A modern description of Rayleigh's criterion

TL;DR

This paper extends the understanding of Rayleigh's criterion by demonstrating that arbitrary incoherent sources can be precisely resolved up to their second moments in the subdiffraction limit, with optimal measurement strategies improving resolution beyond direct imaging.

Contribution

It provides a rigorous proof that second moments can be resolved for any incoherent sources in the subdiffraction limit and constructs optimal measurement bases for 1D and 2D imaging scenarios.

Findings

FI is non-zero for second moments, enabling resolution.

Higher order moments' FI tends to zero polynomially as size decreases.

Optimal measurement bases achieve quadratic scaling improvements.

Abstract

Rayleigh's criterion states that it becomes essentially difficult to resolve two incoherent optical point sources separated by a distance below the width of point spread functions (PSF), namely in the subdiffraction limit. Recently, researchers have achieved superresolution for two incoherent point sources with equal strengths using a new type of measurement technique, surpassing Rayleigh's criterion. However, situations where more than two point sources needed to be resolved have not been fully investigated. Here we prove that for any incoherent sources with arbitrary strengths, a one- or two-dimensional image can be precisely resolved up to its second moment in the subdiffraction limit, i.e. the Fisher information (FI) is non-zero. But the FI with respect to higher order moments always tends to zero polynomially as the size of the image decreases, for any type of non-adaptive measurement. We call this phenomenon a modern description of Rayleigh's criterion. For PSFs under certain constraints, the optimal measurement basis estimating all moments in the subdiffraction limit for 1D weak-source imaging is constructed. Such basis also generates the optimal-scaling FI with respect to the size of the image for 2D or strong-source imaging, which achieves an overall quadratic improvement compared to direct imaging.