Looking down the rabbit hole: Towards quantum optimal estimation of surface roughness

Surface roughness is an important quantity to many engineering and precision manufacturing disciplines. In this paper we investigate the problem of estimating the root-mean-square roughness of a sample by passive linear optical methods. By adopting quantum parameter estimation techniques, we determine the ultimate precision limits on roughness estimation. In particular, we show that the information on the first moment (mean height) and standard deviation (roughness) of the axial profile distribution of multiple incoherent point sources is bounded by a constant. While classical imaging techniques fail to achieve this bound, a quantum inspired imaging technique based on spatial mode demultiplexing is proven to be optimal for estimating the axial standard deviation. Combined with analogous recently investigated methods for estimating radial profiles, this can provide a powerful technique for measuring roughness of nearly smooth surface patches beyond the diffraction limit.