  In this work, the conversion of linear polarization of a laser beam to
circular one through its forward scattering by a TeV order charged lepton beam
in the presence of Lorentz violation correction is explored. We calculate the
ratio of circular polarization to linear one (Faraday Conversion phase
$\Delta\phi_{\rm{FC}}$) of the laser beam interacting with either electron or
the muon beam in the framework of the quantum Boltzmann equation.
  Regarding the experimentally available sensitivity to the Faraday conversion
$\Delta\phi_{\rm{FC}}\simeq 10^{-3}-10^{-2}$, we show that the scattering of a
linearly polarized laser beam with energy $k_0\sim 0.1$ eV and an electron/muon
beam with flux $\bar{\epsilon}_{e,\mu}\sim 10^{10}/10^{12}$ TeV cm$^{-2}$
s$^{-1}$ places an upper bound on the combination of lepton sector Lorentz
violation coefficients $c_{\mu\nu}$ components
$(c_{TT}+1.4~c_{(TZ)}+0.25(c_{XX}+c_{YY}+2~c_{ZZ}))$. The obtained bound on the
combination for the electron beam is at the $4.35\times 10^{-15}$ level and for
the muon beam at the $3.9\times 10^{-13}$ level. It should be mentioned that
the laser and charged lepton beams considered here to reach the experimentally
measurable $\Delta\phi_{\rm{FC}}$ are currently available or will be accessible
in the near future. This study provides a valuable supplementary to other
theoretical and experimental frameworks for measuring and constraining Lorentz
violation coefficients.
