Multi-GeV electron-positron beam generation from laser-electron scattering

The new generation of laser facilities is expected to deliver short (10 fs–100 fs) laser pulses with 10–100 PW of peak power. This opens an opportunity to study matter at extreme intensities in the laboratory and provides access to new physics. Here we propose to scatter GeV-class electron beams from laser-plasma accelerators with a multi-PW laser at normal incidence. In this configuration, one can both create and accelerate electron-positron pairs. The new particles are generated in the laser focus and gain relativistic momentum in the direction of laser propagation. Short focal length is an advantage, as it allows the particles to be ejected from the focal region with a net energy gain in vacuum. Electron-positron beams obtained in this setup have a low divergence, are quasi-neutral and spatially separated from the initial electron beam. The pairs attain multi-GeV energies which are not limited by the maximum energy of the initial electron beam. We present an analytical model for the expected energy cutoff, supported by 2D and 3D particle-in-cell simulations. The experimental implications, such as the sensitivity to temporal synchronisation and laser duration is assessed to provide guidance for the future experiments.

where ξ 2 1 = ξ 2 = (eE 0 /mω 0 ) 2 , f µν 1 = n µ a ν 1 − n ν a µ 1 , a µ 1 = (0, a 0 , 0, 0) T is the normalized vector potential and n µ = (1, n). Here, n is a unit vector parallel to the laser direction of motion n || u 3 and ρ 0 ≡ (nu 0 ). Einstein notation was used (a repeated Greek index is equivalent to a summation over all dimensions and (ab) ≡ a µ b µ ). Functions I 1 (φ ) and h(φ ) carry the imprint of the radiation reaction. They are expressed as: where φ 0 denotes the initial phase, φ current phase of the particle within the field of the electromagnetic wave, α is the fine structure constant, and η 0 = (k 0 u 0 )/m is the wave frequency measured in the electron rest frame in units of the electron mass m. Other quantities denote elementary charge e and electromagnetic field amplitude E 0 . Without radiation reaction, Eqs. (2) reduce to The equations (1)-(3) are presented in natural units, as in Ref. 1 . In those unitsh = c = 1, and the fine-structure constant is defined by α = e 2 /hc 1/137. We now switch to the dimensionless units more common in laser-plasma interactions. Here, all the quantities are normalized to the laser frequency ω 0 , such that t → tω 0 , p → p/mc, E → E /mc 2 , E → eE/mcω 0 and B → eB/mcω 0 . In these units, ω 0 = 1, λ 0 = 2π, c = 1, mass is normalized to the electron mass m and charge is normalized to the elementary charge e. The particle equations of motion by component for the case without radiation reaction starting at arbitrary laser phase with an arbitrary initial momentum can be written as: We can compute the spatial displacement of the particle as a function of the phase φ . We have )dφ , where we used a relation between the phase and time dφ = dt(γ − p 3 )/γ to perform the change of variables. The denominator is an integral of motion, such that γ(φ ) − p 3 (φ ) = γ 0 − p 0,3 . The spatial displacement in the laser direction of motion is then given by: The standard result for a particle starting at rest can be retrieved by taking γ 0 = 1 and p 0,1 = p 0,2 = p 0,3 = 0. However, our particles are not created at rest. Electron-positron pairs are created by scattering an LWFA electron bunch with an intense laser at normal incidence. We can assume as a first approximation that p 0,1 = p 0,3 = 0, but p 0,2 = 0, i.e. the particle initial velocity is perpendicular to both the laser polarization direction x 1 and to the laser propagation direction x 3 .

Maximum energy and plane wave acceleration length for particles initially at 90 degrees
For a particle born at 90 degrees of incidence to the laser axis with an initial Lorentz factor γ 0 , maximum attainable energy according to Eqs. (4) is given by E max = 2a 2 0 /γ 0 . This is achieved when I 1 (φ ) 2 is at the maximum. For simplicity, let us consider a particle with φ 0 = 0, as an example that allows to obtain that maximum. Here, the expression for longitudinal displacement given by Eq. (5) can be simplified: The maximum energy E max is reached for φ = π, after the particle has traveled a distance of l pwa = 3a 2 0 π/(4γ 2 0 ) or l pwa = λ 0 × 3a 2 0 /(8γ 2 0 ). We define this distance as the plane wave acceleration length. If our particle would start at a different φ 0 in otherwise identical conditions, its maximum energy can be smaller or equal to E max . For particles born at normal incidence, the particles with the lowest initial γ 0 are the ones that can attain the highest energy in the interaction with the plane wave (because E max ∝ γ −1 0 ). However, one should note here that the higher energy gain is possible due to the slower dephasing, and the plane wave acceleration length is reversely proportional to the square of the initial energy l pwa ∝ γ −2 0 . Such particles, therefore, propagate a longer distance before reaching the E max .

Correction of the maximum energy due to the pulse focusing
The maximum energy and the acceleration length in the previous section are valid for a plane wave interaction. However, the lasers are not plane waves, and the laser de-focusing affects the acceleration. Particles are expected to attain an energy cutoff similar to E max in a focused laser only if R L l pwa , where R L = πW 2 0 /λ 0 is the Rayleigh range, W 0 and λ 0 are the laser waist and the wavelength respectively. If R L l pwa , we require an estimate on how the laser de-focusing affects the acceleration. To this end, we perform a series of test particle simulations, where the particles are initialized with the most favourable phase for the highest energy gain within the shortest propagation distance. The test particles are born at the laser central axis, in focus, at φ 0 = 0 of the wave. The 3D analytical expression for the fields of the focused laser pulse is taken from Ref. 2 . We varied the laser intensity, and the initial particle energy is chosen according to γ 0 = 5 × 10 4 /a 0 to be consistent with the lowest available energy expected in the system.
The results displayed in Fig. 1 correspond to a range of laser intensities between a 0 = 100 and a 0 = 4000 and three different waist sizes: W 0 = 3.2 µm, W 0 = 5 µm and W 0 = 10 µm. For every example, the maximum energy attained by the particle born in the laser focus with φ 0 = 0 is divided by the maximum energy such a particle could attain in a plane wave. An interesting point to note here is that the differences between the case with and without radiation reaction are not strongly pronounced. This is consistent with the fact that a quickly acquired parallel component of the momentum reduces the χ e and the radiation reaction stops being relevant for the particle dynamics. Figure 1 indicates that the ratio E /E max is a function of R L /l pwa and for R L /l pwa < 1 this can be approximated as: 2/3 Figure 1. Ratio of the maximum attainable energy for a focused laser E and the maximum attainable energy for the same laser intensity in a plane wave E max as a fuction of the ratio of the Rayleigh range R L to the acceleration length l pwa . For 0.05 < R L /l pwa < 1, one can approximate the correction of maximum energy due to the pulse focusing using the Eq. (7).
For R L /l pwa > 10, we consider the condition R L l pwa satisfied and E E max . For l pwa < R L < 10 l pwa , Eq. (7) is not a good estimate, due to its value being on the same order of E max . For l pwa < R L < 10 l pwa , the maximum energy takes values between E max /2 and E max . Equation (7) shows the defocusing correction for the maximum energy of a particle born in the very centre of the laser field. We can assume that for an arbitrary particle born anywhere in the laser field (with the same γ 0 ), the effective interaction length cannot be more than twice the value for the particle born in the centre. As the effective interaction length is proportional to R L , we should not expect to obtain particles with energies higher than Full-scale 3D simulations in the main manuscript are in very good agreement with the predictions for the cutoff of the particle energy spectrum given by Eq. (8).