Fig. 2: Eigen-states of the fractal lattice.

a Energy spectrum with \({\boldsymbol{A}}\left( z \right) = 0\) (straight waveguides). The spectrum of this non-topological system displays a large central gap (large gray region), with a flat band in the mid-gap made up of immobile degenerate states. b Quasi-energy spectrum with \({\boldsymbol{A}}\left( z \right) \ne 0\) and \(A_0 = kR{\mathrm{\Omega }}\). The inset shows an enlarged view of the center box. The shaded regions mark quasi-gaps: regions within which there are no eigen-states. In this topological fractal system, the edge states from either side of the flat band evolve into nondegenerate unidirectional edge states. c Field intensity patterns of two eigen-states localized at the external and internal edges of the fractal lattice (states 93 and 95, gray and red dots, respective√ly). The color bar indicates the intensity (normalized to the peak intensity in each state). The parameters used are the ambient refractive index \(n_0 = 1.45\), coupling strength \(c_0 = 1.9\,{\rm{cm}}^{ - 1}\), wavelength \(\lambda = 0.633\) µm, helix radius \(R = 10\) µm, longitudinal frequency of the helix \({\mathrm{\Omega }} = 2\pi \,{\rm{cm}}^{ - 1}\), and lattice constant \(a = 14\sqrt 3\) µm