  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 size. The number
of non-dispersive 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 non-dispersive 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 the experimentalists to design similar fractal
incorporated kagome or Lieb class of lattices to observe a multifractal
distribution of flat photonic bands.
