A new topic in the studies of topological solitons: Topologically protected higher-order modes in fractal optical systems

This article provides a brief summary of recently reported theoretical and experimental findings which reveal the creation of corner modes in optical waveguides structured as the higher-order topological insulators (HOTIs). In fact, these are second-order HOTIs, in which the transverse dimension of the topologically protected propagating edge modes is smaller by 2 than the transverse dimension (it is 2 for bulk optical waveguides), implying zero dimension of the protected modes. Actually, the zero-dimensional modes are realized as corner or defect ones. Recent works report the theoretical prediction and experimental creation of various forms of the corner modes in optical HOTIs, whose transverse structure is an approximation to a triangular fractal one, theoretically represented by the Sierpiński gasket (SG). The self-focusing nonlinearity of the waveguide's material transforms the linear corner modes into topologically protected corner solitons, almost all of which are stable. The solitons may be attached to external or internal corners created by the underlying SG. A brief discussion of directions for the further work on this topic is presented too.