Topological edge modes with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">PT</mml:mi></mml:math> symmetry in a quasiperiodic structure

Abstract

We investigate topological features of a one-dimensional photonic quasicrystal within the context of $\mathcal{PT}$ symmetry. Via the scattering characteristics, we analyze various properties of a particular mirrored structure, which supports topological edge modes in its band gaps. These interface modes display a nontrivial dependence on the quasiperiodic geometry, even in a passive system. Subsequently, the tailored addition of gain and loss generates curious $\mathcal{PT}$-like features. For example, the quasicrystal high density of modes leads to complicated mode-merging behaviors between edge and band modes, such as the symmetry recovery phenomenon. Furthermore, anisotropic transmission resonances (connected with unidirectional invisibility) are also present, but they display richer patterns in comparison to previously studied periodic structures. Additionally, we examine lasing effects in detail, with numerics and a simple Fabry-P\'erot model. The large variety of mode-merging behaviors opens the way to laser resonance engineering.