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

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.