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arXiv:1805.04116 · Physical Review Letters

Towards Superresolution Surface Metrology: Quantum Estimation of Angular and Axial Separations

Carmine Napoli, Samanta Piano, Richard Leach, Gerardo Adesso, Tommaso Tufarelli

Published Apr 12, 2019• 81 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.122.140505

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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.
The Rayleigh criterion has long served as a fundamental limit for the resolution of optical imaging. Recent advances in multiparameter quantum metrology have led to quantum superresolution that can break this limit and achieve nonvanishing precision in estimating the separation between a pair of closely located incoherent point sources. For two-dimensional optical systems, the quantum superresolution has been studied for the Cartesian components of separation between two incoherent point sources. However, the precision limit of estimating the full distance between two point sources remains unknown so far. In this paper, we investigate the estimation precision of the full distance between two incoherent point sources with arbitrary intensities in a two-dimensional imaging system. Through the multiparameter quantum estimation theory, we obtain the ultimate estimation precision for the distance and show that it remains nonzero when the distance approaches zero, which surpasses the Rayleigh criterion. We further show the dependence of the estimation precision on the relative orientation between the two point sources, which leads to a novel scheme that can enhance the precision by aligning the sources along proper directions if the point-spread functions are not circularly symmetric, and the enhancement is determined by the extent to which the point-spread functions deviate from circular symmetry. Finally, the results are illustrated by incoherent sources with Gaussian point-spread functions.
We establish the multiparameter quantum Cram\'er-Rao bound for simultaneously estimating the centroid, the separation, and the relative intensities of two incoherent optical point sources using a linear imaging system. For equally bright sources, the Cram\'er-Rao bound is independent of their separation, which confirms that the Rayleigh resolution limit is just an artifact of the conventional direct imaging and can be overcome with an adequate strategy. For the general case of unequally bright sources, the amount of information one can gain about the separation falls to zero, but we show that there is always a quadratic improvement in an optimal detection in comparison with the intensity measurements. This advantage can be of utmost importance in realistic scenarios, such as observational astronomy.
Quantum superresolution reveals that the vanishing of separation sensitivity in conventional imaging below the Rayleigh limit does not necessarily indicate a fundamental loss of information in the optical field. However, the quantum limit for estimating the physical distance between two incoherent point sources in three-dimensional imaging systems and its dependence on the spatial structure of the point-spread function remains largely unknown. In this work, we derive the quantum-limited precision for estimating the full distance between two incoherent point sources with arbitrary intensity imbalance in a three-dimensional spatially invariant imaging system. We show that the distance information remains finite in the sub-Rayleigh regime and is governed by the second-order displacement-response tensor of the point-spread function. The eigensystem of this tensor determines the optimal relative orientation between the two sources, and reflection symmetries of the point-spread function can further provide a simplified means of identifying the optimal orientation. This geometric structure is coordinate invariant and provides a direct strategy for improving resolution by physically rotating an anisotropic imaging system to align its optimal principal response direction with the source displacement. For a general three-dimensional Gaussian point-spread function, the response tensor is proportional to the inverse spatial covariance, establishing a direct connection between quantum-limited distance precision and the geometry of Gaussian distribution.
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 obtain the ultimate quantum limit for estimating the transverse separation of two thermal point sources using a given imaging system with limited spatial bandwidth. We show via the quantum Cramér-Rao bound that, contrary to the Rayleigh limit in conventional direct imaging, quantum mechanics does not mandate any loss of precision in estimating even deep sub-Rayleigh separations. We propose two coherent measurement techniques, easily implementable using current linear-optics technology, that approach the quantum limit over an arbitrarily large range of separations. Our bound is valid for arbitrary source strengths, all regions of the electromagnetic spectrum, and for any imaging system with an inversion-symmetric point-spread function. The measurement schemes can be applied to microscopy, optical sensing, and astrometry at all wavelengths.
Superresolution has been demonstrated to overcome the limitation of Rayleigh's criterion and achieve significant precision improvement in resolving the separation of two incoherent optical point sources. However, in recent years, it was found that, if the photon numbers of two incoherent optical sources are unknown, the precision of superresolution vanishes when the photon numbers are actually different. In this work, we first analyze in detail the estimation precision of the separation for two incoherent optical sources with the same point-spread functions and show that, when the two photon numbers are different but sufficiently close, the superresolution can still be realized but with different precisions. We find the condition on how close the photon numbers need to be to realize the superresolution and derive the precision of superresolution in different regimes of the photon number difference. We further consider the superresolution for two incoherent optical sources with different point-spread functions and show that the competition between the difference of photon numbers, the difference of point-spread functions, and the separation of source locations determines the precision of superresolution. The results exhibit precision limits distinct from that of two identical point-spread functions with equal photon numbers and extend the realizable regimes of quantum superresolution. Finally, the results are illustrated by Gaussian point-spread functions. Published by the American Physical Society 2024