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

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