  We develop a Green's function based mean-field theory for coherent mixing of
matter- and light-waves. To demonstrate the utility of this approach, we
analyse a co-propagating Raman matter-wave amplifier. We find that for a given
laser intensity, a significantly faster amplification process can be achieved
employing resonant rather than off-resonance driving. The ratio of the
matter-wave gain to atom loss-rate due to spontaneous emission is given by the
optical depth of the sample, and is the same both on- and off-resonance.
Furthermore, we show that for short-times, the single-mode approximation for
the matter-waves gives exact agreement with the full spatial dynamics. For long
times, the off-resonant case shows suppressed amplification due to a spatially
inhomogenous AC Stark shift associated with laser depletion. This suppression
is absent on-resonance, where the AC Stark shift is absent.
