  It was recently shown that a powerful beam of radio/microwave radiation sent
out to space can produce detectable back-scattering via the stimulated decay of
ambient axion dark matter. This echo is a faint and narrow signal centered at
an angular frequency close to half the axion mass. In this article, we provide
a detailed analytical and numerical analysis of this signal, considering the
effects of the axion velocity distribution as well as the outgoing beam shape.
In agreement with the original proposal, we find that the divergence of the
outgoing beam does not affect the echo signal, which is only constrained by the
axion velocity distribution. Moreover, our findings are relevant for the
optimization of the experimental parameters in order to attain maximal signal
to noise or minimal energy consumption.
