Inducing maximal localization with fractal waveguide arrays

The ability to transmit light through an array of closely packed waveguides while minimizing interwaveguide coupling has important implications for fields such as discrete imaging and telecommunications. Proposals for achieving these effects have involved phenomena ranging from flat bands to Anderson localization. An approach based on Anderson localization is beneficial since every eigenstate localizes and this localization is applicable to arbitrarily large propagation distances, even in the presence of perturbations such as higher-neighbor couplings. However, the localization lengths can be large so that sites within a given finite region can exhibit significant crosstalk. Here, we pose an optimization problem in which we seek to maximally confine the eigenstates. We demonstrate that, for strongly detuned waveguides arranged in an equally spaced lattice in one dimension (1D) and a square lattice in two dimensions (2D), optimal eigenstate localization can be achieved by fractal potentials. We further show that these structures possess a localization-delocalization phase transition in both 1D and 2D. The structures are related to a more general family of self-similar potentials that can be constructed on the class of k-partite lattices satisfying the property that each sublattice forms a rescaled copy of the original lattice. The structures may also be approximated by a family of periodic structures, and we characterize the performance of these approximations in comparison to the full aperiodic structures.