  We derive the in-medium gluon radiation spectrum off a quark within the path
integral formalism at finite energies, including all next-to-eikonal
corrections in the propagators of quarks and gluons. Results are computed for
finite formation times, including interference with vacuum amplitudes. By
rewriting the medium averages in a convenient manner we present the spectrum in
terms of dipole cross sections and a colour decoherence parameter with the same
physical origin as that found in previous studies of the antenna radiation.
This factorisation allows us to present a simple physical picture of the
medium-induced radiation for any value of the formation time, that is of
interest for a probabilistic implementation of the modified parton shower.
Known results are recovered for the particular cases of soft radiation and
eikonal quark and for the case of a very long medium, with length much larger
than the average formation times for medium-induced radiation. Technical
details of the computation of the relevant $n$-point functions in colour space
and of the required path integrals in transverse space are provided. The final
result completes the calculation of all finite energy corrections for the
radiation off a quark in a QCD medium that exist in the small angle
approximation and for a recoilless medium.
