  We present firstly the equation of motion for the photon coupled to a special
bumblebee vector field in a Kerr black hole spacetime and find that the
propagation of light depends on its polarization due to the birefringence
phenomenon. The dependence of black hole shadow on the light's polarization is
dominated by the rotation of black hole. In the non-rotating case, we find that
the black hole shadow is independent of the polarization of light. However, the
status is changed in the rotating case, in which the black hole shadow depends
on the light's polarization and the coupling between bumblebee vector field and
electromagnetic field. These features of black hole shadow casted by polarized
lights could help us to understand the bumblebee vector field with Lorentz
symmetry breaking and its interaction with electromagnetic field.
