# Realistic sub-Rayleigh imaging with phase-sensitive measurements

## TL;DR

This paper investigates phase-sensitive measurement techniques, particularly SPLICE and an adapted hybrid scheme, to improve sub-Rayleigh imaging of two emitters with unequal intensities, surpassing traditional intensity measurements in certain regimes.

## Contribution

It analyzes the performance of SPLICE and proposes an adapted hybrid measurement to handle large intensity imbalances in sub-Rayleigh imaging.

## Key findings

- SPLICE outperforms intensity counting for moderate intensity disparities.
- The adapted SPLICE scheme surpasses IPC for large intensity imbalances, up to 10,000:1.
- Monte Carlo simulations identify optimal measurement strategies based on source parameters.

## Abstract

As the separation between two emitters is decreased below the Rayleigh limit, the information that can be gained about their separation using traditional imaging techniques, photon counting in the image plane, reduces to nil. Assuming the sources are of equal intensity, Rayleigh's "curse" can be alleviated by making phase-sensitive measurements in the image plane. However, with unequal and unknown intensities the curse returns regardless of the measurement, though the ideal scheme would still outperform image plane counting (IPC), i.e., recording intensities on a screen. We analyze the limits of the SPLICE phase measurement scheme as the intensity imbalance between the emitters grows. We find that SPLICE still outperforms IPC for moderately disparate intensities. For larger intensity imbalances we propose a hybrid of IPC and SPLICE, which we call "adapted SPLICE", requiring only simple modifications. Using Monte Carlo simulation, we identify regions (emitter brightness, separation, intensity imbalance) where it is advantageous to use SPLICE over IPC, and when to switch to the adapted SPLICE measurement. We find that adapted SPLICE can outperform IPC for large intensity imbalances, e.g., 10,000:1, with the advantage growing with greater disparity between the two intensities. Finally, we also propose additional phase measurements for estimating the statistical moments of more complex source distributions. Our results are promising for implementing phase measurements in sub-Rayleigh imaging tasks such as exoplanet detection.