2 thus giving a normalized strength parameter a 0 = 1.3. The injection pulse is focused by a 1 m focal length off axis parabola to an intensity of I 1 = 4 × 10 17 W.cm −2 , corresponding to a 1 = 0.4. The two collinear and counterpropagating lasers are focused on the edge of a 2 mm supersonic Helium gas jet whose density profile is represented in figure 1. The electronic density plateau corresponds to n e = 7.5 × 10 18 cm −3 . The spectrum of electrons is recorded by a LANEX screen coupled with a permanent magnet [16]. It is set to record energies from 40 MeV to 400 MeV. It also gives access to the charge and divergence of the electron beam. Finally, a halfwave plate makes it possible to rotate the polarization of the pump pulse during the experiment, thus allowing to study and compare the parallel and crossed polarizations cases. The collision point of the two lasers could be tuned simply by changing the time delay between the two pulses. This is an important feature of the experiment since nonlinear evolution of the laser in the plasma -due to self-focusing [17,18] and self-compression [19,20]changes the laser intensity along the position in the gas jet. We used the particle code WAKE [21] to simulate the nonlinear evolution of the laser pulse and the corresponding wakefield in cylindrical geometry. The results of this simulation are shown in the bottom frame of figure  1: the solid line represents the evolution of a 0 and the dotted line represents the average depth of the plasma wave potential in the accelerating and focusing regions ∆φ (the average was made over 18 µm, i.e. the focal spot FWHM). This parameter characterizes the ability of the wakefield to trap preaccelerated electrons since it is directly related to the minimum momentum that an electron needs to be trapped, in normalized units: The nonlinear evolution of the pulse allows us to indirectly obtain data over a broad range of laser amplitudes a 0 at the collision of the two lasers and thus to change the heating conditions. Simultaneously, the wakefield becomes more suitable for trapping as the pump pulse self-compresses and distorts spatially. The experimental results are summarized in the top frame of figure 1. The graph represents the charge of the monoenergetic component of the spectrum for different collision positions for the cases of parallel and crossed polarizations. The figure shows that in the case of parallel polarizations, injection of a monoernergetic beam starts for z inj > −500 µm. This corresponds to positions where the laser pulse has self-focused ( figure 1, bottom). From that point, the charge increases as injection occurs further inside the gas jet. The increase of the charge can be explained by the evolution of the wakefield: as the laser propagates in the gas jet, ∆φ increases, making the wakefield more suitable for trapping a large amount of electrons (see also Ref. [22] for more detailed simulations). For the case of crossed polarizations, the behavior is different: there is a threshold behavior and monoenergetic electron beams are injected only for z inj > 25 µm. The charge also increases with z, as in the parallel polarizations case, but it is lower by a factor of 5-6. the collision takes place further inside the gas jet, e.g. at z inj = 225 µm, trapping become easier. Thus, for the parallel polarizations case, a 50 MeV beam with 68 pC is injected. The decrease of energy from position (a) (170 MeV) to (b) (50 MeV) can easily be explained by the decrease of the remaining acceleration length after the injection [12]. In the crossed polarizations case, the striking result is the production of a stable quasimonoenergetic electron beam (see figure 2b). The beam is stable in energy E = 72MeV ± 6MeV, in energy spread δE/E = 14% ± 2% (FWHM), divergence 5 ± 2mrad and charge Q = 11 ± 3pC. The injection of such a beam cannot be explained by the beatwave scheme since the residual parallel polarization component due to the possible half-wave plate misalignment is well below the threshold for beatwave injection. Other heating mechanisms that are not polarization dependent such as ponderomotive heating by the colliding envelopes of the lasers [9], phasekick injection where a colliding laser stimulates the wavebreaking of a nonlinear wake [23], or even wake collision [24] could be possible explanations. Here, we will show that the features of this injection of electrons in a cross-polarized scheme can be explained by the concurrence of two physical phenomena: the heating of electrons during the laser collision and the plasma wake inhibition.
First, we will show that heating of electrons occurs even with cross-polarized laser pulses. Here we use a simple 1D model in order to obtain an estimate of the stochastic heating for two lasers with arbitrary linear polarizations: we neglect collective plasma effects and follow test electrons in the laser fields. We use gaussian laser pulses at 800 nm with duration τ = 30 fs at FWHM with normalized strength a 0 = 2 for the pump pulse and a 1 = 0.4 for the injection pulse, which are close to the parameters of the experiment (after some self-focusing has occurred). Electrons are initially randomly distributed in the interval z ∈ [−2τ c, 2τ c], z = 0 corresponding to the position where the two pulse maxima collide. Outside this region, electrons interact with the laser pulses successively and we have checked that they are not significantly heated. Figure 3 shows the electron spectra after the collision: the dashed line represents the heating when the polarizations of the two pulses are parallel, the solid line corresponds to the crossed polarizations case and the dotted line corresponds to the case where electrons experience only the ponderomotive forces ∇(< a 2 0 > + < a 2 1 >). The latter case, where high frequencies are averaged, corresponds to the reference case for which there is no stochastic nor beatwave heating. The first remarkable feature is that electrons are more heated by two crosspolarized pulses (solid line) than by the sum of the two ponderomotive forces (dotted line). This can be explained by the fact that for high laser intensities, the electron motion becomes relativistic (a 0 > 1) which introduces a longitudinal component p z through the v × B force. Thus, the two perpendicular laser fields couple through the relativistic longitudinal motion of electrons. This relativistic coupling makes it possible to heat electrons.
