A compact tunable polarized X-ray source based on laser-plasma helical undulators

Laser wakefield accelerators have great potential as the basis for next generation compact radiation sources because of their extremely high accelerating gradients. However, X-ray radiation from such devices still lacks tunability, especially of the intensity and polarization distributions. Here we propose a tunable polarized radiation source based on a helical plasma undulator in a plasma channel guided wakefield accelerator. When a laser pulse is initially incident with a skew angle relative to the channel axis, the laser and accelerated electrons experience collective spiral motions, which leads to elliptically polarized synchrotron-like radiation with flexible tunability on radiation intensity, spectra and polarization. We demonstrate that a radiation source with millimeter size and peak brilliance of 2 × 1019 photons/s/mm2/mrad2/0.1% bandwidth can be made with moderate laser and electron beam parameters. This brilliance is comparable with third generation synchrotron radiation facilities running at similar photon energies, suggesting that laser plasma based radiation sources are promising for advanced applications.


Supplement materials
Supplementary Discussion 1 Trajectory calculation of the laser pulse's centroid motion In our considered situation, the plasma density distribution only depends on the radical coordinate. According to the eikonal equation and laser group velocity equation v g = c · n(r), in cylindrical coordinates (r, ϕ, z), we have where ω 0 and ⃗ k are the frequency and wavenumber vector of the laser pulse respectively, n c is the critical density of plasma. Rewriting Eq. (S1) and Eq. (S2) in cylinder geometry, we get Based on Eq. (S5) and Eq. (S6), k ϕ r and k z are constant since the laser pulse enters a channel where n e0 = n e (r = √ x 2 0 + y 2 0 ) is the plasma density at the point of laser entrance with x 0 and y 0 are laser initial off-axis distances, θ z = arccos( √ 1 − cos 2 θ x − cos 2 θ y ) with θ x, y, z the angles between laser propagation direction and the space coordinate axes x, y, and z, and b = (y 0 −x 0 cosθ y /cosθ x )/( √ (cosθ y /cosθ x ) 2 + 1) is called the "striking distance", which is defined as the distance between the projection of the incident laser pulse and the point of the plasma channel centre on the incident surface. Substitute k r = √ k 0 · (1 − n e0 /n c ) 1/2 − k 2 ϕ − k 2 z and k ϕ into Eq. (S3) and Eq. (S4), one can obtain the trajectory equation In Eq. (S9), when the expression inside the square brackets equals zero, the electrons are at the points with the largest radial coordinate r and the choice of plus-minus sign should be changed.

Supplementary Discussion 2 Simulation of ionization injected electrons and their radiation from a helical undulator
To justify the feasibility of our scheme adaptable to the self-injected electrons, a non-optimized simulation by using ionization injection mechanism has been carried out. In this simulation, the duration of the laser pulse is L 0 = 6.0 T 0 and the incidence parameters are x 0 = 1 µm, y 0 = 0, θ x = 89 • and θ y = 91 • . The on-axis density of the plasma channel is n 0 = 0.004 n c . The density of the partially ionized nitrogen is n N 5+ = 8.0 × 10 −4 n c which is used for injection purpose. The nitrogen gas is located from x = 30 λ 0 to x = 60 λ 0 with an up-ramp plateau down-ramp (5 λ 0 −20 λ 0 −5 λ 0 ) profile.
In the simulation, we have found that finally about 2 pC electrons are ionization injected and accelerated to about 160 MeV. Typical trajectories of the injected electrons are shown in Fig. S1 and the distribution of the electrons radiation and polarization are plotted in Fig. S2. They are basically similar to the external injected electrons radiation. It should be mentioned that due to the extreme numerical simulation cost for ionization injection in our 3D simulation case, we have not optimized the incidence parameters. However, the current simulation basically justifies the reasonability of using the typical external injection beam in the paper.

Supplementary Discussion 3 Relativistic non-linear effects analysis
To see relativistic non-linear effects, we have made a few more simulations by using small a0. Fig. S3(a) shows the comparison of laser centroid trajectories in theory (black lines), a 0 = 2 simulation (red lines), and a 0 = 0.5 simulation (green lines). It is clear that without the non-linear effect the a 0 = 0.5 simulation trajectory perfectly matches the theoretical trajectory. Whereas in a 0 = 2 situation, simulation trajectory deviates from that of theory as the laser pulse propagating mainly in oscillation period. Fig. S3(b) shows the comparison of electron beam centroid trajectories in a 0 = 2 simulation (blue lines) and a 0 = 0.5 simulation (magenta lines). The electron beam motion in the bubble shaped wakefield has to follow the oscillation process of the laser pulse. So the oscillation amplitude becomes smaller and the oscillation period becomes more constant in the a 0 = 2 simulation. In summary, the relativistic non-linear effect indeed exists in our simulations. It leads to the laser centroid trajectory and the electron beam centroid trajectory deviate to a certain extent from that in theory and in linear regime. However, the general trend remains unchanged and the helical motion of such trajectories are still stable under this parameter.

Supplementary Discussion 4 Further Radiation spectra analysis
The energy spectrum for the point with lowest intensity, which is on the elliptical ring with the azimuth angle ϕ of 45 • (see Fig. 3(a)), is shown in Fig. S4(a). The peak energy at this point is smaller than that of the points with maximum radiation intensities. This is consistent with the peak energy equation E p = 3πγ 3 r os hc/Λ os when γ and r os are smaller. And Fig.  S4(b) shows the energy spectrum integrated over the entire observe region. This spectrum is wider than the radiation spectrum of the marked points in the manuscript (see Fig. 3(c) and (d)). Because the electrons are being accelerated during the helical motion. Electrons with different energy would radiate photons with different energy as well. And the radiation intensity in x direction and y direction are almost the same which is also reasonable to the total integral area.