Relativistic Doppler-boosted γ-rays in High Fields

Abstract

The relativistic Doppler effect is one of the most famous implications of the principles of special relativity and is intrinsic to moving radiation sources, relativistic optics and many astrophysical phenomena. It occurs in the case of a plasma sail accelerated to relativistic velocities by an external driver, such as an ultra-intense laser pulse. Here we show that the relativistic Doppler effect on the high energy synchrotron photon emission (~10 MeV), strongly depends on two intrinsic properties of the plasma (charge state and ion mass) and the transverse extent of the driver. When the moving plasma becomes relativistically transparent to the driver, we show that the γ-ray emission is Doppler-boosted and the angular emission decreases; optimal for the highest charge-to-mass ratio ion species (i.e. a hydrogen plasma). This provides new fundamental insight into the generation of γ-rays in extreme conditions and informs related experiments using multi-petawatt laser facilities.

Figure 1

Schematic illustration of a plasma sail pushed by the radiation pressure of a driver. While accelerated by the radiation pressure of a driver (a) the electrons of the plasma emit intense synchrotron radiation (from MeV to hundreds of MeV energy) which is Doppler-shifted due to the own motion of the sail. The momentum and dynamics of the sail depend on the transverse extent of the driver and the ion inertia, which has a significant impact on the radiation. When the plasma sail becomes transparent to the driver, the range of angles over which γ-rays are produced decreases and the radiation is Doppler-boosted as shown in panels (b and c). Specifically, the larger transverse extent of the driver results in a more Doppler-boosted γ-ray emission with an optimal decrease of the angular range of emission for a plasma layer having the highest ion charge-to-mass ratio, i.e. a hydrogen plasma.

Figure 2

Simulation results of the interaction of an ultra-intense laser pulse with a plasma sail. The laser pulse (I_L = 10²³ Wcm⁻²) interacts at t = 0 with the plasma slab. The transverse extent of the driver (in this case equal to 15 μm) tends to bend the plasma layer, favoring the generation of an intense longitudinal magnetic field (B_x), which guides the radiating electron bunches. (a) Electron density normalized to the critical density, n_c. (b) Longitudinal magnetic field B_x, and (c) total number of emitted photons up to t = 24 T_L.

Figure 3

Two-dimensional particle-in-cell simulation results. Angular distribution of 10 MeV photons over time for several ion charge-to-mass ratio $\left(\overline{\mathcal{Z}}\equiv \frac{Z}{m_{\mathrm{i}}/m_{\mathrm{proton}}}\right)$. (a–c) W₀ = 3 μm; (d–f) W₀ = 6 μm; (g–i) W₀ = 15 μm. (a,d,g), $\overline{\mathcal{Z}}=1$; (b,e,h), $\overline{\mathcal{Z}}=\frac{1}{2}$; (c,f,i), $\overline{\mathcal{Z}}=\frac{1}{3}$. The black dashed lines corresponds to the time t_break when the plasma layer becomes relativistically transparent to the laser pulse.

Figure 4

Characteristic times of the simulation results. (a) Time corresponding to the peak of the radiated intensity (t_γ) and (b) the relativistic transparency time (t_break), as a function of the laser spot size (W₀). Dashed blue lines, orange dashed-pointed lines and green lines represent $\left[\overline{\mathcal{Z}}=1\right]$, $\left[\overline{\mathcal{Z}}=\frac{1}{2}\right]$ and $\left[\overline{\mathcal{Z}}=\frac{1}{3}\right]$ plasmas, respectively.

Figure 5

Electron energy spectra. The total energy electron spectra as well as parallel and perpendicular components are plotted out in maroon lines, red squares and black circles, respectively. (a) $\left[\overline{\mathcal{Z}}=1\right]$ plasma and W₀ = 3 μm; (b) $\left[\overline{\mathcal{Z}}=1\right]$ plasma and W₀ = 15 μm; (c) $\left[\overline{\mathcal{Z}}=\frac{1}{3}\right]$ plasma and W₀ = 3 μm; (d) $\left[\overline{\mathcal{Z}}=\frac{1}{3}\right]$ plasma and W₀ = 15 μm. The spectra have been considered at the time corresponding to the maximum synchrotron radiation emission (t = t_γ), i.e., t = 17 T_L, 20 T_L, 16 T_L and 17 T_L respectively. The parallel and perpendicular component of the temperature are defined compared with the direction of the laser wave propagation, i.e., the x axis.

Figure 6

Evolution of the plasma speed and the mean electron Lorentz factor. (a) Plasma speed as a function of the laser spot size $\left(\frac{W_0}{{\lambda }_{\mathrm{L}}}\right)$ at the transparency time (t_break), computed from simulation results (see methods). Dashed blue lines, $\overline{\mathcal{Z}}=1$; orange dashed lines, $\overline{\mathcal{Z}}=1/2$; green line, $\overline{\mathcal{Z}}=1/3$. (b) Mean electron Lorentz factor (4) as a function of the plasma speed during the transparency regime.

Figure 7

Results of theoretical predictions and numerical simulations. Average angle of the synchrotron radiation 〈|θ_γ|〉 as a function of time, for different values of $\left(\overline{\mathcal{Z}},\frac{W_0}{{\lambda }_{\mathrm{L}}}\right)$. (a) W₀ = 3 μm. (b) W₀ = 6 μm. (c) W₀ = 15 μm. In blue $\left[\overline{\mathcal{Z}}=1\right]$ plasma. In orange $\left[\overline{\mathcal{Z}}=\frac{1}{2}\right]$ plasma. In green $\left[\overline{\mathcal{Z}}=\frac{1}{3}\right]$ plasma. The thick lines and dashed lines correspond to the analytical estimates of Θ (3) defined for t ≥ t_break and the simulation results, respectively.

Figure 8

Electron density and laser field behavior during the transparency regime. (a) The electron density and laser field. (b) The electron energy-angular distribution both for $\frac{W_0}{{\lambda }_L}=15$, $\overline{\mathcal{Z}}=1$ at t = t_break + 13 T_L. (c,d) Same but for $\frac{W_0}{{\lambda }_L}=3$, $\overline{\mathcal{Z}}=\frac{1}{3}$.

Figure 9

Results of numerical simulations. (a) Photon energy spectra above 3 MeV for several values of $\left(\overline{\mathcal{Z}},\frac{W_0}{{\lambda }_{\mathrm{L}}}\right)$. (b) The Doppler-boosted synchrotron factor ${\mathcal{D}}_{\mathrm{synchro}.\mathrm{rad}.}\equiv {[\mathrm{\Gamma }(1-\langle \beta \rangle \cos \mathrm{\Theta })]}^{-1}$ (5) once the plasma is transparent to the electromagnetic wave (i.e. t ≥ t_break), as a function of the plasma velocity ${\langle \beta \rangle }_{t\ge t_{\mathrm{break}}}=\max \langle \beta \rangle$.