One-Way Topological States along Vague Boundaries in Synthetic Frequency Dimensions including Group Velocity Dispersion

We recently proposed a two-dimensional synthetic space including one spatial axis and one synthetic frequency dimension in a one-dimensional ring resonator array [Opt. Lett., Vol. 41, No. 4, 741-744, 2016]. Nevertheless, the group velocity dispersion (GVD) of the waveguides that compose rings was ignored for simplicity. In this paper, we extend the previous work and study the topological one-way edge states in such a synthetic space involving GVD. We show that the GVD brings a natural vague boundary in the frequency dimension, so the topological edge state still propagates at several frequency modes unidirectionally along the spatial axis. Positions of such vague boundary can be controlled by changing the magnitude of the GVD. In particular, a relatively strong GVD can degrade this two-dimensional synthetic space to one-dimensional spatial lattice, but yet the one-way state is still preserved in simulations. Our work therefore exhibits the impact of the GVD on topological photonics in the synthetic space, which will be important for future practical experimental implementations.