Quantum limit for two-dimensional resolution of two incoherent optical point sources

TL;DR

This paper derives the quantum Cramér-Rao bounds for estimating the position and separation of two incoherent optical sources in two dimensions, demonstrating that these bounds can surpass the classical Rayleigh limit and proposing measurement methods to approach these bounds.

Contribution

The paper derives the quantum limits for 2D source separation estimation and introduces measurement schemes that nearly reach these fundamental bounds.

Findings

Quantum bounds are independent of source separation under realistic assumptions.

Proposed measurement methods can approach the quantum limits.

Sub-Rayleigh resolution is achievable with optimal measurement schemes.

Abstract

We obtain the multiple-parameter quantum Cram\'er-Rao bound for estimating the transverse Cartesian components of the centroid and separation of two incoherent optical point sources using an imaging system with finite spatial bandwidth. Under quite general and realistic assumptions on the point-spread function of the imaging system, and for weak source strengths, we show that the Cram\'er-Rao bounds for the $x$ and $y$ components of the separation are independent of the values of those components, which may be well below the conventional Rayleigh resolution limit. We also propose two linear optics-based measurement methods that approach the quantum bound for the estimation of the Cartesian components of the separation once the centroid has been located. One of the methods is an interferometric scheme that approaches the quantum bound for sub-Rayleigh separations. The other method using fiber coupling can in principle attain the bound regardless of the distance between the two sources.