Anomalous localization of light in one-dimensional L\'evy photonic lattices

Localization of coherent propagating waves has been extensively studied over the years, primarily in homogeneous random media. However, significantly less attention has been given to wave localization in inhomogeneous systems, where the standard picture of Anderson localization does not apply, as we demonstrate here. We fabricate photonic lattices with inhomogeneous disorder, modeled by heavy-tailed $\alpha$-stable distributions, and measure the output light intensity profiles. We demonstrate that the spatial localization of light is described by a stretched exponential function, with a stretching parameter $\alpha$, and an asymmetric localized profile with respect to the excitation site. We support our experimental and theoretical findings with extensive tight-binding simulations.