Statistics of Anderson-localized modes in disordered photonic crystal slab waveguides

We present a fully three-dimensional Bloch mode expansion technique and a photon Green function formalism to compute the quality factors, mode volumes, and Purcell enhancement distributions of a disorderedW1 photonic crystal slab waveguide in the slow-light Anderson-localization regime. By considering fabrication (intrinsic) and intentional (extrinsic) disorder we find that the Purcell enhancement statistics are well described by log-normal distributions without any fitting parameters. We also compare directly the effects of hole size fluctuations as well as fluctuations in the hole position. The functional dependence of the mean and standard deviation of the quality factor and Purcell enhancement distributions is found to decrease exponentially with the square root of the extrinsic disorder parameter. The strong coupling probability between a single quantum dot and an Anderson-localized mode is numerically computed and found to exponentially decrease with the squared extrinsic disorder parameter, where low disordered systems give rise to larger probabilities when state-of-the-art quantum dots are considered. The optimal spatial regions to position quantum dots in the W1 waveguide are also discussed. These theoretical results are fundamentally interesting for disordered photonics and connect to recent experimental works on photonic crystal slab waveguides in the slow-light regime. Our three-dimensional slab results also contradict some previous findings that use simpler two-dimensional models to understand these complex planar systems.