  The field of topological photonics studies unique and robust photonic systems
that are immune to defects and disorders due to the protection of their
underlying topological phases. Mostly implemented in static systems, the
studied topological phases are often defined in linear photonic band
structures. In this study, we experimentally demonstrate Floquet Chern
insulators in periodically driven nonlinear photonic crystals, where the
topological phase is controlled by the polarization and the frequency of the
driving field. Mediated by strong material nonlinearity, our system enters what
we call the 'strong Floquet coupling regime', where the photonic Floquet bands
cross and open new energy gaps with non-trivial topology as observed in our
transient sum-frequency generation measurements. Our work offers new
opportunities to explore the role of classical optical nonlinearity in
topological phases and their applications in nonlinear optoelectronics.
