Quantum-limited superresolution of two arbitrary incoherent point sources: beating the resurgence of Rayleigh's curse

Abstract

Abstract Superresolution has been demonstrated to overcome the limitation of the Rayleigh's criterion and achieve significant improvement of the precision in resolving the separation of two incoherent optical point sources. However, in recent years, it was found that if the photon numbers of the two incoherent optical sources are unknown, the precision of superresolution vanishes when the two photon numbers are actually different. In this work, we first analyze the estimation precision of the separation between two incoherent optical sources with the same point-spread functions in detail, and show that when the photon numbers of the two optical sources are different but sufficiently close, the superresolution can still realized but with different precisions. We find the condition on how close the photon numbers of two optical sources need to be to realize the superresolution, and derive the precision of superresolution in different regimes of the photon number difference. We further consider the superresolution for two incoherent optical sources with different point-spread functions, and show that the competition between the difference of photon numbers, the difference of the two point-spread functions and the separation of the two optical sources determines the precision of superresolution. The results exhibit precision limits distinct from the case of two point sources with identical point-spread functions and equal photon numbers, and extend the realizable regimes of the quantum superresolution technique. The results are finally illustrated by Gaussian point-spread functions.