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.
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