Design of Topological Structures with desired-shape mid-gap state

Topological systems with their robust features are of interest in different fields including electronics, acoustics, and photonics. Systems with non-trivial topological configurations are the host of the so-called edge modes with energy eigenfrequency lying at the center of the gap and are protected by the symmetries of the structure. The eigenstates associated with these mid-gap modes are exponentially localized at the boundaries. For practical applications, it is of great interest to design systems that have a protected state not necessarily squeezed towards the edges, and in some cases, it is crucial to have it as a distributed state. Here, by inverse-design programming, we propose a general approach allowing us to generate structures with non-trivial symmetry that have a mid-gap state with the desired shape, which is not restricted to the edge. Although the symmetry in the engineered structure is not obvious, the energy eigenfrequency associated with the mid-gap state in any shape of the distribution is robust against structural imperfections. We demonstrate several examples in 1D and 2D lattices allowing the observation of tunable localization of the mid-gap state and transformed edge state to an extended state. Our work paves the way for new applications of topological effects such as imaging and far-field topological sensing.