This simple model also shows that the heating is much more efficient when the polarizations of the two pulses are parallel (dashed line), the maximal momentum obtained being 4-5 times higher than in the crossed polarizations case. This is why the injection is strongly correlated with the polarization of the two pulses as we have seen in the experimental results.
If the electron injection were only dependent on the preacceleration spectrum of the electrons, the crossed polarizations case should always result in a decrease of the trapped charge by several orders of magnitude compared to the parallel polarizations case. This corresponds to the discussion on polarization influence on the trapped charge made in Ref. [9]. However, in Ref. [9] the electromagnetic fields do not depend self-consistently on the motion of electrons, and therefore the model does not take into account an important physical phenomenon: the wakefield inhibition [25] at the collision of the laser pulses. To exhibit the influence of this phenomenon, we have performed self-consistent 1D PIC simulations. We have used the code CALDER [26] with the same laser parameters as above and an electronic density of n e = 7 × 10 18 cm The solid line corresponds to the case of parallel polarizations whereas the dotted line corresponds to the case of crossed polarizations. The laser fields are also represented by the thin dotted line. When the pulses have the same polarization, electrons are trapped spatially in the beatwave and can not sustain the collective plasma oscillation. Therefore the plasma wave is strongly distorted during and after the collision (see Fig. 4a and 4b). It has been shown in [25] that this process has a dramatic influence on the injected charge: in an inhibited plasma wave, it is harder to trap electrons and the charge is reduced by approximately one order of magnitude compared to trapping in an unaffected plasma wave.
When the polarizations are crossed, the motion of electrons is only perturbatively disturbed compared to the motion under only one laser, and the plasma wave is almost unaffected during the collision, see Fig. 4a and b. This tends to facilitate trapping.
In order to scan different injection conditions we have performed a numerical scan on a 0 , all other parameters of the 1D PIC simulations being the same. Figure 4(c) represents the trapped charge: it shows that the threshold for injection is higher when polarizations are crossed. This is consistent with the fact that electron heating is less efficient with cross-polarized lasers. It also shows that above this threshold, there is only approximately 3 times less charge injected with cross-polarized pulses than in the parallel polarizations case: the charge difference due to the heating efficiency is balanced by the charge reduction due to wake inhibition occurring only in the parallel polarizations case. In these simulations, we have also checked that all trapped electrons witness the overlapping of the two lasers, i.e. electrons located outside the collision region are not trapped. This rules out the wake-wake and wakelaser collisions [23,24] as main injection processes. This has also been confirmed in 2D PIC simulations.
Simulations can also give a partial understanding of the spectra experimentally observed with the cross-polarized pulses. In Fig. 2a, in the crossed polarizations case, the spectrum exhibits a broad component at lower energy, even though the threshold for the crossed polarizations injection is not reached. This could be explained by a staged acceleration mechanism: the electrons are first heated under the collision of the two lasers but as they have not gained enough momentum to be trapped, they slip back in the following buckets of the wakefield. They are then further accelerated in the collision of the two wakefields until they become trapped in the accelerating structure. In that case, injection is not localized as it occurs in multiple buckets, and it leads to the observed broad spectrum at low energy. On the other hand, when operating above the threshold, the volume where the electrons reach the required energy for trapping is very localized. Therefore injection results in a narrow energy distribution beam accelerated in the first plasma wave bucket. Due to the higher threshold, this injection volume is even smaller in the crossed polarizations case than in the parallel polarizations case. Therefore, for a given collision position, injection with crossed polarizations results in a lower charge and a smaller energy spread electron beam (see Fig. 2b). Finally, the energy difference (10 MeV) observed between the two cases (see Fig. 2b) could be explained by beamloading. The wakefield is less distorted by the lower charge injected in the crossed polarizations case and in consequence, electrons are accelerated to higher energies.
In conclusion, we have demonstrated both experimentally and theoretically a new regime of optical injection using the stochastic heating at the collision of two crosspolarized laser beams. This scheme can provide stable monoenergetic electron beams. The injection threshold for crossed polarizations is higher and in our experiment, we found that the beams produced in this case had lower charge and smaller energy spread than those injected by the beatwave scheme. Injection with crossed polarizations is also safer for the laser system when operating in a collinear colliding geometry: two polarizers are sufficient to protect the system from laser feedbacks. This would particularly interesting for experiments using waveguides [27], for which the collinearity of the laser beams is mandatory.
We acknowledge the support of the European Community Research Infrastructure Activity under the FP6 Structuring the European Research Area program (CARE, contract number RII3-CT-2003-506395 and EU-ROLEAP, contract number 028514). X.D. and E.L. acknowledge discussions with A. Bourdier on stochastic heating.