Predominant contribution of direct laser acceleration to high-energy electron spectra in a low-density self-modulated laser wakefield accelerator

The two-temperature relativistic electron spectrum from a low-density ($3\times10^{17}$~cm$^{-3}$) self-modulated laser wakefield accelerator (SM-LWFA) is observed to transition between temperatures of $19\pm0.65$ and $46\pm2.45$ MeV at an electron energy of about 100 MeV. When the electrons are dispersed orthogonally to the laser polarization, their spectrum above 60 MeV shows a forking structure characteristic of direct laser acceleration (DLA). Both the two-temperature distribution and the forking structure are reproduced in a quasi-3D \textsc{Osiris} simulation of the interaction of the 1-ps, moderate-amplitude ($a_{0}=2.7$) laser pulse with the low-density plasma. Particle tracking shows that while the SM-LWFA mechanism dominates below 40 MeV, the highest-energy ($>60$ MeV) electrons gain most of their energy through DLA. By separating the simulated electric fields into modes, the DLA-dominated electrons are shown to lose significant energy to the longitudinal laser field from the tight focusing geometry, resulting in a more accurate measure of net DLA energy gain than previously possible.