Topological oscillated edge states in trimer lattices

We investigate a 1D trimer optical lattice model. Two kinds of topological oscillating optical transmission phenomena at edges are shown. The exact and the approximate solutions of the system's edge states are obtained with and without the inversion symmetry for this system respectively. Based on the solutions, the existence and the periods of the oscillations can be controlled arbitrarily. Moreover, in a system without inversion symmetry, controlling the incident beam can eliminate both types of oscillations, resulting in a more stable edge state compared to the one with inversion symmetry. This prompts us to reconsider topological systems under symmetry protection.