Earth-lens telescope for distant axion-like particle sources with stimulated backward reflection

We propose a novel telescope concept based on Earth's gravitational lensing effect, optimized for the detection of distant dark matter sources, particularly axion-like particles (ALPs). When a unidirectional flux of dark matter passes through Earth at sufficiently high velocity, gravitational lensing can concentrate the flux at a distant focal region in space. Our method combines this lensing effect with stimulated backward reflection (SBR), arising from ALP decays that are induced by directing a coherent electromagnetic beam toward the focal point. The aim of this work is to numerically analyze the structure of the focal region and to develop a framework for estimating the sensitivity to ALP-photon coupling via this mechanism. Numerical calculations show that, assuming an average ALP velocity of 520,km/s -- as suggested by the observed stellar stream S1 -- the focal region extends from $9 \times 10^9$,m to $1.4 \times 10^{10}$,m, with peak density near $9.6 \times 10^9$,m. For a conservative point-like ALP source located approximately 8,kpc from the solar system, based on the S1 stream, the estimated sensitivity in the eV mass range reaches $g/M = \mathcal{O}(10^{-22}),\mathrm{GeV}^{-1}$. This concept thus opens a path toward a general-purpose, space-based ALP observatory that could, in principle, detect more distant sources -- well beyond $\mathcal{O}(10),\mathrm{kpc}$ -- provided that ALP-photon coupling is sufficiently strong, that is, $M \ll M_\mathrm{Planck}$.