  Soon available multi petawatt ultra-high-intensity (UHI) lasers will allow us
to probe high-amplitude electromagnetic fields interacting with either
ultra-relativistic electron beams or hot plasmas in the so-called moderately
quantum regime. The correct modelling of the back-reaction of high-energy
photon emission on the radiating electron dynamics, a.k.a. radiation reaction,
in this regime is a key point for UHI physics. This will lead to both
validation of theoretical predictions on the photon spectrum emitted during the
laser-particle interaction and to the generation of high energy photon sources.
In this paper we analyse in detail such emission using recently developed
models to account for radiation reaction. We show how the predictions on the
spectrum can be linked to a reduced description of the electron distribution
function in terms of the first energy moments. The temporal evolution of the
spectrum is discussed, as well as the parameters for which quantum effects
induce hardening of the spectrum.
