Engineering wave localization in a fractal waveguide network

We present an exact analytical method of engineering the localization of classical waves in a fractal waveguide network. It is shown that a countable infinity of localized eigenmodes with a multitude of localization lengths can exist in a Vicsek fractal geometry built with diamond-shaped monomode waveguides as the "unit cells." The family of localized modes forms clusters of increasing size. The length scale at which the onset of localization for each mode takes place can be engineered at will, following a well-defined prescription developed within the framework of a real space renormalization group. The scheme leads to an exact evaluation of the wave vector for every such localized state, a task that is nontrivial, if not impossible for any random or deterministically disordered waveguide network.