Photon Phase-Space Dynamics in a Plasma Wakefield Accelerator

TL;DR

This work develops a time-dependent photon kinetic theory (PKT) for a witness pulse in a relativistic plasma wakefield, treating the field as quasi-photons with a local Hamiltonian $\omega(\mathbf{r},\mathbf{k},t)=\sqrt{|\mathbf{k}|^2+\omega_p^2(\mathbf{r},t)}$ and a phase-space density $N(\mathbf{r},\mathbf{k},t)$. It introduces both full PKT and a linearized version around the wake-phase-matching point $(\xi_\delta, k_\delta)$, solving via characteristics to reveal a spiral approach to an attractor in phase space and a self-similar evolution under scaled coordinates $\chi_1,\chi_2$. The theory yields concrete scaling relations for pulse parameters with frequency upshift, such as $\tau \sim \sqrt{\omega_0/\omega}$, $\Delta\omega \sim \sqrt{\omega/\omega_0}$, and $U \sim \omega/\omega_0$, and predicts indefinite compression of the witness pulse limited only by drive-beam evolution. Validation against 1D OSIRIS PIC simulations shows good agreement for the bulk phase-space dynamics, supporting the potential of PWPA as a compact, high-intensity XUV light source and providing a computationally efficient framework for photon-acceleration dynamics.

Abstract

Frequency up-shifting of laser light in a beam-driven plasma wakefield has the potential to provide high-intensity sources of short wavelength radiation. Simulations have demonstrated that a laser pulse can undergo large frequency shifts, limited only by the drive beam energy, when the plasma density is tailored to match the accelerating phase of the wake to the group velocity of the pulse. Here, we study the dynamical evolution of photons in the phase-space vicinity of the plasma wake-phase matching condition. Numerical calculations using a photon kinetic model are validated by direct comparison with full electromagnetic particle-in-cell simulations. These calculations form the basis of a linear theory of the photon dynamics which reveals several important results, including scalings for the properties of the witness pulse and a self-similar solution for the photon phase-space dynamics. One prediction of the theory is that the pulse can be compressed indefinitely with no lower bound on the duration. This predication suggests that photon acceleration can provide a novel source of sub-femtosecond, short wavelength radiation.

Figures (9)

• Figure 1: Depiction of PWPA. A charged particle beam excites a large amplitude plasma wave. A trailing laser pulse sits within the negative electron density gradient of the wave, where the associated traveling refractive index gradient causes it to upshift. The fields plotted are contours of the drive beam density (teal), contours of the electric field of the pulse (blue and red), and a surface representing the plasma density in the $xz$ plane (yellow-green). See column A of Table \ref{['tab:simparams']} for a complete list of simulation parameters.
• Figure 2: The plasma wake-phase matching conditions allow for continuous, monotonic frequency shift of the pulse. The plot tracks the centroid $(\xi_\delta, k_\delta)$ along with the spectrum and envelope of the pulse from an OSIRIS simulation. The predicted photon trajectory for the central photon (dashed gray) follows closely the trajectory of the pulse centroid calculated from the OSIRIS simulation (solid white). The initial central frequency was $k_{\delta0}=10$, initial pulse duration $\tau=1$, initial chirp $b_0=0$, drive beam density $n_d=0.8$, and beam length $L_d=1$. See column B of Table \ref{['tab:simparams']} for a complete list of simulation parameters.
• Figure 3: The photon phase space trajectories during PWPA calculated numerically (solid curves) are well approximated by the linearized solutions (dashed curves). Note that all trajectories spiral into the origin in $\Delta\xi$-$\Delta k/k_\delta$ space, as the origin is an attractor. The dots represent the initial conditions. (a) shows the trajectories in $\Delta\xi$-$\Delta k$ space, whereas (b) shows the trajectories in $\Delta\xi$-$\Delta k/k_\delta$ space. The parameters used were $n_d = 0.8$, $L_d = 1$, and $k_{\delta0} = 10$.
• Figure 4: The phase-space density of the accelerating pulse contracts in space, expands in wavenumber, and rotates in phase-space. Contours of the photon phase-space density at $\exp(-1)$ times the maximum are plotted. The photon kinetic model is able to reproduce the phase-space density of the OSIRIS simulation. The linearized model is consistent with the OSIRIS simulation, though it breaks down for large frequency shifts when the a high-frequency tail forms in the OSIRIS simulation. The parameters used were $k_{\delta0}=10$, $\tau=1$, $n_d=0.8$, $b_0=0$, and $L_d=1$. See column B of Table \ref{['tab:simparams']} for a complete list of simulation parameters.
• Figure 5: As the pulse is accelerated, it compresses and spectrally broadens. The plot depicts moments of the phase space densities from Fig. \ref{['fig:wigcontour']} calculated by fits to a Gaussian. The linearized theory is able to predict the duration and bandwidth of the pulse as calculated from and OSIRIS simulation. Due to the phase-space rotation, the pulse initially stretches before compressing. The parameters used were $k_{\delta0}=10$, $\tau=1$, $n_d=0.8$, $b_0=0$, and $L_d=1$. See column B of Table \ref{['tab:simparams']} for a complete list of simulation parameters.