Quantum Limited Superresolution of Extended Sources in One and Two Dimensions

TL;DR

This paper calculates the quantum Fisher information for superresolving extended incoherent sources in one and two dimensions, demonstrating quantum-limited estimation bounds and generalizing to arbitrary shapes and brightness distributions.

Contribution

It introduces a method to compute quantum Fisher information for extended sources using PSWFs and Zernike polynomials, extending to arbitrary shapes with Bessel Fourier functions.

Findings

Quantum Fisher information bounds for line and disk sources.

Classical wavefront projection can achieve quantum-limited estimation.

Generalized approach for arbitrary brightness distributions and shapes.

Abstract

We calculate the quantum Fisher information (QFI) for estimating, using a circular imaging aperture, the length of a uniformly bright incoherent line source with a fixed mid-point and the radius of a uniformly bright incoherent disk shaped source with a fixed center. Prolate spheroidal wavefunctions (PSWFs) on a centered line segment and its generalized version on a centered disk furnish the respective bases for computing the eigenstates and eigenvalues of the one-photon density operator, from which we subsequently calculate QFI with respect to the spatial parameters of the two sources. Zernike polynomials provide a good set into which to project the full source wavefront, and such classical wavefront projection data can realize quantum limited estimation error bound in each case. We subsequently generalize our approach to analyze sources of arbitrary brightness distributions and shapes using a certain class of Bessel Fourier functions that are closely related to the PSWFs. We illustrate the general approach by computing QFI for estimating the lengths of the principal axes of a uniformly bright, centered elliptical disk.