Fundamental precision bounds for three-dimensional optical localization microscopy with Poisson statistics

TL;DR

This paper derives the fundamental quantum limits for 3D optical localization in microscopy, showing current methods are suboptimal and proposing an interferometric approach that approaches the theoretical bound.

Contribution

It provides the first derivation of the quantum Cramér-Rao bound for 3D localization and suggests an interferometric setup to nearly achieve this limit.

Findings

Existing methods exceed the QCRB by a factor >√2.

Proposed interferometer approaches the QCRB.

Two opposed objectives with interferometry reach the bound.

Abstract

Point source localization is a problem of persistent interest in optical imaging. In particular, a number of widely used biological microscopy techniques rely on precise three-dimensional localization of single fluorophores. As emitter depth localization is more challenging than lateral localization, considerable effort has been spent on engineering the response of the microscope in a way that reveals increased depth information. Here we consider the theoretical limits of such approaches by deriving the quantum Cram\'{e}r-Rao bound (QCRB). We show that existing methods for depth localization with single-objective detection exceed the QCRB by a factor $>\sqrt{2}$, and propose an interferometer arrangement that approaches the bound. We also show that for detection with two opposed objectives, established interferometric measurement techniques globally reach the QCRB.