Modern description of Rayleigh's criterion

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 (PSFs), namely, in the subdiffraction limit. Recently, researchers have achieved superresolution for two incoherent point sources with equal strengths, 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 (1D or 2D) image can be precisely resolved up to its second moment in the subdiffraction limit, i.e., the Fisher information (Fl) is nonzero. 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 nonadaptive 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 a 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.