Attaining quantum limited precision of localizing an object in passive imaging

We investigate our ability to determine the mean position, or centroid, of a linear array of equally-bright incoherent point sources of light, whose continuum limit is the problem of estimating the center of a uniformly-radiating object. We consider two receivers: an image-plane ideal direct-detection imager and a receiver that employs Hermite-Gaussian (HG) Spatial-mode Demultiplexing (SPADE) in the image plane, prior to shot-noise-limited photon detection. We compare the Fisher Information (FI) for estimating the centroid achieved by these two receivers, which quantifies the information-accrual rate per photon, and compare those with the Quantum Fisher Information (QFI): the maximum attainable FI by any choice of measurement on the collected light allowed by physics. We find that focal-plane direct imaging is strictly sub-optimal, although not by a large margin. We also find that the HG mode sorter, which is the optimal measurement for estimating the separation between point sources (or the length of a line object) is not only suboptimal, but it performs worse than direct imaging. We study the scaling behavior of the QFI and direct imaging's FI for a continuous, uniformly-bright object in terms of its length, and find that both are inversely proportional to the object's length when it is sufficiently larger than the Rayleigh length. Finally, we propose a two-stage adaptive modal receiver design that attains the QFI for centroid estimation.