Abstract
Improving axial resolution is crucial for three-dimensional optical imaging systems. Here we present a scheme of axial superresolution for two incoherent point sources based on spatial mode demultiplexing. A radial mode sorter is used to losslessly decompose the optical fields into a radial mode basis set to extract the phase information associated with the axial positions of the point sources. We show theoretically and experimentally that, in the limit of a zero axial separation, our scheme allows for reaching the quantum Cram\'er-Rao lower bound and thus can be considered as one of the optimal measurement methods. Unlike other superresolution schemes, this scheme does not require neither activation of fluorophores nor sophisticated stabilization control. Moreover, it is applicable to the localization of a single point source in the axial direction. Our demonstration can be useful to a variety of applications such as far-field fluorescence microscopy.
Highlights
Optical microscopy is one of the most important imaging modalities and has been broadly applied in various areas
One crucial metric for an optical microscope is the spatial resolution, which is typically constrained by the diffraction limit, and the Rayleigh criterion is proposed as the resolution limit of an incoherent imaging system [1,2,3]
Given the non-vanishing quantum Fisher information, there should exist a type of measurement that can outperform the direct imaging method and extract the maximum possible amount of information from each photon. We demonstrate both theoretically and experimentally that the axial superresolution can be achieved at the single-photon level by a radial mode sorter
Summary
Optical microscopy is one of the most important imaging modalities and has been broadly applied in various areas. A widely used approach is to activate each fluorescence molecule individually, and the overlap between neighboring molecules is avoided, and the localization precision can be improved to tens of nanometers [4,5,6] This technique usually requires specially prepared samples, and the reconstruction of an image can take a long time due to the sophisticated activation and detection of individual fluorophores. We demonstrate both theoretically and experimentally that the axial superresolution can be achieved at the single-photon level by a radial mode sorter. While homodyne or heterodyne detection [12,39] provides an easier way to implement the spatial mode projective measurement, the shot noise from the reference beam fundamentally deteriorates the signal-to-noise ratio, and it has been shown that homodyne or heterodyne detection cannot even outperform the direct imaging method when the available photon number is small [53]. The mode sorter does not require any active components, such as the local oscillator in heterodyne detection, and is more favorable in an experiment
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