  The observed time delays between photons with different circular
polarizations from an astrophysical object provide a new, interesting way of
testing the Einstein Equivalence Principle (EEP). In this paper, we constrain
the EEP by considering both Shapiro time delay and Faraday rotation effects. We
continue to search for astronomical sources that are suitable for testing the
EEP accuracy, and obtain 60 extragalactic radio sources with multi-wavelength
polarization angles in three different radio bands (20, 8.6, and 4.8 GHz) and
29 brightest stars within our own Milky Way galaxy with multi-colour linear
polarimetric data in five optical bands ($UBVRI$). We apply the
Metropolis-Hastings Markov Chain to simulate the fit parameters. The final
results show that the values of the parameterized post-Newtonian parameter
$\gamma$ discrepancy ($\Delta \gamma_{p}$) are constrained to be in the range
of $10^{-26}-10^{-23}$ for 60 radio sources and in the range of
$10^{-23}-10^{-20}$ for 29 optical polarization stars. Compared to previous EEP
tests that based on the single polarization measurement in the gamma-ray band,
our results have profound superiority that nearly a few tens of astrophysical
sources with multi-wavelength polarization observations commonly in the optical
and radio bands are available. It ensures that these sources can give more
significantly robust bounds on the EEP. Although the presented method is
straightforward, the resulting constraints on the EEP should be taken as upper
limits as other more complex astrophysical effects affecting a polarization
rotation are hardly considered.
