Probing $\mu$eV ALPs with future LHAASO observation of AGN $\gamma$-ray
  spectra
  Axion-like particles (ALPs) are predicted in some well-motivated theories
beyond the Standard Model. The TeV gamma-rays from active galactic nuclei (AGN)
suffer attenuation by the pair production interactions with the cosmic
background light (EBL/CMB) during its travel to the earth. The attenuation can
be circumvented through photon-ALP conversions in the AGN and Galaxy
magnetic-field, and a flux enhancement is expected to arise in the observed
spectrum. In this work, we study the potential of the AGN gamma-ray spectrum
for energy up to above 100\,TeV to probe ALP-parameter space at around $\mu$eV,
where the coupling $g_{a\gamma}$ is so far relatively weakly constrained.
  We find the nearby and bright sources, Mrk\,501, IC\,310 and M\,87, are
suitable for our objective. Assuming an intrinsic spectrum exponential cutoff
energy, we extrapolate the observed spectra of these sources up to above
100\,TeV by the models with/without ALPs. For $g_{a\gamma}\gtrsim
2\times$$10^{-11} \rm GeV^{-1}$ with $m_{a}\lesssim0.5\,\mu$eV, the flux at
around 100\,TeV predicted by the ALP model can be enhanced more than an order
of magnitude than that from the standard absorption, and could be detected by
LHAASO. Our result is subject to the uncertainty from the intrinsic cutoff
energy and the AGN lobe (or plume) magnetic-field. For an optimistic
estimation, the constraint can be improved to $g_{a\gamma}\gtrsim
2\times$$10^{-11} \rm GeV^{-1}$ with $m_{a}\lesssim1\,\mu$eV. This require
further observations on these sources by the forthcoming CTA, LHAASO, SWGO and
so on.
