Topological $\pi/2$ modes in photonic waveguide arrays

Periodic driving is a powerful tool to generate exotic topological phases without static counterparts, such as the anomalous chiral edge modes from bulk bands with zero Chern number and topological $\pi$ modes exhibiting period-doubled dynamics. Recently, a new class of Floquet topological mode, namely the $\pi/2$ mode, which carries four-period periodicity and has potential applications in quantum computing, was proposed based on a square-root method and realized in an acoustic system. Here we propose a laser-written waveguide array lattice to realize topological $\pi/2$ modes in photonics. Our photonic model simulates a square-root periodically driven Su-Schrieffer-Heeger model and has a rich phase diagram allowing for the co-existence of conventional zero, $\pi$ modes, and the new $\pi/2$ modes. Through numerical simulations of the wave equation, we uncover the unique four-period evolution feature of the $\pi/2$ modes. Our model, which only contains four waveguides per unit cell and two driving steps, is easy to implement with current fabrication techniques and may find applications in quantum optics.