Engineering slow light and mode crossover in a fractal-kagome waveguide network

We present an analytically exact scheme of unraveling a multitude of flat, dispersionless photonic bands in a kagome waveguide strip where each elementary plaquette hosts a deterministic fractal geometry of arbitrary generation. The number of nondispersive eigenmodes grows as higher and higher order fractal geometry is embedded in the kagome motif. Such eigenmodes are found to be localized with finite support in the kagome strip and exhibit a hierarchy of localization areas. The onset of localization can, in principle, be delayed in space by an appropriate choice of frequency of the incident wave. The length scale at which the onset of localization for each mode occurs can be tuned at will as prescribed here using a real-space renormalization method. Conventional methods of extracting the nondispersive modes in such geometrically frustrated lattices fail as a non-translationally-invariant fractal decorates the unit cells in the transverse direction. The scheme presented here circumvents this difficulty, and thus may inspire experimentalists to design similar fractal-incorporated kagome or Lieb classes of lattices to observe a multifractal distribution of flat photonic bands.