Anomalous π modes by Floquet engineering in optical lattices with long-range coupling

Photonic Floquet topological insulators provide a powerful tool to manipulate the optical fields, which have been extensively studied with only nearest-neighbor coupling. Here, we demonstrate that nontrivial Floquet topological phase and photonic π modes are brought from long-range coupling in a one-dimensional periodically driven optical lattice. Interestingly, the long-range coupling is found to give rise to new Floquet π modes that do not exist in the traditional Floquet lattices. We interpret the underlying physics by analyzing the replica bands, which shows quasienergies band crossing and reopening of new nontrivial π gaps due to the long-range coupling. Our results provide a new route in manipulating optical topological modes by Floquet engineering with long-range coupling.