  We calculate a finite momentum-dependent part of the photon polarization
operator in a simple model of Lorentz-violating quantum electrodynamics
nonperturbatively at all orders of Lorentz-violating parameters. We sum
one-particle reducible diagrams into the modified photon propagator, and
determine the physical photon dispersion relation as the location of its pole.
The photon dispersion relation, as well as its group velocity, acquires the
one-loop momentum-dependent radiative correction. We constrain the
Lorentz-violating parameters for heavy charged fermions (muon, $\tau$-lepton,
top-quark) from the photon timing observations.
