Hyperuniform distributions have been extensively studied by Torquato et al. and the subject is now fairly mature on “flat” Euclidean spaces. Stealthy hyperuniform distributions are a sub-class, first studied by Zhang, Stillinger and Torquato, and offer the possibility for the construction of imperfectly ordered artificial materials which do not generate scattering loss. In Euclidean space, infinite stealthy distributions do not scatter over a contiguous frequency band from zero to some finite defined upper frequency.

We have examined several prototype problems of interest to electromagnetic engineers. The first two relate to point distributions on a flat 2D Euclidean space. The later studies relate to stealthy distributions on the surface of a sphere.

- 1. The radar cross section (RCS) of a finite flat surface comprising a finite number of point scatterers with no mutual interactions on a hyperuniform stealthy distribution. The study shows that the RCS generally falls mid-way between a perfectly ordered array of points and a random uniform distribution of points.

- 2. The scattering lobes generated by an infinite flat frequency selective surface comprising dipoles on a hyperuniform stealthy distribution. The study shows that there are no grating lobes generated over the frequency band defined by the distribution. However, this is subject to the assumption that mutual interactions between dipoles are either absent or independent of position.

- 3. More recently we have generalised hyperuniform stealthy distributions to the surface of a sphere. This is the first time this has been done and serves as a first step towards a theory on more general curved surfaces. For this purpose we have employed the definition of the spherical structure factor in the paper by Bozic and Copar [“Spherical structure factor and the classification of hyperuniformity on the sphere”, 2019] and defined a distribution of points on the sphere to be stealthy hyperuniform if the spherical structure factor is zero for modes 1≤l≤L for some integer L>1. We have modified the method of Zhang, Stillinger and Torquato [“Ground states of stealthy hyperuniform potentials”, 2015] and employed molecular dynamics on the sphere for the generation of ensembles of distributions. For statistically isotropic N point distributions on the sphere the order parameter χ=L(L+3)/[4(N-1)]. The scattering statistics for such distributions may then be studied for different order parameters.

Two examples of such distributions for the case N=100 are shown below for L=10 and L=14 and compared with a solution to the Tammes problem for N=100 employed by Bozic and Copar. The Tammes problem is to find point distributions on a sphere that maximise the minimum distance between any two points. For general N this problem is far from trivial!

This research shows distributions with properties very similar to those observed on flat 2D surfaces with similar order parameter values. The similarity between Tammes solutions and stealthy distributions for χ>0.5 is interesting. (Technically only approximately stealthy since for χ>0.5 the structure factor can be made small but not zero with our numerical methods).

We expect several applications of stealthy hyperuniform distributions in engineering and biology and a comparison with Tammes solutions may be of interest to applied mathematicians.