Direct laser acceleration (DLA) of electrons in a plasma of near critical electron density (NCD) and associated synchrotron-like radiation are discussed for moderate relativistic laser intensity (the normalized laser amplitude $a_0$ $leq$ 4.3) and ps-long pulse. This regime is typical for kJ PW-class laser facilities designed for high energy density research. Currently, in experiments at the PHELX laser it was demonstrated that interaction of 10$^{19}$ W/cm$^{2}$ sub-ps laser pulse with sub-mm long NCD plasma results in generation of high-current well-directed super-ponderomotive electrons with effective temperature that is 10$times$ higher than the ponderomotive potential [O. Rosmej et al., PPCF 62, 115024 (2020)]. Three-dimensional Particle-In-Cell simulations provided a good agreement with the measured electron energy distribution and were used in the current work to study synchrotron radiation of the DLA accelerated electrons. The resulting x-ray spectrum with a critical energy of 5 keV reveals an ultra-high photon number of 7$times$10$^{11}$ in the 1-30 keV photon energy range at the focused laser energy of 20 J. Numerical simulations of a betatron x-ray phasecontrast imaging based on the DLA process for the parameters of a PHELIX laser is presented. The results are of interest for applications in high energy density (HED) experiments, which require a picosecond x-ray pulse and a high photon flux.
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