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Physical Review A

Separation estimation of two freely rotating dipole emitters near the quantum limit

Sheng Liu, Sujeet Pani, Sajjad A. Khan, F. E. Becerra, Keith A. Lidke

Published Feb 13, 2026• 1 citation
Publisher

Abstract

According to Rayleigh's criterion, two incoherent emitters with a separation below the diffraction limit are not resolvable with a conventional fluorescence microscope. One method of superresolution microscopy (SRM) circumvents the diffraction-limited resolution by precisely estimating the position of spatiotemporally independent emitters. However, these methods of SRM techniques are not optimal for estimating the separation of two simultaneously excited emitters. Recently, a number of detection methods based on modal imaging have been developed to achieve the quantum Cramér-Rao lower bound (QCRB) to estimate the separations between two nearby emitters. The QCRB determines the minimum achievable precision for all possible detection methods. Current modal imaging techniques assume a scalar field generated from a point source, such as a distant source from an optical fiber or a pinhole. However, for fluorescently labeled samples, point emitters are single fluorophores that are modeled as dipole emitters and, in practice, are often freely rotating. Dipole radiation must be described by vectorial theory, and the assumption of a scalar field no longer holds. Here, we present a method to numerically calculate the QCRB for measuring the separation of two dipole emitters, incorporating the vectorial theory. Furthermore, we propose a near-quantum optimal detection scheme based on one of the modal imaging techniques, superlocalization by image inversion interferometry (SLIVER), for estimating the separation of two freely rotating dipoles. In the proposed method, we introduce a vortex wave plate before the SLIVER detection to separate the radial and azimuthal components of the dipole radiation. With numerical simulations, we demonstrated that our method achieves nondivergent precision at any separation between two dipole emitters. We investigated several practical effects relevant to experimental measurements in superresolution microscopy, including numerical aperture, detection bandwidth, number of estimation parameters, background, and misalignment on separation estimation. Our proposed measurement provides a near-quantum-limited detection scheme for measuring the separation of two freely rotating dipole emitters, such as fluorescently tagged molecules, which are commonly used in superresolution microscopy.

DOI
10.1103/nwrq-4fqv

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According to Rayleigh's criterion, two incoherent emitters with a separation below the diffraction limit are not resolvable with a conventional fluorescence microscope. One method of superresolution microscopy (SRM) circumvents the diffraction-limited resolution by precisely estimating the position of spatiotemporally independent emitters. However, these methods of SRM techniques are not optimal for estimating the separation of two simultaneously excited emitters. Recently, a number of detection methods based on modal imaging have been developed to achieve the quantum Cramér-Rao lower bound (QCRB) to estimate the separations between two nearby emitters. The QCRB determines the minimum achievable precision for all possible detection methods. Current modal imaging techniques assume a scalar field generated from a point source, such as a distant source from an optical fiber or a pinhole. However, for fluorescently labeled samples, point emitters are single fluorophores that are modeled as dipole emitters and, in practice, are often freely rotating. Dipole radiation must be described by vectorial theory, and the assumption of a scalar field no longer holds. Here, we present a method to numerically calculate the QCRB for measuring the separation of two dipole emitters, incorporating the vectorial theory. Furthermore, we propose a near-quantum optimal detection scheme based on one of the modal imaging techniques, superlocalization by image inversion interferometry (SLIVER), for estimating the separation of two freely rotating dipoles. In the proposed method, we introduce a vortex wave plate before the SLIVER detection to separate the radial and azimuthal components of the dipole radiation. With numerical simulations, we demonstrated that our method achieves nondivergent precision at any separation between two dipole emitters. We investigated several practical effects relevant to experimental measurements in superresolution microscopy, including numerical aperture, detection bandwidth, number of estimation parameters, background, and misalignment on separation estimation. Our proposed measurement provides a near-quantum-limited detection scheme for measuring the separation of two freely rotating dipole emitters, such as fluorescently tagged molecules, which are commonly used in superresolution microscopy.
Recent theoretical and experimental work has shown that the quantum Fisher information associated with estimating the separation between two optical point sources remains finite at small separations, effectively opening new routes to super-resolution imaging of simultaneously emitting sources. Most studies to date, however, implicitly invoke the scalar approximation, which is not appropriate in the context of high-numerical-aperture microscopy. Utilizing parameter estimation theory, here we consider the estimation of separation between two closely spaced dipole emitters, a commonly employed model for single-molecule optical beacons. We consider two limiting cases: one in which the orientations of the emitters are fixed and equal, and another in which both dipoles freely sample all of orientation space over the course of the measurement. We quantify precision limits using quantum and classical variants of the Fisher information and Cramér-Rao bound. In all cases, the vectorial nature of the emission complicates the analyses, but with appropriate filtering of the collected light in the azimuthal-radial polarization basis, a previously proposed scheme to saturate the quantum Fisher information via image inversion interferometry can be salvaged.
As a method to extract information from optical systems, imaging can be viewed as a parameter estimation problem. The fundamental precision in locating one emitter or estimating the separation between two incoherent emitters is bounded below by the multiparameter quantum Cramér-Rao bound (QCRB). Multiparameter QCRB gives an intrinsic bound in parameter estimation. We determine the ultimate potential of quantum-limited imaging for improving the resolution of a far-field, diffraction-limited optical field within the paraxial approximation. We show that the quantum Fisher information matrix (QFIm) in about one emitter’s position is independent on its true value. We calculate the QFIm of two unequal-brightness emitters’ relative positions and intensities; the results show that only when the relative intensity and centroids of two-point sources, including longitudinal and transverse directions, are known exactly, the separation in different directions can be estimated simultaneously with finite precision. Our results give the upper bounds on certain far-field imaging technology and will find wide use in applications from microscopy to astrometry.
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 Cramér-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.
We investigate the localization of two incoherent point sources with arbitrary angular and axial separations in the paraxial approximation. By using quantum metrology techniques, we show that a simultaneous estimation of the two separations is achievable by a single quantum measurement, with a precision saturating the ultimate limit stemming from the quantum Cramér-Rao bound. Such a precision is not degraded in the subwavelength regime, thus overcoming the traditional limitations of classical direct imaging derived from Rayleigh's criterion. Our results are qualitatively independent of the point spread function of the imaging system, and quantitatively illustrated in detail for the Gaussian instance. This analysis may have relevant applications in three-dimensional surface measurements.
Resolving light sources below the diffraction limit is a fundamental task both for astronomy and microscopy. Several recent works analysed this problem through the lens of quantum parameter estimation theory and proved that the separation between two point sources can be estimated at the quantum limit using intensity measurements after spatial-mode demultiplexing. We implement this technique and provide an optimal estimator based on a linear combination of demultiplexed intensity measurements [1–3]. Our experimental setup allows for the generation of the images of two sources, with tunable mutual coherence, as well as for spatial mode demultiplexing to estimate their separation [4].
Resolving light sources below the diffraction limit is a fundamental task both for astronomy and microscopy. Several recent works analysed this problem through the lens of quantum parameter estimation theory and proved that the separation between two point sources can be estimated at the quantum limit using intensity measurements after spatial-mode demultiplexing. We implement this technique and provide an optimal estimator based on a linear combination of demultiplexed intensity measurements [1]–[3]. Our experimental setup (see Fig. 1 (left)) allows for the generation of the images of two sources, with tunable mutual coherence, as well as for spatial mode demultiplexing to estimate their separation [4].