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

## TL;DR

This paper develops a 3D Bloch mode expansion and Green function approach to analyze Anderson-localized modes in disordered photonic crystal waveguides, revealing statistical behaviors of quality factors and Purcell enhancements.

## Contribution

It introduces a comprehensive theoretical framework for quantifying disorder effects on localized photonic modes, including statistical distributions and coupling probabilities.

## Key findings

- Quality factor and Purcell enhancement follow log-normal distributions.
- Disorder effects decrease exponentially with the square root of disorder strength.
- Strong coupling probability diminishes exponentially with disorder squared.

## Abstract

We present a fully three-dimensional Bloch mode expansion technique and photon Green function formalism to compute the quality factor, mode volume, and Purcell enhancement distributions of a disordered W1 photonic crystal slab waveguide in the slow-light Anderson localization regime. By considering fabrication (intrinsic) and intentional (extrinsic) disorder we find that the quality factor and 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 on 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 exponential decrease with the squared extrinsic disorder parameter, where low disordered systems give rise to larger probabilities when state-of-art quantum dots are considered. The optimal regions to position quantum dots in the W1 waveguide are also discussed. These theoretical results are fundamental interesting and connect to recent experimental works on photonic crystal slab waveguides in the slow-light regime.