Fast Generation of Spectrally-Shaped Disorder

Media with correlated disorder have recently garnered a lot of attention for their unexpected transport properties. A critical first step towards understanding their complex structure-function relationship is to devise methods for designing structures with desired spectral features at scale. In this work, we introduce an optimal formulation of this inverse problem by means of the non-uniform fast Fourier transform, thus arriving at an algorithm capable of generating systems with arbitrary spectral properties, with a computational cost that scales $O(N \log N)$ with system size. The method is extended to accommodate arbitrary real-space interactions, such as short-range repulsion, to simultaneously control short- and long-range correlations. As an example, we generate the largest-ever stealthy hyperuniform configurations in $2d$ ($N = 10^9$) and $3d$ ($N > 10^7$). By an Ewald sphere construction we then elucidate the link between spectral and optical properties at the single-scattering level, and show that large, yet finite, stealthy hyperuniform structures in $2d$ and $3d$ generically display bandgap-like features in their transmission, thus providing a concrete example of how this method enables fine tuning of a physical property at will. However, we also show that large $3d$ power-law hyperuniformity in particle packings leads to single-scattering properties near-identical to those of simple hard spheres. Our approach for the fast optimization of the spectral properties of point patterns should extend readily beyond materials design, for instance to blue noise sampling and texture generation in computer graphics.