# Fisher information for far-field linear optical superresolution via homodyne or heterodyne detection in a higher-order local oscillator mode

## TL;DR

This paper theoretically demonstrates that homodyne and heterodyne detection using a higher-order TEM mode can significantly improve the estimation of sub-Rayleigh source separation, surpassing direct imaging in Fisher information under certain conditions.

## Contribution

It introduces a theoretical analysis of quantum Fisher information for sub-Rayleigh source separation estimation using higher-order mode detection techniques.

## Key findings

- Per-photon Fisher information exceeds direct imaging when average photon number > 2 (homodyne) or > 4 (heterodyne).
- Homodyne detection with TEM01 mode offers enhanced sensitivity in the sub-Rayleigh regime.
- The approach is effective for thermal sources with specific photon number conditions.

## Abstract

The distance between two point light sources is difficult to estimate if that distance is below the diffraction (Rayleigh's) resolution limit of the imaging device. A recently proposed technique enhances the precision of this estimation by exploiting the source-separation-dependent coupling of light into higher-order $\rm{TEM}$ modes, particularly the $\rm{TEM}_{01}$ mode of the image. We theoretically analyze the estimation of the source separation by means of homodyne or heterodyne detection with a local oscillator in the $\rm{TEM}_{01}$ mode, which is maximally sensitive to the separation in the sub-Rayleigh regime. We calculate the per-photon Fisher information associated with this estimation and compare it with direct imaging. For thermal sources, the per-photon Fisher information depends on the average photon number per thermal mode of the image; it surpasses the Fisher information for direct imaging (in the interesting sub-Rayleigh regime) when the average photon number exceeds two for homodyne detection and four for heterodyne detection.