Quantum Limited Source Localization and Pair Superresolution under Finite Emission Bandwidth

Abstract

Optically localizing a single quasi-monochromatic source to sub-diffractive precisions entails, in the photon-counting limit, a minimum photon cost that scales as the squared ratio of the width, w, of the optical system's point-spread function (PSF) and the sought localization precision, d, i.e., as α(w/d) 2 . For sources with a finite emission-frequency spectrum, while the inverse quadratic scaling is expected to remain unchanged, the coefficient α must increase due to a degrading fidelity of localization as the imaging bandwidth increases and PSF undergoes a frequency-dependent widening. We specifically address how rapidly α must increase with increasing width of a flat-top spectral profile of emission of a point source being localized by an imager with a clear circular aperture by calculating quantum Fisher information (QFI), whose inverse yields the lowest possible unbiasedestimation variance of source-localization error. We subsequently extend our considerations of QFI to treat the finite-bandwidth pair superresolution problem in two dimensions, obtaining similar results. We also consider generalizations to emission power spectra of arbitrary profiles.