  In this paper, we extend the study of the relationship between the photon
sphere and the thermodynamic phase transition, especially the reentrant phase
transition, to this black hole background. According to the number of the
thermodynamic critical points, the black hole systems are divided into four
cases with different values of Born-Infeld parameter b, where the black hole
systems can have no phase transition, reentrant phase transition, or Van der
Waals-like phase transition. For these different cases, we obtain the
corresponding phase structures in pressure-temperature diagram and
temperature-specific volume diagram. The tiny differences between these cases
are clearly displayed. On the other hand, the radius rps and the minimal impact
parameter ups of the photon sphere are calculated via the effective potential
of the radial motion of photons. For different cases, rps and ups are found to
have different behaviors. In particular, with the increase of rps or ups, the
temperature possesses a decrease-increase-decrease-increase behavior for fixed
pressure if there exists the reentrant phase transition. While for fixed
temperature, the pressure will show an increase-decrease-increase-decrease
behavior instead. These behaviors are quite different from that of the Van der
Waals-like phase transition. Near the critical point, the changes of rps and
ups among the black hole phase transition confirm an universal critical
exponent 12. Therefore, all the results indicate that, for the charged
Born-Infeld-AdS black holes, not only the Van der Waals-like phase transition,
but also the reentrant phase transition can be reflected through the photon
sphere.
